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\section{Introduction}\label{sec:introduction}
Large-area sky surveys, both on-going and planned, are revolutionizing our understanding of galaxy evolution. The Dark Energy Survey (DES; \citealt{DES2005}) and upcoming Large Synoptic Survey Telescope (LSST; \citealt{LSST2012}) will scan vast swaths of the sky and create samples of galaxies of unprecedented size. Spectroscopic follow-up of these samples will be instrumental in order to understand their properties. Previously, the Sloan Digital Sky Survey \citep[SDSS;][]{York2000} and its spectroscopic campaign enabled characterization of the mass-metallicity relation (hereafter MZR; \citealt{Tremonti2004}) and the fundamental metallicity relation, (hereafter FMR; e.g., \citealt{Mannucci2010}). As future surveys are accompanied by larger data sets, individual spectroscopic follow-up observations will become increasingly impractical.
Fortunately, the large imaging data sets to be produced are ripe for application of machine learing (ML) methods. ML is already showing promise in studies of galaxy morphology \citeeg{Dieleman2015, Huertas-Company2015, Beck2018, Dai2018, Hocking2018}, gravitational lensing \citeeg{Hezaveh2017, Lanusse2017, Petrillo2017, Petrillo2018}, galaxy clusters \citeeg{Ntampaka2015, Ntampaka2016}, star-galaxy separation \citeeg{Kim2017}, creating mock galaxy catalogs \citeeg{Xu2013}, asteroid identification \citeeg{Smirnov2017}, and photometric redshift estimation \citeeg{Hoyle2016, DIsanto2018, Pasquet2019}, among many others. ML methods utilizing neural networks have grown to prominence in recent years. While neural networks are a relatively old technique \citeeg{LeCun1989}, their recent increase in popularity is driven by the widespread availability of affordable graphics processing units (GPUs) that can be used to do general purpose, highly parallel computing. Also, unlike more ``traditional'' ML methods, neural networks excel at image classification and regression problems.
Inferring spectroscopic properties from the imaging taken as part of a large-area photometric survey is, at a basic level, an image regression problem. These problems are most readily solved by use of convolutions in multiple layers of the network (see, e.g., \citealt{Krizhevsky2012}). Convolutional neural networks (CNNs, or convnets) efficiently learn spatial relations in images whose features are about the same sizes as the convolution filters (or kernels) that are to be learned through training. CNNs are considered \textit{deep} when the number of convolutional layers is large. Visualizing their filters reveals that increased depth permits the network to learn more and more abstract features (e.g., from Gabor filters, to geometric shapes, to faces; \citealt{Zeiler2014}).
In this work, we propose to use supervised ML by training CNNs to analyze pseudo-three color images and predict the gas-phase metallicity. We use predicted metallicities to recover the empirical \cite{Tremonti2004} MZR. This paper is organized as follows: In Section~\ref{sec:data}, we describe the acquisition and cleaning of the SDSS data sample. In Section~\ref{sec:training}, we discuss selection of the network's hyperparameters and outline training the of network. We present the main results in Section~\ref{sec:results}. In Section~\ref{sec:interpretation}, we interpret the CNN's performance and discuss our findings in the context of current literature. In Section~\ref{sec:MZR}, we characterize the MZR using the metallicity predicted by our CNN. We summarize our key results in Section~\ref{sec:summary}.
Unless otherwise noted, throughout this paper, we use a concordance cosmological model ($\Omega_\Lambda = 0.7$, $\Omega_m = 0.3$, and $H_0= 70$ \hbox{km~s$^{-1}$}{} \hbox{Mpc$^{-1}$}), assume a Kroupa initial mass function \citep{Kroupa2001}, and use AB magnitudes \citep{Oke1974}.
\section{Data} \label{sec:data}
To create a large training sample, we select galaxies from the Sloan Digital Sky Survey (SDSS; \citealt{York2000}) DR7 MPA/JHU spectroscopic catalog \citep{Kauffmann2003a, Brinchmann2004, Tremonti2004, Salim2007}. The catalog provides spectroscopically derived properties such as stellar mass (\hbox{$M_{\star}$}) and gas-phase metallicity ($Z$) estimates \citep{Tremonti2004}. We select objects with low reduced chi-squared of model fits (\texttt{rChi2} $< 2$), and median $Z$ estimates available (\texttt{oh\_p50}). We supplement the data from the spectroscopic catalog with photometry in each of the five SDSS photometric bands (\hbox{$u$}, \hbox{$g$}, \hbox{$r$}, \hbox{$i$}, \hbox{$z$}), along with associated errors from SDSS DR14 \citep{Abolfathi2017}.
We require that galaxies magnitudes are $10 < \hbox{$u$} \hbox{$g$} \hbox{$r$} \hbox{$i$} \hbox{$z$} < 25$ mag, in order to avoid saturated and low signal-to-noise detections. We enforce a color cut, $0 < \hbox{$u$}-\hbox{$r$} < 6$, in order to avoid extremely blue or extremely red objects, and require objects to have spectroscopic redshifts greater than $z=0.02$ with low errors ($z_{err} < 0.01$). The median redshift is $0.07$ and the highest-redshift object has $z = 0.38$. We also require that the \hbox{$r$}-band magnitude measured inside the Petrosian radius (\texttt{petroMag\_r}; \citealt{Petrosian1976}) be less than 18 mag, corresponding to the spectroscopic flux limit. With these conditions we construct an initial sample of 142,182 objects (there are four objects with duplicate SDSS DR14 identifiers).
We set aside 25,000 objects for later testing, and use the rest for training and validation.
We create RGB image cutouts of each galaxy with the SDSS cutout service\footnote{\url{http://skyserver.sdss.org/dr14/en/help/docs/api.aspx}}, which converts \hbox{$g$}\hbox{$r$}\hbox{$i$}\ bands to RGB channels according to the algorithm described in \cite{Lupton2004} (with modifications by the SDSS SkyServer team). Since images are not always available, we are left with 116,429 SDSS images with metallicity measurements, including 20,466/25,000 of the test subsample. We create $128\times128$-pixel JPG images with a pixel scale of $0\farcs296$, which corresponds to $38''\times 38''$ on the sky. We do not further preprocess, clean, or filter the images before using them as inputs to our CNN.
\section{Methodology}\label{sec:training}
Before the CNN can be asked to make predictions, it must be trained to learn the relationships between the input data (the images described above) and the desired output (metallicity). The CNN makes predictions using the input images, and the error (or loss) is determined based on the differences between true and predicted values. The CNN then updates its parameters, or weights, in a way that minimizes the loss function. We use the root mean squared error loss function:
\begin{equation} \label{eq:rmse}
\mathrm{RMSE} \equiv \sqrt{\langle |y_{\rm true} - y_{\rm pred}|^2\rangle},
\end{equation}
where $y_{\rm true}$ is the ``true'' and $y_{\rm pred}$ is the predicted value, and $y$ represents the target quantity.
It is worth emphasizing that the $Z_{\rm true}$ is the metallicity estimated by model fits to strong emission lines in the SDSS spectra. \cite{Tremonti2004} determine a likelihood distribution of metallicities based on the model fits, and we define their 50th percentile metallicity estimates to be the \textit{true} metallicity ($Z_{\rm true}$) for the purpose of training our network. The typical systematic uncertainty in their metallicity model fits is about 0.03~dex.
We randomly split our training sample of $\sim 96,953$ images into 80\% (76,711) training and 20\% (19,192) validation data sets, respectively. The test data set of 20,466 images is isolated for now, and is not accessible to the CNN until all training is completed. Images and $Z_{\rm true}$ answers are given to the CNN in ``batches'' of 256 at a time, until the full training data set has been used for training. Each full round of training using all of the data is called an epoch, and we compute the loss using the validation data set at the end of each epoch. We use gradient descent for each batch to adjust weight parameters, and each weight's fractional contribution of loss is determined by the backpropagation algorithm \citep{LeCun1989}, during which finite partial derivatives are computed and propagated backwards through layers (i.e., using the chain rule for derivatives).
We use a 34-layer residual CNN architecture \citep{He2015} initialized to weights pre-trained on the ImageNet data set, which consists of 1.7 million images belonging to 1000 categories of objects found on Earth \citep[e.g., cats, horses, cars, or books;][]{ImageNet}. The CNN is trained for a total of 10 epochs. For more details about the CNN architecture, transfer learning, hyperparameter selection, data augmentation, and the training process, see the Appendix. In total, our training process requires 25-30 minutes on our GPU and uses under 2~GB of memory.
We evaluate predictions using the RMSE loss function, which approaches the standard deviation for Gaussian-distributed data. We also report the NMAD, or the normal median absolute deviation \citeeg{Ilbert2009, Dahlen2013, Molino2017}:
\begin{equation}
{\rm NMAD}(x) \approx 1.4826 \times {\rm median} \big (\big|x - {\rm median }(x) \big|\big ),
\end{equation}
where for a Gaussian-distributed $x$, the NMAD will also approximate the standard deviation, $\sigma$.
NMAD has the distinct advantage in that it is insensitive to outliers and can be useful for measuring scatter.
However, unlike the RMSE, which quantifies the typical scatter distributed about a center of zero, NMAD only describes the scatter around the (potentially non-zero) median.
\section{Results}\label{sec:results}
\subsection{Example predictions}
\begin{figure*}
\includegraphics[width=0.9\textwidth]{01-prediction_examples.pdf}
\caption{\label{fig:examples}
SDSS imaging with predicted and true metallicities from the test data set. Five examples are shown from each of the following categories: (a) lowest predicted metallicity, (b) lowest true metallicity, (c) highest predicted metallicity, (d) highest true metallicity, (e) most under-predicted metallicity, (f) most over-predicted metallicity, and (g) a set of randomly selected galaxies.}
\end{figure*}
In Figure~\ref{fig:examples}, we show examples of $128 \times 128$ pixel \hbox{$g$}\hbox{$r$}\hbox{$i$}\ SDSS images that are evaluated by the CNN. Rows (a) and (b) depict the galaxies with lowest predicted and lowest true metallicities, respectively. The CNN associates blue, edge-on disk galaxies with low metallicities, and is generally accurate in its predictions. In rows (c) and (d), we show the galaxies with highest predicted and highest true metallicities, respectively. Here we find that red galaxies containing prominent nuclei are predicted to be high in metallicity, and that their predictions generally match $Z_{\rm true}$.
Galaxies predicted by our CNN to have high metallicities ($Z_{\rm pred} > 9.0$) tend to be characterized by high $Z_{\rm true}$, and the equivalent is true for low-metallicity galaxies. Conversely, galaxies with the highest (lowest) \textit{true} metallicities in the sample are also predicted to have high (low) metallicities. Note that inclined galaxies tend to be lower in metallicity whereas face-on galaxies appear to be higher in metallicity. \cite{Tremonti2004} explain this correlation by suggesting that the SDSS fiber aperture captures more column of a projected edge-on disk, allowing the metal-poor, gas-rich, and less-extincted outer regions to more easily be detected and depress the integrated $Z_{\rm true}$.
We will now consider examples of the most incorrectly predicted galaxies. In rows (e) and (f), we show instances in which the CNN predicted too low metallicity and too high metallicity, respectively. The two galaxies with the most negative residuals $\Delta Z \equiv Z_{\rm pred} - Z_{\rm true}$ (i.e., most under-predicted metallicities) suffer from artifacts that cause unphysical color gradients, and/or are labeled as quasars on the basis of their SDSS spectra (for which we expect $Z_{\rm true}$ to be biased). It is not unsurprising that the CNN has made mistakes in some of these cases, since they go against astronomers' usual heuristics: blue, disk-dominated sources are generally thought of as lower in metallicity, and redder, more spheroidal objects tend to be higher in metallicity.
In the bottom row (g) of Figure~\ref{fig:examples}, we show five randomly selected galaxies. The random SDSS assortment consists of lenticular, spiral, and possibly even an interacting pair of galaxies. Residuals are low (below 0.15~dex), and we again find that the CNN predictions track with human visual intuition.
\subsection{Comparing predicted and true metallicities}
\begin{figure}
\includegraphics[width=\columnwidth]{03-Z_distribution.pdf}
\caption{\label{fig:distributions}
Distributions of the true (black) and predicted (red) galaxy metallicities. Note that the bin widths are different for the two distributions. See text for details.}
\end{figure}
In Figure~\ref{fig:distributions}, we show histograms of the true and predicted metallicities in black and red, respectively. The histogram bin sizes are chosen according to the \cite{Freedman1981} rule for each distribution. The discreet striping of the \cite{Tremonti2004} and \cite{Brinchmann2004} metallicity estimator appears in the $Z_{\rm true}$ distribution but does not appear in our CNN predictions. This striping should increase the scatter in our distribution of residuals.
The range of $Z_{\rm pred}$ is more limited than the range of $Z_{\rm true}$, which can also be seen from Figure~\ref{fig:examples} for extreme values of $Z_{\rm true}$. Too narrow a domain in $Z_{\rm pred}$ will lead to systematic errors, as the CNN will end up never predicting very high or very low metallicities. Although the two distributions are qualitatively consistent with each other at low metallicities (e.g., $Z < 8.5$), the fraction of galaxies with high $Z_{\rm true} > 9.1$ ($2573/20466 = 12.6\%$) is higher than the fraction with high $Z_{\rm pred} > 9.1$ ($1174/20466 = 5.7\%$).
We find that the mode of the binned predicted metallicity distribution is higher than that of $Z_{\rm true}$. This result may be a consequence of the CNN overcompensating for its systematic under-prediction of metallicity for galaxies with $Z_{\rm true} > 9.1$. However, its effect on the entire distribution is small, and may be remedied simply by increasing the relative fraction of very high-$Z_{\rm true}$ objects. We find overall good qualitative agreement between the $Z_{\rm pred}$ and $Z_{\rm true}$ distributions.
\subsection{Scatter in $Z_{\rm pred}$ and $Z_{\rm true}$}
\begin{figure}
\includegraphics[width=\columnwidth]{02-prediction_summary.pdf}
\caption{\label{fig:predicting-metallicity}
Bivariate distribution of true galaxy metallicity ($Z_{\rm true}$) and CNN prediction ($Z_{\rm pred}$) is shown in the main panel. Overlaid are the median predicted metallicity (solid red line), RMSE scatter (dashed red lines), and NMAD scatter (dashed violet lines), in bins of $Z_{\rm true}$. The solid black line shows the one-to-one relation. The distribution of residuals ($Z_{\rm pred} - Z_{\rm true}$) is shown in the inset plot. In the upper panel, we again show the binned scatter, where the size of each marker is proportional to the number of galaxies in that bin. Each horizontal line corresponds to the average scatter over the entire test data set (and the global value indicated in the upper panel legend).}
\end{figure}
In Figure~\ref{fig:predicting-metallicity}, we compare the distributions of $Z_{\rm true}$ and $Z_{\rm pred}$ using a two-dimensional histogram (shown in grayscale in the main, larger panel). We also show the median predictions varying with binned $Z_{\rm true}$ (solid red line), in addition to the scatter in RMSE (dashed red) and NMAD (dashed violet), and also the one-to-one line (solid black). The running median agrees well with the one-to-one line, although at low metallicity we find that the CNN makes makes overpredictions.
A histogram of metallicity residuals is shown in the inset plot of the Figure~\ref{fig:predicting-metallicity} main panel. The $\Delta Z$ distribution is characterized by an approximately normal distribution with a heavy tail at large positive residuals; this heavy tail is likely due to the systematic over-prediction for low-$Z_{\rm true}$ galaxies.
There is also an overabundance of large negative $\Delta Z$ corresponding to under-predictions for high $Z_{\rm true}$, although this effect is smaller. We do not find significant correlations between $\Delta Z$ and galaxy observables including spectroscopic redshift, any combination of photometric color (including \hbox{$u$}\ and \hbox{$z$}\ bands), emission line signal-to-noise ratios, observed \hbox{$g$}\hbox{$r$}\hbox{$i$}-magnitudes, or axis ratios.
We now turn our attention to the upper panel of Figure~\ref{fig:predicting-metallicity}, which shows how the scatter varies with spectroscopically derived metallicity. The RMSE scatter and outlier-insensitive NMAD are both shown. Marker sizes are proportional in area to the number of samples in each $Z_{\rm true}$ bin, and the horizontal lines are located at the average loss (RMSE or NMAD) for the full test data set.
Predictions appear to be both accurate and low in scatter for galaxies with $Z_{\rm true} \approx 9.0$, which is representative of a typical metallicity in the SDSS sample. Where the predictions are systematically incorrect, we find that the RMSE increases dramatically. However, the same is not true for the NMAD; at $Z_{\rm true} < 8.5$, it asymptotes to $\sim 0.10$~dex, even though the running median is incorrect by approximately the same amount! This discrepancy is because the NMAD determines the scatter about the \textit{median} and not $\Delta Z = 0$, and thus, this metric becomes somewhat unreliable when the binned samples do not have a median value close to zero. Fortunately, the global median of $\Delta Z$ is $-0.006$~dex, or less than 10\% of the RMSE, and thus the global NMAD $= 0.067$~dex is representative of the outlier-insensitive scatter for the entire test data set.
This effect partly explains why the global NMAD ($0.067$~dex) is higher than the weighted average of the binned NMAD ($\sim 0.05$~dex). Also, each binned NMAD is computed using its local scatter, such that the outlier rejection criterion varies with $Z_{\rm true}$. To illustrate this effect with an example: $\Delta Z \approx 0.2$~dex would be treated as an $3\,\sigma$ outlier at $Z_{\rm true} = 9.0$, where the CNN is generally accurate, but the same residual would not be rejected as an outlier using NMAD for $Z_{\rm true} = 8.5$. Since the binned average NMAD depends on choice of bin size, we do not include those results in our analysis and only focus on the global NMAD. RMSE is a robust measure of both local and global scatter (although it becomes biased high by outliers).
\section{Interpreting the CNN} \label{sec:interpretation}
\subsection{Uncertainty in the scatter} \label{sec:scatter-uncertainties}
It is worth examining how reliable our estimate of RMSE$~=0.085$~dex is. Because we have a large data set, we can calculate uncertainties on the RMSE through multiple training/test realizations.
One possible method is by dividing our full data set into cross-validation and nested cross-validation splits in order to see how the RMSE varies.
For the first method (5-fold cross-validation), we take the entire data set from Section~\ref{sec:data} and split it into five 80\%/20\% training/test subsets, each of which is optimized independently. We then compute the mean and standard deviation of the five test samples' $Z_{\rm pred}$, and find that the RMSE~$= 0.0836 \pm 0.0005$. Because there are more training examples here than in our original training set, the mean RMSE is lower than what we have previously found in Section~\ref{sec:results}.
For the second method (nested cross-validation), we split the full data set into five 80\%/20\% training/validation-test splits, and then further divide the training/validation data sets into 75\%/25\% cross-validation splits. We compute the mean and standard deviation of $Z_{\rm pred}$ for the ensemble of 5-fold test splits. When we select the model with the best cross-validation score, the RMSE is $0.0823 \pm 0.0009$. The unweighted average of all training/validation models is RMSE~$=0.0831 \pm 0.0011$.
There is an additional source of scatter due to noise in the SDSS images' pixels. This noise is not uniform across SDSS images, and the \cite{Lupton2004} intensity scaling makes estimating or re-sampling the noise distribution challenging. Therefore, we do not account for the contribution of image noise to our estimate of the uncertainties.
\subsection{Impact of artificially increasing scatter} \label{sec:additional-scatter}
In order to simulate additional uncertainty that may arise from noisier measurements of spectral lines, we add normally distributed scatter to the $Z_{\rm true}$ values. We train our CNN as before (in Section~\ref{sec:results}), except that we add random $\sigma = \{0.03, 0.05, 0.10, 0.20\}$~dex of scatter to the target $Z_{\rm true}$. The smallest value, $0.03~$dex, is the same as the systematic uncertainty in the \cite{Tremonti2004} measurements, and $0.20$~dex represents the standard deviation for the entire $Z_{\rm true}$ distribution. We compare CNN-predicted $Z_{\rm pred}$ with the original $Z_{\rm true}$ (i.e., the underlying values without artificial scatter introduced) using the RMSE metric as before (Equation~\ref{eq:rmse}). For all values of additional scatter, the CNN is able to estimate $Z_{\rm pred}$ to RMSE~$ = \{0.0851, 0.0851, 0.0869, 0.0882\}$~dex respectively. These results show that the CNN is robust to extra scatter added in an unbiased way.
As a second test, we include random scatter drawn from a normal distribution centered at 0 and with standard deviation equal to the galaxy's redshift. This simple model can test how the CNN responds to redshift-dependent scatter in $Z_{\rm true}$. We note that this toy model disproportionately impacts higher-metallicity galaxies because our sample of lower-mass (and thus, lower-metallicity) galaxies is less complete at higher redshifts. After training on the original images with these modified $Z_{\rm true}$ values, we find that the resulting $Z_{\rm pred}$ are not strongly affected: RMSE~$= 0.0862$~dex.
These tests demonstrate that our CNN is robust to normally distributed scatter in $Z_{\rm true}$. Our results show initial promise for cases in which training data are more uncertain, such as at higher redshift. However, more testing is necessary to understand the effects of biased or correlated sources of uncertainty, and to account for the evolving relationships between metallicity and other observed properties \citeeg{Zahid2013,Salim2015}.
\subsection{Resolution and color effects} \label{sec:resolution}
\begin{figure}
\includegraphics[width=\columnwidth]{04-resolution.pdf}
\caption{\label{fig:resolution}
The effects of image resolution and color on CNN performance. Red and violet circular markers indicate scatter in the residual distribution ($\Delta Z$) measured using RMSE and NMAD, respectively, for \textit{gri} imaging. (Each point is analogous to the horizontal lines shown in Figure~\ref{fig:predicting-metallicity}.) We also show predictions from a random forest algorithm as open triangle markers, and constant $\langle Z_{\rm true}\rangle$ predictions as open square markers. Large, labeled squares indicate the scatter for images consisting of only \hbox{$r$}-band imaging and a combination of \hbox{$g$}- and \hbox{$r$}-bands.}
\end{figure}
Because our methodology is so computationally light, we can run the same CNN training and test procedure on images scaled to different sizes in order to understand the effects of image resolution. Our initial results use SDSS $38\arcsec \times 38 \arcsec$ cutouts resized to $128\times 128$ pixels, and we now downsample the same images to $64\times 64$, $32 \times 32$, $\cdots$, $2\times 2$, and even $1\times 1$ pixels via re-binning. All images retain their three channels, so the smallest $1 \times 1$ image is effectively the pixels in each of the \hbox{$g$}\hbox{$r$}\hbox{$i$}\ bands averaged together with the background and possible neighboring sources.
In Figure~\ref{fig:resolution}, we show the effects of image resolution by measuring the global scatter in $\Delta Z$ using the RMSE and NMAD metrics (shown in red and violet circular markers, respectively). Also shown is the scatter in $\Delta Z$ if we always predict the mean value of $Z_{\rm true}$ over the data set (shown using a square marker). This constant prediction effectively delivers the worst possible scatter, and the \cite{Tremonti2004} systematic uncertainty in $Z_{\rm true}$ of $\sim 0.03$~dex yields the best possible scatter. We find that both RMSE and NMAD decrease with increasing resolution, as expected if morphology or color gradients are instrumental to predicting metallicity.
There appears to be little improvement in scatter going from $1 \times 1$ to $2\times 2$ pixel images. $1\times 1$ three-color images contain similar information to three photometric data points (although because background and neighboring pixels are averaged in, they are less information-dense than photometry), which can be used to perform a crude spectral energy distribution (SED) fit. Therefore it unsurprising that the $1 \times 1$ CNN predictions perform so much better than the baseline mean prediction. A $2 \times 2$ three-color image contains four times as many pixels as a $1\times 1$ image, but because the object is centered between all four pixels, information is still averaged among all available pixels. Therefore, the scatter does not improve appreciably going from $1 \times 1$ to $2 \times 2$ resolution.\footnote{There is extra information in the $2\times 2$ pixel images in non-circularly symmetric cases. For an inclined disk, it is possible to roughly determine the orientation in the sky plane, but this information is not very useful. In the case of a major merger or interacting companion, the $2\times 2$ images may be more powerful than $1 \times 1$ images.}
The scatter is a strong function of resolution as the images are resolved from $2 \times 2$ to about $32 \times 32$ pixels. With further increasing resolution, improvement is still evident, although the scaling with scatter is noticeably weaker. Because the angular size of each image cutout stays the same, the pixel scale changes from $1\farcs184$ \hbox{pixel$^{-1}$}\ for $32 \times 32$ images, to $0\farcs592$ \hbox{pixel$^{-1}$}\ for $64 \times 64$ images, to $0\farcs296$ \hbox{pixel$^{-1}$}\ for $128 \times 128$ images. The native SDSS pixel resolution is $0\farcs396$ \hbox{pixel$^{-1}$}, such that the $64 \times 64$ and $128 \times 128$ resolutions result in the oversampling of each image. Thus, scatter is expected to plateau for images larger than $128 \times 128$. It is worth noting, however, that the CNN attempts to learn filters that depend on the size of the input image, so smaller images may result in the CNN training filters that are too low in resolution to be completely effective for prediction. Therefore, it is also not surprising that the CNN makes incremental gains for images with increasing resolution beyond $64 \times 64$ pixels.
We also train the CNN to predict metallicity using only the central $16\times16$-pixel regions of each SDSS $gri$ image. We find that the network is able to predict metallicity to within RMSE~$=0.0965$~dex. Because this value is higher than the scatter found when using the full-sized images, we conclude that the CNN loses valuable information when only the central regions are considered, and that relevant information for predicting global metallicity can be found in the galaxies' outer regions that are not probed by the 3'' SDSS spectroscopic fibers.
As a way of testing how the CNN responds to reduced color information, we have also repeated our training and testing routines using \hbox{$r$}-band and \hbox{$g$}\hbox{$r$}-band $128\times128$ images. In order to make use of our pretrained network, we modify the original, three color JPG images to correspond to either one- or two-band SDSS images. For the single-band imaging, we duplicate the \hbox{$r$}-band data into the blue and red JPG channels. For the \hbox{$g$}\hbox{$r$}-band images, the blue and red channels correspond to the \hbox{$g$}\ and \hbox{$r$}\ filters, while the green channel is the mean of the two.
The large squares labeled ``\textit{r}'' in Figure~\ref{fig:resolution} show that the network trained and tested on single-band images performs relatively poorly (RMSE~$= 0.1381$~dex) compared to $gri$ imaging even at low resolution. The addition of a second color improves CNN performance significantly. When a second band is added to the training images (box labeled ``\textit{gr}'' in Figure~\ref{fig:resolution}), the RMSE improves to a level similar ($0.0915$~dex) to that of the original three-color images ($0.0851$~dex). This enhancement may be due to extra information that the bluer \hbox{$g$}-band provides about younger stellar populations. In both examples, the CNN is able to utilize spatial information about a galaxy to improve metallicity estimates.
\subsection{Random forest predictions for metallicity}\label{sec:RF}
We also construct a random forest (RF) of decision trees in order to predict metallicity using the implementation from \texttt{scikit-learn} \citep{Pedregosa2012}. Hyperparameters are selected according to the optimal RF trained by \cite{Acquaviva2016}. We use exactly the same data labels (i.e., galaxies) to train/validate or test the RF that we have used for training and testing the CNN, so that our measurements of scatter can be directly compared. However, we have used the \hbox{$g$}\hbox{$r$}\hbox{$i$}\ three-band photometry data (given in magnitudes) to train and predict metallicity. Since each galaxy only has three pieces of photometric information, it can be compared to the $1 \times 1$ three-band ``images'' processed by our CNN.
The RF predicts metallicity with RMSE $= 0.130$~dex, which is superior to our CNN trained and tested on $1\times 1$ and $2 \times 2$ images. This result is unsurprising because the RF is supplied aperture-corrected photometry, whereas the CNN is provided $1 \times 1 $ \hbox{$g$}\hbox{$r$}\hbox{$i$}\ ``images'' whose features have been averaged with their backgrounds. $2 \times 2$ images are only marginally more informative. When the resolution is further increased to $4 \times 4$ images, then the CNN can begin to learn rough morphological features and color gradients, which is already enough to surpass the performance (measured by both RMSE and NMAD) of the RF.
This result suggests that the CNN is able to learn a nontrivial representation of gas-phase metallicity based on three-band brightness distributions, even with extremely low-quality data.
\subsection{Comparisons to previous work}\label{sec:previous work}
CNNs have been used for a wide variety of classification tasks in extragalactic astronomy, including morphological classification \citeeg{Dieleman2015, Huertas-Company2015, 2017MNRAS.464.4420S}, distinguishing between compact and extended objects \citep{Kim2017}, selecting observational samples of rare objects based on simulations \citep{Huertas-Company2018, Lanusse2017}, and visualizing high-level morphological galaxy features \citep{Dai2018}. These works seek to improve classification of objects into a discreet number of classes, i.e., visual morphologies. Our paper uses CNNs to tackle the different problem of regression, i.e., predict values from a continuous distribution.
Examples of regressing stellar properties in the astronomical ML literature \citeeg{2000A&A...357..197B, Fabbro2018} train on synthetic stellar spectra and test on real data. Their predicted measurements of stellar properties, e.g., stellar effective temperature, surface gravity, or elemental abundance, can be derived from the available training data set. Our work is novel because we predict metallicity, a spectroscopically determined galaxy property, using only three-color images. Said another way, it is not necessarily the case that $Z$ can be predicted from our training data. However, we find that galaxy shape supplements color information in a way that is useful for predicting metallicity.
A study similar to this work is that of \cite{Acquaviva2016}, who uses a variety of machine learning methods including RFs, extremely random trees (ERTs), boosted decision trees (AdaBoost), and support vector machines (SVMs) in order to estimate galaxy metallicity. The \cite{Acquaviva2016} data set consisted of a $z \sim 0.1$ sample (with $\sim 25,000$ objects) and a $z \sim 0.2$ sample (with $\sim 3,000$ objects), each of which has five-band SDSS photometry ($ugriz$) available as inputs. These samples are sparsely populated at low metallicities, and they contain smaller fractions of objects with $Z_{\rm true} < 8.5$ than our sample, but are otherwise similarly distributed in $Z_{\rm true}$ to ours. Our samples have different sizes because we require SDSS objects to have imaging available, whereas the \cite{Acquaviva2016} criteria impose stronger spectroscopic redshift constraints.
We will first compare RF results, since this technique is common to both of our analyses, and they reveal important differences in our training data. Because outliers are defined differently in both works, we will use the RMSE metric to compare scatter between the two. \cite{Acquaviva2016} obtain RMSE of 0.081 and 0.093~dex when using RFs on the five-band photometry for the $z \sim 0.1$ and $0.2$ subsamples. Using exactly the same RF approach on a larger sample, while working with only \textit{three} bands of photometric information, we find RMSE $= 0.130$~dex. Our scatter is larger than the value reported by \cite{Acquaviva2016} by a factor of $\sim 1.5$. This result may partly be explained by the fact that \cite{Acquaviva2016} $Z_{\rm true}$ distribution is narrower than for our training data set, or the fact that our data set spans a broader range in galaxy redshift; however, some of this advantage is offset by our larger sample size.
Ultimately, it appears that the extra \hbox{$u$}\ and \hbox{$z$}\ bands supply machine learning algorithms with valuable information for predicting metallicity.
Indeed, the \hbox{$u$}\ and \hbox{$z$}-bands convey information about a galaxy's SFR and stellar mass \cite[see, e.g.,][]{Hopkins2003}. For this reason, it is possible that the RF trained on five-band photometry can estimate $Z_{\rm true}$ down to the limit of the FMR, which has very small scatter ($\sim 0.05$~dex) at fixed $M_{\star}$ \textit{and} SFR. The \hbox{$g$}{}, \hbox{$r$}{}, and \hbox{$i$}{} bands are less sensitive to the SFR, but can still provide some information about the stellar mass, and so our RF and CNN results are more linked to the MZR rather than the FMR.
Regardless of these limitations, our CNN is able to estimate metallicity with $\Delta Z = 0.085$~dex, which is comparable to the scatter in residuals using the best algorithms from \cite{Acquaviva2016}. There is evidence that the morphological information provided by using images rather than photometric data is helping the CNN perform so well: (1) the RMSE scatter decreases with increasing image resolution, and (2) it identifies edge-on galaxies as lower-$Z_{\rm pred}$ and face-on galaxies as higher-$Z_{\rm pred}$ (consistent with observational bias). Gradients in color, or identification of mergers \citeeg{Ackermann2018} may also be helpful for predicting metallicity.
\section{The mass-metallicity relation} \label{sec:MZR}
The MZR describes the tight correlation between galaxy stellar mass and nebular gas-phase metallicity. Scatter in this correlation is approximately $\sigma \approx 0.10$~dex in $Z_{\rm true}$ over the stellar mass range $8.5 < \log (\hbox{$M_{\star}$} / \hbox{$M_\odot$}) < 11.5$ \citep{Tremonti2004}, where $\sigma$ is the standard deviation of the metallicity and is equivalent to the RMSE for a normal distribution. The MZR at $z=0$ can be characterized empirically using a polynomial fit:
\begin{equation}\label{eq:mzr}
Z = -1.492 + 1.847 \log (\hbox{$M_{\star}$} / \hbox{$M_\odot$}) - 0.08026 \left [\log(\hbox{$M_{\star}$} / \hbox{$M_\odot$})\right ]^2.
\end{equation}
The physical interpretation of the MZR is that a galaxy's stellar mass strongly correlates with its chemical enrichment. Proposed explanations of this relationship's origin include metal loss through blowout \citep[see, e.g.,][]{2002ApJ...581.1019G,Tremonti2004,Brooks2007,Dave2012}, inflow of pristine gas \cite{Dalcanton2004}, or a combination of the two \citeeg{2013ApJ...772..119L}; however, see also \cite{2013A&A...554A..58S}. Although the exact physical process responsible for the low ($0.10$~dex) scatter in the MZR is not known, its link to SFR via the FMR is clear, as star formation leads to both metal enrichment of the interstellar medium and stellar mass assembly.
The FMR connects the instantaneous ($\sim 10$ Myr) SFR with the gas-phase metallicity \citep[$\sim 1$~Gyr timescales; see, e.g.,][]{2011ApJ...734...48L} and \hbox{$M_{\star}$}{} (i.e., the $\sim 13$~Gyr integrated SFR). Our CNN is better suited for predicting \hbox{$M_{\star}$}{} rather than SFR, using the \hbox{$g$}\hbox{$r$}\hbox{$i$} bands, which can only weakly probe the blue light from young, massive stars. Therefore, we expect the scatter in CNN predictions to be limited by the MZR (with scatter $\sigma \sim 0.10$~dex) rather than the FMR ($\sigma \sim 0.05$~dex). It is possible that galaxy color and morphology, in tandem with CNN-predicted stellar mass, can be used to roughly estimate the SFR, but in this paper we will focus on only the MZR.
\subsection{Predicting stellar mass}
\begin{figure*}
\includegraphics[width=\columnwidth]{05-a-prediction_mass.pdf}
\includegraphics[width=\columnwidth]{05-b-prediction_mzr.pdf}
\caption{\label{fig:mass-metallicity}
In the left panel, we plot the CNN predicted galaxy stellar mass against true stellar mass. Colors and marker or line styles are the same as in Figure~\ref{fig:predicting-metallicity}. In the right panel, we compare the predicted stellar mass converted to metallicity, assuming the \citet{Tremonti2004} MZR, against the true metallicity.
These findings indicate that using the empirical MZR and CNN-predicted $M_{\star,\rm pred}$ yields poor results, unlike what we have observed in Figure~\ref{fig:predicting-metallicity}.
}
\end{figure*}
Since galaxy stellar mass is known to strongly correlate with metallicity, and is easier to predict (than, e.g., SFR) from \hbox{$g$}\hbox{$r$}\hbox{$i$}\ imaging, we consider the possibility that the CNN is simply predicting stellar mass ($M_{\star,\rm pred}$) accurately and then learning the simple polynomial transformation in order to estimate metallicity. We can simulate this scenario by training the CNN on $M_{\star, \rm true}$ and then converting the stellar mass predictions to metallicities using Equation~\ref{eq:mzr}.
We re-run the CNN methodology to train and predict $M_{\star}$ using the 116,394 available images (out of the 142,145/142,186 original objects that have stellar mass measurements). These results are shown in the left panel of Figure~\ref{fig:mass-metallicity}. From the same subsample as before (minus three objects that do not have \hbox{$M_{\star}$} estimates), we verify that $M_{\rm \star, true}$ median agrees with the median of $M_{\rm \star, true}$ for values between $9.0 \lesssim \log \hbox{$M_{\star}$} /\hbox{$M_\odot$} \lesssim 10.5$. The RMSE scatter in the \hbox{$M_{\star}$}{} residuals is $\sim 0.22$~dex, and the NMAD is $\sim 0.20$~dex. The slope of the empirical MZR at $\log (\hbox{$M_{\star}$} / \hbox{$M_\odot$}) \sim 10$ is (0.4~dex in $Z$)/(1.0~dex in \hbox{$M_{\star}$}), implying that the CNN might be able to leverage the MZR and predict metallicity to $\sim 0.08$~dex (plus any intrinsic scatter in the MZR, in quadrature).
We use Equation~\ref{eq:mzr} and $M_{\star,\rm pred}$ to predict metallicity, which we call $Z_{\rm MZR}$. In the right panel of Figure~\ref{fig:mass-metallicity}, we compare $Z_{\rm MZR}$ against $Z_{\rm true}$. The scatter in residuals $Z_{\rm MZR} - Z_{\rm true}$ is $0.12$~dex, which is significantly higher than the $0.085$~dex scatter reported in Section~\ref{sec:results}. If the MZR alone were mediating the CNN's ability to estimate from
\hbox{$g$}\hbox{$r$}\hbox{$i$}\ imaging, then we would expect the scatter for $Z_{\rm pred}$ to be greater than for $Z_{\rm MZR}$; instead we find that the opposite is true. This evidence suggests that the CNN has learned to determine metallicity in a more powerful way than by simply predicting $M_{\rm \star,pred}$ and then effectively applying a polynomial conversion.
\subsection{An unexpectedly strong CNN-predicted mass-metallicity relation}
\begin{figure}
\includegraphics[width=\columnwidth]{06-mzr.pdf}
\caption{\label{fig:mzr}
In the main panel, the predicted MZR comparing true \hbox{$M_{\star}$}\ against CNN-predicted $Z_{\rm pred}$ is shown in grayscale. The running median (solid red) and scatter (dashed red and violet) are shown in 0.2~dex mass bins. For comparison, we also show the \citet{Tremonti2004} observed median and scatter (solid and dashed black lines, respectively), which are binned by 0.1~dex in mass. In the upper panel, we show the scatter in the predicted and empirical MZR. The standard deviation of the scatter for the empirical MZR is shown as a dashed black line, while the red and violet circles respectively show RMSE and NMAD for the predicted MZR. Marker sizes are proportional to the number of galaxies in each stellar mass bin for the test data set. Global scatter in the CNN-predicted MZR appears to be comparable to, or even lower than, scatter in the empirical MZR.}
\end{figure}
The RMSE $= 0.085$~dex difference between the true and CNN-predicted metallicities can be interpreted in one of two ways: (1) the CNN is inaccurate, and $Z_{\rm pred}$ deviates randomly from $Z_{\rm true}$, or (2) the CNN is labeling $Z_{\rm pred}$ according to some other hidden variable, and $\Delta Z$ residuals represent non-random shifts in predictions based on this variable. If the first scenario were true, we would expect the random residuals to increase the scatter of other known correlations such as the MZR when predicted by the CNN. If the second were true, we would expect the scatter of such correlations to remain unchanged or shrink. We can therefore contrast the MZR constructed from $Z_{\rm pred}$ and from $Z_{\rm true}$ in order to test these interpretations.
In the main panel of Figure~\ref{fig:mzr}, we plot CNN-predicted metallicity versus true stellar mass. For comparison, we also overlay the \cite{Tremonti2004} MZR median relation and its $\pm 1~\sigma$ scatter (which is $\sim 0.10$~dex). Their empirical median relation (solid black) matches our predicted MZR median (solid red), and the lines marking observed scatter (dashed black) match $Z_{\rm pred}$ scatter as well (dashed red and violet). Over the range $9.5 \leq \log (M_{\star, \rm true}/\hbox{$M_\odot$}) \leq 10.5$, the RMSE scatter in $Z_{\rm pred}$ appears to be even tighter than the observed $\pm 1\,\sigma$ (dashed black). The same is true for the NMAD, which is even lower over the same interval.
In the upper panel of Figure~\ref{fig:mzr}, we present the scatter in both predicted and \cite{Tremonti2004} MZR binned by mass. We confirm that the CNN predicts a MZR that is at most equal in scatter than one constructed using the true metallicity. The stellar mass bins for which our CNN found tighter scatter than the empirical MZR are the same bins that happen to contain the greatest number of examples ($9.5 \leq \log (M_{\star, \rm true}/\hbox{$M_\odot$}) \leq 10.5$); thus, the strong performance of our network at those masses may be due to a wealth of training examples. If our data set were augmented to include additional low- and high-$M_{\rm \star,true}$ galaxies, then the scatter in the predicted MZR could be even lower overall.
The fact that a CNN trained on only \hbox{$g$}\hbox{$r$}\hbox{$i$}\ imaging is able to predict metallicity accurately enough to reproduce the MZR in terms of median and scatter is not trivial. The error budget is very small: $\sigma = 0.10$~dex affords only, e.g., 0.05~dex of scatter when SFR is a controlled parameter plus a 0.03~dex systematic scatter in $Z_{\rm true}$ measurements, leaving only $\sim 0.08$~dex remaining for CNN systematics, assuming that these errors are not correlated and are added in quadrature. This remaining error budget may barely be able to accommodate our result of RMSE($\Delta Z$) $= 0.085$.
Interpreting the MZR scatter as the combination of intrinsic FMR scatter, $Z_{\rm true}$ systematics, and $\Delta Z$ systematics cannot be correct since it assumes that the CNN is recovering the FMR perfectly.
As we have discussed previously, it is highly unlikely that the CNN is sensitive to the SFR, and therefore cannot probe the MZR at individual values of the SFR.
If we assume that the error budget for the MZR is not determined by the FMR, then the error ``floor'' should be $0.10$~dex.
This is immediately exceeded, as we have found RMSE $\approx 0.10$~dex for the predicted MZR without accounting for the fact that $Z_{\rm pred}$ and $Z_{\rm true}$ differ by RMSE $= 0.085$~dex!
Consider the case in which all $Z_{\rm true}$ values are shifted randomly by a Gaussian noise distribution with $\sigma = 0.085$~dex.
These shifted values should not be able to reconstruct a correlation without introducing additional scatter unless the shifts were not arbitrary to begin with.
We thus find more evidence that the CNN has learned something from the SDSS \hbox{$g$}\hbox{$r$}\hbox{$i$}\ imaging that is different from, but at least as powerful as, the MZR. One possible explanation is that the CNN is measuring some version of metallicity that is more fundamentally linked to the stellar mass, rather than $Z_{\rm pred}$ as derived from oxygen spectral lines. Another possibility is that the MZR is a projection of a correlation between stellar mass, metallicity, and a third parameter, perhaps one that is morphological in nature. If this is the case, then the \cite{Tremonti2004} MZR represents a relationship that is randomly distributed in the yet unknown third parameter, while our CNN would be able to stratify the MZR according to this parameter (much as the FMR does with the SFR). We are unfortunately not able to identify any hidden parameter using the current CNN methodology, but we plan to explore this topic in a future work.
\section{Summary}\label{sec:summary}
We have trained a deep convolutional neural network (CNN) to predict galaxy gas-phase metallicity using only $128 \times 128$-pixel, three-band (\hbox{$g$}\hbox{$r$}\hbox{$i$}), JPG images obtained from SDSS. We characterize CNN performance by measuring scatter in the residuals between predicted ($Z_{\rm pred}$) and true ($Z_{\rm true}$) metallicities.
Our conclusions are as follows:
\begin{enumerate}
\item By training for a half-hour on a GPU, the CNN can predict metallicity well enough to achieve residuals characterized by RMSE $= 0.085$~dex (or outlier-insensitive NMAD $= 0.067$~dex). These findings may be promising for future large spectroscopy-limited surveys such as LSST.
\item We find that the residual scatter decreases in an expected way as resolution or number of channels is increased, suggesting that the CNN is leveraging both the spatial information about a galaxy's light distribution and the color in order to predict metallicity.
\item The CNN outperforms a random forest trained on $gri$ photometry if provided images larger than $4\times 4$ pixels, and is as accurate as a random forest trained on $ugriz$ photometry when given $128 \times 128$ pixel \hbox{$g$}\hbox{$r$}\hbox{$i$}\ images.
\item We find that scatter in the mass-metallicity relation (MZR) constructed using CNN-predicted metallicities is as tight as the empirical MZR ($\sigma = 0.10$~dex). Because predicted metallicities differ from the ``true'' metallicities by RMSE $= 0.085$~dex, the only way that the predicted MZR can have such low scatter is if the CNN has learned a connection to metallicity that is more strongly linked to the galaxies' light distributions than their nebular line emission.
\end{enumerate}
All of the code used in our analysis and for making the figures can be accessed at \url{https://github.com/jwuphysics/galaxy-cnns}.
\section*{Acknowledgements}
SB is supported by NSF Astronomy and Astrophysics Research Program award number 1615657.
The authors thank the anonymous referee for useful comments that have improved this paper, particularly in terms of its scientific content.
The authors thank Andrew Baker, Eric Gawiser, and John Hughes for helpful comments and discussions, and also thank David Shih and Matthew Buckley for the use of their GPU cluster in the Rutgers University Experimental High Energy Physics department.
JW thanks Jeremy Howard, Rachel Thomas, and the development team for creating the \texttt{fastai} on-line courses and deep learning library.\footnote{\url{https://github.com/fastai/fastai}}
JW also thanks Florian Peter for valuable assistance with using the \texttt{fastai} library.
This research made use of the {\tt IPython} package \citep{Perez2007} and {\tt matplotlib}, a Python library for publication quality graphics \citep{Hunter2007}.
Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges
support and resources from the Center for High-Performance Computing at
the University of Utah. The SDSS web site is \url{www.sdss.org}.
SDSS-IV is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS Collaboration including the
Brazilian Participation Group, the Carnegie Institution for Science,
Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics,
Instituto de Astrof\'isica de Canarias, The Johns Hopkins University,
Kavli Institute for the Physics and Mathematics of the Universe (IPMU) /
University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory,
Leibniz Institut f\"ur Astrophysik Potsdam (AIP),
Max-Planck-Institut f\"ur Astronomie (MPIA Heidelberg),
Max-Planck-Institut f\"ur Astrophysik (MPA Garching),
Max-Planck-Institut f\"ur Extraterrestrische Physik (MPE),
National Astronomical Observatories of China, New Mexico State University,
New York University, University of Notre Dame,
Observat\'ario Nacional / MCTI, The Ohio State University,
Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group,
Universidad Nacional Aut\'onoma de M\'exico, University of Arizona,
University of Colorado Boulder, University of Oxford, University of Portsmouth,
University of Utah, University of Virginia, University of Washington, University of Wisconsin,
Vanderbilt University, and Yale University.
\bibliographystyle{mnras}
|
1,941,325,220,489 | arxiv | \section{Introduction}\label{sec:intro}
The way in which atoms vibrate around their equilibrium positions as a function of temperature is of fundamental importance for a range of physical properties of solids, for example thermal expansion, specific heat, thermal conductivity, and superconductivity. These vibrations can be studied computationally using molecular dynamics (MD) simulations, which is nowadays a mature and widely used technique in computational materials science. Phonon spectra and other properties related to the atomic vibrations are nowadays routinely computed. To address systems with millions of atoms with MD, empirical potentials are usually necessary. Only for relatively small systems are MD simulations at the first-principles level feasible.\cite{Car1985,Marx2009aim}
In systems with magnetic order, there also exist collective motion of the spins, in addition to the above mentioned lattice vibrations. The standard approach to simulate the time evolution of the spin texture is to propagate the Landau-Lishitz-Gilbert (LLG) equation. Both continuum models (usually called "micromagnetics") \cite{Fidler2000} and atomistic models, so-called atomistic spin dynamics (ASD),\cite{Antropov1996,Nowak2005,Skubic2008,Evans2014,Eriksson2017asd} have been developed. In principle, spin motion can also be addressed directly at the first-principles level,\cite{Antropov1996,Katsnelson2003,Bhattacharjee2012,Fransson2017} as is possible in the framework of time-dependent density functional theory,\cite{Krieger2015} although this normally requires too much computing resources and time to be a realistic approach for most systems of interest. Also, in order to take dissipation and fluctuations into account in spin-dynamic simulations, a phenomenological stochastic approach is normally employed, for details see e.g. Ref. \onlinecite{Eriksson2017asd}. A full microscopic description of dissipation would require explicit descriptions of all the spin-electron couplings as well as all spin-lattice couplings.
In reality, the atomic magnetic moment and lattice degrees of freedom are always more or less coupled, a coupling which is mediated by the electronic subsystem. These couplings will determine, for instance, how fast it is possible to change the magnetic state of a material, and how relaxation of phonon and electronic subsystems proceed after excitation with ultra-short laser pulses.\cite{Maldonado2017} In addition, these couplings may shine light on angular momentum transfer between the spin and lattice subsystems in pump-probe experiments.
Already in the original work on the equations of motion of atomic moments\cite{Antropov1996} a formalism that allows for coupled spin-lattice simulations was provided. Recently, such coupling was derived from a different ansatz, in which the role of the underlying electronic structure in mediating these couplings is explicitly considered.\cite{Fransson2017} However, the application to real materials remains a challenge for any formulation of spin-lattice simulations.
As already pointed out, dissipation is one of the consequences of these couplings. Since the electron motion is several orders of magnitude faster than both spin and lattice motion, it can, for some purposes, be integrated out. \cite{Antropov1996,Skubic2008,Eriksson2017asd} The spin and lattice degrees of freedom, however, occur at a much slower and roughly equivalent time scale and need to be addressed in a unified way, self-consistently. \cite{Antropov1996,Omelyan2001,Ma2008} Figure~\ref{fig:principle} shows a schematic picture of coupled spin-lattice dynamics. It has been demonstrated that the exchange interactions between atoms in several magnetic materials can depend strongly on the local atomic environment, and vice-versa; the chemical interaction may depend on magnetic configuration.\cite{Shimizu1981,Sabiryanov1995,Sabiryanov1999,Bonetti2016} Hence, both the magnon and phonon spectrum and lifetimes in a material may depend on the configuration of magnetic state or the displacement of atoms.\cite{Hasegawa1987,Koermann2014}
Several studies point at the importance of phonon-magnon coupling in a number of dynamical processes such as demagnetization processes,\cite{Carpene2008,Wietstruk2011,Reid2018} thermal conductivity,\cite{Boona2014,Sanders1977}
magneto-acoustics,\cite{Kim2012prl,Scherbakov2010,Henighan2016}
and the spin-Seebeck effect.\cite{Jaworski2011,Man2017,Maehrlein2017pp}
The interaction between spin and lattice motion is also central for phenomena observed in magnetoelectric and in multiferroic materials,\cite{Pimenov2006,ValdesAguilar2007,Sushkov2008,Takahashi2012,Kubacka2014,Wang2012,Tokura2014,Bossini2018}
magnetocaloric materials,\cite{Tishin2014}
skutterudites,\cite{Rasander2015} and
antiferromagnetic insulator materials for spintronic devices.\cite{Aytan2017} Among recent developments on methods for modelling of phonon-magnon coupling can be mentioned a novel combination of atomistic spin dynamics and ab initio molecular dynamics, applied to the paramagnetic phase of the magnetic insulator CrN,\cite{Stockem2018pp} and a scheme for massively parallel symplectic integration of spin-lattice dynamics equations of motion.\cite{Tranchida2018accepted}
In the present work, we describe a general method for the simulation of coupled spin-lattice dynamics, where all information needed for the effective spin-lattice dynamics Hamiltonian can be obtained from first-principles theory. We demonstrate the accuracy of the method by applying it to bcc Fe, as well as a selection of smaller clusters. The developed method is based on an atomistic approach and draws its strengths from the atomistic spin dynamics framework. In philosophy the method proposed here is similar to the early formulation of Ref. \onlinecite{Antropov1996}, but the practical details are naturally different. The interactions are described using a general Hamiltonian, with parameters computed using density functional theory. This hopefully provides a tool for analysis and even prediction of complex collective modes of magnetic materials that is a complement to experimental activities addressing these questions, for instance inelastic neutron scattering (INS) \cite{Squires2012itt} and resonant inelastic X-ray scattering (RIXS).\cite{2013msr,Ament2011,Schmitt2016} We note that the instrumentation and capabilities of these spectroscopies undergo a rapid development, for instance in form of prismatic analysers for neutron spectrometers \cite{Birk2014} for use in the CAMEA instrument \cite{Groitl2016} at the Paul-Scherrer Institute and in the BIFROST instrument \footnote{BIFROST: https://europeanspallationsource.se/article/bifrost-prismatic-approach-neutron-spectroscopy} commissioned for the European Spallation Source (ESS), and furthermore that INS and RIXS are complementary techniques that enable characterization of excitations throughout large parts of the Brillouin zone.\cite{Toth2016}
\begin{figure}[ht]
\centering
\includegraphics[width=0.50\columnwidth]{principle_SLD.png}
\caption{(Color online) Conceptual figure of the breathing mode in a trimer of magnetic moments. The spins and atoms are represented as green arrows and gray balls, respectively.}
\label{fig:principle}
\end{figure}
Using an empirical potential approach,\cite{Dudarev2005,Derlet2007} spin-lattice dynamics simulations of bcc Fe have been published by several groups \cite{Ma2008,Ma2012,Perera2014,Perera2017}. To put our method in perspective and on firm quantitative ground, we therefore selected to specifically address the spin-lattice interaction in bcc Fe as a test case.
The rest of the paper is organized as follows.
In Section~\ref{sec:methods} we describe the Hamiltonian for coupled spin-lattice dynamics and the associated coupled equations of motion, techniques for calculation of the adiabatic magnon and phonon spectra, and a scheme for numerical integration of the coupled equations of motion. Section~\ref{sec:results} begins with a discussion of the dynamics of magnetic dimers, trimers, and quadmers, and continues with the results for bcc Fe. Finally, we discuss the applicability our of method and give an outlook in Sec.~\ref{sec:conc}.
\section{Model and methods}
\label{sec:methods}
This section is split into five parts. First (Section~\ref{sec:SLDham}), we discuss the underlying Hamiltonian, which includes couplings within the spin and lattice reservoirs, respectively, and interactions between the spins and the lattice. Then, the couplings between the spin and lattice degrees of freedom are discussed in more detail in Section~\ref{sec:exchstr}. We motivate the equations-of-motion and the corresponding observables in Sections~\ref{sec:EOM} and \ref{sec:observ}, respectively. Finally, in Section~\ref{sec:DFT}, we describe how the coupling constants are obtained from density functional theory calculations.
\subsection{The spin-lattice Hamiltonian}
\label{sec:SLDham}
We consider the parametrised Hamiltonian
\bgea
\label{eq:HamSLD}
\mathcal{H}_{\rm SLD}=\mathcal{H}_{\rm S}+\mathcal{H}_{\rm L}+\mathcal{H}_{\rm LS},
\enea
formulated in terms of atomic magnetic moments $\{\bfm_i\}$, ionic displacements $\{\bfu_k\}$ and velocities $\{\bfv_k\}$. The first term is a Hamiltonian describing purely magnetic interactions. The second term contains energies associated with pure lattice displacements, and the third term couples the spin and lattice degrees of freedom. Indices $i,j$ run over atoms $1,\ldots,N_{\rm mag}$ with a finite magnetic moment, whereas indices $k,l$ run over all the atoms in the simulation cell $1,\ldots,N_{\rm all}$, i.e. non-magnetic as well as magnetic ions. Note that for the examples considered in the present paper, all ions are magnetic. Furthermore, $\alpha,\beta\in\{x,y,z\}$ denote the Cartesian components in spin space, and $\mu,\nu\in\{x,y,z\}$ denote the Cartesian components in real space.
In the following, we consider contributions up to a combined order of four. The harmonic approximation to lattice dynamics is described by
\bgea
\label{eq:HamLL}
\mathcal{H}_{\rm LL} = \frac{1}{2}\sum_{kl}\Phi_{kl}^{\mu\nu} u_k^{\mu} u_l^{\nu}
+ \frac{1}{2} \sum_k M_k (v_k^{\mu})^2,
\enea
where the force constant tensor $\Phi_{kl}^{\mu\nu}$ is a rank 2 tensor in real space and $M_k$ is the mass of atom $k$. Since $\Phi_{kl}$ depends also on the magnetic configuration one can extend the expression above to
\bgea
\label{eq:HamLLb}
\mathcal{H}_{\rm LL} = \frac{1}{2}\sum_{kl}\left(\Phi_{kl}^{\mu\nu}
+ \frac {\partial \Phi_{kl}^{\mu\nu}} {\partial m^{\alpha}_i} \delta m^{\alpha}_i \right. \\
+ \left. \frac {\partial^2 \Phi_{kl}^{\mu\nu}} {\partial m^{\alpha}_i \partial m^{\beta}_j} \delta m^{\alpha}_i \delta m^{\beta}_j \right) u_k^{\mu} u_l^{\nu}
+ && \frac{1}{2} \sum_k M_k (v_k^{\mu})^2. \nonumber
\enea
For some materials, the force constants depend significantly on the spin configuration as well as the configuration of nuclei. As an example we note that the traditional explanation for Invar alloys relies on the coupling between the force constants and the spin configuration,\cite{Schilfgaarde1999} as well as materials where many-body terms of the description of nuclear motion are needed. The main purpose of this paper is to outline a general formalism of coupled spin-lattice dynamics and to give examples of how the coupling modifies the dynamical properties in a few selected cases. In the illustration of the method we have for simplicity neglected the second and third terms inside the parenthesis, on the right-hand side of Eq. \ref{eq:HamLLb}, and instead kept a coupling term that originates from the Taylor expansion of exchange parameters (see below). The importance of various contributions to the coupling is materials dependent, but one may note that higher order contributions in the Taylor expansion are expected to be smaller.
The bilinear spin Hamiltonian $\mathcal{H}_{\rm SS}$ contains Heisenberg exchange, Dzyaloshinskii-Moriya interaction and symmetric, anisotropic interactions that in a compact form can be expressed as
\bgea
\label{eq:HamMM}
\mathcal{H}_{\rm SS} = -\frac{1}{2}\sum_{ij} \mathcal{J}_{ij}^{\alpha\beta}(\{u_k^{\mu}\}) m_i^{\alpha} m_j^{\beta}.
\enea
The exchange tensor $\mathcal{J}_{ij}^{\alpha\beta}$ is a rank 2 tensor in spin space, with elements that in general have a dependence on the atomic displacements $\{u_k^{\mu}\}$ as well as the magnetic configuration. For clarity we specify in Eqn.4, the explicit dependence of tensor $\mathcal{J}_{ij}^{\alpha\beta}$ on $\{u_k^{\mu}\}$. The contributions to the mixed spin-lattice Hamiltonian $\mathcal{H}_{\rm LS}$ can then be obtained by expanding the bilinear magnetic Hamiltonian $\mathcal{H}_{\rm SS}$ up to second order in displacement, i.e.
\bgea
\label{eq:HamMMexp}
\mathcal{H}_{\rm SS}
&=& -\frac{1}{2}\sum_{ij} \mathcal{J}_{ij}^{\alpha\beta} m_i^{\alpha} m_j^{\beta} - \frac{1}{2}\sum_{ijk} \frac{\partial \mathcal{J}_{ij}^{\alpha\beta}}{\partial u_k^{\mu}} u_k^{\mu} m_i^{\alpha} m_j^{\beta} \nonumber\\
&& - \frac{1}{4}\sum_{ijkl} \frac{\partial^2 \mathcal{J}_{ij}^{\alpha\beta}}{\partial u_k^{\mu}\partial u_l^{\nu}} u_k^{\mu} u_l^{\nu} m_i^{\alpha} m_j^{\beta}.
\enea
Note that both in Eq. \ref{eq:HamMMexp} and Eq. \ref{eq:HamLLb} a term enters, that contains bilinear couplings in both spin- and lattice displacement. In one case it appears due to Taylor expansion of the Heisenberg exchange parameter in lattice displacement, and in the other it appears due to a Taylor expansion of the force constant in magnetic moments. In general both contributions are additive, as they have similar mathematical form. The relative importance of these two contributions is materials dependent and should preferably be calculated for the material one wants to investigate. The four-body interaction accounts for a renormalization of the phonon dispersion due to spin configuration. It also results in a renormalization of the magnon dispersion due to atomic displacements, and enables photon absorption by phonon-assisted multimagnon excitation.\cite{Lorenzana1995,Lorenzana1995a,Hellsvik2016}
Introducing the coupling constant
$\Gamma_{ijk}^{\alpha\beta\mu}=\nicefrac{\partial \mathcal{J}_{ij}^{\alpha\beta}}{\partial u_k^{\mu}}$
we may write the three-body interaction as
\bgea
\label{eq:HamMML}
\mathcal{H}_{\rm SSL} &=& -\frac{1}{2}\sum_{ijk}\Gamma_{ijk}^{\alpha\beta\mu} m_i^{\alpha} m_j^{\beta} u_k^{\mu}.
\enea
This represents a spin-lattice coupling which is bilinear in spin and linear in displacement, i.e. $\Gamma_{ijk}^{\alpha\beta\mu}$ is a rank 3 tensor given by the direct product of a rank 2 tensor in spin space and a rank 1 tensor in orbital space.
Defining
$\Lambda_{ijkl}^{\alpha\beta\mu\nu} = \nicefrac{\partial^2 \mathcal{J}_{ij}^{\alpha\beta}}{\partial u_k^{\mu}\partial u_l^{\nu}}$
the four-body interaction reads
\bgea
\label{eq:HamMMLL}
\mathcal{H}_{\rm SSLL} &=& -\frac{1}{4}\sum_{ijkl}\Lambda_{ijkl}^{\alpha\beta\mu\nu} m_i^{\alpha} m_j^{\beta} u_k^{\mu} u_l^{\nu},
\enea
where the factor $1/4$ is due to that this interaction is bilinear both in spin and in displacements.
\begin{comment}
\anna{Is the next paragraph really needed? (I tried to reformulate it to connect it logically to the previous text but it still looks off topic.) In the method section, it's a good idea to stick to just the method stuff.}
Long range magnetic order manifests itself as ferromagnetism and other, more complex magnetic orders or textures such as antiferromagnetism, ferrimagnetism, spiral and helical order.
However, magnetic correlations have much larger implications than simply leading to long-range magnetic order.
In the paramagnetic phase, short-range spin correlations can be present and have important and unexpected consequences.
In some materials, the dominating interatomic force constants are short-ranged, wherefore local modulation of the force constants, here due to short range spin correlations, can play a role.
An example is the geometrically frustrated spinel ZnCr$_2$O$_4$ \cite{Fennie2006,Sushkov2005}, for which it was shown that spin correlations in the paramagnetic phase affect the phonon dispersion.
\end{comment}
Taken together, the combined spin-lattice Hamiltonian reads
\bgea
\label{eq:HamSLDexp}
\mathcal{H}_{\rm SLD}
&=& -\frac{1}{2}\sum_{ij} \mathcal{J}_{ij}^{\alpha\beta} m_i^{\alpha} m_j^{\beta} - \frac{1}{2}\sum_{ijk} \Gamma_{ijk}^{\alpha\beta\mu} u_k^{\mu} m_i^{\alpha} m_j^{\beta} \nonumber\\
&& -\frac{1}{4}\sum_{ijkl} \Lambda_{ijkl}^{\alpha\beta\mu\nu} u_k^{\mu} u_l^{\nu} m_i^{\alpha} m_j^{\beta} + \frac{1}{2}\sum_{kl}\Phi_{kl}^{\mu\nu} u_k^{\mu} u_l^{\nu} \nonumber\\
&& + \frac{1}{2} \sum_k M_k v_k^{\mu}v_k^{\mu}.
\enea
We note that in order to be even more general, higher coupling such as biquadratic exchange,\cite{Hayden2010,Hellsvik2014,Hellsvik2016} four-ring exchange,\cite{Fedorova2015} as well as third and fourth order\cite{Dove1993itl} interatomic lattice potential can be added, which is relatively straight-forward to do, and that for a very accurate description of the magnetic exchange dependence on the magnetic configuration needs to be considered.\cite{Szilva2013} Electrostatic contributions to the interatomic force field is also a relevant generalization to consider, since they can be important in polar materials, especially for excitations to the zone center. Likewise, magnetostatic interactions are sometimes of relevance.
\subsection{Exchange striction}
\label{sec:exchstr}
The third-order spin-lattice coupling is considered usually in insulating magnets where the spin texture simultaneously breaks time and spatial reversion. This occurs, for instance, when describing ferroelectric polarization and multiferroic phases,\cite{Tokura2014} and it drives the magnetoelectric response in the electromagnetic field driven dynamics in the GHz and THz regime.\cite{Pimenov2006,ValdesAguilar2007,Sushkov2008,Takahashi2012,Kubacka2014}
The isotropic (with regard to spin space) part of the $\Gamma_{ijk}^{\alpha\beta\mu}$ tensor is the exchange striction where the Heisenberg coupling between magnetic moments at $i$ and $j$ is modulated by the displacement of ion $k$. The antisymmetric anisotropic (with regard to spin space) part of the tensor represents the Dzyaloshinskii-Moriya interaction. This coupling will not be further discussed in the present paper, although the implementation discussed here does in principle allow for it.
In this paper we focus on the non-relativistic correction term to Heisenberg exchange
\begin{eqnarray}
\label{eq:HamExchStr}
\mathcal{H}_{\rm SSL}=-\frac{1}{2}\sum_{ijk} \vec{A}_{ijk}\cdot \bfu_k(\bfm_i\cdot\bfm_j),
\end{eqnarray}
in Eq.~\eqref{eq:HamMML} and for the simulations presented in Sec.~\ref{sec:results}, where atom $k$ could either coincide with one of the atoms $i$ and $j$ or be a distinct atom. The components of the exchange striction vector $\vec{A}_{ijk}$ relates to $\Gamma_{ijk}^{\alpha\beta\mu}$ according to
$\vec{A}_{ijk}^\mu=\Gamma_{ijk}^{xx\mu}=\Gamma_{ijk}^{yy\mu}=\Gamma_{ijk}^{zz\mu}$. The exchange striction coupling parameters are governed by certain symmetry rules: From the symmetry of Heisenberg exchange interaction
\bgea
\label{eq:exchsym}
\vec{A}_{ijk}=\vec{A}_{jik},
\enea
and in order to obey Newton's third law (forces are defined in Sec. \ref{sec:EOM}), the sum rule
\bgea
\label{eq:sumrule}
\vec{A}_{iji} = -\sum_{k\neq i}\vec{A}_{ijk}
\enea
has to hold. In different model approximations for the Heisenberg exchange, for instance the Ruderman–Kittel–Kasuya–Yosida (RKKY) type interaction \cite{Kundu:2015dn} or the effective model used by Ma \textit{et al.},\cite{Ma2017} the exchange interaction depends only on the distance $\vec{r}_{ij}$ between site $i$ and $j$, $\mathcal{J}_{ij}(\bfrr)=\mathcal{J}_{ij}(\bfrr_j+\bfu_j-\bfrr_i-\bfu_i)$, and, consequently, $\vec{A}_{iji}=|\vec{A}_{iji}| \bfr_{ij}$.
It obviously reflects the symmetry rule
$\vec{A}_{iji}=-\vec{A}_{ijj}$,
which is also valid in general for
$\vec{A}_{ijk}$.
It guarantees that the exchange striction force on atom $i$ and $j$ will cancel each other. The direction of the force caused by the exchange striction coupling is contained in
$\vec{A}_{ijk}$ and is constrained by the point group symmetry of the crystal.
\subsection{The SLD Equations of motion}
\label{sec:EOM}
The coupled equations of motion for the spin-lattice system reads \cite{Ma2008}
\bgea
\frac{d\bfm_i}{dt} &=& -\frac{\gamma}{(1+{\alpha}^2)}\bfm_i \times \left(\bfbb_{i}+\bfbf_{i}\right) \label{eq:SLDeom1} \\
&&-\frac{\gamma}{(1+{\alpha}^2)}\frac{\alpha}{m_i}\bfm_i\times\left(\bfm_i\times[\bfbb_{i}+\bfbf_{i}]\right),\nonumber\\
\frac{d\bfu_k}{dt} &=& \bfv_k, \label{eq:SLDeom2}\\
\frac{d\bfv_k}{dt} &=& \frac{\bfff_k}{M_k} + \frac{\bffff_k}{M_k} - \nu\bfv_k, \label{eq:SLDeom3}
\enea
when expressed in the form of Langevin equations. Here, the effective magnetic field is obtained from the SLD Hamiltonian in Eq. \eqref{eq:HamSLD} as $\bfbb_i=-\partial{\mathcal{H}_{\rm SLD}}/{\partial \bfm_i}$ and the effective interatomic force field is determined from $\bfff_k=-\partial{\mathcal{H}_{\rm SLD}}/{\partial\bfu_k}$. $M_k$ is the mass of the atom at site $k$, $m$ is the saturation magnetization, $\gamma$ is the gyromagnetic ratio while $\alpha$ and $\nu$ are scalar (isotropic) damping constants. The stochastic fields $\bfbf_i$ and $\bffff_k$ are assumed to obey white noise properties, which implies that $\langle\bfbf_i(t)\bfbf_j(t^\prime)\rangle=2 D_M\delta_{ij}\delta(t-t^\prime)$ and $\bffff_k(t)\bffff_l(t^\prime)=2 D_L\delta_{kl}\delta(t-t^\prime)$. From the fluctuation-dissipation theorem it follows that $D_M = \alpha k_B T / \gamma m$
and $D_L=\nu M k_B T$.\cite{Ma2008}
The coupled partial differential equations are numerically solved using the method by Mentink \textit{et al.}~\cite{Mentink2010} combined with the Gr{\o}nbech-Jensen-Farago Verlet-type method,\cite{Gronbech-Jensen2013} or with a fixed-point scheme for implicit midpoint method. The methods require numerical step width of the order $\unit[10^{-15}]{s}$ down to $\unit[10^{-16}]{s}$. Details about the algorithms are provided in Appendix \ref{append:numint}.
Finally, we emphasize the fundamental difference between the here presented method and the one proposed by Ma \textit{et al.} \cite{Ma2008,Ma2012} and Perera \textit{et al.} \cite{Perera2017}: \textit{i)} All sets of parameters $\{J\}$,$\{\phi\}$, and $\{\Gamma_{ijk}^{\alpha\beta\mu}\}$
are determined from first principles and are not obtained from an effective potential or exchange model; \textit{ii)} we consider an established parametrisation of the lattice potential as presented in Eq.~\eqref{eq:HamLL}, which is directly available from standard first-principles tools (see Section~\ref{sec:DFT}); \text{iii)} the exchange striction term contains also couplings for $k\neq i,j$ and, consequently, will be highly applicable for magnets showing super- and double exchange mechanism for the magnetic coupling.
\subsection{Observables}
\label{sec:observ}
The primary output of a spin-lattice dynamics simulation are trajectories in time of the system variables $\{\bfm_i\}$, $\{\bfu_k\}$ and $\{\bfv_k\}$. In order to sample spatial and temporal fluctuations of the spins and the ions, we define the space- and time-displaced pair correlation functions
\bgea
C_{(\bfm)}^{\alpha \beta}(\mathbf{r},t-t_0)
&=& \frac{1}{N}\sum_{\substack{i,j ~ \text{where} \\\ ~\mathbf{r}_i-\mathbf{r}_j=\mathbf{r}}} \langle m_i^\alpha(t) m_j^\beta(t_0) \rangle, \label{eq:cmrt}\\
C_{(\bfu)}^{\mu \nu}(\mathbf{r},t-t_0) &=&\frac{1}{N}\sum_{\substack{i,j ~ \text{where} \\\ ~\mathbf{r}_i-\mathbf{r}_j=\mathbf{r}}} \langle u_i^\mu(t) u_j^\nu(t_0) \rangle, \label{eq:curt}\\
C_{(\bfv)}^{\mu \nu}(\mathbf{r},t-t_0) &=&\frac{1}{N}\sum_{\substack{i,j ~ \text{where} \\\ ~\mathbf{r}_i-\mathbf{r}_j=\mathbf{r}}} \langle v_i^\mu(t) v_j^\nu(t_0) \rangle .\label{eq:cvrt}
\enea
Equations~\ref{eq:cmrt}-\ref{eq:cvrt} can thus describe how the magnetic-, displacement- and velocity correlations evolves both in space and over time. It would of course be natural to investigate coupled modes, via coupled correlations, e.g., between moment and displacement in a similar way as outlined in Eqs.~\ref{eq:cmrt}-\ref{eq:cvrt}.
In the context of the simulations of bcc Fe in the present paper, the more relevant property is however obtained by a Fourier transform over space and time to give the dynamic structure factor for spin, displacement, and velocities. Defining relative time $\tau=t-t_0$ we obtain
\bgea
S_{(\bfm)}^{\alpha \beta}(\mathbf{Q},\omega)=\frac{1}{N\sqrt{2\pi}}\sum_{\mathbf{r}} e^{i\mathbf{Q}\cdot\mathbf{r}} \int_{-\infty}^\infty C_{(\bfm)}^{\alpha \beta}(\mathbf{r},\tau) \,d\tau, \label{eq:msqw} \\
S_{(\bfu)}^{\alpha \beta}(\mathbf{Q},\omega)=\frac{1}{N\sqrt{2\pi}}\sum_{\mathbf{r}} e^{i\mathbf{Q}\cdot\mathbf{r}} \int_{-\infty}^\infty C_{(\bfu)}^{\alpha \beta}(\mathbf{r},\tau) \,d\tau, \label{eq:usqw} \\
S_{(\bfv)}^{\alpha \beta}(\mathbf{Q},\omega)=\frac{1}{N\sqrt{2\pi}}\sum_{\mathbf{r}} e^{i\mathbf{Q}\cdot\mathbf{r}} \int_{-\infty}^\infty C_{(\bfv)}^{\alpha \beta}(\mathbf{r},\tau) \,d\tau, \label{eq:vsqw}
\enea
which are closely related to what is measured by inelastic neutron or electron scattering experiments. The dynamic structure factors are naturally analyzed in terms of the differential cross section \cite{Squires2012itt} which for many materials is proportional to the dynamical structure function. This means that by simulating the dynamical structure factor, the relation between momentum transfer $\mathbf{Q}$ and frequency $\omega$ for magnons and phonons in the material can be obtained.
For systems lacking periodicity, such as the small dimer, trimer, and quadmer clusters that are also considered in this work, the $\mathbf{Q}$-dependance of the structure factor is in principle undefined. Instead, the spatially dependent excitation spectra of these finite systems can be obtained by performing the Fourier transform over time only. Summing the resulting correlation function over all sites results in the total excitation spectra of the system.
The spin dynamical structure factor accurately describe magnon dispersions, especially in thin films,\cite{Bergqvist2013} since it properly takes into account magnon-magnon scattering properties and damping at finite temperatures. On the other hand, in the limit of very low temperatures and damping, the magnon dispersion is more easily obtained through the adiabatic linear spin wave theory.\cite{Singer:2007hi}
Let us first focus on the spin-degree of freedom and the adiabatic magnon spectra for the collinear magnetic case with a system consisting of 1 atom/cell such as bcc Fe. Then the spatial Fourier transform of the exchange interactions reads
\bgea
\mathcal{J}(\mathbf{Q})=\sum_{j\ne0}
\mathcal{J}_{0j} e^{i\bfqq\cdot(\bfrr_j-\bfrr_0)},
\enea
where $\mathcal{J}_{0j}$ is the exchange interaction between magnetic atoms at site $0$ and $j$, respectively. Note that here the index $j$ runs over all magnetic sites with the origin at $\bfrr_0$. The spin wave energies $\omega(\mathbf{Q})$ will then be given by the following expression \cite{Halilov1998}
\bgea
\label{eq:AMS}
\omega(\mathbf{Q})=\frac{4}{M}\left( \mathcal{J}(\mathbf{0})-\mathcal{J}(\mathbf{Q}) \right).
\enea
Generalization can be done towards multi-sublattice systems, see e.g. Ref.~\onlinecite{Eriksson2017asd}, and using Bogoliubov transformation \cite{Colpa1978} towards general non-collinear formulation.\cite{Toth2015,Yadav2017pp}
For lattice degree-of-freedom, the reciprocal space dynamical matrix $D_{s\beta t\nu}(\bfqq)$ is related to the force constant matrix in real space by the mass normalised Fourier transform
\bgea
\label{eq:dynmat}
D^{\mu\nu}_{st}(\bfqq)=\frac{1}{\sqrt{M_sM_t}}\sum_l\Phi^{\mu\nu}_{0slt}e^{[i\bfqq\cdot(\bfrr_l-\bfrr_0)]}
\enea
where $M_s$ is the mass of atom $s$ in the unit cell. Given the translational symmetry of the crystal, it is enough to sum only over $l$ in all the $N_P$ primitive cells in the supercell. For the $\Gamma$ point, the Fourier transform is a plain sum over all repetitions of the primitive cell, both the ones contained in the Wigner-Seitz cell of the simulation supercell and, due to the periodic boundary conditions, the ones outside it.\cite{Wang2010a} Note that in linear response density functional perturbation theory computation of phonons, it is actually the elements of the dynamical matrix that are calculated from which the interatomic force constants can be calculated by the inverse Fourier transform.\cite{Baroni2001} Solving the eigenvalue problem
\bgea
\label{eq:phondisp}
{\rm det}[D_{s\mu t\nu}(\bfqq)-\omega^2(\bfqq)]=0
\enea
the $3N$ phonon modes (eigenvectors) and frequencies (the square roots of the occasionally degenerate eigenvalues $\omega^2$) are obtained for a given $\bfqq$ vector.
Opposite to the dynamic structure factor (Eqs. \ref{eq:msqw}-\ref{eq:vsqw}), the adiabatic spectra described above do not directly account for the coupling between the spin- and lattice reservoir. A possible way forward in this regard is to replace the exchange interaction and force constant matrix in Eqns.21 and 23, with corresponding Taylor expanded entities defined in Eqns.5 and 3, respectively. This can also be handled as described e.g in Refs. \onlinecite{White:1965ji,Kim2007,Ruckriegel2014,Toth2016}, which is, however, beyond the
scope of the present paper.
\subsection{DFT calculations}\label{sec:DFT}
Aiming for a first-principles description of the coupled spin-lattice dynamics in bcc Fe, we calculated the coupling constants that occur in Eq.~\ref{eq:HamSLDexp} by means of density functional theory (DFT) calculations.
The harmonic force constants $\Phi_{kl}^{\mu\nu}$ were calculated with the finite displacement method using the Vienna {\it ab initio} Vienna simulation package (VASP)\cite{Kresse1996a,Kresse1996} and the Phonopy \cite{Togo2015} software. The VASP calculations were performed using the projector augmented wave method\cite{Bloechl1994,Kresse1999} and the local density approximation as exchange-correlation functional. A $6\times6\times6$ supercell was used for bcc Fe. We employed a plane-wave energy cutoff of 600 eV and a $\Gamma$-centered $\textbf{\emph{k}}$-points mesh of size $4\times4\times4$.
In order to achieve a complete first-principles spin-lattice model, we also approach the Heisenberg interaction and the exchange striction from DFT. To this end, we applied the full-potential linear muffin-tin orbitals (FP-LMTO) method as implemented in the RSPt software~\cite{Wills2010fes}. The maximum value of the angular momentum used for the angular ($l$) decomposition of the charge density and the potential inside the muffin-tin spheres was taken equal to $l_{max}$ = 12. Three kinetic energy tails were used for the description of the states in the interstitial region: -0.3, -2.3, and -1.5 Ryd. Within this setup, we calculated the Heisenberg exchange coupling $\mathcal{J}_{ij}$ via the Liechtenstein-Katsnelson-Antropov-Gubanov (LKAG) formalism \cite{Liechtenstein1984,Liechtenstein1987}. For the actual implementation into the RSPt code we refer the reader to Ref. \onlinecite{Kvashnin2015}. Recently, this method has been successfully applied to strongly correlated systems such as NiO\cite{{Kvashnin2015}} and BiFeO$_3$~\cite{Paul2018}.
The magnetic exchange interactions $\mathcal{J}_{ij}$ have been calculated with RSPt for a primitive cell and for a $2\times2\times2$ supercell for which identical $\mathcal{J}_{ij}$ came out. In order to calculate first order exchange striction, $\mathcal{J}_{ij}$ have been calculated
for the same $2\times2\times2$ cell but with one atom displaced with a finite displacement $\Delta$ along $\vec{e}_{\Delta}$. A $\textbf{\emph{k}}$-point mesh of $30 \times 30 \times 30$ was employed for the supercell calculations. The so obtained set of exchange couplings have a lower symmetry. For instance, the set of eight equivalent nearest neighbor couplings in the bcc structure are broken up to sets of 1, 3, 3, 1 degenerate couplings for a distortion along the [100] axis (in bcc lattice vectors, i.e. along the [111] for Cartesian axis), and sets of 4, 4 couplings for a distortion along the $[110]$ axis (Cartesian $[100]$). For the bcc structure we have compared carefully that for different distortion directions, the symmetry lowering for the $\mathcal{J}_{ij}$ up to the fourth coordination shell is identical to the symmetry lowering of the crystal itself.
Since anisotropic exchange parts are not considered, the tensor $\mat{\Gamma}_{ijk}$ becomes a vector $\vec{A}_{ijk}$. Similar to the force constants and magnetic exchange, $\vec{A}_{ijk}$ fulfill point group symmetries, in particular for bcc Fe the 48 symmetry operations of space group number 229 (Im$\bar{3}$m). Furthermore, the exchange striction energy $E_{ijk}$ related to the sites $i$, $j$, and $k$ is isotropic. Consequently, we obtain $\vec{A}_{ijk}=\vec{A}_{jik}$ from $E_{ijk}=E_{jik}$, which is furthermore caused by the isotropic properties of the magnetic exchange. In our spin-lattice dynamics simulations we even have to guarantee that the centre of mass is not drifting, which is guaranteed when also the spin-lattice couplings fulfill Newton's third law.
Using a finite difference method involving the non-displaced set $\{\mathcal{J}^0\}$ and a displaced set $\{\mathcal{J}^{\vec{e}_{\Delta}}\}$, we obtain the directional derivative $\Gamma^{\Delta}_{ijk}$. The gradient $\nabla_k\mat{\mathcal{J}}_{ij}$ is finally constructed from the directional derivative definition and out of three different sets $\{\mathcal{J}^{\vec{e}^{\nu}_{\Delta}}\}$ of independent directions $\vec{e}^{\nu}_{\Delta}$, where $\nu=1,2,3$, but the same displacement strength $\Delta$. We have chosen $\Delta$ to be 0.003$a_0$, 0.002$a_0$,0.001$a_0$, and 0.00001$a_0$, where $a_0$ is the Bohr radius. In order to fulfill the finite displacement criteria, $\Delta$ is interpolated to zero by Hermite interpolation. Numerical noise is reduced by applying various symmetries, as discussed above. We apply iteratively the above mentioned symmetry operations until we reach convergence. It is important to mention that in the last step Newton's third law has to be enforced.
\section{Results}\label{sec:results}
This section is divided into two parts. In Section~\ref{sec:exch234} we discuss the application of our method to low-dimensional model systems and discuss symmetry-related issues of the three-body exchange coupling.
(Note that we have selected, for simplicity, to neglect four-body interaction in the present work.) Section~\ref{sec:bccFe} deals with applications to real materials, in particular to bcc Fe. All required parameters are calculated from first principles. We present quasi-particle dispersion relations and discuss the role of the three-body interaction in the spectra at various temperatures.
\subsection{Exchange striction in 2-, 3- and 4-site systems}
\label{sec:exch234}
As conceptual examples for our method, we perform coupled spin-lattice dynamics simulations for systems consisting of two (dimer), three (trimer), and four (quadmer) atoms. If not mentioned, we neglect energy dissipation in our model and, consequently, the total energy has to be conserved. Furthermore, we account only for the isotropic part of the magnetic exchange tensor, namely the Heisenberg exchange, but note that anisotropy in general is of significant importance in low-dimensional systems \cite{Bergqvist2013,Bergman2016,Smogunov2007,LeBacq2002,Delin2004,Menendez2000}.
\subsubsection{Dimer}
\label{sec:exchstrdimer}
In this model system, we consider a dimer where the two sites are denoted by 1 and 2 (see the inset of Fig.~\ref{fig:sldenergy1}) and we have set $M_1=M_2=$ atomic mass units, and $m_1=m_2=1$ $\mu_B$. The simplicity of this system allows to provide explicit expressions for the Hamiltonian, the effective magnetic fields and the interatomic forces. The four parts of the dimer Hamiltonian
\begin{eqnarray}\label{eq:HamDimer}
\mathcal{H}^{\mathrm{dimer}} &=& \mathcal{H}_{\mathrm{LL}}^{\mathrm{dimer}} + \mathcal{H}_{\mathrm{SS}}^{\mathrm{dimer}} +
\mathcal{H}_{\mathrm{SSL}}^{\mathrm{dimer}} +
\mathcal{H}_{\mathrm{KIN}}^{\mathrm{dimer}}
\end{eqnarray}
reads
\begin{eqnarray}\label{eq:HamDimer2}
\mathcal{H}_{\mathrm{LL}}^{\mathrm{dimer}} &=& \frac{1}{2}\Phi_{11}^{\mu\nu} u_1^{\mu} u_1^{\nu} + \frac{1}{2}\Phi_{12}^{\mu\nu} u_1^{\mu} u_2^{\nu}, \nonumber\\
&& + \frac{1}{2}\Phi_{21}^{\mu\nu} u_2^{\mu} u_1^{\nu} + \frac{1}{2}\Phi_{22}^{\mu\nu} u_2^{\mu} u_2^{\nu}, \\
\mathcal{H}_{\mathrm{SS}}^{\mathrm{dimer}} &=& -\frac{1}{2}J_{12}\bfm_1\cdot\bfm_2 - \frac{1}{2}J_{21}\bfm_2\cdot\bfm_1, \\
\mathcal{H}_{\mathrm{SSL}}^{\mathrm{dimer}}
&=& \frac{1}{2} A_{121}^{\mu}(\bfm_1\cdot\bfm_2)u_1^{\mu} + \frac{1}{2} A_{122}^{\mu}(\bfm_1\cdot\bfm_2)u_2^{\mu} \\
&& + \frac{1}{2} A_{211}^{\mu}(\bfm_2\cdot\bfm_1)u_1^{\mu} + \frac{1}{2} A_{212}^{\mu}(\bfm_2\cdot\bfm_1)u_2^{\mu}, \nonumber\\
\mathcal{H}_{\mathrm{KIN}}^{\mathrm{dimer}} &=& \frac{1}{2} M_1 v_1^{\mu}v_1^{\mu} + \frac{1}{2} M_2 v_2^{\mu}v_2^{\mu}.\label{eq:HamDimer3}
\end{eqnarray}
In particular for the dimer, we choose the magnetic interaction to be $J=\unit[0.1]{mRyd}$ and the harmonic atomic force constants uniaxial with $\phi_{12}=-\unit[1]{Ryd \mbox{\normalfont\AA}^{-2}}$. The three-body interaction is introduced along the bond and is set to $\unit[1]{mRyd \mbox{\normalfont\AA}^{-1}}$. The scalar product of the two moments in the dimer is, when damping is ignored, a constant of motion and hence also the exchange energy will be constant (see Fig.~\ref{fig:sldenergy1} blue line).
The evolution of the energy origins from the corresponding harmonic interatomic forces
\begin{eqnarray}
F_{\mathrm{LL},1}^{\mathrm{dimer},\mu} &=& -\Phi_{11}^{\mu\nu} u_1^{\nu} -\Phi_{12}^{\mu\nu} u_2^{\nu},
\label{eq:FDimerLL2}
\end{eqnarray}
and interatomic forces from the three-body exchange
\begin{align}
F_{\mathrm{SLL},1}^{\mathrm{dimer},\mu} = -A_{121}^{\mu}(\bfm_1\cdot\bfm_2),
\label{eq:FDimerMML2}
\end{align}
where we used the symmetry relation of the force constants $\Phi_{kl}^{\mu\nu}=\Phi_{lk}^{\nu\mu}$ and the exchange striction terms $A_{121}^{\mu}=A_{211}^{\mu}$.
\begin{figure}
\includegraphics[width=0.55\textwidth]{DimerEnergyCombined.jpg}
\caption{(Color online) Energy trajectories at $T=0$ K of a dimer oriented along the $x$-direction with Hamiltonian according to Eqs. \ref{eq:HamDimer}-\ref{eq:HamDimer3}. The inset shows a conceptual figure of the exchange striction coupling constants $A_{12k}$, $k=1,2$ (black arrows) in a dimer. The atoms are represented by golden balls, where the initial magnetic moment configuration in the dimer is given by purple arrows.}
\label{fig:sldenergy1}
\end{figure}
\begin{figure}
\includegraphics[width=0.50\textwidth]{dispmagtraj1}
\caption{(Color online) Trajectories at $T=0$ K of a dimer oriented along the $x$-direction. The slow ionic motion couples over the exchange striction to the spin system and induces a modulation of the frequencies of spin precession. (upper panel) The Cartesian components of $\bfm_1$, labeled in the figure with $X$, $Y$, and $Z$. (lower panel) The displacement of ion 1 and 2 along the dimer bond axis.}
\label{fig:dispmagtraj1}
\end{figure}
Likewise, the magnetic degree of freedom is driven by the magnetic exchange field
\begin{align}
\bfbb_{\mathrm{SS},1} = J_{12}\bfm_2,
\end{align}
and exchange striction field
\begin{align}
\bfbb_{\mathrm{SSL},1} = -A_{121}^{\mu}\bfm_2 u_1^{\mu} -A_{122}^{\mu}\bfm_2 u_2^{\mu}.
\label{eq:BDimerMML}
\end{align}
Here, we applied the isotropy property of the magnetic exchange $J_{12}=J_{21}$. Equations~\eqref{eq:FDimerLL2}-\eqref{eq:BDimerMML} show that the direction of $\vec{F}_{\mathrm{SLL},1}^{\mathrm{dimer}}$ is only dictated by the coupling constant, where the amplitude is also related to the relative angle between the magnetic moments. Hence, in the absence of Gilbert damping that strives to align the spins $\vec{F}_{\mathrm{SLL},1}^{\mathrm{dimer}}$ will be also a constant of motion in the case of a dimer. For the effective magnetic field, the three-body term only scales the field strength. The exchange striction term conserves the center of mass and, consequently, $\vec{u}_1=-\vec{u}_{2}$ and $\vec{A}_{121}=-\vec{A}_{122}$. Thus, the case $J_{12}<2\vec{A}_{121}\cdot\vec{u}_1$ is of high interest, since the effective exchange switches from a ferromagnetically to an antiferromagnetically coupled system. For the dimer, however, this will only change the direction of precession locally in time.
The fields and forces in Eqs.~\eqref{eq:FDimerLL2}-\eqref{eq:BDimerMML} finally lead to the evolution of the spin and lattice degree-of-freedom as shown in Fig.~\ref{fig:sldenergy1}, for the different contributions to the energy, and in Fig.\ref{fig:dispmagtraj1} for the displacement and moment direction. The simulations reproduce the conservation of the center of mass (Fig.~\ref{fig:dispmagtraj1} lower panel) as well as the relative angle between the magnetic moments. The initial conditions for the displacement in the dimer are set to be $\unit[0.02]{\mbox{\normalfont\AA}}$ ($2\%$ of the lattice constant). It should be noted that the oscillation is not around the equilibrium position ($\vec{u}=0$), but around a shifted position along the bond axis ($x$-axis). On the other hand, the magnetic moments move only in the $yz$-plane as a result of the initial conditions (see Fig. \ref{fig:sldenergy1}, inset, purple arrows). The precession of the magnetic moment is seen to vary in time and to be largest when the displacement $\vec{u}_1$ is significant\footnote{See the supplementary movie}. This is a natural consequence of the exchange striction field, defined in Eqn.33.
To further analyze the reliability of the method we also simulate the dynamics in the presence of a viscous damping $\nu=10^{-14}$ kg$/$s. The analytical solution of the damped 1D-harmonic oscillator (see e.g. Ref. \cite{Meriam1993em}) is compared with the numerical one and we obtain perfect agreement as shown in Fig.~\ref{fig:xdisp}. The ions oscillate around the center of mass and the envelope of the trajectories decays exponentially in time.
\begin{figure}
\includegraphics[width=0.50\textwidth]{xdisp}
\caption{(Color online)
Trajectories at $T=0$ K of the damped lattice motion of a dimer oriented along the $x$-direction. The analytical solution for atom 1 (black symbols) lies on top of the simulated trajectory (red line).}
\label{fig:xdisp}
\end{figure}
\subsubsection{Trimer}
\label{sec:exchstrtrimer}
The three sites of a trimer are mutually nearest neighbors, which enables for a total of 18 possible $\vec{A}_{ijk}$ couplings. Respecting that symmetry under exchange of spin sites (Eq. \ref{eq:exchsym}), point group symmetry $D_{3h}$, and the sum rule (Eq. \ref{eq:sumrule}) should always hold, we consider the following case for the exchange striction (Fig.~\ref{fig:trimer}): Exchange striction vectors tilt away from the bond by angle $\theta_1$ (see Fig.~\ref{fig:trimer}): $\vec{A}_{iji}\nparallel\bfr_{ij}$ and, consequently, $\vec{A}_{ijk(k\neq i,j)}\neq\vec{0}$. Furthermore, $\vec{A}_{ij1} = -\sum_{l\neq 1}\vec{A}_{ijl}= -\sum_{l\neq 1}\vec{A}_{jil} = \vec{A}_{ji1}$. Note, that only for $\theta=\nicefrac{\pi}{6}$ we have $|\vec{A}_{iji}|=|\vec{A}_{ijk(k\neq i,j)}|$.
\begin{figure}[ht]
\centering
\includegraphics[width=0.50\columnwidth]{principle_symmetry_trimer_2}
\caption{(Color online) Conceptual figure of the exchange striction coupling constants $A_{12k}$, $k=1,2,3$ (black arrows) in a trimer. By symmetry, we have $A_{123}=-A_{121}-A_{122}$, and one free angle $\theta_1$ as a parameter.
The atoms are represented by red balls, where the initial magnetic moment configuration in the trimer is given by blue arrows.}
\label{fig:trimer}
\end{figure}
Note that we here refer to the amplitude of $\vec{A}_{iji}$ as $A$. Just as for the dimer case, we assume nearest neighbor coupling of $J=\unit[1]{mRyd}$. The irreducible part of the atomic force constants $\phi$ for $D_{3h}$ symmetry are in the notation of Refs.~\onlinecite{Cserti2004,Toth2016}, i.e. $\Phi_2^{xx}=\unit[-0.25]{Ryd \mbox{\normalfont\AA}^{-2}}$, $\Phi_2^{xy}=\unit[-0.43]{Ryd \mbox{\normalfont\AA}^{-2}}$, and $\Phi_2^{yy}=\unit[-0.25]{Ryd \mbox{\normalfont\AA}^{-2}}$. The mass of each atom is put to 1 atomic mass unit.
In order to see direct effects of the $A$ coupling, we plot in Fig.~\ref{fig:trimer_powdisp}, the measured excitation spectra given by Eqn.18 and 19, obtained from simulations for three choices of $\theta_1$ (defined in Fig.5) with $A=\unit[0.1] {mRyd}$. For comparison, the excitation spectra for a reference system with $A=0$, i.e. without any coupling between the spin and lattice systems is also included in the figure. For the decoupled trimer, two significant peaks are present. The peak that is lower in energy, at $\unit[3.2] {THz}$, represents the lattice vibrations while magnetic fluctuations cause the other peak at $\unit[8.2]{THz}$. In the case of a finite spin-lattice coupling we find that for two of the considered angles ($\theta_1=0^\circ$ and $\theta_1=70^\circ$), a very fine splitting of the magnetic energy level is noticable while a low-energy peak at $\unit[2.2] {THz}$ occurs for the lattice vibrations. Interestingly, for the third choice of angle, $\theta_1=30^\circ$, the difference compared to the decoupled system is found to be minimal. Frequencies such as the ones shown in Fig.6 are available through Raman spectroscopy, and an experimental detection of a spin-lattice coupling, via the split peaks in Fig.6 would be interesting.
\begin{figure}[ht]
\centering
\includegraphics[width=0.48\textwidth]{trimer_dispmom_ps.png}
\caption{(Color online) Excitation spectra for lattice displacements (red/thick lines) and magnetic oscillations (green/thin lines), for the trimer configurations. The top three panels corresponds to SLD simulations with $\theta_1=0^\circ$, $\theta_1=30^\circ$ and $\theta_1=70^\circ$, as described in the text. The bottom panel corresponds to decoupled lattice-dynamics and spin-dynamic simulations.
}
\label{fig:trimer_powdisp}
\end{figure}
In our simulation we varied both the strength $A$ and the angle $\theta_1$ of the exchange striction coupling. The strength $A$ affects the frequency $\omega_A$ of an enveloping oscillation on top of the spin precession frequency $\omega_p$: $\omega_A$ scales quadratically with the strength $A$ (see Fig.~\ref{fig:trimerprinciple}(upper panel). Without energy dissipation, the magnetic energy is conserved, just as for the dimer case. The variations of the various energy contributions will be similar to the one in Fig.~\ref{fig:sldenergy1}. The total energy increases linearly with the strength $A$, but oscillates with $\theta_1$ which is related to the fixed initial spin configuration (see Figs.~\ref{fig:trimerprinciple}(lower panel)).
\begin{figure}[t]
\centering
\includegraphics[width=0.35\textwidth]{enclose_time}
\includegraphics[width=0.35\textwidth]{energy_angle}
\caption{(Color online) Spin-lattice dynamics of trimer. (upper panel) Envelope frequency of spin oscillations as function of $|A|$. (lower panel) Total energy as function of tilt angle of $A$.}
\label{fig:trimerprinciple}
\end{figure}
\subsubsection{Quadmer}
\label{sec:exchstrquadmer}
\begin{figure}[ht]
\centering
\includegraphics[width=0.80\columnwidth]{principle_symmetry_quadmer}
\caption{\label{fig:quadmers}(Color online) Conceptual figure of the exchange striction coupling constants $A_{23k}$, $k=1,2,3,4$ (black arrows) in a quadmer. By symmetry, $A_{232}=-A_{233}$ and $A_{234}=-A_{231}$ are allowed. The atoms are represented by red balls. The initial, random magnetic moment configuration in the quadmer is given by blue arrows, where the length and orientation is related to the in-plane component, and the color to the $z$-component of the moment. The dotted line indicates the bond axis.
}
\end{figure}
Although the trimer offers already rich phenomena, it addresses only nearest neighbor couplings, which are symmetry related and not independent. Contrary to the trimer, the sites of a 4-site system with periodic boundary condition (chain of atoms, see Fig.~\ref{fig:quadmers}(a), has both nearest neighbor (NN) and next nearest neighbors (NNN). Consequently, it is possible to have finite couplings $\vec{A}_{132}=\vec{A}_{312}=-\vec{A}_{134}=\vec{A}_{314}$ also when $\vec{A}_{iji}\parallel\bfr_{ij}$.
Here, we consider three different cases for the three-body interaction in the chain of four atoms.
\begin{enumerate}
\item Exchange striction vectors parallel to bonds between spin $i$ and $j$ (Fig.~\ref{fig:quadmers}(a)): $\vec{A}_{iji}\parallel\bfr_{ij},\vec{A}_{ijk(k\neq i,j)}=\vec{0}$.
\item Exchange striction vectors parallel to bonds between spin $i$ and $j$, but the second nearest neighbor coupling is different from zero (Fig.~\ref{fig:quadmers}(b)): $\vec{A}_{iji}\parallel\bfr_{ij},\vec{A}_{ijk(k\neq i,j)}\neq\vec{0}$. Thus, the sum over the second-nearest neighbor couplings has to cancel: $\sum_{k\in {\rm NNN}}\vec{A}_{ijk}=0$.
\item Exchange striction vectors not parallel to bonds for all indices (Fig.~\ref{fig:quadmers}(c)): $\vec{A}_{iji}\nparallel\bfr_{ij},\vec{A}_{ijk(k\neq i,j)}\neq\vec{0}$
\end{enumerate}
For the Heisenberg exchange $J_{ij}$ and force constants $\phi_{ij}$ we include only nearest neighbor interactions of $J_{\rm NN}=\unit[1]{mRyd}$ and $\Phi_1^{xx}=\unit[-0.07]{Ryd \mbox{\normalfont\AA}^{-2}}$. The atomic mass is put to 10 atomic mass units. In Fig.~\ref{fig:quadmer_powdisp} we show simulated excitation spectra for the tree sets of quadmer configurations discussed above, together with results from a decoupled simulation where all $\vec{A}_ijk=0$. Starting with the decoupled results in the lower panel of Fig.~\ref{fig:quadmer_powdisp} it is found that the lattice excitations are about one order of magnitude lower in energy compared to the magnetic excitations. The effect of finite spin-lattice couplings, shown in the top three panels in Fig.~\ref{fig:quadmer_powdisp}, is directly visible as lattice vibrations then contribute to high energy peaks, while magnetic fluctuations contribute to the lower energies. Compared to the pure lattice and magnetic excitation energies, the spectral weights of these coupled, or hybridized, excitations are however small. It can also be noticed that for the two sets of quadmer configurations with higher symmetry, (Sets 1 and 2), the excitation spectra has well defined peaks while for the lower symmetry configuration, set 3, certain peaks are more diffuse. This is most clearly visible for the lattice excitations at $\unit[2]{THz}$ and at $\unit[10]{THz}$ in the third panel of Fig.~\ref{fig:quadmer_powdisp}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.48\textwidth]{quadmer_dispmom_ps.png}
\caption{(Color online) Excitation spectra for lattice displacements (red/thick lines) and magnetic oscillations (green/thin lines) for the considered quadmer clusters. The top three panels show the excitations for the SLD simulations corresponding to the three different cases discussed in the text. The bottom panel corresponds to decoupled LD and ASD excitations.
}
\label{fig:quadmer_powdisp}
\end{figure}
In effective pair-interaction models, as used for instance in Refs.~\onlinecite{Ma2012a,Perera2016}, only interactions between site $i$ and $j$ are possible. This situation corresponds to the system depicted in Fig.~\ref{fig:quadmers}(a) with its excitation spectra displayed in the top panel of Fig.~\ref{fig:quadmer_powdisp}. Allowing for three-body interactions, as in the system shown in Fig.~\ref{fig:quadmers}(b) results in the excitation spectra displayed in the second panel from the top in Fig.~\ref{fig:quadmer_powdisp}. Here, we have chosen that the NN and NNN three-body exchange have the same strength, e.g. $\unit[1]{mRyd\mbox{\normalfont\AA}^{-1}}$. Comparing these two models we find for the spin excitations, that even though there is a difference in the distribution of spectral weight between the largest peaks, the actual spin excitation energies do not change much. There is however a larger difference noticeable for the lattice excitation energies, where both the position of the peaks and the spectral weights changes significantly when including NNN three-body exchange couplings.
\subsection{Exchange striction in bcc Fe}
\label{sec:bccFe}
\subsubsection{Coupling constants for bcc Fe}
\label{sec:resJijPhiiijAijk}
In this Section we discuss the coupling parameters that we obtain from DFT, namely Heisenberg exchange $J_{ij}$, force constants $\phi_{ij}$, and exchange striction $\vec{A}_{ijk}$.
The force constants are obtained from supercell calculations and are related by point-group symmetry \cite{Ackland1997,Finkenstadt2006,Flocken1969,Frank1995}. Hence, we list only the irreducible values in Table \ref{tab:phiij}. In the table, $j = 1$ denotes the nearest neighbor shell, $j = 2$ the next nearest neighbor shell, and so forth. The indices $\mu$ and $\nu$ stand for the Cartesian coordinates.
\begin{table}[t]
\centering
\caption{
The nonvanishing elements $\phi^{\mu\nu}_{0j}$ of the force-constant matrix in Fe. The values are given in $\unit[]{mRyd \mbox{\normalfont\AA}^{-2}}$.}
{\begin{tabularx}{0.35\textwidth}{X X p{0.5cm} p{0.5cm} X r}
$j$ & & $\mu$ & $\nu$ & & $\phi_{0j}^{\mu \nu}$ ($\unit[]{mRyd \mbox{\normalfont\AA}^{-2}}$)\\
\hline
\hline
1 & & x & x & & -4360 \\
& & x & y & & -2670 \\
\hline
2 & & x & x & & -4850 \\
2 & & y & y & & -2230 \\
\hline
3 & & x & x & & -459 \\
& & x & y & & -209 \\
& & z & z & & 403 \\
\hline
4 & & x & x & & 55 \\
& & x & y & & -48 \\
& & y & y & & -126 \\
& & y & z & & -251 \\
\hline
5 & & x & x & & 122 \\
& & x & y & & -169 \\
\hline
\hline
\end{tabularx}
}
\label{tab:phiij}
\end{table}
The Heisenberg exchange for bcc Fe is well studied and our calculations agree well with previous studies \cite{Pajda2001,Kvashnin2015,Szilva2017,Cardias2017}. The $J$'s are isotropic, long ranged, oscillating with decay typically as $r^{-3}$. The calculations reveal that $J\approx 0$ at $\unit[10]{nm}$. In Figure~\ref{fig:aijk})(b) the exchange is plotted against distance $r_{ij}$. The nearest neighbor interaction is about $\unit[1]{mRyd}$ and with also the second nearest neighbor interaction positive, bcc Fe is ferromagnetic. For the simulations in Sections \ref{sec:phasediag} and \ref{sec:sqwres} we use the first three coordination shells of exchange coupling.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{ExchangeStrictionPrinciple}
\includegraphics[width=0.9\columnwidth]{ExchandExchstr}
\caption{\label{fig:aijk}(Color online)
a) The nearest neighbor (NN) and next-nearest neighbor (NNN) coordination shells in bcc structure.
b) The Heisenberg exchange coupling $J_{ij}$ as a function of the distance $r_{ij}$.
c) Absolute value of the exchange striction $\vec{A}_{ijk}$ as a function of the distance $r_{ij}$, for on-site contribution (red cross), NN contribution (green plus), and NNN contribution (blue circle).
}
\end{figure}
From our first-principles calculations we obtained certain magnitude $\left|\vec{A}\right|=A$ and direction $\vec{e}^A$ of the spin-lattice coupling. We distinguish between the set of couplings $\{\vec{A}\}$ in a subset where the site $k$ is equal to site $i$ or $j$, e.g. $\vec{A}_{iji}$, and where $k\neq i,j$. The first subset refers to the couplings between two magnetic moments, where the lattice displacement happen on one of these two magnetic sites. Note that only these couplings were used, e.g. in Ref. \onlinecite{Ma2017}, although the many body property solved by density functional theory emphasize couplings at $k$ where $k\neq i,j$.
The DFT calculations of $A_{iji}$ (red dots, Fig.~\ref{fig:aijk}) reveal a nearest-neighbor (NN) coupling of $A_{\mathrm{NN}}=\unit[0.34]{mRyd \mbox{\normalfont\AA}^{-1}}$, where next-nearest neighbor (NNN) couplings are $A_{\mathrm{NNN}}=\unit[-0.26]{mRyd \mbox{\normalfont\AA}^{-1}}$. Both align along the distance vector $\vec{r}_{ij}$. This implies that for this set of couplings the Heisenberg exchange $J_{ij}$ can be assumed to be a function only of the distance $\vec{r}_{ij}$.
Here, $\nabla_i J_{ij}(\vec{r}_{ij})=\nicefrac{\partial J_{ij}}{\partial r_{ij}} \vec{e}^r_{ij}$. It also implies that the magnitude of the $\vec{A}_{iji}$ coupling is isotropic, and that the sum rule in Eq.~\ref{eq:sumrule} simplifies to $\vec{A}_{iji}=-\vec{A}_{ijj}$.
The strength of the exchange striction coupling differs in different direction. For instance, for the nearest and next-nearest neighbor coupling in both $\vec{r}_{ij}$ and $\vec{r}_{ik}$, the coupling $\vec{A}_{ijk}$ are bigger for the cases where $k\neq j,-j$. These couplings are $A_{\mathrm{NN,NN}}=\unit[0.62]{mRyd \mbox{\normalfont\AA}^{-1}}$, $A_{\mathrm{NNN,NN}}=\unit[0.57]{mRyd \mbox{\normalfont\AA}^{-1}}$, and $A_{\mathrm{NNN,NN}}=\unit[0.62]{mRyd \mbox{\normalfont\AA}^{-1}}$. The $\vec{A}_{\mathrm{NNN,NNN}}$ couplings are only different from zero for the cases $k=j$.
\subsubsection{Thermodynamic properties of bcc Fe}
\label{sec:phasediag}
In order to investigate how the presence of exchange striction affect the magnetization order parameter at finite temperature we have performed Langevin dynamics spin-lattice dynamics (SLD), and uncoupled spin-dynamics (SD) and lattice dynamics (LD), simulations in the temperature range 0 to 1500 K for simulation cells with size $N\times N\times N$ and periodic boundary conditions.
For low and high temperatures the magnetic order parameters deviate negligibly when comparing SLD and SD data but, as shown in Fig. \ref{fig:bccfeN20}(a), in the vicinity of the phase transition temperature the order parameter takes a lower value in the SLD simulation than in the SD simulation. The inset shows the crossing of the $U_4$ Binder cumulant for cell sizes $N=16,20$ and 24 uncoupled LD/SD simulations from which a critical temperature $T_c=1020$ K can be read out.
The energies of the different terms in the spin-lattice Hamiltonian are shown as a function of temperature in Fig. \ref{fig:bccfeN20}(b), and we can here observe that the difference in the energies for SLD simulation to energies for the uncoupled LD and SD simulation (not shown in figure), are smaller than the line-size in the graph, apart from, naturally, the exchange striction energy (SSL) which is identically zero in the uncoupled simulation and finite for the coupled system, this however changing the total energy (E) with a small fraction. The harmonic lattice potential (LL) and the kinetic energy (KIN) coincide as expected given the equipartition theorem, and are linear in temperature. Unlike the lattice energies, the magnetic energy (SS) has an upper bound and flattens out above the phase transition temperature, and is taking a concave shape with values smaller than the lattice energy at lower temperature.
In Figure \ref{fig:relaxTemp} is shown exchange striction mediated relaxation of the temperature and energy of spin and lattice subsystem in microcanonical evolution of the $N_{\rm atoms}=8000$ cell with edge length $N=20$, for undamped simulations. At $t=0$ ps, the spin system and lattice system are in thermal equilibrium with heat baths at different temperature. At $t>0$ the system evolve in Hamiltonian dynamics simulated with the fixed-point scheme for implicit mid-point method (see Appendix \ref{append:numint}) using a time step $dt=10^{-16}$ s. The total energy is conserved but is redistributed between the degrees of freedom. For initial conditions $T_S=800$ K and $T_L=300$ K are shown the time trajectories of temperature in \ref{fig:relaxTemp}(a) and energies in \ref{fig:relaxTemp}(b). As expected from the equipartition theorem, the lattice harmonic potential energy and kinetic energy take the same values. Complete equilibration of the spin and lattice subsystem to the same temperature does not happen during the displayed time interval of 400 ps. We attribute this incomplete relaxation to a kinematic constraint due to not only total energy, but also the total spin angular momentum, being constants of motion. At $t=0$ the average magnetization is $M\approx 1.3$ $\mu_B$ and as no net torque act on the spin system, energy can be transferred from the spin system to the lattice only in dynamics in which the magnetization is preserved. The situation is different in Figs. \ref{fig:relaxTemp}(c) and \ref{fig:relaxTemp}(d) where the initial condition is $T_S=2000$ K and $T_L=800$ K, and relaxation to a common temperature $T_S=T_L\approx 1000$ K occur, a process which is possible given that the system is paramagnetic at $T_S=2000$ K, and that the net equilibrium magnetization is very small at $T_S=1000$ K. At this temperature the harmonic potential, kinetic energy, and Heisenberg exchange coincide, in agreement with the thermal equilibrium data in Fig. \ref{fig:bccfeN20}. Finally, we note that in contrast to the presently discussed results for a ferromagnetic system, antiferromagnetic dynamics allow for relaxation without an angular momentum bottleneck, see, e.g., the relaxation in Hamiltonian dynamics reported in Ref.~\onlinecite{Hellsvik2016}.
\begin{figure}
\includegraphics[width=0.50\textwidth]{avrgMTN20ins}
\includegraphics[width=0.50\textwidth]{avrgET}
\caption{\label{fig:bccfeN20} (Color online) (a) The magnetic order parameter M(T)/M(T=0) sampled in SLD and in SD simulation for a $N=20$ simulation cell.
The inset shows the crossing of the $U_4$ Binder cumulant for cell sizes $N=16,20$ and 24 uncoupled SD simulations.
(b) Energies of the spin-lattice dynamics Hamiltonian as a function of temperature.}
\end{figure}
\begin{figure}
\includegraphics[width=0.50\textwidth]{relaxTempEnergyT800T300B}
\includegraphics[width=0.50\textwidth]{relaxTempEnergyT2000T800B}
\caption{\label{fig:relaxTemp} (Color online)
Exchange striction mediated relaxation of the temperature and energy of spin and lattice subsystem in microcanonical evolution for two sets of initial temperatures of the subsystems. In a) and b) the initial spin temperature was 800 K and the initial lattice temperature was 300 K. In c) and d) the initial spin temperature was 2000 K and the initial lattice temperature was 800 K. At $t=0$ ps, the spin system and lattice system are each in thermal equilibrium with these heat baths. At $t>0$, the system evolves in Hamiltonian dynamics, with the total energy being conserved but being redistributed between the degrees of freedom.}
\end{figure}
\subsubsection{bcc Fe Magnon and phonon dispersions}
\label{sec:sqwres}
The simulations that we have performed to sample the dynamic structure factor for bcc Fe were performed in two stages:
i) Equilibration stage with Langevin dynamics simulated with the combined velocity-Verlet and SIB solver algorithm, as described in the Appendix~\ref{append:numint}, subdivided into phases with first a longer time step ($dt=10^{-15}$~s) and high damping, followed by gradually shorter time steps and lower damping. In the fourth and final phase we used $N_t=10^4$ time steps of $dt=10^{-16}$~s and the damping parameters $\alpha=0.01$ and $\nu=10^{-14}$ kg$/$s,
ii) Measurement stage done in Hamiltonian evolution of the system over $N_t=2\cdot 10^5$ time steps of $dt=5\cdot10^{-16}$~s with the fixed-point iteration implicit midpoint method, see Appendix~\ref{append:numint}. The sampling step $t_{\rm samp}=5\cdot10^{-15}$~s for the correlation functions defined in Eqs.~\ref{eq:cmrt}-\ref{eq:cvrt} is used for a sampling window of $t_{\rm win}=5\cdot10^{-11}$~s and combined with averaging of the correlations by moving the time window over $t_0=(0,5,10,\ldots,5\cdot10^4)10^{-15}$~s. The corresponding frequency range for the dynamic structure factors in Eqs.~\ref{eq:msqw}-\ref{eq:vsqw} is $\nicefrac{\omega}{2\pi}=[0.02, 0.04,\ldots,200]$~THz (0.0827 meV to 827 meV).
In order to investigate the impact of the exchange striction on the magnon and phonon spectra at finite temperature, we pursued both spin-lattice dynamics simulations and uncoupled spin dynamics and lattice dynamics simulations. In Figure \ref{fig:sqwuSLDT300} is shown the displacement-displacement dynamic structure factor $S(\bfqq,E)$ sampled in SLD simulation at $T=300$ K, as well as the $T=0$ K adiabatic phonon dispersion $\omega(\bfqq)$ obtained from use of Eq. \ref{eq:phondisp}.
Similarly, in Figure \ref{fig:sqwspindiffT300} we display the $x$-component of the spin-spin dynamic structure factor $S(\bfqq,E)$ and the $T=0$ K adiabatic magnon dispersion $\omega(\bfqq)$ (black) calculated using Eq. \ref{eq:AMS}. The insets show the shape of the Lorentzian functions at the H and N points for SLD and SD simulation respectively. The presence of exchange striction in the SLD simulation causes a broadening of the resonance peaks as compared to the SD simulation.
Shown in the upper panel of Fig. \ref{fig:bccfeSQW} are the peak positions of the magnon dispersions at different temperature, obtained by means of fitting to a Lorentzian function. In the very detailed investigations by Perera \emph{et al.} \cite{Perera2017} of magnon and phonon spectra of the Dudarev-Derlet potential \cite{Dudarev2005,Derlet2007} potential for bcc Fe, the measure $(\omega_{\rm SLD}(Q)-\omega_{\rm SD}(Q))/\omega_{\rm SD}(Q)$ was used to analyze the temperature-dependent influence of exchange striction on magnon dispersion. Similarly, the quantity $(\omega_{\rm SLD}(Q)-\omega_{\rm LD}(Q))/\omega_{\rm LD}(Q)$ was defined for the phonon dispersions. In the lower panel of Fig. \ref{fig:bccfeSQW} we show results for the ratio $(\omega_{\rm SLD}(Q)-\omega_{\rm SD}(Q))/\omega_{\rm SD}(Q)$ and note that our results for a Hamiltonian constructed by means of first principles density functional theory methods, compare well with the results obtained by Perera \emph{et al.} for the Dudarev-Derlet potential.
Overall our results compare well with \cite{Brockhouse1967,Minkiewicz1967,Klotz2000} (experimental) and theoretical \cite{Muller2007,Perera2017} (theory) phonon dispersions at finite temperature, and with \cite{Lynn1975} (experiment) and \cite{Kvashnin2015} (theory) magnon dispersions.
\begin{figure}
\includegraphics[width=0.50\textwidth]{uqwfitT300B}
\includegraphics[width=0.50\textwidth]{uqwHNT300}
\caption{\label{fig:sqwuSLDT300}
(Color online)
(a) The $T=0$ K adiabatic phonon dispersion $\omega(\bfqq)$ (black curve) calculated using Eq. \ref{eq:phondisp}, and the peaks of the
displacement-displacement dynamic structure factor $S(\bfqq,E)$ (red symbols) sampled in SLD simulations of bcc Fe at $T=300$~K using a 20$\times$20$\times$20 supercell with periodic boundary conditions.
The dynamic structure factor for selected q-points (b)
H/2 , (c) H and (d) N points.
}
\end{figure}
\begin{figure}
\includegraphics[width=0.50\textwidth]{sqwspindiffT300D}
\caption{\label{fig:sqwspindiffT300}
(Color online) The $T=0$ K adiabatic magnon dispersion $\omega(\bfqq)$ (black) calculated using Eq. \ref{eq:AMS}, and the peak of the $x$-component of the spin-spin dynamic structure factor $S(\bfqq,E)$ sampled in SLD (blue) and SD (red) simulations of bcc Fe at $T=300$~K using a 20$\times$20$\times$20 supercell with periodic boundary conditions. The insets shows the shape of the Lorentzian functions at the H and N points for SLD and SD simulation respectively.}
\end{figure}
\begin{figure}
\includegraphics[width=0.50\textwidth]{sqwspinDiffTB}
\includegraphics[width=0.50\textwidth]{sqwspinRatioTB}
\caption{\label{fig:bccfeSQW}
(Color online) The dispersions fitted from the spin-spin dynamic structure factor $S(\bfqq,E)$ sampled in SLD simulations and in SD simulations and lattice dynamics simulations of bcc Fe at different temperatures using a 20$\times$20$\times$20 supercell with periodic boundary conditions.
(a) The finite temperature dispersion $\omega(\bfqq)$ fitted with a Lorentzian function from $S(\bfqq,E)$ SLD (full lines) and SD (dashed lines) simulations at $T=300$ K (red), $T=500$ K (green), and $T=800$ K (blue).
(b) The ratio $(\omega(\bfqq)_{SLD}-\omega(\bfqq)_{SD})/\omega(\bfqq)_{SD}$, at $T=300$ K (red), $T=500$ K (green), and $T=800$ K (blue).}
\end{figure}
\section{Conclusion}
\label{sec:conc}
In conclusion, we have demonstrated a computationally efficient general method for performing spin-lattice coupled simulations. The method is, in short, based on a Taylor expansion of the bilinear magnetic term of the Hamiltonian with respect to motion of atmic moments and nuclear positions.
To test the reliability of our method, we checked it against available analytical results, obtaining excellent agreement. In our conceptual simulations for small magnetic clusters, we observe new modes emerging as a result of strong interaction between atomic and spin motion. We propose that these coupled modes should be detectable in Raman spectroscopy. We also performed simulations of bulk bcc iron obtaining very good agreement with previous simulations based on an empirical Hamiltonian.
In general, the interaction between the spin and lattice degrees of freedom can be expected to lead to significant changes in both the magnon and phonon spectra, and our simulations indeed demonstrate this. Also, as expected, the changes tend to become more pronounced as the temperature is increased.
\section{Acknowledgements}
We acknowledge fruitful discussions with Pavel F. Bessarab, Jonas Fransson, Pablo Maldonado, Johan H. Mentink, and Lars Nordström.
Financial support from Vetenskapsrådet (grant numbers VR 2015-04608 and VR 2016-05980), and Swedish Energy Agency (grant number STEM P40147-1) is acknowledged. J.H. is partly funded by the Swedish Research Council (VR) through a neutron project grant (BIFROST, Dnr. 2016-06955). O.E. acknowledges the support from Swedish Research Council (VR), eSSENCE, the Knut and Alice Wallenberg (KAW) Foundation (Grants No. 2012.0031 and No. 2013.0020), STandUPP, and the foundation for strategic research (SSF). L.B. acknowledges the support from Swedish Research Council (VR), Grant. No. 2017-03763.
The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the Uppsala Multidisciplinary Center for Advanced Computational Science (UPPMAX), the National Supercomputer Center (NSC), Linköping University, the PDC Centre for High Performance Computing (PDC-HPC), KTH, and the High Performance Computing Center North (HPC2N), Umeå University.
\section{Numerical integration of the spin-lattice dynamics equations of motion}
\label{append:numint}
In this appendix, the schemes we use for numerical integration of the coupled equation of motions expressed in Eqs.~(\ref{eq:SLDeom1}-\ref{eq:SLDeom3}) are described.
Explicit methods for integrating the stochastic LLG equation are commonly two-step numerical integration as is the case for the Heun method~\cite{Rumelin1982}, the Depondt-Merten's method~\cite{Depondt2009}, and the semi-implicit SIB method by Mentink \emph{et al.}~\cite{Mentink2010}. A description of these methods, including benchmarks, can be found in Ref.~\cite{Eriksson2017asd}.
The Depondt-Merten's method and the semi-implicit SIB method can be extended with a suitable explicit or semi-implicit solver for the lattice degrees of freedom, such as the velocity-Verlet method. Note that integration with Heun or other explicit Runge-Kutta schemes is well known to have poor stability for molecular dynamics.
For the Hamiltonian simulations we use a fixed-point iteration of the implicit midpoint scheme. For the simulations in the canonical ensemble, we use a combination of the Gr{\o}nbech-Jensen and Farago (G-JF) \cite{Gronbech-Jensen2013} Verlet-type methods for simulation of Langevin molecular dynamics and the Mentink \emph{et al.} semi-implicit SIB method for the stochastic LLG equation~\cite{Mentink2010}. The combined algorithm for the canonical simulation is written out in pseudocode below.
\\
\begin{algorithmic}
\For {$k \leq K$} \Comment{Loop over time step}\\
\For {$i \in N_{\rm mag}$} \Comment{The first step}
\State calculate $\bfbb_k^i (\bfu_k,\bfm_k)$
\EndFor
\For {$i \in N_{\rm all}$}
\State calculate $\bfff_k^i (\bfu_k,\bfm_k)$
\EndFor
\For {$i \in N_{\rm mag}$}
\State calculate $\tilde{\bfm}_{k+1}^i (\bfm_k^i,\tilde{\bfm}_{k+1}^i,\bfbb_k^i)$ \Comment{Implicit in $\tilde{\bfm}_{k+1}^i$}
\EndFor
\For {$i \in N_{\rm all}$}
\State calculate $\bfu_{k+1}^i (\bfu_k^i,\bfv_k^i,\bfff_k^i)$ \Comment{One-shot}
\EndFor\\
\For {$i \in N_{\rm mag}$} \Comment{The second step}
\State calculate $\bfbb_{k+1}^i (\bfu_{k+1},\tilde{\bfm}_{k+1})$
\EndFor
\For {$i \in N_{\rm mag}$}
\State calculate $\bfm_{k+1}^i (\bfm_k^i,\bfbb_k^i,\bfm_{k+1}^i,\bfbb_{k+1}^i)$ \Comment{Implicit in $\bfm_{k+1}^i$}
\EndFor
\For {$i \in N_{\rm all}$}
\State calculate $\bfff_{k+1}^i (\bfu_{k+1},\bfm_{k+1})$
\EndFor
\For {$i \in N_{\rm all}$}
\State calculate $\bfv_{k+1}^i(\bfv_k^i,\bfff_k^i,\bfff_{k+1}^i)$ \Comment{One-shot}
\EndFor\\
\EndFor
\end{algorithmic}
|
1,941,325,220,490 | arxiv | \section{Introduction}
\label{Intro}
A system in thermodynamic equilibrium with its surroundings can be described using three macroscopic variables
corresponding to the thermal, mechanical, and the chemical equilibrium. For each fixed value of these macroscopic
variables (macrostate) we have many possible microscopic configurations (microstates).
A collection of systems existing in the various possible microstates, but characterized by the same macroscopic
variables is called an ensemble. The thermal, mechanical, and the chemical parameters can be choosen
between either an extensive variable or an intensive variable and so, we have, in total eight different ensembles.
The eight ensembles are further divided into two classes namely the isothermal class for which the thermal
equilibrium variable is the temperature and the adiabatic class for which the thermal equilibrium variable is the
heat function. The isothermal class comprises of the the canonical $(N,V,T)$, isothermal-isobaric $(N,P,T)$,
grandcanonical $(\mu,V,T)$ and the generalized ensemble $(\mu,P,T)$. All the individual members of the adiabatic
class of ensembles have same value for the heat function. The heat function is defined through the relation
\begin{equation}
\mathfrak{H} = E + \sum_{\{\ell\}} x_{\ell} X_{\ell},
\label{hf_def}
\end{equation}
where $E$ is the internal energy and the $x$ is an intensive thermodynamic variable whose conjugate extensive
variable is $X$. Defining $\mathfrak{X}_{1}$ and $\mathfrak{X}_{2}$ as the variables corresponding to the chemical
and mechanical equilibrium the specific form of each ensemble, its heat function and the corresponding entropy are
listed in the Table 1.
\begin{table}[!ht]
{\small
\begin{center}
\caption{List of Adiabatic ensembles}
\vspace{0.2cm}
\begin{tabular}{|c|c|c|c|}
\hline
& & Heat function & Entropy \\
$\ell$ &Ensemble & $\mathfrak{H} = E + \displaystyle{\sum_{\{\ell\}}}\, x_{\ell} X_{\ell}$ &
$S(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H})$\\
\hline
$0$ & Microcanonical & $E$ & $S(N,V,E)$\\
& $(N,V,E)$ & (Internal energy) & \\
\hline
$1$ & Isoenthalpic - isobaric & $H = E + P V$ & $S(N,P,H)$\\
& $(N,P,H)$ & (Enthalpy) & \\
\hline
$1$ & Third adiabatic ensemble & $\mathsf{L} = E - \mu N $ & $S(\mu,V,{\mathsf{L}})$\\
& $(\mu,V,\mathsf{L})$ & (Hill energy) & \\
\hline
$2$ & Fourth adiabatic ensemble & $\mathsf{R} = E + P V - \mu N$ & $S(\mu,P,{\mathsf{R}})$\\
& $(\mu,P,\mathsf{R})$ & (Ray energy) & \\
\hline
\end{tabular}
\label{adiabatic_tab}
\end{center}
}
\end{table}
Of the four adiabatic ensembles, the microcanonical and the isoenthalpic-isobaric ensembles
described in [\cite{EG}-\cite{JR1}] are well known, but the other two ensembles namely $(\mu,V,\mathsf{L})$ and $(\mu,P,\mathsf{R})$
introduced through References [\cite{JR2},\cite{JR4}] are relatively less known. But they are important
in studying adiabatically confined systems with variable number of particles.
For example in Ref. [\cite{JR93}] a Monte Carlo simulation
of a system of liquid palladium has been carried out in the $(\mu,P,\mathsf{R})$ ensemble.
The simulation was much more convenient in this ensemble and the results agreed with those
obtained in the grandcanonical ensemble. Realizing the importance of classifying the ensembles,
a unified treatment of the adiabatic ensembles was carried out in [\cite{HG1}]. Later on the
physical realizations corresponding to the eight ensembles and their interrelations through
Laplace-Legendre transforms was studied in [\cite{HG2}].
Tsallis proposed a generalization of the existing Boltzmann Gibbs statistical mechanics [\cite{CT1}] through
the introduction of $q$-deformed logarithm and exponential functions. The entropic expression in the generalized
statistical mechanics was based on the $q$-deformed logarithm. Many other deformed entropies like the $\kappa$-entropy
[\cite{GK2001}], the basic deformed entropies [\cite{AL1}] etc., were also proposed. The equilibrium formulation
of statistical mechanics based on the deformed entropies were developed and applied to a wide variety of systems
like long range interacting rotators [\cite{BJC02}], relativistic gases [\cite{AL2},\cite{CCN2}], and systems with long range microscopic
memory [\cite{MT12}]. Classification of the eight ensembles in to two different classes namely, the isothermal
and the adiabatic class, and a unified description for each class in the framework of generalized statistical mechanics
based on Tsallis $q$ entropy has been carried out in [\cite{CM2011}].
Later it was shown in [\cite{SW05}] that an information theoretic entropy known as the
Sharma-Taneja-Mittal entropy [\cite{ST75},\cite{M75}] based on a two parameter logarithm was a natural
generalization of both the Tsallis $q$-entropy and the $\kappa$-entropy. A study of nonlinear Fokker Planck
equation corresponding to the two parameter $(r,s)$ Sharma-Taneja-Mittal entropy was carried out in [\cite{TF99}] and the
role played by the two parameters was investigated. One of the parameter `$r$' was found to describe
the nature of the stationary solution i.e., whether it is a weak stationary or a strong stationary solution and
hence $r$ determines the degree of distortion of the usually obtained Gaussian distribution. The other parameter
`$s$' characterizes the dynamical properties of the transient solution and thus distinctly differentiates the
regions of subdiffusion and superdiffusion. Investigation using the Lie symmetries [\cite{AS09}] proved that the localized
initial states are well approximated by the two parameter Gaussian rather than the standard Gaussian. The Sharma-Taneja-Mittal
entropy is nonsymmetric with respect to its parameters, so with a view to
provide a symmetric generalization of the Tsallis $q$ entropy, a two parameter logarithm and its inverse function the two
parameter exponential were introduced in [\cite{S2007}]. The doubly deformed $(q,q^{\prime})$ logarithm and its
inverse the $(q,q^{\prime})$-exponential for a real variable $x$ are
\begin{eqnarray}
\ln_{q,q^{\prime}} (x) &=& \frac{1}{1-q^{\prime}} \left[\exp \left(\frac{1-q^{\prime}}{1-q} (x^{1-q} -1)\right) -1\right],
\label{2plog}\\
\exp_{q,q^{\prime}} (x) &=& \left[ 1 + \frac{1-q}{1-q^{\prime}} \ln\left(1+(1-q^{\prime})x \right) \right]^\frac{1}{1-q}.
\label{2pexp}
\end{eqnarray}
The two parameter algebra based on the doubly deformed logarithm and the exponential was found [\cite{EB08}] to be
nondistributive in nature. Based on the definition of the two parameter logarithm $(\ref{2plog})$ the generalized entropy is
\begin{eqnarray}
S_{q,q^{\prime}} &=& k_{{}_{TS}} \, \sum_{i=1}^{w} p_{i} \, \ln_{q,q^{\prime}} \left(\frac{1}{p_{i}}\right), \nonumber \\
&=& \frac{k_{{}_{TS}}}{1-q^{\prime}} \, \sum_{i=1}^{w} p_{i} \,
\left(\exp \left(\frac{1-q^{\prime}}{1-q} (p_{i}^{q-1} -1)\right) -1 \right).
\label{2p_entropy}
\end{eqnarray}
Under the assumption of equiprobability i.e., $p_{i} = 1/w$ the two parameter entropy (\ref{2p_entropy}) reads:
\begin{equation}
S_{q,q^{\prime}} = \frac{k_{{}_{TS}}}{1-q^{\prime}} \, \left(\exp \left(\frac{1-q^{\prime}}{1-q} (w^{1-q} -1)\right) -1 \right).
\label{2p_entropy_mc}
\end{equation}
Both the doubly deformed logarithm (\ref{2plog}) and the exponential (\ref{2pexp})
reduce to the respective one parameter logarithm or exponential defined in [\cite{CT1}], when we let either
$q$ or $q^{\prime}$ tend to unity. Due to this property, the Tsallis generalized entropy can be recovered by allowing
either of the deformation parameters to approach it limiting value.
Nonextensivity has been proved to introduce effective interactions in ideal systems. In [\cite{CCN1}],
we observed that an effective interaction was introduced between different degrees of freedom
in an ideal diatomic gas molecule and in Ref. [\cite{AL1}] it was clearly proved that the effect of basic
deformation of a free ideal gas can be viewed as an effective interaction described by an
Hamiltonian with logarithmic interactions. Recent simulational studies [\cite{RGD09}] show that classical
collisions of a single ion in a radio frequency ion trap with an ideal gas generates non-Maxwellian
distribution functions. Thus nonextensive statistical mechanics may be considered
as a method to study interacting systems using noninteracting Hamiltonians, wherein the interactions
are introduced through the nonextensivity in statistical mechanics. Taking this point of view, we feel
that using a two parameter entropy gives us more freedom in introducing different forms of
interactions both in terms of variety and their strengths. Throughout this article we take this spirit
in investigating the ideal systems. Investigating the effect of a gravitational field on a system of
ideal gas and one dimensional hard rods will help us in understanding the motion of a interacting gas cloud under the
influence of gravity. We expect these calculations to correspond to some scenarios in which a massive object
gravitationally attracts gas molecules from a gas cloud.
A canonical formulation of generalized statistical mechanics based on the two parameter entropy (\ref{2p_entropy}) was
carried out in Ref. [\cite{As2008}] and the authors found that the distribution could be obtained in terms
of the Lambert's $W$-function. This can be formally extended to describe the other members of the isothermal class of ensembles.
In our current work we investigate the adiabatic class of ensembles of the generalized statistical mechanics
based on the two parameter entropy. We provide a unified description of the adiabatic class and provide a
two parameter generalization of the equipartition theorem, virial theorem and the adiabatic theorem. All the
four ensembles are illustrated using the classical nonrelativistic ideal gas and their corresponding heat functions
and heat capacities are obtained analytically. For the ideal gas, the phase space volume in the microcanonical and the
isoenthalpic-isobaric ensemble can be found easily. To compute the respective heat function we need to adopt the
large $N$ limit and the results are obtained in terms of the Lambert's $W$-function. A brief introduction to the
$W$-function is provided in Section II of this article. The phase space volume corresponding to the $(\mu,V,\mathsf{L})$
and the $(\mu,P,\mathsf{R})$ ensembles could not be evaluated exactly and we overcome this
by using the large $N$ limit. Since the large $N$ limit has been already used in the computation of the phase space volume
no further approximations are needed in the calculation of the heat function and the specific heat. The effect of
gravity on a classical nonrelativistic ideal gas, and a system of one dimensional hard rod gas has been investigated
[\cite{Ro1997},\cite{Ro1999}] in the extensive Boltzmann-Gibbs statistical mechanics. The authors found that the entropy is
a decreasing function of the gravitational field indicating that gravity has an ordering effect in a thermodynamic system.
In the present work we study the effect of a gravitational field in the framework of two parameter $(q,q^{\prime})$-entropy.
First a $D$-dimensional nonrelativistic classical ideal gas confined in a finite region of space, subjected to
an external gravitational field is investigated. The entropy and the specific heat are obtained exactly as a function of the internal
energy. In the infinite height limit, the internal energy could be found only when we assume that the number of
particles is very large. Next we study a system of one dimensional hard rods in the presence of gravity. Analogous to
the classical ideal gas, the entropy and the specific heat are obtained as a function of internal energy. But in the
infinite height limit the internal energy could be obtained as a function of temperature subject to the condition that
the number of particles is very large. The heat capacities were found to admit both positive and negative values, in contrast
to the previous results in [\cite{SA99}] where the specific heat of a system of gas molecules in thermodynamic limit permits only
negative values. Specific heats are supposed to characterize the amount of energy change when temperature is varied. Energy can both
be liberated or absorbed depending on the interactions in the system as $T$ is varied. Our contention here is that with
the $(q,q^{\prime})$ dependence the system exhibits effective interactions. The fact that specific heat can become negative
for some range of $(q,q^{\prime})$ and may give clues about {\it(i)} Choosing the range of $(q,q^{\prime})$ for systems described
by positive specific heat. {\it(ii)} If for a non-Boltzmann system with negative specific heat, it can help us to infer
about interacting systems.
The plan of the article is as follows: Following the introduction in Section I we give a brief summary of the
Lambert's $W$ function in Section II. A unified description of the thermostatistical structure of all the four adiabatic
ensembles is provided in Section III. In Section IV, we study all the four ensembles using classical nonrelativistic
ideal gas, and each ensemble is treated separately in a subsection. The effect of an external
gravitational field on thermodynamic system is studied in Section V. The first part of the section deals with the
effect of gravity on a $D$-dimensional classical ideal gas. A system of one dimensional hard rod gas is examined in the
second part of the section. The investigations are carried out in the microcanonical ensemble. We present our conclusions
in Section V.
\setcounter{equation}{0}
\section{Lambert's W-function: a primer}
\label{wfunction}
A brief summary of the Lambert's $W$-function and its mathematical properties is presented
in the current section. Lambert's $W$-function represented by $W(z)$ is defined as the multivalued
inverse of the function $w \ e^{w} = z$ and satisfies the relation
\begin{equation}
W(z) \ e^{W(z)} = z, \qquad z \in \mathbb{C}.
\label{Wfunc_def}
\end{equation}
The various branches corresponding to the $W$-function are indexed by $k=0,\pm1,\pm2,...,$
and for real $z<-1/e$, the function $W(z)$ is always complex and multivalued. For real
$z$ in the range $-1/e \leq z < 0$, the function $W(z)$ comprises of two real branches namely
$W_{0}(z)$ and $W_{-1}(z)$ represented by the solid line and the dotted line respectively in
Figure 1. The branch $W_{0}(z)$ satisfies the condition $W(z) \geq -1$,
and is generally known as the principal branch of the $W$ function. Correspondingly, when
$W(z) \leq -1$ we have the $W_{-1}(z)$ branch.
\begin{figure}
\begin{center}
\scalebox{1.0}{\input{wfunction.tex}}
\label{wfunc} \\
Figure 1: A plot of the Lambert's $W$ function $W(x)$ as a function of $x$.
\end{center}
\end{figure}
The principal branch is analytic at $z=0$ and has the series expansion
\begin{equation}
W_{0}(z) = \sum_{n=1}^{\infty} \frac{(-n)^{n-1}}{n!} \, z^{n},
\end{equation}
with a radius of convergence $e^{-1}$. Similarly the series corresponding
to the branch $W_{-1}$ is
\begin{equation}
W_{-1}(z) = \sum_{n=0}^{\infty} c_{n} \, f^{n}(z);
\qquad \qquad f(z) = \sqrt{2 \ (ez + 1)},
\end{equation}
where the expansion coefficients can be computed from the recurrence relations
\begin{eqnarray}
c_{k} &=& \frac{k-1}{k+1} \,\left(\frac{c_{k-2}}{2} + \frac{d_{k-2}}{4} \right)
- \frac{d_{k}}{2} - \frac{c_{k-1}}{k+1},
\qquad
d_{k} = \sum_{j=2}^{k-1} c_{j} \, c_{k+1-j},
\label{exp_coeff} \\
c_{0} &=& -1, \qquad c_{1} = 1, \qquad d_{0} = 2, \qquad d_{1} = -1.
\label{ini_coeff}
\end{eqnarray}
The series converges for $-1/e \leq z < 0$, which covers the whole domain
of the $W_{-1}$ branch. From (\ref{Wfunc_def}) the derivative of the $W$
function is found
\begin{equation}
W^{\prime}(z) = \frac{W(z)}{z \ (1+W(z))}.
\end{equation}
In the current article physical requirements restricts our choice of the $W$ function to the principal branch.
An excellent introduction to the Lambert's $W$-function and its applications to engineering
problems is discussed in [\cite{C96}]. Recently in signal processing the role of the $W_{-1}$
branch was analyzed [\cite{FCB02}]. The Lambert's function has been used in the study of
both classical [\cite{Ca2004}] and quantum statistical mechanics [\cite{Va2009},\cite{Va2010}].
It also occurs [\cite{EL05}] in the study of Fokker Planck equation in the small noise limit.
\setcounter{equation}{0}
\section{Adiabatic ensemble: Generalized formulation}
\label{micro}
The individual members of an adiabatic ensemble have the same value of the heat
function though they can be at different temperatures. For a system in
thermodynamic equilibrium, there are four different adiabatic ensembles namely
the microcanonical ensemble $(N,V,U)$, the isoenthalpic-isobaric ensemble
$(N,P,H)$, the adiabatic ensemble with number fluctuations $(\mu,V,\mathsf{L})$,
and the adiabatic ensemble with both number and volume fluctuations $(\mu,P,\mathsf{R})$.
The microstate of a system of $N$ particles can be represented by a single point in
the $2 DN$ dimensional phase space. Corresponding to a particular value of the heat
function which is a macrostate, we have a huge number of microstates. We need to compute
the total number of microstates since it is a measure of the entropy. The points
denoting the microstate of the system lie so close to each other that the surface area
of the constant heat function curve in the phase space is regarded as a measure of the
total number of microstates. For a system described by a Hamiltonian $\mathcal{H}$ the
surface area corresponding to a constant heat function
$\mathfrak{H}$ curve can be calculated from
\begin{eqnarray}
\Omega (\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H})
= \sum_{X_{\{\ell\}}} \, \frac{1}{N! \, h^{DN}}
\int_{r_{i}} \int_{p_{i}}
\delta \Big(\mathcal{H}+\sum_{\{\ell\}}
x_{\ell}X_{\ell} - {\mathfrak{H}}\Big)
\prod_{i=1}^{N} {\rm d}^{D} r_{i} \, {\rm d}^{D} p_{i}.
\label{sds_en_TS}
\end{eqnarray}
Similarly the phase space volume enclosed by the constant heat function ${\mathfrak{H}}$ curve is
\begin{equation}
\varSigma(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H}) =
\sum_{X_{\{\ell\}}} \, \frac{1}{ N! \, h^{DN}}
\int_{r_{i} } \int_{p_{i}}
\Theta \Big(\mathcal{H} + \sum_{\{\ell\}}
x_{\ell}X_{\ell} - {\mathfrak{H}} \Big)
\prod_{i=1}^{N} {\rm d}^{D} r_{i}\, {\rm d}^{D} p_{i},
\label{vds_en_TS}
\end{equation}
where $\Theta$ is the Heaviside step function.
Computation of the volume of the constant heat function curve assumes significance because
of the difficulty in calculating the area of the curve. The volume enclosed by the phase space curve
and its surface area are related via the expression
\begin{equation}
\Omega (\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H}) = \frac{\partial}{\partial \mathfrak{H}}
\varSigma(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H}).
\label{sds_vds_rel}
\end{equation}
Since the phase space volume is a measure of the number of the microstates of the system,
the two parameter entropy can be directly obtained from the knowledge of
$\varSigma(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H})$ via the relation
\begin{equation}
S_{q,q^{\prime}}(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H}) = k \; \ln_{q,q^{\prime}}
\varSigma(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H}).
\label{entr_def_rel}
\end{equation}
For a given adiabatic ensemble the temperature is defined via the relation
\begin{equation}
T = \left(\frac{\partial S_{q,q^{\prime}}}{\partial {\mathfrak{H}}}\right)^{-1}
= \frac{(\varSigma(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H}))^{{}^{q}}}
{k \;\Omega(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H}) \,
\exp\left(\frac{1-q^{\prime}}{1-q}((\varSigma(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H}))^{1-q} -1) \right)}.
\label{temp_def_TS}
\end{equation}
Using the definition of the temperature (\ref{temp_def_TS}), the phase space volume (\ref{vds_en_TS}) and
the surface area (\ref{sds_en_TS}), we can calculate the expression corresponding to
the heat function. From the expression for the heat function the specific heat can be calculated through
the relation
\begin{equation}
C_{x_{\{\ell\}}} = \frac{\partial {\mathfrak{H}}_{q}}
{\partial T} \bigg|_{x_{\{\ell\}}}.
\label{psh_def_TS}
\end{equation}
For any adiabatic ensemble the expectation value of an observable $O$ is defined as
\begin{equation}
\langle O \rangle = \frac{1}{\Omega(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H})} \,
\sum_{X_{\{\ell\}}}
\frac{1}{N! \, h^{DN}} \,
\int_{r_{i} } \int_{p_{i}} O \;\;
\delta \Big(\mathcal{H} + \sum_{\{\ell\}}
x_{\ell} X_{\ell} - {\mathfrak{H}}\Big)
\prod_{i=1}^{N} {\rm d}^{D} r_{i} \, {\rm d}^{D} p_{i}.
\label{oex_sds_TS}
\end{equation}
The above expression is extremely useful in computing the average energy in the $(N,P,H)$,
$(\mu,V,\mathsf{L})$ and the $(\mu,P,\mathsf{R})$ ensembles. By using the suitable Legendre
transformations, the average energy can also be obtained from the heat functions corresponding to
these ensembles.
From the entropy the extensive thermodynamic quantities whose conjugate intensive variables are held fixed
can be computed through the relation
\begin{equation}
X_{\ell}(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H}) = - \frac{1}{\beta} \;
\frac{\partial}{\partial x_{\ell}}\,
S(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H}).
\label{vol_N_def_TS}
\end{equation}
The equipartition theorem is derived in a unified manner for all the four adiabatic ensembles in the framework
of generalized statistical mechanics based on the Schw\"{a}mmle-Tsallis $(q,q^{\prime})$ entropy. Representing the
phase space variables $r_{i}$ and $p_{i}$ $(i=1,2,...,3N)$ by a common variable $y_{i}$, we find the
following expectation value
\begin{equation}
\bigg\langle y_{i}\; \frac{\partial {\mathcal{H}}}
{\partial y_{j}} \bigg\rangle = k T \;
\left(\varSigma(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H})\right)^{1-q} \;
\exp\left(\frac{1-q^{\prime}}{1-q}((\varSigma(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H}))^{1-q} -1) \right)
\delta_{ij},
\label{ept_uf_TS}
\end{equation}
which is a generalization of the equipartition function. We notice that the $(q,q^{\prime})$ generalized
equipartition theorem is dependent on the phase space volume in contrast to the extensive Boltzmann-Gibbs statistics.
This observation is in line with the earlier result obtained for the Tsallis entropy in Ref. [\cite{CM2011}] and
yields the Boltzmann-Gibbs limit when both the nonextensive parameters $q$ and $q^{\prime}$ are set to unity.
When we let the variable $y_{i}$ to be the coordinate $r_{i}$ and invoke the Dirac delta function, we get a
specific form of the equipartition theorem:
\begin{eqnarray}
\bigg\langle r_{i}\; \frac{\partial {\mathcal{H}}}
{\partial r_{i}} \bigg\rangle \equiv - \langle r_{i} \, \dot{p}_{i} \rangle
= \langle r_{i} \, F_{i} \rangle
&=& k T \; \left(\varSigma(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H})\right)^{1-q} \nonumber \\
& & \exp\left(\frac{1-q^{\prime}}{1-q}((\varSigma(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H}))^{1-q} -1) \right).
\label{ept_pos_TS}
\end{eqnarray}
Similarly when we set $y_{i}$ to be equal to $p_{i}$ the momentum variable we obtain
\begin{equation}
\bigg\langle p_{i}\, \frac{\partial {\mathcal{H}}}{\partial p_{i}} \bigg\rangle \equiv
\langle p_{i} \, \dot{r}_{i} \rangle = k T \, \left(\varSigma(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H})\right)^{1-q} \,
\exp\left(\frac{1-q^{\prime}}{1-q}((\varSigma(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H}))^{1-q} -1) \right).
\label{ept_mtm_TS}
\end{equation}
Through a canonical transformation, sometimes the Hamiltonian of a system can be written in the following form
$\mathcal{H} = \sum_{i} (A_{i} \ P_{i}^{2} + B_{i} \ Q_{i}^{2})$. Aided by equations (\ref{ept_pos_TS}) and
(\ref{ept_mtm_TS}) we find the expectation value of such an Hamiltonian to be
\begin{equation}
\langle \mathcal{H} \rangle = DN \, k T\, \left(\varSigma(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H})\right)^{1-q} \,
\exp\left(\frac{1-q^{\prime}}{1-q}((\varSigma(\mathfrak{X}_{1},\mathfrak{X}_{2},
\mathfrak{H}))^{1-q} -1) \right).
\end{equation}
A two parametric generalization of the virial theorem could be obtained from the relation (\ref{ept_pos_TS})
and reads:
\begin{equation}
\bigg \langle \sum_{i}^{DN} r_{i} \, \dot{p}_{i} \bigg \rangle = - DN \, k T \,
\left(\varSigma(\mathfrak{X}_{1},\mathfrak{X}_{2},
\mathfrak{H})\right)^{1-q} \,
\exp\left(\frac{1-q^{\prime}}{1-q}((\varSigma(\mathfrak{X}_{1},
\mathfrak{X}_{2},\mathfrak{H}))^{1-q} -1) \right).
\label{vt_TS}
\end{equation}
Below we verify the adiabatic theorem for the generalized statistical mechanics based on the
two parameter entropy. Let us consider the Hamiltonian to contain an external parameter
$a$, in addition to the phase space co-ordinates. An expectation value of the derivative of
the Hamiltonian with respect to the external parameter yields the thermodynamic conjugate
variable $f$ corresponding to the parameter $a$
\begin{eqnarray}
\bigg\langle \frac{\partial \mathcal{H}} {\partial a} \bigg\rangle &=&
\frac{1}{\Omega(\mathfrak{X}_{1},\mathfrak{X}_{2},\mathfrak{H})}
\sum_{X_{\ell}} \frac{1}{N! \, h^{DN}} \, \int_{r_{i}} \int_{p_{i}}
\left(\frac{\partial \mathcal{H}}{\partial a} \right)
\delta(\mathcal{H} + \sum_{X_{\ell}} x_{\ell} X_{\ell} - \mathfrak{H})
\prod_{i=1}^{N} d^{D}r_{i} \, d^{D} p_{i}, \nonumber \\
&=&\frac{\partial \mathfrak{H}} {\partial a} = f.
\label{adb_thr_TS}
\end{eqnarray}
The description of the adiabatic ensembles given above has been illustrated through
certain examples. First we consider a classical ideal gas and provide an analytic
solution to the specific heat in all the four ensembles by using a large $N$
approximation. Later on we study a microcanonical $D$-dimensional classical
ideal gas under the influence of gravity and compute its specific heat. Finally
we investigate a hard rod gas under gravity in the framework of microcanonical ensemble.
\setcounter{equation}{0}
\section{Application - Classical ideal gas}
\label{Ideal gas}
The Hamiltonian of a nonrelativistic classical ideal gas in $D$ dimensions is
\begin{equation}
\mathcal{H} = \sum_{i} \frac{p_{i}^{2}}{2m}, \qquad
p_{i}=|{\bf{p}}_{i}|,
\label{ham_ig}
\end{equation}
where ${\bf{p}}_{i}$ for $(i = 1,2,...,N)$ represent the $D$-dimensional momenta of the gas molecules.
In this section we find the phase space volume corresponding to this Hamiltonian for all the
four ensembles. From the phase space volume we derive
the relevant thermodynamic quantities like the entropy, the heat function and
the heat capacity.
\subsection{Microcanonical ensemble}
The classical nonrelativistic ideal gas described by (\ref{ham_ig}) is studied in the microcanonical
ensemble. In order to compute the entropy of the system, we calculate the phase space volume
enclosed by the constant energy curve. Substituting the expression of the Hamiltonian in
(\ref{vds_en_TS})
\begin{equation}
\varSigma(N,V,E) = \frac{1}{N! \, h^{DN}} \; \int_{r_{i}} \int_{p_{i}}
\Theta\left(\sum_{i} \frac{p_{i}^{2}}{2m} - E\right)\;
\prod_{i=1}^{N} {\rm d}^{D} r_{i}\, {\rm d}^{D} p_{i}.
\label{mc_vs}
\end{equation}
The phase space integral is computed in the following manner: First we notice that
the momentum integration is the volume of a $DN$-dimensional sphere of radius
$\sqrt{2mE}$. Next we integrate over the position co-ordinates to obtain the
phase space volume in the microcanonical ensemble
\begin{equation}
\varSigma(N,V,E) = \frac{V^{N}}{N!} \; \frac{\mathcal{M}^{N}}{\Gamma\left(\frac{D N}{2}+1\right)} \;
E^{\frac{DN}{2}},
\label{mc_vs_fe}
\end{equation}
where we define $\mathcal{M} = (2 \pi m/h^{2})^{D/2}$ for the sake of convenience.
The corresponding surface area enclosed by the phase space curve is
\begin{equation}
\Omega(N,V,E) = \frac{V^{N}}{N!} \; \frac{\mathcal{M}^{N}}{\Gamma\left(\frac{D N}{2}\right)} \;
E^{\frac{DN}{2}-1}.
\end{equation}
The microcanonical entropy of a classical ideal gas obtained from the knowledge of the
phase space volume (\ref{mc_vs_fe}) reads:
\begin{equation}
S_{q,q^{\prime}} = \frac{k}{1-q^{\prime}} \; \left[\exp \left(\frac{1-q^{\prime}}{1-q}
\big((\Xi_{mc} \; E^{\frac{DN}{2}})^{1-q} -1\big)\right) -1\right],
\label{entr_mc_ig}
\end{equation}
where the factor $\Xi_{mc}$ is defined as
\begin{equation}
\Xi_{mc} = \frac{V^{N}}{N!} \; \frac{\mathcal{M}^{N}}{\Gamma\left(\frac{D N}{2}+1\right)}.
\label{xi_def_ig}
\end{equation}
In the limit $(q,q^{\prime}) \rightarrow 1$, we recover the extensive Boltzmann Gibbs entropy.
From the definition of temperature (\ref{temp_def_TS}), we arrive at the following expression
\begin{equation}
\frac{1}{T} = \frac{DN}{2} \, k \; \Xi_{mc}^{1-q} \, E^{(1-q)\,\frac{DN}{2} - 1} \,
\exp \left(\frac{1-q^{\prime}}{1-q}
\big((\Xi_{mc} \; E^{\frac{DN}{2}})^{1-q} -1\big)\right).
\label{temp_inte_rel}
\end{equation}
Inversion of the above relation (\ref{temp_inte_rel}), to obtain the internal energy is intractable and so,
we look into the naturally occurring large $N$ limit. In this limit we can safely neglect the factor of one
in comparison with $(1-q) DN/2$ and this enables us to make the following approximation
$E^{(1-q) \, {\frac{DN}{2}} -1} \approx E^{(1-q) \, {\frac{DN}{2}}}$. For very large values of $N$ the
deformation parameter $(1-q)$ should be $O(1/N)$ for the factor $1$ to make a reasonable contribution.
Using the assumption outlined above we get
\begin{equation}
\exp\left( \frac{1-q^{\prime}}{1-q} \right) \frac{2}{DN \, k \, T} =
\Xi_{mc}^{1-q} \, E^{(1-q) \,\frac{DN}{2}} \,
\exp\left(\frac{1-q^{\prime}}{1-q} \, (\Xi_{mc} \; E^{\frac{DN}{2}})^{1-q} \right).
\label{lamb_eqn}
\end{equation}
The inversion of the above function leads to an expression of the internal energy in terms of the
Lambert's $W$ function as
\begin{equation}
E = \left[\frac{1-q}{1-q^{\prime}} \; \frac{1}{\Xi_{mc}^{1-q}} \;
W_{0} \left(\frac{1-q^{\prime}}{1-q} \, \exp\left( \frac{1-q^{\prime}}{1-q}\right)
\frac{2 \beta}{DN} \right)\right]^
{\frac{2}{(1-q)\, DN}}.
\label{mc_ig_fe}
\end{equation}
The requirement that the entropy be concave decides the range of the deformation parameters. From
the discussion in Ref. [\cite{S2007}], we notice that the entropy is concave in the region $q + q^{\prime} \geq 1$
excluding the region $0 \leq (q,q^{\prime}) \leq 1$, where the entropy does not have fixed curvature sign. The argument
of the $W$ function in (\ref{mc_ig_fe}) is positive in the region (a) for which $q > 1$ and $q^{\prime} > 1$, and hence the energy
in this region is characterized by the principal branch. Under the restriction $q + q^{\prime} < 1$, there are
two other major regions namely, {(b)} $q > 1$ and $-\infty < q^{\prime} < 1$ and {(c)} $q^{\prime} > 1$
and $-\infty < q < 1$ in which the argument of the $W$ function is negative. Though we have a choice between the
$W_{0}$ and the $W_{-1}$ branch, the continuity requirement on energy restricts our choice to the $W_{0}$ branch.
Thus we conclude that only the principal branch contributes in the definition of energy. In all the subsequent
discussions in this article, we maintain the definition of the regions (a), (b) and (c) as described above.
The specific heat at constant volume evaluated from the internal energy is
\begin{equation}
C_{V} \equiv \frac{\partial E}{\partial T} = - \frac{k \, \beta}{(1-q) \, DN/2} \;
\frac{W_{0} (\mathsf{b} \, \beta)}{1 + W_{0} (\mathsf{b} \, \beta)} \;
\left[\mathsf{a}\, W_{0} (\mathsf{b} \, \beta)\right]^{\frac{1-(1-q)DN/2}{(1-q) DN/2}},
\label{mc_ig_spht}
\end{equation}
where the factors $\mathsf{a}$ and $\mathsf{b}$ are as defined below:
\begin{equation}
\mathsf{a} = \frac{1-q}{1-q^{\prime}} \; \frac{1}{\Xi_{mc}^{1-q}}, \qquad
\mathsf{b} = \frac{1-q^{\prime}}{1-q} \, \exp\left(\frac{1-q^{\prime}}{1-q}\right)
\frac{2}{DN}.
\label{mc_ab_def}
\end{equation}
Invoking the large $N$ limit in the final expression of the specific heat we arrive at
\begin{equation}
C_{V} = - \frac{k \, \beta}{(1-q) \, DN/2} \; \frac{1}{\mathsf{a}(1 + W_{0} (\mathsf{b} \, \beta))}.
\label{mc_ig_spht_app}
\end{equation}
From the expression of the specific heat (\ref{mc_ig_spht_app}), we notice that it can be either
positive or negative depending on the values of the deformation parameters $q$ and $q^{\prime}$. In the
region (a) and region (c) the specific heat is positive, whereas in the region (b) it is negative.
We do not recover the Boltzmann Gibbs statistics from our above calculations, because in the
computation of energy (\ref{mc_ig_fe}), we have made use of the large $N$ limit which does not commute with extensive
$(q,q^{\prime}) \rightarrow 1$ limit.
\subsection{Isoenthalpic-isobaric ensemble}
A system which exchanges internal energy and volume with its surroundings in such a way that its
enthalpy remains constant is described by the isoenthalpic-isobaric ensemble. In order to calculate
the thermodynamic quantities we first find the phase space volume enclosed by the constant
enthalpy curve. The integral expression corresponding to the phase space volume is
\begin{equation}
\varSigma(N,P,H) = \sum_{V} \frac{1}{N! \, h^{DN}} \; \int_{r_{i}} \int_{p_{i}}
\Theta\left(\sum_{i} \frac{p_{i}^{2}}{2m} + P V - H\right)\;
\prod_{i=1}^{N} {\rm d}^{D} r_{i}\, {\rm d}^{D} p_{i}.
\label{VS_nrg}
\end{equation}
Using the fact that the momentum integration is nothing but geometrically equivalent to a
$DN$ dimensional sphere of radius $\sqrt{2m(H - PV)}$, and integrating over the
position co-ordinates we arrive at
\begin{equation}
\varSigma(N,P,H) = \frac{\mathcal{M}^{N}}{N!} \; \frac{1}{\Gamma\left(\frac{D N}{2}+1\right)} \;
\sum_{V} V^{N} (H-P V)^{\frac{DN}{2}}.
\label{V_sum}
\end{equation}
In the next step we consider a summation over the volume eigenstates. Since the volume states
are very closely spaced the summation is replaced by an integration. But an integral over the
volume leads to over counting of the eigenstates. To overcome this we employ the shell particle
method of counting volume states developed in [\cite{DC1},\cite{DC2}]. In this method we take
into account only the minimum volume needed to confine a particular configuration. The minimum
volume needed to confine a particular configuration is found by imposing a condition wherein we
require atleast one particle to lie on the boundary of the system. All the equivalent ways of
choosing a minimum volume for a particular configuration is treated as the same volume eigenstate
and is considered only once. Using this shell particle technique to reject the redundant volume
states we arrive at
\begin{equation}
\varSigma(N,P,H) = \mathcal{M}^{N} \;
\left(\frac{1}{P}\right)^{N} \;
\frac{H^{\mathfrak{D}}}
{ \Gamma({\mathfrak{D}} + 1 )}, \qquad
\mathfrak{D} = \frac{DN}{2} + N.
\label{vds_nr_fe}
\end{equation}
A similar evaluation of the surface area enclosed by the curve of constant enthalpy yields
\begin{equation}
\Omega(N,P,H) = \mathcal{M}^{N} \;
\left(\frac{1}{P}\right)^{N} \;
\frac{H^{{\mathfrak{D}}-1}}
{\Gamma({\mathfrak{D}})}.
\label{sds_nr_fe}
\end{equation}
From the knowledge of the phase space volume (\ref{vds_nr_fe}), the entropy of a classical ideal gas in the
isoenthalpic-isobaric ensemble is found:
\begin{equation}
S_{q,q^{\prime}} = \frac{k}{1-q^{\prime}} \; \left[\exp \left(\frac{1-q^{\prime}}{1-q}
\big((\Xi_{ie} \; {H^{\mathfrak{D}}})^{1-q} -1\big)\right) -1\right].
\label{entr_ie_ig}
\end{equation}
The Boltzmann Gibbs entropy is recovered in the limit $(q,q^{\prime}) \rightarrow 1$.
The factor $\Xi_{ie}$ used in the above equation is
\begin{equation}
\Xi_{ie} = \mathcal{M}^{N} \;
\left(\frac{1}{P}\right)^{N} \;
\frac{1}{ \Gamma({\mathfrak{D}} + 1)}.
\end{equation}
The partial derivative of the entropy (\ref{entr_ie_ig}) with respect to the enthalpy gives the temperature of the
isoenthalpic-isobaric ensemble
\begin{equation}
\frac{1}{T} = {\mathfrak{D}} \, k \, \Xi_{ie}^{1-q} \, H^{(1-q)\, {\mathfrak{D}} - 1} \,
\exp \left(\frac{1-q^{\prime}}{1-q}
\big((\Xi_{ie} \; {H^{\mathfrak{D}}})^{1-q} -1\big)\right).
\label{temp_def_ie}
\end{equation}
In order to invert the above equation we assume the large $N$ limit and this consequently
leads to the approximation that
$H^{(1-q) \, {\mathfrak{D}}-1} \approx H^{(1-q) \, {\mathfrak{D}}}$.
Rewriting the equation (\ref{temp_def_ie}) based on the assumption yields the expression
\begin{equation}
\exp\left( \frac{1-q^{\prime}}{1-q} \right) \frac{1}{{\mathfrak{D}} \, k \, T} =
\Xi_{ie}^{1-q} \, H^{(1-q) \, {\mathfrak{D}}} \,
\exp\left(\frac{1-q^{\prime}}{1-q} \, (\Xi_{ie} \; {H^{\mathfrak{D}}})^{1-q} \right).
\label{mve_lambw_func}
\end{equation}
The solution of the above equation yields the enthalpy in terms of the Lambert's $W$-function
\begin{equation}
H = \left[\frac{1-q}{1-q^{\prime}} \; \frac{1}{\Xi_{ie}^{1-q}} \;
W_{0} \left(\frac{1-q^{\prime}}{1-q} \, \exp\left( \frac{1-q^{\prime}}{1-q}\right)
\frac{\beta}{{\mathfrak{D}}} \right)\right]^
{\frac{1}{(1-q) {\mathfrak{D}}}}.
\label{enth_expr_ie}
\end{equation}
The requirement that the entropy should be concave, along with the fact that the enthalpy is a
continuous function restricts our choice of the $W$ function to the principal branch.
The specific heat at constant pressure computed from the enthalpy (\ref{enth_expr_ie}) is
\begin{equation}
C_{P} \equiv \frac{\partial H}{\partial T}= - \frac{k \, \beta}{(1-q) \, {\mathfrak{D}}}
\; \frac{W_{0}(\mathfrak{b} \, \beta)}{1 + W_{0} (\mathfrak{b} \, \beta)} \;
\left[\mathfrak{a}\, W_{0} (\mathfrak{b} \, \beta)\right]^{\frac{1-(1-q){\mathfrak{D}}}{(1-q) {\mathfrak{D}}}},
\label{spht_ie_expr}
\end{equation}
where, the factors $\mathfrak{a}$ and $\mathfrak{b}$ are as defined below:
\begin{equation}
{\mathfrak{a}} = \frac{1-q}{1-q^{\prime}} \; \frac{1}{\Xi_{ie}^{1-q}}, \qquad
{\mathfrak{b}} = \frac{1-q^{\prime}}{1-q} \, \exp\left( \frac{1-q^{\prime}}{1-q}\right)
\frac{1}{{\mathfrak{D}}}.
\label{mfab_def}
\end{equation}
In the large $N$ limit the expression for the heat capacity reads:
\begin{equation}
C_{P} = - \frac{k \, \beta}{(1-q) \, {\mathfrak{D}}} \; \frac{1}{{\mathfrak{a}}(1 + W_{0} (\mathfrak{b} \, \beta))}.
\label{spht_lgN}
\end{equation}
The specific heat is positive in the regions (a) and (c) and negative in the region (b). Since we adopted
the large $N$ limit in evaluating the enthalpy (\ref{enth_expr_ie}) we do not recover the corresponding Boltzmann Gibbs result,
because the extensive limit and the large $N$ limit do not commute with each other.
\subsection{$(\mu,V,\mathsf{L})$ ensemble}
The Hill energy $\mathsf{L}$ is the heat function corresponding to the $(\mu,V,\mathsf{L})$ ensemble. The
phase space volume enclosed by the curve of constant Hill energy $\mathsf{L}$ is
\begin{equation}
\varSigma(\mu,V,\mathsf{L}) = \sum_{} \frac{1}{N! \, h^{DN}} \; \int_{r_{i}} \int_{p_{i}}
\Theta\left(\sum_{i} \frac{p_{i}^{2}}{2m} -\mu N - \mathsf{L}\right)\;
\prod_{i=1}^{N} {\rm d}^{D} r_{i}\, {\rm d}^{D} p_{i}.
\label{nad_psv_ig}
\end{equation}
Integrating over the phase space variables $p_{i}$ and $r_{i}$, we arrive at
\begin{equation}
\varSigma(\mu,V,\mathsf{L}) = \sum_{N=0}^{\infty} \frac{V^{N}}{N!} \; \frac{\mathcal{M}^{N}}{\Gamma\left(\frac{D N}{2}+1\right)} \;
(\mathsf{L} + \mu N)^{\frac{DN}{2}}.
\label{nad_psv_nsum}
\end{equation}
An exact evaluation of the summation given in (\ref{nad_psv_nsum}) could not be achieved. In order to calculate an approximate value
in the large $N$ limit we carry out the following procedure. First we approximate the factor $(\mathsf{L} + \mu N)^{\frac{DN}{2}}$
as follows
\begin{eqnarray}
(\mathsf{L} + \mu N)^{\frac{DN}{2}} &=& (\mu N)^{\frac{DN}{2}} \, \left(1 + \frac{\mathsf{L}}{\mu N}\right)^{\frac{DN}{2}}, \nonumber \\
&=& (\mu N)^{\frac{DN}{2}} \, \exp\left(\frac{DN}{2}\ln\left(1 + \frac{\mathsf{L}}{\mu N}\right)\right).
\label{npn_approx}
\end{eqnarray}
For very small values of $x$ we can make use of the approximation $\ln(1+x) \approx x$ and this leads to
\begin{equation}
(\mathsf{L} + \mu N)^{\frac{DN}{2}} = (\mu N)^{\frac{DN}{2}} \exp\left(\frac{D}{2} \, \frac{\mathsf{L}}{\mu}\right).
\label{npn_approx_fe}
\end{equation}
In the next step we use the Stirling's approximation for the gamma function as given below:
\begin{equation}
\Gamma\left(\frac{DN}{2} +1\right) \approx \left(\frac{DN}{2}\right)^{\frac{DN}{2}} \exp\left(- \frac{DN}{2}\right).
\label{str_approx}
\end{equation}
Substituting the relations (\ref{npn_approx_fe}), and (\ref{str_approx}) in (\ref{nad_psv_nsum}) we arrive at
\begin{equation}
\varSigma(\mu,V,\mathsf{L}) = \exp\left(\frac{D}{2} \, \frac{\mathsf{L}}{\mu}\right) \; \sum_{N=0}^{\infty}\; \frac{V^{N}}{N!} \;
\frac{\mathcal{M}^{N}}{\left(\frac{DN}{2}\right)^{\frac{DN}{2}} \exp\left(- \frac{DN}{2}\right)} \;
(\mu N)^{\frac{DN}{2}}.
\label{psv_sum_expr}
\end{equation}
Now the summation over $N$ can be carried out in (\ref{psv_sum_expr}), enabling us to write the approximate
expression for the phase space volume
\begin{equation}
\varSigma(\mu,V,\mathsf{L}) = \exp\left(\frac{D}{2} \, \frac{\mathsf{L}}{\mu}\right) \;
\exp\left(\frac{V \, \mu^{D/2} \, e^{D/2} \, \mathcal{M}}{(D/2)^{D/2}}\right).
\label{psv_4e_fe}
\end{equation}
The approximate computation used by us to calculate the phase space volume (\ref{psv_4e_fe}) can be
considered as a first order approximation of the summation in (\ref{nad_psv_nsum}).
Similarly the surface area enclosed by the curve of constant ${\mathsf{L}}$ is also found
\begin{equation}
\Omega(\mu,V,\mathsf{L}) = \frac{D}{2 \mu} \; \exp\left(\frac{D}{2} \, \frac{\mathsf{L}}{\mu}\right) \;
\exp\left(\frac{V \, \mu^{D/2} \, e^{D/2} \, \mathcal{M}}{(D/2)^{D/2}}\right).
\label{psa_4e_fe}
\end{equation}
The entropy of the classical ideal gas in this adiabatic ensemble found from (\ref{psv_4e_fe}) is
\begin{equation}
S_{q,q^{\prime}} = \frac{k}{1-q^{\prime}} \; \left[\exp \left(\frac{1-q^{\prime}}{1-q}
\left(\left(\Xi_{he} \; \exp\left(\frac{D}{2} \, \frac{\mathsf{L}}{\mu}\right)\right)^{1-q} -1\right)\right) -1\right],
\end{equation}
where, for the sake of convenience we define:
\begin{equation}
\Xi_{he} = \exp\left(\frac{V \, \mu^{D/2} \, e^{D/2} \, \mathcal{M}}{(D/2)^{D/2}}\right).
\end{equation}
The temperature of this ensemble calculated from the defining relation (\ref{temp_def_TS}) is
\begin{equation}
\frac{1}{T} = \frac{D}{2 \mu} \, k \, \Xi_{he}^{1-q} \, \left(\exp\left(\frac{D}{2} \, \frac{\mathsf{L}}{\mu}\right)\right)^{1-q}
\exp \left(\frac{1-q^{\prime}}{1-q}
\left(\left(\Xi_{he} \; \exp\left(\frac{D}{2} \, \frac{\mathsf{L}}{\mu}\right)\right)^{1-q} -1\right)\right).
\label{temp_4e_fe}
\end{equation}
Rewriting (\ref{temp_4e_fe}) we immediately recognize it as the equation whose solution is given by the
Lambert's $W$-function. Thus the Hill energy in terms of the $W$-function reads:
\begin{equation}
\mathsf{L} = \frac{2 \, \mu}{(1-q) \, D} \; \ln \left[\frac{1-q}{1-q^{\prime}} \; \frac{1}{\Xi_{he}^{1-q}} \;
W_{0} \left(\frac{1-q^{\prime}}{1-q} \, \exp\left( \frac{1-q^{\prime}}{1-q}\right)
\frac{2 \, \beta \mu}{D} \right)\right].
\label{he_ig_fe}
\end{equation}
The principal branch of the $W$ function is chosen based on the concavity requirement on the entropy
and the continuity of the Hill energy. The Hill energy $\mathsf{L}$ is uniformly positive in the region
(c) for which $q^{\prime} > 1$ and $-\infty < q < 1$, whereas in the other two regions (b) and (c) it is positive
only when the argument of the logarithm is less than $1$. From (\ref{he_ig_fe}) the specific heat at constant
volume is found
\begin{equation}
C_{V} \equiv \frac{\partial \mathsf{L}}{\partial T} = - \frac{\mu}{(1-q) D/2} \, \frac{k \beta}{1 + W_{0} (\mathtt{b} \, \beta)}, \qquad
\mathtt{b} = \frac{1-q^{\prime}}{1-q} \, \exp\left(\frac{1-q^{\prime}}{1-q}\right) \, \frac{\mu}{D/2}.
\label{sph_he_fe}
\end{equation}
Subject to the condition $q + q^{\prime} > 1$ and excluding the region $0 \leq (q,q^{\prime}) \leq 1$,
the specific heat is positive in the region $q > 1$ and negative in the region $q < 1$. Due to the noncommutative
nature of the extensive limit and the large $N$ limit, we are unable to recover the Boltzmann Gibbs result
corresponding to the Hill energy (\ref{he_ig_fe}) and the heat capacity (\ref{sph_he_fe}).
\subsection{$(\mu,P,\mathsf{R})$ ensemble}
The adiabatic ensemble with both the number and volume fluctuations is illustrated using the classical ideal gas
in this section. The phase space volume of the classical ideal gas in this ensemble is
\begin{equation}
\varSigma(\mu,P,\mathsf{R}) = \sum_{N} \sum_{V} \frac{1}{N! \, h^{DN}} \; \int_{r_{i}} \int_{p_{i}}
\Theta\left(\sum_{i} \frac{p_{i}^{2}}{2m} + PV -\mu N - \mathsf{R}\right)\;
\prod_{i=1}^{N} {\rm d}^{D} r_{i}\, {\rm d}^{D} p_{i}.
\label{psv_re_ig}
\end{equation}
Integrating over the phase space co-ordinates namely $p_{i}$ and $r_{i}$ we arrive at
\begin{equation}
\varSigma(\mu,P,\mathsf{R}) = \sum_{N=0}^{\infty} \sum_{V} \frac{V^{N}}{N!} \; \frac{\mathcal{M}^{N}}{\Gamma\left(\frac{D N}{2}+1\right)} \;
(\mathsf{R} + \mu N - PV)^{\frac{DN}{2}}.
\label{psv_re_vnsum}
\end{equation}
The sum over volume eigenstates is calculated by approximating it to an integral. Since a direct integration
will lead to over counting of the volume states, we use the shell particle method of counting
which was developed in [\cite{DC1},\cite{DC2}]. Using this technique eliminates the redundant volume states
and the obtained expression for the phase space volume reads:
\begin{equation}
\varSigma(\mu,P,\mathsf{R}) = \sum_{N=0}^{\infty} \mathcal{M}^{N} \;
\left(\frac{1}{P}\right)^{N} \;
\frac{(\mathsf{R} + \mu N)^{\mathfrak{D}} }
{ \Gamma \left(\mathfrak{D} + 1 \right) }.
\label{psv_nsum_re}
\end{equation}
Finally we evaluate the summation over the number of particles in (\ref{psv_nsum_re}) using a large $N$ approximation.
The large $N$ limit of the $(\mathsf{R} + \mu N)^{\mathfrak{D}}$ factor is calculated as follows
\begin{equation}
(\mathsf{R} + \mu N)^{\mathfrak{D}} = (\mu N)^{\mathfrak{D}} \, \left(1 + \frac{\mathsf{R}}{\mu N}\right)^{\mathfrak{D}}
= (\mu N)^{\mathfrak{D}} \, \exp\left(\mathfrak{D}
\ln\left(1 + \frac{\mathsf{R}}{\mu N}\right)\right).
\label{Nbinom_approx}
\end{equation}
Using the approximation $\ln(1+x) \approx x$ for small values of $x$ we get
\begin{equation}
(\mathsf{R} + \mu N)^{\mathfrak{D}} = (\mu N)^{\mathfrak{D}} \, \exp\left(\frac{\mathcal{D} \, \mathsf{R}}{\mu}\right).
\label{Nbinom_approx_fe}
\end{equation}
Based on the Stirling's approximation the Gamma function in (\ref{psv_nsum_re}) is written as
\begin{equation}
\Gamma\left(\mathfrak{D} + 1\right) \approx \mathfrak{D}^{\mathfrak{D}} \, \exp\left(-\mathfrak{D}\right).
\label{gf_str_re}
\end{equation}
Substituting (\ref{Nbinom_approx_fe}) and (\ref{gf_str_re}), we can rewrite (\ref{psv_nsum_re}) as follows
\begin{equation}
\varSigma(\mu,P,\mathsf{R}) = \exp\left(\frac{\mathcal{D} \, \mathsf{R}}{\mu}\right) \, \sum_{N=0}^{\infty}\,
\left(\frac{\mathcal{M}}{P}\right)^{N} \left(\frac{\mu}{\mathcal{D}}\right)^{\mathfrak{D}}
\exp\left(\mathfrak{D}\right),
\label{psv_nsum_fe}
\end{equation}
where $\mathcal{D} = \mathfrak{D}/N$.
Carrying out the summation in (\ref{psv_nsum_fe}), we get the final expression of the phase space volume
in the large $N$ limit
\begin{equation}
\varSigma(\mu,P,\mathsf{R}) = \exp\left(\frac{\mathcal{D} \, \mathsf{R}}{\mu}\right)
\left[ 1 - \frac{\mathcal{M}}{P} \, \left(\frac{\mu}{\mathcal{D}}\right)^{\mathcal{D}} \,
\exp\left(\mathcal{D}\right) \right]^{-1}.
\label{psv_fe_LNL}
\end{equation}
The computed phase space volume (\ref{psv_fe_LNL}) can be considered as a first order approximation of
(\ref{psv_nsum_re}). The surface area of the phase space curve is
\begin{equation}
\Omega(\mu,P,\mathsf{R}) = \frac{\mathcal{D}}{\mu} \;
\exp\left(\frac{\mathcal{D} \, \mathsf{R}}{\mu}\right)
\left[1 - \frac{\mathcal{M}}{P} \left(\frac{\mu}{\mathcal{D}}\right)^{\mathcal{D}} \,
\exp\left(\mathcal{D}\right) \right]^{-1}.
\end{equation}
The two parameter entropy of the classical ideal gas in the $(\mu,P,\mathsf{R})$ ensemble can be immediately obtained from
the phase space volume (\ref{psv_fe_LNL}) and reads:
\begin{equation}
S_{q,q^{\prime}} = \frac{k}{1-q^{\prime}} \; \left[\exp \left(\frac{1-q^{\prime}}{1-q}
\left(\left(\Xi_{re} \; \exp\left(
\frac{\mathcal{D} \, \mathsf{R}}{\mu}\right)\right)^{1-q} -1\right)\right) -1\right],
\label{entr_re_ig}
\end{equation}
where the temperature independent factor $\Xi_{re}$ of (\ref{entr_re_ig}) is
\begin{equation}
\Xi_{re} = \left[1 - \frac{\mathcal{M}}{P} \, \left(\frac{\mu}{\mathcal{D}}\right)^{\mathcal{D}} \,
\exp\left(\mathcal{D}\right) \right]^{-1}.
\label{xi_re_def}
\end{equation}
The temperature of the $(\mu,P,\mathsf{R})$ ensemble is
\begin{equation}
\frac{1}{T}\equiv \frac{\partial S_{q,q^{\prime}}}{\partial \mathsf{R}}
= \frac{\mathcal{D}}{\mu} \, \Xi_{re}^{1-q} \,
\left(\exp\left(\frac{ \mathcal{D} \, \mathsf{R}}{\mu}\right)\right)^{1-q}
\exp \left(\frac{1-q^{\prime}}{1-q}
\left(\left(\Xi_{re} \; \exp\left( \frac{\mathcal{D} \, \mathsf{R}}{\mu}\right) \right)^{1-q} -1\right)\right).
\label{temp_re_def}
\end{equation}
Inverting the expression for the temperature in (\ref{temp_re_def}) helps us in computing the Ray energy,
the heat function corresponding to the $(\mu,P,\mathsf{R})$ ensemble. The expression for the Ray energy of the
classical ideal gas reads:
\begin{equation}
\mathsf{R} = \frac{\mu}{(1-q) \, \mathcal{D}} \; \ln \left[\frac{1-q}{1-q^{\prime}} \; \frac{1}{\Xi_{re}^{1-q}} \;
W_{0} \left(\frac{1-q^{\prime}}{1-q} \, \exp\left( \frac{1-q^{\prime}}{1-q}\right)
\frac{ \beta \mu}{\mathcal{D}} \right)\right].
\label{re_ig_fe}
\end{equation}
The continuity requirement of the Ray energy $\mathsf{R}$, and the concavity conditions on the entropy
restricts our choice of the $W$-function to the principal branch. In the region (c) for which
$q^{\prime} > 1$ and $-\infty < q < 1$ the Ray energy is uniformly positive whereas in the other
two regions {\it viz} (a) and (b) it is positive only when the argument of the logarithm is less than
$1$. From (\ref{re_ig_fe}) the specific heat at constant pressure is found:
\begin{equation}
C_{P} \equiv \frac{\partial R}{\partial T} = - \frac{\mu}{(1-q) \, \mathcal{D}} \, \frac{k\, \beta} {1 + W_{0} (\mathbf{b} \, \beta)},
\qquad
\mathbf{b} = \frac{1-q^{\prime}}{1-q} \, \exp\left(\frac{1-q^{\prime}}{1-q} \right) \, \frac{\mu}{\mathcal{D}}.
\label{spht_re}
\end{equation}
Analogous to the $(\mu,V,\mathsf{L})$ the specific heat is positive in the region $q < 1$ and negative in the
region $q > 1$ under the condition that $q + q^{\prime} > 1$ and excluding the region $0 \leq (q,q^{\prime}) \leq 1$.
Further the Boltzmann Gibbs results could not be recovered from (\ref{re_ig_fe}) and (\ref{spht_re}) since we have used
the large $N$ limit in the computation of the density of states and it does not commute with the extensive
$(q,q^{\prime}) \rightarrow 1$ limit.
\setcounter{equation}{0}
\section{Gas molecules under gravity}
\label{gravity}
In order to investigate the effect of an external field on a thermodynamic
system we study a system of gas molecules in the presence of a gravitational field in
the current section. A system of $D$-dimensional classical ideal gas under the effect
of gravity is explored in the first part of the section. To understand an interacting
system in an external field, we, in the later part of the section,
examine a system of hard rods confined in a linear box of length $L$ in the
presence of an external gravitational field.
\subsection{D-dimensional ideal gas under gravity}
A system of $N$ noninteracting classical point-like particles of mass $m$ is considered
in $D$-dimensions in the presence of a uniform gravitational field $g$. The acceleration
due to gravity acts along one of the dimensions labelled
as the $z$ co-ordinate and the force is oriented in a direction opposite to that of the
co-ordinate axis. To conform to reality we consider that the gas molecules are located
within a fixed region in space and so the position co-ordinates take values over a finite
interval. Of the $D$ position co-ordinates denoted by $r_{i} \; \; (i= 1,2,...,D)$,
the $z$ co-ordinate ranges over the interval $(0,\mathsf{H})$ and the remaining
$D-1$ co-ordinates range over the values $(0,L)$. The Hamiltonian of the $D$-dimensional
ideal gas under gravity is
\begin{equation}
\mathcal{H} = \sum_{i=1}^{DN} \frac{p_{i}^{2}}{2m} + \sum_{j = 1}^{N} mgz_{j} + U.
\label{ham_ug_D}
\end{equation}
The fact that the gas molecules are to exist within a finite region in space is indicated
in the Hamiltonian through a potential which mimics the presence of a wall
\begin{eqnarray}
U = \begin{cases}
0 & {\text{within the allowed region}}, \\
\infty & {\text{outside the allowed region}}.
\end{cases}
\label{pot_cond}
\end{eqnarray}
Substituting the expression for the Hamiltonian (\ref{ham_ug_D}) in the integral for
the phase space volume enclosed by a constant energy curve, we arrive at
\begin{equation}
\varSigma(N,V,E) = \frac{1}{N! \, h^{DN}} \; \int_{r_{i}} \int_{p_{i}}
\Theta\left(\sum_{i=1}^{DN} \frac{p_{i}^{2}}{2m}
+ \sum_{j=1}^{N} mgz_{j} + U - E\right)\;
\prod_{i=1}^{N} {\rm d}^{D} r_{i}\, {\rm d}^{D} p_{i}.
\label{psv_int_grg}
\end{equation}
Integrating over the phase space co-ordinates, the volume enclosed by the constant energy curve is
\begin{equation}
\varSigma(N,V,E) = \frac{V^{N}}{N! \, h^{DN}} \, \frac{\mathcal{M}^{N}}
{\Gamma\left(\mathfrak{D}+1\right) \, (mg\mathsf{H})^{N}} \
\mathfrak{S}_{D} \left(E,mg \mathsf{H}\right),
\label{psv_fe_grg}
\end{equation}
where $\mathfrak{S}(a,b)$ is a finite sum defined as below:
\begin{equation}
\mathfrak{S}_{D} \left(a,b \right) =
\sum_{k=0}^{N} (-1)^{k} \, \binom{N}{k} \, (a - kb)^{\mathfrak{D}} \,
\Theta(a - k b).
\label{sum_def}
\end{equation}
The surface area of the constant energy curve in the phase space is
\begin{equation}
\Omega(N,V,E) = \frac{V^{N}}{N! \, h^{DN}} \, \frac{\mathcal{M}^{N}}{\Gamma\left(\mathfrak{D}\right) \, (mg\mathsf{H})^{N}} \
\mathfrak{S}^{\prime}_{D} \left(E,mg \mathsf{H}\right),
\label{psa_fe_grg}
\end{equation}
where the prime over the summation (\ref{sum_def}) denotes a partial derivative with respect to the energy.
Substituting the phase space volume (\ref{psv_fe_grg}) in equation (\ref{entr_def_rel}) we obtain the entropy of
a $D$-dimensional ideal gas in the presence of gravity
\begin{equation}
S_{q,q^{\prime}} = \frac{k}{1-q^{\prime}} \; \left[\exp \left(\frac{1-q^{\prime}}{1-q}
\left(\left(\Xi_{g} \; \mathfrak{S}_{D} \left(E,mg \mathsf{H}\right)
\right)^{1-q} -1\right)\right) -1\right],
\label{entr_grg_fe}
\end{equation}
where, the factor $\Xi_{g}$ is defined as
\begin{equation}
\Xi_{g} = \frac{A^{N}}{N! \, h^{DN}} \, \frac{\mathcal{M}^{N}}{\Gamma\left(\mathfrak{D}+1\right) \, (mg)^{N}},
\qquad
A = \prod_{\alpha=1}^{D-1} L_{\alpha}.
\label{xi_def}
\end{equation}
Using the entropic expression (\ref{entr_grg_fe}), and from the
definition of temperature (\ref{temp_def_TS}) we get
\begin{eqnarray}
\frac{1}{T} = k \; \Xi_{g}^{1-q} \,
\frac{\mathfrak{S}^{\prime}_{D} \left(E,mg \mathsf{H}\right)}
{\left(\mathfrak{S}_{D} \left(E,mg \mathsf{H}\right)\right)^{q}} \;
\exp \left(\frac{1-q^{\prime}}{1-q}
\left(\left(\Xi_{g} \; \mathfrak{S}_{D} \left(E,mg \mathsf{H}\right) \right)^{1-q} -1\right)\right).
\label{temp_fe_grg}
\end{eqnarray}
Looking at the structure of the relation (\ref{temp_fe_grg}) given above, we realize that an inversion
to obtain the internal energy as a function of temperature is not feasible. But the specific heat of
the system can also be computed from the knowledge of the phase space volume (\ref{psv_fe_grg}) and the
surface area (\ref{psa_fe_grg}) and the temperature (\ref{temp_fe_grg}) via the relation:
\begin{equation}
C_{V} \equiv \left( \frac{\partial T}{\partial E} \right)^{-1}
= \left[ T \, \left(\frac{q \ \Omega}{\varSigma} - \frac{1}{\Omega} \; \frac{\partial \Omega}{\partial E}
- (1-q^{\prime}) \frac{\Omega}{\varSigma^{q}}\right) \right]^{-1}.
\label{spht_psv_psa_temp}
\end{equation}
Substituting the phase space volume (\ref{psv_fe_grg}), area of the curve (\ref{psa_fe_grg}) and
the expression for the temperature (\ref{temp_fe_grg}) in (\ref{spht_psv_psa_temp}) the specific heat
as a function of energy reads:
\begin{eqnarray}
C_{V} &=& \mathfrak{D} \, k\, \exp \left(\frac{1-q^{\prime}}{1-q}
\left(\left(\Xi_{g} \; \mathfrak{S}_{D} \left(E,mg \mathsf{H}\right) \right)^{1-q} -1\right)\right) \,
\bigg[\mathfrak{D} \, q \; (\Xi_{g} \, \mathfrak{S}_{D} \left(E,mg \mathsf{H}\right))^{q-1} \nonumber \\
& & - \Xi_{g}^{q-1} \, \frac{(\mathfrak{S}_{D} \left(E,mg \mathsf{H}\right))^{q}}
{\mathfrak{S}_{D}^{\prime} \left(E,mg \mathsf{H}\right)} \,
\frac{\partial}{\partial E} \ln \mathfrak{S}_{D}^{\prime} \left(E,mg \mathsf{H}\right)
- (1-q^{\prime}) \, \mathfrak{D} \bigg]^{-1}.
\label{spht_ie_fe}
\end{eqnarray}
From (\ref{spht_ie_fe}) it can be noticed that the specific heat can have both positive and negative values
depending on the term in square bracket.
We investigate the relevant limiting cases: {\it(i)} First is the $g \rightarrow 0$ limit in which
the gas molecules behave like an ideal gas. {\it(ii)} The second case is the $\mathsf{H} \rightarrow \infty$
limit in which eqn. (\ref{temp_fe_grg}) can be inverted in the large $N$ limit to obtain the energy as a
function of temperature. We present the relevant calculation of this limiting case in the discussion below:
In the infinite height limit the limiting value of the summation (\ref{sum_def}) and its derivative are
\begin{equation}
\lim_{\mathsf{H} \rightarrow \infty}
\; \mathfrak{S}_{D} \left(E,mg \mathsf{H}\right) = E^{\mathfrak{D}},
\qquad
\lim_{\mathsf{H} \rightarrow \infty}
\; \mathfrak{S}_{D}^{\prime} \left(E,mg \mathsf{H}\right) = \mathfrak{D} \, E^{\mathfrak{D}-1}.
\label{sum_lt_ht}
\end{equation}
Substitution of the limiting value (\ref{sum_lt_ht}) in the expression for the temperature
(\ref{temp_fe_grg}) leads to:
\begin{equation}
\frac{1}{k \, T} = \mathfrak{D} \, k \; \Xi_{g}^{1-q} \, E^{(1-q)\, \mathfrak{D} - 1} \,
\exp \left(\frac{1-q^{\prime}}{1-q}
\big((\Xi_{g} \; E^{\mathfrak{D}})^{1-q} -1\big)\right).
\label{temp_rel_ig}
\end{equation}
In the large $N$ limit using the approximation
$E^{(1-q) \,\mathfrak{D} - 1} \approx E^{(1-q) \, \mathfrak{D}}$
in (\ref{temp_rel_ig}), we can invert it to obtain the
internal energy as a function of temperature in terms of the Lambert's W-function
\begin{equation}
E = \left[\frac{1-q}{1-q^{\prime}} \; \frac{1}{\Xi_{g}^{1-q}} \;
W_{0} \left(\frac{1-q^{\prime}}{1-q} \, \exp\left( \frac{1-q^{\prime}}{1-q}\right)
\frac{\beta}{\mathfrak{D}} \right)\right]^
{\frac{1}{(1-q)\, \mathfrak{D}}}.
\label{IE_rel_ig}
\end{equation}
The preconditions that the entropy should be concave, and the energy should be a continuous function
helps us to conclude that only the principal branch of the $W$ function occurs in (\ref{IE_rel_ig}).
The specific heat of the classical ideal gas under gravity computed from the
internal energy is
\begin{equation}
C_{V} = - \frac{k \, \beta}{(1-q) \, \mathfrak{D}} \;
\frac{W_{0} (\mathfrak{b} \, \beta)}{1 + W_{0} (\mathfrak{b} \, \beta)} \;
\left[\mathsf{a}_{g} \, W_{0} (\mathfrak{b} \, \beta)\right]^{\frac{1-(1-q)\mathfrak{D}}{(1-q)\mathfrak{D}}},
\label{spht_fe_ig}
\end{equation}
where the factor $\mathfrak{b}$ was introduced in (\ref{mfab_def}) and $\mathsf{a}_{g}$ is
\begin{equation}
{\mathsf{a}}_{g} = \frac{1-q}{1-q^{\prime}} \, \frac{1}{\Xi_{g}^{1-q}}.
\label{a_def}
\end{equation}
The large $N$ limit of the specific heat for the ideal gas under gravity reads:
\begin{equation}
C_{V} = - \frac{k \beta}{(1-q) \mathfrak{D}} \; \frac{1}{{\mathsf{a}}_{g} (1 + W_{0} (\mathfrak{b} \beta))}.
\label{spht_lgN_ig}
\end{equation}
Investigating the expression for the specific heat, we find that it can be either positive in the
regions (a) and (c) or negative in the region (b). Since we have already made use of the large $N$
limit we do not recover the Boltzmann Gibbs results because the thermodynamic limit and the
extensive limit do not commute with each other.
The one parameter limits corresponding to the temperature relation (\ref{temp_fe_grg})
can be obtained by allowing either $q$ or $q^{\prime}$ to take the limiting value. We notice
that the expression for the temperature is the same when we set either of the parameter
to unity and reads:
\begin{equation}
\frac{1}{T} = k \, \Xi_{g}^{1-q} \,
\frac{\mathfrak{S}_{D} \left(E,mg \mathsf{H}\right)}
{\mathfrak{S}_{D}^{\prime} \left(E,mg \mathsf{H}\right)}.
\label{tmp_rel_1p}
\end{equation}
The definition of temperature (\ref{tmp_rel_1p}) can be inverted to obtain the internal energy
and the specific heat in the $\mathsf{H} \rightarrow \infty$ limit, but there is no need to invoke
the large $N$ limit.
\subsection{1D hard rod gas under gravity}
The generalized statistical mechanics based on the two parameter $(q,q^{\prime})$ entropy
is applied to a system comprising of $N$ one dimensional hard rods under gravity.
Initially we consider a system of hard rods of mass $m$ and length $\sigma$ in a finite
region of the space under the influence of a uniform gravitational field of strength $g$.
The position of the centers and the momentum of the hard rods are denoted by
the set of values $(z_{1},...,z_{N};p_{1},...,p_{N})$. Two rods with centers given by
$z_{0}$ and $z_{N+1}$ are considered so that their edges define the boundaries of the region, in
such way that $z_{N+1} - z_{0} = L + \sigma$.
The relevant thermodynamic quantities like the entropy, the temperature, and
the heat capacity are then computed. Later we assume the $L \rightarrow \infty$ limit of the
system and obtain the internal energy in this limiting case. The Hamiltonian of a one
dimensional hard rod gas under gravity reads:
\begin{equation}
\mathcal{H} = \sum_{i=1}^{N} \, \frac{p_{i}^{2}}{2m} + \sum_{i=1}^{N} mgz_{i}
+ \sum_{i<j} u(|z_{j}-z_{i}|) + \sum_{i=1}^{N} U(z_{i};z_{0},z_{N+1}).
\label{ham_hrdg}
\end{equation}
The distance between the centers of any two particles in the system cannot be less than
the length of the rod $\sigma$ and the potential corresponding to it is
\begin{eqnarray}
u(|z_{j}-z_{i}|) = \begin{cases}
\infty, & |z_{j}-z_{i}| < \sigma, \\
0, & |z_{j}-z_{i}| \geq \sigma.
\end{cases}
\label{pot_int_cond}
\end{eqnarray}
Similarly the existence of the gas molecules within the finite region of the space is
indicated in the Hamiltonian (\ref{ham_hrdg}) via the potential
\begin{eqnarray}
U = \begin{cases}
0, & z_{0} + \sigma \leq z_{i} \leq z_{N+1} - \sigma, \\
\infty, & {\text{otherwise}}.
\end{cases}
\label{pot_hrdg_cond}
\end{eqnarray}
Using the Hamiltonian (\ref{ham_hrdg}) in (\ref{vds_en_TS}), the integral expression corresponding to
the phase space volume is
\begin{equation}
\varSigma(N,L,E) = \frac{1}{N! \, h^{N}} \; \int_{z_{i}} \int_{p_{i}}
\Theta\left(\sum_{i=1}^{N} \frac{p_{i}^{2}}{2m}
+ \sum_{j=1}^{N} mgz_{j} + U + u(|z_{j}-z_{i}|)- E\right)\;
\prod_{i=1}^{N} {\rm d} z_{i}\, {\rm d} p_{i}.
\label{psv_int_ghrg}
\end{equation}
To evaluate the phase space volume we first integrate over the momentum variables. In the next
step we use the substitution $r_{k} = z_{k} - (z_{0} + k \sigma)$ for the position
integrals which helps us in integrating over the position variables. The final expression for the
phase space volume thus computed is
\begin{equation}
\varSigma(N,L,E) = \frac{\mathcal{M}_{1}^{N}}{N! \; \Gamma\left(\frac{3 N}{2}+1\right)\; (mg)^{N}} \;
\mathfrak{S}_{1}(E - mg \alpha,mg \bar{L}).
\label{psv_hrdg_fe}
\end{equation}
where the $\mathcal{M}_{1}$ is obtained by setting $D=1$ in the expression for $\mathcal{M}$. For convenience
we have also defined the following quantities
\begin{equation}
\alpha = N z_{0} + \frac{N(N+1)}{2} \, \sigma, \qquad
\bar{L} = L - N \sigma.
\end{equation}
The surface area of the constant energy curve is
\begin{equation}
\Omega(N,L,E) = \frac{\mathcal{M}_{1}^{N}}{N! \; \Gamma\left(\frac{3 N}{2}\right) \; (mg)^{N}} \;
\mathfrak{S}_{1}^{\prime}(E - mg \alpha,mg \bar{L}).
\label{psa_hrdg_fe}
\end{equation}
The knowledge of the phase space volume (\ref{psv_hrdg_fe}) enables us to obtain the entropy of the
system defined through the relation (\ref{entr_def_rel}). Combining this with the definition of temperature
(\ref{temp_def_TS}), we arrive at
\begin{equation}
\frac{1}{T} = k\, \Xi_{gh}^{1-q} \, \frac{\mathfrak{S}_{1}^{\prime}(E - mg \alpha,mg \bar{L})}
{(\mathfrak{S}_{1}(E - mg \alpha,mg \bar{L}))^{q}}
\exp\left(\frac{1-q^{\prime}}{1-q}\left((\Xi_{gh} \,
\mathfrak{S}_{1}(E - mg \alpha,mg \bar{L}))^{1-q} - 1 \right)\right),
\label{temp_hrdg_rel}
\end{equation}
where the factor $\Xi_{gh}$ is defined as
\begin{equation}
\Xi_{gh} = \frac{\mathcal{M}_{1}^{N}}{N! \; \Gamma\left(\frac{3 N}{2}+1\right)\; (mg)^{N}}.
\end{equation}
The form of (\ref{temp_hrdg_rel}) makes us realize that an exact inversion to obtain the internal energy is
not feasible. To overcome this we use (\ref{spht_psv_psa_temp}) in conjunction with (\ref{psv_hrdg_fe}),
(\ref{psa_hrdg_fe}) and (\ref{temp_hrdg_rel}) to compute the specific heat as a function of the internal energy
and the free length $(\bar{L})$. The specific heat thus evaluated is
\begin{eqnarray}
C_{V} &=& \frac{3N}{2} k\, \exp \left(\frac{1-q^{\prime}}{1-q}
\left(\left(\Xi_{gh} \; \mathfrak{S}_{1} \left(E - mg \alpha,mg \bar{L}\right) \right)^{1-q} -1\right)\right) \,
\label{spht_ie_hdg} \\
& & \bigg[\frac{3N}{2} \, q \; (\Xi_{gh} \, \mathfrak{S}_{1} \left(E - mg \alpha,mg \bar{L}\right))^{q-1} \nonumber \\
& & - \Xi_{gh}^{q-1} \, \frac{(\mathfrak{S}_{1} \left(E - mg \alpha,mg \bar{L}\right))^{q}}
{\mathfrak{S}_{1}^{\prime} \left(E - mg \alpha,mg \bar{L}\right)} \,
\frac{\partial}{\partial E} \ln \mathfrak{S}_{1}^{\prime} \left(E - mg \alpha,mg \bar{L}\right)
- (1-q^{\prime}) \, \frac{3N}{2} \bigg]^{-1}. \nonumber
\end{eqnarray}
Analyzing (\ref{spht_ie_hdg}) we realize that both positive and negative values of specific heat are permissible
depending on the argument within the square bracket. There are two pertinent limiting cases namely {\it (i)}
the $g \rightarrow 0$ limit wherein it becomes a system of hard rods moving in a length $L$, and {\it (ii)} the
$L \rightarrow \infty$ limit. In the second limit we present the expression for temperature below:
\begin{equation}
\frac{1}{T} = \frac{3N}{2} k \, \Xi_{gh}^{1-q} \, (E - mg\alpha)^{(1-q)\frac{3N}{2} - 1} \,
\exp\left(\frac{1-q^{\prime}}{1-q}\left(\left(\Xi_{gh} \; (E - mg\alpha)^{\frac{3N}{2}}\right)^{1-q} - 1 \right) \right),
\label{temp_hrdg_lNlt}
\end{equation}
which can be inverted in the large $N$ limit by assuming $(3N/2)-1 \approx (3N/2)$. This approximation
yields the internal energy in terms of the Lambert's $W$-function
\begin{equation}
E = \left[{\mathsf{a}}_{gh} \;
W_{0} \left(\frac{1-q^{\prime}}{1-q} \, \exp\left( \frac{1-q^{\prime}}{1-q}\right)
\frac{2 \beta}{3N} \right)\right]^
{\frac{2}{(1-q)\, 3N}} + mg \alpha,
\qquad
{\mathsf{a}}_{gh} = \frac{1-q}{1-q^{\prime}} \; \frac{1}{\Xi_{gh}^{1-q}}.
\label{ie_hrdg_rel}
\end{equation}
The stipulations that the entropy should be concave and the energy should be a continuous function limits
us to the principal branch of the $W$-function. The specific heat of the hard rod gas in the large $N$ limit
can be obtained from (\ref{ie_hrdg_rel}) and reads:
\begin{equation}
C_{V} = - \frac{2 k \beta}{(1-q) 3N} \; \frac{1}{{\mathsf{a}}_{gh} (1 + W_{0} (\mathfrak{b}_{h} \beta))},
\qquad
{\mathfrak{b}}_{h} = \frac{1-q^{\prime}}{1-q} \, \exp\left( \frac{1-q^{\prime}}{1-q}\right) \ \frac{2}{3N}.
\label{spht_lgN_hrdg}
\end{equation}
This has the same regimes of positive and negative specific heat values as the classical ideal gas.
\begin{figure}[!ht]
\begin{center}
\resizebox{75mm}{!}{\includegraphics{5.eps}}
\resizebox{75mm}{!}{\includegraphics{7.eps}}
\resizebox{75mm}{!}{\includegraphics{1.eps}}
\resizebox{75mm}{!}{\includegraphics{2.eps}}
\end{center}
\caption{In the following graphs we plot the specific heat (\ref{spht_lgN_hrdg}).
In the first set of graphs above we plot the variation of the specific heat with respect to the
temperature at a fixed value of the acceleration due to gravity. The second set of graphs consists
of the variation of the specific heat with respect to the acceleration due to gravity at a fixed
temperature. The values of $q$ and $q^{\prime}$ made use of are given in the graphs.}
\end{figure}
The variation of the specific heat with respect to the temperature and the acceleration due to gravity have been
plotted in the set of graphs above. Maintaining $g$ at a constant value, we first investigate the variation of
the specific heat with the temperature. Depending on the value of $q$ and $q^{\prime}$ the specific heat can take
positive and negative values and show a saturating behaviour at high temperatures. Similarly, we study the
variation of specific heat with respect to the acceleration due to gravity $g$. The specific heat increases
(decreases) exponentially when $q$ is greater (lesser) than one. Previous studies in nonextensive statistical
mechanics have pointed out that nonextensivity can induce effective interactions [\cite{CCN1},\cite{AL1}] and can also
describe systems in which collisions generate non-Maxwellian distribution functions [\cite{RGD09}]. A comparison of the
results obtained in our work, with systems in which a cloud of interacting gas molecules is influenced by
gravity will help us to appreciate the use of nonextensive statistical mechanics in studying interacting systems.
The one parameter limits of the temperature (\ref{temp_hrdg_rel}) is investigated. It can be noticed that
whether we set $q$ or $q^{\prime}$ to unity is not relevant and we always arrive at
\begin{equation}
\frac{1}{T} = k\, \Xi_{gh}^{1-q} \, \frac{\mathfrak{S}_{1}^{\prime}(E - mg \alpha,mg \bar{L})}
{(\mathfrak{S}_{1}(E - mg \alpha,mg \bar{L}))^{q}},
\label{temp_1p_rel}
\end{equation}
the temperature relation corresponding to the Tsallis $q$-entropy. Though the internal energy and the
specific heat can be obtained from (\ref{temp_1p_rel}) in the $L \rightarrow \infty$ limit, there is no need
to use the large $N$ limit unlike the two parameter case.
\setcounter{equation}{0}
\section{Conclusions}
In the current work we investigate the adiabatic class of ensembles in the framework of generalized statistical mechanics
based on Schw\"{a}mmle-Tsallis $(q,q^{\prime})$ entropy. We do not study the isothermal class of ensembles since the
canonical treatment carried out in [\cite{As2008}] can be extended to the other members of the class which includes
the isothermal-isobaric, the grandcanonical and the generalized ensembles. We provide a unified description of the
adiabatic class of ensembles which includes the microcanonical $(N,V,E)$, isoenthalpic-isobaric $(N,P,H)$, the
$(\mu,V,\mathsf{L})$ ensemble, and the $(\mu,P,\mathsf{R})$ ensemble. A generalized form of the equipartition theorem,
the virial theorem, and the adiabatic theorem are obtained. We investigate the nonrelativistic classical ideal gas in
all the four ensembles. In the microcanonical, and the isoenthalpic-isobaric ensemble, the entropy could be found for
an arbitrary number of particles. Using the large $N$-limit, the heat functions are obtained in terms of the temperature
and expressed in terms of the Lambert's $W$-function. From the heat functions, the respective specific heats are evaluated.
To the best of our knowledge an exact evaluation of the phase space volume in the $(\mu,V,\mathsf{L})$ and the
$(\mu,P,\mathsf{R})$ ensembles has not been done so far. We, in our current work assume a large $N$ limit and compute
the approximate phase space volume. From the phase space volume the entropy, the heat function and the specific heat of
the classical ideal gas is found. The heat function and the specific heat are obtained in terms of the Lambert's
$W$-function without any further approximation. The preconditions that the entropy should be concave and the heat function should be a
continuous function of the deformation parameter restricts our choice of the $W$-function to the principal branch. The
two parameter entropy is concave only in the region where $q+q^{\prime} > 1$ excluding the zone $0 < (q,q^{\prime}) < 1$
where the entropy is not always concave and so we analyze the specific heats only in the above mentioned region.
For the microcanonical and the isoenthalpic-isobaric ensemble the specific heat is positive in the region for which
both $q$ and $q^{\prime}$ are greater than one and in the region where $q^{\prime} >1$ and $-\infty < q < 1$. But it is
negative in the region $q > 1$ and $-\infty < q^{\prime} < 1$. In the regime where the entropy is concave, the heat capacities
of the $(\mu,V,\mathsf{L})$ and the $(\mu,P,\mathsf{R})$ ensembles are positive when $q > 1$ and negative if $q < 1$.
The microcanonical specific heat of classical ideal gas and a system of hard rod gas confined in a finite region of
space and subjected to gravity is also analyzed. The entropy and the specific heat are initially calculated as a
function of internal energy exactly. But when we assume the height to be infinite and the number of particles to be
very large, the internal energy could be obtained as a function of temperature in terms of the Lambert's $W$-function.
Again only the principal branch of the $W$-function contributes to the internal energy and the specific heat.
In the case of the hard rod gas, the free length $\bar{L}$ and the factor $E-mg \alpha$ plays a role analogous to the
height and the energy in the classical ideal gas. It has been proved in [\cite{AL1}] that the
study of free ideal gas in deformed statistical mechanics can describe an interacting gas in the ordinary statistical
mechanics, and the deformation parameter provides information about interactions. Also we notice from our current study
that the two parameter entropy allows regions of both positive and negative specific heat unlike the case of the
Tsallis $q$-entropy in which the large $N$-limit permits only regions of negative heat capacity [\cite{SA99}].
Thus the generalized statistical mechanics based on the two parameter entropy shows more rich features and accommodates
much variations due to the presence of one more deformation parameter. A further application of this generalized
statistical mechanics to study self gravitating systems which exhibit negative specific heat will be worth pursuing.
Application of the large $N$ limit developed in the current article to study the adiabatic ensembles of other entropies
like the $\kappa$-entropy [\cite{GK2001}] and the two parameter Sharma-Mittal-Taneja entropy [\cite{ST75},\cite{M75}]
may help us in understanding the similarities and differences in the thermostatistical structure of different entropies.
Such an understanding may help us to choose the generalized entropy which describes a given system in an appropriate way.
The construction of a Laplace transform based on the two parameter exponential may be of great help in establishing
the connection between the isothermal and the adiabatic class of ensembles. The repeated occurrence of the Lambert's
$W$-function in the field of generalized statistical mechanics [\cite{As2008},\cite{FS04},\cite{FS09},\cite{TO07}]
points to an important connection between them which is worthy of further investigation.
\section*{Acknowledgements}
The authors would like to thank Professor Ranabir Chakrabarti for discussions.
|
1,941,325,220,491 | arxiv | \section{Introduction}
Instance segmentation is one of the fundamental tasks in computer vision.
While significant progress has been witnessed in instance segmentation of images
\cite{he2017mask,tian2020conditional,wang2020solo,SoloV22020,chen2020blendmask,wang2020centermask},
much less effort was spent on segmenting instances in videos.
Here we propose a new video instance segmentation framework built upon Transformers.
Video instance segmentation (VIS), recently proposed in \cite{vis2019},
requires one to simultaneously classify,
segment and track object instances of interest in a video sequence.
It is more challenging in that one needs
to perform instance segmentation for each individual frame and at the same time to establish data association of instances across consecutive frames, \textit{a.k.a.}, tracking.
State-of-the-art methods typically develop sophisticated pipelines to tackle this task.
Top-down approaches~\cite{vis2019,bertasius2020classifying} follow the tracking-by-detection paradigm, relying heavily on image-level instance segmentation models \cite{he2017mask,chen2019hybrid} and complex human-designed rules to associate the instances.
Bottom-up approaches \cite{Athar_Mahadevan20ECCV} separate object instances by clustering learned pixel embeddings.
Due to heavy reliance on the dense prediction quality, these methods often need multiple steps to generate the masks iteratively, which makes them slow.
Thus, a simple, end-to-end trainable VIS framework is highly desirable.
\iffalse
\begin{figure*}[!t]
\centering
\includegraphics[width=.64\linewidth]{figs/1.pdf}
\caption{\textbf{Overall pipeline of {VisTR}\xspace.}
The model takes a sequence of images as input and outputs a sequence of
instance predictions.
Here same shapes represent predictions in one image, and same colors represent predictions of the same object instance. Note that the overall predictions follow the input frame order, and the order of object predictions for different images keeps the same (Best viewed on screen).
}
\label{fig:1}
\end{figure*}
\fi
Here, we take a deeper look at the video instance segmentation task.
%
%
Video frames contain richer information than single images such as motion patterns and temporal consistency of instances, offering useful cues for instance segmentation, and classification.
At the same time, the better learned instance features can help tracking of instances. In essence, the instance segmentation and instance tracking are both concerned with similarity learning:
instance segmentation is to learn the pixel-level similarity and instance tracking is to learn the similarity between instances.
Thus, it is natural to solve these two sub-tasks in a single framework and benefit each other.
Here we aim to develop such an end-to-end VIS framework. The framework needs to be simple and achieves strong performance without whistles and bells.
To this end,
we propose to %
employ
the Transformers \cite{vaswani2017attention}.
Importantly, for the first time we demonstrate that, as the Transformers provide
building blocks, \textit{it enables one to design a simple and clean framework for VIS, and possibly for a much wider range of video processing tasks in computer vision}. Thus potentially, it is possible to unify most vision tasks of different input modalities---such as image, video and point clouds processing---into the Transformer framework.
Transformers are widely used for sequence to sequence learning in NLP \cite{vaswani2017attention}, and start to show promises in vision \cite{Detr,dosovitskiy2020}.
Transformers are capable of modeling long-range dependencies,
and thus can be naturally applied to video for learning temporal information across multiple frames.
In particular, the core mechanism of Transformers, self-attention, is designed to learn and update the features based on all pairwise similarities between them.
The above characteristics of Transformers make them great candidates for the VIS task.
In this paper, we propose the Video Instance Segmentation TRansformer ({VisTR}\xspace), which views the VIS task as a parallel sequence decoding/prediction problem.
Given a video clip that consists of multiple image frames as input, the {VisTR}\xspace outputs the sequence of masks for each instance in the video in order directly. The output sequence for each instance is referred to as \textit{instance sequence} in this paper.
The overall {VisTR}\xspace pipeline is illustrated in \figref{fig:1}. In the first stage, given a sequence of video frames,
a standard CNN module extracts features of individual image frames, then the multiple image features are concatenated in the frame order to form the clip-level feature sequence. In the second stage, the Transformer takes the clip-level feature sequence as input, and outputs a sequence of object predictions in order. In
\figref{fig:1}
same shapes represent predictions for the same image, and the same colors
represent the same instance of different images. The sequence of predictions follow the order of input images, and the predictions of each image follows the same instance order.
Thus, instance tracking is achieved \textit{seamlessly and naturally} in the same framework of instance segmentation.
To achieve this goal, there are two main challenges: 1) how to maintain the order of outputs and 2) how to obtain the mask sequence for each instance out of the Transformer network. Correspondingly, we introduce the \textit{instance sequence matching} strategy and the \textit{instance sequence segmentation} module. The instance sequence matching performs bipartite graph
matching between the output instance sequence and the ground-truth instance sequence, and supervises the sequence as a whole.
Thus, the order can be maintained directly. The instance sequence segmentation accumulates the mask features for each instance across multiple frames through self-attention and segments the mask sequence for each instance through 3D convolutions.
Our main contributions are summarized as follows.
\begin{itemize}
\itemsep -0.17cm
\item
We propose a new video instance segmentation framework built upon Transformers,
termed
\textbf{{VisTR}\xspace},
%
which
views the VIS task as a \textit{direct end-to-end}
%
%
parallel
sequence decoding/prediction problem. The framework is significantly different from existing approaches, considerably simplifying the overall pipeline.
\item
{VisTR}\xspace solves the %
VIS
from a new perspective of similarity learning. Instance segmentation is to learn the pixel-level similarity and instance tracking is to learn the similarity between instances. %
Thus,
instance tracking is achieved \textit{seamlessly and naturally} in the same framework of instance segmentation.
\item
%
The key to the success of {VisTR}\xspace is
a new strategy for \textit{instance sequence matching and segmentation},
%
which is tailored for our framework.
%
This carefully-designed strategy enables us to
supervise and segment instances at the sequence level as a whole.
\item
{VisTR}\xspace achieves strong
results on the YouTube-VIS dataset, achieving 40.1\% in mask mAP at the speed of 57.7 FPS
, which is the best and fastest among methods that use a single model.
%
%
%
%
%
%
\end{itemize}
\section{Related work}
\myparagraph{Video object segmentation.} VOS \cite{DAVIS16} is closely related to VIS.
Analogue to object tracking, which is detecting boxes of foreground objects
in a class-agnostic fashion, VOS is segmenting masks of foreground
class-agnostic objects.
Same as in tracking, usually one is allowed to use only the first few frames' annotations for training. In contrast, VIS requires to segment and track all instance masks of a fixed category set of objects in a video sequence.
\myparagraph{Video instance segmentation.}
The VIS task \cite{vis2019} requires classifying, segmenting instances in each frame and linking the same instance across frames. State-of-the-art methods typically develop sophisticated pipelines to tackle it. MaskTrack R-CNN\cite{vis2019} extends the Mask R-CNN \cite{he2017mask} with a tracking branch and external memory that saves the features of instances across multiple frames. Maskprop \cite{bertasius2020classifying} builds on the Hybrid Task Cascade Network \cite{chen2019hybrid}, and re-uses the predicted masks to crop the extracted features, then propagates them temporally to improve the segmentation and tracking. STEm-Seg \cite{Athar_Mahadevan20ECCV} proposes to model video clips as 3D space-time volumes and then separates object instances by clustering learned embeddings.
Note that the above approaches either rely on complex heuristic rules to associate the instances or require multiple steps to generate and optimize the masks iteratively.
In contrast, here
we aim to build a \textit{simple and end-to-end trainable} VIS framework.
\myparagraph{Transformers.}
Transformers were first proposed in \cite{vaswani2017attention} for the sequence-to-sequence machine translation task, and since then have become the \textit{de facto}
method in most NLP tasks.
The core mechanism of Transformers, self-attention, makes it particularly suitable for modeling long-range dependencies.
Very recently,
Transformers start to show promises in solving computer vision tasks. DETR \cite{Detr} builds an object detection systems based on Transformers, which largely simplifies the traditional detection pipeline, and achieves \textit{on par}
performances compared with highly-optimized CNN based detectors\cite{ren2015faster}. Our work here is inspired by DETR.
ViT \cite{dosovitskiy2020} introduces the Transformer to image recognition and models an image as a sequence of patches, which attains excellent results compared to state-of-the-art convolutional networks. The above works show the effectiveness of Transformers in image understanding tasks.
To our knowledge, thus far there are
no prior applications of Transformers to video instance segmentation.
It is intuitive to see that the Transformers' advantage of modeling long-range dependencies makes it an ideal candidate for learning temporal information across multiple frames for video understanding tasks. Here,
we propose the {VisTR}\xspace method and provide an affirmative answer to that.
As the original Transformers are auto-regressive models, which generate output tokens one by one,
for efficiency, {VisTR}\xspace employs a non-auto-regressive variant of the Transformer to achieve parallel sequence generation.
\section{Our Method: {VisTR}\xspace}
\begin{figure*}[h]
\centering
\includegraphics[width=.95\linewidth]{figs/2.pdf}
\caption{\textbf{The overall architecture of {VisTR}\xspace.} It contains four main components: 1)
a CNN backbone that extracts feature representation of multiple images;
2) an encoder-decoder Transformer that models the relations of pixel-level features and decodes the instance-level features; 3) an instance sequence matching module
that supervises the model; and 4) an instance sequence segmentation module that outputs the final mask sequences (Best viewed on screen).
}
\label{fig:2}
\end{figure*}
We tackle the video instance segmentation task by modeling it as a direct sequence prediction problem.
Given a video clip that consists of multiple image frames as input, the {VisTR}\xspace outputs the sequence of masks for each instance in the video in order.
To achieve this goal, we introduce the instance sequence matching and segmentation strategy to supervise and segment the instances at the sequence level as a whole.
In this section, we first introduce the overall architecture of the proposed {VisTR}\xspace in \secref{sec:3.1}, then the details of the instance sequence matching and segmentation module will be described in \secref{sec:3.2} and \secref{sec:3.3} respectively.
\subsection{{VisTR}\xspace Architecture}\label{sec:3.1}
The overall {VisTR}\xspace architecture is depicted in \figref{fig:2}. It contains four main components: a CNN backbone to extract compact feature
representations
of multiple frames,
an encoder-decoder Transformer to model the similarity of pixel-level and instance-level features, an instance sequence matching module for supervising the model, and an instance sequence segmentation module.
\myparagraph
{Backbone.}
The backbone %
extracts the original pixel-level feature sequence of the input video clip.
Assume that the initial video clip with $T$ frames of resolution $H_{0}\times W_{0}$ is denoted by $x_{clip}\in\mathbb{R}^{T\times 3 \times H_{0}\times W_{0}}$.
First, a standard CNN backbone generates a lower-resolution activation map for each frame, then the features for each frame are concatenated to form the clip level feature map $f_{0}\in\mathbb{R}^{T\times C \times H\times W}$.
\myparagraph
{Transformer encoder.}
The Transformer encoder is
employed
to model the similarities among all the pixel level features in the clip.
First, a 1$\times$1 convolution is applied to the above feature map, reducing the dimension from $C$ to %
$d$ ($ d < C$), %
resulting in
a new feature map $f_{1}\in\mathbb{R}^{T\times d \times H\times W}$.
To form a clip level feature sequence that can be fed into the Transformer encoder, we flatten the spatial and temporal dimensions of $f_{1}$ into one dimension, resulting in a 2D feature map of size $d\times ( T \cdot H \cdot W )$.
Note that the temporal order is always in accordance with that of the initial input. Each encoder layer has a standard architecture that consists of a multi-head self-attention module and a fully connected feed forward network (FFN).
\myparagraph
{Temporal and spatial positional encoding.} The Transformer architecture is permutation-invariant, while the segmentation task requires precise position information. To compensate for this, we supplement the features with fixed positional encodings information that contains the three dimensional (temporal, horizontal and vertical) positional information in the clip.
Here we adapt the positional encoding in the original Transformer \cite{vaswani2017attention} for our 3D case.
Specifically, for the coordinates of each dimension we independently use $\nicefrac{d}{3}$ sine and cosine functions with different frequencies:
\def{\rm PE}{{\rm PE}}
\def{\rm pos}{{\rm pos}}
\begin{equation}
{\rm PE}({\rm pos},i)=
\begin{cases}
\sin \bigl( {\rm pos} \cdot \omega_{k} \bigr), ~~~~\text{for } i=2k,
\\
\cos \bigl( {\rm pos} \cdot \omega_{k} \bigr), ~~~~\text{for } i=2k+1;
\end{cases}
\label{eq:pe}
\end{equation}
where $ \omega_{k}= 1/ 10000^{2k/ \frac{d}{3}}$;
`${\rm pos}$' is the position in the corresponding dimension. Note that the $d$ should be divisible by 3, as the positional encodings of the three dimensions should be concatenated to form the final $d$ channel positional encoding. These encodings are added to the input of each attention layer.
\myparagraph
{Transformer decoder.}
The Transformer decoder aims to decode the top pixel features that can represent the instances of each frame, which is called instance level features. Motivated by DETR \cite{Detr}, we also introduce a
fixed number of input embeddings to query the instance features from pixel features, termed as \textit{instance queries}. Suppose that the model decodes $n$ instances each frame, then for $T$ frames the instance query number is $N=n \cdot T$. The instance queries are learned by the model and have the same dimension with the pixel features.
Taking the output of encoder $E$ and $N$ instance queries $Q$ as input, the Transformer decoder outputs $N$ instance features, denoted by $O$ in \figref{fig:2}. The overall predictions follow the input frame order, and the order of instance predictions for different images is the same.
Thus, the tracking of instances in different frames could be realized by linking the items of the corresponding indices directly.
\subsection{Instance Sequence Matching}\label{sec:3.2}
{VisTR}\xspace infers a fixed-size sequence of $N$ predictions, in a single pass through the decoder. One of the main challenges for this framework is to maintain the relative position of predictions for the same instance in different images, \textit{a.k.a.}, \textit{instance sequence}. In order to find the corresponding ground truth and supervise the instance sequence as a whole, we introduce the instance sequence matching strategy.
As the {VisTR}\xspace decode $n$ instances each frame, the number of instance sequence is also $n$. Let us denote by $\hat y=\left \{{ \hat y_{i}} \right \}_{i=1}^{n}$ the predicted instance sequences, and $y$ the ground truth set of instance sequences. Assuming $n$ is larger than the number of instances in the video clip, we consider $y$ also as a set of size $n$ padded with $\emptyset$. In order to find a bipartite
graph
matching between the two sets, we search for a permutation of $n$ elements $\sigma \in S_{n}$ with the lowest cost:
\begin{equation}
\hat{\sigma}=\underset{\sigma \in S_{n}}{\arg \min } \sum_{i}^{n} \mathcal{L}_{\text{match}}\left(y_{i}, \hat{y}_{\sigma(i)}\right)
\label{eq:1}
\end{equation}
where $\mathcal{L}_{\operatorname{match}}\left(y_{i}, \hat{y}_{\sigma(i)}\right)$ is a pair-wise \textit{matching cost} between ground truth $y_{i}$ and an instance sequence prediction with index $\sigma(i)$. The optimal assignment could be computed efficiently by the Hungarian algorithm\cite{kuhn1955hungarian}, following prior work (\textit{e.g.}, \cite{stewart2016end}).
As computing the mask sequence similarity directly is computationally intensive, we find a surrogate, the box sequence to perform the matching. To obtain the box predictions, we apply a 3-layer feed forward network (FFN) with ReLU activation function and a linear projection layer to the object predictions $O$ of Transformer decoder. Following the same practice of DETR \cite{Detr}, the FFN
predicts the normalized center coordinates, height and width of the box w.r.t.\ input image, and the linear layer predicts the class label using a softmax function. We also add a ``background" class to represent that no object is detected.
Given the $N=n \cdot T$ bounding box predictions for the object predictions sequence, we could associate $n$ box sequences for each instance by their indices, referred to as ins1 box seq...ins4 box seq in \figref{fig:2}. The matching loss takes both the class predictions and the similarity of predicted and ground truth boxes into account. Each element $i$ of the ground truth set can be seen as
\begin{equation}
y_{i} = \{
(c_{i},c_{i} ...,c_{i}), (b_{i,0},b_{i,1}...,b_{i,T})
\}
\end{equation}
where $c_{i}$ is the target class label (which may be $ \emptyset$) for this instance, and $b_{i,t}\in[0,1]^{4}$ is a vector that defines ground truth box center coordinates and its relative height and width in the frame $t$. $T$ represent the number of input frames. Thus, for the predictions of instance with index $\sigma(i)$ we denoted the probability of class $c_{i}$ as
\begin{equation}
{\hat p}_{(\sigma(i))}(c_{i}) = \{{\hat p}_{(\sigma(i),0)} (c_{i})...,\hat p_{(\sigma(i),T)}(c_{i}) \}
\end{equation}
and the predicted box sequence as
\begin{equation}
\hat b_{\sigma(i)}= \left \{ \hat b_{(\sigma(i),0)},\hat b_{(\sigma(i),1)}..., \hat b_{(\sigma(i),T)} \right \}
\end{equation}
With the above notation, we define
\begin{equation}
\mathcal{L}_{\text{match}}\left(y_{i}, \hat{y}_{\sigma(i)}\right)=- \hat{p}_{\sigma(i)}\left(c_{i}\right)+ \mathcal{L}_{\mathrm{box}}
\bigl(b_{i}, \hat{b}_{\sigma(i)}
\bigr),
\label{eq:2}
\end{equation}
where $c_{i} \neq \emptyset$.
Based on the above criterion, we could find the one-to-one matching of the sequences by the Hungarian algorithm. Given the optimal assignment, we could compute the loss function, the \textit{Hungarian loss} for all pairs matched in the previous step. The loss is a linear combination of a negative log-likelihood for class prediction, a box loss and mask loss for the instance sequences:
\begin{align}
\mathcal{L}_{\text{Hung}}(y, \hat{y})
&=
\sum_{i=1}^{N} \Bigl[
( -\log \hat{p}_{\hat{\sigma}(i)}(c_{i}))
+ \mathcal{L}_{\text {box }}(b_{i}, \hat{b}_{\hat{\sigma}}(i))
\notag
\\
&+
\mathcal{L}_{\text {mask }}(m_{i}, \hat{m}_{\hat{\sigma}}(i))
\Bigr].
\label{eq:3}
\end{align}
Here $c_{i} \neq \emptyset$,
and $\hat \sigma$ is the optimal assignment computed in Eq.~(\ref{eq:1}). The \textit{Hungarian loss} is used to train the whole framework.
The second part of the matching cost and the Hungarian loss is $\mathcal{L}_{\rm box}$ that scores the bounding boxes. We use a linear combination of the sequence level $\mathcal{L}_{1}$ loss and the generalized IOU loss \cite{rezatofighi2019generalized}:
\begin{align}
\mathcal{L}_{\mathrm{box}}\bigl(b_{i}, \hat{b}_{\sigma(i)}
\bigr)
& =
\frac{1}{T} \sum_{t=1}^{T}
\Bigl[
\lambda_{\mathrm{iou}} \cdot \mathcal{L}_{\mathrm{iou}}\bigl( b_{i,t}, \hat{b}_{\sigma(i),t} \bigr)
\notag
\\
& +
\lambda_{\mathrm{L} 1}\left\|b_{i,t}-\hat{b}_{\sigma(i), t}\right\|_{1}
\Bigr].
\label{eq:4}
\end{align}
Here $\lambda_{\mathrm{iou}}, \lambda_{\mathrm{L} 1} \in \mathbb{R}$ are hyper-parameters. These two losses are normalized by the number of instances inside the batch.
In the sequel, we present the details.
\subsection{Instance Sequence Segmentation}\label{sec:3.3}
The instance sequence segmentation module aims to predict the mask sequence for each instance. To realize that, the model needs to accumulate the mask features of multiple frames for each instance firstly, then the mask sequence segmentation is performed on the accumulated features.
The mask features are obtained by computing the similarity map between the object predictions $O$ and the Transformer encoded features $E$. To simplify the calculation, we only compute with the features of its corresponding frame for each object prediction. For each frame, the object predictions $O$ and the corresponding encoded feature maps $E$ are fed into the self-attention module to obtain the initial attention maps. Then the attention maps will be fused with the initial backbone features $B$ and the transformed encoded features $E$ of the corresponding frames, following a similar practice with the DETR\cite{Detr}. The last layer of the fusion is a deformable convolution layer\cite{dai2017deformable}. In this way, the mask features for each instance of different frames are obtained.
Following the same spirit of taking the instance sequence as a whole, the mask features of the same instance in different frames should be propagated and reinforce each other. We propose to utilize the 3D convolution to realize that.
Assume that the mask feature for instance $i$ of frame $t$ is $g_{i,t}\in \mathbb{R}^{1 \times a \times \nicefrac{H_{0}}{4} \times \nicefrac{W_{0}}{4}}$, where $a$ is the channel number, then we concatenate the features of $T$ frames to form the $G_{i}\in \mathbb{R}^{1 \times a\times T \times \nicefrac{H_{0}}{4} \times \nicefrac{W_{0}}{4}}$.
The \textit{instance sequence segmentation} module takes the instance sequence mask feature $G_{i}$ as input, and output the mask sequence $m_{i}\in \mathbb{R}^{1 \times 1\times T \times \nicefrac{H_{0}}{4} \times \nicefrac{W_{0}}{4}}$ for the instance directly. This module contains three 3D convolutional layers and Group Normalization\cite{wu2018group} layers with ReLU activation function. No normalization or activation is performed after the last convolution layer, and the output channel number of the last layer is 1. In this way, the masks of the instance for $T$ frames are obtained. The mask loss for supervising the predictions in Eq.~\eqref{eq:3} is defined as a combination of the Dice \cite{milletari2016v} and Focal loss \cite{Lin_2017_ICCV}:
\def{ {\cal L}_{ \text{Dice}}} {{ {\cal L}_{ \text{Dice}}} }
\def{ {\cal L}_{ \text{Focal}}} {{ {\cal L}_{ \text{Focal}}} }
\begin{align}
\mathcal{L}_{\text{mask}}\left(m_{i}, \hat{m}_{\sigma(i)}\right)
& =
\lambda_{\text{mask}}\frac{1}{T}\sum_{t=1}^{T}
\Bigl[
{ {\cal L}_{ \text{Dice}}} (m_{i,t}, \hat{m}_{\sigma(i),t}
)
\notag
\\
&+
{ {\cal L}_{ \text{Focal}}} (m_{i,t}, \hat{m}_{\sigma(i),t})
\Bigr].
\label{eq:5}
\end{align}
\section{Experiments}
In this section, we conduct experiments on the YouTube-VIS \cite{vis2019} dataset, which contains 2238 training, 302 validation and 343 test video clips. Each video of the dataset is annotated with per pixel segmentation mask, category and instance labels. The object category number is 40. As the test set evaluation is closed, we evaluate our method in the validation set. The evaluation metrics are average precision (AP) and average recall (AR), with the video Intersection over Union (IoU) of the mask sequences as the threshold.
\subsection{Implementation Details}
\myparagraph{Model settings.}
As the largest number of the annotated video length for YouTube-VIS \cite{vis2019} is 36, we take this value as the default input video clip length $T$.
Thus, no post-processing is needed to associate different clips from one video, which makes our model totally end-to-end trainable. The model predicts 10 objects for each frame, thus the total object query number is 360. For the Transformer we use 6 encoder, 6 decoder layers of width 384 with 8 attention heads. Unless otherwise specified, ResNet-50 \cite{he2016deep} is used as our backbone networks and the same hyper-parameters of
DETR \cite{Detr} are used.
\myparagraph{Training.} The model is implemented with PyTorch-1.6 \cite{paszke2019pytorch}, trained with AdamW \cite{loshchilov2018decoupled} of initial Transformer’s learning rate being $10^{-4}$ , the backbone’s learning rate being $10^{-5}$. The models are trained for 18 epochs, with the learning rate decays by 10x at 12 epochs. We initialize our backbone networks with the weights of DETR pretrained on COCO \cite{lin2014microsoft}. The models are trained on 8 V100 GPUs of 32G RAM, with 1 video clip per GPU. The frame sizes are downsampled to 300$\times$540 to fit the GPU memory.
\myparagraph{Inference.} During inference, we follow the same scale setting as training. No post-processing is needed for associating instances. Instances with scores larger than 0.001 are kept. The mean score for all the frames is used as the instance score. For instances that have been classified to different categories in different frames, we use the most frequently predicted category as the final instance category.
\subsection{Ablation Study}
In this section we conduct extensive ablation experiments to study the core factors of {VisTR}\xspace.
Comparison results are
reported
in Table \ref{tab:1}.
\begin{table*}[!t]
\small
\begin{subtable}{0.5\linewidth}
\centering
\captionsetup{width=0.9\linewidth}
\begin{tabular}{c|c|cccc}
Length & AP & $\rm AP_{50}$ & $\rm AP_{75}$ & $\rm AR_{1}$ & $\rm AR_{10}$ \\
\hline
18&29.7&50.4&31.1&29.5&34.4\\
24&30.5&47.8&33.0&29.5&34.4\\
30&31.7&53.2&32.8&31.3&36.0\\
36&33.3&53.4&35.1&33.1&38.5
\end{tabular}
\caption{\textbf{Video sequence length.}
The performance improves as the sequence length increases.
}
\label{tab:length}
\end{subtable}%
\begin{subtable}{0.5\linewidth}
\centering
\captionsetup{width=0.9\linewidth}
\setlength{\tabcolsep}{1mm}{
\begin{tabular}{ r |c|c|cccc}
&
\#
&AP & $\rm AP_{50}$ & $\rm AP_{75}$ & $\rm AR_{1}$ & $\rm AR_{10}$ \\
\hline
video level& 1&8.4&13.2&9.5&20.0&20.8\\
frame level& 36&13.7&23.3&14.5&30.4&35.1\\
ins. level& 10&32.0&52.8&34.0&31.6&37.2\\
pred. level& 360&33.3&53.4&35.1&33.1&38.5
\end{tabular}
}
\caption{\textbf{Instance query embedding}.
Instance-level query is only 1.3\% lower in AP than the prediction-level query with 36$\times$ fewer embeddings. }
\label{tab:query}
\end{subtable}%
\begin{subtable}{0.5\linewidth}
\centering
\captionsetup{width=0.9\linewidth}
\begin{tabular}{c|c|cccc}
time order & AP & $\rm AP_{50}$ & $\rm AP_{75}$ & $\rm AR_{1}$ & $\rm AR_{10}$ \\
\hline
random&32.3&52.1&34.3&33.8&37.3\\
in order&33.3&53.4&35.1&33.1&38.5
\end{tabular}
\caption{\textbf{Video sequence order.} Sequence in time order is 1.0\% better in AP than sequence in random order.}
\label{tab:order}
\end{subtable}%
\vspace{-0.3cm}
\begin{subtable}{0.5\linewidth}
\centering
\captionsetup{width=0.9\linewidth}
\begin{tabular}{c|c|cccc}
& AP & $\rm AP_{50}$ & $\rm AP_{75}$ & $\rm AR_{1}$ & $\rm AR_{10}$ \\
\hline
w/o&28.4&50.1&29.5&29.6&33.3\\
w &33.3&53.4&35.1&33.1&38.5
\end{tabular}
\caption{\textbf{Position encoding}. Position encoding brings about 5\% AP gains to {VisTR}\xspace.}
\label{tab:pos}
\end{subtable}%
\begin{subtable}{0.5\linewidth}
\centering
\captionsetup{width=0.9\linewidth}
\begin{tabular}{c|c|cccc}
& AP & $\rm AP_{50}$ & $\rm AP_{75}$ & $\rm AR_{1}$ & $\rm AR_{10}$ \\
\hline
CNN &32.0&54.5&31.5&31.6&37.7\\
Transformer&33.3&53.4&35.1&33.1&38.5
\end{tabular}
\caption{\textbf{CNN-encoded feature vs.\ Transformer-encoded feature} for mask prediction. The transformer improves the feature quality.}
\label{tab:encoded}
\end{subtable}%
\begin{subtable}{0.5\linewidth}
\centering
\captionsetup{width=0.9\linewidth}
\begin{tabular}{c|c|cccc}
& AP & $\rm AP_{50}$ & $\rm AP_{75}$ & $\rm AR_{1}$ & $\rm AR_{10}$ \\
\hline
w/o &33.3&53.4&35.1&33.1&38.5\\
w&34.4&55.7&36.5&33.5&38.9
\end{tabular}
\caption{\textbf{Instance sequence segmentation module.} The module with 3D convolutions brings 1.1\% AP gains.}
\label{tab:refine}
\end{subtable}
\caption{Ablation experiments for {VisTR}\xspace. All models are trained on YouTubeVIS \texttt{train} for 10 epochs and tested on YouTubeVIS \texttt{val}, using the ResNet-50 backbone.}\label{tab:1}
\vspace{-0.7cm}
\end{table*}
The main difference between video and image is that video contains temporal information.
How to effectively learn and exploit temporal information is the key to video understanding.
Firstly, we study the importance of temporal information to {VisTR}\xspace in two dimensions: the amount and the order.
\myparagraph{Video sequence length.}
To evaluate the importance of the amount of temporal information to {VisTR}\xspace, we experiment with models trained with different input video sequence lengths. As reported in Table~\ref{tab:length}, with the length varying
from 18 to 36, the AP increases monotonically from 29.7\% to 33.3\%. This result shows that more temporal information indeed helps the model learn better. As the largest video length of the dataset is 36, we argue that, if with a larger dataset, {VisTR}\xspace can achieve even better results.
Note that for this experiment, if the clip length is less than the video length, instance matching in overlapping frames is used for associating them from different clips.
\myparagraph{Video sequence order.}
As the movement of objects in %
real scenes
are continuous, we %
believe
that the order of temporal information is also important. To evaluate, we perform a comparison of the model trained with input video sequence in random order vs.\ time order. Results in Table~\ref{tab:order} show that the model learned %
according to the
time order information achieves 1 point higher, which verifies the importance of the temporal order.
\myparagraph{Positional encoding.}
Position information is important for the dense prediction problem of VIS. As the original feature sequence contains no positional information, we supplement with the spatial and temporal positional encodings,
which indicate the relative positions in the video sequence. Experiments of models with and without positional encoding are %
presented
in Table~\ref{tab:pos}. The model without positional encoding %
manages to
achieve 28.4\% AP.
Our explanation is that the ordered format of the sequence supervision and the correspondence between the input and output order of the Transformer provide some relative positional information \textit{implicitly}. In the second experiment, the performance improves by about 5 points, which verifies the necessity of explicit positional encoding.
\myparagraph{Instance queries.}
The instance queries are learned embeddings for decoding the representative instance predictions.
In this experiment, we study the effect of instance queries and attempt to exploit the inner connections among them by varying the embedding number. Suppose the model decode $n$ instances each frame, and the frame number is $T$. The input instance query number should be $n \times T$ to decode the same number for predictions.
In the default setting, one embedding is responsible for one prediction, the model directly learns $n \times T$ unique embeddings, termed as `prediction level' in Table~\ref{tab:query}.
In the `video level setting', one embedding is learned for all the instance predictions, \textit{i.e.}, the same embedding is repeated $n\times T$ times as the input of decoder. In the `frame-level' setting, the model only learns $T$ unique embeddings and repeats them by $n$ times.
In the `instance level' setting, the model only learns $n$ unique embeddings and repeats them by $T$ times.
The n and T corresponds to the value of 10 and 36 in the table respectively. The result is 8.4\% AP and 13.7\% AP for `video level' and `frame level' settings respectively.
Surprisingly, the `instance level' queries can achieve 32.0\% AP, which is only 1.3 points lower than the default setting. The result shows that the queries for one instance can be shared for the {VisTR}\xspace model, which makes the tracking natural. But the queries for one frame can not be shared.
\myparagraph{Transformers for feature encoding.}
As illustrated in the
`instance sequence segmentation' module of \figref{fig:2}.
The module takes three types of features as input:
the features `B' from the backbone, the feature `E' from the encoder and the attention map computed by the feature `E' and `O'.
To show the superiority of Transformers in feature encoding, we compare the results of using the original input `O' vs.\ output `E' of the encoder for the second feature, \textit{a.k.a.}, CNN-encoded features vs.\ Transformer-encoded features. As reported in Table~\ref{tab:encoded}, the CNN-encoded features achieves 32.0\% AP, and the Transformer-encoded features achieve 1.3 points higher. This demonstrates that features are learned better after the Transformer updates them based on all pairwise similarities between them through self-attention. The result also shows the superiority of modeling the spatial and temporal features as a whole.
\myparagraph{Instance sequence segmentation.}
The segmentation process contains both the instance mask feature accumulation and instance sequence segmentation modules. The instance sequence segmentation module takes the instance sequence as a whole. We expect that it can strengthen the mask prediction by learning the temporal information through 3D convolutions. Thus, when objects are in challenging situations such as occlusions or motion blurs, the module can learn to propagate information from other frames to help the segmentation. Besides, the features of the same instance from multiple frames could help the network recognize the instance better. In this experiment, we perform a study of models with or without the 3D instance sequence segmentation module. For the former case, we apply a 2D convolutional layer with the output channel being 1 to the mask features for each instance of each frame to obtain the masks. The comparison is shown in Table~\ref{tab:refine}.
The instance sequence segmentation module improves the result by 1.1 points, which verifies the effectiveness of the proposed module.
With
these ablation studies, we conclude that in
{VisTR}\xspace design: the temporal information, positional encodings, instance queries, global self-attention in the encoder and the instance sequence segmentation module,
all play important roles \textit{w.r.t.}\ the final performance.
\subsection{Main Results}
\begin{table*}
\small
\centering
\begin{tabular}{ r|l|c|c|cccc}
Method&backbone& FPS& AP & $\rm AP_{50}$ & $\rm AP_{75}$ & $\rm AR_{1}$ & $\rm AR_{10}$ \\
\hline
DeepSORT\cite{wojke2017simple}&ResNet-50&-&26.1&42.9&26.1&27.8&31.3\\
FEELVOS\cite{voigtlaender2019feelvos}&ResNet-50&-&26.9&42.0&29.7&29.9&33.4\\
OSMN\cite{yang2018efficient}&ResNet-50&-&27.5&45.1&29.1&28.6&33.1\\
MaskTrack R-CNN\cite{vis2019}&ResNet-50&20.0&30.3&51.1&32.6&31.0&35.5\\
STEm-Seg\cite{Athar_Mahadevan20ECCV}&ResNet-50&-&30.6&50.7&33.5&31.6&37.1\\
STEm-Seg\cite{Athar_Mahadevan20ECCV}&ResNet-101&2.1&34.6&55.8&37.9&34.4&41.6\\
MaskProp\cite{bertasius2020classifying}&ResNet-50&-&40.0&-&42.9&-&-\\
MaskProp\cite{bertasius2020classifying}&ResNet-101&-&42.5&-&45.6&-&-\\
\hline
\textbf{{VisTR}\xspace}&ResNet-50&30.0/69.9&36.2&59.8&36.9&37.2&42.4\\
\textbf{{VisTR}\xspace}&ResNet-101&27.7/57.7&40.1&64.0&45.0&38.3&44.9\\
\end{tabular}
\caption{\textbf{Video instance segmentation} AP (\%) on the YouTube-VIS \cite{vis2019} validation dataset. Note that,
for the first three methods, we have cited the results reported by the re-implementations in \cite{vis2019} for VIS. Other results are adopted from their original paper.
For the speed of {VisTR}\xspace we report the FPS results with and without the data loading process.
Here we naively load the images serially, taking unnecessarily long time.
The data loading process can be much faster by parallelizing.
}
\label{tab:sota}
\vspace{-0.4cm}
\end{table*}
\begin{figure*}[h]
\centering
\includegraphics[width=\linewidth]{figs/3.pdf}
\caption{\textbf{Visualization of {VisTR}\xspace} on the YouTube-VIS \cite{vis2019} validation dataset. Each row contains images from the same video. For each video, here the same colors
depict the mask sequences of the same instances (Best viewed on screen).
}
\label{fig:3}
\vspace{-0.5cm}
\end{figure*}
We compare {VisTR}\xspace against some state-of-the-art methods in video instance segmentation in Table~\ref{tab:sota}. The comparison is performed in terms of both accuracy and speed.
The methods in the first three rows are originally proposed for tracking or VOS.
We have cited the results reported by the re-implementations in
\cite{vis2019} for VIS. Other methods including the MaskTrack RCNN, MaskProp \cite{bertasius2020classifying} and STEm-Seg \cite{Athar_Mahadevan20ECCV} are originally proposed for the VIS task in the temporal order.
For the accuracy measured by AP, \textit{{VisTR}\xspace achieves the best result among methods using a single model without any bells and whistles}. Using the same backbone of ResNet-50 \cite{he2016deep}, {VisTR}\xspace achieves about 6 points higher %
in AP
than the MaskTrack R-CNN and the recently proposed STEm-Seg method. Besides, we argue the AP gap between {VisTR}\xspace and MaskProp mainly comes from its combination of multiple networks, \textit{i.e.}, Spatiotemporal Sampling Network \cite{bertasius2018object}, Feature Pyramid Network \cite{lin2017feature}, Hybrid Task Cascade Network \cite{chen2019hybrid} and the High-Resolution Mask Refinement post-processing.
Since our aim is to design a conceptually simple and end-to-end framework, many improvements methods, such as complex video data augmentation and multi-stage mask refinement are beyond the scope of this work. For the speed measured by FPS (frames per second), {VisTR}\xspace shows a
significant
advantage among all the reported results, achieving 27.7 FPS with the ResNet-101 backbone. If excluding the data loading process of multiple images, the speed
can
achieve 57.7 FPS. Note that,
as we load the images in serial, the data loading process can be easily parallelized.
The fast speed of {VisTR}\xspace owes to its design of parallel decoding and no post-processing.
The visualization of {VisTR}\xspace on the YouTube-VIS\cite{vis2019} validation dataset is shown in \figref{fig:3}, with each row containing images sampled from the same video.
{VisTR}\xspace can track and segment instances well in challenging situations
such as: (a) instances overlapping, (b) changes of relative positions between instance, (c) confusion by the same category instances
that are close together and (d) instances in %
various
poses.
\section{Conclusion}
In this paper, we have proposed a new video instance segmentation framework built upon Transformers, which views the VIS task as a direct end-to-end parallel sequence decoding/prediction problem. In this way, instance tracking is achieved \textit{seamlessly and naturally} in the same framework of instance segmentation, which is significantly different
from and simpler than existing approaches, considerably simplifying the overall pipeline.
Without bells and whistles, {VisTR}\xspace achieves
the best result and the highest speed among methods using a single model on the YouTube-VIS dataset.
To our knowledge, our work is the first one that applies the Transformer
to video instance segmentation. We hope that similar approaches can
be applied to many more video understanding tasks in the future.
\textbf{Acknowledgement
}
This work was in part supported by Beijing Science and Technology Project (No.\ Z1\-8\-1\-1\-0\-0\-0\-0\-8\-9\-1\-8\-0\-1\-8).
CS and his employer received no financial
support for the research, authorship, and/or publication of this article.
{\small
\bibliographystyle{ieee_fullname}
|
1,941,325,220,492 | arxiv | \section{Introduction}
Since their introduction in 1974 \cite{Nygren}, time projection chambers (TPCs) have proved to be one of the most effective ways of detecting particles and reconstructing their trajectories. The versatility of these devices, being compatible with $B$-fields, allowing readout flexibility and a wide range of density media (from some 10's of mbar up to liquid or even solid phase), makes them the perfect tool to study many different phenomena in particle physics \cite{DiegoReview}.
In particular, TPCs making use of their scintillation signal (either primary or secondary) are being used in different experiments, such as direct dark matter searches \cite{1,2}, neutrinoless double-beta decay ($\beta\beta0\nu$) \cite{3} or double electron capture \cite{4}, but also in future neutrino experiments such as DUNE \cite{5}. Depending on which noble element is used, they usually rely on well-known scintillation bands, such as the “second continuum” bands in xenon or argon, which are around 172~nm and 128~nm in gas, respectively \cite{Takahashi}.
Until very recently, it was believed that secondary scintillation in noble element TPCs was produced solely from the excited states created by the primary ionization electrons when drifting across a high field region (conveniently situated at the anode plane). These states result largely on excimer production through three body processes, giving rise to those well-known emission bands \cite{AprileBook}. This notion was challenged three years ago \cite{Buzu} by showing that there was light emission below the classical electroluminescence (EL) threshold of argon and proposing a mechanism to explain it: neutral bremsstrahlung (NBrS). Since then further studies were conducted, showing its presence in Xe and Xe-C$_2$H$_6$ mixtures and helping to firmly establish the phenomenon \cite{PRX, TalkLIDINE}.
Regarding the potential impact in real-world experiments, the readout of a dual-phase TPC demonstrator in the visible-band by making a partial use of this emission was performed in \cite{DarSide_NBrS}. Despite the yields being much lower than those achieved through excimer emission, a number of advantages like the immunity of NBrS to quenching by impurities, fast nature, broad-band characteristics and low electric fields needed for operation make it a subject of ongoing technological interest. At the same time, the phenomenon has been shown to have an impact in ongoing experiments, causing background scintillation in the 'buffer' regions of single-drift (asymmetric) TPCs, i.e. between the cathode and photosensor plane \cite{PRX}.
In this work we explore the possibilities of the framework used to calculate NBrS in \cite{PRX}. First we will introduce briefly the theoretical foundations of NBrS and explain how the calculations can be performed, giving results for a number of weakly quenched noble gases, conditions under which the current framework is expected to apply. Finally we will discuss the density scalings of the NBrS yields and its verification in transport.
\section{Theory and calculation}
In a process analogous to nuclear bremsstrahlung, NBrS can be viewed as the interaction of an electron with the dipole field of a neutral atom thereby leading to the emission of a photon, while conserving energy and momentum thanks to the atom recoil \cite{Dalgarno,Johnston,Ohmura,Geltman}. This is possible even in the absence of permanent dipole moments of the gas species, as those can still interact with the incoming electron via induced dipole moments.
In order to study NBrS, Fermi's golden rule can be used to calculate the transition probability between the initial/final state of the incoming/scattered electron. This leads to the following result for the differential cross section per unit of photon frequency \cite{Dalgarno}:
\begin{equation}
\frac{d\sigma}{d\nu} = \frac{8 \pi e^2 \nu^3 m_e^2 k_f}{3 \hbar^3 c^3 k_i} |M|^2
\label{FermiG}
\end{equation}
being ${\hbar}k_{i(f)}$ the initial (final) momentum of the electron and $M$ a matrix element involving the initial and final states:
\begin{equation}
|M|^2 \equiv |\langle \Psi_f|\vec{r}|\Psi_i \rangle|^2 \label{matrix1}
\end{equation}
Those may be approximated by using a scattering formalism based on partial waves and keeping the first terms. If approximating those by the ones used for computations in the case of the hydrogen ion (with $1\%$ accuracy in that case, according to \cite{Ohmura}), the following expression for the matrix element can be derived \cite{Dalgarno}:
\begin{equation}
|M|^2 = \frac{64\pi^2}{(k_i^2 - k_f^2)^4} [k_i^2 Q(k_f) + k_f^2 Q(k_i)] \label{Mcalc}
\end{equation}
thus displaying explicit proportionality with the elastic scattering cross section $Q$. By plugging this differential cross section directly into equation \ref{FermiG} and recalling the relationships $\varepsilon_{i,f} = (\hbar^2/2m_e)k_{i,f}^2$, $h\nu = \varepsilon_i - \varepsilon_f$, we can obtain the final expression for the cross section per unit of photon frequency:
\begin{equation}
\frac{d\sigma}{d\nu} = \frac{8}{3}\frac{r_e}{c}\frac{1}{h\nu}\left(\frac{\varepsilon_i-h\nu}{\varepsilon_i}\right)^{1/2} \cdot \left[\varepsilon_i \cdot{Q_{(m)}}(\varepsilon_i-h\nu) + (\varepsilon_i - h\nu)\cdot{Q_{(m)}}(\varepsilon_i) \right]\label{nBr_eq4}
\end{equation}
Here we make explicit an ambiguity in the existing analytical derivations, through which the elastic and momentum transfer (subindex $m$) cross sections can be used interchangeably (see \cite{PRX} for a discussion). In the following we will assume proportionality with $Q_m$ for simplicity, although the differences are generally small and at the level of 10's of a $\%$.
In order to compute the NBrS emission rate in a TPC, a swarm of ionization electrons needs to be considered, characterized through an energy distribution $dP/d\varepsilon$. This is done by simply averaging the product of the energy-dependent velocity of the electrons ($v(\epsilon)$), and the differential cross section for emission ($d\sigma/d\nu$) over the energy distribution of the electrons:
\begin{equation}
\frac{dN_\gamma}{d\nu dt} = \int_0^{\infty} N \frac{d\sigma}{d\nu} v(\varepsilon) \frac{dP}{d\varepsilon} d{\varepsilon} \label{nBr_eq2}
\end{equation}
\begin{figure}[h!!!]
\centering
\includegraphics[width=0.9\textwidth,origin=c,angle=0]{Nobles_and_spectrum_noborders2.eps}
\caption{\label{fig:spectrum} Left: pressure-reduced scintillation yields for NBrS in noble gases as a function of electric field (integrated over the practical detection range 120-1000~nm). Right: example of emission spectra (rate per electron, second and nanometer) at a pressure-reduced electric field of $E/P=1200$ V/cm/bar, $T=300$~K and $P=1$~bar. }
\end{figure}
The yield per unit length, a magnitude easier to determine experimentally than the scintillation rate, can be calculated in the following way:
\begin{equation}
\label{nBr_eq6}
Y \equiv \frac{dN_\gamma}{dz} = \frac{1}{v_d} \int_{\nu_{min}}^{\nu_{max}} \frac{dN_\gamma}{d\nu dt} = \frac{1}{v_d} \int_{\nu_{min}}^{\nu_{max}}\int_0^{\infty} N \frac{d\sigma}{d\nu} v(\varepsilon) \frac{dP}{d\varepsilon} d{\varepsilon} d{\nu}
\end{equation}
where $v_d$ is the drift velocity of the electrons in the gas. The above formalism can be easily extended for a weakly quenched gas (meaning in this context that NBrS emission involving inelastic degrees of freedom is expected to be subdominant). The yield calculation can be performed in the following way for a two-component mixture with concentrations $f_{A}$ and $f_{B}$, and cross sections $\sigma_{A}$ and $\sigma_{B}$:\footnote{An arbitrary number of compounds can be considered, but for simplicity we discuss only binary mixtures in this work.}
\begin{equation}
\label{nBr_mixt}
Y_{AB} \left(dP/d\varepsilon \right) =f_A \cdot Y_{A}\left( dP/d\varepsilon, \sigma_{A}\right)
+ (1-f_A) \cdot Y_{B}\left( dP/d\varepsilon, \sigma_{B}\right)
\end{equation}
In order to have access to both the drift velocity and energy distributions, Monte Carlo simulations with a locally modified version of the Pyboltz transport code \cite{Pyboltz} have been realized.
\section{Spectrum}
NBrS is a broadband emission with a blue-wing depending on the maximum electron energy (equivalently, on the density-reduced electric field). Spectral emission rates obtained for noble gases based on expression \ref{nBr_eq2} can be found in figure \ref{fig:spectrum}. A reduced field roughly corresponding to the maximum of the emission was chosen.
\section{Noble and weakly-quenched gas mixtures}
The integrated yields of NBrS (considered here in the practical range of interest of 120-1000~nm) can be `tuned' to a considerable extent in noble element-mixtures, through the interplay between the different position of the Ramsauer minima, the impact of the cross sections on the energy distribution of the electrons (e.g., through the higher cooling power of helium) as well as the overall magnitude of the cross sections, that are directly proportional to the emission probability (e.g., with xenon cross section being generally higher at low energies). Figure \ref{fig:yields} shows for illustration tbe NBrS yields in mixtures of xenon-argon (left) and xenon-helium (right). In particular, as Xe-He mixtures are being considered for electroluminescence-based time projection chambers due to their low electron diffusion \cite{XeHeTPCs}, it is interesting to note that the spurious presence of NBrS in the chamber would be beneficially suppressed.
\begin{figure}[htbp]
\centering
\includegraphics[width=1\textwidth,origin=c,angle=0]{Nobles_and_nobles_mixtures_noborders.eps}
\caption{\label{fig:yields} NBrS yield for mixtures based on xenon-argon (left) and xenon-helium (right), integrated over the wavelength range from 120 to 1000~nm, at $T=300~$K and $P=1~$bar.}
\end{figure}
Weakly quenched gas mixtures can be simulated as well, as it is seen in figure \ref{fig:CH4} where Xe-CH$_4$ has been chosen for the purpose of illustration. In this case, the EL simulations from \cite{CH4yields} have been added to illustrate the contributions of both electron cooling (shifting the curves towards higher $E/P$ values) and scintillation quenching (suppressing the yields in the EL region) to the overall gas scintillation.
\begin{figure}[h!!!]
\centering
\includegraphics[width=.7\textwidth,origin=c,angle=0]{Xe_CH4_yield_noborders2.eps}
\caption{\label{fig:CH4} Scintillation yields of Xe-CH$_4$ mixtures as a function of the pressure-reduced electric field (integrated over the practical detection range 120-1000~nm). The chosen dopings are $0.11\%$, $0.37\%$ and $1\%$ (molar), and the gas operating conditions are 300~K and 1~bar. EL simulations have been taken from \cite{CH4yields}.}
\end{figure}
\section{Density scalings}
NBrS is expected to show scalings with the number density, $N$, (or pressure, at fixed temperature). The $N$-scaling on the yield, $Y$, can be understood from the fact that the probability of NBrS emission per collision is constant, so the yields are bound to depend linearly on the gas gap and pressure. Concerning the electric field, as long as the energy distribution depends on its density-reduced value ($E/N$), $Y/N$ can be expected to be a function of $E/N$ too. The validity of these scalings in Monte Carlo is demonstrated over almost three orders of magnitude in figure \ref{fig:scaling}.
\begin{figure}[h!!!]
\centering
\includegraphics[width=.8\textwidth,origin=c,angle=0]{Xenon_scalings_test_resize_noborders2.eps}
\caption{\label{fig:scaling} NBrS yield calculated for different densities of gaseous xenon in the range of 120 to 1000 nm at $T=300$~K. The green line corresponds to a reference simulation performed for $P=1$~bar and with a large number of collisions. At higher fields and densities the absolute yield increases (left plot) but the reduced yield stays the same at the same reduced electric field (right plot).}
\end{figure}
\section{Conclusions}
In this paper we have shown how to easily perform NBrS calculations for a wide variety of mixtures and elements, resorting to the open-source code Pyboltz. The predictive capability anticipated from the data-benchmarking performed with xenon in \cite{PRX}, should allow future experiments to take informed decisions on regard to this new effect. Implementation of the above algorithms in the Pyboltz repository \cite{PyBoltzGithub} is currently being undertaken and expected to happen within weeks of the time of publication of this work.
|
1,941,325,220,493 | arxiv | \section{Introduction}\label{introduction section}
\subsection{Preliminary description of results}
Let $d\ge2$ be an integer and let $X\overset{\on{def}}{=} \operatorname{SL}_d(\mathbb{R})/\operatorname{SL}_d(\mathbb{Z})$ denote the space of unimodular lattices in $\mathbb{R}^d$. Let $A<\operatorname{SL}_d(\mathbb{R})$ be the group
of diagonal matrices with positive diagonal entries. The aim of this paper is to establish the following result.
\begin{theorem}\label{t.0932}
There exists a sequence of distinct compact $A$-orbits $Ax_k$ such that any limit point of the form $x=\lim a_k x_k$, with $a_k\in A$,
satisfies $x\in A\mathbb{Z}^d$.
\end{theorem}
We note that in dimension $d=2$ Peter Sarnak points out in~\cite{Sarnakreciprocal} a construction yielding the above result. In higher
dimensions the analysis is more intricate.
Given a compact orbit $Ax\subset X$, let us denote by $\mu_{Ax}$ the unique $A$-invariant probability measure supported on it.
Theorem~\ref{t.0932} implies
the following result which sharpens~\cite[Theorem 1.10]{ELMV-Duke} in which only partial escape of mass is claimed.
\begin{corollary}\label{t.0901}
Let $Ax_k$ be a sequence of compact orbits satisfying the conclusion of Theorem~\ref{t.0932}. Then the probability measures
$\mu_{Ax_k}$ converge to the zero measure.
\end{corollary}
\begin{proof}
If $\mu$ is an accumulation point of $\mu_{Ax_k}$ then by Theorem~\ref{t.0932}, we conclude that $\mu$ is
supported on the divergent orbit of the standard lattice.
By Poincar\'e recurrence theorem, the only finite $A$-invariant measure supported on $A\mathbb{Z}^d$ is the zero measure.
\end{proof}
Our results are much more precise than Theorem~\ref{t.0932} but as a consequence, more lengthy to state and depend on various notation to be presented
later. We roughly describe their features. For $\epsilon>0$ we set
\eqlabel{eq length of vectors}{
X^{\ge\epsilon}\overset{\on{def}}{=}\set{x\in X:\ell(x)\ge \epsilon}\textrm{ where }\ell(x)\overset{\on{def}}{=}\min\set{\norm{v}:0\ne v\in x}.
}
In \S\ref{escape and geometry section} we isolate a family of compact orbits $\mathcal{F}$, (which in~\S\ref{construction section} is shown to be infinite),
for which there exists a function $\eta:\mathcal{F}\to(0,\infty)$ with $\eta(Ax)\to 0$ as $Ax\in \mathcal{F}$ varies, such that for any $\delta>0$ and for all but
finitely many
$Ax\in\mathcal{F}$, the following statements are satisfied (see Propositions~\ref{eomprop}, \ref{accprop}, and Theorem~\ref{index theorem} for precise statements):
\begin{enumerate}
\item $\mu_{Ax}(X^{\ge\delta})\le \eta(Ax)$.
\item $\forall y \in Ax\cap X^{\ge\delta},\; \on{d}(y, A\mathbb{Z}^d)\le \eta(Ax)$.
\item The number of connected components of $Ax\cap X^{\ge \delta}$ is at least $(d-1)!$.
\end{enumerate}
In fact, the results are sharper in the sense that $\eta(Ax)$ is explicit and more interestingly, $\delta$ could be chosen to be an explicit function of
the orbit such that $\delta(Ax)\to 0$ as $Ax\in\mathcal{F}$ varies.
From the point of view of algebraic number theory,
Theorem~\ref{index theorem}, which gives new tools for showing that certain collections of units in an order $\mathcal{O}$ generate
the group of units $\mathcal{O}^\times$ up to torsion\footnote{This result is only applicable under certain assumptions relating the discriminant of the order
to the geometry of the collection of units.}, might be of interest on its own. This result is (on the face of it) unrelated to
the discussion on escape of mass but its proof uses the analysis yielding Theorem~\ref{t.0932} in a fundamental way.
For instance,
given a vector of distinct integers $\mb{m}\in\mathbb{Z}^d$, let $\theta_k$ be a root of the polynomial
$p_k(x)\overset{\on{def}}{=} \prod_1^d(x-km_i)-1$. Then, it follows from Theorem~\ref{index theorem} and
the analysis in \S\ref{construction section}, that
for all large enough $k$, the collection $\set{\theta_k-km_i}_{i=1}^d$
generates the group of units of the order $\mathbb{Z}[\theta_k]$ up to roots of unity.
\subsection{Cassels' work on Minkowski's conjecture}
This paper originated from a simple observation made while reading Cassles' paper~\cite{Casselsnforms}
showing that the value $2^{-d}$ is not isolated in the Minkowski spectrum. Given a lattice $x\in X$ let us define
$$\mu(x)=\sup_{\mb{v}\in\mathbb{R}^d}\inf\set{\prod_1^d\av{w_i-v_i}:\mb{w}\in x},$$
and let us define the the \textit{Minkowski spectrum} in dimension $d$ to be $\crly{M}\overset{\on{def}}{=}\set{\mu(x) :x\in X}$. A
famous conjecture attributed to Minkowski asserts that $\crly{M}\subset [0,2^{-d}]$ and that $\mu(x) = 2^{-d}$
if and only if $x\in A\mathbb{Z}^d$.
In~\cite{Casselsnforms} Cassels gives a construction of a sequence $\set{x_k}$ of lattices having compact $A$-orbits
with the property that $\lim_k\mu(x_k)=2^{-d}$. Taking into account the fact that $\mu$ is $A$-invariant and upper semi-continuous
(in the sense that if $x_k\to x$ then $\limsup\mu(x_k)\le \mu(x)$), one arrives at the inevitable conclusion that, assuming Minkowski's conjecture
holds in dimension $d$, the sequence $x_k$ from Cassels' construction must satisfy the conclusion of Theorem~\ref{t.0932}; thus proving it
in any dimension in which Minkowski's conjecture is known to hold. To this date Minkowski's conjecture has been validated up to dimension $n=9$
(see for example~\cite{McMullenMinkowski},\cite{Hans-Gill-dim8},\cite{Hans-Gill-dim9},\cite{SW2} for recent accounts).
As it turns out, in order to prove Theorem~\ref{t.0932} one does not need Minkowski's conjecture as an input and it could be derived by
a careful analysis of Cassels' construction which we slightly generalize in \S\ref{construction section}.
\subsection{Homogeneous dynamics context}
Another context for interpreting the results is the comparison between unipotent dynamics and (higher-rank) diagonalizable dynamics. See
\cite{ELMV-Duke} for a thorough discussion explaining the fundamental differences between these two worlds. In this spirit, we remark that
Theorem~\ref{t.0932} and its corollary are in sharp contrast to the rigidity exhibited by sequences of periodic orbits $Hx_k$
if one assumes $H<\operatorname{SL}_d(\mathbb{R})$ is generated by unipotent elements. It is a consequence of a result
of Mozes and Shah~\cite{MozesShah} (relying on results by Dani and Margulis), that if $Hx_k\subset X$ is such a sequence and there is a fixed compact set $K\subset X$ such that
$Hx_k\cap K\ne\varnothing$, then any accumulation point of the sequence $\mu_{Hx_k}$ is a probability measure\footnote{The
information on such accumulation points is much more informative but we state here only the part relevant for us.}.
We note that there exists
a compact set $K$ such that $Ax\cap K\ne\varnothing$ for any $x\in X$ (see Theorem~\ref{thm del0} and the discussion preceding it for references).
Having this comparison in mind, we find the following open questions (for $d\ge 3$) natural and interesting.
\begin{enumerate}[(Q1)]
\item Does there exists a sequence of compact orbits $Ax_k$ such that $\mu_{Ax_k}$ converges to a non-ergodic measure?
\item Does there exists a sequence of compact orbits $Ax_k$ such that $\mu_{Ax_k}$ exhibits strictly partial escape
of mass (i.e.\ which converges to a non-zero measure of total mass $<1$)?
\item Does there exists a sequence of compact orbits $Ax_k$ such that $\mu_{Ax_k}$ converges to a
limit whose ergodic decomposition contains a periodic $A$-invariant measure with positive weight?
\end{enumerate}
Another interesting question is whether or not one can construct a sequence of compact orbits $Ax_k$ with $\mu_{Ax_k}$ exhibiting
full (or even partial) escape of mass such that the lattices $x_k$ are geometric embeddings of full modules in a fixed number field.
We note though that
in dimension 2 for example,
due to the results in~\cite{AS}, the arithmetic relations between the full modules in such an example
must involve infinitely many primes. Interestingly, in positive characteristics this situation breaks and there are examples in dimension
2 of sequences of compact orbits arising from a fixed quadratic field and sharing arithmetic relation involving a single prime which
produce full escape of mass. For details see~\cite{KPS}.
\subsection{Acknowledgments} This paper was under writing for quite some time during which it evolved to its present shape.
This evolution was greatly influenced by conversing with others and for that I am grateful. Thanks are due to Elon Lindenstrauss, Manfred Einsiedler, Shahar Mozes, Barak Weiss, and Ofir David.
\section{Intuition}\label{section intuition}
As often happens, the idea behind the proofs is simple but it is not unlikely that without clarifications
it may appear quite hidden behind
the details. We therefore try to explain it briefly and along the way present some of the notation that we shall use.
The diagonal group is naturally identified with the hyperplane
$$\mathbb{R}^d_0\overset{\on{def}}{=}\set{\mb{t}\in\mathbb{R}^d:\sum_1^d t_i=0}.$$
As the dimension of this hyperplane will appear frequently in our
formulas, we henceforth denote
$$n\overset{\on{def}}{=} \dim \mathbb{R}^d_0=d-1.$$
Given a compact orbit $Ax$, it corresponds in a natural way\footnote{For the sake of the current discussion
the reader might envision $\Delta$ as the image under the logarithm of the stabilizer or $x$ but in fact,
the lattice we consider could potentially be slightly larger.
See \S\ref{escape and geometry section} for the definition.}
to a lattice $\Delta<\mathbb{R}^d_0$ and
on choosing a fundamental domain $F$ (say a parallelepiped) for this lattice, we may identify the orbit with $F$ via
the map $\mb{t}\mapsto a(\mb{t})x$, where
\eqlabel{eq at}{
a(\mb{t})\overset{\on{def}}{=} \diag{e^{t_1},\dots, e^{t_d}}.
}
We
think of $\mb{t}\in\mathbb{R}^d_0$ as the `time parameter', and as it varies, the point on the orbit corresponding to it
varies as well.
If it so happens that $x$ contains a very short vector, say of length $\ell$, at time $0$ (and hence
at any time from $\Delta$),
then one can estimate quite easily a radius $r=r(\ell)$ such that for any time in a ball of radius $r$ around a
corner of $F$
the vector is still short. A strategy for deducing that the orbit spends most
of the time near infinity is to establish that
the volume of these neighbourhoods occupies most of the volume of $F$. The success of this strategy depends therefore
first and foremost on the relationship between the volume of $F$ and the length $\ell$ but also on the `geometry' of $F$
as is illustrated in Figure~\ref{figure basic idea} below.
\begin{figure}[h]
\centering
\fbox{\includegraphics[scale = 0.15]{fund1}
\includegraphics[scale = 0.20]{fund2}}
\caption{The square on the left is a tamed fundamental domain of area 1 and the grey circles cover most of its area.
The rectangle on the right is a skewed fundamental domain and the circles fail to cover a significant proportion of it.
The grey areas represents times in which the orbit is near infinity.}\label{figure basic idea}
\end{figure}
This strategy suffices to establish partial escape of mass in certain examples but in order to obtain the sharper
results of this paper, and in particular, full escape of mass,
one needs to modify it slightly.
The first modification has to do with the norm. In the above description we mentioned
an estimate of the radius of a euclidean ball around points of $\Delta$ in which we have control on the length of some short vector. It turns out that this estimate becomes sharper when we replace the euclidean norm with a slight modification of it
which takes advantage of the fact that $\mathbb{R}^d_0$ sits in $\mathbb{R}^d$ .
Given $\mb{t}\in\mathbb{R}^d_0$, we denote
$$\ceil{\mb{t}}=\max\set{t_i : 1\le i\le d}.$$
Note that there exists a constant $c=c(d)>0$ such that for $\mb{t}\in\mathbb{R}^d_0$,
$c^{-1}\norm{\mb{t}}\le \ceil{\mb{t}}\le c\norm{\mb{t}}$. The ball of radius $r$ is then replaced by the set
$\set{\mb{t}\in\mathbb{R}^d_0: \ceil{\mb{t}}\le r}$ which as the reader can easily convince himself, is a simplex. Another
modification to the above strategy, and which is tightly related to the switch from euclidean balls to simplexes,
is as follows. Since we have good estimates for the length of vectors when the time parameter is confined to a simplex, it
will be more convenient not to work with a fundamental domain given by the
parallelepiped obtained from a basis of $\Delta$, but with a certain simplex which one can cook up
from this basis. This brings us to a situation where the balls and the rectangles
in Figure~\ref{figure basic idea} are replaced respectively by simplexes centered around points of $\Delta$ and a simplex of the same type
which contains a fundamental domain as illustrated in Figure~\ref{figure passing to simplexes} below.
\begin{figure}[h]
\centering
\fbox{
\includegraphics[scale = 0.30]{passingtosimplexes}
}
\caption{The black dots are the points of $\Delta$, the grey simplexes are the areas where we have control on the short vector,
and the indicated blue triangle border a simplex which contains a fundamental domain for $\Delta$.}
\label{figure passing to simplexes}
\end{figure}
The structure of the paper is as follows.
In~\S\ref{section simplex sets} we introduce the concept of a \textit{simplex set} $\Phi$ of a lattice $\Delta<\mathbb{R}^d_0$
and various quantities
attached to it. This section culminates in Proposition~\ref{covering prop} which is fundamental for the analysis. In a nutshell, it
gives the appropriate mathematics content to the illustration in
Figure~\ref{figure passing to simplexes}.
In \S\ref{escape and geometry section}
we study lattices with compact orbits of the diagonal group possessing nice simplex sets (which we refer to as $M$-tight simplex sets).
Having an $M$-tight simplex set allows us translate Proposition~\ref{covering prop} and establish the main results which are Propositions~\ref{eomprop},
\ref{accprop}, and Theorem~\ref{index theorem}. In particular, for these lattices we will control the white area inside the blue simplex
in Figure~\ref{figure passing to simplexes} and show that it is negligible compared to the covolume of $\Delta$
(see Proposition~\ref{eomprop}\eqref{eq1630}).
In \S\ref{construction section} we give a construction and carefully analyze it,
showing that the results of \S\ref{escape and geometry section} may be applied. This construction is a slight generalization
of the one given by Cassels.
\section{Simplex sets and a covering radius calculation}\label{section simplex sets}
\subsection{}
We begin by fixing some notation and terminology. For concreteness we fix the supremum norm on $\mathbb{R}^d$ and work only with this norm and the operator norm corresponding to it.
\begin{comment}
that will be used while discussing lattices in $\mathbb{R}^d_0\overset{\on{def}}{=}\set{\mb{t}\in\mathbb{R}^d:\sum_1^d t_i=0}$ (which arise naturally when
discussing compact orbits of the diagonal group). Since the dimension of $\mathbb{R}^d_0$ will appear frequently
in our discussion we henceforth denote
$$n\overset{\on{def}}{=} \dim \mathbb{R}^d_0=d-1.$$
For concreteness we fix the supremum norm on $\mathbb{R}^d$ and work only with this norm and the operator norm corresponding to it. For $\mb{v}\in\mathbb{R}^d$
we write
$$\ceil{\mb{v}}=\max\set{v_i : 1\le i\le d}.$$
\end{comment}
By $A\ll B$ we mean that there exists a constant $c$ such that $A\le cB$. If the constant depends on some parameter $M$, we indicate this by writing $\ll_M$. An
exception to this rule is the dependency of the constant in the dimension which we do not record in the notation (thus if not indicated otherwise, implicit constants
are universal but may depend on $d$). Similar conventions are used with big $O$ and small $o$ notation. We write $A\asymp B$ to indicate that $A\ll B$ and $B\ll A$. For example, as noted in~\S\ref{section intuition},
\eqlabel{eq1054}{
\forall \mb{t}\in\mathbb{R}^d_0,\;\;\norm{\mb{t}}\asymp \ceil{\mb{t}}.
}
\begin{definition}
A \textit{simplex set} $\Phi\subset \mathbb{R}^d_0$ is a spanning set of cardinality $d$
such that $\sum_{\mb{t}\in\Phi}\mb{t}=0$. The \textit{associated simplex} is defined as the convex hull of $\Phi$; $S_\Phi\overset{\on{def}}{=}\on{conv}(\Phi)$, and the \textit{associated lattice} is $\Delta_\Phi\overset{\on{def}}{=}\on{span}_\mathbb{Z}\Phi$.
\end{definition}
Although the concept of a simplex set
is almost equivalent to a choice of a basis for the associated lattice, it will be more convenient to work with it.
Similar to the standard lattice and standard basis in euclidean space it will be convenient to introduce a certain ``standard'' simplex set:
Let
\begin{equation*}
\Phi_*\overset{\on{def}}{=}\set{\mb{b}_j}_{j=1}^d,\textrm{ where, }
\mb{b}_j\overset{\on{def}}{=}(1,\dots, \underbrace{-n}_{\textrm{\tiny{ $j$'th place}}},\dots,1)^t,
\end{equation*}
and denote $\Del_*\overset{\on{def}}{=} \Delta_{\Phi_*}$. As will be shown in the proof of Lemma~\ref{reduction lemma}, $\Delta_*$ is a
dilated copy of the orthogonal projection of $\mathbb{Z}^d$ to $\mathbb{R}^d_0$.
\begin{definition}
Given a simplex set $\Phi$, we define its \textit{distortion map} to be an element $h_\Phi \in \operatorname{GL}(\mathbb{R}^d_0)$
satisfying
$h_\Phi\Phi_*=\av{\Delta_\Phi}^{-\frac{1}{n}}\Phi$
(and as a consequence $h_\Phi\Del_*=\av{\Delta_\Phi}^{-\frac{1}{n}}\Delta_\Phi$). We define the \textit{distortion} of
$\Phi$ to be $\norm{h_\Phi}$.
\end{definition}
Note that there is no canonical choice of $h_\Phi$ but the $d!$ possible choices differ by precomposing with a permutation matrix and
thus this ambiguity will cause no harm.
Most of the assertions regarding a general simplex set $\Phi$ will be proved by establishing them first to $\Phi_*$ and then applying
the linear map $\av{\Delta_\Phi}^{\frac{1}{n}} h_\Phi$ that maps $\Phi_*$ to $\Phi$.
A related quantity to the distortion is the following.
\begin{definition}
Given a simplex set $\Phi$ we set
$$\xi_\Phi\overset{\on{def}}{=} \max_{\mb{t}\in\Phi} \ceil{\mb{t}}.$$
\end{definition}
We collect a few elementary facts about the concepts introduced above.
\begin{lemma}\label{dist lemma}
\begin{enumerate}
\item\label{dist1}For any simplex set $\Phi$, the distortion of $\Phi$ satisfies $\norm{h_\Phi}\asymp\av{\Delta_\Phi}^{-\frac{1}{n}}\xi_\Phi$.
\item\label{dist1.5} If $h_\Phi$ is a distortion map with $\norm{h_\Phi}\le T$ then for any $\mb{t}\in\mathbb{R}^{d}_0$,
$\norm{\mb{t}}\ll_T \norm{h_\Phi \mb{t}}$.
\item\label{dist1.6} If $h_\Phi$ is a distortion map with $\norm{h_\Phi}\le T$, then $\xi_\Phi\ll_T \min\set{\norm{\mb{t}}:\mb{t}\in\Phi}$.
\end{enumerate}
\end{lemma}
\begin{proof}
\eqref{dist1}. It is straightforward that
$\av{\Delta_\Phi}^{\frac{1}{n}}\norm{h_\Phi}\asymp \max\set{\norm{\mb{t}}:\mb{t}\in \Phi}$ and so the results follows from the definition of $\xi_\Phi$
and~\eqref{eq1054}.
\eqref{dist1.5} Because the determinant of a distortion map $h_\Phi$ is a constant, (independent of $\Phi$), we have
that $\norm{h_\Phi^{-1}}\ll \norm{h_\Phi}^n$ and the claim follows.
\eqref{dist1.6}. This follows from~\eqref{dist1}, \eqref{dist1.5}.
\end{proof}
\begin{lemma}\label{xi bound}
Let $\Phi$ be a simplex set. Then, $\av{\Delta_\Phi}^{\frac{1}{n}}\ll \xi_\Phi.$
\end{lemma}
\begin{proof}
For each $\mb{t}\in \Phi$ we have that
$\norm{\mb{t}}\ll\ceil{\mb{t}}\le\xi_\Phi$ so that $\Delta_\Phi$ is spanned by a basis of vectors whose lengths are $\ll\xi_\Phi$. It follows that
$\av{\Delta_\Phi}\ll \xi_\Phi^n$. Extracting the $n$'th root we obtain the desired claim.
\end{proof}
\subsection{}
We proceed by introducing a collection of points $\mb{W}_\Phi$ associated to a simplex set $\Phi$ which will play an important
role in our considerations.
These points are indicated in red on the boundary of the blue triangle in Figure~\ref{figure W phi} below, which represents $\frac{n}{2}S_\Phi$. The black points represent $\Delta_\Phi$ and the grey simplexes represent $\Delta_\Phi +(1-\rho)\frac{n}{2}S_\Phi$ for some
$0\le \rho\le 1$.
\begin{figure}[h]
\centering
\fbox{
\includegraphics[scale = 0.22]{Wtau}
\includegraphics[scale = 0.22]{Wtau2}
\includegraphics[scale = 0.22]{Wtau3}
}
\caption{Envision the grey triangles being dilated simultanuously around their centers. The red points are the critical
points which are covered last (so called \textit{deep holes}). See Proposition~\ref{covering prop}.}\label{figure W phi}
\end{figure}
Let $\crly{S}$ denote the group of permutation matrices in $\operatorname{GL}_d(\mathbb{R})$ and let $\theta\in\crly{S}$ denote the standard cycle permutation; that is, $\theta\mb{e}_i=\mb{e}_{i+1 \on{mod} d}$, where $\set{\mb{e}_i}_{i=1}^d$ denotes the standard basis for $\mathbb{R}^d$.
\begin{definition}
For any $\tau\in \crly{S}$ let
$$\mb{w}_\tau^{\Phi_*}\overset{\on{def}}{=}\tau (d^{-1}\sum_{\ell=1}^d (\ell-1) \mb{b}_\ell)$$
and in turn, for a general simplex set $\Phi$ with distortion map $h_\Phi$, let
\eqlabel{def:w}{
\mb{w}_\tau^\Phi\overset{\on{def}}{=} \av{\Delta_\Phi}^{\frac{1}{n}} h_\Phi \mb{w}_\tau^{\Phi_*}.
}
\end{definition}
Note that the indexation in~\eqref{def:w} depends on the choice of the distortion map but this should cause no confusion. We further let $\mb{W}_\Phi\overset{\on{def}}{=} \set{\mb{w}^\Phi_\tau}_{\tau\in\crly{S}}$ and choosing a set $\crly{S}'\subset \crly{S}$ of representatives of the quotient $\crly{S}/\idist{\theta}$ we
set\footnote{The collections $\crly{S}', \mb{W}_\Phi'$ are not canonically defined.}
$\mb{W}_\Phi'\overset{\on{def}}{=}\set{\mb{w}_\tau^\Phi}_{\tau\in\crly{S}'}$.
When $\Phi$ is understood we usually write $\mb{w}_\tau$. We remark that given a simplex set $\Phi$, the collection $\mb{W}_\Phi$ lies on the boundary of the dilation $\frac{n}{2}S_\Phi$ because $\frac{2}{n}\mb{w}_\tau$ is a convex
combination of the elements in $\Phi$ with one of the coefficients being $0$.
\begin{comment}
\begin{figure}[h]
\centering
\fbox{\includegraphics[scale = 0.15]{figure1}}
\fbox{\includegraphics[scale = 0.15]{figure2}}
\fbox{\includegraphics[scale = 0.15]{figure3}}
\caption{This figure illustrates in grey the set $(1-\rho)\frac{n}{2}S_\Phi$ for $\rho = 0.3, 0.1, 0$ respectively. The points of $\Delta_\Phi$ are marked in black and the other triangles are translates of the grey one by the lattice points. The red points are the critical points,
$\mb{W}_\Phi+\Delta_\Phi$. What makes them critical is that only when $\rho=0$ the are covered by the translates of the grey triangle (see Proposition~\ref{covering prop}).
These critical points decompose into $n!$ cosets of $\Delta_\Phi$ (see Lemma~\ref{reduction lemma}). See also Figure~\ref{figure2}.}
\label{figure1}
\end{figure}
\end{comment}
The following lemma analyzes the reduction of $\mb{W}_\Phi$ modulo $\Delta_\Phi$.
\begin{lemma}\label{reduction lemma}
Let $\Phi$ be a simplex set.
\begin{enumerate}
\item\label{claim4} For $\tau,\sigma\in\crly{S}$, $\mb{w}_\tau+\Delta_\Phi=\mb{w}_\sigma+\Delta_\Phi$ if and only if $\tau^{-1}\sigma\in\idist{\theta}$.
\item\label{claim5} If $\tau^{-1}\sigma\notin\idist{\theta}$ then
there exists $\tau'\in\crly{S}$ such that for some $1\le k\le n$,
$\mb{w}_{\tau'}+k(\mb{w}_\tau-\mb{w}_\sigma)\notin
\mb{W}_\Phi+\Delta_\Phi.$ Moreover, if $d$ is prime then any $\tau'$ satisfies this.
\end{enumerate}
\end{lemma}
Statement~\eqref{claim4} is illustrated in Figure~\ref{figure W phi} where one can see that the six red points on the boundary of the blue
simplex decompose into two classes modulo the black points.
Statement~\eqref{claim5} will be used in the proof of Theorem~\ref{index theorem}
to show that a certain lattice that a priori contains
$\Delta_\Phi$ is actually equal to it. This will allow us to prove in~\S\ref{construction section} that certain collections of units in certain
orders generate the corresponding group of units modulo torsion.
\begin{proof}
It suffices to prove the lemma for the standard simplex set $\Phi_*$.
Let us denote by $\on{p}:\mathbb{R}^d\to \mathbb{R}^d_0$ the orthogonal projection.
Note that if $\set{\mb{e}_\ell}$ denotes the standard basis for $\mathbb{R}^d$ then for any $\ell$ and for any $\sigma\in\crly{S}$
\begin{align}
\nonumber \mb{b}_\ell &= \on{p}(-d\mb{e}_\ell),\\
\label{eq1750}\Del_* &=\on{p}(d\mathbb{Z}^d),\\
\nonumber \mb{w}_\sigma&=\on{p}((\sigma(1,\dots, d))^t).
\end{align}
We first show that for any $\tau$, $\mb{w}_\tau - \mb{w}_{\tau\textrm{{\tiny $\circ$}} \theta^j}\in\Del_*$. By the above, this is equivalent to
showing
that the difference $\tau(1,\dots,d)-\tau\textrm{{\tiny $\circ$}}\theta^j(1,\dots,d)\in d\mathbb{Z}^d+\ker\on{p}$. Clearly, it is enough to deal
with the case $j=1$ and moreover, post-composing with $\tau^{-1}$ we may assume $\tau=\on{id}$ (here we are using
the fact that both $\mathbb{Z}^d$ and $\ker\on{p}$ are $\crly{S}$-invariant).
Indeed,
$$ \smallmat{1\\2\\ \vdots \\d} -\theta \smallmat{1\\2\\ \vdots \\ d}
=\smallmat{1\\1\\ \vdots \\1}-\smallmat{d\\0\\ \vdots \\ 0}\in\ker\on{p}+d\mathbb{Z}^d.$$
We prove the second part of~\eqref{claim4}. Let $\tau,\sigma\in\crly{S}$ be such that $\tau^{-1}\sigma\notin\idist{\theta}.$
We show that $\mb{w}_\tau-\mb{w}_\sigma\notin\Del_*$, which is equivalent to $\tau(1,\dots,d)-\sigma(1,\dots,d)\notin d\mathbb{Z}^d+\ker\on{p}$. Post-composing with $\tau^{-1}$ and rewriting $\tau^{-1}\sigma$ as $\sigma$ we are reduced to the case $\tau=\on{id}$. Moreover, using the already established first part of~\eqref{claim4}, we may replace $\sigma$ with $\sigma\textrm{{\tiny $\circ$}}\theta^j$ and we choose $j$ so that $\sigma\textrm{{\tiny $\circ$}}\theta^j$ fixes the $d$'th coordinate. To this end, assume by way of contradiction
the existence of a non-trivial permutation $\sigma$ of $1,\dots,n$ such that
the difference $(1,\dots,n,d)-(\sigma(1),\dots,\sigma(n),d)$ belongs to
$d\mathbb{Z}^d+\ker\on{p}$. Alternatively, all the differences between the coordinates of $(1-\sigma(1),\dots,n-\sigma(n),0)$ are
equal modulo $d$. As the last coordinate is $0$ this is equivalent to saying that $(1,\dots,n)=(\sigma(1),\dots,\sigma(n))$ modulo
$d$. But since this equality involves only numbers between 1 to $n$ it is equivalent to $\sigma$ being trivial which contradicts our
assumption and concludes the proof of~\eqref{claim4}.
The proof of~\eqref{claim5} follows similar lines. First observe that because $\ker\on{p}\cap \mathbb{Z}^d=\set{k\mb{1}:k\in\mathbb{Z}}$, where
$\mb{1}\overset{\on{def}}{=} (1,\dots,1)^t$, it follows from~\eqref{eq1750} that $\set{\mb{v}\in\mathbb{Z}^d:\on{p}(\mb{v})\in\mb{W}_\Phi}$ can be characterized as the set of integer vectors having all residues modulo $d$ as coordinates.
Similarly to the consideration above we
reduce to the case where $\tau=\on{id}$ and $\sigma$ is non-trivial and fixes the $d$'th axis. Thus we need to find $\tau'$ and $k$ such that
$\tau'(1,\dots,d)+k(1-\sigma(1),\dots, n-\sigma(n),0)$ does not contain all the residues modulo $d$ as coordinates. Since $\sigma$ is non-trivial we know that for some $1\le j\le n$, $j-\sigma(j)\ne 0$. If we choose
$\tau'$ so that $\tau'$ fixes the $d$ axis and also, $\tau'(j) =-(j-\sigma(j))$ modulo $d$ then we see that for $k=1$ the above vector will contain the residue $0$ twice and therefore will not contain all residues. If $d$ is assumed to be prime, then
for any $\tau'$ we could solve the equation $\tau'(d) = \tau'(j)+k(j-\sigma(j))$ modulo $d$ with some $1\le k\le d$, and then
the above vector will contain the residue $\tau'(d)$ twice yielding the same contradiction.
We note that when $d$ is not prime it
is possible to find $\tau',\tau,\sigma$ such that $\tau^{-1}\sigma\notin \crly{S}$ but
$\mb{w}_{\tau'}+k(\mb{w}_\tau-\mb{w}_\sigma)\in\mb{W}_\Phi+\Delta$ for all $k$. For example,
when $d=4$ one can take
$\mb{w}_{\tau'}=\mb{w}_\tau = (3,2,1,4), \mb{w}_\sigma = (1,2,3,4)$.
\end{proof}
We now state a covering result which plays a crucial role in the analysis below and shows the significance of the
dilation $\frac{n}{2}S_\Phi$ and the collection
$\mb{W}_\Phi$ (see Figure~\ref{figure W phi}). This is roughly a restatement of \cite[Lemma 1]{Casselsnforms} but we give the full proof for
completeness.
\begin{comment}
\begin{figure}[h]
\centering
\fbox{\includegraphics[scale = 0.2]{figure11}}
\fbox{\includegraphics[scale = 0.2]{figure21}}
\caption{In this figure the grey area represents $\Delta_\Phi + (1-\rho)\frac{n}{2}S_\Phi$ (for $\rho=0.3,0.1,$ respectively), and the white area represent the complement which equals, as is illustrated, to the union of simplexes centred at the points in
$\Delta_\Phi +\mb{W}_\Phi$ and which have side-length proportional to $\rho$. See also Figure~\ref{figure1}.}
\label{figure2}
\end{figure}
\end{comment}
\begin{proposition}\label{covering prop}
Let $\Phi$ be a simplex.
\begin{enumerate}
\item\label{claim1} $\mathbb{R}^d_0= \frac{n}{2} S_\Phi +\Delta_\Phi$.
\item\label{claim2} $\mathbb{R}^d_0\smallsetminus(\frac{n}{2} S_\Phi^\circ+\Delta_\Phi)\subset \mb{W}_\Phi+\Delta_\Phi=\mb{W}_\Phi'+\Delta_\Phi$, where
$S^\circ$ denotes the interior of $S$.
\item\label{claim3} There exists a universal constant $c>0$
such that for any $\rho\in(0,1)$,
$\mathbb{R}^d_0\smallsetminus((1-\rho)\frac{n}{2} S_\Phi+\Delta_\Phi)\subset \mb{W}_\Phi' + B_{c\rho\xi_\Phi}+\Delta_\Phi.$
\end{enumerate}
\end{proposition}
As explained in Remark~\ref{equality remark}, the inclusion in~\eqref{claim2} is in fact equality. This shows that the covering radius of $\Delta_\Phi$ by $S_\Phi$
is exactly $\frac{n}{2}$.
\begin{proof}
We prove the proposition for the standard simplex set $\Phi_*$.
In this case, it follows from the definition (or from~\eqref{eq1750}), that $\mb{w}_\tau=\tau(\mb{w})$, where $\mb{w}=(n/2,n/2-1,...,-n/2)$.
The statement for general simplex sets follows by applying the linear map $\av{\Delta_\Phi}^{\frac{1}{n}}h_\Phi$ which takes
$\Phi_*$ to $\Phi$. Indeed, it is clear that
the statements \eqref{claim1}, \eqref{claim2}, follow immediately and \eqref{claim3} follows from the fact that the
above linear transformation has norm $\ll\xi_\Phi$ by Lemma~\ref{dist lemma}\eqref{dist1}.
We prove~\eqref{claim1} and \eqref{claim2}.
One of the benefits of working in the hyperplane $\mathbb{R}^d_0$ is the following observation:
\eqlabel{eq1444}{
\frac{n}{2}S_{\Phi_*}=\set{\mb{v}\in\mathbb{R}^d_0: \ceil{ \mb{v}}\le \frac{n}{2}}.
}
Given $\mb{u}\in\mathbb{R}^d_0$, if $\ceil{\mb{u}}\ge\frac{n}{2}$ we describe a procedure by which
\begin{enumerate}[(i)]
\item\label{pos2} either $\mb{u}=\mb{w}_\tau$ for some $\tau\in\crly{S}$,
\item\label{pos1} or we find $\mb{v}\in\Del_*$ such that
$\ceil{\mb{u}+\mb{v}}<\ceil{\mb{u}}.$
\end{enumerate}
As long as \eqref{pos1} occurs, we replace $\mb{u}$ by
$\mb{u+v}$ and iterate.
As the set $\Del_*+\mb{u}$ is discrete and the sets $\set{\mb{v}\in\mathbb{R}^d_0: \ceil{\mb{v}}\le t}$ are compact, we cannot iterate this procedure indefinitely, which means that after finitely many iterations we will arrive at a
vector $\mb{u}'\in\mb{u}+\Del_*$ with $\mb{u}'\in \frac{n}{2}S^\circ$, or to one of the points $\mb{w}_\tau$.
This will conclude the
proof of~\eqref{claim1} and \eqref{claim2} (the equality $\mb{W}_{\Phi_*}+\Delta_*=\mb{W}'_{\Phi_*}+\Delta_*$ follows from Lemma~\ref{reduction lemma}\eqref{claim4}).
The procedure is as follows:
Let $\mb{u}\in\mathbb{R}^d_0$ be such that $\max \mb{u} \ge \frac{n}{2}$ and assume for concreteness that $u_1\ge u_2\dots \ge u_d$.
Successively for $k=1,\dots, n$ look at the vector
$$ \mb{u}^{(k)}\overset{\on{def}}{=} \mb{u}+\sum_{j=1}^{k} \mb{b}_j =
\smallmat{u_1\\ \vdots\\ u_k\\ u_{k+1}\\ \vdots \\ u_d}+
\smallmat{-d+k\\ \vdots \\ -d+k\\ k\\ \vdots\\ k}=
\smallmat{u_1 - (d-k)\\ \vdots\\ u_k - (d-k)\\ u_{k+1}+k\\ \vdots \\ u_{d}+k}.$$
If for
some $1\le k\le n$ we have that $\max \mb{u}^{(k)}<\max \mb{u}$ then we are done. Otherwise,
by our assumption on the order of the coordinates of $\mb{u}$ we conclude that $\ceil{ \mb{u}^{(k)}}$ must be obtained at its
$k+1$'th coordinate $u_{k+1}+k$ and moreover that $u_1\le u_{k+1}+k$. Summing these $n$ inequalities and taking into account that
$\mb{u}\in\mathbb{R}^{n+1}_0$ we arrive at $(n+1)u_1\le \frac{n(n+1)}{2}$, and so $u_1\le \frac{n}{2}$. Together with our assumption that
$\ceil{ \mb{u}}\ge \frac{n}{2}$ we conclude that in fact $u_1=\frac{n}{2}$. Thus all the inequalities in the above consideration must be
equalities. In particular, $u_1= u_{k+1}+k$ for all relevant $k$ which implies that $\mb{u}=\mb{w}$ as desired.
Claim \eqref{claim3} follows from \eqref{claim2} and the following elementary statement about intersections of half-spaces:
Given $d$ linear functionals $\psi_i$ on $\mathbb{R}^d_0$ and a point $\mb{w}\in\mathbb{R}^d_0$, consider the half spaces
$$H_i^\rho\overset{\on{def}}{=}\set{\mb{v}\in\mathbb{R}^d_0:\psi_i(\mb{v})\le \psi_i(\mb{w})+\rho}$$ for $\rho\ge 0$. If the $\psi_i$ are in
general position (in fact, if and only if), then the intersection $\cap_1^dH_i^\rho$ is a simplex which is contained in
the ball $B_{c\rho}(\mb{w})$ for some constant $c$ that depends only on the functionals $\psi_i$ (this constant deteriorates when
$n$ of these functionals are almost linearly dependent). In the case at hand the linear functionals are those giving rise for the faces
of $S_{\Phi_*}$ and $\mb{w}$ is a point of $\mb{W}_{\Phi_*}+\Delta_*$. In Figure~\ref{figure W phi} these are the white simplexes
centered at the red points.
\end{proof}
\begin{remark}\label{equality remark}
A nice indirect way to see why there must be equality in Proposition~\ref{covering prop}\eqref{claim2} is as follows. When considering $\Phi_*$,
since all the objects involved are invariant under $\crly{S}$, it is clear that the established inclusion implies that the left hand side of~\eqref{claim2}
is either empty or there is equality. Assuming it is empty, we deduce the existence of a positive $\rho>0$ such that the left hand side of~\eqref{claim3} is empty
as well. As will be shown in \S\ref{escape and geometry section}, \ref{construction section}, there are compact $A$-orbits whose visits to $X^{\ge\delta}$ occur
only at times $\mb{t}$ belonging to the left hand side of~\eqref{claim3}. We arrive at the desired contradiction as this set is never empty by Theorem~\ref{thm del0}.
\end{remark}
\section{Escape and geometry of compact $A$-orbits}\label{escape and geometry section}
\subsection{}
Denote
\eqlabel{Omega}{
\Omega\overset{\on{def}}{=}\set{x\in X_d: Ax \textrm{ is compact}}.\\
}
Let $U$ denote the group of diagonal matrices with $\pm 1$ diagonal entries so that $UA$ is the group of all diagonal
matrices of determinant $\pm 1$. Let $\varphi:UA\to \mathbb{R}^d_0$ be the homomorphism given by $\varphi(\diag{a_1,\dots,a_d})=(\log\av{a_1},\dots,\log\av{a_d})^t$. Finally, for $x\in\Omega$ we let\footnote{Note that $\av{\Delta_x}$
is (up to a constant factor) the regulator of some order in a totally real number field.
}
\eqlabel{deltax}{
\Delta_x\overset{\on{def}}{=} \varphi(\on{stab}_{UA}(x)).}
Note that $\Delta_x$ might strictly contain $\log(\on{stab}_A(x))$.
In practice, $\mb{u}\in\Delta_x$ if and only if there is a $\pm 1$ diagonal matrix $u$ such that $ua(\mb{u})x=x$ (see~\eqref{eq at} for the notation), and since $U$ acts
by isometries on $\mathbb{R}^d$ and we will only be interested in estimating length of vectors, the presence of $u$ will have no effect.
The results to be proved are stated in an effective way with respect to the covolume $\av{\Delta_x}$ and thus they become meaningful
only when $\av{\Delta_x}\to\infty$. That this is indeed the case when $Ax$ varies is a consequence of the basic fact from algebraic
number theory that says that the regulator of a number field goes to infinity when the number field varies. We take this fact for granted.
A simplex set
$\Phi\subset \Delta_x$ will be referred to as a \textit{simplex set
for} $x$. Note that we do not assume that $\Phi$ generates $\Delta_x$ and in fact, one of our goals (see Theorem~\ref{index theorem}), is to develop
tools that will enable to prove that under certain assumptions $\Phi$ generates $\Delta_x$; a significant fact from the number
theoretical point of view as it allows to compute regulators.
\subsection{}
In this section, we extract effective information about a compact orbit $Ax$ assuming that (i) $x$ has a very short vector,
(ii) there exists a simplex set for $x$ satisfying various
assumptions related to the length of the short vector.
The results originates from comparing the results of \S\ref{section simplex sets} and the following basic fact
(recall the notation of~\eqref{eq length of vectors}).
\begin{theorem}\label{thm del0}
There exists a universal constant $\delta_0>0$ such that for any $x\in X$, $Ax\cap X^{\ge\delta_0}\ne\varnothing$.
\end{theorem}
The proof of Theorem~\ref{thm del0} goes back to the solution of Mordell's inverse problem by Siegel and by
Hlawka~\cite{Hlawka-inverse-problem} (see also~\cite{DavenportNote}). The largest possible value of $\delta_0$
(which changes with the choice of norm on $\mathbb{R}^d$) is an interesting quantity in the geometry of numbers. A proof
of Margulis to Theorem~\ref{thm del0} may be found in the appendix of~\cite{TW} and recently it was proved
in~\cite{McMullenMinkowski}, \cite{SW2}, that for the euclidean norm one can take $\delta_0\ge 1$.
We begin with interpreting Proposition~\ref{covering prop}\eqref{claim1} in this context.
\begin{lemma}\label{lemma orbit}
Let $x\in\Omega$ and $\Phi$ be a simplex set for $x$. Then $Ax\subset U\set{a(\mb{t}x):\mb{t}\in \frac{n}{2}S_\Phi}.$
\end{lemma}
\begin{proof}
Given any $\mb{t}\in\mathbb{R}^d_0$, by Proposition~\ref{covering prop}\eqref{claim1} there exists $\mb{u}\in\Delta_x$
with $\mb{t}'\overset{\on{def}}{=} \mb{t}-\mb{u}\in\frac{n}{2}S_\Phi$ and so $a(\mb{t})x=a(\mb{t}')a(\mb{u})x=ua(\mb{t}')x$ for some $u\in U$.
\end{proof}
Recall the notation of~\eqref{eq length of vectors}. The following observation gives a lower bound for $\ell(x)$ in terms of the geometry of a simplex set.
\begin{proposition}\label{length bound prop}
Let $x\in\Omega$ and let $\Phi\subset \Delta_x$ be a simplex set. Then, $e^{-\frac{n}{2}\xi_\Phi}\ll \ell(x)$.
\end{proposition}
\begin{proof}
Let $v\in x$ attain the length $\ell(x)$
and let $\mb{t}\in\frac{n}{2}S_\Phi$. We give an upper bound
for $\norm{a(\mb{t})v}$. Write
$\mb{t}=\frac{n}{2}\sum_1^{n+1} \alpha_j\mb{t}_j$ with $\alpha_j$ coefficients of a convex combination.
Using the definition of $\xi_\Phi$ and the convexity of the exponential map we get
\begin{align}\label{eq238}
\nonumber\norm{a(\mb{t})v}&\le\ell(x) \max_i \pa{e^{\sum_j \alpha_jt_{ij}}}^{\frac{n}{2}}\\
&\le \ell(x)\max_i \pa{\sum_j\alpha_j e^{t_{ij}}}^{\frac{n}{2}}\le \ell(x)e^{\frac{n}{2}\xi_\Phi}.
\end{align}
Theorem~\ref{thm del0} and Lemma~\ref{lemma orbit} imply that there exist $u\in U$ and $\mb{t}\in \frac{n}{2}S_\Phi$
such that $\delta_0\le\norm{ua(\mb{t})v}=\norm{a(\mb{t})v}$. This together with \eqref{eq238} gives $ e^{-\frac{n}{2}\xi_\Phi}\ll\ell(x)$ as desired.
\end{proof}
Motivated by Proposition~\ref{length bound prop} we make the following definition.
\begin{definition}\label{defomegaM}
For $M\ge1$ say that a simplex set $\Phi\subset \Delta_x$ is $M$-\textit{tight} for $x$, if $\ell(x)\le M e^{-\frac{n}{2}\xi_\Phi}$.
We let
\begin{align}\label{OmegaM}
\Omega_M\overset{\on{def}}{=}\set{x\in \Omega:\textrm{ there exists an $M$-tight simplex set for $x$}}.
\end{align}
\end{definition}
The following reinterprets Proposition~\ref{covering prop} in this context and establishes an effective escape of
mass statement for lattices in $\Omega_M$ in terms of the regulator $\av{\Delta_x}$. It is the tool with which one can
prove Theorem~\ref{t.0901} directly and not as a corollary to Theorem~\ref{t.0932}.
\begin{proposition}\label{eomprop}
Fix $M>1$ and $\kappa\in(0,1)$ and for $x\in \Omega_M$ with an $M$-tight simplex set $\Phi$, set
\begin{align}\label{functions}
\nonumber
&\delta_\kappa(x)=\delta \overset{\on{def}}{=} Me^{-\frac{n}{2}\av{\Delta_x}^{\frac{\kappa}{n}}};\\
&\rho_\kappa(x)=\rho\overset{\on{def}}{=}\frac{\av{\Delta_x}^{\frac{\kappa}{n}}}{\xi_\Phi} \ll\av{\Delta_x}^{-\frac{1-\kappa}{n}};\\
\nonumber
&r_\kappa(x)=r\overset{\on{def}}{=} c\av{\Delta_x}^{\frac{\kappa}{n}} \textrm{ ($c$ being as in Proposition~\ref{covering prop})}.
\end{align}
Then,
\begin{enumerate}
\item\label{eq1151}
$\set{a(\mb{t})x : \mb{t}\in\frac{(1-\rho)n}{2}S_\Phi+\Delta_\Phi}\subset X^{<\delta}$.
\item\label{eq1651}
$\set{\mb{t}\in\mathbb{R}^{n+1}_0:a(\mb{t}x)\in X^{\ge \delta}} \subset \mb{W}'_\Phi+B_{r}+\Delta_\Phi$.
\item\label{eq1630} $\mu_{Ax}(X^{\ge\delta})\ll \av{\Delta_{x}}^{-1+\kappa}$.
\end{enumerate}
\end{proposition}
\begin{remark}
Hereafter we use the symbols $\delta,\rho,r$ as in~\eqref{functions} recording implicitly the dependence on $x,\kappa$.
\end{remark}
\begin{proof}
Let $x\in\Omega_M$ and $\Phi$ an $M$-tight simplex set for $x$.
Note that by Lemma~\ref{xi bound},
$\rho\ll\av{\Delta_x}^{-\frac{1-\kappa}{n}}$.
We prove~\eqref{eq1151}. By the discussion following~\eqref{deltax} it suffices to prove
$\set{a(\mb{t})x : \mb{t}\in\frac{(1-\rho)n}{2}S_\Phi}\subset X^{<\delta}$.
Let
$\mb{t}=(1-\rho)\frac{n}{2}\sum_1^{n+1} \alpha_j\mb{t}_j\in (1-\rho)\frac{n}{2}S_\Phi$ (with $\alpha_j$ coefficients of a convex combination).
Let $v\in x$ be a vector attaining $\ell(x)$. Similarly to \eqref{eq238}
we get
\begin{align}\label{eq05}
\nonumber \norm{a(\mb{t})v}&\le \ell(x)\max_i \pa{e^{\sum_j \alpha_jt_{ij}}}^{(1-\rho)\frac{n}{2}}
\le Me^{-\frac{n}{2}\xi_\Phi}\max_i \pa{\sum_j\alpha_j e^{t_{ij}}}^{(1-\rho)\frac{n}{2}} \\
&\le M e^{-\frac{n}{2}\xi_\Phi}\pa{e^{\xi_\Phi}}^{(1-\rho)\frac{n}{2}}= Me^{-\frac{n}{2}\av{\Delta_x}^{\frac{\kappa}{n}}}=\delta_\kappa(x),
\end{align}
which shows that $a(\mb{t}x)\in X^{<\delta}$.
We prove~\eqref{eq1651}. By~\eqref{eq1151} we have that
$$\set{\mb{t}\in\mathbb{R}^{n+1}_0:a(t)x\in X^{\ge\delta}}\subset \mathbb{R}^{n+1}_0\smallsetminus\pa{(1-\rho)\frac{n}{2}S_\Phi+\Delta_\Phi},$$
and by
Proposition~\ref{covering prop}\eqref{claim3},
$$\mathbb{R}^{n+1}_0\smallsetminus\pa{ (1-\rho)\frac{n}{2}S_\Phi+\Delta_\Phi}\subset \pa{\mb{W}'_\Phi+B_{c\rho\xi_\Phi}+\Delta_\Phi},$$
which implies~\eqref{eq1651} as $r=c\rho\xi_\Phi$.
Statement~\eqref{eq1630} now follows from~\eqref{eq1651}:
\footnote{Note that the index of $\log(\on{stab}_A(x))$ in $\Delta_x$ is
bounded by $2^d.$.}
$$\mu_{Ax}( X^{\ge \delta})\ll r^n/\av{\Delta_x}\ll \av{\Delta_x}^{-1+\kappa}.$$
\end{proof}
The following result is the one that we use in order to prove Theorem~\ref{t.0932}. It assumes that we have accurate knowledge regarding the position of the orbit $Ax$ at the special times $\mb{W}_\Phi$ and concludes that the orbit collapses in an effective manner on the orbit $A\mathbb{Z}^d$. In order to state it we need the following definition:
\begin{definition}\label{omega epsilon c}
For $\epsilon>0, M,C>1$ we denote by
$\Omega_M(\epsilon,C)$ the set of $x\in \Omega_M$ such that there exists an $M$-tight simplex set $\Phi$ for $x$ with respect to which, for any $\tau\in\crly{S}$
\eqlabel{eq1156}{
a(\mb{w}_\tau)x = g_\tau a(\mb{s}_\tau)\mathbb{Z}^{n+1} .
}
where $g_\tau$ satisfies $\norm{g_\tau - I}\le Ce^{-\av{\Delta_x}^\epsilon}$ and $s_\tau\in\mathbb{R}^d_0$.
\end{definition}
\begin{proposition}\label{accprop}
Fix $M,C>1$ and $\epsilon>0$.
Then, for any $\kappa<n\epsilon$ there exists $C'>0$ such that for any $x\in\Omega_M(\epsilon,C) $,
$$\forall y\in Ax\cap X^{\ge\delta},\; \on{d}(y,A\mathbb{Z}^d)\le C'e^{-\frac{1}{2}\av{\Delta_x}^\epsilon}.$$
\end{proposition}
\begin{proof}
Let $x\in\Omega_M(\epsilon,C)$ and let $\Phi$ be an $M$-tight simplex set as in Definition~\ref{omega epsilon c}. By Proposition \ref{eomprop}\eqref{eq1651}
for any point $y\in Ax\cap X_{n+1}^{\ge\delta}$ there exists $\tau\in\crly{S}$ such that $y=a(\mb{t}+\mb{w}_\tau)x$
for some $\mb{t}\in
\mb{w}_\tau+B_r$. By~\eqref{eq1156} it follows that
\eq{
y=a(\mb{t})a(\mb{w}_\tau)x=a(\mb{t})g_\tau a(-\mb{t})a(\mb{t}+\mb{s}_\tau)\mathbb{Z}^{n+1}
}
and so the distance between $y$ and $A\mathbb{Z}^{n+1}$ is
$\le \on{d}(I, a(\mb{t})g_\tau a(-\mb{t}))\le Ce^{2r_x-\av{\Delta_x}^\epsilon}=Ce^{2 c\av{\Delta_x}^{\frac{\kappa}{n}}-\av{\Delta_x}^\epsilon}=
Ce^{(-1+2c\av{\Delta_x}^{\frac{\kappa}{n}-\epsilon})\av{\Delta_x}^\epsilon}$,
and if $\kappa<n\epsilon$ then $2c\av{\Delta_x}^{\frac{\kappa}{n}-\epsilon}<\frac{1}{2}$ for large values of $\av{\Delta_x}$ and so for
a suitable $C'$ the latter expression is $\le C'e^{-\frac{1}{2}\av{\Delta_x}^\epsilon}$ as desired.
\end{proof}
\subsection{}
In applications one exhibits a simplex set $\Phi$ for $x\in\Omega$ and it is desirable to be able to determine that it generates $\Delta_x$.
In this section we prove Theorem~\ref{index theorem} establishing this in some cases. This theorem also discusses the
topological complexity of $Ax\cap X^{\ge\delta_1}$.
Our results rely on an assumption regarding the lattice $\Delta_\Phi$ when considered as a point in the space of lattices in
$\mathbb{R}^d_0$ (identified up to homothety). More precisely, we will have to assume that $\Delta_\Phi$ belongs to a given compact set
in that space (it is not clear at the moment if this assumption is necessary). Below we refer to the aforementioned space of lattices
as the space of \textit{shapes} in order to distinguish it from $X$.
It might be worth mentioning here that understanding the set of shapes $\set{\Delta_x:x\in\Omega}$ is a highly non-trivial question with virtually no results (although in the next section we will see that the shape $\Delta_*$ is an accumulation
point of this set).
\begin{definition}\label{omegamk}
Given $M>1$ and $K$ a compact set of shapes we denote by,
$\Omega_{M,K}$ the set of $x\in \Omega_M$ possessing an $M$-tight simplex set $\Phi$ for $x$ with shape $\Delta_\Phi\in K$.
\end{definition}
Because $\Delta_\Phi$ is obtained from $\Del_*$ by applying a distortion map $h_\Phi$ followed by a dilation, it is obvious from the definition that the set $\set{\Delta_\Phi:\norm{h_\Phi}\le T}$ is contained
in some compact set $K=K_T$ of shapes. Lemma~\ref{dist lemma} below is a converse statement of some sort for tight simplex sets
with associated lattice in a given compact set of shapes.
\begin{lemma}\label{lemmadistortion}
For any $M>0$ and any compact set of shapes $K$, if $x\in\Omega_{M,K}$ and $\Phi\subset
\Delta_x$ is an $M$-tight simplex set then the distortion of $\Phi$ satisfies $\norm{h_\Phi}\ll_{M,K} 1$.
\end{lemma}
We note that in applications, most often one knows that $x\in\Omega_M$ belongs to $\Omega_{M,K}$ because one exhibits
a simplex set $\Phi$ of bounded distortion, which makes the conclusion of the lemma automatic. Nonetheless we
chose to give it here as it seems reasonable to expect an application in which one knows that $\Delta_\Phi$ belongs to a compact set of shapes without knowing that $\Phi$ is responsible for that.
\begin{proof}
Let $\Phi$ be as in the statement. The assumption $\Delta_\Phi\in K$, implies that there exists a
basis $\set{\mb{t}_\ell}_1^n$ of $\Delta_\Phi$ such
that the linear map $h$ sending $\mb{b}_\ell$ to $\av{\Delta_\Phi}^{-\frac{1}{n}}\mb{t}_\ell$, ($\ell=1\dots n$), satisfies
$\norm{h}\ll_K1$ . Denote $\Phi'\overset{\on{def}}{=} \set{\mb{t}_\ell}_1^{n+1}$ where $\mb{t}_{n+1}=-\sum_1^n\mb{t}_\ell$ so that
$h_{\Phi'}=h$. Let $g$ be a linear map sending $\Phi'$ to $\Phi$ and note that $h_\Phi=gh_{\Phi'}$. It follows that we
would be done if we show that $\norm{g}\ll_{M,K}1$. To see this we note that on the one hand by Lemma~\ref{dist lemma}\eqref{dist1.6},
for any $\mb{t}\in \Phi'$,
$\norm{\mb{t}}\gg_K \xi_{\Phi'}$ and therefore $\norm{g}\ll_K \frac{\xi_\Phi}{\xi_{\Phi'}}$. On the other hand, by Proposition~\ref{length bound prop} and our assumption we have that $e^{-\frac{n}{2}\xi_{\Phi'}}\ll \ell(x)\le
M e^{-\frac{n}{2}\xi_\Phi}$ and therefore $\xi_\Phi-\xi_{\Phi'}\ll_M1$. We conclude that $\norm{g}\ll_{M,K} 1+\frac{1}{\xi_{\Phi'}}$.
Since $\xi_{\Phi'}\gg 1$ by Lemma~\ref{xi bound}, we get $\norm{g}\ll_{M,K}1$ as desired.
\end{proof}
The following lemma gives us the tool with which we bound the index of one lattice in another.
\begin{lemma}\label{index lemma}
Let $\Delta<\Sigma$ be lattices and suppose that there is a ball $B$ and vectors $\mb{v}_i$, $i=1,\dots,\ell,$ such that $\on{(i)}$
$\Sigma \subset \set{\mb{v}_i}_{i=1}^\ell +B +\Delta$, and $\on{(ii)}$ $\av{\Sigma\cap (\mb{v}_i+B+\mb{u})}\le 1$ for any $\mb{u}\in\Delta$, $1\le i\le \ell$. Then
$\br{\Sigma:\Delta}\le \ell$.
\end{lemma}
\begin{proof}
Using assumption (i) we may define a map $\alpha:\Sigma\to\Delta$ by the following procedure: Given $\mb{t}\in\Sigma$ we choose $1\le i\le \ell$ such that $\mb{t}\in \mb{v}_i+B+\mb{u}$ (with $\mb{u}\in\Delta$), and set $\alpha(\mb{t})=\mb{u}$. Assumption (ii) implies that $\alpha$ is at most $\ell$ to 1.
Note also that there is a constant $T_0$ such that $\norm{\alpha(\mb{t})}\le \norm{\mb{t}}+T_0$. We conclude that
if we denote by $C_T$ the cube of side length $T$ centered at $0$, then for any $T>0$,
$\av{\Sigma\cap C_T}\le \ell\av{\alpha(\Sigma)\cap C_{T+T_0}}$.
Recall that for any lattice $\Delta'$,
$\av{\Delta'}^{-1}=\lim_T \frac{1}{T^n}\av{\Delta'\cap C_T}$. It follows that
$$\br{\Sigma:\Delta}=\frac{\av{\Delta}}{\av{\Sigma}}=\lim_T\frac{\av{\Sigma\cap C_T}}{\av{\Delta\cap C_T}}
\le \limsup_T \ell \frac{\av{\alpha(\Sigma)\cap C_{T+T_0}}}{\av{\Delta\cap C_T}}\le \ell.
$$
\end{proof}
We use the following definition to discuss what we referred to above as `topological complexity'.
\begin{definition}
Given $\delta_1>0$ and a lattice $x$ we
say that $\mb{t}_1,\mb{t}_2\in\mathbb{R}^d_0$ are
two \textit{distinct visits times} to $X^{\ge\delta_1}$ if $a(\mb{t}_i)x$ belong to two distinct connected components of $Ax\cap X^{\ge\delta_1}$.
\end{definition}
The following lemma will help us distinguish between visit times.
\begin{lemma}\label{distinct visits lemma}
Let $x\in \Omega$, $B_1,B_2$ two balls in $\mathbb{R}^d_0$ and $\delta_1>0$. Assume that
$\on{(i)}$ $\exists \mb{t}_i\in B_i$ with $a(\mb{t}_i)x\in X^{\ge\delta_1}$, and $\on{(ii)}$
$\set{a(\mb{t})x :\mb{t}\in\partial B_1\cup\partial B_2}\subset X^{<\delta_1}$. Then, if $\mb{t}_i$ are not distinct visit times to $X^{\ge\delta_1}$ then there exist $\mb{u}_i\in B_i$ such that $\mb{u}_1-\mb{u}_2\in\Delta_x$.
\end{lemma}
\begin{proof}
If $a(\mb{t}_i)x$ are in the same component of $Ax\cap X^{\ge \delta_1}$ then we may join them by a path within that component. As
the map $\mb{t}\mapsto a(\mb{t})x$ is a covering map, this path can be lifted to a one starting at $\mb{t}_1$. Because of assumption (ii), this lift is contained in $B_1$ and in particular, its end point $\mb{t}_2'$, which satisfies $a(\mb{t}_2')x=a(\mb{t}_2)x$,
is in $B_1$. We conclude that the difference $\mb{t}_2'-\mb{t}_2$ belongs to $\Delta_x$ and so we may take $\mb{u}_1=\mb{t}_2'$
and $\mb{u}_2=\mb{t}_2$.
\end{proof}
\begin{theorem}\label{index theorem}
Fix $M>1$, $K$ a compact set of shapes, and $\delta_1\in(0,\delta_0)$. For all but finitely many $x\in \Omega_{M,K}$, if
$\Phi\subset \Delta_x$ is an $M$-tight simplex set and we let
$$\mb{W}_\Phi''\overset{\on{def}}{=}\set{\mb{w}_\tau\in\mb{W}_\Phi': \exists \mb{t}\in \mb{w}_\tau+B_r, \;a(\mb{t})x\in X^{\ge\delta_1}},$$
then,
\begin{enumerate}
\item\label{eq1524} $\br{\Delta_x:\Delta_\Phi}\le \av{\mb{W}_\Phi''}$ (and of course, $\av{\mb{W}_\Phi''}\le n!$).
\item\label{eq1525} If $d$ is prime then $\br{\Delta_x:\Delta_\Phi}=1$.
\item\label{eq1526} If $\mb{W}_\Phi''=\mb{W}_\Phi'$ then
$\br{\Delta_x:\Delta_\Phi}=1$.
\item\label{eq1046} Under assumption~\eqref{eq1525} or \eqref{eq1526}, there are at least $\av{\mb{W}_\Phi''}$ distinct visits to $X^{\ge\delta_1}$.
\end{enumerate}
\end{theorem}
\begin{remark}
We note that the assumption in~\eqref{eq1526} yields the best results, namely, we both get that $\Phi$ generates $\Delta_x$ and that
the orbit $Ax$ is as complex it could be. We point out two ways to verify this assumption in practice:
\begin{enumerate}
\item If the lattice $x$ belongs to $\Omega_M(\epsilon,C)$ and the times $\mb{s}_\tau$ from Definition~\ref{omega epsilon c} are assumed
to be bounded then it follows that for any $\tau\in\mb{W}_\Phi$, $a(\mb{w}_\tau)x\in X^{\ge\delta_1}$ for some fixed $\delta_1$. This
is what happens in the construction discussed in~\S\ref{construction section} below.
\item Sometimes one can exploit symmetries of $x$ to bootstrap the existence of one $\mb{w}_\tau$ such that $a(\mb{w}_\tau)x\in X^{\ge\delta_1}$ and upgrade it into the same statement for all $\mb{w}_\tau$. Such symmetries might exist if one assumes $x$
arises from a Galois extension.
\end{enumerate}
\end{remark}
\begin{proof}
Let $x\in \Omega_{M,K}$ and $\Phi$ an $M$-tight simplex set for $x$.
We use the notation of Proposition~\ref{eomprop} fixing some $\kappa\in(0,1)$.
Consider a time $\mb{t}$ such that $a(\mb{t})x\in X^{\ge\delta_0}$. By Proposition \ref{eomprop}\eqref{eq1651},
(and the definition of $\mb{W}_\Phi''$), for all but finitely many $x\in\Omega_M$
we have that
\eqlabel{eq1514}{
\mb{t}+\Delta_x\subset \mb{W}''_\Phi+ B_{r}+\Delta_\Phi,
}
where $r\asymp \av{\Delta_x}^{\frac{\kappa}{n}}$ is given in~\eqref{functions}.
Statement~\eqref{eq1524} will follow from Lemma~\ref{index lemma}, once
we prove that for any $\tau$, the coset $\mb{t}+\Delta_x$ can contain at most one point in each ball $\mb{w}_\tau+B_r+\mb{v}$ (for $\mb{v}\in\Delta_\Phi$).
Assume such a ball contains two points of a coset of $\Delta_x$. Then $\Delta_x$ contains a non-trivial vector of size $\le 2r$. The assumption
that the shape of $\Delta_\Phi$ is in $K$ implies, by Lemma~\ref{lemmadistortion}, that the distortion map $h_\Phi$ satisfies $\norm{h_\Phi}\ll_{M,K}1$, which in turn, by Lemma~\ref{dist lemma}\eqref{dist1.5}, implies that the distances between the balls composing $\mb{W}_\Phi+B_r+\Delta_\Phi$
is $\gg_{M,K} \av{\Delta_\Phi}^{\frac{1}{n}}-2r$. Since the latter expression is $>2r$ as soon as $\av{\Delta_x}$ is big enough, we arrive at
a contradiction to the containment~\eqref{eq1514}.
We now establish~\eqref{eq1525} and \eqref{eq1526}. Let $\mb{t}$ be such that $a(\mb{t})x\in X^{\ge\delta_0}$ as above. By \eqref{eq1514}, we may assume that
\eqlabel{eq1550}{
\mb{t}\in \mb{w}_\tau+B_r,
}
for some $\mb{w}_\tau\in\mb{W}_\Phi''$. We will show that
$\mb{t}+\Delta_x\subset \mb{w}_\tau+B_r+\Delta_\Phi$ which together with the already established fact by which each ball in
the latter set contains at most one point of $\mb{t}+\Delta_x$, will imply, by Lemma~\ref{index lemma}, that $\Delta_x=\Delta_\Phi$ as claimed.
Assuming otherwise, there exists $\mb{w}_\tau\ne \mb{w}_\sigma\in\mb{W}_\Phi''$ such that $(\mb{t}+\Delta_x)\cap\pa{\mb{w}_\sigma+B_r}\ne\varnothing$, which together
with~\eqref{eq1550} implies the existence of $\mb{s}\in B_{2r}$ such that
\eqlabel{eq1157}{
\mb{v}\overset{\on{def}}{=}\mb{w}_\tau-\mb{w}_\sigma+\mb{s}\in\Delta_x.
}
To establish~\eqref{eq1525}, assume $d$ is prime and conclude from Lemma~\ref{reduction lemma}\eqref{claim5} that there
exists an integer $1\le k\le n$ such that $\mb{w}_\tau+k(\mb{w}_\tau-\mb{w}_\sigma)\notin \mb{W}_\Phi+\Delta_\Phi.$ As these objects are obtained from the corresponding ones for $\Phi_*$ by applying a distortion map $h_\Phi$ followed by a dilation by a factor of
$\av{\Delta_\Phi}^{\frac{1}{n}}$, we conclude by Lemma~\ref{lemmadistortion}, Lemma~\ref{dist lemma}\eqref{dist1.5}, that
$$\on{d}( \mb{w}_\tau+k(\mb{w}_\tau-\mb{w}_\sigma), \mb{W}_\Phi+\Delta_\Phi)\gg_{M,K} \av{\Delta_x}^{\frac{1}{n}}.$$
Since
$\mb{w}_\tau+k(\mb{w}_\tau-\mb{w}_\sigma) = \mb{t}+k\mb{v} +\mb{s}'$ for $\mb{s}'$ with $\norm{\mb{s}'}\ll r$ we have that
$\on{d}(\mb{t}+k\mb{v},\mb{W}_\Phi+\Delta_\Phi)\gg_{M,K} \av{\Delta_x}^{\frac{1}{n}}$ which shows in particular, that for all but finitely many $x\in\Omega_{M,K}$, $\mb{t}+k\mb{v}\notin\mb{W}_\Phi+B_r+\Delta_\Phi$ and in turn, by Proposition~\ref{eomprop}\eqref{eq1651} gives that
$a(\mb{t}+k\mb{v})x\in X^{<\delta}$. We arrive at the desired contradiction because
$a(\mb{t}+k\mb{v})x=ua(\mb{t})x\in X^{\ge\delta_0}$ (for some $u\in U$).
To establish~\eqref{eq1526} we proceed similarly. By Lemma~\ref{reduction lemma}\eqref{claim5} there exists $\tau'\in\crly{S}$ and
$1\le k\le n$ such that $\mb{w}_{\tau'}+k(\mb{w}_\tau-\mb{w}_\sigma)\notin \mb{W}_\Phi+\Delta_\Phi$.
As above, this implies that
$\on{d}( \mb{w}_{\tau'}+k(\mb{w}_\tau-\mb{w}_\sigma), \mb{W}_\Phi+\Delta_\Phi)\gg_{M,K} \av{\Delta_x}^{\frac{1}{n}}.$
By assumption there exists $\mb{t}'\in \mb{w}_{\tau'}+B_{r}$ such that $a(\mb{t}')x\in X^{\ge \delta_1}$. As above, we conclude
that $\on{d}(\mb{t}'+k\mb{v},\mb{W}_\Phi+\Delta_\Phi)\gg_{M,K} \av{\Delta_x}^{\frac{1}{n}}$ which implies that
$\mb{t}'+k\mb{v}\notin \mb{W}_\Phi+B_{r}+\Delta_\Phi$ (except for finitely many possible exceptions). By Proposition~\ref{eomprop}\eqref{eq1651} we conclude that $a(\mb{t}'+k\mb{v})x\in X^{<\delta}$ which contradicts the fact that for some $u\in U$, $a(\mb{t}'+k\mb{v})x=ua(\mb{t}')x\in X^{\ge\delta_1}$ (as soon as $\av{\Delta_x}$ is big enough).
Statement~\eqref{eq1046} follows from Lemma~\ref{distinct visits lemma} and the above considerations:
For any $\mb{w}_\tau\in\mb{W}_\Phi''$ let $\mb{s}_\tau\in B_r$ be such that $a(\mb{w}_\tau+\mb{s}_\tau)x\in X^{\ge\delta_1}$.
By Lemma~\ref{distinct visits lemma}, if amongst these times are two non-distinct visit times, then $\Delta_x$ must contain
a vector as in~\eqref{eq1157} which results in a contradiction as shown above.
\end{proof}
\subsection{} We summarize the results above for convenience of reference:
\begin{corollary}\label{maincor}
Fix $M,C\ge 1$, $\epsilon>0$ and $K$ a compact set of shapes. Then the following hold:
\begin{enumerate}
\item Any sequence of (distinct) orbits $Ax_k$ of lattices $x_k\in\Omega_M$ satisfies the conclusion of Corollary~\ref{t.0901}.
\item Any sequence of (distinct) orbits $Ax_k$ of lattices $x_k\in\Omega_M(\epsilon,C)$ satisfies the conclusion of Theorem~\ref{t.0932}.
\item For any sequence of (distinct) orbits $Ax_k$ of lattices $x_k\in\Omega_M(\epsilon,C)\cap \Omega_{M,K}$, if $\Phi_k$ is an $M$-tight
simplex set for $x_k$ satisfying Definition~\ref{omega epsilon c} such that the times $\mb{s}_\tau$ appearing there are uniformly bounded, then, for any $\delta_1\in (0,\delta_0)$, and for all large $k$, $\on{(i)}$ $\Phi_k$ generates $\Delta_x$ and, $\on{(ii)}$
$Ax_k\cap X^{\ge \delta_1}$ has at least $n!$ distinct connected components.
\end{enumerate}
\end{corollary}
\section{The construction}\label{construction section}
In this section we construct infinite families of lattices satisfying the requirements of Corollary~\ref{maincor} and doing so, we
prove Theorem~\ref{t.0932}.
\subsection{The polynomials}
Fix $\eta>0$ and define
\eqlabel{Zr}{
\mathbb{Z}^d(\eta)\overset{\on{def}}{=}\set{\mb{m}\in\mathbb{Z}^d: \forall i,j, \;\frac{\av{m_i-m_j}}{\norm{\mb{m}}}\ge \eta}.
}
Thus, $\mathbb{Z}^d(\eta)$ consists of those integer vectors which are projectively bounded away from the ${d \choose 2}$ hyperplanes of equal coordinates. For $\mb{m}\in\mathbb{Z}^d(\eta)$ define
$$p_{\mb{m}}(x)\overset{\on{def}}{=}\prod_1^d(x-m_j)-1.$$
\begin{lemma}\label{basic lemma}
Fix $\eta>0$. For all but finitely many $\mb{m}\in\mathbb{Z}^d(\eta)$, the polynomial $p_{\mb{m}}$ is irreducible over $\mathbb{Q}$ and has $d$
real roots $\theta_j=\theta_j(\mb{m})$, $j=1\dots d$
which when put in the correct order satisfy $\theta_{j}= m_j+O_\eta(\norm{\mb{m}}^{-n})$.
\end{lemma}
\begin{proof}
Let $\mb{m}\in\mathbb{Z}^d(\eta)$ and let $\theta\in\mathbb{C}$ be a root of $p_{\mb{m}}$; that is,
\begin{equation}\label{productis1}
\prod_{j=1}^{d}(\theta-m_j)=1.
\end{equation}
As $\av{m_i-m_j}\ge \eta\norm{m}$, for all but finitely many $\mb{m}$,
$\theta$ can satisfy $\av{\theta-m_j}<1$ for at most one choice of $j$. In turn, because of the above
equation, this inequality must hold for exactly one such $j\overset{\on{def}}{=} j_\theta$.
It follows that for $j\ne j_\theta$ we have $\av{\theta- m_j}= \av{m_{j_\theta}-m_j}+O(1)=O_\eta(\norm{\mb{m}})$.
Going back to the equation $\av{\theta-m_{j_\theta}}=\prod_{j\ne j_\theta} \av{\theta-m_j}^{-1}$
we conclude that $\av{\theta-m_{j_\theta}}=O_\eta(\norm{\mb{m}}^{-n})$.
We claim that the map $\theta\mapsto j_\theta$ is one to one and onto from the set of complex zeros of $p_\mb{m}$
to $\set{1,\dots,d}$. Assume this is not the case and write $p_{\mb{m}}(x)=\prod_{j=1}^{d}(x-\theta_{j})$. Then
there are $j_1\ne j_2$ such that
$\ell\overset{\on{def}}{=} j_{\theta_{j_1}}=j_{\theta_{j_2}}$. We conclude that the value
\begin{equation}\label{e1010}
\av{p_{\mb{m}}(m_\ell)}=\prod_{j=1}^{d}\av{m_\ell-\theta_{j}}
\end{equation}
is a non-zero integer on the one hand, but on the other hand, the two factors in the product~\eqref{e1010} that correspond to
$j_1,j_2$ are $O_\eta(\norm{\mb{m}}^{-n})$ while all the other $n-1$ factors are $O_\eta(\norm{\mb{m}})$. We conclude that
for all but finitely many $\mb{m}$, the value in~\eqref{e1010} is $<1$ which gives a contradiction.
Thus, we may order the roots $\theta_{j}$ of $p_{\mb{m}}$ in such a way so that
\begin{align}
\label{e14351} &\av{\theta_{j}- m_j}=O_\eta(\norm{\mb{m}}^{-n}),\\
\label{e14352}\textrm{for $\ell\ne j$, }&\av{\theta_{j}-m_\ell}=O_\eta(\norm{\mb{m}}).
\end{align}
This immediately implies that (as soon as $\norm{\mb{m}}$ is large enough), all the roots are real because $p_\mb{m}$ is real so complex roots come in conjugate pairs.
A similar argument gives the irreducibility of $p_\mb{m}$: If $p_\mb{m}$ is reducible, then there must exist a proper subset $I$ of $\set{1\dots d}$ such that
$q(x)= \prod_{j\in I}(x-\theta_{j})$ is a polynomial over the integers.
We choose $\ell\in I$ and consider the value
$
\av{q(m_\ell)}=\prod_{j\in I}\av{m_\ell-\theta_{j}}.
$
This number should be a (non-zero) integer by assumption but
on the other hand,
the term in the product which corresponds to $j=\ell$ is $O_\eta(\norm{\mb{m}}^{-n})$ while all the other terms
are $O_\eta(\norm{\mb{m}})$. We derive a contradiction (as soon as $\norm{\mb{m}}$ is large enough), because there are at most $n-1$ such terms
(as the cardinality of $I$ is assumed to be $\le n$).
\end{proof}
\subsection{The lattices}
Fix $\eta>0$ and $\mb{m}\in\mathbb{Z}^d(\eta)$ with $\norm{\mb{m}}$ large enough so that Lemma~\ref{basic lemma} applies.
Fix an abstract root $\theta=\theta(\mb{m})$ of $p_\mb{m}$ and let $F_\mb{m}\overset{\on{def}}{=}\mathbb{Q}(\theta)$ be the number field generated by it. By Lemma~\ref{basic lemma}, $F_\mb{m}$ is a totally real number field of degree $d$ over $\mathbb{Q}$. By Lemma~\ref{basic lemma}, we can order
the embeddings $\sigma_1\dots \sigma_d$ of $F_\mb{m}$ into $\mathbb{R}$ in such a
way so that $\theta_{j}\overset{\on{def}}{=} \sigma_j(\theta)$ satisfies $\theta_{j}=m_j+O_\eta(\norm{\mb{m}}^{-n})$. Let
$$\Lambda_\mb{m}=\on{span}_\mathbb{Z}\set{1,\theta,\dots,\theta^n}\subset F_\mb{m}.$$
Then, $\Lambda_\mb{m}$ is a full module in $F_\mb{m}$; that is, it is a free $\mathbb{Z}$-module of rank $d$ that spans $F_\mb{m}$ over $\mathbb{Q}$ (in fact it is
the order $\mathbb{Z}[\theta]$). Let $\bm{\sigma}:F_\mb{m}\hookrightarrow\mathbb{R}^d$ denote the map
whose $j$'th coordinate is $\sigma_j$.
Then it is a well known fact that the image under $\bm{\sigma}$ of a full module in $F_\mb{m}$ is a lattice in $\mathbb{R}^d$. In particular, if $D_\mb{m}\overset{\on{def}}{=} \on{covol}(\bm{\sigma}(\Lambda_\mb{m}))$ then
$$x_\mb{m}\overset{\on{def}}{=} D_\mb{m}^{-\frac{1}{d}}\bm{\sigma}(\Lambda_\mb{m})$$
is a unimodular lattice in $\mathbb{R}^d$; i.e.\ $x_\mb{m}\in X$.
\subsection{} We now show that Corollary~\ref{maincor} applies for the lattices constructed above and by that complete
the proof of the statements in \S\ref{introduction section}.
\begin{proposition}\label{prop1238}
For any $\eta>0$ there exists $M,C>1$, $\epsilon>0$ and a compact set of shapes $K$ such that for all but finitely many
$\mb{m}\in\mathbb{Z}^d(\eta)$, the lattice $x_\mb{m}$ belongs to $\Omega_M(\epsilon,C)\cap \Omega_{M,K}$. Moreover, we exhibit an $M$-tight simplex set $\Phi_\mb{m}$
for $x_\mb{m}$ satisfying Definition~\ref{omega epsilon c} such that the times $\mb{s}_\tau$ appearing there are uniformly bounded.
\end{proposition}
\begin{proof}
Let $\mb{m}\in\mathbb{Z}^d(\eta)$ be of large enough norm so that Lemma~\ref{basic lemma} applies.
Our goal is to find a simplex set $\Phi$ for $x_\mb{m}$ that will be $M$-tight and will satisfy the requirements of $\Omega_M(\epsilon,C)$ and $\Omega_{M,K}$. We start by estimating $\ell(x_\mb{m})$.
As $\mb{1}\in\bm{\sigma}(\Lambda_\mb{m})$ we have that $v_\mb{m}\overset{\on{def}}{=} D_\mb{m}^{-\frac{1}{d}}\mb{1}\in x_\mb{m}$.
Let us calculate $D_\mb{m}$. By definition $\bm{\sigma}(\Lambda_\mb{m})$ is the lattice spanned by the columns of the (Vandermonde)
matrix $\pa{\sigma_i(\theta^{j-1})}=\pa{\theta_i^{j-1}}$;
\begin{align}
\nonumber D_\mb{m} = \det\smallmat{1&\theta_{1}&\theta_{1}^2&\dots& \theta_{1}^{n}\\
1&\theta_{2}&\theta_{2}^2&\dots&\theta_{2}^{n}\\
& &\ddots& & \\
1&\theta_d&\theta_d^2&\dots&\theta_d^{n}
}&=\prod_{i<j}(\theta_{j}-\theta_{i})
=\prod_{i<j}(m_j-m_i +O_\eta(\norm{\mb{m}}^{-n}))\\
\nonumber &=\norm{\mb{m}}^{{d\choose 2}}\prod_{i<j}(\frac{m_j-m_i}{\norm{\mb{m}}}+O_\eta(\norm{\mb{m}}^{-d}))\\
\label{eqdiscest}&=\norm{\mb{m}}^{d\choose 2}O(1).
\end{align}
We conclude that
\eqlabel{eq:lovk}{
\ell(x_\mb{m})\le \norm{v_\mb{m}}\ll \norm{\mb{m}}^{-\frac{n}{2}}.
}
We now turn to investigate $\Delta_{x_\mb{m}}$ in order to find a suitable simplex set $\Phi$ in it.
The link with the action of the diagonal group on $X$ originates from the following observation.
Let $\alpha\in F_\mb{m}$ and let $R_\alpha:F_\mb{m}\to F_\mb{m}$ denote multiplication by $\alpha$.
Then the following diagram commutes:
\begin{equation}\label{diagram1}
\xymatrix{
F_\mb{m} \ar[d]_{R_\alpha}\ar[r]^{\bm{\sigma}}&\mathbb{R}^d\ar[d]^{\diag{\bm{\sigma}(\alpha)}}\\
F_\mb{m}\ar[r]_{\bm{\sigma}}& \mathbb{R}^d
}
\end{equation}
It is clear that for any $1\le \ell\le d$, ${\omega}_\ell\Lambda_\mb{m}\subset\Lambda_\mb{m}$, where
\eqlabel{omegahat}{
\omega_\ell=\omega_\ell(\mb{m})\overset{\on{def}}{=} \theta-m_\ell.
}
In fact, we have that $\omega_\ell\Lambda_\mb{m}=\Lambda_\mb{m}$ because $\omega_\ell$ is a unit in the ring of integers. To see this,
note that~\eqref{productis1} shows that both $\omega_\ell$ and its inverse are algebraic integers.
It follows from~\eqref{diagram1} that $\diag{\bm{\sigma}(\omega_\ell)}\in UA$ stabilizes $x_\mb{m}$ and therefore, by~\eqref{deltax},
\eqlabel{eqtl}{
\mb{t}_\ell\overset{\on{def}}{=}\log(\av{\diag{\bm{\sigma}(\omega_\ell)}})\in\Delta_{x_\mb{m}}.
}
We claim that $\Phi_\mb{m}=\Phi\overset{\on{def}}{=}\set{\mb{t}_\ell}_{\ell=1}^d$ forms the desired simplex set.
The fact that $\sum_\ell \mb{t}_\ell=0$ follows from~\eqref{productis1}. From~\eqref{e14351}, \eqref{e14352} we
conclude several things: First we note
that the set $\frac{1}{\log(\norm{\mb{m}})}\Phi_\mb{m}$ converges as $\norm{\mb{m}}\to\infty$ to $\Phi_*$. This implies in particular, that
$\Phi$ spans $\mathbb{R}^d_0$ and in turn implies that it is a simplex set. Moreover, it shows that the distortion map $h_\Phi$ converges
to the identity and so there exists a compact set $K$ of shapes such that $\Delta_\Phi\in K$. Second, we conclude that
\eqlabel{eq1056}{
\xi_\Phi = \max_{1\le \ell\le d}\max \mb{t}_\ell\le \log \norm{\mb{m}}+ O_\eta(1).
}
Combining~\eqref{eq:lovk} and \eqref{eq1056} we conclude one can choose $M=M(\eta)$ such that
$\ell(x_\mb{m})\le Me^{-\frac{n}{2}\xi_\Phi}$. Summarizing, we have established that for all but finitely many $\mb{m}\in\mathbb{Z}^d(\eta)$,
$x_\mb{m}\in \Omega_{M,K}$ and that $\Phi_\mb{m}$ is an $M$-tight simplex set for $x_\mb{m}$ with $\Delta_\Phi$ in $K$.
We now turn to study the lattices $a(\mb{w}_\tau)x_\mb{m}$ for $\tau\in\crly{S}$ in order to verify Definition~\ref{omega epsilon c}.%
It would be convenient to introduce the following notation:
Given two matrices $A=(a_{ij}),B=(b_{ij})$ (say, depending on the parameter $\eta$),
we write $$A\ll^{\on{av}}_\eta B$$ to indicate that there exists a constant $c=c(\eta)>0$, such that $\av{a_{ij}}\le c\av{b_{ij}}$.
For simplicity let us assume that $\tau$ is the identity permutation and denote $\mb{w}\overset{\on{def}}{=} \mb{w}_{\on{id}}^{\Phi_\mb{m}}.$
We begin by estimating $a(\mb{w})$ and then continue to find a suitable matrix representing the lattice
$a(\mb{w})x_\mb{m}$:
By definition~\eqref{def:w},
\eq{
\mb{w}=\sum_{\ell= 1}^{d}\frac{\ell-1}{d}\cdot\mb{t}_\ell
}
and so by~\eqref{eqtl},
\begin{align}
\nonumber a(\mb{w})&=\textstyle\prod_{\ell=1}^{d}\pa{ \diag{\av{\sigma_1(\omega_\ell)},\dots,\av{\sigma_{d}(\omega_\ell)}}}^{\frac{\ell-1}{d}}\\
\nonumber &=\textstyle\diag{\av{\sigma_1( \prod_{\ell=1}^{d}\omega_\ell^{\ell-1})},\dots,\av{\sigma_{d}(\prod_{\ell=1}^{d}\omega_\ell^{\ell-1})}}^{\frac{1}{d}}\\
\label{eq2336}&\ll^{\on{av}}_\eta \norm{\mb{m}}^{-\frac{n}{2}}\diag{\norm{\mb{m}}^n, \norm{\mb{m}}^{n-1},\dots, \norm{\mb{m}}, 1},
\end{align}
where the last estimate follows from~\eqref{omegahat}, \eqref{e14351}, \eqref{e14352}.
Recall that the lattice $x_\mb{m}$ has a basis corresponding to the basis of $\Lambda_\mb{m}$ given by $\set{\theta^j}$, $j=0\dots n$.
We make the following change of base of $\Lambda_\mb{m}$
\eq{
\set{1,\theta,\dots,\theta^n}\to\set{1,\omega_1,\omega_1\omega_2,\dots,\omega_1\cdots\omega_n}.
}
The corresponding basis of $x_\mb{m}$ is given by the columns of the
matrix
\eq{
f_\mb{m}\overset{\on{def}}{=}
D_\mb{m}^{-\frac{1}{d}}\mat{
1&\sigma_1(\omega_1)&\dots&\sigma_1(\omega_1\cdots \omega_n)\\
1&\sigma_2(\omega_1)&\dots&\sigma_2(\omega_1\cdots \omega_n)\\
\vdots& & & \vdots \\
1&\sigma_{d}(\omega_1)&\dots&\sigma_{d}(\omega_1\cdots \omega_n)
}
}
Recall that by~\eqref{eqdiscest}, $D_\mb{m}^{-\frac{1}{d}}= O(\norm{\mb{m}}^{-\frac{n}{2}})$ and by~\eqref{e14351}, \eqref{e14352} that
\eq{
\sigma_i(\omega_\ell)=\Bigg\{
\begin{array}{ll}
O_\eta(\norm{\mb{m}}) & \textrm{ if }\ell\ne i;\\
O_\eta(\norm{\mb{m}}^{-n}) & \textrm{ if }\ell=i
\end{array}
\Longrightarrow\;
\sigma_i(\prod_1^{j-1} \omega_\ell)=\Bigg\{
\begin{array}{ll}
O_\eta(\norm{\mb{m}}^{j-n-1}) & \textrm{ if } i< j;\\
O_\eta(\norm{\mb{m}}^{j-1}) & \textrm{ if } i\ge j.
\end{array}
}
In other words,
\eqlabel{eq2335}{f_\mb{m}\ll^{\on{av}}_\eta
\norm{\mb{m}}^{-\frac{n}{2}}\smallmat{
1 & \norm{\mb{m}}^{-n} & \norm{\mb{m}}^{-(n-1)} & \norm{\mb{m}}^{-(n-2)} &\dots & \norm{\mb{m}}^{-1}\\
1 & \norm{\mb{m}} & \norm{\mb{m}}^{-(n-1)} & \norm{\mb{m}}^{-(n-2)} &\dots & \norm{\mb{m}}^{-1}\\
1 & \norm{\mb{m}} & \norm{\mb{m}}^2 & \norm{\mb{m}}^{-(n-2)} &\dots & \norm{\mb{m}}^{-1}\\
1 & \norm{\mb{m}} & \norm{\mb{m}}^2 & \norm{\mb{m}}^3 &\dots & \norm{\mb{m}}^{-1}\\
\vdots & \; & \vdots & \vdots &\ddots& \vdots \\
1 & \norm{\mb{m}} & \norm{\mb{m}}^2 &||\mb{m}||^3 &\dots &\norm{\mb{m}}^n
}.
}
So, by~\eqref{eq2336}, \eqref{eq2335} we see that the matrix $a(\mb{w})f_\mb{m}$ which represents $a(\mb{w})x_\mb{m}$ satisfies
\eqlabel{eq1413}{
a(\mb{w})f_\mb{m}\ll^{\on{av}}_\eta
\smallmat{
1 & \norm{\mb{m}}^{-n} & \norm{\mb{m}}^{-(n-1)} & \norm{\mb{m}}^{-(n-2)} &\dots & \norm{\mb{m}}^{-1}\\
\norm{\mb{m}}^{-1} & 1 & \norm{\mb{m}}^{-n} & \norm{\mb{m}}^{-(n-1)} &\dots & \norm{\mb{m}}^{-2}\\
\norm{\mb{m}}^{-2} & \norm{\mb{m}}^{-1} & 1 & \norm{\mb{m}}^{-n} &\dots & \norm{\mb{m}}^{-3}\\
\norm{\mb{m}}^{-3} & \norm{\mb{m}}^{-2} & \norm{\mb{m}}^{-1} & 1 &\dots & \norm{\mb{m}}^{-4}\\
\vdots & \; & \vdots & \vdots &\ddots& \vdots \\
\norm{\mb{m}}^{-n} & \norm{\mb{m}}^{-(n-1)} & \norm{\mb{m}}^{-(n-2)} & \norm{\mb{m}}^{-(n-3)} &\dots & 1
}.
}
Write $a(\mb{w})f_\mb{m}=(c_{ij})$. Working with the inequalities in~\eqref{eq1413} and the fact that $\det a(\mb{w})f_\mb{m}=1$
it is straightforward to deduce that in the brut-force decomposition
$$(c_{ij}) =
\smallmat{
\frac{c_{11}}{\av{c_{11}}}(\prod_1^d c_{\ell\ell})^{\frac{1}{d}}&\cdots& \frac{c_{1d}}{\av{c_{dd}}}(\prod_1^d c_{\ell\ell})^{\frac{1}{d}}\\
\vdots & \frac{c_{ij}}{\av{c_{jj}}}(\prod_1^d c_{\ell\ell})^{\frac{1}{d}}&\vdots\\
\frac{c_{d1}}{\av{c_{11}}}(\prod_1^d c_{\ell\ell})^{\frac{1}{d}}&\cdots& \frac{c_{dd}}{\av{c_{dd}}}(\prod_1^d c_{\ell\ell})^{\frac{1}{d}}
}
\smallmat{
\frac{\av{c_{11}}}{(\av{\prod_1^d c_{\ell\ell}})^{\frac{1}{d}}}&\cdots&0\\
\vdots&\ddots&\vdots\\
0&\cdots& \frac{\av{c_{dd}}}{(\av{\prod_1^d c_{\ell\ell}})^{\frac{1}{d}}}}$$
if we denote the matrices on the right by $g,a(\mb{s})$ respectively, then $\norm{g-I}\ll_\eta \norm{\mb{m}}^{-1}$, $\norm{\mb{s}}\ll_\eta 1$.
Combining \eqref{eq1056} and Lemma~\ref{dist lemma} we get
that $\Delta_{x_\mb{m}}\le \Delta_\Phi\ll \xi_\Phi^n\ll(\log\norm{\mb{m}})^n$ and so choosing $\epsilon, C$ appropriately we can make sure that
$\norm{g-I}\le Ce^{-\av{\Delta_x}^\epsilon}$. This concludes the proof that $x_\mb{m}\in\Omega_M(\epsilon,C)$ and hence the proof of the Proposition.
\end{proof}
\bibliographystyle{plain}
|
1,941,325,220,494 | arxiv | \section{Introduction}
\paragraph{}
After the initial discoveries of superconductivity in Fe-based superconductors, first in LaFePO with $T_c\approx4$ K \cite{Kamihara2006LaFePO} and then in LaFeAs(O$_{1-x}$F$_x$) with $T_c \approx 26$ K \cite{Kamihara2008LaFeAsFO}, $T_c$ as high as 55 K has been reported in SmFeAs(O$_{1-x}$F$_x$) \cite{Ren2008Sm1111}. These findings have triggered an intense research aimed at understanding the fundamental physics governing this new family of superconductors. Since the original discoveries, at least seven different classes of iron-based superconductors have been identified, all exhibiting unconventional physical properties \cite{Johnston2010review,Mazin2010Nature,Paglione2010review,Stewart2011RMP}. Among those, perhaps the most diverse family is based on (AE)Fe$_2$As$_2$ parent compounds (where AE=alkaline earth, denoted the ``122" system) in which electron \cite{Sefat2008FeCo122,Ni2010FeT122,Ni2010PdRh122,Canfield2010review122},
hole \cite{Rotter2008BaK122,Ni2008BaK122} and isovalent \cite{Ren2009EuAsP122,Jiang2009BaAsP122} dopants induce a superconductivity ``dome'' with maximum $T_c$ achieved at fairly high concentrations of the dopant (up to 45 \% in case of BaK122) and showing coexistence of superconductivity and long-range magnetic order on the underdoped side.
Three plus years into the intense study of the Fe-based superconductors, several experimental conclusions supported by a large number of reports can be made. Among those, relevant to our discussion of London penetration depth, are: 1) two distinct superconducting gaps of the magnitude ratio of about 2; 2) power-law behavior of thermodynamic quantities at low temperatures presumably due to pair-breaking scattering and gap anisotropy; 3) doping dependent three-dimensional gap structure possibly with nodes. For some reviews of various experiments and theories the reader is referred to
\cite{Johnston2010review,Mazin2010Nature,Paglione2010review,Stewart2011RMP,Mizuguchi2010review11,Wilson2010JPCM,Maiti2011,Canfield2010review122,Hosono2008review}. Here we focus on London penetration depth as a sensitive tool to study superconducting gap structure, albeit averaged over the Fermi surface \cite{Prozorov2006SUSTreview}.
\section{Measurements of London Penetration Depth}
\paragraph{}There are several ways to measure magnetic penetration depth in superconductors \cite{Prozorov2006SUSTreview}.
These include muon-spin rotation ($\mu$SR) \cite{Luetkens2008La1111,Williams2010FeCouSR,Sonier2010}, frequency dependent conductivity \cite{ValdesAguilar2010THzFeCo122,Wu2010IR}, magnetic force and SQUID microscopy \cite{Luan2010FeCo122Opt,Luan2011FeCoL0}, measurements of the first critical field using either global \cite{Prozorov2009physC,Song2010LiFeAsHc1} or local probes \cite{Okazaki2009Pr1111,Klein2010FeSeTe}, microwave cavity perturbation \cite{Hashimoto2009BaK122,Hashimoto2009Pr1111,Shibauchi2009review,Bobowski2010MW,Takahash2011MWFeSeTe}, mutual inductance (especially suitable for thin films) \cite{Yong2011FeCo122films} and self-oscillating tunnel-diode resonator (TDR) \cite{Prozorov2009physC,Malone2009Sm1111,Gordon2010FeCoL0vsX,Gordon2010Pairbreaking,Kim2011LiFeAsLambda,Martin2010FeNinodes,Martin2009NdLa1111,Shibauchi2009review}. Each technique has its advantages and disadvantages and only by a combination of several different measurements performed on different samples and at different experimental conditions, one can obtain a more or less objective picture. In the case of Fe-based superconductors, most of the reports agree in general, but still differ on the details. Most difficulties come from the necessity to resolve very small variations of the penetration depth at the lowest temperatures (below 1 K) while maintaining extreme temperature resolution and the ability to measure statistically sufficient number of data points over the full temperature range. In addition, it is important to look at a general picture by measuring many samples of each composition and over extended doping regime. This review is based on the results obtained with a self-oscillating tunnel-diode resonator (TDR) that has the advantage of providing stable resolution of about 1 part per 10$^9$ \cite{Prozorov2006SUSTreview}. Indeed, comparison of TDR with other techniques, especially bulk probes, such as thermal conductivity, is important to come up with a consensus regarding the gap structure in pnictide superconductors \cite{Luo2009BaK122TC,Tanatar2010FeCoTCvsX,Reid2010FeCo122TC}.
\subsection{Tunnel - diode resonator (TDR)}
\subsubsection{Frequency - domain measurements:}\label{SEC:TDR}
\paragraph{}When a non-magnetic conducting sample is inserted into a coil of a tank circuit of quality factor $Q$ and resonating at a frequency $f_{0}=1/2\pi\sqrt{L_0C}$, it causes changes in the resonant frequency and the quality factor. For a flat slab of width $2w$ and volume $V_s$ in parallel field \cite{Hardy1993YBCO}:
\begin{equation}
\frac{\Delta f}{f_0} = \frac{f_0-f\left( \alpha \right)}{f_0}=\frac{V_{s}}{2V_{c}}\left( 1-\Re\frac{\tanh \left(\alpha w\right) }{\alpha w}\right)
\end{equation}%
\begin{equation}
\Delta \left( \frac{1}{Q}\right) =\frac{V_{s}}{V_{c}}\Im\left(\frac{\tanh{\left( \alpha w\right)} }{\alpha w}\right)
\end{equation}%
\noindent where $\alpha =(1-i)/\delta$ in a normal metal and $\alpha =1/\lambda$ in a superconductor. Here, $\delta $ is the normal skin depth and $\lambda $ is the London penetration depth. $V_{c}$ is volume of the inductor. Solving explicitly, we have for a normal metal:
\begin{equation}
\frac{\Delta f}{f}=\frac{V_{s}}{2V_{c}}\left( 1-\frac{\delta }{2w}\frac{%
\sinh \frac{2w}{\delta }+\sin \frac{2w}{\delta }}{\cosh \frac{2w}{\delta }%
+\cos \frac{2w}{\delta }}\right)
\end{equation}
\noindent and, in a superconductor, we obtain the known ``infinite slab'' solution:
\begin{equation}
\frac{\Delta f}{f}=\frac{V_{s}}{2V_{c}}\left( 1-\frac{\lambda }{w}\tanh
\left( \frac{w}{\lambda }\right) \right)
\end{equation}%
\noindent which allows us to measure both superconducting penetration depth and normal - state skin depth (which gives the contact-less measure of resistivity).
In the case of a finite sample with magnetic susceptibility $\chi$, we can still write: $\Delta f =-4 \pi\chi(T)G$, where $G\simeq f_{0}V_{s}/2V_{c}\left( 1-N\right) $ is a geometric calibration constant that includes the effective demagnetization factor $N$ \cite{Prozorov2000Meissner}. Constant $G$ is measured directly by pulling the sample out of the coil \cite{Prozorov2000Meissner,Prozorov2006SUSTreview}. The susceptibility in the Meissner state can be written in terms of London penetration depth $\lambda(T)$ and normal state paramagnetic permeability (if local - moment magnetic impurities are present), $\mu(T)$, as \cite{Prozorov2000Meissner}:
\begin{equation}
4\pi\chi=\frac{\sqrt{\mu(T)} \lambda(T)}{R}\tanh{\left(\frac{\sqrt{\mu(T)} R}{ \lambda(T)}\right)}-1
\label{TDR}
\end{equation}
\noindent where $R$ is the characteristic sample dimension. For a most typical in the experiment slab geometry with $2b \geq 2a$ planar dimensions and thickness $2d$ (magnetic field excitation is along the $d-$ side), $R$ is approximately given by \cite{Prozorov2000Meissner}:
\begin{equation}
\frac{w}{R}\approx 2 \left[1+\left(1+\left(\frac{2d}{w}\right)^2 \right)\arctan{\left(\frac{w}{2d}\right)}-\frac{2d}{w} \right]
\label{R}
\end{equation}
\noindent with $w \approx ab/(a+b)$.
Unlike microwave cavity perturbation technique that requires scanning the frequency \cite{Hardy1993YBCO}, TDR is a self-oscillating resonator always locked onto its resonant frequency \cite{Degrift1975TDR}. A sample to be studied is mounted on a sapphire rod and inserted into the inductor coil of the tank circuit. Throughout the measurement the temperature of the circuit (and of the coil) is stabilized at $\pm 1$ mK. This is essential for the stability in the resonant frequency, which is resolved to about 0.01 Hz. This translates to the ability to detect changes in $\lambda(T)$ in the range of a few {\AA}ngstrom. The ac magnetic excitation field, $H_{ac}$, in the coil is about 20 mOe, which
is small enough to ensure that no vortices are created and London penetration depth is measured.
\subsubsection{Measurements of the absolute value of $\lambda(T)$:}\label{SEC:L0}
\paragraph{}The described TDR technique provides precise measurements of the \emph{variation} of the penetration depth, $\Delta\lambda(T)$, but not the absolute value for the reasons described in detail elsewhere \cite{Prozorov2000L0}. However, the TDR technique can be extended to obtain the absolute value of the penetration depth, $\lambda(T)$. The idea is to coat the entire surface of the superconductor under study with a thin film of a conventional superconductor with lower $T_c$ and a known value of $\lambda(0)$. In this work we used aluminum, which is convenient, since its $T_c^{Al} \approx 1.2$ K is quite low for most of the discussed materials, so it is possible to extrapolate to $T = 0$ and obtain $\lambda(0)$. While the Al film is superconducting it screens the magnetic field and the effective penetration depth is \cite{Gordon2010FeCoL0vsX}:
\begin{equation}
\label{LeffL0}
\lambda_{eff}(T)=\lambda_{Al}(T)\frac{\lambda(T)+\lambda_{Al}(T)\tanh{\left(\frac{t}{\lambda_{Al}(T)}\right)}}
{\lambda_{Al}(T)+\lambda(T)\tanh{\left(\frac{t}{\lambda_{Al}(T)}\right)}},
\end{equation}
\noindent However, when it becomes normal, Al layer does not cause any screening because its thickness, $t$, is much less than the normal state skin depth at the TDR operating frequency of 14 MHz, where $\delta_{Al} \approx$ 75 $\mu$m for $\rho^{Al}_0$=10 $\mu \Omega$-cm \cite{Hauser1972Al}. By measuring the frequency shift upon warming from the base temperature, $T = T_{min}$, to $T = T_c^{Al}$ where $\lambda_{Al} \rightarrow \infty$ and Eq.~(\ref{LeffL0}) gives $\lambda_{eff}(T_c^{Al}) = t+\lambda(T_c^{Al})$, we can calculate $\lambda(T_c^{Al})$. We also note that if aluminum coating is damaged (i.e., cracks and/or holes), the result will \emph{underestimate} $\lambda(0)$. This method, therefore, provides the lower boundary estimate.
\subsubsection{Out-of-plane Penetration Depth:}
\label{SEC:Lc}
\paragraph{}Finally, the TDR technique can be used to measure out-of-plane component of the penetration depth, $\lambda_c$. For the excitation field $H_{ac}\parallel c$, screening currents flow only in the $ab$-plane and $\Delta f$ is only related to the in-plane penetration depth, $\Delta\lambda_{ab}$. However, when the magnetic field is applied along the $ab$-plane ($H_{ac}\parallel ab$), screening currents flow both in the plane and between the planes, along the $c$-axis. In this case, $\Delta f^{\perp}$ contains contributions from both $\Delta\lambda_{ab}$ and $\Delta\lambda_{c}$. For a rectangular sample of thicknesses $2t$, width $2w$ and length $l$, $\Delta f^{\perp}$ is approximately given by Eq.~(\ref{eqmix})
\begin{equation}
\frac{\Delta f^{\perp}}{\Delta f^{\perp}_{0}} \approx \frac{\Delta\lambda_{ab}}{t}+\frac{\Delta\lambda_{c}}{w}=\frac{\Delta\lambda_{mix}}{R_b}
\label{eqmix}
\end{equation}
\noindent where $R_b$ is the effective dimension that takes into account finite size
effects \cite{Prozorov2000Meissner}, Eq.~\ref{R}. Knowing $\Delta\lambda_{ab}$ from the measurements with $H_{ac}\parallel c$ and the sample dimensions, one can obtain $\Delta\lambda_{c}$ from Eq.~(\ref{eqmix}). However, because $2w\geq 4\times 2t$ in most cases, $\Delta f^{\perp}$ is in general dominated by the contribution from $\Delta\lambda_{ab}$. The subtraction of $\Delta\lambda_{c}$ becomes therefore prone to large errors. The alternative, more accurate approach is to measure the sample twice \cite{Fletcher2007NbSe2}. After the first measurement with the field along the longest side $l$ ($H_{ac}\parallel l$), the sample is cut along this $l$ direction in two halves, so that the width (originally $2w$) is reduced to $w$. Since the thickness $2t$ remains the same, we can now use Eq.~(\ref{eqmix}) to calculate $\Delta\lambda_{c}$ without knowing $\Delta\lambda_{ab}$. Note that the length $l$ and width $w$ are in the crystallographic $ab-$plane, whereas the thickness $2t$ is measured along the $c-$axis. In our experiments, both approaches to estimate $\Delta \lambda_c(T)$ produced similar temperature dependence, but the former technique had a larger data scatter, as expected.
\section{London Penetration Depth and Superconducting Gap}\label{SEC:theory}
\paragraph{}In order to describe London penetration depth in multi-gap superconductors and to take into account pair-breaking scattering, we use the weak coupling Eilenberger quasi-classical formulation of the superconductivity theory that holds for a general anisotropic Fermi surface and for any gap symmetry \cite{Eilenberger1968}. This method suitable for the analysis of the experimental data is described in a few publications of one of us\cite{Kogan2002anisotropy,Kogan2009pairBreaking,Kogan2009gamma,Gordon2010Pairbreaking}; here we briefly summarize the scheme.
In the clean case the Eilenberger equations read:
\begin{eqnarray}
{\bm v} {\bm \Pi}f=2\Delta g/\hbar -2\omega f \,,
\label{eil1}\\
-{\bm v} {\bm \Pi}^*f^+=2\Delta^* g/\hbar -2\omega f^+ \,,
\label{eil2}\\
g^2=1-ff^{+}\,, \label{eil3}\\
\Delta({\bm r},{\bm v})=2\pi TN(0) \sum_{\omega >0}^{\omega_D} \Big\langle
V({\bm v},{\bm v}^{\prime\,}) f({\bm v}^{\prime},{\bm
r},\omega)\Big\rangle_{{\bm v}^{\prime\,}}, \label{eil4}\\
{\bm j}=-4\pi |e|N(0)T\,\, {\rm Im}\sum_{\omega >0}\Big\langle {\bm v}g\Big\rangle\,.
\label{eil5}
\end{eqnarray}
\noindent Here, ${\bm v}$ is the Fermi velocity, ${\bm \Pi} =\nabla +2\pi i{\bm
A}/\phi_0$, $\phi_0$ is the flux quantum. $\Delta ({\bm r})$ is the gap
function (the order parameter) which may depend on the position ${\bm k}_F$ at
the Fermi surface (or on ${\bm v}$). Eilenberger functions $f({\bm r},{\bm v},\omega),\,\, f^{+} $, and $g$ originate from Gor'kov Green's functions integrated over the energy variable near the Fermi
surface to exclude fast spatial oscillations on the scale $1/k_F$; instead
$f,g$ vary on the relevant for superconductivity scale of the
coherence length $\xi$. Functions $f, f^+$ describe the superconducting
condensate, whereas $g$ represents normal excitations. $N(0)$ is the total density of states at the Fermi level per spin. The Matsubara frequencies are defined by $\hbar\omega=\pi T(2n+1)$
with an integer $n$, and $\omega_D$ being the Debye frequency (for phonon-mediated superconductivity or a relevant energy scale for other mechanisms). The averages over the Fermi surface weighted with the local density of states
$\propto 1/|{\bm v}|$ are defined as
\begin{equation}
\Big\langle X \Big\rangle = \int \frac{d^2{\bm k}_F}{(2\pi)^3\hbar N(0)|{\bm v}|}\,
\,X\,.
\label{<>}
\end{equation}
The order parameter $\Delta$ is related to $f$ in the self-consistency
equation Eq.~(\ref{eil4}). Often, the effective coupling $V$ is assumed to be factorizable \cite{MarkowitzKadanoff1963}, $ V({\bm v},{\bm v}^{\prime\,})=V_0 \,\Omega({\bm v})\,\Omega({\bm v}^{\prime\,})$; this assumption is not always justifiable but it makes the algebra much simpler. One looks for the order parameter in the form $\Delta (
{\bm r},T;{\bm v})=\Psi ({\bm r},T)\, \Omega({\bm v})$.
Then, the self-consistency Eq.\,(\ref{eil4}) takes the form:
\begin{equation}
\Psi( {\bm r},T)=2\pi T N(0)V_0 \sum_{\omega >0}^{\omega_D} \Big\langle
\Omega({\bm v} ) f({\bm v} ,{\bm r},\omega)\Big\rangle \,.
\label{gap}
\end{equation}
The function $\Omega({\bm v})$ (or $\Omega({\bm k}_F)$), which describes the
variation of $\Delta$ along the Fermi surface, is conveniently normalized \cite{Pokrovskii1961anisoSC}:
\begin{equation}
\Big\langle \Omega^2 \Big\rangle=1\,.
\label{norm}
\end{equation}
In the absence of currents and fields ${\bm \Pi}=0$, and the Eilenberger
equations give for the uniform ground state:
\begin{equation}
f _0 =f ^+_0={\Delta_0 \over
\beta },\quad g _0 = {\hbar\omega\over
\beta } ,\quad
\beta ^2=\Delta_0 ^2+ \hbar^2\omega^2\,; \label{f_0}
\end{equation}
Note that in general, both $\Delta_0=\Psi_0(T) \Omega(\bm k_F)$ and
$\beta$ depend on the position ${\bm k}_F$ at the Fermi surface.
Instead of dealing with the effective microscopic electron-electron
interaction $V$, one can use in this formal scheme the measurable critical
temperature $T_c$ utilizing the identity
\begin{equation}
\frac{ 1}{N(0) V_0}= \ln \frac{T}{T_{c}}+2\pi T\sum_{\omega >0}^{\omega_D}
\frac{ 1}{\hbar \omega} \,,
\label{1/NV}
\end{equation}
which is equivalent to the famous relation $\Delta(0)=\pi T_{c}
e^{-\gamma}=2\hbar\omega_D
\exp(-1/N(0)V_0)$; $\gamma=0.577$ is the Euler constant. Substitute Eq.\,(\ref{1/NV}) in
Eq.\,(\ref{gap}) and replace
$\omega_D$ with infinity due to fast convergence:
\begin{equation}
\frac{\Psi }{2\pi T} \ln \frac{T_{c}}{T}= \sum_{\omega
>0}^{\infty}\left(\frac{\Psi}{\hbar\omega}-\Big\langle \Omega \, f
\Big\rangle
\right)\,.
\label{gap1}
\end{equation}
Now equation for $\Psi(T)$ reads:
\begin{equation}
\frac{1}{2\pi T} \ln \frac{T_{c}}{T}= \sum_{\omega
>0}^{\infty}\left(\frac{1}{\hbar\omega}-\Big\langle
\frac{\Omega^2}{\sqrt{\Psi^2\Omega^2+\hbar^2\omega^2}}\Big\rangle
\right)\,.
\label{gap2}
\end{equation}
\subsection{London Penetration Depth}
\paragraph{}
Within microscopic theory, the penetration of weak magnetic fields into
superconductors is treated perturbatively.
Weak supercurrents and fields leave the order parameter modulus unchanged,
but cause the condensate, i.e., $\Delta$ and the amplitudes $f$ to acquire an
overall phase $\theta({\bm r})$. Using the method of perturbations, one obtains
corrections to $f_0,g_0$ among which we need only
\begin{equation}
g _1 =i\hbar\,\frac{\Delta_0 ^ 2 }{2\beta ^3}\,
{\bm v}\cdot(\nabla\theta+ 2\pi{\bm A}/\phi_0 ) \,.
\label{g_corr}
\end{equation}
Substituting this in the general expression (\ref{eil5}) for the current
density, one obtains
the London relation between the current and the ``gauge invariant vector potential" ${\bm a}= \phi_0\nabla\theta/2\pi + {\bm A}$:
\begin{equation}
4\pi j_i/c=- (\lambda^2)_{ik}^{-1} a_k\,.
\end{equation}
Then, the general expression (\ref{eil5}) for the current gives the inverse tensor of the squared penetration depth:
\begin{equation}
(\lambda^2)_{ik}^{-1}= \frac{16\pi^2 e^2T}{
c^2}\,N(0) \sum_{\omega} \Big\langle\frac{
\Delta_0^2v_iv_k}{\beta ^{3}}\Big\rangle \,.
\label{lambda-tensor}
\end{equation}
This result holds at any temperature for clean materials with arbitrary Fermi surface and order parameter anisotropies \cite{Kogan2002anisotropy}. The temperature dependence of $\Delta_0=\Psi\Omega$ and $\beta$ should be obtained solving
Eq.\,(\ref{gap2}).
Thus, the general scheme for evaluation of $\lambda(T)$ in clean superconductors consists of two major steps: first evaluate the order
parameter $\Delta_0(T)$ in uniform zero-field state for a given gap anisotropy $\Omega({\bm v})$, then use Eq.\,(\ref{lambda-tensor}) with a proper averaging over the Fermi surface. The sum over Matsubara frequencies is rapidly convergent and is easily done numerically (except in a few limiting situations for which analytic evaluation is possible).
\subsubsection{Isotropic $\Delta$ on a general Fermi surface: }
For majority of materials with electron-phonon interaction responsible for superconductivity, the relevant phonons have frequencies on the order of $\omega_D$. One can see that in exchanging such phonons, the electron momentum transfer is of the order of $\hbar k_F$; this leads to considerable smoothing of $\Delta({\bm k}_F)$ \cite{Abrikosov1988metalsBOOK}. Indeed, in such materials $\Delta({\bm k}_F)$ is nearly constant along the Fermi surface, the strong possible anisotropy of the latter notwithstanding. In this case, commonly called ``$s-$wave", $\Delta$ is taken as a constant at the Fermi surface:
\begin{equation}
(\lambda^2)_{ik}^{-1}= \frac{8\pi e^2 N(0)\langle v_iv_k
\rangle}{ c^2}\,2\pi T\Delta^2 \sum_{\omega} \frac{1
}{\beta ^{3}} \,. \label{lambda-is}
\end{equation}
We obtain in the Ginzburg-Landau domain:
\begin{equation}
(\lambda^2)_{ik}^{-1}(T\to T_c)= 2(\lambda^2)_{ik}^{-1}(0)\,\,(1-T/T_c)\,,\qquad t=T/T_c \,.
\label{lambda-isGL}
\end{equation}
At low temperatures:
\begin{eqnarray}
\frac{\Delta (T)}{T_c} =\frac{\Delta (0)}{T_c} - \sqrt{\frac{2\pi T
\Delta (0)}{T_c^2} } \,e^{- \Delta(0) /T}
\approx 1.764 - 3.329
\sqrt{t}\, e^{-1.764/t}\,.
\label{D(T)}
\end{eqnarray}
The low-temperature behavior of the penetration depth is given by:
\begin{equation}
(\lambda^2)_{ik}^{-1}= (\lambda^2)_{ik}^{-1}(0)\,
\left(1-2\sqrt{\frac{\pi\Delta(0)
}{ T }}e^{- \Delta(0)/T} \right) \,.
\label{eq37}
\end{equation}
\subsubsection{2D $d-$wave: }
As an example of anisotropic $\Delta$, let us take a relatively simple but important case of a $d-$wave order parameter on the two-dimensional cylindrical Fermi surface: $\Omega = \Omega_0\cos 2\varphi$ where $\varphi$ is the properly chosen azimuth angle on the Fermi cylinder. With this choice, the gap nodes are at $\varphi=\pm \pi/4, \pm 3\pi/4$. The normalization Eq.(\ref{norm}) gives $\Omega_0= \sqrt{2}$, so that $\Delta_0 =\Psi\sqrt{2}\cos 2\varphi$. The order parameter at $T=0$ is now given by:
\begin{equation}
\frac{\Delta_{max}(0)}{T_c} = \frac{2\pi}{e^{ \gamma+0.5}} \approx
2.139\,.
\label{d-ratio}
\end{equation}
and at $T \ll T_c$:
\begin{equation}
\frac{\Delta_{max}(T)}{T_c} \approx 2.139 - 0.927\, t^3\,.
\label{****}
\end{equation}
After averaging over Fermi cylinder we obtain for the in-plane penetration depth:
\begin{eqnarray}
\lambda ^{-2}(T) &=& \lambda ^{-2}(0)
\left(1-\sqrt{2}\,\frac{T}{\Delta_m}\right)\,,\label{lambda-d}\\
\lambda ^{-2}(0) &=&\frac{4\pi e^2 N(0)v^2}{c^2 }\, .
\label{lambda-d0}
\end{eqnarray}
\subsection{Eilenberger two-gap Scheme: The $\gamma$ - model}
\label{SEC:gamma-model}
The full-blown microscopic approach based on the Eliashberg theory is quite involved and not easy for the data analysis \cite{Golubov1997impurities,Brinkman2002mgB2Eliashberg,Dolgov2005MgB2,Nicol2005twoGaps}. Hence, the need for a relatively simple but justifiable, self-consistent and effective scheme experimentalists could employ. The weak-coupling model is such a scheme. Over the years, the weak-coupling theory had proven to describe well multitude of superconducting phenomena. Similar to the weak coupling is the ``renormalized BCS" model \cite{Nicol2005twoGaps} that incorporates the Eliashberg corrections in the effective coupling constants. We call our approach a ``weak-coupling two-band scheme" and refer the reader to original papers where it is clarified that the applicability of the model for the analysis of the superfluid density and specific heat data is broader than the traditional weak coupling \cite{Kogan2002anisotropy,Kogan2009gamma}.
The Eilenberger approach can be used to describe self-consistently two-gap situation, in which
\begin{equation}
\Delta ({\bf k})= \Delta_{1,2}\,,\quad {\bf k}\in F_{1,2} \,,
\label{e50}
\end{equation}
where $F_1,F_2$ are two separate sheets of the Fermi surface. Denoting the densities of states on the two parts as $N_{1,2}$, we have for a quantity $X$ constant at each Fermi sheet:
\begin{equation}
\langle X \rangle = (X_1 N_1+X_2 N_2)/N(0) = n_1X_1+n_2X_2\,,
\label{norm2}
\end{equation}
where $n_{1,2}= N_{1,2}/N(0)$; clearly, $n_1+n_2=1$.
The self-consistency equation (\ref{eil4}) now takes the form:
\begin{eqnarray}
\Delta_\nu= 2\pi T\sum_{ \mu=1,2} n_\mu \lambda_{\nu\mu}
f_\mu
= \sum_{\mu} n_\mu \lambda_{\nu\mu}
\Delta_\mu \sum_{\omega }^{\omega_D}\frac{2\pi T}{\beta_\mu} ,
\label{self-cons1}
\end{eqnarray}
where $\nu=1,2$ is the band index and $\lambda_{\nu\mu} = N(0)V(\nu,\mu)$ are dimensionless effective interaction constants. Note that the notation commonly used in literature for $\lambda^{(lit)}_{\nu\mu}$ differs from ours: $\lambda^{(lit)}_{\nu\mu}=n_\mu \lambda_{\nu\mu}$.
Turning to the evaluation of $\Delta_\nu(T)$, we note that the sum over $\omega$ in Eq.\,(\ref{self-cons1}) is logarithmically divergent. To deal with this difficulty, we employ Eilenberger's idea of replacing $\hbar\omega_D$ with the measurable $T_c$. Introducing dimensionless quantities
\begin{eqnarray}
\delta_\nu= \frac{\Delta_\nu}{2\pi T} = \frac{\Delta_\nu}{ T_c}\, \frac{1}{2\pi t}\,,
\label{deltas}
\end{eqnarray}
with $t=T/T_c$, we obtain:
\begin{eqnarray}
\delta_\nu= \sum_{ \mu=1,2} n_\mu \lambda_{\nu\mu} \delta_\mu \left( \frac{1}{{\tilde\lambda}}+\ln\frac{T_c}{T}
-A_\mu\right)\,, \nonumber\\ A_\mu = \sum_{n=0}^{\infty}\left(\frac{1}{ n+1/2 }- \frac{1} {\sqrt{\delta_\mu^2+(n+1/2)^2}} \right)\,.
\label{.pdf}
\end{eqnarray}
where $\tilde\lambda$ is defined as:
\begin{eqnarray}
1.76\,T_c= 2\hbar\omega_D \exp (-1/{\tilde\lambda})\,,
\label{Tc}
\end{eqnarray}
or
\begin{eqnarray}
\frac{1}{{\tilde\lambda}}= \ln\frac{T}{T_c}+\sum_{\omega }^{\omega_D}\frac{2\pi T}{\hbar\omega}\,.
\label{Eil0}
\end{eqnarray}
In terms of $\lambda_{\nu\mu}$, $\tilde\lambda$ is expressed as:
\begin{equation}
\tilde\lambda=\frac{2n_1n_2( \lambda_{11} \lambda_{22}-\lambda_{12}^2
)}{ n_1 \lambda_{11} + n_2 \lambda_{22} - \sqrt{ (n_1 \lambda_{11} - n_2 \lambda_{22})^2 -4n_1n_2\lambda^2_{12}}}\,.
\label{S1}
\end{equation}
For the given coupling constants $\lambda_{\nu\mu}$ and densities of states $n_\nu$, this system can be solved numerically for $\delta_\nu$ and therefore provide the gaps $\Delta_\nu=2\pi T\delta_\nu(t)$. Example calculations are shown in Fig.~\ref{diffLambdas}. First graph in the top row is calculated assuming no interband coupling. Naturally, we obtain material with two different transition temperatures. The second graph shows the gaps with $\lambda_{12}=0.05$, which features a single $T_c$ and quite non-single-BCS-gap temperature dependence of the smaller gap. The single BCS gap is shown for comparison.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig1.pdf}%
\caption{Top row: self-consistent gaps calculated for the indicated $\lambda_{\nu\mu}$ in a compensated metal with $\mu = 1$. First graph is for no inter-band pairing, $\lambda_{12}=0$. The second for $\lambda_{12}=0.05$. Bottom row: corresponding partial and total superfluid densities.}
\label{diffLambdas}
\end{figure}
\paragraph{Superfluid density: }
Having formulated the way to evaluate $\Delta(T)$, we turn to the London penetration depth given for general anisotropies of the Fermi surface and of $\Delta$ by Eq.~(\ref{lambda-tensor}) \cite{Kogan2002anisotropy}. We consider here only the case of currents in the $ab$ plain of uniaxial or cubic materials with two separate Fermi surface sheets, for which the superfluid density,
$\rho=\lambda_{ab}^2(0)/\lambda_{ab}^2(T)$, is:
\begin{eqnarray}
\rho &=& \gamma {\rho _1} + \left( {1 - \gamma } \right){\rho_2}\,,
\nonumber\\
\rho _\nu &=& \delta_\nu^2\sum_{n=0}^\infty
\left[\delta_\nu^2+(n+1/2)^2\right]^{-3/2}\,,
\nonumber\\
\gamma &=& \frac{{{n_1}v_{1}^2}}{{{n_1}v_{1}^2 +
{n_2}v_{2}^2}}\,.
\label{rhogamma}
\end{eqnarray}
\noindent where $v_{\nu}^2$ are the averages of the in-plane Fermi velocities over the corresponding band.
With the discovery of two-gap superconductivity in a number of materials, including
MgB$_2$ \cite{Bouquet2001CpMgB2,Fletcher2005MgB2}, Nb$_2$Se \cite{Fletcher2007NbSe2}, V$_3$Si \cite{Nefyodov2005V3Si}, Lu$_2$Fe$_3$Si$_5$ \cite{Gordon2008Lu2Fe3Si5} and ZrB$_{12}$ \cite{Gasparov2006ZrB12} one of the most popular approaches to analyze the experimental results has been the so called ``$\alpha$-model" \cite{Bouquet2001CpMgB2}. Developed originally to renormalize a single weak-coupling BCS gap to account for strong - coupling corrections \cite{Padamsee1973StrongCoupling}, it was used to introduce two gaps, $\Delta_{1,2}$, each having a BCS temperature dependence, but different amplitudes \cite{Bouquet2001CpMgB2}. This allowed for a simple way to fit the data on the specific heat \cite{Bouquet2001CpMgB2} and the superfluid density, $\rho=x\rho_1+(1-x)\rho_2$ \cite{Fletcher2005MgB2,Prozorov2006SUSTreview}. Here, $\rho_{1,2}$ are evaluated with $\Delta_{1,2}=(\alpha_{1,2}/1.76)\Delta_{BCS}(T)$ and $x$ takes into account the relative band contributions. The fitting is usually quite good (thanks to a smooth and relatively ``featureless'' $\rho(T)$ and the parameters $\alpha$ where found to be one larger and one smaller than unity (unless they are both equal to one in the single-gap limit). Although the $\alpha$-model had played an important and timely role in providing convincing evidence for the two-gap superconductivity, it is \emph{intrinsically inconsistent} as applied to the full temperature range. The problem is that one cannot assume \textit{a priory} temperature dependencies for the gaps in the presence of however weak interband coupling (required to have single $T_c$). In unlikely situation of zero interband coupling, two gaps would have single-gap BCS-like $T-$ dependencies, but will have two different transition temperatures. The formal similarity in terms of additive partial superfluid densities prompted to name our scheme the ``$\gamma$-model". We note, however, that these models are quite different: our $\gamma$ that determines partial contributions from each band is not just a partial density of states $n_1$ of the $\alpha$-model, instead it involves the band's Fermi velocities. The gaps, $\Delta_{1,2}(T)$, are calculated self-consistently during the fitting procedure.
\begin{figure}[tbh]
\centering
\includegraphics[width=9cm]{fig2.pdf}%
\caption{\label{fig4}Symbols: superfluid density, $\rho_s(T)$ calculated with $\lambda(0)=200$ nm. Solid lines represent the fit to a two-gap $\gamma-$model, $\rho_s = \gamma \rho_1 + (1-\gamma) \rho_2$. Dashed line is a single-gap BCS solution. Upper inset: superconducting gaps, $\Delta_1(T)$ and $\Delta_2(T)$ calculated self-consistently during the fitting. Lower inset: $\Delta_1/\Delta_2$ as a function of temperature.}
\label{LiFeAs}
\end{figure}
The $\gamma$-models can be simplified for a compensated metal, such as clean stoichiometric superconductor LiFeAs \cite{Kim2011LiFeAsLambda}. To reduce the number of fitting parameters, yet capturing compensated multiband structure, we consider a simplest model of two cylindrical bands with the mass ratio, $\mu = m_1/m_2$, whence the partial density of states of the first band, $n_1=\mu/(1+\mu)$. The total superfluid density is $\rho_s = \gamma \rho_1 + (1-\gamma) \rho_2$ with $\gamma=1/(1+\mu)$. We also use the Debye temperature of 240 K \cite{Wei2010CpMultigap} to calculate $T_c$, which allows fixing one of the in-band pairing potentials, $\lambda_{11}$. This leaves three free fit parameters: the second in-band potential, $\lambda_{22}$, inter-band coupling, $\lambda_{12}$, and the mass ratio, $\mu$. Indeed, we found that $\rho_s(T)$ can be well described in the entire temperature range by this clean-limit weak-coupling BCS model \cite{Kim2011LiFeAsLambda}. Figure \ref{LiFeAs} shows the fit of the experimental superfluid density to the $\gamma-$model. The inserts show temperature - dependent superconducting gaps obtained as a solution of the self-consistency equation, Eq.~(\ref{self-cons1}) and the lower inset show the gap ration as a function of temperature. Evidently, the smaller gap does not exhibit a BCS temperature dependence emphasizing the failure of the commonly used $\alpha-$model.
Lastly we note that in order to have two distinct gaps as observed in many experiments \cite{Johnston2010review} one has to have significant in-band coupling constants, $\lambda_{11}$ and $\lambda_{11}$. To support this conclusion we calculate the gap ratio $\gamma_{\Delta}=\Delta_1(0)/\Delta_2(0)$ for fixed $\lambda_{12}$ and varying $\lambda_{11}$ and $\lambda_{22}$. Red lines in each graph show $\gamma_{\Delta}=2$. For all cases, one needs significant $\lambda_{12}$ and varying $\lambda_{11}$ to reach this gap ratio. Therefore, the original simplified $s_{\pm}$ model with two identical Fermi surfaces and only interband coupling, $\lambda_{12} \neq 0$ and $\lambda_{11}=\lambda_{22}=0$ does not describe the experimentally found two distinct gaps in Fe-based superconductors \cite{Mazin2008spm}.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig3.pdf}%
\caption{Gap ratio, $\gamma_{\Delta}=\Delta_1(0)/\Delta_2(0)$ for indicated $\mu$ (to check the effect of different partial densities of states) and $\lambda_{12}$. The magnitude of $\gamma_{\Delta}$ is given by the intensity shown on the right. Red lines in each graph show $\gamma_{\Delta}=2$. }
\label{GapRatio}
\end{figure}
\section{Effects of Scattering}\label{SEC:pairbreaking}
\paragraph{}The scattering by impurities strongly affects the London penetration depth even in the simplest case of non-magnetic impurities in materials with isotropic gap parameter and for scattering processes which can be characterized by the scalar (isotropic) scattering rate. The $\Delta$ anisotropy and magnetic impurities complicate the analysis of
$\lambda(T)$, among other reasons due to the $T_c$ suppression in these
cases. For extended treatment the reader is referred to \cite{Kogan2009pairBreaking,Gordon2010Pairbreaking} and here we only summarize properties related to the discussion of our results. Introducing the non-magnetic scattering rate, $1/\tau$, and magnetic scattering rate (spin - flip), $1/\tau_m$, London penetration depth is expressed as:
\begin{equation}
(\lambda^2)_{ik}^{-1}= \frac{16\pi^2 e^2TN(0)
\langle v_iv_k\rangle}{
c^2}\, \sum_{\omega} \frac{f_0^2
}{\Delta/f_0+\hbar/2\tau^-} \,. \label{lambda-m}
\end{equation}
\noindent here $1/\tau^{\pm} = 1/\tau \pm 1/\tau_m$. Clearly, the evaluation of $\lambda(T)$ in the presence of magnetic impurities is quite involved. There is one limit, however, for which we have a simple analytic answer:
\paragraph{Gapless limit:}
This is the case when $\tau_m$ is close to $ 2\hbar/\Delta_0(0)$, the critical value for which $T_c=0$, i.e. $\tau_m\Delta \ll 1$.
The resulting expression for the order parameter is remarkably simple \cite{AbrikosovGorkov1960}:
\begin{equation}
\Delta^2 = 2\pi^2(T_c^2-T^2)
\label{AG_Delta}
\end{equation}
The result for superfluid density is valid in the entire temperature domain, $T<T_c$.
\begin{equation}
(\lambda^2)_{ik}^{-1}= \frac{8\pi e^2N(0)
\langle v_iv_k\rangle}{
c^2 (\rho^-)^2}\left( \ln\frac{2\rho_m}{\rho^+}+\frac{\rho^-}{2\rho_m
}\right)(1-t^2). \label{lambda-gapless}
\end{equation}
\noindent where $\rho=\hbar/(2\pi T_c \tau)$, $\rho_m=\hbar/(2\pi T_c \tau_m)$ and $\rho^{\pm}=\rho \pm \rho_m$. For a short transport mean-free path $\rho\gg\rho_m$ we have
Abrikosov-Gor'kov's result:
\begin{equation}
(\lambda^2)_{ik}^{-1}= \frac{8\pi^3 e^2N(0)
\langle v_iv_k\rangle}{
c^2 \rho\rho_m}\,(1-t^2). \label{lambda-gaplessAG}
\end{equation}
The idea of strong pair-breaking and, perhaps, gapless superconductivity finds experimental evidence in Fe-based superconductors in form of scaling relations for the specific heat jump and a pre-factor of the quadratic temperature variation of $\lambda(T)$ \cite{Kogan2009pairBreaking,Gordon2010Pairbreaking}.
To summarize, in the case of superconductor with line nodes, impurity scattering will change linear temperature dependence of the penetration depth at $T \ll T_c$ to become quadratic \cite{Hirschfeld1993dwave}. Disorder will also lift the c-axis line nodes in case of extended $s-$wave and induce a change from effective $T^2$ to exponentially activated behavior for the in-plane penetration depth \cite{Mishra2009nodesLifting}. However, if we start in the clean limit of a fully - gapped superconductor and introduce pair-breaking scattering (either due to magnetic impurities or due to unconventional gap structure, such as $s_{\pm}$) \cite{Mazin2008spm}, penetration depth will also exhibit a power-law behavior approaching $T^2$ variation in the gapless limit. Therefore, with increasing impurity scattering, if $\lambda(T) \sim T^n$, we expect the exponent $n$ to change from 1 to 2 in case of nodal superconductor (or even from 1 to $\sim\exp$ in case of non symmetry - imposed nodes) and from $\sim\exp$ to 2 in case of fully-gapped material with pair-breaking. This is schematically illustrated by shaded areas in Fig.~\ref{n_vs_Tc} (where we used large values of the exponent $n$ to designate $\sim\exp$ behavior).
\section{Experimental results}\label{SEC:DATA}
\paragraph{}
Ba(Fe$_{1-x}$T$_x$)$_2$As$_2$ is one of the most studied systems among all Fe-based superconductors and we have collected extensive data that illustrate general features often common to many other members of the diverse pnictide family. Here we focus on the electron doped Ba(Fe$_{1-x}T_x$)$_2$As$_2$ with $T=$ Co and Ni. One reason why these series were chosen is because large, high quality single crystals are available \cite{Canfield2010review122}.
All samples were grown from the self-flux and were extensively characterized by transport, structural, thermal and magneto-optical analysis. They all exhibited uniform superconductivity at least at the 1 $\mu m$ scale and dozens of samples were screened before entering into the resulting discussion \cite{Ni2008FeCoHc2,Canfield2010review122}. To demonstrate sample quality we show magneto-optical images in Fig.~\ref{FeCo122MO1} and Fig.~\ref{FeCo122MO2}. Details of this visualization technique are described elsewhere \cite{Prozorov2009vortices}. In the images intensity is proportional to the local magnetic induction. All samples show excellent Meissner screening \cite{Prozorov2009vortices}. Fig.~\ref{FeCo122MO1} shows penetration of the magnetic field into the optimally doped sample at 20 K. A distinct "Bean oblique wedge" shape \cite{Bean1964} of the penetrating flux with the current turn angle of 45$^o$ (implying isotropic in-plane current density) is observed.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig4.pdf}%
\caption{Magnetic flux penetration at 20 K into a crystal with x=0.07. The last frame shows a schematic overlay of the expected "Bean oblique wedge" shape with isotropic in-plane current density.}
\label{FeCo122MO1}
\end{figure}
Furthermore, to look for possible mesoscopic faults and inhomogeneities, we show the trapped magnetic flux obtained after cooling in a 1.5 kOe magnetic field and turning field off. The vortex distribution is quite homogeneous indicating robust uniform superconductivity for various doping levels. This is shown in Fig.~\ref{FeCo122MO1} for four different doping levels.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig5.pdf}%
\caption{Magnetic flux trapped in samples after cooling in a 1.5 kOe magnetic field to 5 K and turning field off. Doping levels are indicated in right top corners.}
\label{FeCo122MO2}
\end{figure}
The parent compound, BaFe$_2$As$_2$ is a poor metal \cite{Tanatar2009parent122} having a high temperature tetragonal phase with no long range magnetic order and undergoes structural and magnetic transitions around 140 K into a low temperature orthorhombic phase with long range antiferromagnetic spin density wave order \cite{Canfield2010review122}. Transition metal doping onto the iron site serves to suppress the structural and magnetic transition temperatures and superconductivity emerges after magnetism has sufficiently been weakened. Doping into barium site with potassium results in hole doped superconductivity. Properties of this hole - doped system, at least as far as penetration depth is concerned, are quite similar to the electron - doped FeT122 \cite{Hashimoto2009BaK122,Martin2009BaK}. On the other hand, properties materials obtained by isovalent doping of phosphorus into the arsenic site seems to induce superconductivity without introducing significant scattering and seems to result in a superconducting gap with line nodes \cite{Hashimoto2010AsPnodesPD}.
In the charge - doped systems, angle-resolved spectroscopy (ARPES) \cite{Ding2008ARPESBaK122,Evtushinsky2009ARPESBaK,Liu2009ARPES,Xu2011BaK3DgapARPES}
and thermal conductivity \cite{Reid2010FeCo122TC,Tanatar2010FeCoTCvsX} consistently show fully gapped Fermi surfaces (however with gap anisotropy increasing upon departure from optimal doping in case of thermal conductivity \cite{Tanatar2010FeCoTCvsX}.) Furthermore, the $c-$axis behavior is quite different and it is possible that a nodal state develops upon doping beyond optimal level \cite{Martin2010FeNinodes,Reid2010FeCo122TC}. The in-plane penetration depth consistently shows non-exponential power-law behavior \cite{Bobowski2010MW,Gofryk2010Cp,Gordon2010Pairbreaking,Gordon2009FeCo122vsX,Gordon2009FeCo122Opt,Hashimoto2009BaK122,Kim2010irr,Luan2010FeCo122Opt,Luan2011FeCoL0,Martin2009BaK,Martin2010SUST,Prozorov2010SUST,Prozorov2009physC,Williams2010FeCouSR,Williams2009uSRL0}, which will be discussed in detail below. It seems that such behavior can be explained by the pair-breaking scattering \cite{Bang2009,Dolgov2009,Senga2008,Vorontsov2009b,Mishra2011}. This is supported experimentally by the observed variation of the low-temperature $\lambda(T)$ within nominally the same system (and even in pieces of the same sample) \cite{Hashimoto2009BaK122} as well as deliberately introduced defects \cite{Kim2010irr}. In this review we also discuss the case of a substantial variation of $\lambda(T)$ between various samples, probably due to edge effect.
In the following analysis we use two ways to represent the power law behavior: $\lambda(T) = \lambda(0) + A(T/T_{c})^2$ at low temperatures (below 0.3 $T_c$) with $A$ being the only free parameter, because at a gross level, all samples follow the $\lambda(T) \sim T^{2}$ behavior rather well and $\lambda(T) = \lambda(0) + CT^{n}$, leaving the exponent $n$ as free parameter to analyze its evolution with doping or artificially introduced defects. In the case of vertical line nodes, we expect a variation from $n=1$ to $n=2$ upon increase of pair-breaking scattering \cite{Hirschfeld1993dwave}, but in the case of a fully gapped $s_{\pm}$ state we expect an opposite trend to approach $n=2$ in the dirty limit from clean - limit exponential behavior \cite{Bang2009,Dolgov2009,Senga2008,Vorontsov2009b}. If, however, nodes are formed predominantly along the c-axis in the extended $s-$wave scenario, the effect of scattering on the in-plane penetration depth would be opposite - starting from roughly $n=2$ in the clean limit and approaching exponential in the dirty limit \cite{Mishra2011}.
\subsection{In-plane London penetration depth}
\paragraph{}
Figure \ref{FeCoFig1} shows normalized differential TDR magnetic susceptibility of several single crystals of Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ across the superconducting ``dome''. All but one samples were grown at the same conditions and with similar starting chemicals. All, but one had thicknesses in the range of 100 - 400 nm. One of the samples ($x=0.074$, denoted batch \#2) was cleaved for the irradiation experiments and had thickness of 20 nm. We use it for comparison with the ``thick'' batch \#1 and, also, to study the effects of deliberately induced defects. It turns out that the edges of the thicker samples are not quite smooth and, when imaged in SEM, look like a used book (see Fig.~\ref{FeCoFig5}(a)). Since calibration of the TDR technique relies on the volume penetrated by the magnetic field, the thinner samples should be closer to the idealization of the sample geometry (top and bottom surfaces are always very flat and mirror-like), thus producing a more reliable calibration. On the other hand, this would only lead to a change of the amplitude (due to geometric mis-calibration) of the penetration depth variation (i.e., pre-factor $A$) and would not change its functional temperature dependence (i.e., the exponent $n$). Thinner samples, on the other hand, have better chance to be more chemically uniform, thus have reduced scattering. We observed these effects comparing samples from different batches.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig6.pdf}%
\caption{$4\pi\chi\left( T\right)$ in single crystals of Ba(Fe$_{1-x}$Co$_{x}$)$_{2}$As$_{2}$ for different $x$. Sample from batch \#1 is shown by a solid line.}
\label{FeCoFig1}
\end{figure}
Low-temperature variation of London penetration depth is presented in Fig.~\ref{FeCoFig2} as a function of $(T/T_c)^n$ obtained by fitting the data to $\Delta \lambda(T) = \lambda(0) + C(T/T_c)^n$. Each curve reveals a robust power law behavior with the exponent $n$ shown in the inset in Fig.~\ref{FeCoFig2}. The fitted exponent $n$ varies from $n = 2 \pm 0.1$ for underdoped samples to $n=2.5\pm0.1$ for the overdoped samples within the batch \#1 and reaches $n=2.83$ in batch \#2. If the superconducting density itself follows a power law with a given $n$, then $C \sim f_s(c/\omega_{p}) S$, where $f_{s}$ is the superconducting fraction at zero temperature, $c$ is the speed of light and $S$ is defined by the fraction of the Fermi surface that is gapless (which may reflect a multigap character of the superconductivity, possible nodal structure, unitary impurity scattering strength, etc) and $\omega_{p}$ is the plasma frequency.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig7.pdf}%
\caption{Low-temperature behavior of $\Delta \lambda(T)$ vs. $(T/T_c)^n$ for all
studied concentrations. Inset shows exponent $n$ as function of concentration. Sample from batch \#2 is shown for comparison.}
\label{FeCoFig2}
\end{figure}
Clearly the sample of batch \#2 shows behavior much closer to exponential compared to batch \#1. As discussed above, this could be due to the variation of scattering between the batches. We analyze low-temperature $\lambda(T)$ in Fig.~\ref{FeCoFig3}. We attempted to fit the data with three functions: the power-law with free pre-factor $C$ and exponent $n$, the standard single-gap BCS behavior, Eq.~(\ref{eq37}), with a fixed value of $\lambda(0) = 200$ nm and $\Delta(0)$ as a free parameter and to a BCS - like function where both $\lambda(0)$ and $\Delta(0)$ were free parameters. The resulting values are shown in Fig.~\ref{FeCoFig3}.
The power-law fit yields quite high exponent $n \approx 2.83$ and the best fit quality. The BCS - like fit yields a reasonable fit quality, but produces impossible values of both $\lambda(0) \approx 48$ nm and $\Delta(0)\approx 0.78T_c$ (the latter cannot be less than a weak coupling BCS value of 1.76. Finally, the fixed $\lambda(0)$ BCS fit does not agree with the data and also produces unreasonable $\Delta(0)\approx 1.25T_c$. One strong conclusion follows from this exercise - we are dealing with a multi-gap superconductor.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig8.pdf}%
\caption{Low-temperature behavior of $\Delta \lambda(T)$ for the sample batch \#2. Solid lines are the fits to three functions described in the text. Insert shows full temperature variation indicating very sharp superconducting transition.}
\label{FeCoFig3}
\end{figure}
In order to understand the validity of the empirical power-law behavior, Fig.~\ref{FeCoFig4} shows low-temperature behavior of $\Delta \lambda(T)$ vs. $(T/T_c)^{2.83}$ for the sample from batch \#2. Arrows show actual reduced temperature. Inset zooms at below of the commonly accepted ``low-temperature limit'' of $\approx T_c/3$. Clearly, power-law behavior is robust and persists down to the lowest temperature of the experiment of $T \approx 0.02 T_c$.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig9.pdf}%
\caption{Low-temperature behavior of $\Delta \lambda(T)$ vs. $(T/T_c)^{2.83}$ for the sample from batch \#2. Marks show actual reduced temperature. Insets show the behavior below commonly accepted ``low-temperature limit'' of $\approx T_c/3$.}
\label{FeCoFig4}
\end{figure}
We now summarize the observed power-law behavior of the in-plane penetration depth for the electron - doped 122 family of superconductors. Figure \ref{n_vs_Tc} shows the experimental low-temperature limit power-law exponent for different dopants on the Fe site and at different doping regimes. The shaded areas show the expectations for the pair-breaking scattering effects in $s-$ and $d-$wave scenario. It seems that statistically $d-$wave pairing (more generally - vertical line nodes) cannot explain the in-plane variation of the penetration depth. However, if the nodes appear somewhere predominantly along the c-axis, they may induce an apparent power law behavior of $\lambda_{ab}(T)$ with the effective $n \approx 2$ in the clean limit and going towards exponential behavior with the increase of the scattering rate \cite{Mishra2011}.
\begin{figure}[h]
\centering
\includegraphics[width=12cm]{fig10.pdf}%
\caption{Power-law exponent of the low-temperature variation of in-plane $\lambda_{ab}(T)$ is several electron-doped Ba122 superconductors at various doping levels. Shaded areas show the influence of pair-breaking scattering with $n=2$ being the limiting value of $n$ approaching from either the $s-$wave side (nodeless exponential) or $d-$wave side (vertical line nodes). In case of extended $s-$wave with nodes predominantly along the c-axis, effective $n$ increasing from 2 in the clean limit towards the exponential behavior when the nodes are lifted by scattering.}
\label{n_vs_Tc}
\end{figure}
\subsection{Absolute value of the penetration depth}
\paragraph{}
To further investigate the effects of doping and the difference between the batches, we use the method described in section \ref{SEC:L0}, which involves measuring the sample, coating it with a uniform layer of Al and re-measuring \cite{Prozorov2000L0,Gordon2010FeCoL0vsX}. The Al film was deposited onto each sample while it was suspended from a rotating stage by a fine wire in an argon atmosphere of a magnetron sputtering system. Film thickness was checked using a scanning electron microscope in two ways, both of which are shown in Fig.~\ref{FeCoFig5}. The first method involved breaking a coated sample after all measurements had been performed to expose its cross section. After this, it was mounted on an SEM sample holder using silver paste, shown in Fig.~\ref{FeCoFig5}(a). The images of the broken edge are shown for two different zoom levels in Fig.~\ref{FeCoFig5}(b) and (c). The second method used a focused ion beam (FIB) to make a trench on the surface of a coated sample, with the trench depth being much greater than the Al coating thickness, shown in Fig.~\ref{FeCoFig5}(d). The sample was then tilted and imaged by the SEM that is built into the FIB system, shown in Fig.~\ref{FeCoFig5}(e).
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig11.pdf}%
\caption{Scanning electron microscope images of the Al coated samples. (a) Large scale view. The broken side is on top. (b) and (c) are zoomed in on the Al film on the edge of the broken side. (d) A trench produced by a focused-ion beam (FIB). (e) Close-up view of the FIB trench showing the Al film and its thickness.}
\label{FeCoFig5}
\end{figure}
Example of the penetration depth measurements before and after coating are shown in Fig.~\ref{FeCoFig6}. Notice how small is the effect of Al coating when presented on a large scale of a full superconducting transition of the coated sample. However, TDR technique is well suited to resolve the variation due to aluminum layer \cite{Prozorov2000L0,Gordon2010FeCoL0vsX}.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig12.pdf}%
\caption{\underline{Main frame}: Full superconducting transition of an optimally doped FeCo122 crystal from batch \#1 before and after Al coating. \underline{Inset}: Zoomed in low-temperature region, $T_{min} \lesssim T \lesssim T_c^{Al}$, before (green triangles) and after (brown circles) the Al coating on the same sample. (Notice how small is the effect on a full scale of the main frame).}
\label{FeCoFig6}
\end{figure}
Obtained values of $\lambda_{ab}(0)$ are summarized in the top panel of Fig.~\ref{FeCoFig7} for doping levels, $x$, across the superconducting region of the phase diagram, shown schematically in the bottom panel of Fig.~\ref{FeCoFig7}. The size of the error bars for the $\lambda_{ab}(0)$ points was determined by considering the film thickness to be $t=100 \pm10$ nm and $\lambda_{Al}(0)=50\pm 10$ nm. The scatter in the $\lambda_{ab}(0)$ values shown in the upper panel of Fig.~\ref{FeCoFig7} has an approximately constant value of $\pm$ 0.075 $\mu$m for all values of $x$, which probably indicates that the source of the scatter is the same for all samples. For comparison, Fig.~\ref{FeCoFig7} also shows $\lambda_{ab}(0)$ obtained from $\mu$SR measurements (red stars) \cite{Williams2010FeCouSR}, the MFM technique (open stars) \cite{Luan2010FeCo122Opt,Luan2011FeCoL0}
and optical reflectivity (purple open triangles) \cite{Nakajima2010opticsL0}. Given statistical uncertainty these measurements are consistent with our results within the scatter. It may also be important to note that the $\lambda_{ab}(0)$ values from other experiments are all on the higher side of the scatter that exists within the TDR $\lambda_{ab}(0)$ data set. As discussed above, our TDR techniques give a low bound of $\lambda(0)$, consistent with this observation.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig13.pdf}%
\caption{\underline{Top panel}: The zero temperature London penetration depth, $\lambda_{ab}(0)$, as a function of the Co concentration, $x$. The three dashed blue lines are theoretical curves obtained using Eq.~(\ref{SC+SDW}) for three different values of $\lambda_{ab}(0)$ in the pure superconducting state. The solid gray line is a fit to the TDR data only of the form A+B/$x^n$. Also shown are values of $\lambda_{ab}(0)$ obtained by other experiments for comparison explained in the text. \underline{Bottom panel}: Schematic phase diagram for FeT122 system showing the coexisting region \cite{Fernandes2010unconvPairing,Nandi2010OrthoDist}.}
\label{FeCoFig7}
\end{figure}
In order to provide a more quantitative explanation for the observed increase in $\lambda_{ab}(0)$ as $x$ decreases in the underdoped region, we have considered the case of $s^{\pm}$ superconductivity coexisting with itinerant antiferromagnetism \cite{Fernandes2010unconvPairing}. For the case of particle hole symmetry (nested bands), the zero temperature value of the in-plane penetration depth in the region where the two phases coexist is given by \cite{Fernandes2010unconvPairing,Gordon2010FeCoL0vsX}:
\begin{equation}
\label{SC+SDW}
\lambda^{SC+SDW}_{ab}(0)=\lambda^0_{ab}(0)\sqrt{1+\frac{\Delta^2_{AF}}{\Delta^2_0}}
\end{equation}
\noindent where $\lambda^0_{ab}(0)$ is the value for a pure superconducting system with no magnetism present, and $\Delta_{AF}$ and $\Delta_0$ are the zero temperature values of the antiferromagnetic and superconducting gaps, respectively. Deviations from particle hole symmetry lead to a smaller increase in $\lambda^{SC+SDW}_{ab}(0)$, making the result in Eq.~(\ref{SC+SDW}) an upper estimate \cite{Fernandes2010unconvPairing}.
The three blue dashed lines shown in the top panel of Fig.~\ref{FeCoFig7}, which were produced using Eq.~(\ref{SC+SDW}), show the expected increase in $\lambda_{ab}(0)$ in the region of coexisting phases below $x\approx0.047$ by normalizing to three different values of $\lambda_{ab}(0)$ in the pure superconducting state, with those being 120 nm, 180 nm and 270 nm to account for the quite large dispersion of the experimental values. This theory does not take into account changes in the pure superconducting state, so for $x>0.047$ the dashed blue lines are horizontal. These theoretical curves were produced using parameters that agree with the phase diagram in the bottom panel of Fig.~\ref{FeCoFig7} \cite{Fernandes2010unconvPairing,Nandi2010OrthoDist}. While the exact functional form was not provided by any physical motivation and merely serves as a guide to the eye, the solid gray line in Fig.~\ref{FeCoFig7} is a fit of the TDR $\lambda_{ab}(0)$ data to a function of the form A+B/$x^n$, which does indeed show a dramatic increase of $\lambda_{ab}(0)$ in the coexistence region and also a relatively slight change in the pure superconducting phase. It should be noted that a dramatic increase in $\lambda_{ab}(0)$ below $x\approx0.047$ cannot be explained by the impurity scattering, which would only lead to relatively small corrections in $\lambda(T)$.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig14.pdf}%
\caption{\underline{Top panel}: $\lambda^{-2} \propto n_s$ vs. $T$ for underdoped, optimally doped and overdoped concentrations for batch \#1. A sample from batch \#2 is shown for comparison. \underline{Bottom panel}: normalized superfluid density, $\rho_s \equiv n_s(T)/n_s(0) = (\lambda(0)/\lambda(T))^2$.}
\label{FeCoFig8}
\end{figure}
With the experimental values of $\lambda(0)$ we can now analyze the superfluid density. In general $\lambda^{-2}(T)$ is given by Eq.~(\ref{lambda-tensor}) and depends on the averaging over the particular Fermi surface. For example, in the simplest cylindrical case, Eq.~(\ref{lambda-d}) and $\lambda^{-2}(0)=4 \pi e^2 N(0)v^2/c^2$, where $N(0)$ is the density of states at the Fermi level and $v$ is Fermi velocity. However, it is instructive to analyze the behavior from a two-fluid London theory point of view looking at the density of superconducting electrons, $n_s= (mc^2/4\pi e^2)\lambda^{-2}$ as function of temperature. The zero value, $n_s(0)$ will in general be less than total density of electrons due to pair-breaking scattering, so the magnitude of $n_s$ is useful when comparing different samples.
Figure \ref{FeCoFig8} shows the data for underdoped, optimally doped and overdoped samples from batch \#1 and also a sample from batch \#2 for comparison. For this sample \#2 $\lambda(0) = 200$ nm was used. There is a clear, but expected, asymmetry with respect to doping. Underdoped samples show quite low density, because not all electrons are participating in forming the Cooper pairs and parts of the Fermi surface are gapped by the SDW as was discussed above. The overdoped sample, $x=0.01$, despite having smaller $T_c$ than the ones with $x = 0.074$, shows the highest $n_s$. Obviously, the data scatter is significant. Therefore, the only reliable conclusion is that penetration depth increases dramatically upon entering the coexisting region. The overdoped side has to be studied more to acquire enough statistics. Furthermore, comparing two samples with $x=0.074$ from two different batches reveals an even more striking difference. Not only does the sample from batch \#2 has larger $n$, but the temperature dependence of $\rho_s$ in the full temperature range is also quite different. The pronounces convex shape (positive curvature) of $\rho_s(T)$ observed in all samples from batch \#1 at the elevated temperatures becomes much less visible in sample \#2. The bottom panel of \ref{FeCoFig8} clearly demonstrates this difference, which is hard to understand based purely on the geometrical consideration (different thicknesses). It seems that thicker samples of the batch \#1 had higher chance of being chemically inhomogeneous across the layers. On the other hand, the convex shape of $\rho_s(T)$ at elevated temperatures is a sign of the two-gap superconductivity \cite{Kogan2009gamma}, which depend sensitively on the interaction matrix, $\lambda_{\nu\mu}$, see section \ref{SEC:gamma-model} and Eq.~(\ref{self-cons1}). This feature becomes more pronounced when the interband coupling, $\lambda_{12}$, becomes smaller compared to the in-band coupling potentials, $\lambda_{11}$ and $\lambda_{22}$. If our interpretation that the difference between $x=0.074$ samples from batch \#1 and batch \#2 is due to enhanced pair-breaking scattering in \#1, this would indicate that this \emph{scattering} is primarily of inter-band character, so it disrupts the \emph{inter-band pairing}. This gives indirect leverage to the $s_{\pm}$ scenario where inter-band coupling plays the major role.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig15.pdf}%
\caption{Normalized superfluid density for sample \#2. Symbols show the data and the solid lines represent partial and total $\rho(T)$ obtained from a fit to Eq.(\ref{rhogamma}). Also shown are the clean (dashed grey lines) and dirty (dotted grey lines) single gap $s-$ and $d-$wave cases. Inset shows superconducting gaps obtained self-consistently during the fitting, see Eq.(\ref{.pdf})}.
\label{FeCoFig9}
\end{figure}
The normalized superfluid density for sample \#2 is analyzed in Fig.~\ref{FeCoFig9} by using a two-band $\gamma-$model described is section \ref{SEC:gamma-model}. Symbols show the data and the solid lines represent partial, $\rho_1(T)$ and $\rho_2(T)$, as well as total $\rho_s(T)$ obtained from the fit using Eq.(\ref{rhogamma}). The fit requires solving the self-consistent coupled ``gap'' equations, Eq.~(\ref{.pdf}), which are shown in the inset in Fig.~\ref{FeCoFig9}. Parameters of the fit are as follows: $\lambda_11 = 0.80$, $\lambda_22 = 0.49$, $\lambda_12 = 0.061$, $\gamma = 0.87$. We used a Debye temperature of 250 K \cite{Johnston2010review} to obtain the correct $T_c$ via Eq.~\ref{Tc}) that fixed $\lambda_{11}$ and gave $\tilde\lambda = 0.41$. We also assumed equal partial densities of states on the two bands, so the value of $\gamma = 0.87$ most likely comes from the difference in the $k-$ dependent Fermi velocities, but may also reflect the fact that densities of states are not equal. Indeed, presented fitting parameters should not be taken too literally. The superfluid density is calculated from the temperature - dependent gaps (inset in Fig.~\ref{FeCoFig9}) and close temperature dependencies can be obtained with quite different fitting parameters. However, the gaps fully determine the experimental $\rho_s$ and this is the main result. We find that $\Delta_1(0) = 1.883 T_c = 3.73$ meV and $\Delta_2(0) = 0.754 T_c = 1.49$ meV, which are in good agreement with specific heat \cite{Hardy2010CpFeCo,Gofryk2010Cp} and $\mu$SR penetration depth measurements \cite{Williams2009uSRL0} done on the samples of similar composition. Also shown in Fig.~\ref{FeCoFig9}) are the clean (dashed grey lines) and dirty (dotted grey lines) single gap $s-$ and $d-$wave cases. (Note that while the gap does not depend on the non-magnetic impurities in isotropic $s-$wave case (Anderson theorem), the superfluid density does.) Clearly, $\rho_s(T)$ for sample \#2 (and, of course for samples of batch \#1) does not come even close to any of these single-gap scenarios.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig16.pdf}%
\caption{In - plane, $\lambda_{ab}(T)$, and out - of - plane, $\lambda_{c}(T)$ in a single crystal of FeCo122 with $x=0.075$. Inset - temperature - dependent anisotropy, $\gamma_{\lambda} = \lambda_{c}/\lambda_{ab}$.}
\label{FeCoFig10}
\end{figure}
\subsection{Anisotropy of London penetration depths}
\paragraph{}
Let us now discuss the electromagnetic anisotropy in the superconducting state, parameterized by the ratio $\gamma_{\lambda} = \lambda_{c}/\lambda_{ab}$. To determine $\lambda_{c}$ we used method described in section \ref{SEC:Lc} and the results are presented in Fig.~\ref{FeCoFig10}. The problem is that we do not know the absolute value of $\lambda_{c}(0)$, so we could only obtain $\Delta \lambda_{c}(T)$ with the help of knowing $\Delta \lambda_{ab}(T)$, which was measured on the same crystal independently. To find the total $\lambda_{c}(T)$ we use the fact that close to $T_c$ (in the region of validity of Ginzburg-Landau theory),
we should have \cite{Kogan2002anisotropy}:
\begin{equation}
\gamma_\lambda \left( T_c \right) = \sqrt{\gamma_{\rho}\left(T_c \right)}
\end{equation}
\noindent where anisotropy of normal state resistivity, $\gamma_{\rho} = \rho_c/\rho_{ab}$ is taken right above $T_c$. With $\gamma \rho \approx 4 \pm 1$ \cite{Tanatar2009anisotropyRHO}, so that $\gamma_\lambda \left( {T_c} \right) \approx 2$. The results is show in the inset in Fig.~\ref{FeCoFig10}. Of course, there is some ambiguity in determining the exact anisotropy value, but the qualitative behavior does not change - the anisotropy \emph{increases} upon cooling. With our estimate it reaches a modest value of 5 at low temperatures, which makes pnictides very different from high-$T_c$ cuprates. This is opposite to a two-gap superconductor MgB$_2$ where $\gamma_{\lambda}$ \emph{decreases} upon cooling \cite{Kogan2004MgB2,Fletcher2005MgB2}, which may be due to different dimensionality of the Fermi sheets. In any case, temperature - dependent $\gamma_{\lambda}$ can only arise in the case of a multi-gap superconductor.
Next we examine the anisotropy of $\lambda(T)$ at different doping levels. This study was performed on FeNi122 samples and is reported in detail elsewhere \cite{Martin2010FeNinodes}. Figure~\ref{FeNi122c1}(a) summarizes the $T(x)$ phase diagram showing structural/magnetic ($T_{sm}$) and superconducting ($T_{c}$) transitions. The inset shows TDR measurements in a full temperature range for all concentrations used in this study. Figure~\ref{FeNi122c1}(b) shows the low-temperature ($T\leq 0.3T_{c}$) behavior of $\lambda_{ab}(T)$ for several Ni concentrations. The data plotted versus $(T/T_c)^2$ are linear for underdoped compositions and show a clear deviation towards a smaller power-law exponent (below temperatures marked by arrows in Fig.~\ref{FeNi122c1}(b)) for overdoped samples. While at moderate doping levels the results are fully consistent with our previous measurements in FeCo-122 \cite{Gordon2009FeCo122vsX,Gordon2009FeCo122Opt}, the behavior in the overdoped samples is clearly less quadratic. It should be noticed that in order to suppress $T_c$ by the same amount, one needs a two times lower Ni concentration compared to Co. In previously FeCo-122 \cite{Gordon2009FeCo122vsX}, the samples never reached highly overdoped compositions equivalent to $x=0.072$ of Ni shown in Fig.~\ref{FeNi122c1}(b). Therefore, Ni doping has the advantage of spanning the phase diagram with smaller concentrations of dopant ions, which may act as the scattering centers. The evolution of the exponent $n$ with $x$ is summarized in the lower inset in Fig.~\ref{FeNi122c1}.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig17.pdf}%
\caption{(a) Temperature-doping phase diagram of Ba(Fe$_{1-x}$Ni$_x$)$_2$As$_2$. The inset shows full-temperature range superconducting transitions. (b) $\Delta \lambda _{ab} (T)$ for different doping levels vs. $(T/T_{c})^{2}$. The arrows mark the temperature below which $n$ becomes less than 2. The curves are shifted vertically for clarity. The inset shows the power-law exponent $n(x)$ obtained by fitting to $\lambda _{ab} (T)= a+bT^n$ from the base temperature up to $T/T_c$=0.3.}
\label{FeNi122c1}
\end{figure}
Now we apply a technique described in Sec.~\ref{SEC:Lc} to estimate $\lambda_{c}(T)$. Figure~\ref{FeNi122c3} shows the effective penetration depth, $\lambda_{mix}$ (see Eq.~\ref{eqmix}), for overdoped, $x$=0.072 (main panel), and underdoped, $x$=0.033 (inset), samples before (A) and after (B) cutting in half along the longest side ($l$-side) as illustrated schematically at the top of the figure. Already in the raw data, it is apparent that the overdoped sample exhibits a much smaller exponent $n$ compared to the $\lambda_{ab}(T)$, while underdoped samples show a tendency to saturate below 0.13T$_{c}$. Using Eq.~\ref{eqmix} we can now extract the true temperature dependent $\Delta \lambda_c(T)$. The result is shown in Fig.~\ref{FeNi122c4} for two different overdoped samples of the same composition, $x=0.072$ having $T_c=7.5$ K and $T_c=6.5$ K, and for an underdoped sample with $x=0.033$ having $T_c=15$ K. Since the thickness of the sample is smaller than its width, we estimate the resolution of this procedure for $\Delta\lambda _{c}$ to be about 10 nm, which is much lower than $0.2$ nm for $\Delta\lambda_{ab}$. Nevertheless, the difference between the samples is obvious. The overdoped samples show a clear linear temperature variation up to $T_c/3$, strongly suggesting nodes in the superconducting gap. The average slope is large, about $d\lambda_{c}/dT\approx 300$ nm/K indicating a significant amount of thermally excited quasiparticles. By contrast, in the underdoped sample the inter-plane penetration depth saturates indicating a fully gapped state. If fitted to the power-law the exponent in the underdoped sample $2\leq n\leq 3$, depending on the fitting range.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig18.pdf}%
\caption{Schematics of magnetic field penetration in the case of $H_{ac} \parallel l$ for the whole sample [A] and after cutting in half along length $l$ [B]. (Main panel) The mixed penetration depth $\Delta\lambda_{mix}(T)$ before [A] and after [B] cutting for the overdoped sample $x$=0.072, $T_c=7.5$ K. Inset shows similar data for the underdoped $x$=0.033, $T_c=15$ K.}
\label{FeNi122c3}
\end{figure}
Nodes, if present somewhere on the Fermi surface, affect the temperature dependence of both components of $\Delta\lambda(T)$. However, the major contribution still comes from the direction of the supercurrent flow, thus placing the nodes in the present case at or close to the poles of the Fermi surface. The nodal topologies that are consistent with our experimental results are latitudinal circular line nodes located at the finite $k_z$ wave vector or a point (or extended area) polar node with a nonlinear ($\Delta(\theta) \sim \theta^p$, $p>1$) variation of the superconducting gap with the polar angle, $\theta$. It is interesting to note a close similarity to the results of thermal conductivity measurements in overdoped FeCo122 that have reached the same conclusions - in - plane state is anisotropic, but nodeless \cite{Tanatar2010FeCoTCvsX}, whereas out of plane response is nodal \cite{Reid2010FeCo122TC}. Still, we emphasize that the apparent power-law behavior of the in-plane penetration depth, $\lambda_{ab}(T)$, in a heavily overdoped samples could be induced by the out-of-plane nodes \cite{Graser2010,Mishra2011}. To summarize, it appears that not only is the gap not universal across different pnictide families \cite{Hashimoto2010AsPnodesPD}, it is not universal even within the same family over an extended doping range. Similar conclusion has been reached for the hole-doped BaK122 pnictides \cite{Thomale2011,Kim2011BaKLambda,Reid2011BaK122lineNodes}.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig19.pdf}%
\caption{$\Delta\lambda _{c}(T)$ for the underdoped, $x=0.033$, $T_c=15$ K (UNDER), and for two overdoped, $x=0.072$, $T_c=7.5$ K and $T_c=6.5$ K (OVER), samples. Dashed lines are linear fits.}
\label{FeNi122c4}
\end{figure}
\subsection{Pair-breaking}
\paragraph{}
Although the natural variation in the scattering rates between samples provides a good hint towards importance of pair-breaking scattering, for more quantitative conclusions we need to introduce additional disorder. This is can be achieved with the help of heavy-ion irradiation. To examine the effect of irradiation, $\sim 2\times 0.5\times 0.02-0.05$ mm$^3$ single crystals were
selected and then cut into several pieces preserving the width and the thickness. We compare sets of
samples, where the samples in each set are parts of the same original large crystal and had identical temperature-dependent penetration depth in unirradiated state. (These samples is what we call batch \#2 in this review with unirradiated reference piece appearing in the discussion and figures of the previous sections). Irradiation with 1.4 GeV $^{208}$Pb$^{56+}$ ions was performed at the Argonne Tandem Linear Accelerator System (ATLAS) with an
ion flux of $\sim 5\times 10^{11}$ ions$\cdot$s$^{-1}$$\cdot$m$^{-2}$. The actual total dose was
recorded in each run. Such irradiation usually produces columnar defects or the elongated pockets of disturbed superconductivity along the ions propagation direction. The density of defects, $d$, per unit area is usually expressed in terms of so-called ``matching field'', $B_\phi=\Phi_0 d$, which is obtained assuming one flux quanta,
$\Phi_0\approx 2.07\times 10^{-7}$ G$\cdot$cm$^2$ per ion track. Here we studied samples with
$B_\phi=0.5$, 1.0 and 2.0 T corresponding to $d=2.4\times 10^{10}$ cm$^{-2}$, $4.8\times 10^{10}$
cm$^{-2}$ and $9.7\times 10^{10}$ cm$^{-2}$. The sample thickness was chosen in the range of
$\sim 20 -50 \mu$m to be smaller than the ion penetration depth, $60 - 70~\mu$m. The same samples were studied by
magneto-optical imaging. The strong Meissner screening and large uniform enhancement of pinning have
shown that the irradiation has produced uniformly distributed defects \cite{Prozorov2010irr}.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig20.pdf
\caption{Detailed comparison of the functional form of $\Delta\lambda(T)$ for irradiated FeCo122
and FeNi122. In the main panels $\Delta\lambda(T)$ is plotted vs. $(t=T/T_c)^{n_0}$ with the
exponents $n$ taken from the best fits of \emph{unirradiated} samples: $n_0 =$ 2.8 and 2.5 for
FeCo122 and FeNi122, respectively. Apparently, irradiation causes
low-temperature deviations, which are better seen in the derivatives, $d \Delta\lambda(t)/dt^{n_0}$,
plotted in the insets.}
\label{FeCo122irr2}
\end{figure}
Indeed, to see the effect, we need to start with the best (largest exponent) sample we have. We irradiated the sample designated as batch \#2 with $n=2.83$, which was discussed in detail above (see Figures \ref{FeCoFig3}, \ref{FeCoFig4} and \ref{FeCoFig9}).
To analyze the power-law behavior and its variation with irradiation, we plot $\Delta\lambda$ as
a function of $(t=T/T_c)^{n_0}$ in Fig.~\ref{FeCo122irr2}, where the $n_0$ values for FeCo122 and FeNi122 were
chosen from the best power-law fits of the unirradiated samples (see Fig.~\ref{FeCo122irr3}). While the data
for unirradiated samples appear as almost perfect straight lines showing robust power-law behavior, the
curves for irradiated samples show downturns at low temperatures indicating smaller exponents. This
observation, emphasized by the plots of the derivatives $d\Delta\lambda(t)/dt^{n_0}$ in the inset of
Fig.~\ref{FeCo122irr2}, points to a significant change in the low-energy excitations with radiation.
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig21.pdf
\caption{Top panel: The suppression of $T_c$ with disorder relative to
\emph{unirradiated} $T^{un}_c$. The vertical bars denote the width of the transition corresponding the diamagnetic signal change
from 90\% (onset) to 20\% (end). Symbols are shown at the mean values between onset and end of the transition. Lower panel: exponent $n$ vs. $B_{\phi}$.}
\label{FeCo122irr3}
\end{figure}
The variations of $T_c$ and $n$ upon irradiation are illustrated in
Fig.~\ref{FeCo122irr3}. Dashed lines and circles show FeCo122, while solid lines and
triangles show FeNi122. The upper panel shows the variation of $T_c$ and the width of
the transition. Since $B_{\phi}$ is directly proportional to the area density of the ions, $d$, we can say that $T_c$ decreases roughly linearly with $d$. The same trend is evident for the exponent $n$ shown in the lower panel of Fig.~\ref{FeCo122irr3}.
The influence of impurities, assuming $s_{\pm}$ pairing, has been analyzed numerically in a T-matrix approximation \cite{Kim2010irr}. Figure \ref{FeCo122irr3}(a) shows calculated superfluid densities for different values of the scattering rate. Figure \ref{FeCo122irr3}(b) shows corresponding densities of states. Finally, Fig.~\ref{FeCo122irr3}(c) shows the central result: the correlation between $T_c$ and $n$. Note that these two quantities are obtained independently of each other. Assuming that the unirradiated samples
have some disorder due to doping, and scaling $T^{un}_c$ to lie on the theoretical curve, we find that the $T_c(B_{\phi})$ of the irradiated samples also follows this curve. The assumption of similarity between doping and radiation-induced disorder, implied in this comparison, while not unreasonable, deserves further scrutiny. More recent discussion on the variation of $T_c$ with disorder is found in Ref.~(\cite{Efremov2011TcVsDisorder}).
\begin{figure}[h]
\centering
\includegraphics[width=9cm]{fig22.pdf}%
\caption{(a) Superfluid density and (b) the density of states,
computed for the $s_{\pm}$ state with sign-changing isotropic gaps
and strong interband impurity scattering, between the Born and unitary limits.
The dashed line in (a) is an example of a power-law fit
$\rho(T)/\rho_0 = \rho(0)/\rho_0 - a (T/T_{c0})^n$ for $0<T <0.4 \, T_{c0}$;
(b) As the impurity concentration $n_{imp} \sim \Gamma$)) increases,
the band of mid-gap states approaches the Fermi level and the exponent $n$ is reduced.
(c) $T_c$ vs. power $n$, from the theoretical model (triangles)
and experiment (squares and circles).}
\label{FeCo122irr4}
\end{figure}
\section{Conclusions}
\paragraph{}
It was not possible to include in this review many interesting results obtained for various members of the diverse family of iron-based superconductors. However, we may provide some general conclusions based on our work as well as on results by others.
\begin{enumerate}
\item The superconducting gap in optimally doped pnictides is isotropic and nodeless.
\item Pair-breaking scattering changes the clean - limit low temperature asymptotics (exponential for nodeless and $T-$linear for line nodes) to a $\sim T^2$ behavior. Therefore, additional measurements (such as deliberately introduced disorder) are needed to make conclusions about the order parameter symmetry. In anisotropic superconductors in general and in $s_{\pm}$, in particular, even the non-magnetic impurities are pair-breaking.
\item The materials can be described within a self-consistent two-band $\gamma-$model with two gaps with the ratio of magnitudes of about $\Delta_1(0)/\Delta_2(0) \approx 2 - 3$.
\item Upon doping, the power-law exponent, $n$ for the in-plane penetration depth, $\lambda_ab(T)$, decreases reaching values below 2 signaling of developing significant anisotropy, whereas out-of-plane $\lambda_c(T)$ shows a linear~$-T$ behavior signaling of line nodes with Fermi velocity predominantly in the $c-$direction.
\item There is a fairly large region of coexisting superconductivity and long-range magnetic order, albeit with suppressed superfluid density.
\item Overall, the observed behavior is consistent with the $s_{\pm}$ pairing, but realistic 3D calculations are required to achieve agreement with experiments.
\end{enumerate}
\section*{Acknowledgments}
This review is based on the experimental results obtained by the members of R.P. laboratory: Makariy Tanatar, Catalin Martin, Kyuil Cho, Ryan Gordon and Hyunsoo Kim during 2008 - 2010. More details can be found in Ryan Gordon's Ph.D. thesis \cite{RyanThesis2011}. Our colleague, Makariy Tanatar, was responsible for all sample preparation and handling.
The samples were grown in the group of Paul Canfield and Sergey Bud'ko. We are grateful to many colleagues for insightful discussions - too many to be listed in the limited space of this review. This research was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under contract No. DE-AC02-07CH11358. R.P. acknowledges support from the Alfred P. Sloan Foundation.
\section*{References}
\bibliographystyle{unsrt}
|
1,941,325,220,495 | arxiv | \section{Relationship with local and global spectral graph partitioning}
\label{sxn:local-partitioning}
\vspace{-2mm}
In this section, we briefly describe the connections between our results
and global and local versions of spectral partitioning, which were a
starting point for this work.
The idea of spectral clustering is to approximate the best partition of the
vertices of a connected graph into two pieces by using the nontrivial
eigenvectors of a Laplacian (either the combinatorial or the normalized
Laplacian).
This idea has had a long history~\cite{Donath:1973,fiedler75B,spielman96_spectral,guatterymiller98,ShiMalik00_NCut}.
The simplest version of spectral clustering involves computing the first
nontrivial eigenvector (or another vector with Rayleigh quotient close to
that of the first nontrivial eigenvector) of $L$, and ``sweeping'' over
that vector.
For example,
one can take that vector, call it $x$, and, for a given threshold $K$,
define the two sets of the partition as
\vspace{-2mm}
\begin{align*}
C(x,K) &= \{ i : x_i \ge K \}, \quad \text{and} \\
\bar C(x,K) &= \{ i : x_i < K \}.
\end{align*}
Typically, this vector $x$ is computed by calling a black-box solver, but it
could also be approximated with an iteration-based method (such as the Power
Method or Lanczos Method) or a random walk-based method (such as running a
diffusive procedure or PageRank-based procedure to the asymptotic state).
Far from being a ``heuristic,''
this procedure provides a \emph{global
partition} that (via Cheeger's inequality and for an appropriate choice of
$K$) satisfies provable quality-of-approximation guarantees with respect to
the combinatorial optimization problem of finding the best ``conductance''
partition in the entire
graph~\cite{Mihail,spielman96_spectral,guatterymiller98}.
Although spectral clustering reduces a graph to a single vector---the
smallest nontrivial eigenvector of the graph's Laplacian---and then
clusters the nodes using the information in that vector,
it is possible to obtain much more information about the graph by looking
at more than one eigenvector of the Laplacian.
In particular, the elements of the pseudoinverse of the combinatorial
Laplacian, $L_0^{+}$, give local (\emph{i.e.}, node-specific) information
about random walks on the graph.
The reason is that the pseudoinverse $L_0^+$ of the Laplacian is closely
related to random walks on the graph.
See, \emph{e.g.}~\cite{chebotarev1998proximity} for details.
For example,
it is known that the
quantity $L_0^{+} (u,u) + L_0^{+} (v,v) - L_0^{+} (u,v) - L_0^{+} (v,u)$ is
proportional to the commute time, a symmetrized version of the length of
time before a random walker started at node $u$ reaches node $v$, whenever
$u$ and $v$ are in the same connected component~\cite{chandra1989electrical}.
Similarly, the elements of the pseudoinverse of the \emph{normalized}
Laplacian give degree-scaled measures of proximity between the nodes of a
graph.
It is likely that $L^{+}(u,v)$ has a probabilistic interpretation in terms
of random walks on the graph, along the lines of our methodology, but we
are not aware of any such interpretation.
From this perspective, given $L^{+}$ and a cutoff value, $K$, we can
define a \emph{local partition} around node $u$ via
$P_K(u) = \{ v : L^{+}(u,v) > K \}$.
(Note that if $v$ is in $P_K(u)$, then $u$ is in $P_K(v)$; in addition, if
the graph is disconnected, then there exists a $K$ such that $u$ and $v$
are in the same connected component iff $v \in P_K(u)$.)
We call clustering procedures based on this idea \emph{local spectral
partitioning}.
Although the na\"{i}ve way of performing this local spectral partitioning,
\emph{i.e.}, to compute $L^{+}$ explicitly, is prohibitive for anything but
very small graphs, these ideas form the basis for very fast local spectral
clustering methods that employ truncated diffusion-based procedures to
compute localized vectors with which to partition.
For example, this idea can be implemented by performing a diffusion-based
procedure with an input seed distribution vector localized on a node $u$ and
then sweeping over the resulting vector.
This idea was originally introduced in~\cite{Spielman:2004} as a
diffusion-based operational procedure that was local in a very strong sense
and that led to Cheeger-like bounds analogous to those obtained with the
usual global spectral partitioning; and this was extended and improved
by~\cite{andersen06local,Chung07_heatkernelPNAS}.
In addition, an optimization perspective on this was provided
by~\cite{MOV09_TRv2}.
Although~\cite{MOV09_TRv2} is local in a weaker sense, it does obtain local
Cheeger-like guarantees from an explicit locally-biased optimization
problem, and it provides an optimization ansatz that may be interpreted as
a ``local eigenvector.''
See~\cite{Spielman:2004,andersen06local,Chung07_heatkernelPNAS,MOV09_TRv2}
for details.
Understanding the relationship between the ``operational procedure versus
optimization ansatz'' perspectives was the origin
of~\cite{MO11-implementing} and thus of this work.
\vspace{-2mm}
\section{Heuristic justification for the Wishart density}
\label{sxn:justification}
\vspace{-2mm}
In this section, we describe a sampling procedure for $L$ which, in a very
crude sense, leads approximately to a conditional Wishart density for
$p(L \mid \mathcal{L})$.
Let $G$ be a graph with vertex set $V = \{ 1, 2, \dotsc, n \}$, edge set
$E = V \times V$ equipped with the equivalence relation $(u,v) = (v,u)$.
Let $\omega$ be an edge weight function, and let $\mathcal{L}_0$ and
$\mathcal{L}$ be the corresponding combinatorial and normalized Laplacians.
Let $\Delta$ be a diagonal matrix with
$\Delta(u,u) = \sum_{v} \omega(u,v)$, so that
$\mathcal{L} = \Delta^{-1/2} \mathcal{L}_0 \Delta^{-1/2}$. Suppose
the weights are scaled such that $\sum_{(u,v) \in E} \omega(u,v) = 1$,
and suppose further that $\Delta(u,u) > 0$.
We refer to $\omega(u,v)$ as the population weight of edge $(u,v)$.
A simple model for the sample graph is as follows: we sample $m$ edges
from $E$, randomly chosen according to the population weight function.
That is, we see edges $(u_1, v_1), (u_2, v_2), \dotsc, (u_m, v_m)$,
where the edges are all drawn independently and identically such that
the probability of seeing edge $(u,v)$ is determined by $\omega$:
\[
\mathbb{P}_\omega\{ (u_1, v_1) = (u,v) \} = \omega(u,v).
\]
Note that we will likely see duplicate edges and not every edge with a
positive weight will get sampled.
Then, we construct a weight function from the sampled edges, called the
sample weight function, $w$, defined such that
\vspace{-2mm}
\[
w(u,v) = \frac{1}{m} \sum_{i=1}^{m} 1\{ (u_i, v_i) = (u,v) \} ,
\vspace{-2mm}
\]
where $1\{\cdot\}$ is an indicator vector.
In turn, we construct a sample combinatorial Laplacian, $L_0$, defined such
that
\[
L_0(u,v)
=
\begin{cases}
\sum_{w} w(u,w) &\text{when $u = v$,} \\
-w(u,v) &\text{otherwise.}
\end{cases}
\]
Let $D$ be a diagonal matrix such that
$D(u,u) = \sum_{v} w(u,v)$, and define $L = D^{-1/2} L_0 D^{-1/2}$.
Letting $\mathbb{E}_\omega$ denote expectation with respect to the probability
law $\mathbb{P}_\omega$, note that $\mathbb{E}_\omega[w(u,v)] = \omega(u,v)$,
that $\mathbb{E}_\omega L_0 = \mathcal{L}_0$, and that $\mathbb{E}_\omega D = \Delta$.
Moreover, the strong law of large numbers guarantees that as $m$ increases,
these three quantities converge almost surely to their expectations.
Further, Slutzky's theorem guarantees that $\sqrt{m} (L - \mathcal{L})$ and
$\sqrt{m} \Delta^{-1/2} (L_0 - \mathcal{L}_0) \Delta^{-1/2}$ converge in
distribution to the same limit.
We use this large-sample behavior to
approximate the the distribution of $L$ by the distribution of
$\Delta^{-1/2} L_0 \Delta^{-1/2}$. Put simply, we treat the degrees as known.
The distribution of $L_0$ is completely determined by the edge
sampling scheme laid out above. However, the exact form for the
density involves an intractable combinatorial sum.
Thus, we appeal to a
crude approximation for the conditional density.
The approximation works as follows:
\begin{enumerate}
\item For $i = 1, \dotsc, m$, define $x_i \in \mathbb{R}^n$ such that
\[
x_i(u)
=
\begin{cases}
+s_i &\text{when $u = u_i$,} \\
-s_i &\text{when $u = v_i$,} \\
0 &\text{otherwise,}
\end{cases}
\]
where $s_i \in \{ -1, +1 \}$ is chosen arbitrarily.
Note that $L_0 = \frac{1}{m} \sum_{i=1}^m x_i x_i'$.
\item Take $s_i$ to be random, equal to $+1$ or $-1$ with probability
$\tfrac{1}{2}$. Approximate the distribution of $x_i$ by the
distribution of a multivariate normal random variable, $\tilde x_i$,
such that $x_i$ and $\tilde x_i$ have the same first and second
moments.
\item Approximate the distribution of $L_0$ by the distribution of $\tilde L_0$, where
\(
\tilde L_0 = \frac{1}{m} \sum_{i=1}^m \tilde x_i \tilde x_i'.
\)
\item Use the asymptotic expansion above to approximate the
distribution of $L$ by the distribution of
$\Delta^{-1/2} \tilde L_0 \Delta^{-1/2}$.
\end{enumerate}
\noindent
The next two lemmas derive the distribution of $\tilde x_i$ and
$\tilde L_0$ in terms of $\mathcal{L}$, allowing us to get an
approximation for $p(L \mid \mathcal{L})$.
\begin{lemma}
With $x_i$ and $\tilde x_i$ defined as above,
\[
\mathbb{E}_\omega[ x_i ] = \mathbb{E}_\omega[ \tilde x_i ] = 0,
\]
and
\[
\mathbb{E}_\omega[ x_i x_i' ] = \mathbb{E}_\omega [ \tilde x_i \tilde x_i' ] = \mathcal{L}_0.
\]
\end{lemma}
\begin{proof}
The random variable $\tilde x_i$ is defined to have the same first
and second moments as $x_i$.
The first moment vanishes since $s_i \overset{d}{=} -s_i$ implies
that $x_i \overset{d}{=} -x_i$. For the second moments, note that
when $u \neq v$,
\[
\mathbb{E}_\omega[x_i(u) \, x_i(v)]
= -s_i^2 \, \mathbb{P}_\omega\{ (u_i,v_i) = (u,v) \} = -\omega(u,v)
= \mathcal{L}_0(u,v).
\]
Likewise,
\[
\mathbb{E}_\omega[\{x_i(u)\}^2]
= \sum_{v} \mathbb{P}_\omega\{ (u_i,v_i) = (u,v) \}
= \sum_{v} \omega(u,v)
= \mathcal{L}_0(u,u). \qedhere
\]
\end{proof}
\begin{lemma}\label{L:approx-wishart}
The random matrix $\tilde L_0$ is distributed as $\tfrac{1}{m}
\mathrm{Wishart }(\mathcal{L}_0, m)$ random variable.
This distribution is supported on the set of
positive-semidefinite matrices with the same nullspace as $\mathcal{L}_0$. When
$m \geq \rank(\mathcal{L}_0)$, the distribution has a density on this space
given by
\begin{equation}\label{E:wishart-density}
f( \tilde L_0 \mid \mathcal{L}_0, m)
\propto
\frac{|\tilde L_0|^{(m - \rank(\mathcal{L}) - 1)/2}
\exp\{-\tfrac{m}{2} \Tr(\tilde L_0 \mathcal{L}_0^+) \}}
{|\mathcal{L}_0|^{m/2}}
\end{equation}
where the constant of proportionality depends only on $m$ and $n$
and where $|\cdot|$ denotes pseudodeterminant (product of nonzero
eigenvalues).
\end{lemma}
\begin{proof}
Since $m \tilde L$ is a sum of $m$ outer products of multivariate
$\mathrm{Normal}(0, \mathcal{L}_0)$, it is Wishart distributed
(by definition).
Suppose $\rank(\mathcal{L}_0) = r$ and
$U \in \mathbb{R}^{n \times r}$ is a matrix whose columns are the
eigenvectors of $\mathcal{L}_0$. Note that
$U' \tilde x_i \overset{d}{=} \mathrm{Normal}(0, U' \mathcal{L}_0 U)$,
and that $U' \mathcal{L}_0 U$ has full rank. Thus,
\(
U' \tilde L_0 U
\)
has a density over the space of $r \times r$ positive-semidefinite
matrices whenever $m \geq r$. The density of $U' \tilde L U$ is
exactly equal to $f(\tilde L_0 \mid \mathcal{L}_0, m)$,
defined above.
\end{proof}
Using the previous lemma, the random variable $\tilde L = \Delta^{-1/2} \tilde L_0
\Delta^{-1/2}$ has density
\[
f(\tilde L \mid \mathcal{L}, m)
\propto
\frac{|\Delta^{1/2} \tilde L \Delta^{1/2}|^{(m - \rank(\mathcal{L}) - 1)/2}
\exp\{-\tfrac{m}{2} \Tr(\Delta^{1/2} \tilde L \Delta^{1/2} \mathcal{L}_0^+) \}}
{|\Delta^{1/2} \mathcal{L}_0 \Delta^{1/2}|^{m/2}},
\]
where we have used that $\rank(\mathcal{L}_0) = \rank(\mathcal{L})$, and
the constant of proportionality depends on $m$, $n$,
$\rank(\mathcal{L})$, and $\Delta$.
Then, if we approximate
$| \Delta^{1/2} \tilde L \Delta^{1/2}| \approx |\Delta| |\tilde L|$ and
$\Delta^{1/2} \mathcal{L}_0^+ \Delta^{1/2} \approx \mathcal{L}^+$,
then $f$ is ``approximately'' the density of a $\tfrac{1}{m}
\mathrm{Wishart }(\mathcal{L}, m)$ random variable. These last
approximations are necessary because $\tilde L$ and $\mathcal{L}_0$
are rank-degenerate.
To conclude, we do not want to overstate the validity of this heuristic
justification.
In particular, it makes three key approximations:
\begin{enumerate}
\item the true degree matrix $\Delta$ can be
approximated by the observed degree matrix $D$;
\item the distribution of $x_i$, a sparse vector, is well approximated
$\tilde x_i$, a Gaussian (dense) vector;
\item the quantities
$| \Delta^{1/2} \tilde L \Delta^{1/2}|$
and
$\Delta^{1/2} \mathcal{L}_0^+ \Delta^{1/2}$
can be replaced with
$|\Delta| |\tilde L|$
and $\mathcal{L}^+$.
\end{enumerate}
None of these approximations hold in general, though as argued above, the
first is plausible if $m$ is large relative to $n$.
Likewise, since $\tilde L$ and $\mathcal{L}$ are nearly full rank, the third
approximation is likely not too bad.
The biggest leap of faith is the second approximation.
Note, \emph{e.g.}, that despite their first moments being equal, the
second moments of $\tilde x_i \tilde x_i'$ and $x_i x_i'$ differ.
\vspace{-2mm}
\section{Other priors and the relationship to Heat Kernel and Lazy Random Walk}
\label{sxn:other-priors}
\vspace{-2mm}
There is a straightforward generalization of Proposition~\ref{P:map-sdp} to
other priors.
In this section, we state it, and we observe connections with the Heat
Kernel and Lazy Random Walk procedures.
\begin{proposition}
\label{prop:map-generaliz}
Suppose the conditional likelihood for $L$ given $\mathcal{L}$ is as
defined in \eqref{E:density} and the prior density for $\mathcal{L}$
is of the form
\begin{equation}
p(\mathcal{L})
\propto
p(\tau)
|\Theta|^{-m/2}
\exp\{ -q(\tau) \, G(\Theta) \},
\label{eqn:prior-app}
\end{equation}
where
$\tau = \Tr(\mathcal{L}^{+})$,
$\Theta = \tau^{-1} \mathcal{L}^{+}$,
and $p$ and $q$ are functions with $q(\tau) > 0$ over the support
of the prior.
Define
$\mathcal{\hat L}$ to be the MAP estimate of $\mathcal{L}$. Then,
$[\Tr(\mathcal{\hat L}^+)]^{-1} \mathcal{\hat L}^+$ solves the
Mahoney-Orecchia regularized SDP \eqref{eqn:mo-reg-sdp}, with $G$ the same
as in the expression (\ref{eqn:prior-app}) for $p(\mathcal{L})$ and with
\[
\eta = \frac{m \hat \tau}{2 \, q(\hat \tau)},
\]
where $\hat \tau = \Tr(\mathcal{\hat L}^+)$.
\end{proposition}
The proof of this proposition is a straightforward generalization of the
proof of Proposition~\ref{P:map-sdp} and is thus omitted.
Note that we recover the result of Proposition~\ref{P:map-sdp} by setting
$G(\Theta) = - \log |\Theta|$ and $q(\tau) = \frac{m}{2} + \alpha - 1$.
In addition, by choosing $G(\cdot)$ to be the generalized entropy or the matrix
$p$-norm penalty of~\cite{MO11-implementing}, we obtain variants of the
Mahoney-Orecchia regularized SDP \eqref{eqn:mo-reg-sdp} with the
regularization term $G(\cdot)$.
By then combining Proposition~\ref{prop:map-generaliz} with their result, we
get that the MAP estimate of $\mathcal{L}$ is related to the Heat Kernel and
Lazy Random Walk procedures, respectively, in a manner analogous to what we
saw in Section~\ref{sxn:priors} with the PageRank procedure.
In both of these other cases, however, the prior $p(\mathcal{L})$ is
data-dependent in the strong sense that it explicitly depends on the number
of data points; and, in addition, the priors for these other cases do not
correspond to any well-recognizable parametric distribution.
\vspace{-2mm}
\section{Regularization performance with respect to the relative spectral error}
\label{S:spectral-performance}
\vspace{-2mm}
In this section, we present Figure~\ref{fig:perf-spec}, which shows
the regularization performance for our empirical evaluation, when the
performance criterion is the relative spectral norm error, \emph{i.e.},
$\|\Theta - \hat \Theta_\eta\|_\mathrm{2} / \|\Theta - \hat \Theta
\|_\mathrm{2}$, where $\|\cdot\|_\mathrm{2}$ denotes spectral norm of a
matrix (which is the largest singular value of that matrix).
See Section~\ref{S:estimation} for details of the setup.
Note that these results are very similar to those for the relative Frobenius
norm error that are presented in Figure~\ref{fig:perf-frob}.
\begin{figure}[h]
\centering
\subfigure[$m/\mu = 0.2$ and $s = 0$.]{
\makebox{\includegraphics[scale=0.34]{plots/estimation-spec-s0-p020}}
}
\subfigure[$m/\mu = 0.2$ and $s = 4$.]{
\makebox{\includegraphics[scale=0.34]{plots/estimation-spec-p020}}
}
\subfigure[$m/\mu = 0.2$ and $s = 32$.]{
\makebox{\includegraphics[scale=0.34]{plots/estimation-spec-s32-p020}}
} \\
\subfigure[$m/\mu = 2.0$ and $s = 0$.]{
\makebox{\includegraphics[scale=0.34]{plots/estimation-spec-s0-p200}}
}
\subfigure[$m/\mu = 2.0$ and $s = 4$.]{
\makebox{\includegraphics[scale=0.34]{plots/estimation-spec-p200}}
}
\subfigure[$m/\mu = 2.0$ and $s = 32$.]{
\makebox{\includegraphics[scale=0.34]{plots/estimation-spec-s32-p200}}
}
\vspace{-1em}
\caption{Regularization performance, as measured with the relative
spectral norm error, versus the (normalized) regularization parameter
$\eta / \bar \tau$.
Shown are plots for various values of the (normalized) number of edges,
$m/\mu$, and the edge-swap parameter, $s$.
Recall that the regularization parameter in the regularized
SDP~(\ref{eqn:mo-reg-sdp}) is $1/\eta$, and thus smaller values along the
X-axis correspond to stronger regularization.}
\label{fig:perf-spec}
\end{figure}
\section{Introduction}
\label{sxn:intro}
Approximation algorithms and heuristic approximations are commonly used to
speed up the running time of algorithms in machine learning and data
analysis.
In some cases, the outputs of these approximate procedures are ``better''
than the output of the more expensive exact algorithms, in the sense that
they lead to more robust results or more useful results for the downstream
practitioner.
Recently, Mahoney and Orecchia formalized these ideas in the context of
computing the first nontrivial eigenvector of a graph
Laplacian~\cite{MO11-implementing}.
Recall that, given a graph $G$ on $n$ nodes or equivalently its $n \times n$
Laplacian matrix $L$, the top nontrivial eigenvector of the Laplacian
\emph{exactly} optimizes the Rayleigh quotient, subject to the usual
constraints.
This optimization problem can equivalently be expressed as a vector
optimization program with the objective function $f(x) = x^TLx$,
where $x$ is an $n$-dimensional vector, or as a Semi-Definite Program (SDP)
with objective function $F(X)=\mathrm{Tr}(L X)$, where $X$ is an $n \times n$
symmetric positive semi-definite matrix.
This first nontrivial vector is, of course, of widespread interest in
applications due to its usefulness for graph partitioning, image
segmentation, data clustering, semi-supervised learning, etc.~\cite{spielman96_spectral,guatterymiller98,ShiMalik00_NCut,BN03,Joa03,LLDM09_communities_IM}.
In this context, Mahoney and Orecchia asked the question: do popular
diffusion-based procedures---such as running the Heat Kernel or performing a
Lazy Random Walk or computing the PageRank function---to compute a quick
\emph{approximation} to the first nontrivial eigenvector of $L$ solve some
other regularized version of the Rayleigh quotient objective function
\emph{exactly}?
Understanding this algorithmic-statistical tradeoff is clearly of interest
if one is interested in very large-scale applications, where performing
statistical analysis to derive an objective and then calling a black box
solver to optimize that objective exactly might be too expensive.
Mahoney and Orecchia answered the above question in the affirmative, with
the interesting twist that the regularization is on the SDP formulation
rather than the usual vector optimization problem.
That is, these three diffusion-based procedures exactly optimize a
regularized SDP with objective function $F(X)+ \tfrac{1}{\eta} G(X)$, for
some regularization function $G(\cdot)$ to be described below, subject to
the usual constraints.
\vspace{-1mm}
In this paper, we extend the Mahoney-Orecchia result by providing a
statistical interpretation of their approximation procedure.
Our interpretation will be analogous to the manner in which
$\ell_2$-regularized or $\ell_1$-regularized $\ell_2$-regression (often
called Ridge regression and Lasso regression, respectively) can be
interpreted in terms of a Gaussian prior or a Laplace prior, respectively,
on the coefficient vector of the regression problem.
In more detail, we will set up a sampling model, whereby the graph Laplacian
is interpreted as an observation from a random process; we will posit the
existence of a ``population Laplacian'' driving the random process; and we
will then define an estimation problem: find the inverse of the population
Laplacian.
We will show that the maximum a posteriori probability (MAP) estimate of
the inverse of the population Laplacian leads to a regularized SDP, where
the objective function $F(X)=\mathrm{Tr}(L X)$ and where the role of the
penalty function $G(\cdot)$ is to encode prior assumptions about the
population Laplacian.
In addition, we will show that when $G(\cdot)$ is the log-determinant
function then the MAP estimate leads to the Mahoney-Orecchia regularized
SDP corresponding to running the PageRank heuristic.
Said another way, the solutions to the Mahoney-Orecchia regularized SDP can
be interpreted as regularized estimates of the pseudoinverse of the graph
Laplacian.
Moreover, by Mahoney and Orecchia's main result, the solution to this
regularized SDP can be computed very quickly---rather than solving
the SDP with a black-box solver and rather computing explicitly the
pseudoinverse of the Laplacian, one can simply run the fast diffusion-based
PageRank heuristic for computing an approximation to the first nontrivial
eigenvector of the Laplacian $L$.
\vspace{-1mm}
The next section describes some background.
Section~\ref{snx:framework} then describes a statistical framework for graph
estimation; and Section~\ref{sxn:priors} describes prior assumptions that
can be made on the population Laplacian.
These two sections will shed light on the computational implications
associated with these prior assumptions; but more importantly they will shed
light on the implicit prior assumptions associated with making certain
decisions to speed up computations.
Then, Section~\ref{sxn:empirical} will provide an empirical evaluation,
and
Section~\ref{sxn:conc} will provide a brief conclusion.
Additional discussion is available in the Appendix.
\vspace{-4mm}
\section{Background on Laplacians and diffusion-based procedures}
\label{S:introduction}
\vspace{-3mm}
A weighted symmetric graph $G$ is defined by a vertex set
$V = \{ 1, \dotsc, n \}$, an edge set $E \subset V \times V$,
and a weight function $w : E \to \mathbb{R}_+$, where $w$ is assumed to
be symmetric
(\emph{i.e.}, $w(u,v) = w(v,u)$).
In this case, one can construct a matrix, $L_0 \in \mathbb{R}^{V \times V}$,
called the combinatorial Laplacian of $G$:
\[
L_0(u,v)
=
\begin{cases}
- w(u,v) & \text{when $u \neq v$,} \\
d(u) - w(u,u) & \text{otherwise,}
\end{cases}
\]
where $d(u) = \sum_{v} w(u,v)$ is called the degree of $u$.
By construction, $L_0$ is positive semidefinite.
Note that the all-ones vector, often denoted $1$, is an eigenvector of
$L_0$ with eigenvalue zero, \emph{i.e.}, $L 1 = 0$.
For this reason, $1$ is often called trivial eigenvector of $L_0$.
Letting $D$ be a diagonal matrix with $D(u,u) = d(u)$, one can also define
a normalized version of the Laplacian: $L = D^{-1/2} L_0 D^{-1/2}$.
Unless explicitly stated otherwise, when we refer to the Laplacian of a
graph, we will mean the normalized Laplacian.
\vspace{-1mm}
In many situations, \emph{e.g.}, to perform spectral graph partitioning,
one is interested in computing the first \emph{nontrivial} eigenvector of
a Laplacian.
Typically, this vector is computed ``exactly'' by calling a black-box
solver; but it could also be approximated with an iteration-based method
(such as the Power Method or Lanczos Method) or by running a random
walk-based or diffusion-based method to the asymptotic state.
These random walk-based or diffusion-based methods assign positive and
negative ``charge'' to the nodes, and then they let the distribution of
charge evolve according to dynamics derived from the graph structure.
Three canonical evolution dynamics are the following:
\begin{description}
\item[Heat Kernel.]
Here, the charge evolves according to the
heat equation
$\frac{\partial H_t}{\partial t} = - L H_t$.
Thus, the vector of charges evolves as
$ H_t = \exp ( -tL ) = \sum_{k=0}^{\infty} \frac{(-t)^k}{k!}L^k $,
where $t \ge 0$ is a time parameter, times an input seed distribution vector.
\item[PageRank.]
Here, the charge at a node evolves by either moving to a neighbor of
the current node or teleporting to a random node.
More formally, the vector of charges evolves as
\begin{equation}
\label{eqn:page-rank}
R_{\gamma} = \gamma \left(I-\left(1-\gamma \right)M \right)^{-1} ,
\end{equation}
where $M$ is the natural random walk transition matrix associated
with the graph and
where $\gamma \in (0,1)$ is the so-called teleportation parameter,
times an input seed vector.
\item[Lazy Random Walk.]
Here, the charge either stays at the current node or moves to a neighbor.
Thus, if $M$ is the natural random walk transition matrix associated
with the graph, then the vector of charges evolves as some power of
$ W_{\alpha}= \alpha I + (1-\alpha)M $,
where $\alpha \in (0,1)$ represents the ``holding probability,'' times
an input seed vector.
\end{description}
In each of these cases, there is a parameter ($t$, $\gamma$, and the number
of steps of the Lazy Random Walk) that controls the ``aggressiveness'' of
the dynamics and thus how quickly the diffusive process equilibrates; and
there is an input ``seed'' distribution vector.
Thus, \emph{e.g.}, if one is interested in global spectral graph
partitioning, then this seed vector could be a vector with entries drawn
from $\{-1,+1\}$ uniformly at random, while if one is interested in local
spectral graph
partitioning~\cite{Spielman:2004,andersen06local,Chung07_heatkernelPNAS,MOV09_TRv2},
then this vector could be the indicator vector of a small ``seed set'' of
nodes.
See Appendix~\ref{sxn:local-partitioning} for a brief discussion of local
and global spectral partitioning in this context.
Mahoney and Orecchia showed that these three dynamics arise as solutions to
SDPs of the~form
\begin{equation}
\begin{aligned}
& \underset{X}{\text{minimize}}
& & \mathrm{Tr}(L X) + \tfrac{1}{\eta} G(X) \\
& \text{subject to}
& & X \succeq 0, \\
& & & \mathrm{Tr}(X) = 1, \\
& & & X D^{1/2} 1 = 0,
\end{aligned}
\label{eqn:mo-reg-sdp}
\end{equation}
where $G$ is a penalty function (shown to be the
generalized entropy, the log-determinant, and a certain matrix-$p$-norm,
respectively~\cite{MO11-implementing}) and where $\eta$ is a
parameter related to the aggressiveness of the diffusive
process~\cite{MO11-implementing}.
Conversely, solutions to the regularized SDP of~(\ref{eqn:mo-reg-sdp}) for
appropriate values of $\eta$ can be computed \emph{exactly} by running one
of the above three diffusion-based procedures.
Notably, when $G = 0$, the solution to the SDP
of~(\ref{eqn:mo-reg-sdp}) is $u u'$, where $u$ is the smallest nontrivial
eigenvector of $L$.
More generally and in this precise sense, the Heat Kernel, PageRank, and Lazy
Random Walk dynamics can be seen as ``regularized'' versions of spectral
clustering and Laplacian eigenvector computation.
Intuitively, the function $G(\cdot)$ is acting as a penalty function, in a
manner analogous to the $\ell_2$ or $\ell_1$ penalty in Ridge regression or
Lasso regression, and by running one of these three dynamics one is
\emph{implicitly} making assumptions about the form of $G(\cdot)$.
In this paper, we provide a statistical framework to make that intuition
precise.
\vspace{-2mm}
\section{A statistical framework for regularized graph estimation}
\label{snx:framework}
\vspace{-1mm}
Here, we will lay out a simple Bayesian framework for estimating a graph
Laplacian.
Importantly, this framework will allow for regularization by incorporating
prior information.
\vspace{-2mm}
\subsection{Analogy with regularized linear regression}
\label{S:regression}
It will be helpful to keep in mind the Bayesian interpretation of
regularized linear regression.
In that context, we observe $n$ predictor-response pairs in
$\mathbb{R}^p \times \mathbb{R}$, denoted $(x_1, y_1), \dotsc, (x_n, y_n)$;
the goal is to find a vector $\beta$ such that $\beta' x_i \approx y_i$.
Typically, we choose $\beta$ by minimizing the residual sum of squares,
\emph{i.e.}, $F(\beta)=\RSS(\beta) = \sum_i \| y_i - \beta' x_i \|_2^2$, or
a penalized version of it.
For Ridge regression, we minimize $F(\beta) + \lambda \|\beta\|_2^2$; while
for Lasso regression, we minimize $F(\beta) + \lambda \|\beta\|_1$.
The additional terms in the optimization criteria (\emph{i.e.},
$\lambda \|\beta\|_2^2$ and $\lambda \|\beta\|_1$) are called penalty
functions; and adding a penalty function to the optimization criterion can
often be interpreted as incorporating prior information about $\beta$.
For example, we can model $y_1, \dotsc, y_n$ as independent random
observations with distributions dependent on $\beta$.
Specifically, we can suppose $y_i$ is a Gaussian random variable with mean
$\beta' x_i$ and known variance $\sigma^2$.
This induces a conditional density for the vector $y = (y_1, \dotsc, y_n)$:
\begin{equation}\label{E:regression-density}
p(y \mid \beta)
\propto
\exp\{ -\tfrac{1}{2 \sigma^2} F(\beta) \},
\end{equation}
where the constant of proportionality depends only on $y$ and $\sigma$.
Next, we can assume that $\beta$ itself is random, drawn from a
distribution with density $p(\beta)$.
This distribution is called a prior, since it encodes prior knowledge about
$\beta$.
Without loss of generality, the prior density can be assumed to take the form
\begin{equation}\label{E:regression-prior}
p(\beta) \propto \exp\{ -U(\beta) \}.
\end{equation}
Since the two random variables are dependent, upon observing $y$, we have
information about $\beta$.
This information is encoded in the posterior density, $p(\beta \mid y)$,
computed via Bayes' rule as
\begin{equation}\label{E:regression-posterior}
p(\beta \mid y)
\propto p(y \mid \beta) \, p(\beta)
\propto \exp\{ -\tfrac{1}{2 \sigma^2} F(\beta) - U(\beta) \}.
\end{equation}
The MAP estimate of $\beta$ is the value that maximizes $p(\beta \mid y)$;
equivalently, it is the value of $\beta$ that minimizes
$-\log p(\beta \mid y)$.
In this framework, we can recover the solution to Ridge regression or Lasso
regression by setting
$U(\beta) = \tfrac{\lambda}{2 \sigma^2} \| \beta \|_2^2$ or
$U(\beta) = \tfrac{\lambda}{2 \sigma^2} \| \beta \|_1$, respectively.
Thus, Ridge regression can be interpreted as imposing a Gaussian prior on
$\beta$, and Lasso regression can be interpreted as imposing a
double-exponential prior on $\beta$.
\vspace{-1mm}
\subsection{Bayesian inference for the population Laplacian}
\label{S:bayesian-laplacian}
For our problem, suppose that we have a connected graph with $n$ nodes; or,
equivalently, that we have $L$, the normalized Laplacian of that graph.
We will view this observed graph Laplacian, $L$, as a ``sample'' Laplacian,
\emph{i.e.}, as random object whose distribution depends on a true
``population'' Laplacian, $\mathcal{L}$.
As with the linear regression example, this induces a conditional density
for $L$, to be denoted $p(L \mid \mathcal{L})$.
Next, we can assume prior information about the population Laplacian in the
form of a prior density, $p(\mathcal{L})$; and,
given the observed Laplacian, we can estimate the population Laplacian by
maximizing its posterior density, $p(\mathcal{L} \mid L)$.
Thus, to apply the Bayesian formalism, we need to specify the conditional
density of $L$ given $\mathcal{L}$.
In the context of linear regression, we assumed that the observations
followed a Gaussian distribution.
A graph Laplacian is not just a single observation---it is a positive
semidefinite matrix with a very specific structure.
Thus, we will take $L$ to be a random object with expectation~$\mathcal{L}$,
where $\mathcal{L}$ is another normalized graph Laplacian.
Although, in general, $\mathcal{L}$ can be distinct from $L$, we will
require that the nodes in the population and sample graphs have the same
degrees.
That is, if $d = \big(d(1), \dotsc, d(n)\big)$ denotes the ``degree
vector'' of the graph, and $D = \diag\big(d(1), \dotsc, d(n)\big)$, then we
can define
\begin{equation}
\mathcal{X} = \{ X : X \succeq 0, \, X D^{1/2} 1 = 0, \, \rank(X) = n - 1 \} ,
\label{def:chi-set}
\end{equation}
in which case the population Laplacian and the sample Laplacian will both
be members of $\mathcal{X}$.
To model $L$, we will choose a distribution for positive semi-definite
matrices analogous to the Gaussian distribution: a scaled Wishart matrix
with expectation $\mathcal{L}$.
Note that, although it captures the trait that $L$ is positive
semi-definite, this distribution does not accurately model every feature of
$L$.
For example, a scaled Wishart matrix does not necessarily have ones along
its diagonal.
However, the mode of the density is at $\mathcal{L}$, a Laplacian; and for
large values of the scale parameter, most of the mass will be on matrices
close to $\mathcal{L}$.
Appendix~\ref{sxn:justification} provides a more detailed heuristic
justification for the use of the Wishart distribution.
To be more precise, let $m \geq n - 1$ be a scale parameter, and suppose
that $L$ is distributed over $\mathcal{X}$ as a
$\tfrac{1}{m} \mathrm{Wishart}(\mathcal{L}, m)$ random variable.
Then, $\mathbb{E}[L \mid \mathcal{L}] = \mathcal{L}$, and $L$ has conditional density
\begin{equation}\label{E:density}
p(L \mid \mathcal{L})
\propto
\frac{\exp\{ -\frac{m}{2} \Tr(L \mathcal{L}^+)\}}
{|\mathcal{L}|^{m/2}},
\end{equation}
where $|\cdot|$ denotes pseudodeterminant (product of nonzero eigenvalues).
The constant of proportionality depends only on $L$, $d$, $m$, and $n$; and
we emphasize that the density is supported on $\mathcal{X}$.
Eqn.~\eqref{E:density} is analogous to Eqn.~\eqref{E:regression-density}
in the linear regression context, with $1/m$, the inverse of the sample
size parameter, playing the role of the variance parameter $\sigma^2$.
Next, suppose we have know that $\mathcal{L}$ is a random object drawn from
a prior density $p(\mathcal{L})$.
Without loss of generality,
\begin{equation}
\label{E:prior}
p(\mathcal{L}) \propto \exp\{ -U(\mathcal{L}) \},
\end{equation}
for some function $U$, supported on a subset
$\mathcal{\bar X} \subseteq \mathcal{X}$.
Eqn.~\eqref{E:prior} is analogous to Eqn.~\eqref{E:regression-prior}
from the linear regression example.
Upon observing $L$, the posterior distribution for $\mathcal{L}$ is
\begin{equation}\label{E:posterior}
p(\mathcal{L} \mid L)
\propto p(L \mid \mathcal{L}) \, p (\mathcal{L})
\propto \exp\{ -\tfrac{m}{2} \Tr(L \mathcal{L}^+)
+ \tfrac{m}{2} \log |\mathcal{L}^+|
- U(\mathcal{L}) \},
\end{equation}
with support determined by $\mathcal{\bar X}$.
Eqn.~\eqref{E:posterior} is analogous to Eqn.~\eqref{E:regression-posterior}
from the linear regression example.
If we denote by $\mathcal{\hat L}$ the MAP estimate of $\mathcal{L}$, then
it follows that $\mathcal{\hat L}^+$ is the solution to the program
\begin{equation}
\begin{aligned}
& \underset{X}{\text{minimize}}
& & \Tr(L X) + \tfrac{2}{m} U(X^+) - \log |X| \\
& \text{subject to}
& & X \in \mathcal{\bar X} \subseteq \mathcal{X} .
\end{aligned}
\end{equation}
Note the similarity with Mahoney-Orecchia regularized SDP
of~(\ref{eqn:mo-reg-sdp}).
In particular, if
\(
\mathcal{\bar X} = \{ X : \Tr(X) = 1 \} \cap \mathcal{X},
\)
then the two programs are identical except for the factor of
$\log |X|$ in the optimization criterion.
\vspace{-2mm}
\section{A prior related to the PageRank procedure}
\label{sxn:priors}
\vspace{-1mm}
Here, we will present a prior distribution for the population Laplacian
that will allow us to leverage the estimation framework of
Section~\ref{snx:framework}; and we will show that the MAP estimate
of $\mathcal{L}$ for this prior is related to the PageRank procedure via
the Mahoney-Orecchia regularized SDP.
Appendix~\ref{sxn:other-priors} presents priors that lead to the Heat
Kernel and Lazy Random Walk in an analogous way; in both of these cases,
however, the priors are data-dependent in the strong sense that they
explicitly depend on the number of data points.
\vspace{-2mm}
\subsection{Prior density}
The prior we will present will be based on neutrality and invariance
conditions; and it will be supported on $\mathcal{X}$, \emph{i.e.}, on the
subset of positive-semidefinite matrices that was the support set for the
conditional density defined in Eqn.~(\ref{E:density}).
In particular, recall that, in addition to being positive semi-definite,
every matrix in the support set has rank $n - 1$ and satisfies
$X D^{1/2} 1 = 0$.
Note that because the prior depends on the data (via the orthogonality
constraint induced by $D$), this is not a prior in the fully Bayesian
sense; instead, the prior can be considered as part of an empirical or
pseudo-Bayes estimation procedure.
The prior we will specify depends only on the eigenvalues of the normalized
Laplacian, or equivalently on the eigenvalues of the pseudoinverse of the
Laplacian.
Let $\mathcal{L}^+ = \tau \, O \Lambda O'$ be the spectral decomposition of
the pseudoinverse of the normalized Laplacian $\mathcal{L}$, where $\tau \geq 0$
is a scale factor, $O \in \mathbb{R}^{n \times n -1}$ is an orthogonal matrix,
and $\Lambda = \diag\big(\lambda(1), \dotsc, \lambda({n-1})\big)$, where
$\sum_v \lambda(v) = 1$.
Note that the values $\lambda(1), \dotsc, \lambda(n-1)$ are unordered and
that the vector $\lambda = \big(\lambda(1), \dotsc, \lambda(n-1)\big)$ lies
in the unit simplex.
If we require that the distribution for $\lambda$ be exchangeable
(invariant under permutations) and neutral ($\lambda(v)$ independent of the
vector $\big(\lambda(u) / (1 - \lambda(v)) : u \neq v\big)$, for all $v$),
then the only non-degenerate possibility is that $\lambda$ is
Dirichlet-distributed with parameter vector
$(\alpha, \ldots, \alpha)$~\cite{fabius1973two}.
The parameter $\alpha$, to which we refer as the ``shape'' parameter, must
satisfy $\alpha > 0$ for the density to be defined.
In this case,
\begin{equation}\label{E:dirichlet-prior}
p(\mathcal{L})
\propto p(\tau) \prod_{v=1}^{n-1} \lambda(v)^{\alpha - 1},
\end{equation}
where $p(\tau)$ is a prior for $\tau$.
Thus, the prior weight on $\mathcal{L}$ only depends on $\tau$ and $\Lambda$.
One implication is that the prior is ``nearly'' rotationally invariant,
in the sense that $p(P' \mathcal{L} P) = p(\mathcal{L})$ for any
rank-$(n-1)$ projection matrix $P$ satisfying $P D^{1/2} 1 = 0$.
\vspace{-2mm}
\subsection{Posterior estimation and connection to PageRank}
\label{S:posterior-density}
To analyze the MAP estimate associated with the prior of
Eqn.~(\ref{E:dirichlet-prior}) and to explain its connection with the
PageRank dynamics, the following proposition is crucial.
\begin{proposition}\label{P:map-sdp}
Suppose the conditional likelihood for $L$ given $\mathcal{L}$ is as
defined in \eqref{E:density} and the prior density for $\mathcal{L}$ is as
defined in \eqref{E:dirichlet-prior}.
Define $\mathcal{\hat L}$ to be the MAP estimate of $\mathcal{L}$.
Then, $[\Tr(\mathcal{\hat L}^+)]^{-1} \mathcal{\hat L}^+$ solves the
Mahoney-Orecchia regularized SDP \eqref{eqn:mo-reg-sdp},
with $G(X) = -\log |X|$ and $\eta$ as given in Eqn.~(\ref{E:eta}) below.
\end{proposition}
\begin{proof}
For $\mathcal{L}$ in the support set of the posterior,
define $\tau = \Tr(\mathcal{L}^+)$ and $\Theta = \tau^{-1}
\mathcal{L}^+$, so that $\Tr(\Theta) = 1$. Further, $\rank(\Theta) = n - 1$.
Express the prior in the form of Eqn.~\eqref{E:prior} with function
$U$ given by
\[
U(\mathcal{L})
= -\log \{ p(\tau) \, |\Theta|^{\alpha - 1} \}
= -(\alpha - 1) \log |\Theta| - \log p(\tau),
\]
where, as before, $|\cdot|$ denotes pseudodeterminant. Using
\eqref{E:posterior} and the relation
$|\mathcal{L}^+| = \tau^{n-1} |\Theta|$, the posterior
density for $\mathcal{L}$ given $L$ is
\[
p(\mathcal{L} \mid L)
\propto
\exp\Big\{
-\tfrac{m \tau}{2} \Tr(L \Theta)
+\tfrac{m + 2(\alpha - 1)}{2} \log | \Theta |
+ g(\tau)\Big\},
\]
where
\(
g(\tau)
= \tfrac{m (n-1)}{2} \log \tau
+ \log p(\tau).
\)
Suppose $\mathcal{\hat L}$ maximizes the
posterior likelihood. Define $\hat \tau = \Tr(\mathcal{\hat L}^+)$
and $\hat \Theta = [\hat \tau]^{-1} \mathcal{\hat L}^{+}$.
In this case, $\hat \Theta$ must minimize the quantity
\(
\Tr(L \hat \Theta) - \tfrac{1}{\eta} \log |\hat \Theta|,
\)
where
\begin{equation}\label{E:eta}
\eta
= \frac{m \hat \tau}{m + 2(\alpha - 1)}.
\end{equation}
Thus $\hat \Theta$ solves the regularized SDP \eqref{eqn:mo-reg-sdp} with
$G(X) = - \log |X|$.
\end{proof}
Mahoney and Orecchia showed that the solution to
\eqref{eqn:mo-reg-sdp} with $G(X) = - \log |X|$ is closely related to
the PageRank matrix, $R_\gamma$, defined in Eqn.~\eqref{eqn:page-rank}.
By combining Proposition~\ref{P:map-sdp} with their result, we get that the
MAP estimate of $\mathcal{L}$ satisfies
\( \mathcal{\hat L}^+ \propto D^{-1/2} R_\gamma D^{1/2}; \)
conversely,
\(
R_\gamma
\propto
D^{1/2} \mathcal{\hat L}^+ D^{-1/2}.
\)
Thus, the PageRank operator of Eqn.~(\ref{eqn:page-rank}) can be viewed as a
degree-scaled regularized estimate of the pseudoinverse of the Laplacian.
Moreover, prior assumptions about the spectrum of the graph Laplacian have
direct implications on the optimal teleportation parameter.
Specifically Mahoney and Orecchia's Lemma~2 shows how $\eta$ is related to
the teleportation parameter $\gamma$, and Eqn. \eqref{E:eta} shows how the
optimal $\eta$ is related to prior assumptions about the Laplacian.
\vspace{-2mm}
\section{Empirical evaluation}
\label{sxn:empirical}
\vspace{-1mm}
In this section, we provide an empirical evaluation of the performance of
the regularized Laplacian estimator, compared with the unregularized
estimator.
To do this, we need a ground truth population Laplacian $\mathcal{L}$ and a
noisily-observed sample Laplacian $L$.
Thus, in Section~\ref{S:prior-evaluation}, we construct a family of
distributions for $\mathcal{L}$; importantly, this family will be able to
represent both low-dimensional graphs and expander-like graphs.
Interestingly, the prior of Eqn.~\eqref{E:dirichlet-prior} captures some of
the qualitative features of both of these types of graphs (as the shape
parameter is varied).
Then, in Section~\ref{S:estimation}, we describe a sampling procedure for
$L$ which, superficially, has no relation to the scaled Wishart conditional
density of Eqn.~\eqref{E:density}.
Despite this model misspecification, the regularized estimator
$\hat L_{\eta}$ outperforms $L$ for many choices of the regularization
parameter $\eta$.
\vspace{-2mm}
\subsection{Ground truth generation and prior evaluation}
\label{S:prior-evaluation}
The ground truth graphs we generate are motivated by the Watts-Strogatz
``small-world'' model~\cite{watts98collective}.
To generate a ground truth population Laplacian,
$\mathcal{L}$---equivalently, a population graph---we start with a
two-dimensional lattice of width $w$ and height $h$, and thus $n = w h$ nodes.
Points in the lattice are connected to their four nearest neighbors, making
adjustments as necessary at the boundary.
We then perform $s$ edge-swaps: for each swap, we choose two edges
uniformly at random and then we swap the endpoints.
For example, if we sample edges $i_1 \sim j_1$ and $i_2 \sim j_2$, then we
replace these edges with $i_1 \sim j_2$ and $i_2 \sim j_1$.
Thus, when $s = 0$, the graph is the original discretization of a
low-dimensional space; and as $s$ increases to infinity, the graph becomes
more and more like a uniformly chosen $4$-regular graph (which is an
expander~\cite{HLW06_expanders} and which bears similarities with an
Erd\H{o}s-R\'{e}nyi random graph~\cite{Bollobas85}).
Indeed, each edge swap is a step of the Metropolis algorithm toward a
uniformly chosen random graph with a fixed degree sequence.
For the empirical evaluation presented here, $h = 7$ and $w = 6$; but the
results are qualitatively similar for other values.
\begin{figure}[t]
\centering
\subfigure[Dirichlet distribution order statistics.]{
\makebox{\label{F:dirichlet}\includegraphics[scale=0.6]{plots/dirichlet}}
}
\subfigure[Spectrum of the inverse Laplacian.]{
\makebox{\label{F:interpolate}\includegraphics[scale=0.6]{plots/interpolate}}
}
\vspace{-0.5em}
\caption{Analytical and empirical priors.
\ref{F:dirichlet} shows the Dirichlet distribution order statistics versus the shape parameter; and
\ref{F:interpolate} shows the spectrum of $\Theta$ as a function of the
rewiring parameter. }
\vspace{-1em}
\label{fig:priors}
\end{figure}
Figure~\ref{fig:priors} compares the expected order statistics (sorted
values) for the Dirichlet prior of Eqn.~\eqref{E:dirichlet-prior} with the
expected eigenvalues of $\Theta = \mathcal{L}^+ / \Tr(\mathcal{L}^+) $ for
the small-world model.
In particular, in Figure~\ref{F:dirichlet}, we show the behavior of the
order statistics of a Dirichlet distribution on the $(n-1)$-dimensional
simplex with scalar shape parameter $\alpha$, as a function of $\alpha$.
For each value of the shape $\alpha$, we generated a random
$(n-1)$-dimensional Dirichlet vector, $\lambda$, with parameter vector
$(\alpha, \dotsc, \alpha)$; we computed the $n-1$ order statistics of
$\lambda$ by sorting its components; and we repeated this procedure for
$500$ replicates and averaged the values.
Figure~\ref{F:interpolate} shows a corresponding plot for the ordered
eigenvalues of $\Theta$.
For each value of $s$ (normalized, here, by the number of edges $\mu$,
where $\mu = 2 w h - w - h=71$), we generated the normalized Laplacian,
$\mathcal{L}$, corresponding to the random $s$-edge-swapped grid; we
computed the $n-1$ nonzero eigenvalues of $\Theta$; and we performed $1000$
replicates of this procedure and averaged the resulting eigenvalues.
Interestingly, the behavior of the spectrum of the small-world model as
the edge-swaps increase is qualitatively quite similar to the behavior of
the Dirichlet prior order statistics as the shape parameter $\alpha$
increases.
In particular, note that for small values of the shape parameter $\alpha$
the first few order-statistics are well-separated from the rest; and that
as $\alpha$ increases, the order statistics become concentrated around
$1/(n-1)$.
Similarly, when the edge-swap parameter $s = 0$, the top two eigenvalues
(corresponding to the width-wise and height-wise coordinates on the grid)
are well-separated from the bulk; as $s$ increases, the top eigenvalues
quickly merge into the bulk; and eventually, as $s$ goes to infinity, the
distribution becomes very close that that of a uniformly chosen $4$-regular
graph.
\vspace{-2mm}
\subsection{Sampling procedure, estimation performance, and optimal regularization behavior}
\label{S:estimation}
\begin{figure}[h]
\centering
\subfigure[$m/\mu = 0.2$ and $s = 4$.]{
\makebox{\label{fig:perf-frob:a}
\includegraphics[scale=0.34]{plots/estimation-frob-p020}}
}
\subfigure[$m/\mu = 1.0$ and $s = 4$.]{
\makebox{\label{fig:perf-frob:b}
\includegraphics[scale=0.34]{plots/estimation-frob-p100}}
}
\subfigure[$m/\mu = 2.0$ and $s = 4$.]{
\makebox{\label{fig:perf-frob:c}
\includegraphics[scale=0.34]{plots/estimation-frob-p200}}
} \\
\vspace{-1em}
\subfigure[$m/\mu = 2.0$ and $s = 0$.]{
\makebox{\label{fig:perf-frob:d}
\includegraphics[scale=0.34]{plots/estimation-frob-s0-p200}}
}
\subfigure[$m/\mu = 2.0$ and $s = 32$.]{
\makebox{\label{fig:perf-frob:e}
\includegraphics[scale=0.34]{plots/estimation-frob-s32-p200}}
}
\subfigure[Optimal $\eta^\ast / \bar \tau$.]{
\makebox{\label{F:optimal-frob}
\includegraphics[scale=0.34]{plots/optimal}}
}
\vspace{-1em}
\caption{Regularization performance.
\ref{fig:perf-frob:a} through \ref{fig:perf-frob:e} plot the relative
Frobenius norm error, versus the (normalized) regularization parameter
$\eta / \bar \tau$.
Shown are plots for various values of the (normalized) number of edges,
$m/\mu$, and the edge-swap parameter, $s$.
Recall that the regularization parameter in the regularized
SDP~(\ref{eqn:mo-reg-sdp}) is $1/\eta$, and thus smaller values along the
X-axis correspond to stronger regularization.
\ref{F:optimal-frob} plots the optimal regularization parameter
$\eta^\ast / \bar \tau$ as a function of sample proportion for different
fractions of edge swaps. }
\label{fig:perf-frob}
\end{figure}
Finally, we evaluate the estimation performance of a regularized estimator
of the graph Laplacian and compare it with an unregularized estimate.
To do so, we construct the population graph $\mathcal{G}$ and its Laplacian
$\mathcal{L}$, for a given value of $s$, as described in
Section~\ref{S:prior-evaluation}.
Let $\mu$ be the number of edges in $\mathcal{G}$.
The sampling procedure used to generate the observed graph $G$ and its
Laplacian $L$ is parameterized by the sample size $m$.
(Note that this parameter is analogous to the Wishart scale parameter in
Eqn.~\eqref{E:density}, but here we are sampling from a different
distribution.)
We randomly choose $m$ edges with replacement from $\mathcal{G}$; and we
define sample graph $G$ and corresponding Laplacian $L$ by setting the weight
of $i \sim j$ equal to the number of times we sampled that edge.
Note that the sample graph $G$ over-counts some edges in $\mathcal{G}$ and
misses others.
We then compute the regularized estimate $\mathcal{\hat L}_\eta$, up to a
constant of proportionality, by solving (implicitly!) the Mahoney-Orecchia
regularized SDP~\eqref{eqn:mo-reg-sdp} with $G(X) = -\log|X|$.
We define the unregularized estimate $\hat L$ to be equal to the observed
Laplacian, $L$.
Given a population Laplacian $\mathcal{L}$, we define
$\tau = \tau(\mathcal{L}) = \Tr(\mathcal{L}^+)$ and
$\Theta = \Theta(\mathcal{L}) = \tau^{-1} \mathcal{L}^+$.
We define $\hat \tau_\eta$, $\hat \tau$, $\hat \Theta_\eta$, and
$\hat \Theta$ similarly to the population quantities.
Our performance criterion is the relative Frobenius error
$\|\Theta - \hat \Theta_\eta\|_\mathrm{F} / \|\Theta - \hat \Theta
\|_\mathrm{F}$, where $\|\cdot\|_\mathrm{F}$ denotes the Frobenius norm
($\|A\|_\mathrm{F} = [\Tr(A' A)]^{1/2}$).
Appendix~\ref{S:spectral-performance} presents similar results when the
performance criterion is the relative spectral norm error.
Figures~\ref{fig:perf-frob:a}, \ref{fig:perf-frob:b},
and~\ref{fig:perf-frob:c} show the regularization performance when $s=4$
(an intermediate value) for three different values of $m/\mu$.
In each case, the mean error and one standard deviation around it are
plotted as a function of $\eta/\bar \tau$, as computed from 100 replicates;
here, $\bar \tau$ is the mean value of $\tau$ over all replicates.
The implicit regularization clearly improves the performance of the
estimator for a large range of $\eta$ values.
(Note that the regularization parameter in the regularized
SDP~(\ref{eqn:mo-reg-sdp}) is $1/\eta$, and thus smaller values along the
X-axis correspond to stronger regularization.)
In particular, when the data are very noisy, \emph{e.g.}, when $m/\mu=0.2$,
as in Figure~\ref{fig:perf-frob:a}, improved results are seen only for very
strong regularization;
for intermediate levels of noise, \emph{e.g.}, $m/\mu=1.0$, as in
Figure~\ref{fig:perf-frob:b}, (in which case $m$ is chosen such that $G$
and $\mathcal{G}$ have the same number of edges counting multiplicity),
improved performance is seen for a wide range of values of $\eta$; and for
low levels of noise, Figure~\ref{fig:perf-frob:c} illustrates that improved
results are obtained for moderate levels of implicit regularization.
Figures~\ref{fig:perf-frob:d} and~\ref{fig:perf-frob:e} illustrate similar
results for $s=0$
and
$s=32$.
As when regularization is implemented explicitly, in all these cases, we
observe a ``sweet spot'' where there is an optimal value for the implicit
regularization parameter.
Figure~\ref{F:optimal-frob} illustrates how the optimal choice of $\eta$
depends on parameters defining the population Laplacians and sample
Laplacians.
In particular, it illustrates how $\eta^\ast$, the optimal value of
$\eta$ (normalized by $\bar \tau$), depends on the sampling proportion
$m / \mu$ and the swaps per edges $s / \mu$.
Observe that as the sample size $m$ increases, $\eta^\ast$ converges
monotonically to $\bar \tau$; and, further, that higher values of $s$
(corresponding to more expander-like graphs) correspond to higher values
of $\eta^\ast$.
Both of these observations are in direct agreement with Eqn.~\eqref{E:eta}.
\vspace{-2mm}
\section{Conclusion}
\label{sxn:conc}
\vspace{-1mm}
We have provided a statistical interpretation for the observation that
popular diffusion-based procedures to compute a quick approximation to the
first nontrivial eigenvector of a data graph Laplacian exactly solve a
certain regularized version of the problem.
One might be tempted to view our results as ``unfortunate,'' in that it is
not straightforward to interpret the priors presented in this paper.
Instead, our results should be viewed as making explicit the implicit prior
assumptions associated with making certain decisions (that are \emph{already}
made in practice) to speed up computations.
Several extensions suggest themselves.
The most obvious might be to try to obtain Proposition~\ref{P:map-sdp} with
a more natural or empirically-plausible model than the Wishart distribution;
to extend the empirical evaluation to much larger and more realistic data
sets; to apply our methodology to other widely-used approximation
procedures; and to characterize when implicitly regularizing an eigenvector
leads to better statistical behavior in downstream applications where that
eigenvector is used.
More generally, though, we expect that understanding the
algorithmic-statistical tradeoffs that we have illustrated will become
increasingly important in very large-scale data analysis applications.
\newpage
|
1,941,325,220,496 | arxiv |
\subsection{Contracts with Trace Vulnerabilities}
\label{sec:bugs}
While trace vulnerabilities are a broader class, we our focus
attention on three example properties to check of contract
traces. Specifically, we flag contracts which (a) can be killed by
arbitrary addresses, (b) have no way to release Ether after a certain
execution state, and (c) release Ether to arbitrary addresses
carelessly.
Note that any characterization of bugs must be taken with a grain of
salt, since one can always argue that the exposed behavior embodies
intent --- as was debated in the case of the DAO bug~\cite{dao}. Our
characterization of vulnerabilities is based, in part, on anecdotal
incidents reported publicly~\cite{Governmental,parity-bug,dao}. To the
best of our knowledge, however, our characterization is the first to
precisely define checkable properties of such incidents and measure
their prevalence. Note that there are several valid reasons for
contracts for being killable, holding funds indefinitely under certain
conditions, or giving them out to addresses not known at the time of
deployment.
For instance, a common security best practice is that when under
attack, a contract should be killed and should return funds to a
trusted address, such as that of the owner.
Similarly, benign contracts such as bounties or games, often hold
funds for long periods of time (until a bounty is awarded) and release
them to addresses that are not known statically.
Our characterization admits these benign behaviors and flags egregious
violations described next, for which we are unable to find justifiable
intent.
\input{codes/example_leak.tex}
\paragraph{Prodigal Contracts}
Contracts often return funds to owners (when under attack), to
addresses that have sent Ether to it in past (\emph{e.g.}\xspace, in lotteries), or
to addresses that exhibit a specific solution (\emph{e.g.}\xspace, in bounties).
However, when a contract gives away Ether to an arbitrary address---
which is not an owner, has never deposited Ether in the contract, and
has provided no data that is difficult to fabricate by an arbitrary
observer---we deem this as a vulnerability. We are interested in
finding such contracts, which we call as \emph{prodigal}.
Consider the \code{Bounty} contract with code fragment given in Figure~\ref{fig:buggyPayout}.
This contract collects Ether from different
sources and rewards bounty to a selected set of recipients.
In the contract, the function \code{payout} sends to a list of
recipients specified amounts of Ether. It is clear from the function
definition that the recipients and the amounts are provided as inputs,
and anybody can call the function (\emph{i.e.}\xspace, the function does not have
restrictions on the sender). The message sender of the transaction is
not checked for; the only check is on the size of lists. Therefore,
any user can invoke this function with a list of recipients of her
choice, and completely drain its Ether.
The above contract requires a single function invocation to leak its
Ether. However, there are examples of contracts which need two or more
invocations (calls with specific arguments) to cause a leak. Examples
of such contracts are presented in Section~\ref{sec:eval}.
\input{codes/example_kill.tex}
\paragraph{Suicidal Contracts}
A contract often enables a security fallback option of being killed by
its owner (or trusted addresses) in emergency situations like when being
drained of its Ether due to attacks, or when malfunctioning. However,
if a contract can be killed by \emph{any} arbitrary account, which
would make it to execute the \opn{SUICIDE} instruction, we consider it
vulnerable and call it \emph{suicidal}.
The recent \emph{Parity} fiasco\cite{Parity} is a concrete example of
such type of a contract. A supposedly innocent Ethereum
user~\cite{Parityinnocent} killed a library contract on which the main
\emph{Parity} contract relies, thus rendering the latter
non-functional and locking all its Ether. To understand the
\emph{suicidal} side of the library contract, focus on the shortened
code fragment of this contract given in Figure~\ref{fig:parity}.
To kill the contract, the user invokes two different functions: one to
set the ownership,\footnote{The bug would have been prevented has the
function \texttt{initMultiowned} been properly initialized by the
authors.} and one to actually kill the contract.
That is, the user first calls \code{initMultiowned}, providing empty array for \code{_owners}, and zero for \code{_required}.
This effectively means that the contract has no owners and that nobody has to agree to execute a specific contract function.
Then the user invokes the function \code{kill}. This function needs
\code{_required} number of owners to agree to kill the contract,
before the actual \code{suicide} command at line 22 is
executed. However, since in the previous call to
\code{initMultiowned}, the value of \code{_required} was set to
zero, \code{suicide} is executed, and thus the contract is killed.
\paragraph{Greedy Contracts}
We refer to contracts that remain alive and lock Ether indefinitely,
allowing it be released under no conditions, as {\em greedy}.
\begin{comment}
\prateek{Why don't we take the example of the Parity bug here, first?
The following text is put here to be approved by others}.
\end{comment}
In the example of the \code{Parity} contract, many other
\code{multisigWallet}-like contracts which held Ether, used functions
from the \code{Parity} library contract to release funds to their
users. After the \code{Parity} library contracts was killed, the wallet contracts could
no longer access the library, thus became greedy. This
vulnerability resulted in locking of \$200M US worth of
Ether indefinitely!
Greedy contracts can arise out of more direct errors as well. The
most common such errors occur in contracts that accept Ether but
either completely lack instructions that send Ether out
(e.g. \code{send, call, transfer}), or such instructions are not
reachable. An example of contract that lacks commands that release
Ether, that has already locked Ether is given in
Figure~\ref{fig:examplelock}.
\input{codes/example_lock.tex}
\begin{comment}
Consider
the anecdotal example of the \emph{GovernMental contract}\cite
{GovernmentalContract}, which takes investment from many adresses in a
time-bound round, and pays out the last investor the full value of
jackpot. The payout is delivered by the function called
\emph{lendGovernmentMoney}, which has two instructions shown in
Figure~\ref{governmental}. \prateek{Fix this example, please}.
These instructions compile to a code which removes storage entries for
each creditor one by one, iteratively~\cite{Governmental}. Hence a
certain minimum amount of gas is required to execute these
instructions depending on the number of creditors.
To execute this payout function, the total gas required exceeded the
maximum permissible gas per transaction;\footnote{This is true as of
block height 1,405,425.\is{I don't understand this remark -- why is
it important? Ans:since the gas limit increases with every new
block}} hence, no trace can successfully execute the payout
function. \prateek{Aashish, can you precisely say in Sec 2.1 when
the gas limits increase? Who decides that? If it's programmatic,
then this contract doesn't seem sensible to me, because it is NOT
THE CASE that Ether is locked indefinitely}
This mistake led to $1,100$ Ether being locked in this contract for
$52$ days on blockchain, until the maximum permissible gas limit was
increased. Had the external modification for gas limit remained
constant, this contract would have locked Ether indefinitely. This
example shows the importance of reasoning about unreachable execution
states, which are dependent here on the gas costs of instruction,
execution and gas limits.
Our results in Section~\ref{sec:eval} highlight several other cases.
For instance, in Figure~\ref{fig:storage} we present a contract that
accepts Ether but it has no instructions to send Ether out.
\end{comment}
\paragraph{Posthumous Contracts}
When a contract is killed, its code and global variables are cleared
from the blockchain, thus preventing any further execution of its
code. However, all killed contracts continue to receive transactions.
Although such transactions can no longer invoke the code of the
contract, if Ether is sent along them, it is added to the contract
balance, and similarly to the above case, it is locked
indefinitely. Killed contract or contracts that do not contain any
code, but have non-zero Ether we call \emph{posthumous}.
It is the onus of the sender to check if the contract is alive before
sending Ether, and evidence shows that this is not always the
case. Because posthumous contracts require no further static analysis beyond
that for identifying suicidal contracts, we do not treat this as a separate
class of bugs. We merely list all posthumous contracts on the live Ethereum
blockchain we have found in Section~\ref{sec:eval}.
{\setlength{\belowcaptionskip}{-15pt}{
\begin{figure}[t]
\centering
\def0.5\textwidth{0.5\textwidth}
\input{tool_svg.pdf_tex}
\caption{\codename}
\label{fig:tool}
\end{figure}
}}
\subsection{Our Approach}
Each run of the contract, called an invocation, may exercise an
execution path in the contract code under a given input context. Note
that prior works have considered bugs that are properties of
\emph{one} invocation, ignoring the chain of effects across a
\emph{trace} of
invocations~\cite{Luu-al:CCS16,Chen-al:SANER17,mythril,mueller-z3,securify,manticore}.
We develop a tool that uses systematic techniques to find contracts
that violate specific properties of traces. The violations are either:
(a) of \emph{safety} properties, asserting that there \emph{exists} a
trace from a specified blockchain state that causes the contract to
violate certain conditions; and
(b) of \emph{liveness} properties, asserting whether some actions
\emph{cannot} be taken in \emph{any} execution starting from a
specified blockchain state.
We formulate the three kinds of vulnerable contracts as these safety
and liveness trace properties in Section~\ref{sec:properties}.
Our technique of finding vulnerabilities, implemented as a tool called
\codename and described in Section~\ref{sec:tool}, consists of two
major components: symbolic analysis and concrete validation. The
symbolic analysis component takes contract bytecode and analysis
specifications as inputs. The specifications include vulnerability
category to search for and depth of the search space, which further we
refer to as \emph{invocation depth}, along with a few other analysis
parameters we outline in Section~\ref{sec:tool}. To develop our
symbolic analysis component, we implement a custom Ethereum Virtual
Machine, which facilitates symbolic execution of contract
bytecode~\cite{Oyente}. With every contract candidate, our component
runs possible execution traces symbolically, until it finds a trace
which satisfies a set of predetermined properties. The input context
to every execution trace is a set of symbolic variables. Once a
contract is flagged, the component returns concrete values for these
variables.
Our final step is to run the contract concretely and validate the
result for true positives; this step is implemented by our concrete
validation component. The concrete validation component takes the
inputs generated by symbolic analysis component and checks the exploit
of the contract on a private fork of Ethereum blockchain. Essentially,
it is a testbed environment used to confirm the correctness of the
bugs. As a result, at the end of validation the candidate contract is
determined as true or false positive, but the contract state on main
blockchain is not affected since no changes are committed to the
official Ethereum blockchain.
\section{Conclusion}
\label{sec:conclusion}
\vspace{-5pt}
We characterize vulnerabilities in smart contracts that are checkable
as properties of an entire execution trace (possibly infinite sequence
of their invocations). We show three examples of such trace
vulnerabilities, leading to greedy, prodigal and suicidal contracts.
Analyzing $970,898$ contracts, our new tool \codename flags thousands
of contracts vulnerable at a high true positive rate.
\section{Discussion}
\label{sec:discussion}
\is{Discuss, how the properties can be strengthened.}
\section{Evaluation}
\label{sec:eval}
We analyzed $970,898$ smart contracts, obtained by downloading the
Ethereum blockchain from the first block utill block number
$4,799,998$, which is the last block as of December $26$,
2017. Ethereum blockchain has only contract bytecodes. To obtain the
original (Solidity) source codes, we refer to the Etherscan
service~\cite{etherscanSourceCodes} and obtain source for $9,825$
contracts. Only around $1\%$ of the contracts have source code,
highlighting the utility of \codename as a bytecode analyzer.
Recall that our concrete validation component can analyze a contract
from a particular block height where the contract is alive (\emph{i.e.}\xspace,
initialized, but not killed). To simplify our validation process for a
large number of contracts flagged by the symbolic analysis component,
we perform our concrete validation at block height of $4,499,451$,
further denoted as \texttt{BH}. At this block height, we find that
most of the flagged contracts are alive, including the Parity library
contract~\cite{Parity} that our tool successfully finds. This contract
was killed at a block height of $4,501,969$. All contracts existing on
blockchain at a block height of $4,499,451$ are tested, but only
contracts that are alive at \texttt{BH} are concretely
validated.\footnote{We also concretely validate the flagged candidates
which were killed before \texttt{BH} as well.}
\paragraph{Experimental Setup and Performance}
\codename supports parallel analysis of contracts, and scales linearly
in the number of available cores. We run it on a Linux box, with
64-bit Ubuntu 16.04.3 LTS, 64GB RAM and 40 CPUs Intel(R) Xeon(R)
E5-2680 [email protected]. In most of our experiments we run the tool on
$32$ cores.
On average, \codename requires around 10.0 seconds to analyze a
contract for the three aforementioned bugs: 5.5 seconds to check if a
contract is prodigal, 3.2 seconds for suicidal, and $1.3$ seconds for
greedy.
\paragraph{Contract Characteristics}
The number of contracts has increased tenfold from Dec, $2016$ to Dec,
$2017$ and $176$-fold since Dec, $2015$. However, the distribution of
Ether balance across contracts follows a skewed distribution.
Less than $1\%$ of the contracts have more than $99\%$ of the Ether in
the ecosystem. This suggests that a vulnerability in any one of these
high-profile contracts can affect a large fraction of the entire Ether
balance. Note that contracts interact with each other, therefore, a
vulnerability in one contract may affect many others holding Ether, as
demonstrated by the recent infamous Parity library which was used by
wallet contracts with $\$200$ million US worth of Ether~\cite{Parity}.
\input{resultTable}
\subsection{Results}
Table~\ref{fig:table} summarizes the contracts flagged by
\codename. Given the large number of flagged contracts, we select a
random subset for concrete validation, and report on the true positive
rates obtained. We report the number of distinct contracts, calculated
by comparing the hash of the bytecode; however, all percentages are
calculated on the original number of contracts (with duplicates).
\paragraph{Prodigal contracts}
Our tool has flagged $1,504$ candidates contracts ($438$ distinct)
which may leak Ether to an arbitrary Ethereum address, with a true
positive rate of around $97\%$. At block height \texttt{BH}, $46$ of
these contracts hold some Ether.
The concrete validation described in Section~\ref{sec:validation}
succeeds for exploits for $37$ out of $46$ --- these are true
positives, whereas $7$ are false positives. The remaining $2$
contracts leak Ether to an address different from the caller's
address. Note that all of the 37 true positive contracts are alive as
of this writing. For ethical reasons, no exploits were done on
the main blockchain.
Of the remaining $1,458$ contracts which presently do not have Ether
on the public Ethereum blockchain, $24$ have been killed and $42$ have
not been published (as of block height \texttt{BH}). To validate the
remaining alive contracts (in total $1392$) on a private fork, first
we send them Ether from our mining account, and find that $1,183$
contracts can receive Ether.\footnote{These are live and we could
update them with funds in testing.} We then concretely
validate whether these contract leak Ether to an arbitrary address. A
total of $1,156$ out of $1,183$ ($97.72\%$) contracts are confirmed to
be true positives; $27$ ($2.28\%$) are false positives.
For each of the $24$ contracts killed by the block height \texttt{BH},
the concrete validation proceeds as follows. We create a private test
fork of the blockchain, starting from a snapshot at a block height
where the contract is alive. We send Ether to the contract from one
of our addresses address, and check if the contract leaks Ether to an
arbitrary address. We repeat this procedure for each contract, and
find that all $24$ candidate contracts are true positives.
\paragraph{Suicidal contracts}
\codename flags $1,495$ contracts ($403$ distinct), including the
\code{ParityWalletLibrary} contract, as found susceptible to being
killed by an arbitrary address, with a nearly $99\%$ true positive
rate. Out of $1,495$ contracts, $1,398$ are alive at \texttt{BH}.
Our concrete validation engine on a private fork of Ethereum confirm
that $1,385$ contracts (or $99.07\%$) are true positives, \emph{i.e.}\xspace, they
can be killed by any arbitrary Ethereum account, while $13$ contracts
(or $0.93\%$) are false positives. The list of true positives
includes the recent \code{ParityWalletLibrary} contract which was
killed at block height $4,501,969$ by an arbitrary account. Of the
$1,495$ contracts flagged, $25$ have been killed by \texttt{BH}; we
repeat the procedure described previously and cofirmed all of them as
true positives.
\paragraph{Greedy contracts}
Our tool flags $31,201$ greedy candidates ($1,524$ distinct), which
amounts to around $3.2\%$ of the contracts present on the blockchain.
The first observation is that \codename deems all but these as
accepting Ether but having states that release them (not locking
indefinitely). To validate a candidate contract as a true positive one
has to show that the contract does not release/send Ether to any
address for any valid trace. However, concrete validation may not
cover all possible traces, and thus it cannot be used to confirm if a
contract is greedy. Therefore, we take a different strategy and divide
them into two categories:
\begin{enumerate}[label=(\roman*)~]
\item Contracts that accept Ether, but in their bytecode do not have any of the instructions that release Ether
(such instructions include \opn{CALL}, \opn{SUICIDE}, or \opn{DELEGATECALL}).
\item Contracts that accept Ether, and in their bytecode have at
least one of \opn{CALL}, \opn{SUICIDE}
or \opn{DELEGATECALL}.
\end{enumerate}
\codename flagged $1,058$ distinct contracts from the first
category. We validate that these contracts can receive Ether (we send
Ether to them in a transaction with input data according to the one
provided by the symbolic execution routine). Our experiments show that
$1,057$ out of $1,058$ (\emph{e.g.}\xspace, $99.9\%$) can receive Ether and thus are
true positives. On the other hand, the tool flagged $466$ distinct
contracts from the second category, which are harder to confirm by
testing alone. We resort to manual analysis for a subset of these
which have source code. Among these, only $25$ have Solidity source
code. With manual inspection we find that none of them are true
positive --- some traces can reach the \texttt{CALL} code, but
\codename failed to reach it in its path exploration. The reasons for
these are mentioned in the Section~\ref{sec:falsepos}. By
extrapolation (weighted average across $1,083$ validated), we obtain
true positive rate among greedy contracts of $69\%$.
\paragraph{Posthumous Contracts}
Recall that posthumous are contracts that are dead on the blockchain
(have been killed) but still have non-zero Ether balance. We can find
such contracts by querying the blockchain, \emph{i.e.}\xspace, by collecting all
contracts without executable code, but with non-zero balance. We
found $853$ contracts at a block height of $4,799,998$ that do not
have any compiled code on the blockchain but have positive Ether
balance. Interestingly, among these, $294$ contracts have received
Ether after they became dead.
\input{codes/eval_leak1}
\vspace{-10pt}
\subsection{Case Studies: True Positives}
Apart from examples presented in section~\ref{sec:bugs}, we now
present true and false postive cases studies. Note that we only
present the contracts with source code for readability. However, the
fraction of flagged contracts with source codes is very low ($1\%$).
\paragraph{Prodigal contracts}
In Figure ~\ref{fig:leaky}, we give an example of a prodigal contract.
The function \code{tap} seems to lock Ether because the condition at line $4$, semantically, can never be true. However, the compiler optimization
of Solidity allows this condition to pass when an input greater than 20 bytes is used to
call the function \code{tap}.
Note, on a bytecode level, the EVM can only load chunks of 32 bytes of input data.
At line 3 in \code{tap} the first 20 bytes of \code{nickname}
are assigned to the global variable \code{prev}, while neglecting the remaining 12 bytes. The
error occurs because EVM at line 4, correctly nullifies the 12 bytes in \code{prev}, but not in \code{nickname}.
Thus if \code{nickname} has non-zero values in these 12 bytes then the inequality is true.
This contract so far has lost $5.0001$ Ether to different addresses on real Ethereum blockchain.
\input{codes/eval_leak2}
A contract may also leak Ether by getting killed
since the semantic of \texttt{SUICIDE} instruction enforce it to send all of its
balance to an address provided to the instruction. In Figure \ref{fig:mortal}, the
contract \code{Thing}\cite{MortalContract} is inherited from a base
contract \code{Mortal}. The contract implements a review system in which
public reviews an ongoing topic. Among others, the contract has a
\code{kill} function inherited from its base contract which is used to send its balance
to its \code{owner} if its killed. The function \code{mortal}, supposedly a constructor, is misspelled, and thus
anyone can call \code{mortal} to become the \code{owner} of the
contract. Since the derived contract \code{Thing} inherits functions from
contract \code{Mortal}, this vulnerability in the base contract allows an
arbitrary Ethereum account to become the \code{owner} of the derived
contract, to kill it, and to receive its Ether.
\input{codes/eval_kill}
\paragraph{Suicidal contracts} A contract can be killed by exploiting an
unprotected \texttt{SUICIDE} instruction. A trivial example is a public kill function which
hosts the suicide instruction. Sometimes, \texttt{SUICIDE} is protected
by a weak condition, such as in the contract \code{Dividend} given in Figure~\ref{fig:dividend}.
This contract allows users to buy shares or withdraw their investment. The
logic of withdrawing investment is implemented by the \code{withdraw} function.
However, this function has a \code{self_destruct} instruction which can be
executed once the last investment has been made more than $4$ weeks ago.
Hence, if an investor calls this function after $4$ weeks of the last investment,
all the funds go to the owner of the contract and all the records of
investors are cleared from the blockchain. Though the ether is safe with the owner
, there would be no record of any investment for the owner to return
ether to investors.
In the previous example, one invocation of \code{withdraw} function was
sufficient to kill the contract. There are, however, contracts which require two
or more function invocations to be killed. For instance, the contract
\code{Mortal} given in Figure~\ref{fig:mortal} checks whether it is the owner that calls the
\code{kill} function. Hence, it requires an attacker to become the owner of
the contract to kill it. So, this contract requires two invocations to be killed: one call to the function \code{mortal} used to become an owner of the
contract and one call to the function \code{kill} to kill the contract.
A more secure contract would leverage
the \code{mortal} function to a constructor so that the function is called
only once when the contract is deployed. Note, the recent Parity bug similarly also
requires two invocations~\cite{Parity}.
\paragraph{Greedy contracts}
The contract \code{SimpleStorage}, given in Figure~\ref{fig:storage},
is an example of a contract that locks Ether indefinitely. When an
arbitrary address sends Ether along with a transaction invoking the
\code{set} function, the contract balance increases by the amount of
Ether sent. However, the contract does not have any instruction to
release Ether, and thus locks it on the blockchain.
The \code{payable} keyword has been introduced in Solidity recently to
prevent functions from accepting Ether by default, \emph{i.e.}\xspace, a function not
associated with \code{payable} keyword throws if Ether is sent in a
transaction. However, although this contract does not have any
function associated with the \code{payable} keyword, it accepts Ether
since it had been compiled with an older version of Solidity compiler
(with no support for \code{payable}).
\input{codes/eval_lock}
\subsection{Case Studies: False Positives}
\label{sec:falsepos}
We manually analyze cases where \codename's concrete validation fails
to trigger the necessary violation with the produced concrete values,
if source code is available.
\paragraph{Prodigal and Suicidal contracts}
In both of the classes, false positives arise due to two reasons:
\begin{enumerate}[label=(\roman*)~]
\item Our tool performs inter-procedural analysis within a contract,
but does not transfer control in cross-contract calls. For calls
from one contract to a function of another contract, \codename
assigns symbolic variables to the return values. This is imprecise,
because real executions may only return one value (say
\texttt{true}) when the call succeeds.
\item \codename may assign values to symbolic variables related to
block state (\emph{e.g.}\xspace, \code{timestamp} and \code{blocknumber}) in cases
where these values are used to decide the control flow.
%
Thus, we may get false positives because those values may be
different at the concrete validation stage. For instance, in
Figure~\ref{fig:rng}, the \code{_guess} value depends on the values
of block parameters, which cannot be forced to take on the concrete
values found by our analyzer.
\end{enumerate}
\input{codes/eval_FP2}
\input{codes/eval_FP1}
\paragraph{Greedy contracts}
The large share of false positives is attributed to two causes:
\begin{enumerate}[label=(\roman*)~]
\item Detecting a trace which leads to release of Ether may need three
or more function invocations. For instance, in
Figure~\ref{fig:multisig}, the function \code{confirmTransaction}
has to be executed by the majority of owners for the contract to
execute the transaction. Our default invocation depth is the reason
for missing a possible reachable state.
\item Our tool is not able to recover the subtype for the generic
\code{bytes} type in the EVM semantics.
\item Some contracts release funds only if a random number (usually
generated using transaction and block parameters) matches a
predetermined value unlike in the case of the contract in
Figure~\ref{fig:rng}. In that contract the variable \code{_guess} is
also a symbolic variable, hence, the solver can find a solution for
condition on line~$7$. If there is a concrete value in place of
\code{_guess}, the solver times out since the constraint involves a
hash function (hard to invert by the SMT solver).
\end{enumerate}
\vspace{-10pt}
\subsection{Summary and Observations}
The symbolic execution engine of \codename flags \totalbuggy
contracts.
With concrete validation engine or manual inspection, we have
confirmed that around $97\%$ of prodigal, $97\%$ of suicidal and
$69\%$ of greedy contracts are true positive. The importance of
analyzing the bytecode of the contracts, rather than Solidity source
code, is demonstrated by the fact that only $1\%$ of all contracts
have source code. Further, among all flagged contracts, only
\withsource have verified source codes according to the widely used
platform Etherscan, or in percentages only $1.06\%$, $0.47\%$ and
$0.49\%$, in the three categories of prodigal, suicidal, and greedy,
respectively. We refer the reader to Table~\ref{fig:table} for the
exact summary of these results.
Furthermore, the maximal amount of Ether that could have been
withdrawn from prodigal and suicidal contracts, before the block
height \texttt{BH}, is nearly $4,905$ Ether, or $5.9$ million US
dollars\footnote{Calculated at
$1,210$ USD/Eth~\cite{EtherscanWebsite}.}
according to the exchange rate at the time of this writing. In
addition, $6,239$ Ether ($7.5$ million US dollars) is locked inside
posthumous contracts currently on the blockchain, of which $313$ Ether
($379,940$ US dollars) have been sent to dead contracts after they
have been killed.
Finally, the analysis given in Table~\ref{fig:table2} shows the number
of flagged contracts for different invocation depths from $1$ to
$4$. We tested $25,000$ contracts being for greedy, and $100,000$ for
remaining categories, inferring that increasing depth improves results
marginally, and an invocation depth of $3$ is an optimal tradeoff
point.
{\setlength{\belowcaptionskip}{-10pt}{
\begin{table}[t]
\centering
\footnotesize
\begin{tabular}{|>{\centering\arraybackslash}m{0.55 in}||>{\centering\arraybackslash}m{0.55 in}|>{\centering\arraybackslash}p{0.55 in}|>{\centering\arraybackslash}m{0.55 in}|}
\hline
Inv. depth & Prodigal & Suicidal & Greedy \\
\hline
1 & 131 & 127 & 682 \\
\hline
2 & 156 & 141 & 682 \\
\hline
3 & 157 & 141 & 682 \\
\hline
4 & 157 & 141 & 682 \\
\hline
\end{tabular}
\caption{The table shows number of contracts flagged
for various invocation depths. This analysis is done on a
random subset of $25,000$--$100,000$ contracts.
}
\label{fig:table2}
\end{table}
}}
\section{Introduction}
\label{sec:introduction}
Cryptocurrencies feature a distributed protocol for a set of computers
to agree on the state of a public ledger called the blockchain.
Prototypically, these distributed ledgers map accounts or addresses
(the public half of a cryptographic key pair) with quantities of
virtual ``coins''. Miners, or the computing nodes, facilitate
recording the state of a payment network, encoding transactions that
transfer coins from one address to another. A significant number of
blockchain protocols now exist, and as of writing the market value of
the associated coins is over \$300 billion US, creating a lucrative
attack target.
{\em Smart contracts} extend the idea of a blockchain to a compute
platform for decentralized execution of general-purpose applications.
Contracts are programs that run on blockchains: their code and state
is stored on the ledger, and they can send and receive coins. Smart
contracts have been popularized by the Ethereum blockchain. Recently,
sophisticated applications of smart contracts have arisen, especially
in the area of token management due to the development of the ERC20
token standard. This standard allows the uniform management of custom
tokens, enabling, \emph{e.g.}\xspace, decentralized exchanges and complex
wallets. Today, over a million smart contracts operate on the Ethereum
network, and this count is growing.
Smart contracts offer a particularly unique combination of security
challenges. Once deployed they cannot be upgraded or
patched,\footnote{Other than by ``hard forks'', which are essentially
decisions of the community to change the protocol and are extremely
rare.} unlike traditional consumer device software. Secondly, they are
written in a new ecosystem of languages and runtime environments, the
de facto standard for which is the Ethereum Virtual Machine and its
programming language called Solidity.
Contracts are relatively difficult to test, especially since their
runtimes allow them to interact with other smart contracts and
external off-chain services; they can be invoked repeatedly by
transactions from a large number of users.
Third, since coins on a blockchain often have significant value,
attackers are highly incentivized to find and exploit bugs in
contracts that process or hold them directly for profit. The attack
on the DAO contract cost the Ethereum community \$60 million
US; and several more recent ones have had impact of a similar
scale~\cite{Parity}.
In this work, we present a systematic characterization of a class of
vulnerabilities that we call as {\em trace} vulnerabilities. Unlike
many previous works that have applied static and dynamic analyses to
find bugs in contracts
automatically~\cite{Luu-al:CCS16,Oyente,Kalra-al:NDSS18,securify}, our
work focuses on detecting vulnerabilities across a long sequence of
invocations of a contract. We label vulnerable contracts with three
categories --- greedy, prodigal, and suicidal --- which either lock
funds indefinitely, leak them to arbitrary users, or be susceptible to
by killed by any user. Our precisely defined properties capture many
well-known examples of known anecdotal
bugs~\cite{dao,Governmental,Parity}, but broadly cover a class of
examples that were not known in prior work or public reports. More
importantly, our characterization allows us to concretely check for
bugs by running the contract, which aids determining confirmed true
positives.
We build an analysis tool called \codename for finding these
vulnerabilities directly from the bytecode of Ethereum smart
contracts, without requiring source code access. In total, across the
three categories of vulnerabilities, \codename has been used to
analyze $970,898$ contracts live of the public Ethereum blockchain.
Our techniques are powerful enough to find the infamous Parity bug
that indirectly caused $200$ million dollars worth of Ether, which is
not found by previous analyses. A total of \totalbuggy
(\distinctbuggy distinct) contracts are flagged as potentially buggy,
directly carry the equivalent of millions of dollars worth of Ether.
As in the case of the Parity bug, they may put a larger amount to
risk, since contracts interact with one another. For \validated
contracts we tried to concretely validate, \codename has found
over \confirmedTP confirmed vulnerabilities with \tpweight\% true
positive rate. All vulnerabilities are uncovered on average within 10
seconds of analysis per contract.
\paragraph{Contributions}
We make the following contributions:
\begin{itemize}
\item We identify three classes of {\em trace} vulnerabilities, which
can be captured as properties of a execution traces --- potentially
infinite sequence of invocations of a contract. Previous techniques
and tools~\cite{Oyente} are not designed to find these bugs because
they only model behavior for a single call to a contract.
\item We provide formal high-order properties to check which admit a
mechanized symbolic analysis procedure for detection. We fully
implement \codename, a tool for symbolic analysis of smart contract
bytecode (without access to source code).
\item We test close to one million contracts, finding thousands of
confirmed true positives within a few seconds of analysis time per
contract. Testing trace properties with \codename is practical.
\end{itemize}
\section{Appendix Title}
{\footnotesize
\bibliographystyle{IEEEtran}
\section{Problem}
We define a new class of trace vulnerabilities, showing three specific
examples of properties that can be checked in this broader class. We
present our approach and tool to reason about the class of trace
vulnerabilities.
\subsection{Background on Smart Contracts}
\label{sec:background}
Smart contracts in Ethereum run on Ethereum Virtual Machine (EVM), a
stack-based execution runtime~\cite{Gavin-al:yellow-paper}. Different
source languages compile to the EVM semantics, the predominant of them
being Solidity~\cite{Solidity}. A smart contract embodies the
concept of an autonomous agent, identified by its program logic, its
identifying address, and its associated balance in Ether.
Contracts, like other addresses, can receive Ether from external
agents storing it in their \code{balance} field; they can can also
send Ether to other addresses via transactions. A smart contract is
created by the {\em owner} who sends an initializing transaction,
which contains the contract bytecode and has no specified recipient.
Due to the persistent nature of the blockchain, once initialized, the
contract code cannot be updated. Contracts live perpetually unless
they are explicitly terminated by executing the \opn{SUICIDE}
bytecode instruction, after which they are no longer invocable or
called {\em dead}. When alive, contracts can be invoked many times.
Each invocation is triggered by sending a transaction to the contract
address, together with input data and a fee (known as
\emph{gas})~\cite{Gavin-al:yellow-paper}. The mining network executes
separate instances of the contract code and agrees on the outputs of
the invocation via the standard blockchain consensus protocol, \emph{i.e.}\xspace,
Nakamoto consensus~\cite{Nakamoto:08,Pirlea-Sergey:CPP18}. The result
of the computation is replicated via the blockchain and grants a
transaction fee to the miners as per block reward rates established
periodically.
The EVM allows contract functions to have local
state, while the contracts may have global variables stored on the
blockchain. Contracts can invoke other contracts via message calls;
outputs of these calls, considered to be a part of the same
transaction, are returned to the caller during the runtime.
Importantly, calls are also used to send Ether to other contracts and
non-contract addresses.
The \code{balance} of a contract can be read by anyone, but is only
updated via calls from other contracts and externally initiated
transactions.
Contracts can be executed repeatedly over their lifetime. A
transaction can run one {\em invocation} of the contract and an
execution \emph{trace} is a (possibly infinite) sequence of runs of a
contract recorded on the blockchain. Our work shows the importance of
reasoning about execution traces of contracts with a class of
vulnerabilities that has not been addressed in prior works, and
provides an automatic tool to detect these issues.
\input{bugs}
\section{Execution Model and Trace Properties}
\label{sec:properties}
A life cycle of a smart contract can be represented by a sequence of
the contract's states, which describe the values of the contract's
fields, as well as its balance, interleaved with instructions and
\emph{irreversible} actions it performs modifying the global context
of the blockchain, such transferring Ether or committing suicide.
One can consider a contract to be \emph{buggy} with respect to a
certain class of unwelcome high-level scenarios (\emph{e.g.}\xspace, ``leaking''
funds) if some of its finite execution traces fail to satisfy a
certain condition.
Trace properties characterised this way are traditionally qualified as
\emph{trace-safety} ones, meaning that ``during a final execution
nothing bad happens''.
Proving the absence of some other high-level bugs will, however,
require establishing a statement of a different kind, namely,
``something good must eventually happen''. Such properties are known
as \emph{liveness} ones and require reasoning about progress in
executions.
An example of such property would be an assertion that a contract can
always execute a finite number of steps in order to perform an action
of interest, such as tranferring money, in order to be considered
non-\emph{greedy}.
In this section, we formally define the execution model of Ethereum
smart contracts, allowing one to pinpoint the vulnerabilities
characterised in Section~\ref{sec:bugs}.
The key idea of our bug-catching approach is to formulate the
erroneous behaviours as predicates of observed contract \emph{traces},
rather than individual configurations and instruction invocations,
occurring in the process of an execution.
By doing so, we are able to (a) capture the prodigal/suicidal
contracts via conditions that relate the unwelcome agents gaining, at
some point, access to a contract's funds or suicide functionality by
finding a way around a planned semantics, and (b) respond about
repeating behavioural patterns in the contract life cycles, allowing
us to detect greedy contracts.
\subsection{EVM Semantics and Execution Traces}
\label{sec:semantics}
We begin with defining cotnract execution traces by adopting a low-level
execution semantics of an EVM-like language in the form of \tname{EtherLite}-like
calculus~\cite{Luu-al:CCS16}. \tname{EtherLite} implements a small-step stack
machine, operating on top of a global configuration of the blockchain,
which used to retrieve contract codes and ascribe Ether balance to
accounts, as well as manipulations with the local contract
configuration.
As customary in Ethereum, such agent is represented by its address
$\mathit{id}$, and might be a contract itself.
For the purpose of this work, we simplify the semantics of \tname{EtherLite} by
eliding the executions resulting in exceptions, as reasoning about
such is orthogonal to the properties of interest. Therefore, the
configurations $\delta$ of the \tname{EtherLite} abstract machine are defined as follows:
{\small{
\[
\begin{array}{l@{\ \ \ \ }r@{\ \ }c@{\ \ }l}
\text{Configuration} & \delta & \triangleq & \angled{A, \sigma}
\\[2pt]
\text{Execution stack} & A &\triangleq & \angled{M, \mathit{id}, \mathit{pc}, s, m} \cdot A ~~|~~ \epsilon
\\[2pt]
\text{Message} & m & \triangleq & \set{\mathit{sender} \mapsto \mathit{id};~\mathit{value} : \mathbb{N};~ \mathit{data} \mapsto \ldots}
\\[2pt]
\text{Blockchain state} & \sigma & \triangleq & \many{\mathit{id} \mapsto \set{\mathit{bal} : \mathbb{N};~\mathit{code}? \mapsto
M;~ \many{f? \mapsto v}}}
\end{array}
\]
}}
That is, a contract execution configuration consists of an activation
record stack $A$ and a blockchain context $\sigma$. An activation
record stack $A$ is a list of tuples $\angled{M, \mathit{id}, \mathit{pc}, s, m}$,
where $\mathit{id}$ and $M$ are the address and the code of the contract
currently being executed, $\mathit{pc}$ is a program counter pointing to the
next instruction to be executed, $s$ is a local operand stack, and $m$
is the last message used to invoke the contract execution. Among other
fields, $m$ stores the identity of the $\mathit{sender}$, the amount $\mathit{value}$
of the ether being transferred (represented as a natural number), as
well as auxiliary fields ($\mathit{data}$) used to provide additional
arguments for a contract call, which we will be omitting for the sake
of brevity.
Finally, a simplified context $\sigma$ of a blockchain is encoded as
a finite partial mapping from an account $\mathit{id}$ to its balance and
contract code $M$ and its mutable state, mapping the field names $f$
to the corresponding values,\footnote{For simplicity of presentation,
we treat all contract state as \emph{persistent}, eliding operations
with \emph{auxiliary} memory, such as
\opn{MLOAD}/\opn{MSTORE}.} which both are optional (hence,
marked with ?) and are only present for contract-storing blockchain
records.
We will further refer to the union of a contract's fields entries
$\many{f \mapsto v}$ and its balance entry $\mathit{bal} \mapsto z$ as a
\emph{contract state}~$\rho$.
{
\setlength{\abovecaptionskip}{-10pt}
\setlength{\belowcaptionskip}{-10pt}
{
\begin{figure}
{\footnotesize{
\begin{mathpar}
\inferrule*[Lab=SStore]
{
M[\mathit{pc}] = \opn{SSTORE}
\and
\sigma' = \sigma[\mathit{id}][f \mapsto v]
}
{
\angled{\angled{M, \mathit{id}, \mathit{pc}, f \cdot v \cdot s, m} \cdot A, \sigma}
\stepc{\Sstore{f,~v}}
\angled{\angled{M, \mathit{id}, \mathit{pc}+1, s, m} \cdot A, \sigma'} }
\and
\inferrule*[Lab=SLoad]
{
M[\mathit{pc}] = \opn{SLOAD}
\and
v = \sigma[\mathit{id}][f]
}
{
\angled{\angled{M, \mathit{id}, \mathit{pc}, f \cdot s, m} \cdot A, \sigma}
\stepc{\Sload{f,~v}}
\angled{\angled{M, \mathit{id}, \mathit{pc} + 1, v \cdot s, m} \cdot A, \sigma}
}
\and
\inferrule*[Lab=Call]
{
M[\mathit{pc}] = \opn{CALL}
\and
\sigma[\mathit{id}][\mathit{bal}] \geq z
\\
s = \mathit{id}' \cdot z \cdot \mathit{args} \cdot s'
\and
a = \angled{M, \mathit{id}, \mathit{pc} + 1, s', m}
\\
m' = \set{\mathit{sender} \mapsto \mathit{id}; \mathit{value} \mapsto z; \mathit{data} \mapsto \mathit{args}}
\and
M' = \sigma[\mathit{id}'][\mathit{code}]
\\
\sigma' = \sigma[\mathit{id}][\mathit{bal} \mapsto \sigma[\mathit{id}][\mathit{bal}] - z]
\quad
\sigma'' = \sigma'[\mathit{id}'][\mathit{bal} \mapsto \sigma'[\mathit{id}'][\mathit{bal}] + z]
}
{
\angled{\angled{M, \mathit{id}, \mathit{pc}, s, m} \cdot A, \sigma}
\stepc{\Call{\mathit{id}',~m'}}
\angled{\angled{M', \mathit{id}', 0, \epsilon, m'} \cdot a \cdot A, \sigma''}
}
\and
\inferrule*[Lab=SuicideNonEmptyStack]
{
M[\mathit{pc}] = \opn{SUICIDE}
\and
s = \mathit{id}' \cdot s'
\and
a = \angled{M', \mathit{pc}', s'', m'}
\\
\sigma' = \sigma[\mathit{id}'][\mathit{bal} \mapsto (\sigma[\mathit{id}'][\mathit{bal}] +
\sigma[\mathit{id}][\mathit{bal}])]
\quad
\sigma'' = \sigma'[\mathit{id}][\mathit{bal} \mapsto 0]
}
{
\angled{\angled{M, \mathit{id}, \mathit{pc}, s, m} \cdot a \cdot A, \sigma}
\stepc{\Suicide{\mathit{id}'}}
\angled{\angled{M', \mathit{id}', \mathit{pc}', 1 \cdot s'', m'} \cdot A, \sigma''} }
\end{mathpar}
}}
\caption{Selected execution rules of \tname{EtherLite}.}
\label{fig:opsem}
\end{figure}
}}
Figure~\ref{fig:opsem} presents selected rules for a smart contract
execution in \tname{EtherLite}.\footnote{The remaining rules can be found in the
work by Luu~\emph{et~al.}\xspace~\cite{Luu-al:CCS16}.}
The rules for storing and loading values to/from a contract's field
$f$ are standard.
Upon calling another account, a rule \rname{Call} is executed, which
required the amount of Ether $z$ to be transferred to be not larger
than the contract $\mathit{id}$'s current balance, and changes the activation
record stack and the global blockchain context accordingly.
Finally, the rule \rname{SuicideNonEmptyStack} provides the semantics
for the $\opn{SUICIDE}$ instruction (for the case of a non-empty
activation record stack), in which case all funds of the terminated
contract $\mathit{id}$ are transferred to the caller's $\mathit{id}'$.
An important addition we made to the semantics of \tname{EtherLite} are execution
\emph{labels}, which allow to distinguish between specific transitions
being taken, as well as their parameters, and are defined as follows:
{\footnotesize{
\[
\begin{array}{rcl}
\ell & \triangleq & \Sstore{f,~v} ~|~ \Sload{f,~v} ~|~ \Call{\mathit{id},~m}
~|~ \Suicide{\mathit{id}} ~|~ \ldots
\end{array}
\]
}}
\noindent
For instance, a transition label of the form $\Call{\mathit{id},~m}$ captures
the fact that a currently running contract has transferred control to
another contract $\mathit{id}$, by sending it a message $m$, while the label
$\Suicide{\mathit{id}}$ would mean a suicide of the current contract, with
transfer of all of its funds to the account (a contract's or not)
$\mathit{id}$.
With the labelled operational semantics at hand, we can now provide a
definition of partial contract execution \emph{traces} as sequences of
interleaved contract states $\rho_i$ and transition labels
$\ell_j$ as follows:
\begin{definition}[Projected contract trace]
\label{def:ttrace}
A \emph{partial} projected trace $t = \widehat{\tau}_{\mathit{id}}(\sigma, m)$ of a
contract $\mathit{id}$ in an initial blockchain state $\sigma$ and an
incoming message $m$ is defined as a sequence
$[\angled{\rho_0, \ell_0}, \ldots, \angled{\rho_n, \ell_n}]$,
such that for every $i \in \set{0 \ldots n}$,
$\rho_i = \sigma_i[\mathit{id}]|_{\mathit{bal}, \many{f}}$, where $\sigma_i$ is
the blockchain state at the $i^{\text{th}}$ occurrence of a
configuration of the form,
$\angled{\angled{\bullet, \mathit{id}, \bullet, \bullet, \bullet}, \sigma_i}$
in an execution sequence starting from the configuration
%
$\angled{\angled {\sigma[\mathit{id}][\mathit{code}], \mathit{id}, 0, \epsilon, m} \cdot
\epsilon, \sigma} $,
and $\ell_i$ is a label of an immediate next transition.
\end{definition}
In other words, $\widehat{\tau}_{\mathit{id}}(\sigma, m)$ captures the states of a
contract $\mathit{id}$, interleaved with the transitions taken ``on its
behalf'' and represented by the corresponding labels, starting from the
initial blockchain $\sigma$ and triggered by the message $m$.
The notation $\sigma[\mathit{id}]|_{\mathit{bal}, \many{f}}$ stands for a projection
to the corresponding components of the contract entry in~$\sigma$.
States and transitions of contracts \emph{other} than $\mathit{id}$ and
involved into the same execution are, thus, ignored.
Given a (partial) projected trace $\widehat{\tau}_{\mathit{id}}(\sigma, m)$, we say
that it is \emph{complete}, if it corresponds to an execution, whose
last configuration is $\angled{\epsilon, \sigma'}$ for some
$\sigma'$.
The following definition captures the behaviors of multiple subsequent
transactions with respect to a contract of interest.
\begin{definition}[Multi-transactional contract trace]
\label{def:trace}
A contract trace $t = \trace{\mathit{id}}{\sigma, \many{m_i}}$, for a
sequence of messages $\many{m_i} = m_0, \ldots, m_n$, is a
concatenation of single-transaction traces
$\widehat{\tau}_{\mathit{id}}(\sigma_i, m_i)$, where $\sigma_0 = \sigma$,
$\sigma_{i+1}$ is a blockchain state at the end of an execution
starting from a configuration
$\angled{\angled {\sigma[\mathit{id}][\mathit{code}], \mathit{id}, 0, \epsilon, m_i} \cdot
\epsilon, \sigma_i}$,
and all traces $\widehat{\tau}_{\mathit{id}}(\sigma_i, m_i)$ are complete for
$i \in \set{0, \ldots, n - 1}$.
\end{definition}
As stated, the definition does \emph{not} require a trace
to end with a complete execution at the last transaction.
For convenience, we will refer to
the last element of a trace $t$ by $\last{t}$ and to its length as
$\length{t}$.
\subsection{Characterising Safety Violations}
\label{sec:trace-safety-prop}
The notion of contract traces allows us to formally capture the
definitions of buggy behaviors, described previously in
Section~\ref{sec:bugs}. First, we turn our attention to the
prodigal/suicidal contracts, which can be uniformly captured by the
following higher-order trace predicate.
\begin{definition}[Leaky contracts]
\label{def:leaky}
A contract with an address $\mathit{id}$ is considered to be \emph{leaky} with
respect to predicates $\mathit{P}$, $\mathit{R}$ and $\mathit{Q}$, and a
blockchain state~$\sigma$ (denoted
$\leaky{\mathit{P}, \mathit{R}, \mathit{Q}}(\mathit{id}, \sigma)$) \emph{iff}\xspace \emph{there
exists} a sequence of messages $\many{m_i}$, such that for a trace
$t = \trace{\mathit{id}}{\sigma, \many{m_i}}$:
\begin{enumerate}
\item the \emph{precondition} $\mathit{P}(\sigma[\mathit{id}][\mathit{code}], t_0, m_0)$
holds,
\item the \emph{side condition} $\mathit{R}(t_i, m_0)$ holds for all
$i < \length{t}$,
\item the \emph{postcondition} $\mathit{Q}(t_n, m_0)$ holds for
$t_n = \last{t}$.
\end{enumerate}
\end{definition}
Definition~\ref{def:leaky} of leaky contracts is \emph{relative} with
respect to a current state of a blockchain: a contract that is
currently leaky may stop being such in the future.
Also, notice that the ``triggering'' initial message $m_0$ serves as
an argument for all three parameter predicates.
We will now show how two behaviors observed earlier can be encoded via
specific choices of $\mathit{P}$, $\mathit{R}$, and $\mathit{Q}$.\footnote{In most
of the cases, it is sufficient to take $R \triangleq \mathsf{True}$, but in
Section~\ref{sec:related} we hint certain properties that require a
non-trivial side condition.}
\paragraph{Prodigal contracts}
A contract is considered prodigal if it sends Ether, immediately or
after a series of transitions (possibly spanning multiple
transactions), to an arbitrary sender. This intuition can be
encoded via the following choice of $\mathit{P}$, $\mathit{R}$, and $\mathit{Q}$
for Definition~\ref{def:leaky}:
{\footnotesize{
\[
\begin{array}{lcl@{\ \ \ }l}
\mathit{P}(M, \angled{\rho, \ell}, m) & \triangleq
&&
m[\mathit{sender}] \notin \im{\rho} \wedge
m[\mathit{value}] = 0
\\[2pt]
\mathit{R}(\angled{\rho, \ell}, m) & \triangleq && \mathsf{True}
\\[2pt]
\mathit{Q}(\angled{\rho, \ell}, m) & \triangleq &&
\ell = \Call{m[\mathit{sender}], m'} \wedge m'[\mathit{value}] > 0
\\[2pt]
&&\vee& \ell = \Delegatecall{m[\mathit{sender}]}
\\[2pt]
&& \vee&\ell = \Suicide{m[\mathit{sender}]}
\end{array}
\]
}}
According to the instantiation of the parameter predicates above,
a prodigal contract is exposed by a trace that is triggered by a
message $m$, whose sender does \emph{not} appear in the contract's
state ($m[\mathit{sender}] \notin \im{\rho}$), \emph{i.e.}\xspace, it is not the owner,
and the Ether payload of $m$ is zero.
To expose the erroneous behavior of the contract, the postcondition
checks that the transition of a contract is such that it transfer
funds or control (\emph{i.e.}\xspace, corresponds to \opn{CALL}, \opn{DELEGATECALL}
or \opn{SUICIDE} instructions~\cite{Gavin-al:yellow-paper}) with the
recipient being the sender of the initial message. In the case of
sending funds via \opn{CALL}
we also check that the
amount being transferred is non zero.
In other words, the initial caller $m[\mathit{sender}]$, unknown to the
contract, got himself some funds without any monetary contribution!
In principle, we could ensure minimality of a trace, subject to the
property, by imposing a non-trivial side condition $\mathit{R}$, although
this does not affect the class of contracts exposed by this
definition.
\paragraph{Suicidal contracts}
A definition of a suicidal contract is very similar to the one of a
prodigal contract. It is delivered by the following choice of
predicates:
{\footnotesize{
\[
\begin{array}{lcl}
\mathit{P}(M, \angled{\rho, \ell}, m) & \triangleq
&
\opn{SUICIDE} \in M \wedge m[\mathit{sender}] \notin \im{\rho}
\\[2pt]
\mathit{R}(\angled{\rho, \ell}, m) & \triangleq & \mathsf{True}
\\[2pt]
\mathit{Q}(\angled{\rho, \ell}, m) & \triangleq & \ell = \Suicide{m[\mathit{sender}]}
\end{array}
\]
}}
That is, a contract is suicidal if its code $M$ contains the
$\opn{SUICIDE}$ instruction and the corresponding transition can be
triggered by a message sender, that does not appear in the contract's
state at the moment of receiving the message, \emph{i.e.}\xspace, at the initial
moment $m[\mathit{sender}] \notin \im{\rho}$.
\subsection{Characterising Liveness Violations}
\label{sec:liveness-properties}
A contract is considered \emph{locking} at a certain blockchain state
$\sigma$, if at any execution originating from $\sigma$ prohibits
certain transitions to be taken. Since disproving liveness properties
of this kind with a finite counterexample is impossible in general, we
formulate our definition as an \emph{under-approximation} of the
property of interest, considering only final traces up to a certain
length:
\begin{definition}[Locking contracts]
\label{def:locking}
A contract with an address $\mathit{id}$ is considered to be \emph{locking}
with respect to predicates $\mathit{P}$ and $\mathit{R}$, the transaction
number $k$, and a blockchain state~$\sigma$ (denoted
$\locking{\mathit{P}, \mathit{R}, k}(\mathit{id}, \sigma)$) \emph{iff}\xspace \emph{for all}
sequences of messages $\many{m_i}$ of length less or equal than $k$,
the corresponding trace $t = \trace{\mathit{id}}{\sigma, \many{m_i}}$
satisfies:
\begin{enumerate}
\item the \emph{precondition} $\mathit{P}(\sigma[\mathit{id}][\mathit{code}], t_0, m_0)$,
\item the \emph{side condition} $\mathit{R}(t_i, m_0)$ for all
$i \leq \length{t}$.
\end{enumerate}
\end{definition}
Notice that, unlike Definition~\ref{def:leaky}, this Definition does
not require a postcondition, as it is designed to under-approximate
potentially infinite traces, up to a certain length $k$,\footnote{We
discuss viable choices of $k$ in Section~\ref{sec:eval}.} so the
``final state'' is irrelevant.
\paragraph{Greedy contracts}
In order to specify a property asserting that in an interaction with
up to $k$ transactions, a contract does not allow to release its
funds, we instantiate the predicates from Definition~\ref{def:locking}
as follows:
{\footnotesize{
\[
\begin{array}{l@{\ \ }c@{\ \ }l}
\mathit{P}(M, \angled{\rho, \ell}, m) & \triangleq
&\rho[\mathit{bal}] > 0
\\[2pt]
\mathit{R}(\angled{\rho, \ell}, m) & \triangleq &
\neg\left(\!\!
\begin{array}{l@{\ \ }l}
&
\ell = \Call{m[\mathit{sender}], m'} \wedge m'[\mathit{value}] > 0
\\[2pt]
\vee& \ell = \Delegatecall{m[\mathit{sender}]}
\\[2pt]
\vee&\ell = \Suicide{m[\mathit{sender}]}
\end{array}
\!\!\right)
\end{array}
\]
}}
Intuitively, the definition of a greedy contract is dual to the notion
of a prodigal one, as witnessed by the above formulation: at any trace
starting from an initial state, where the contract holds a non-zero
balance, no transition transferring the corresponding funds (\emph{i.e.}\xspace,
matched by the side condition $\mathit{R}$) can be taken, no matter what
is the $\mathit{sender}$'s identity. That is, this definition covers the case
of contract's \emph{owner} as well: \emph{no one} can withdraw any
funds from the contract.
\section{Related Work}
\label{sec:related}
\paragraph{Dichotomy of smart contract bugs}
\label{sec:dich-smart-contr}
The early work by Delmolino~\emph{et~al.}\xspace~\cite{Delmolino-al:FC16}
distinguishes the following classes of problems: (a) contracts that do
not refund their users, (b) missing encryptions of sensitive user data
and (c) lack of incentives for the users to take certain actions. The
property (a) is the closest to our notion of \emph{greedy}. While that
outlines the problem and demonstrates it on series of simple examples
taught in a class, they do not provide a systematic approach for
detection of smart contracts prone to this issue.
Later works on contract safety and security identify potential bugs,
related to the concurrent transactional
executions~\cite{Sergey-Hobor:WTSC17}, mishandled
exceptions~\cite{Luu-al:CCS16}, overly extensive gas
consumption~\cite{Chen-al:SANER17} and implementations of fraudulent
financial schemes~\cite{Bartoletti-al:Arxiv17}.\footnote{See the
works~\cite{Atzei-al:POST17,consensys-practices} for a survey of
known contract issues.}
In contrast to all those work, which focus on bad implementation
practices or misused language semantics, we believe, our
characterisation of several classes of contract bugs, such as greedy,
prodigal, \emph{etc}, is novel, as they are stated in terms of properties
execution traces rather than particular instructions taken/states
reached.
\vspace{15pt}
\paragraph{Reasoning about smart contracts}
\tname{Oyente}~\cite{Luu-al:CCS16,Oyente} was the first symbolic
execution-based tool that provided analysis targeting several specific
issues: (a) mishandled exceptions, (b) transaction-ordering
dependence, (c)~timestamp dependence and (d)
reentrancy~\cite{reentrancy}, thus remedying the corner cases of
Solidity/EVM semantics (a) as well as some programming anti-patterns
(b)--(d).
Other tools for symbolic analysis of EVM and/or EVM have been
developed more recently: \tname{Manticore}~\cite{manticore},
\tname{Mythrill}~\cite{mythril,mueller-z3},
\tname{Securify}~\cite{securify}, and
\tname{KEVM}~\cite{Hildenbrandt:KEVM17,erc20-k}, all focusing on
detecting \emph{low-level} safety violations and vulnerabilities, such
as integer overflows, reentrancy, and unhandled exceptions, \emph{etc},
neither of them requiring reasoning about contract execution traces.
A very recent work by Grossman~\emph{et~al.}\xspace~\cite{Grossman-al:POPL18} similar
to our in spirit and providing a dynamic analysis of execution traces,
focuses exclusively on detecting \emph{non-callback-free} contracts
(\emph{i.e.}\xspace, prone to reentrancy attacks)---a vulnerability that is by now
well studied.
Concurrently with our work, Kalra~\emph{et~al.}\xspace developed
\tname{Zeus}~\cite{Kalra-al:NDSS18}, a framework for automated
verification of smart contracts using abstract interpretation and
symbolic model checking, accepting user-provided \emph{policies} to
verify for.
Unlike \codename, \tname{Zeus} conducts policy checking at a level of
LLVM-like intermediate representation of a contract, obtained from
Solidity code, and leverages a suite of standard tools, such as
off-the-shelf constraint and SMT
solvers~\cite{Gurfinkel-al:CAV15,McMillan:VMCAI07,deMoura-Bjorner:TACAS08}.
\tname{Zeus} does not provide a general framework for checking trace
properties, or under-approximating liveness properties.
Various versions of EVM semantics~\cite{Gavin-al:yellow-paper} were
implemented in Coq~\cite{Hirai:EVMCoq},
Isabelle/HOL~\cite{Hirai:WTSC17,Amani-al:CPP18},
$\text{F}^{\star}$~\cite{Bhargavan-al:PLAS16},
Idris~\cite{ethereum-idris:16}, and
Why3~\cite{eth-why3,Filliatre-Paskevich:ESOP13}, followed by
subsequent mechanised contract verification efforts. However, none of
those efforts considered trace properties in the spirit of what we
defined in Section~\ref{sec:properties}.
Several contract languages were proposed recently that distinguish
between \emph{global actions} (\emph{e.g.}\xspace, sending Ether or terminating a
contract) and \emph{instructions} for ordinary
computations~\cite{bamboo,scilla}, for the sake of simplified
reasoning about contract executions.
For instance, the work on the contract
language~\tname{Scilla}~\cite{scilla} shows how to encode in
Coq~\cite{Coq-manual} and formally prove a property, which is very
similar to a contract being non-\emph{leaky}, as per
Definition~\ref{def:leaky} instantiated with a non-trivial side
condition~$R$.
\section{The Algorithm and the Tool}
\label{sec:tool}
\codename is a symbolic analyzer for smart contract execution traces,
for the properties defined in Section~\ref{sec:properties}.
It operates by taking as input a contract in its bytecode form and a
concrete starting block value from the Ethereum blockchain as the
input context, flagging contracts that are outlined in
Section~\ref{sec:bugs}.
When reasoning about contract traces, \codename follows the \tname{EtherLite}
rules, described in Section~\ref{sec:semantics}, executing them
symbolically.
During the execution, which starts from a contract state satisfying
the precondition of property of interest (\textit{cf.}\xspace
Definitions~\ref{def:leaky} and~\ref{def:locking}), it checks if there
exists an execution trace which violates the property and a set of
candidate values for input transactions that trigger the property
violation.
For the sake of tractability of the analysis, it does not keep track
of the entire blockchain context~$\sigma$ (including the state of
other contracts), treating only the contract's transaction inputs and
certain block parameters as symbolic.
To reduce the number of false positives and confirm concrete exploits
for vulnerabilities, \codename calls its {\em concrete validation}
routine, which we outline in Section~\ref{sec:validation}.
\subsection{Symbolic Analysis}
\label{sec:symbolic}
Our work concerns finding properties of traces that involve multiple
invocations of a contract. We leverage static symbolic analysis to
perform this step in a way that allows reasoning across contract calls
and across multiple blocks. We start our analysis given a contract
bytecode and a starting concrete context capturing values of the
blockchain.
\codename reasons about values read from input transaction fields and
block parameters\footnote{Those being \opn{CALLVALUE}, \opn{CALLER},
\opn{NUMBER}, \opn{TIMESTAMP}, \opn{BLOCKHASH}, \opn{BALANCE},
\opn{ADDRESS}, and \opn{ORIGIN}.} in a symbolic way---specifically,
it denotes the set of all concrete values that the input variable can
take as a symbolic variable. It then symbolically interprets the
relationship of other variables computed in the contract as a symbolic
expression over symbolic variables. For instance, the code \code{y :=
x + 4} results in a symbolic value for \code{y} if \code{x} is a
symbolic expression; otherwise it is executed as concrete
value. Conceptually, one can imagine the analysis as maintaining two
memories mapping variables to values: one is a symbolic memory mapping
variables to their symbolic expressions, the other mapping variables
to their concrete values.
\paragraph{Execution Path Search}
The symbolic interpretation searches the space of all execution paths
in a trace with a depth-first search. The search is a best effort to
increase coverage and find property violating traces. Our goal is
neither to be sound, \emph{i.e.}\xspace, search all possible paths at the expense of
false positives, nor to be provably complete, \emph{i.e.}\xspace, have only true
positives at the expense of
coverage~\cite{Godefroid:2011}. From a practical perspective,
we make design choices that strike a balance between these two goals.
The symbolic execution starts from the entry point of the contract,
and considers all functions which can be invoked externally as an
entry point. More precisely, the symbolic execution starts at the first instruction in the bytecode,
proceeding sequentially until the execution path ends in terminating instruction.
Such instruction can be valid (\emph{e.g.}\xspace, \opn{STOP, RETURN}), in which case we assume to have reached the end of some contract function, and thus restart the symbolic execution again from the first bytecode instruction to simulate the next function call.
On the other hand, the terminating instruction can be invalid (\emph{e.g.}\xspace, non-existing instruction code or invalid jump destination), in which case we terminate the search down this path and backtrack in the depth-first search procedure to try another path.
When execution reaches a branch, \codename
concretely evaluates the branch condition if all the variables used in
the conditional expression are concrete. This uniquely determines the
direction for continuing the symbolic execution. If the condition
involves a symbolic expression, \codename queries an external SMT
solver to check for the satisfiability of the symbolic conditional
expression as well as its negation. Here, if the symbolic conditional
expression as well as its negation are satisfiable, both branches are
visited in the depth-first search; otherwise, only the satisfiable
branch is explored in the depth first search. On occasions, the
satisfiability of the expression cannot be decided in a pre-defined
timeout used by our tool; in such case, we terminate the search down
this path and backtrack in the depth-first search procedure to try
another path. We maintain a symbolic path constraint which captures
the conditions necessary to execute the path being analyzed in a
standard way. \codename implements support for 121 out of the 133 bytecode
instructions in Ethereum's stack-based low-level language.
At a call instruction, control follows transfer to the target. If the
target of the transfer is a symbolic expression, \codename backtracks
in its depth-first search. Calls outside a contract, however, are not
simulated and returns are marked symbolic. Therefore, \codename
depth-first search is inter-procedural, but not inter-contract.
\paragraph{Handling data accesses}
The memory mappings, both symbolic and concrete, record all the
contract memory as well blockchain storage. During the symbolic
interpretation, when a global or blockchain storage is accessed for
the first time on a path, its concrete value is read from the main Ethereum blockchain into local
mappings. This ensures that subsequent reads or writes are kept local to
the path being presently explored.
The EVM machine supports a flat byte-addressable memory,
and each address has a bit-width of 256 bits. The accesses are in
32-byte sized words which \codename encodes as bit-vector constraints
to the SMT solver. Due to unavailability of source code, \codename does
not have any prior information about higher-level datatypes in
the memory. All types default to 256-bit integers in the encoding used
by \codename. Furthermore, \codename attempts to recover more advanced
types such as dynamic arrays by using the following heuristic:
if a symbolic variable,
say $x$, is used in constant arithmetic to create an expression (say
$x + 4$) that loads from memory (as an argument to the
\opn{CALLDATALOAD} instruction), then it detects such an access as a
dynamic memory array access. Here, \codename uses the SMT solver to
generate $k$ concrete values for the symbolic expression, making the
optimistic assumption that the size of the array to be an integer in
the range $[0,k]$. The parameter $k$ is configurable, and
defaults to $2$.
Apart from this case, whenever accesses in the memory involve a
symbolic address, \codename makes no attempt at alias analysis and
simply terminates the path being search and backtracks in its
depth-first search.
\paragraph{Handling non-deterministic inputs}
Contracts have several sources of non-deterministic inputs such as the
block timestamp, \emph{etc}. While these are treated as symbolic, these are
not exactly under the control of the external users. \codename does
not use their concrete values as it needs to reason about invocations
of the contract across multiple invocations, \emph{i.e.}\xspace, at different blocks.
\paragraph{Flagging Violations}
Finally, when the depth-first search in the space of the contract
execution reaches a state where the desired property is violated, it
flags the contract as a buggy {\em candidate}. The symbolic path
constraint, along with the necessary property conditions, are asserted
for satisfiability to the SMT solver. We
use~\tname{Z3}~\cite{deMoura-Bjorner:TACAS08} as our solver, which
provides concrete values that make the input formula satisfiable. We
use these values as the concrete data for our symbolic inputs,
including the symbolic transaction data.
\paragraph{Bounding the path search space}
\codename takes the following steps to bound the search in the
(potentially infinite) path space. First, the call depth is limited to
the constant called \code{max_call_depth}, which defaults to 3 but can
be configured for empirical tests. Second, we limit the total number
of jumps or control transfers on one path explored to a configurable
constant \code{max_cfg_nodes}, default set to $60$. This is
necessary to avoid being stuck in loops, for instance. Third, we set a
\code{timeout} of $10$ seconds per call to our SMT solver.
Lastly, the total time spent on a contract is limited to configurable
constant \code{max_analysis_time}, default set to $300$ seconds.
\paragraph{Pruning}
To speed up the state search, we implement pruning with
memorization. Whenever the search encounters that the particular
configuration (\emph{i.e.}\xspace, contract storage, memory, and stack) has been seen
before, it does not further explore that part of the path space.
\subsection{Concrete Validation}
\label{sec:validation}
In the concrete validation step, \codename creates a private fork of
the original Ethereum blockchain with the last block as the input
context. It then runs the contract with the concrete values of the
transactions generated by the symbolic analysis to check if the
property holds in the concrete execution. If the concrete execution
fails to exhibit a violation of the trace property, we mark the
contract as a {\em false positive}; otherwise, the contract is marked
as a {\em true positive}.
To implement the validating framework, we added a new functionality to
the official go-ethereum package~\cite{goEthereum} which allows us to
fork the Ethereum main chain at a block height of our choice.
Once we fork the main chain, we mine on that fork without connecting to any peers on the Ethereum network,
and thus we are able to mine our own transactions without
committing them to the main chain.
\paragraph{Prodigal Contracts} The validation framework checks if a
contract indeed leaks Ether by sending to it the transactions with
inputs provided by the symbolic analysis engine. The transactions are
sent by one of our accounts created previously.
Once the transactions are executed, the validation framework checks
whether the contract has sent Ether to our account.
If a verifying contract does not have Ether, our framework first sends
Ether to the contract and only then runs the exploit.
\paragraph{Suicidal Contracts}
In a similar fashion, the framework checks if a
contract can be killed after executing the transactions provided by the symbolic analysis engine
on the forked chain.
Note, once a contract is killed,
its bytecode is reset to \texttt{'0x'}.
Our framework uses precisely this test to confirm the correctness of the exploit.
\paragraph{Greedy Contracts} A strategy similar to the above two
cannot be used to validate the exploits on contracts that lock
Ether. However, during the bug finding process, our symbolic execution
engine checks firsthand whether a contract accepts Ether.
The validation framework can, thus, check if a contract is true
positive by confirming that it accepts Ether and does not have
\texttt{CALL}, \texttt{DELEGATECALL}, or
\texttt{SUICIDE} opcodes in its bytecode. In Section~\ref{sec:eval} we
give examples of such contracts.
|
1,941,325,220,497 | arxiv |
\section{Introduction and statement of results}
To state our results, we first recall from \cite{GMTW} the
definition of the $d$-dimensional cobordism category. Let us give the definition in outline here, and in full detail in \S\ref{sec:CobordismCategoryDefinition}.
A tangential structure is a fibration $\theta: \mathscr{X} \to
BO(d)$, and we write $\gamma \to BO(d)$ for the canonical vector bundle. A $\theta$-structure on a vector bundle is a bundle map (i.e.\ fibrewise linear isomorphism) from the vector bundle to $\theta^* \gamma$, and a $\theta$-structure on a manifold is a $\theta$-structure on its tangent bundle.
The objects of the cobordism category $\mathcal{C}_\theta$ are pairs $(M,\ell)$, where $M
\subseteq \bR^\infty$ is a closed $(d-1)$-manifold and $\ell$ is a $\theta$-structure on $\epsilon^1 \oplus TM$. The non-identity morphisms of $\mathcal{C}_\theta$, $(M_0, \ell_0) \to (M_1, \ell_1)$, are pairs $(t,W)$ with $t > 0$ and $W \subseteq [0,t] \times
\bR^\infty$ an embedded cobordism, together with a $\theta$-structure $\ell$ on $W$ that agrees with $\ell_0$ and $\ell_1$ over the boundary. There is a ``collar'' condition on $(W,\ell)$ near $\partial W$, and the
space of all such $W$ has a nice topology, described in detail in \S\ref{sec:CobordismCategoryDefinition} (and also \cite{GMTW}). The homotopy type of morphism spaces is given by
\begin{align}\label{eq:MorphismsInCobCat}
\mathcal{C}_\theta(M_0, M_1) \simeq \coprod_{W}
B\Diff_\theta(W,\partial W),
\end{align}
where $W$ ranges over connected compact cobordisms from $M_0$ to $M_1$, one in each diffeomorphism class rel $\partial W$. $\Diff(W,\partial W)$ is the topological
group of diffeomorphisms of $W$ which restrict to the identity on a
neighbourhood of $\partial W$, and $B\Diff_\theta(W,\partial W)$ denotes
the homotopy quotient
\begin{align*}
B\Diff_\theta(W,\partial W) = \Bun^\partial(TW, \theta^*\gamma_d)
/\!\!/ \Diff(W,\partial W),
\end{align*}
where $\Bun^\partial(TW,\theta^*\gamma_d)$ is the space of bundle maps
fixed near the boundary. We will say that $B \Diff_\theta(W, \partial W)$ is the moduli space of $d$-manifolds with $\theta$-structure, with underlying manifold diffeomorphic to $W$.
Finally, we recall the main theorem of \cite{GMTW} (reproved below as Theorem \ref{thm:MainThmGMTW}, largely following Chapter 6 of \cite{galatius-2006}). It determines the
homotopy type of the classifying space of the cobordism category,
\begin{align
B\mathcal{C}_\theta \simeq \Omega^{\infty-1} \bold{MT \theta},
\end{align}
where $\bold{MT \theta} = \mathscr{X}^{-\theta}$ is the Thom spectrum of the inverse of
the vector bundle classified by $\theta: \mathscr{X} \to BO(d)$.
To state our results, let us consider a version of
$\mathcal{C}_\theta$ where manifolds have ``basepoints''. Pick once and for all a $\theta$-structure on the vector space $\mathbb{R}^d$. This induces a $\theta$-structure on any framed manifold, which we shall call the \textit{standard $\theta$-structure} on that manifold.
\begin{definition}\label{defn:ThetaDotCategory}
Let $d \geq 2$. Let $\mathcal{C}^\bullet_\theta$ be the subcategory of
$\mathcal{C}_\theta$ where objects $M \subseteq \bR^\infty$ are
required to contain the interval $(-\epsilon,\epsilon)^{d-1} \times
\{0\} \subseteq \bR^\infty$ for some $\epsilon > 0$ and to be
\emph{connected}.
Morphisms $W \subseteq [0,t] \times \bR^\infty$ are required to be
connected and to contain the strip $[0,t] \times
(-\epsilon,\epsilon)^{d-1} \times \{0\}$ for some $\epsilon > 0$.
Furthermore, the tangential structure $l: TW \to \theta^*\gamma$ is
required to be standard on the strip $[0,t] \times (-\epsilon,\epsilon)^{d-1} \times
\{0\}$.
\end{definition}
Homotopically, the result of this condition is to replace \eqref{eq:MorphismsInCobCat}
by
\begin{align}\label{eq:CdotDefn}
\mathcal{C}_\theta^\bullet(M_0, M_1) \simeq \coprod_{W}
B\Diff_\theta(W,L \cup \partial W),
\end{align}
where $W$ ranges over connected compact cobordisms between connected manifolds $M_0$ and $M_1$, containing an embedded arc $L$ connecting the two boundary components, one such $W$ in each diffeomorphism class rel to $\partial W \cup L$.
Tangential structures and diffeomorphisms are
required to be fixed near the line $L \subseteq W$. Thus we can cut
open the surface $W$ along $L$, so the spaces in~(\ref{eq:CdotDefn}) are
effectively moduli spaces of surfaces with \emph{one} boundary
component.
\begin{MainThm}\label{thm:SubcategoryConnected}
Let $d \geq 2$ and $\theta : \mathscr{X} \to BO(d)$ be a tangential structure such that $\mathscr{X}$ is path connected and $S^d$ admits a $\theta$-structure. Then the inclusion
$$B \mathcal{C}_\theta^\bullet \to B \mathcal{C}_\theta \simeq \Omega^{\infty-1}\bold{MT \theta}$$
is a weak homotopy equivalence.
\end{MainThm}
Our main result is the following. Let us say that two objects in
$\mathcal{C}^\bullet_\theta$ are \textit{cobordant} if they give the same element in $\pi_0 B \mathcal{C}_\theta^\bullet = \pi_{-1}\bold{MT \theta}$. In section \ref{sec:ReversingMorphisms} we show that this happens precisely when there is a morphism between the objects (not just a zig-zag).
\begin{MainThm}\label{thm:SubcategoryMonoids}
Let $\theta: \mathscr{X} \to BO(2)$ be a tangential structure such that $\mathscr{X}$ is
path connected and that $S^2$ admits a $\theta$-structure. Let
$\mathcal{D} \subseteq \mathcal{C}_\theta^\bullet$ be a full
subcategory. Then the inclusion
\begin{align*}
B\mathcal{D} \to B\mathcal{C}_\theta^\bullet
\end{align*}
is a weak homotopy equivalence of each component of $B\mathcal{D}$ onto a component of $B\mathcal{C}_\theta^\bullet$.
If $\mathcal{D}$ has exactly one object, then it is homotopy commutative monoid; if it has at most one object in each cobordism class, then it is a disjoint union of homotopy commutative monoids.
\end{MainThm}
The assumption that $\mathscr{X}$ be path connected is innocent, since
a disconnected $\mathscr{X}$ could be considered one path component at a time.
The assumption that $S^2$ admits a $\theta$-structure is necessary for
our proof, which uses surgery techniques. Under that assumption the
connected sum of two surfaces with $\theta$-structure will again have
a $\theta$-structure (in contrast, the connected sum of e.g.\
\emph{framed} surfaces is not framable; this corresponds to $\mathscr{X} =
EO(2)$).
Let us spell out our result explicitly in the case of ordinary orientation, (although it is not new in this case). Then $W$ in (\ref{eq:CdotDefn}) ranges over connected oriented
surfaces, and $B\Diff_\theta(W,\partial W \cup L)$ is homotopy
equivalent to $B\Gamma_{g,1}$, where $\Gamma_{g,1}$ is the mapping
class group of an oriented genus $g$ surface with one boundary
component. The monoid $\mathcal{D}$ is homotopy equivalent to the
disjoint union
\begin{align*}
\mathcal{D} \simeq \coprod_{g \geq 0} B\Gamma_{g,1},
\end{align*}
and the composition is the ``pair of pants'' composition of
\cite{Miller}. Our result then says that this monoid has the same
classifying spaces as the full cobordism category of \cite{GMTW}. Our proof is entirely geometric and does not rely on Harer stability.
\subsection{Stabilisation and Madsen--Weiss' theorem}\label{sec:harer-stab-mads}
Let us explain an application of Theorem \ref{thm:SubcategoryMonoids} that highlights the advantage of homotopy commutativity. First we briefly discuss group completion of a homotopy commutative monoid $\mathcal{M}$, following McDuff and Segal \cite{McDuff-Segal}. There is a canonical map $\mathcal{M} \to \Omega B \mathcal{M}$ which is a homotopy equivalence if and only $\mathcal{M}$ is grouplike, i.e.\ $\pi_0\mathcal{M}$ is a group. There is an induced map in homology,
$$H_*(\mathcal{M}) \to H_*(\Omega B \mathcal{M})$$
which sends the multiplicative subset $\pi_0\mathcal{M} \subset H_*(\mathcal{M})$ to units of $H_*(\Omega B \mathcal{M})$, so there is an induced map from the localisation
\begin{align}\label{eq:GpCompletion}
H_*(\mathcal{M})[\pi_0 \mathcal{M}^{-1}] \to H_*(\Omega B \mathcal{M}).
\end{align}
The main theorem about group completion for homotopy commutative monoids \cite{McDuff-Segal} is that (\ref{eq:GpCompletion}) is an isomorphism of rings.
In many cases of interest, $\pi_0 \mathcal{M}$ is a finitely generated monoid, so the localisation (\ref{eq:GpCompletion}) can be formed as a sequential direct limit
$$H_*(\mathcal{M}) \to H_*(\mathcal{M}) \to H_*(\mathcal{M}) \to \cdots$$
over multiplication by an element $m$ which is the product of a set of generators. In fact, $\pi_0\mathcal{M}$ need not be finitely generated, it is only necessary that $\pi_0\mathcal{M}$ may be group completed by inverting finitely many elements. This direct limit is the homology of the space $\mathcal{M}_\infty$ obtained as the mapping telescope of the analogous direct system of spaces $\mathcal{M} \to \mathcal{M} \to \mathcal{M} \to \cdots$. Consequently we get a map $\mathcal{M}_\infty \to \Omega B \mathcal{M}$, inducing an isomorphism
$$H_*(\mathcal{M}_\infty) \to H_*(\Omega B \mathcal{M}).$$
If $k$ is a field, the isomorphism (\ref{eq:GpCompletion}) can be reinterpreted in terms of invariants of the action of $\pi_0\mathcal{M}$ on $H^*(\mathcal{M} ; k)$, namely that
\begin{align}\label{eq:GpCompletionCohomology}
H^*(\mathcal{M} ; k)^{\pi_0 \mathcal{M}} \cong H^*(\Omega_0 B \mathcal{M} ; k),
\end{align}
where $\Omega_0$ means the basepoint component of the loop space. Here the monoid $\pi_0\mathcal{M}$ acts on homology and cohomology of both spaces by translation. To deduce (\ref{eq:GpCompletionCohomology}) we take coinvariants of the isomorphism (\ref{eq:GpCompletion}) in $k$-homology and algebraically dualise, and note that the invariants of the action on $H^*(\Omega B \mathcal{M} ; k)$ is isomorphic to the cohomology of $\Omega_0 B \mathcal{M}$.
\vspace{2ex}
We now apply this to the homotopy commutative monoid $\mathcal{D}$ produced by Theorem \ref{thm:SubcategoryMonoids}. The assumption that $\pi_0\mathcal{D}$ may be group completed by inverting finitely many elements holds in many cases of interest. The homology of a component of $\mathcal{D}_\infty$ can be interpreted as the homology of $B\Diff_\theta(W_\infty)$, the moduli space of infinite genus $\theta$-surfaces with parametrised germ at infinity.
The cohomology $H^*(\mathcal{D})$ is the ring of characteristic classes of bundles of $\theta$-surfaces with one parametrised boundary component. Then $H^*(\Omega_0^\infty \bold{MT \theta})$ is the ring of characteristic classes that are invariant under gluing a trivial bundle onto the boundary.
Ebert \cite{Ebert09} has recently proved that a similar result cannot hold for odd dimensional manifolds. Indeed he shows there are non-trivial rational classes in $H^*(\Omega^\infty \MT{SO}{2n+1} ; \mathbb{Q} {} )$ which vanish in $H^*(B\Diff^+(M) ; \mathbb{Q} {} )$ for all oriented $(2n+1)$-manifolds $M$.
\vspace{2ex}
We now give some particular cases of interest. Returning to the case of ordinary orientations, we reproduce the Madsen--Weiss theorem \cite{MW}, that there is a homology equivalence
\begin{equation}\label{thm:MW}
\mathbb{Z} \times B\Gamma_{\infty,1} \to \Omega^\infty \MT{SO}{2},
\end{equation}
where $\Gamma_{\infty,1}$ is the limit of the mapping class groups
$\Gamma_{g,1}$ as $g \to \infty$.
Again we point out that we prove this result \emph{without} quoting the homological stability results of Harer \cite{H} and Ivanov \cite{Ivanov}. The case of unoriented and spin surfaces can be treated similarly, cf.\ \S\ref{sec:Applications}.
For an oriented surface $F$, Cohen and Madsen \cite{CM} introduced spaces
$$\mathcal{S}_{g,n}(Y) = \mathrm{Map}^\partial(F_{g,n}, Y) /\!\!/ \Diff^+(F_{g,n})$$
where $\mathrm{Map}^\partial$ is the space of maps taking $\partial F$ to the basepoint of $Y$, and $\Diff^+$ denotes diffeomorphisms that preserve orientation and boundary. These are morphism spaces in the category $\mathcal{C}_\theta$, for $\theta : BSO(2) \times Y \to BO(2)$, and it follows from \cite{CM} that, if $Y$ is simply connected, there is a homology equivalence
$$\mathbb{Z} \times \mathcal{S}_{\infty,1}(Y) \to \Omega^\infty \MT{SO}{2} \wedge Y_+.$$
Here, $\pi_0(\mathcal{D}) = \mathbb{N} \times H_2(Y;\mathbb{Z})$ and $\mathbb{Z} \times \mathcal{S}_{\infty,1}(Y)$ is the direct limit $\mathcal{D} \to \mathcal{D} \to \dots$ over multiplication by an element corresponding to $(1,0) \in \mathbb{N} \times H_2(Y,\mathbb{Z})$. Our result provides an analogue to this for all connected spaces $Y$ and also to surfaces with any tangential structure satisfying the assumption of Theorem \ref{thm:SubcategoryMonoids}, such as unoriented or spin surfaces.
The special case $Y = B\mathbb{Z} = S^1$ can be interpreted in complete analogy with (\ref{thm:MW}). Let $v \in
H^1(F_{g,1};\mathbb{Z})$ be a ``primitive'' cohomology class, i.e.\ one that can
be extended to a symplectic basis of $H^1$ (equivalently, the Poincar\'{e}
dual to an embedded non-separating curve). Let $\Gamma'_{g,1} \leq
\Gamma_{g,1}$ denote the stabilizer of $v$. Then our results give a
homology isomorphism
\begin{equation*}
\mathbb{Z} \times B\Gamma'_{\infty,1} \to
\Omega^\infty \MT{SO}{2} \wedge S^1_+.
\end{equation*}
Similarly the case $X = B\mathbb{Z}/n$ gives a homology equivalence
\begin{equation*}
\mathbb{Z} \times B\Gamma'_{\infty,1}(n) \to \Omega^\infty \MT{SO}{2} \wedge B\mathbb{Z}/n_+,
\end{equation*}
where $\Gamma'_{g,1}(n) \leq \Gamma_{g,1}$ denotes
the subgroup that stabilizes the mod $n$ reduction of the primitive
vector $v$. The space $B\Gamma'_{g,1}(n)$ has the same homotopy type
as the moduli space of pairs $(\overline \Sigma, \Sigma)$ where
$\Sigma$ is a genus $g$ Riemann surface with one boundary component
and $\overline \Sigma \to \Sigma$ is an unbranched $n$-fold cyclic Galois cover.
For a completely general space $X$ the monoid $\pi_0 \mathcal{D}$
is difficult to understand explicitly, and our result must be stated
as an algebraic isomorphism $H_*(\mathcal{D})[\pi_0 \mathcal{D}^{-1}] \simeq H_*(\Omega^\infty
\bold{MT \theta})$.
\section{Spaces of Manifolds}\label{SoM}
\subsection{Definitions}\label{sec:definitions}
\begin{definition}
Let $U\subseteq \bR^N$ be an open set. Let $\Psi_d(U)$ be
the set of subsets $M \subseteq U$ which are smooth $d$-dimensional
submanifolds without boundary, and such that $M$ is closed as a
subset. When the dimension is not important, we may simply write $\Psi(U)$.
\end{definition}
If $V \subseteq U$, there is a restriction map $\Psi(U) \to \Psi(V)$
given by $M \mapsto M \cap V$. This makes $\Psi(U)$ into a sheaf of
sets. We now define three topologies on $\Psi(U)$. The first two are
used only as intermediate steps for defining the third. In Theorem \ref{thm:sheaf} below we prove that the third topology makes $\Psi(-)$ into a \emph{sheaf of topological spaces}.
\textbf{Step 1.} We first define the \emph{compactly supported
topology}. We will write $\Psi(U)^{\mathit{cs}}$ for $\Psi(U)$
equipped with this topology. In fact, $\Psi(U)^{\mathit{cs}}$ will
be an infinite-dimensional smooth manifold, in which a neighbourhood of
$M \in \Psi(U)^{\mathit{cs}}$ is homeomorphic to a neighbourhood of
the zero-section in the vector space $\Gamma_c(NM)$ consisting of
compactly supported sections of the normal bundle $NM$ of $M \subseteq
U$.
Let $C^\infty_c(M)$ denote the set of compactly supported smooth
functions on $M$. Given a function $\epsilon: M \to (0,\infty)$ and
finitely many vector fields $X = (X_1, \dots, X_r)$ on $M$, let
$B(\epsilon, X)$ denote the set of functions such that $|(X_1 X_2
\dots X_r f)(x)| < \epsilon(x)$ for all $x$. Declare the family of
sets of the form $f + B(\epsilon,X)$ a subbasis for the topology on
$C^\infty_c(M)$, as $f$ ranges over $C^\infty_c(M)$, $\epsilon$ over
functions $M \to (0,\infty)$, and $X$ over $r$-tuples of vector
fields, and $r$ over non-negative integers. This makes
$C^\infty_c(M)$ into a locally convex vector space.
We define the normal bundle $NM$ to be the subbundle of $\epsilon^n$
which is the orthogonal complement to the tangent bundle $T M
\subseteq \epsilon^n$. This identifies $\Gamma_c(NM)$ with a linear
subspace of $C^\infty_c(M)^{\oplus n}$. We topologise it as a
subspace.
By the tubular neighbourhood theorem, the standard map $NM \to \bR^n$
restricts to an embedding of a neighbourhood of the zero section.
Associating to a section $s$ its image $s(M)$ gives a partially
defined injective map
\begin{align
\Gamma_c(NM) \stackrel{c_M}{\dasharrow} \Psi(U)^{\mathit{cs}}
\end{align}
whose domain is an open set. Topologise $\Psi(U)^{\mathit{cs}}$ by
declaring the maps $c_M$ to be homeomorphisms onto open sets. This makes $\Psi(U)^{\mathit{cs}}$ into
an \emph{infinite dimensional manifold}, modelled on the topological
vector spaces $\Gamma_c(NM)$.
\textbf{Step 2.} For each compact set $K$, we define a topology on
$\Psi(U)$, called the \textit{$K$-topology}. We will write $\Psi(U)^K$ for
$\Psi(U)$ equipped with this topology.
Let
\begin{align*}
\Psi(U)^{\mathit{cs}} \stackrel{\pi_K}\to \Psi(K\subseteq U)
\end{align*}
be the quotient map that identifies elements of
$\Psi(U)^{\mathit{cs}}$ if they agree on a neighbourhood of $K$. The
image of a manifold $M \in \Psi(U)^{\mathit{cs}}$ is the germ of $M$
near $K$, and we shall also write $\pi_K(M) = M|_{K}$. Give
$\Psi(K\subseteq U)$ the quotient topology.
Now, let $\Psi(U)^K$ be the topological space with the same underlying
set as $\Psi(U)^{\mathit{cs}}$, and with the coarsest topology making
$\pi_K: \Psi(U)^K \to \Psi(K\subseteq U)$ continuous. It is a formal
consequence of the universal properties of initial and quotient
topologies that the identity map $\Psi(U)^{L} \to \Psi(U)^K$ is
continuous when $K\subseteq L$ are two compact sets. That is, the
$L$-topology is finer than the $K$-topology.
\textbf{Step 3.} Finally, let $\Psi(U)$ have the coarsest topology
finer than all the $K$-topologies. In other words, $\Psi(U)$ is the
inverse limit of $\Psi(U)^K$ over larger and larger compact sets.
\begin{example}
The simplest example is taking $\{ t \} \times \mathbb{R} \subseteq \mathbb{R}^2$ as a function of $t$. This a a path in $\Psi^1(\mathbb{R}^2)$, and as $t \to \infty$ it converges to the empty manifold $\emptyset$. This is because for each compact subset $K \subseteq \mathbb{R}^2$, $K \cap
\{t\}\times \mathbb{R}$ is empty for all sufficiently large $t$, so it converges to $\emptyset$ in the $K$-topology, for all $K$.
\end{example}
\subsection{Elementary properties and constructions}\label{sec:properties}
Let $V \subseteq U$. The restriction map $\Psi(U)^{\mathit{cs}} \to
\Psi(V)^{\mathit{cs}}$ is not continuous. We have the following
result instead.
\begin{lemma}\label{lem:r-open}
The restriction map $r: \Psi(U)^{\mathit{cs}} \to
\Psi(V)^{\mathit{cs}}$ is an open map.
\end{lemma}
\begin{proof}
Let $M \in \Psi(U)^{\mathit{cs}}$. We have the following
commutative diagram
\begin{align*}
\xymatrix{
{\Gamma_c(NM)} \ar@{-->}[r]^{c_M} & {\Psi(U)^{\mathit{cs}}} \ar[d] \\
{\Gamma_c(N(M)|_V)} \ar@{-->}[r]^-{c_{M\cap V}}\ar[u]^{z} &
\Psi(V)^{\mathit{cs}} }
\end{align*}
where $z$ denotes extension by 0, which is continuous. This induces a
partially defined right inverse $\Psi(V)^{\mathit{cs}} \to
\Psi(U)^{\mathit{cs}}$ to the restriction map. The right inverse is
defined in an open neighbourhood of every point in the image of the
restriction; this proves the claim.
\end{proof}
The following technical result will be used several times.
\begin{lemma}\label{lem:lambda}
Let $K \subseteq U$ be compact. Let $0 < 3\epsilon \leq \mathrm{dist}(K,
\bR^n - U)$ and let $\lambda: \bR^n \to [0,1]$ be a smooth function
such that $\lambda(x) = 1$ if $\mathrm{dist}(x,K) \leq \epsilon$ and
$\lambda(x) = 0$ if $\mathrm{dist}(x,K) \geq 2\epsilon$. If the support of
$\lambda$ is contained in an open set $V\subseteq U$, then multiplication
by $\lambda$ gives a (continuous!) map $\overline \lambda:
\Gamma_c(NM) \to \Gamma_c(N(M\cap V))$. If we let $z: \Gamma_c(N(M
\cap V)) \to \Gamma_c(NM)$ denote extension by zero, we have the
following diagram
\begin{align*}
\xymatrix{ \Gamma_c(N(M\cap V)) \ar[r]^-z &
\Gamma_c(NM)\ar@{-->}[d]^{c_M}
\ar@{-->}[rd]^{\pi_K \circ c_M} &\\
\Gamma_c(NM)\ar[u]^{\overline \lambda} \ar@{-->}[r]_{c_M} & \Psi(U)^{\mathit{cs}}
\ar[r]_-{\pi_K} & \Psi(K\subseteq U) }
\end{align*}
in which the triangle and the outer pentagon commute after possibly
restricting to a smaller neighbourhood of $0 \in \Gamma_c(NM)$
\end{lemma}
\begin{proof}
The pentagon commutes for sections $h \in \Gamma_c(NM)$ satisfying
$|h(x)| < \epsilon$, and this is an open condition.
\end{proof}
\begin{lemma}\label{lem:open-map}
The quotient map $\pi_K: \Psi(U)^{\mathit{cs}} \to \Psi(K\subseteq
U)$ is an open map.
\end{lemma}
\begin{proof}
We are claiming that $(\pi_K)^{-1}(\pi_K(A))$ is open for all
open $A\subseteq \Psi(U)^{\mathit{cs}}$. Let $M \in
(\pi_K)^{-1}(\pi_K(A))$. This means that $M \cap V = M'
\cap V$ for some open $V \supseteq K$ and some $M' \in A$. Now
the composition $\pi_K \circ c_M: \Gamma_c(NM)
\dashrightarrow \Psi(U)^{\mathit{cs}} \to \Psi(K \subseteq U)$ can
be factored as in Lemma~\ref{lem:lambda}. Thus, if we want to check
that some point $M \in (\pi_K)^{-1}(\pi_K(A))$ is
interior, it suffices to check that the inverse image of
$\pi_K(M)$ in $\Gamma_c(N(M\cap V))$ is a neighbourhood of
0.
This holds for $M' \in A \subseteq
(\pi_K)^{-1}(\pi_K(M))$, and hence also holds for $M$
since they agree inside $V$.
\end{proof}
\begin{lemma}\label{lem:restriction}
Let $V\subseteq U$. The injection $\rho: \Psi(K\subseteq U) \to
\Psi(K\subseteq V)$ is a homeomorphism onto an open subset.
\end{lemma}
\begin{proof}
Continuity follows from Lemma~\ref{lem:lambda}. Indeed, we get a diagram
\begin{align*}
\xymatrix{ \Gamma_c(NM) \ar[d]_{\overline{\lambda}}\ar@{-->}[r]^-{c_M} &
{\Psi(U)^{\mathit{cs}}} \ar[d]_{r} \ar[r]^-{\pi_K} &
\Psi(K\subseteq U) \ar[d]^\rho \\
\Gamma_c(N(M\cap V)) \ar@{-->}[r]^-{c_{M \cap V}} & \Psi(V)^{\mathit{cs}}\ar[r]^-{\pi_K}
& \Psi(K\subseteq V) }
\end{align*}
where the outer rectangle and right hand square commute.
This proves that $\rho\circ \pi_K\circ c_M$ is continuous. Since
$\pi_K$ is a quotient map and $c_M$ is a local homeomorphism, this
proves that $\rho$ is continuous.
To see that $\rho(A)$ is open when $A\subseteq \Psi(K \subseteq U)$
is open, let $B = (\pi_K)^{-1}(A)$. Then $\rho(A) = \pi_K\circ
r(B)$. Since $\pi_K$ and $r$ are both open maps, by
Lemmas~\ref{lem:open-map} and~\ref{lem:r-open}, $\rho(A)$ is open.
\end{proof}
\begin{theorem}\label{thm:restr-cont}
For $V \subseteq U$, the restriction map $\Psi(U) \to \Psi(V)$ is
continuous.
\end{theorem}
\begin{proof}
We need to show that the composition $\Psi(U) \to \Psi(V) \to \Psi(K
\subseteq V)$ is continuous for all compact $K \subseteq V$. This
follows from Lemma~\ref{lem:restriction}.
\end{proof}
\begin{lemma}\label{lem:cover}
Let $K_i \subseteq U$ be compact, $i = 1, \dots, r$, and let $K =
\cup_i K_i$. Then the diagonal map $\delta: \Psi(U)^K \to \prod
\Psi(U)^{K_i}$ is a homeomorphism onto its image.
\end{lemma}
\begin{proof}
Continuity of $\delta$ follows from continuity of each $\Psi(U)^K
\to \Psi(U)^{K_i}$. Let $\Delta = \delta(\Psi(U)^K)$. We need to
see that $\delta: \Psi(U)^K \to \Delta$ is open. We have a diagram
\begin{align*}
\xymatrix{
\Psi(U)^K \ar[r]^-\delta & {\Delta} \ar[d]_\pi\ar[r] & {\prod
\Psi(U)^{K_i}} \ar[d]^{\prod \pi_{K_i}}\\
&{D} \ar[r] & {\prod \Psi(K_i \subseteq U)},
}
\end{align*}
where $D = (\prod \pi_{K_i})(\Delta)$, the horizontal maps are
inclusions of subsets, and $\pi$ is the restriction of $\prod
\pi_{K_i}$. A set-theoretic consideration shows that for any subset
$A \subseteq \Psi(U)$ we have
\begin{align}
\label{eq:100}
\pi^{-1} \pi \delta(A) = \big(\prod \pi_{K_i}^{-1}
\pi_{K_i}(A)\big) \cap \Delta.
\end{align}
Now consider the diagram
\begin{align}\label{eq:101}
\begin{aligned}
\xymatrix{
{\Delta} \ar[r]^\pi & D\ar[d]^p\\
{\Psi(U)^K}\ar[u]^\delta \ar[r]^-{\pi_K} & \Psi(K\subseteq U), }
\end{aligned}
\end{align}
where $p$ is defined by commutativity of the diagram:
$p(\pi\delta(M)) = \pi_K(M)$. This is well defined because both
$\pi$ and $\delta$ are surjective, and because if $\pi \delta(M) =
\pi\delta(M')$, then $M$ and $M'$ agree in a neighbourhood of each
$K_i$ and hence agree in a neighbourhood of $K$, so $\pi_K(M) =
\pi_K(M')$. By Lemma~\ref{lem:open-map}, the open neighbourhoods of
$M \in \Psi(U)^K$ are precisely of the form $\pi_K^{-1}\pi_K(A)$,
for $M \in A\subseteq \Psi(U)^{\mathit{cs}}$ open. We need to prove
that the set
\begin{align*}
\delta (\pi_K^{-1}\pi_K(A)) \subseteq \Delta
\end{align*}
is a neighbourhood of $\delta(M)$. By diagram~(\ref{eq:101}), we can
replace $\pi_K(A)$ by $p\pi\delta(A)$, and replace
$\delta\pi_K^{-1}$ by $\pi^{-1}p^{-1}$. Using~(\ref{eq:100}) this
gives
\begin{align*}
\delta (\pi_K^{-1}\pi_K(A)) = \pi^{-1}p^{-1}p\pi\delta(A)
\supseteq \pi^{-1}\pi \delta(A) = \big( \prod \pi_{K_i}^{-1}
\pi_{K_i}(A) \big) \cap \Delta,
\end{align*}
which is an open subset of $\Delta$, containing $\delta(M)$.
\end{proof}
\begin{theorem}\label{thm:sheaf}
Let $f: X \to \Psi(U)$ be a map such that each $x \in U$ has an open
neighbourhood $U_x\subseteq U$ such that the composition $X \to
\Psi(U) \to \Psi(U_x)$ is continuous. Then $X \to \Psi(U)$ is
continuous.
\end{theorem}
\begin{proof}
We need to show that $f: X \to \Psi(U)^{K}$ is continuous for any
compact $K \subseteq U$. $K$ is covered by finitely many of the
$U_x$'s, say $U_1, \dots, U_r$. Pick $K_i \subseteq U_i$ compact
with $K \subseteq \cup K_i$. Then the composition
\begin{align*}
X \xrightarrow{f} \Psi(U)^{K_i} \to \Psi(K_i \subseteq U) \to
\Psi(K_i \subseteq U_i)
\end{align*}
is continuous for each $i$ by assumption. By Lemma~\ref{lem:restriction}, the composition $X \to \Psi(U) \to \Psi(K_i
\subseteq U)$ is continuous, and hence each $X \to \Psi(U)^{K_i}$ is
continuous. It now follows from Lemma~\ref{lem:cover} that $X \to
\Psi(U)^K$ is continuous as required.
\end{proof}
\begin{lemma}
Let $\Emb(U,V)$ denote the space of embeddings of one open subset of $\mathbb{R}^N$ into another, and let $j_0 \in
\Emb(U,V)$. Then there exists a partially defined map
\begin{align*}
\Emb(U,V) \stackrel{\varphi}{\dashrightarrow} \Diff_c(U),
\end{align*}
defined in an open neighbourhood of $j_0$, such that $j(x) = j_0
\circ (\varphi(j))(x)$ for $x$ in a neighbourhood of $K$.
\end{lemma}
\begin{proof}
Pick a compactly supported function $\lambda: U \to [0,1]$ with $K
\subseteq \mathrm{int}(\lambda^{-1}(0))$. Then let
\begin{align*}
(\varphi(j))(x) = (1-\lambda(x))x + \lambda(x)(j_0^{-1} \circ j)(x),
\end{align*}
which defines a compactly supported diffeomorphism $U \to U$ for all
$j$ sufficiently close to $j_0$.
\end{proof}
\begin{theorem}\label{thm:action-cont}
The map
\begin{align*}
\Emb(U,V) \times \Psi(V) \to \Psi(U),
\end{align*}
given by $(j,M) \mapsto j^{-1}(M)$, is continuous.
\end{theorem}
\begin{proof}
It suffices to see that the composition
\begin{align*}
\Emb(U,V) \times \Psi(V) \to \Psi(U) \to \Psi(K \subseteq U)
\end{align*}
is continuous in a neighbourhood of each $\{j_0\} \times \Psi(V)$, for each
$K\subseteq U$ compact. But this follows from the previous lemma,
which implies that the map factors through $\Diff_c(U)$, which acts continuously on $\Psi(U)$.
\end{proof}
\subsection{Tangential structures}
We will study analogues of the point-set topological results from the
previous section, where all manifolds $M \in \Psi_d(U)$ are equipped
with some tangential structure. Examples are orientations, spin structures, almost complex structures, etc.
\begin{definition}
Let $\theta: \mathscr{X} \to BO(d)$ be a Serre fibration. A $\theta$-structure on $M \in \Psi_d(U)$ is a bundle map (i.e.\ fibrewise linear
isomorphism) $\ell: TM \to \theta^*\gamma$. Let $\Psi_\theta(U)$
denote the set of pairs $(M,\ell)$, where $M \in \Psi_d(U)$ and $\ell$ is a
$\theta$-structure on $M$.
More generally, if $\theta : \mathscr{X} \to BO(d+k)$ is a Serre fibration, a $\theta_d$-structure on $M \in \Psi_d(U)$ is a bundle map $\ell : \epsilon^k \oplus TM \to \theta^*\gamma$. Let $\Psi_{\theta_d}(U)$ denote the set of pairs $(M, \ell)$, where $M \in \Psi_d(U)$ and $\ell$ is a
$\theta$-structure on $M$.
\end{definition}
Often we will omit the tangential structure and just write $M \in \Psi_\theta(\mathbb{R}^n)$ instead of $(M, \ell) \in \Psi_\theta(\mathbb{R}^n)$. The second case is a simple generalisation of the first, so we will only discuss the face of a fibration over $BO(d)$.
Clearly $\Psi_\theta$ is again a sheaf of sets. We wish to endow it
with a topology so that it is a sheaf of topological spaces. We start by
defining the analogue of the compactly supported topology. For
$M \in \Psi(U)^{\mathit{cs}}$, let $\Bun(TM,\theta^*
\gamma)$ be the space of bundle maps, topologised in the
compact-open topology, and let $\Gamma_c(NM) \times
\Bun(TM,\theta^*\gamma)$ have the product topology. For $s \in
\Gamma(NM)$ close to the zero section, we have $c_M(s) =
s(M) \in\Psi(U)^{\mathit{cs}}$. We also get the bundle
isomorphism $Ds: TM \to T(c_M(s))$ and hence $(Ds)^{-1}
\circ l$ is a $\theta$-structure on $c_M(s)$. This gives a map
\begin{align*}
c_M^\theta: \Gamma_c(NM)\times \Bun(TM,\theta^*\gamma)
\dashrightarrow \Psi_\theta(U)^{\mathit{cs}}
\end{align*}
viz.\ $(s,\ell) \mapsto (s(M), (Ds)^{-1} \circ \ell)$. It is clear that
$c_M^\theta$ is injective and that the image is $u^{-1}(\mathrm{Im}(c_M))$,
where $u: \Psi_\theta(U) \to \Psi(U)$ is the forgetful map $(M,\ell)
\mapsto M$. Topologise $\Psi_\theta(U)^{\mathit{cs}}$ by declaring
the maps $c_M^\theta$ homeomorphisms onto open sets.
Then define $\Psi_\theta(K\subseteq U)$, $\Psi_\theta(U)^K$ and
$\Psi_\theta(U)$ from $\Psi_\theta(U)^{\mathit{cs}}$ as in ``Step 2''
and ``Step 3'' in \S\ref{sec:definitions}. It is not hard to
modify the proofs of Theorems~\ref{thm:restr-cont}, \ref{thm:sheaf}
and \ref{thm:action-cont} to work also in the presence of $\theta$
structures. We summarise the result in the following theorem.
\begin{theorem}
$\Psi_\theta$ is a sheaf of topological spaces on $\bR^n$. The map
\begin{align*}
\Emb(U,V)\times \Psi_\theta(U) \to \Psi_\theta(V)
\end{align*}
is continuous.
\end{theorem}
\begin{proof}[Proof sketch]
First we have, with the same proof, an analogue of
Lemma~\ref{lem:lambda} where the two $\Psi$ are replaced by
$\Psi_\theta$ and all three spaces of sections of normal bundles in
the diagram are replaced with their product with $\Bun(TM,
\theta^*\gamma)$. Once that is established, the $\theta$
analogues of Theorems~\ref{thm:restr-cont} and \ref{thm:sheaf} are
proved in the exact same way. This proves that $\Psi_\theta$
is a sheaf of spaces.
An embedding $j: U \to V$ gives a map $\Psi_\theta(U) \to
\Psi_\theta(V)$ by $(M,\ell) \mapsto (j^{-1}M, \ell \circ
(Dj|_{j^{-1}(M)}))$. This defines the map in the theorem, and its
continuity is proved exactly as in Theorem~\ref{thm:action-cont}.
\end{proof}
\subsection{Smooth maps}
In practice it can be tedious to check continuity of a map $X \to \Psi_\theta(\mathbb{R}^n)$. If $X$ is smooth there is an easier property to verify.
\begin{definition}
Let $f: X \lra \Psi_\theta(U)$ be a continuous map, and write $f(x) = (M_x, \ell_x)$. Define the \textit{graph} of $f$ to be the space of pairs
$$\Gamma(f) = \cup_{x \in X} \{x\} \times M_x \subset X \times U$$
and the vertical tangent bundle to be
$$T^v \Gamma(f) = \cup_{x \in X} \{x\} \times TM_x \subset X \times TU,$$
both with the subspace topology. The $\ell_x$ determine a bundle map $\ell(f) : T^v \Gamma(f) \lra \theta^*\gamma$.
\end{definition}
\begin{definition}
If $X$ is a manifold, say a continuous map $f : X \to \Psi_\theta(\mathbb{R}^n)$ is \textit{smooth} if $\Gamma(f) \subseteq X \times U$ is a smooth submanifold and $\pi: \Gamma(f) \lra X$ is a submersion.
More generally say it is \textit{smooth near} $(x, u) \in X \times U$ if there are neighbourhoods $A \subset X$ of $x$ and $B \subset U$ of $u$ such that $A \to X \to \Psi_\theta(U) \to \Psi_\theta(B)$ is smooth. For a closed $C \subset X \times U$ say $f$ is \textit{smooth near} $C$ if it is smooth near every point of $B$.
\end{definition}
\begin{lemma}\label{lem:SmoothMapsAreContinuous}
If $X$ is a $k$-dimensional smooth manifold, there is a bijection between the set of smooth maps $f:X \to \Psi_\theta(U)$ and the set of pairs $(\Gamma, \ell)$ where $\Gamma \subseteq X \times U$ is a smooth $(d+k)$-dimensional submanifold that is closed as a subspace and such that $\pi_X : \Gamma \to X$ is a submersion, and $\ell : \Ker(D\pi_X : T \Gamma \to T X) \to \theta^*\gamma$ is a bundle map.
\end{lemma}
\begin{proof}
A smooth map $f$ determines a graph $\Gamma(f)$, and a bundle map $\ell(f): T^v\Gamma(f) \to \theta^*\gamma$, having the required properties.
Given a pair $(\Gamma, \ell)$, there is certainly a map of sets $f : X \to \Psi_\theta(U)$ given by $x \mapsto (\pi_X^{-1}(x),\ell|_{\pi_X^{-1}(x)} : T\pi_X^{-1}(x) \to \theta^*\gamma)$. Then $\Gamma(f) = \Gamma$, $T^v\Gamma(f) = T^v \Gamma$ and $\ell(f) = \ell$. We must just check that this $f$ is continuous.
By the local submersion theorem, we can choose local coordinates $\mathbb{R}^k \to X$ and $\mathbb{R}^n \to U$ inside which $\Gamma \subseteq X \times U$ is the image of an embedding $\mathbb{R}^{d+k} \to \mathbb{R}^k \times \mathbb{R}^n$ such that $\mathbb{R}^{d+k} \to \mathbb{R}^k$ is the projection onto the first $k$ coordinates. This produces a continuous map $\mathbb{R}^k \to \Emb(\mathbb{R}^d, \mathbb{R}^n)\times \Bun(T\mathbb{R}^d, \theta^*\gamma)$, which is smooth on the first coordinate. We may suppose that the origin in $\mathbb{R}^k$ is sent to the standard embedding $e:\mathbb{R}^d \hookrightarrow \mathbb{R}^n$. There is a map, defined near $e$, from $\Emb(\mathbb{R}^d, \mathbb{R}^n)$ to $\Emb(\mathbb{R}^d, \mathbb{R}^d) \times C^\infty(\mathbb{R}^d, \mathbb{R}^{n-d})$, and we identify the second factor with the space $\Gamma(Ne(\mathbb{R}^d))$. Let us write $B_k(r)$ for the ball of radius $r$ inside $\mathbb{R}^k$. There is also a map $(-)^{-1} : \Emb(\mathbb{R}^d, \mathbb{R}^d) \dashrightarrow \Emb(B_d(2), \mathbb{R}^d)$ given on a neighbourhood of the identity by $e \mapsto e^{-1} : B_d(2) \to \mathbb{R}^d$, as on a neighbourhood of the identity embeddings $e$ contain the ball $B_d(2)$ in their image.
Composing with the action via pullback gives a map
\begin{eqnarray*}
\Emb(\mathbb{R}^d, \mathbb{R}^n) &\dashrightarrow & \Emb(\mathbb{R}^d, \mathbb{R}^d) \times \Gamma(Ne(\mathbb{R}^d)) \\ &\dashrightarrow & \Emb(B_d(2), \mathbb{R}^d) \times \Gamma(Ne(\mathbb{R}^d)) \to \Gamma(Ne(B_d(2)))
\end{eqnarray*}
which sends an embedding $f$ to a section of $Ne(B_d(2))$ having the same image inside $B_n(1) \subset \mathbb{R}^n$.
Choosing a compactly supported function $\varphi : B_d(2) \to [0,1]$ that is identically 1 inside the unit ball and multiplying sections by it gives a continuous map to $\Gamma_c(Ne(B_d(2)))$, so we have a sequence of continuous maps
$$\mathbb{R}^k \to \Emb(\mathbb{R}^d, \mathbb{R}^n) \dashrightarrow \Gamma(Ne(B_d(2))) \overset{\cdot \varphi}\to \Gamma_c(Ne(B_d(2))).$$
In total, we obtain a continuous map
$$\mathbb{R}^k \dashrightarrow \Gamma_c(Ne(B_d(2))) \times \Bun(TB_d(2), \theta^*\gamma) \dashrightarrow \Psi_\theta(B_n(2))$$
defined near the origin, that agrees with $\mathbb{R}^k \to X \overset{f}\to \Psi_\theta(U) \to \Psi_\theta(\mathbb{R}^n)$ after restricting both to the unit ball $B_n(1) \subset \mathbb{R}^n$. In particular, by Theorem \ref{thm:sheaf} $f$ is continuous.
\end{proof}
\begin{lemma}
Let $X$ be a smooth manifold and $f: X \to \Psi(U)$ be a continuous
map. Let $V \subseteq X \times U$ be open, and $W \subseteq X
\times U$ be such that $\overline{V} \subseteq \mathrm{int}(W)$. Then there
exists a homotopy $F: [0,1]\times X \to \Psi(U)$ starting a $f$,
which is smooth on $(0,1] \times V \subseteq [0,1] \times X \times
U$ and is the constant homotopy outside $W$. Furthermore, if $f$ is
already smooth on an open set $A \subseteq V$, then the homotopy can
be assumed smooth on $[0,1] \times A$.
\end{lemma}
\begin{proof}
Let us say that the open set $W \subseteq X \times U$ is
\emph{small} if there are closed sets $K \subseteq X$, $L \subseteq
U$, such that $W \subseteq K \times L$ and the composition
\begin{align*}
K \to X \to \Psi(U) \to \Psi(L)
\end{align*}
factors through a continuous map $K \to \Psi(U)^\mathit{cs}$. Then
we can use the manifold structure on $\Psi(U)^\mathit{cs}$ to find a
homotopy $[0,1] \times K \to \Psi(U)^\mathit{cs}$ which is smooth on
$(0,1] \times K$. Using a suitable bump function we can ensure that
it is the constant homotopy outside $W \subseteq K \times U$, and
hence extends to a homotopy $[0,1]\times X \to \Psi(U)$ which is
constant outside $W$.
Next we consider the case where $W$ is the disjoint union of small
open sets. This is easy, since we can superimpose the corresponding
homotopies. After restricting the homotopy to $[0,\epsilon] \times
X$ and composing with the linear diffeomorphism $[0,1] \times
[0,\epsilon]$, we can assume that the homotopy is arbitrarily small.
For general W, we first triangulate $X \times U$ such that every
simplex is contained in a small open set. For each $p$-simplex
$\sigma \subseteq X \times U$, let $\mathrm{st}(\sigma) \subseteq X \times
U$ be the open star of $\sigma$, thought of as a vertex of the
barycentric subdivision. Let $W^p \subseteq X \times U$ be the
union of $\mathrm{st}(\sigma)$ over all $p$-simplices $\sigma$. Then $X
\times U$ is the (finite) union of the $W^p$. We can pick slightly
smaller open sets $V^p \subseteq W^p$ such that $\overline{V^p}
\subseteq \mathrm{int}(W^p)$ and such that the $V^p$ still cover $X \times
U$. Using the previous cases we can now proceed by induction as
follows: First find a homotopy $F^0: [0,1] \times X \to \Psi(U)$
which is constant outside $W^0 \subseteq X \times U$ and smooth on
$(0,1] \times V^0$. By construction, $[0,1]\times W^1$ is again a disjoint union of small sets, so we may find a homotopy $F^1: [0,1]^2 \times X \to
\Psi(U)$ starting at $F^0$ which is constant outside $[0,1] \times
W^1$ and smooth on $(0,1] \times [0,1] \times V^1$, and so on. In
the end we get a map $[0,1]^k \times X \to \Psi(U)$, and we can let
$F: [0,1] \times X \to \Psi(U)$ be the restriction to the diagonal.
\end{proof}
\section{The homotopy type of spaces of manifolds}\label{sec:SoaM}
This section is a self-contained proof of the main theorem of \cite{GMTW}.
\subsection{Constructions with tangential structures}
We describe certain constructions that can be made relating $\theta$-manifolds and their submanifolds, using the tangential structure $\theta_{d-1}$ on $(d-1)$-manifolds.
\begin{definition}\label{defn:cross-with-R}
If $M \in \Psi_{\theta_{d-1}}(\bR^{n-1})$, let $\bR \times M \subseteq
\Psi(\bR^n)$ have the $\theta$-structure induced by composing the obvious
bundle map
\begin{align*}
T(\bR^1\times M) \to \epsilon^1 \oplus TM
\end{align*}
with the bundle map $\epsilon^1 \oplus TM \to \theta^*\gamma$ defining the $\theta_{d-1}$-structure on $M$.
This defines a map
\begin{align}\label{eq:7:defn:cross-with-R}
\begin{aligned}
\Psi_{\theta_{d-1}}(\bR^{n-1}) & \to \Psi_\theta(\bR^n) \\
M &\mapsto \mathbb{R} \times M.
\end{aligned}
\end{align}
\end{definition}
The construction in Definition~\ref{defn:cross-with-R} can be
generalised a bit. Let $f : \mathbb{R} \to \Psi_{\theta_{d-1}}(\mathbb{R}^{n-1})$ be a smooth map, having graph $\Gamma(f) \subset \mathbb{R} \times \mathbb{R}^{n-1}$. Differentiating the first coordinate $x_1 : \Gamma(f) \to \mathbb{R}$ gives a short exact sequence of vector bundles
\begin{align*}
T^v(\Gamma(f)) \to T(\Gamma(f)) \to \epsilon^1
\end{align*}
and the standard inner product on $\mathbb{R}^n$ gives an
inner product on $T(\Gamma(f))$, and so a
canonical splitting
\begin{align}\label{eq:20}
\epsilon^1 \oplus T^v(\Gamma(f)) \xrightarrow{\cong} T(\Gamma(f)).
\end{align}
The $\theta_{d-1}$-structure on each fibre of $\Gamma(f)$ is described by a vector bundle map
\begin{align}
\label{eq:21}
\epsilon^1 \oplus T^v(\Gamma(f)) \to \theta^*\gamma.
\end{align}
\begin{definition}\label{defn:take-theta-graph}
Let
\begin{align*}
C^\infty(\bR,\Psi_{\theta_{d-1}}(\bR^{n-1})) &\to
\Psi_\theta(\bR^n)\\
f & \mapsto \Gamma(f)
\end{align*}
be the map described above, where $\Gamma(f)$ is given the $\theta$
structure obtained by composing (\ref{eq:21}) with the inverse of~(\ref{eq:20}).
\end{definition}
The process in Definition~\ref{defn:cross-with-R} is the special case
of Definition~\ref{defn:take-theta-graph} where the path $f$ is constant.
There is a partially defined reverse process which decreases
dimensions of manifolds by one. Let $M \in
\Psi_\theta(\bR^n)$, and again let $x_1: M \to \bR$ be the projection to the
first coordinate in $\mathbb{R}^n$. If $a \in \bR$ is a regular value of $x_1$, then
$M_a = M \cap x_1^{-1}(a)$ is a smooth manifold and we have a short exact
sequence
\begin{align*}
TM_a \to TM|_{M_a} \to \epsilon^1.
\end{align*}
Using the inner product on $TM$ induced by $M \subseteq \bR^n$ we get a
splitting
\begin{align}
\label{eq:22}
\epsilon^1 \oplus TM_a \xrightarrow{\cong} TM|_{M_a}.
\end{align}
\begin{definition}\label{defn:IntersectionMap}
Let
\begin{align}\label{eq:6:defn:IntersectionMap}
\begin{aligned}
\Psi_\theta(\bR^n) & \dashrightarrow
\Psi_{\theta_{d-1}}(\bR^{n-1})\\
M & \mapsto M_a = M \cap x_1^{-1}(a)
\end{aligned}
\end{align}
be the partially defined map which gives $M_a$ the
$\theta$-structure induced by composing~(\ref{eq:22}) with the
bundle map $TM \to \theta^*\gamma$ defining the $\theta$-structure
on $M$.
\end{definition}
Finally we discuss an extended functoriality of $\Psi_\theta$. Let $M \in \Psi_\theta(\bR^n)$ and let $F: \bR^n \to \bR^n$ be a
map which is transverse to $M$. Then $W = F^{-1}M$ is again a
$d$-manifold, but in general has no induced $\theta$-structure. For example if $F: \bR^3
\to \bR^3$ given by $F(x) = (0,0,|x|^2)$ then $F^{-1}(\bR^2 \times
\{1\}) = S^2$, but the canonical framing of $\bR^2 \times \{1\}$ cannot
induce a framing of $S^2$.
It is, however, possible to define such an induced structure in special cases. The following will be needed below. Let $\varphi: \bR \to \bR$ be a
smooth map with $\varphi' \geq 0$. Let $F = \varphi \times \mathrm{Id}: \bR^n \to
\bR^n$. If $M \in \Psi(\bR^n)$ and $F$ is transverse to $M$, then $W =
F^{-1}(M)$ is again a $d$-manifold. Both bundles $TW$ and $F^*(TM)$
are subbundles of $\epsilon^n$, and they are related via the
endomorphism $DF: \epsilon^n \to \epsilon^n$ by the equation
\begin{align*}
TW = DF^{-1}(F^*TM).
\end{align*}
An exercise in linear algebra shows that the composition
\begin{align*}
TW \hookrightarrow \epsilon^n \twoheadrightarrow F^*TM,
\end{align*}
the inclusion followed by orthogonal projection, defines an
isomorphism $TW \to F^*TM$. Using this isomorphism to give $W$ an
induced $\theta$-structure, we get a partially defined map
\begin{align}
\begin{aligned}
\label{eq:10}
\Psi_\theta(\bR^n) & \dashrightarrow \Psi_\theta(\bR^n)\\
M & \mapsto F^{-1}M = (\varphi \times \mathrm{Id})^{-1}(M).
\end{aligned}
\end{align}
A typical application of this construction is given by the following
lemma. Let us say that an $M \in \Psi_\theta(\bR^n)$ is
\emph{cylindrical} in $x_1^{-1}(a,b)$ if there is an $N \in
\Psi_{\theta_{d-1}}(\bR^{n-1})$ such that
\begin{align*}
(\bR \times N)|_{x_1^{-1}(a,b)} = M |_{x_1^{-1}(a,b)} \in \Psi_\theta(
x_1^{-1}(a,b)).
\end{align*}
\begin{lemma}\label{Cylindrical}
Let $f: X \to \Psi_\theta(\bR^n)$ be continuous, and let $U,V
\subseteq X$ be open, with $\overline{U} \subseteq V$. Let $a \in
\bR$ be a regular value for $x_1: f(x) \to \bR$ for all $x \in V$.
Let $\epsilon > 0$. Then there is a homotopy
\begin{align*}
f_t: X \to \Psi_\theta(\bR^n),\quad t \in [0,1]
\end{align*}
with $f_0 = f$, and
\begin{enumerate}[(i)]
\item $f_1(x)$ cylindrical in $x_1^{-1}([a-\epsilon,a+\epsilon])$
for $x \in U$.
\item The restriction to $[0,1] \times (X - V) \to
\Psi_\theta(\bR^n)$ is a constant homotopy.
\item The composition $[0,1]\times X \to \Psi_\theta(\bR^n) \to
\Psi_\theta(\bR^n - x_1^{-1}([a-2\epsilon,a+2\epsilon]))$ is a
constant homotopy.
\end{enumerate}
\end{lemma}
\begin{proof}
Choose once and for all a smooth function $\lambda: \bR \to \bR$ with
$\lambda(s) = 0$ for $|s| \leq 1$, $\lambda(s) = s$ for $|s| \geq
2$, and $\lambda'(s) > 0$ for $|s|> 1$. For $\tau \in [0,1]$, let
$\varphi_\tau(s) = (1-\tau) s + \tau \epsilon\lambda(\frac{s-a}{\epsilon})$. Then
pick a function $\rho: X \to [0,1]$ with $\mathrm{supp}(\rho) \subseteq V$
and $U \subseteq \rho^{-1}(1)$ and define the homotopy by
\begin{align*}
[0,1] \times X & \to \Psi_\theta(\bR^n)\\
(t,x) &\mapsto (\varphi_{t\rho(x)} \times \mathrm{Id})^{-1} (f(x)).\qedhere
\end{align*}
\end{proof}
Finally, let us introduce subspaces of $\Psi_\theta(\bR^n)$ which are
very important in the sequel.
\begin{definition}\label{defn:SpacesOfManifolds}
Let $\psi_\theta(n,k) \subseteq \Psi_\theta(\bR^n)$ be the subspace
defined by the condition $M \subseteq \bR^k \times (-1,1)^{n-k}$. Let $\psi_{\theta_{d-1}}(n-1,k-1) \subseteq \Psi_{\theta_{d-1}}(\mathbb{R}^{n-1})$ be the subspace defined by the condition $M \subseteq \mathbb{R}^{k-1} \times (-1,1)^{n-k}$.
\end{definition}
We see that the map~(\ref{eq:7:defn:cross-with-R}) and the partially defined
map~(\ref{eq:6:defn:IntersectionMap}) restrict to maps
\begin{align}
\begin{aligned}\label{eq:17}
\psi_{\theta_{d-1}}(n-1,k-1) &\to \psi_\theta(n,k)\\
M &\mapsto \mathbb{R} \times M,
\end{aligned}
\end{align}
\begin{align}
\begin{aligned}\label{eq:19}
\psi_\theta(n,k) &\dashrightarrow \psi_{\theta_{d-1}}(n-1,k-1) \\
M &\mapsto M_a = M \cap x_k^{-1}(a).
\end{aligned}
\end{align}
\begin{proposition}\label{prop:IdentifyingComponents}
For $k > 1$, the map~(\ref{eq:17}) induces an isomorphism on
$\pi_0$, with inverse induced by~(\ref{eq:19}). Consequently $\pi_0
\big(\psi_\theta(n,k)\big) \cong \pi_0 \big( \psi_{\theta_{d-k+1}}(n-k+1,1)\big)$.
\end{proposition}
\begin{proof}
By Sard's theorem, every element $M \in \psi_\theta(n,k)$ is in the domain of the map (\ref{eq:19}) for some $a$. We will show that the element $[M_a] \in \pi_0 \big(\psi_{\theta_{d-1}}(n-1,k-1)\big)$ is independent of $a$.
Let $M \in \psi_\theta(n,k)$ and $a < b$ be two regular values of $x_1 : M \to \mathbb{R}$. Near $x_1^{-1}(a)$ and $x_1^{-1}(b)$, the function $x_1$ has no critical points. We can perturb it, relative to these subsets, to be a Morse function (as they are dense in the space of smooth functions) and still give an embedding (as the space of embeddings is open in the space of all smooth maps). We then obtain a manifold $M'$, such that $x_1$ has isolated critical points. We may then perturb $M'$ slightly so that there are no critical points in $[a, b] \times \{0\}^{k-1} \times [-1,1]^{n-k}$. Let us still call this $M'$. As critical points are isolated, there is an $\epsilon > 0$ such that there are no critical points in $[a,b] \times (-\epsilon, \epsilon)^{k-1} \times [-1,1]^{n-k}$. Choosing an isotopy of embeddings $e_t : \mathbb{R} \to \mathbb{R}$ from the identity to a diffeomorphism onto $(-\epsilon, \epsilon)$, we can form $h_t = \mathbb{R} \times e_t^{k-1} \times [-1,1]^{n-k}$ and let $M'(t) = h_t^{-1}(W')$.
This gives a path from $M' = M'(0)$ to a manifold $M'(1)$ such that $x_1$ has no critical values in $[a,b]$. Thus there are paths
$$M_a = M'_a \overset{M_a'(t)}\leadsto M_a'(1) \overset{M_t'(1)}\leadsto M_b'(1) \overset{M_b'(t)}\leadsto M'_b = M_b.$$
We have shown that (\ref{eq:19}) gives a well defined map $\psi_\theta(n,k) \to \pi_0 \big(\psi_{\theta_{d-1}}(n-1,k-1)\big)$. This map is locally constant so it factors through $\pi_0\big( \psi_{\theta}(n,k) \big)$. The composition
$$\pi_0\big(\psi_{\theta_{d-1}}(n-1, k-1)\big) \to \pi_0\big(\psi_\theta(n,k)\big) \to \pi_0\big(\psi_{\theta_{d-1}}(n-1, k-1)\big)$$
is the identity map, so the first map is injective. To see surjectivity let $M \in \psi_\theta(n,k)$ and $a$ be a regular value of $x_1 : M \to \mathbb{R}$. Similarly to the proof of Lemma \ref{Cylindrical}, let $\varphi_t : \mathbb{R} \to \mathbb{R}$, $t \in [0,1]$ be given by $\varphi_t(s) = (1-t)\cdot (s-a) + a$. Then $M(t) = (\varphi_t \times \mathrm{Id})^{-1}(W)$ gives a path from $M$ to the cylindrical manifold $\mathbb{R} \times M_a$, where tangential structures are handled using the extended functoriality of (\ref{eq:10}). Thus the first map is also surjective, and so a bijection.
\end{proof}
\subsection{The cobordism category}\label{sec:CobordismCategoryDefinition}
Let us give a definition of the embedded cobordism category
$C_\theta(\bR^n)$ from \cite{GMTW}, using the topological sheaf $\Psi_\theta$
from the previous section. There are several versions of the
definition, but they all give homotopy equivalent categories, where we say a functor $F: C \to D$ is a \textit{homotopy equivalence of categories} if $N_kF: N_k C \to N_k D$
is a homotopy equivalence for all $k$.
\begin{definition}\label{defn:CobordismCategory}
Let $C_\theta(\mathbb{R}^n)$ have object space $\psi_{\theta_{d-1}}(n-1,0)$. The set of
non-identity morphisms from $M_0$ to $M_1$ is the set of $(t,W) \in
\bR \times \psi_{\theta}(n,1)$ such that $t > 0$ and such that there exists an
$\epsilon > 0$ such that
\begin{align*}
W|_{(-\infty,\epsilon) \times \bR^{n-1}} &= (\bR \times
M_0)|_{(-\infty,\epsilon)\times\bR^{n-1}} \in \Psi_\theta((-\infty, \epsilon)\times \mathbb{R}^{n-1})\\
W|_{(t-\epsilon,\infty) \times \bR^{n-1}} &= (\bR \times
M_1)|_{(t-\epsilon,\infty)\times\bR^{n-1}} \in \Psi_\theta((t-\epsilon, \infty)\times \mathbb{R}^{n-1}),
\end{align*}
where $\bR \times M_\nu \in \Psi_\theta(\mathbb{R}^n)$ is as explained in~(\ref{eq:7:defn:cross-with-R}). Composition in the category is defined by
\begin{align*}
(t,W) \circ (t',W') = (t + t', W''),
\end{align*}
where $W''$ agrees with $W$ near $(-\infty,t] \times \bR^{n-1}$ and
with $t \cdot e_1 + W'$ near $[t,\infty) \times \bR^{n-1}$. The total space
of morphisms is topologised as a subspace of $(\{0\} \amalg
(0,\infty)) \times \psi_\theta(n,1)$, where $(0,\infty)$ is given
the usual topology.
\end{definition}
Of course, the important part of a morphism $(a_0 < a_1, M)$ is the
part of $M$ that lies in $[a_0,a_1]\times \bR^{n-1}$, since it uniquely determines the rest of $M$.
The second version of the definition of the cobordism category is a topological
poset.
\begin{definition}
Let
\begin{align*}
D_\theta(\mathbb{R}^n) \subseteq \bR \times \psi_\theta(n,1)
\end{align*}
denote the space of pairs $(t, M)$ such that $t$ is a regular value of
$f: M \to \bR$. Topologise $D_\theta(\mathbb{R}^n)$ as a subspace and order it by
declaring $(t,M) \leq (t',M')$ if and only if $M = M'$ and $t \leq
t'$.
\end{definition}
\begin{theorem}\label{thm:CobordismCategoryEquivalence}
There is a zig-zag
$$C_\theta(\mathbb{R}^n) \overset{c}\lla D_\theta^\perp(\mathbb{R}^n) \overset{i}\lra D_\theta(\mathbb{R}^n)$$
of homotopy equivalences of categories. In particular, the classifying spaces $BC_\theta(\mathbb{R}^n)$ and $BD_\theta$ are homotopy equivalent.
\end{theorem}
\begin{proof}
We must first describe $D_\theta^\perp(\mathbb{R}^n)$. It is defined as $D_\theta(\mathbb{R}^n)$, except that we only allow pairs $(t, M)$ such that $M$ is cylindrical in $x_1^{-1}(t-\epsilon, t+\epsilon)$ for some $\epsilon>0$. The functor $D_\theta^\perp(\mathbb{R}^n) \lra D_\theta(\mathbb{R}^n)$ is by inclusion. It is a levelwise homotopy equivalence on simplicial nerves by Lemma \ref{Cylindrical}.
Let us digress for a moment to a construction similar to (\ref{eq:10}). Let $\varphi_s(a,b)$ be the function as in the following diagram, where we allow $s=\infty$ to be the obvious limit. Note that it is smooth away from the set of points $\{a-s, a, b, b+s\}$, and a diffeomorphism on $(-\infty, a-s] \cup [a,b] \cup [b+s, \infty)$.
\begin{center}
\includegraphics[bb = 152 608 274 718, scale=1]{figures/graph.eps}
\end{center}
Suppose $M \in \psi_\theta(n, 1)$ is cylindrical near $a$ and $b$, then $(\varphi_s(a,b) \times \mathrm{Id})^{-1}(M)$ defines an element of $\psi_\theta(n, 1)$ as follows. It certainly defines a smooth submanifold of $\mathbb{R} \times (-1,1)^{n-1}$ that agrees with the underlying manifold of $M$ inside $x_1^{-1}(a,b)$, is cylindrical in $x_1^{-1}(a-s,a)$ and $x_1^{-1}(b, b+s)$, and agrees with translated copies of the underlying manifold $M$ on $x_1^{-1}(-\infty, a-s]$ and $x_1^{-1}[b+s, \infty)$. The $\theta$-structure is defined to be that of $M$ on $x_1^{-1}((-\infty, a-s) \cup (a,b) \cup (b+s, \infty))$ where there are diffeomorphisms to $M$. On $x_1^{-1}[a-s,a]$ the underlying manifold is cylindrical and so agrees with $\mathbb{R} \times x_1^{-1}\{a\}$. The $\theta$-structure on $x_1^{-1}(\{a-s,a\})$ also agrees, so we can use this identification to give a $\theta$-structure on $x_1^{-1}[a-s,a]$. Similarly $x_1^{-1}[b,b+s]$.
\vspace{2ex}
The functor $D_\theta^\perp(\mathbb{R}^n) \lra C_\theta(\mathbb{R}^n)$ sends an object $(t, M)$ to $x_1^{-1}(t)$ as in (\ref{eq:6:defn:IntersectionMap}). On non-identity morphisms it is
$$(t_0 < t_1, M) \mapsto (t_1-t_0, (\varphi_\infty(t_0, t_1) \times \mathrm{Id})^{-1}(M) -t_0 \cdot e_1),$$
where $e_1 \in \mathbb{R}^n$ is the first basis vector and $-t_0\cdot e_1$ denotes the parallel translation.
On simplicial nerves there is a map in the reverse direction, $N_l C_\theta(\mathbb{R}^n) \to N_l D_\theta^\perp(\mathbb{R}^n)$, that is the inclusion of those $(0 = t_0 < t_1 < \cdots < t_{l}, M)$ such that $M$ is cylindrical on $x_1^{-1}(-\infty, 0)$ and on $x_1^{-1}(t_{l}, \infty)$. The functor can then be viewed as the self map $h_1$ of $N_l D_\theta^\perp(\mathbb{R}^n)$ that sends $(t_0 < t_1 < \cdots < t_{l}, M)$ to
$$(0 < t_1-t_0 < \cdots < t_{l}-t_0, (\varphi_\infty(t_0, t_{l}) \times \mathrm{Id})^{-1}(M) - t_0 \cdot e_1).$$
This is isotopic to the identity via the homotopy $h_s$, where $h_s$ sends $(t_0 < t_1 < \cdots < t_{l}, M)$ to
$$(t_0 - s \cdot t_0 < t_1-s \cdot t_0 < \cdots < t_{l}-s\cdot t_0, \left(\varphi_{\tfrac{s}{1-s}}(t_0, t_{l}) \times \mathrm{Id}\right)^{-1}(M) - s\cdot t_0 \cdot e_1).$$
\end{proof}
Finally we determine the homotopy type of $BC_\theta(\mathbb{R}^n) \simeq
BD_\theta(\mathbb{R}^n)$.
\begin{theorem}\label{thm:PosetModel}
The forgetful map induces a weak equivalence
\begin{align*}
BD_\theta(\bR^n) \stackrel{u}{\to} \psi_\theta(n,1).
\end{align*}
\end{theorem}
\begin{proof}
The inverse image of a point $M \in \psi_\theta(n,1)$ is a subspace
of the infinite simplex $B(\bR,\leq)$. It is the simplex whose
vertices is the space of regular values of $x_1: M \to \bR$. Points
in the inverse image of $M$ can be represented as formal affine
combinations (i.e.\ formal linear combinations where the
coefficients are non-negative and sum to 1) of regular values of
$x_1$. This inverse image is contractible (the space of vertices is
non-empty by Sard's theorem) which suggests the map might be a weak
equivalence. To give a rigorous proof we calculate the relative
homotopy groups.
Let
\begin{align*}
f: D^m &\to \psi_\theta(n,1) \\
\hat f: \partial D^m &\to BD_\theta(\bR^n)
\end{align*}
be continuous maps with $u \circ \hat f = f|_{\partial D^m}$.
For $a \in \bR$, let $U_a \subseteq D^m$ be the set of points $x$
such that $a$ is a regular value of $x_1: f(x) \to \bR$. This is an
open subset of $D^m$, so by compactness we can pick finitely many
$a_1, \dots, a_k \in \bR$ such that the $U_{a_i}$ cover $D^m$. Pick
a partition of unity $\lambda_1, \dots, \lambda_k: D^m \to [0,1]$
subordinate to the cover. Using $\lambda_i$ as a formal coefficient
of $a_i$ gives a map
\begin{align*}
g: D^m \to BD_\theta(\bR^n)
\end{align*}
which lifts $f$, i.e.\ $u \circ g = f$. Finally we produce a
homotopy between the two maps $g |_{\partial D^m}$ and $\hat f$.
Since they are both lifts of $f|_{\partial D^m}$, we can just use the affine
structure on the fibres of $u$ to give the straight-line homotopy.
This proves that the relative homotopy groups (of $BD_\theta(\bR^n)$
as a subspace of the mapping cylinder of $u$) vanish and hence the
map $u$ is a weak equivalence.
\end{proof}
We can now calculate the set of path components of $\psi_\theta(n,1)$.
We can define a product of two elements $W_1$ and $W_2$ of the space
$\psi_\theta(n,1)$ in the following way. Take the union of the
disjoint manifolds $W_1$ and $W_2 + e_2$ and scale the second
coordinate by $1/2$. This product
\begin{align*}
\psi_\theta(n,1) \times \psi_\theta(n,1) \to \psi_\theta(n,1)
\end{align*}
makes $\psi_\theta(n,1)$ into an $H$-space (in fact it is an $E_{n-1}$
space) and hence $\pi_0 (\psi_\theta(n,1))$ is a monoid. We have the
following corollary.
\begin{corollary}\label{cor:grouplike}
The monoid $\pi_0 \psi_\theta(n,1)$ is isomorphic to the monoid of
cobordism classes of $\theta_{d-1}$ manifolds in $\bR^{n-1}$. In
particular the monoid is a group.
\end{corollary}
\begin{proof}
By Theorem \ref{thm:PosetModel} and Proposition \ref{prop:IdentifyingComponents} we have
\begin{align*}
\pi_0(\psi_\theta(n,1)) = \pi_0(BD_\theta(\bR^n)) =
\pi_0(BC_\theta(\bR^n)),
\end{align*}
which can be identified by the set objects of $C_\theta$, modulo the
equivalence relation generated by the morphisms. This proves the
first claim.
The monoid structure on $\pi_0(BC_\theta(\bR^n))$ comes from an
$H$-space structure on the objects and morphisms of $C_\theta$,
defined by disjoint union in the second coordinate direction of
$\bR^n$, as for $\psi_\theta(n,1)$. To see that it is a group, let
$M \in \psi_{\theta_{d-1}}(n-1,0)$ be an object, and let $\bR \times
M \in \psi_\theta(n,1)$ be the corresponding cylindrical element.
There is another object ``$-M$'', such that $\bR \times (-M)$ is
obtained from $\bR \times M$ by changing signs of the first two
coordinates in $\bR^n$.
There is an embedding $e$ of $\mathbb{R} \times I$ into itself given by the following picture. Then $(e \times I^{n-2}) (\mathbb{R} \times M)$ gives a morphism from $M \coprod -M$ to the empty manifold, showing that $[M]$ and $[-M]$ are inverse points in the monoid structure on $\pi_0(B\mathcal{C}_\theta(\mathbb{R}^n))$.
\begin{center}
\includegraphics[bb = 145 597 264 721, scale=1]{figures/bend.eps}
\end{center}
\end{proof}
\subsection{The homotopy type of the space of all $\theta$-manifolds}
In this section we finish our proof of the main theorem of \cite{GMTW}. There is a space $\mathscr{X}(\mathbb{R}^n)$ is defined by the fibre product
\begin{diagram}
\mathscr{X}(\mathbb{R}^n) & \rTo & \mathscr{X} \\
\dTo^{\theta_{n}} & & \dTo^{\theta} \\
Gr_d(\mathbb{R}^{n}) & \rTo^{inc} & BO(d)
\end{diagram}
where we consider $BO(d)$ as the infinite Grassmannian $Gr_d(\mathbb{R}^\infty)$ and the inclusion map as that induced by $\mathbb{R}^n \subset \mathbb{R}^\infty$. Thus $\mathscr{X}(\mathbb{R}^n)$ is simply that part of $\mathscr{X}$ which lies over those $d$-planes which are contained in $\mathbb{R}^n \subset \mathbb{R}^\infty$.
\begin{theorem}\label{thm:MainThmGMTW}
There is a weak homotopy equivalence
$$B \mathcal{C}_\theta(\mathbb{R}^n) \simeq \psi_\theta(n, 1) \overset{\simeq} \lra \Omega^{n-1}\Th\big(\theta_{n}^*(\gamma_{d,n}^\perp) \to \mathscr{X}(\mathbb{R}^n)\big).$$
Letting $n$ go to infinity, we get the main theorem of \cite{GMTW},
$$B \mathcal{C}_\theta \simeq \psi_\theta(\infty, 1) \overset{\simeq} \lra \Omega^{\infty-1} \bold{MT \theta}.$$
\end{theorem}
Theorem \ref{thm:MainThmGMTW} will be proved in Theorems \ref{thm:main-cpt-3} and \ref{thm:Scanning} below. First we construct the relevant maps. Consider the map
\begin{align}\label{eq:16}
\begin{aligned}
\bR \times \psi_\theta(n,k-1) &\to \psi_\theta(n,k)\\
(t,M) &\mapsto M - t \cdot e_k,
\end{aligned}
\end{align}
where $e_k \in \bR^n$ is the $k$th standard basis vector and $M - t \cdot e_k$
denotes the inverse image of $M$ under the diffeomorphism $x \mapsto x
+ t \cdot e_k$. This is a continuous map of spaces as it is induced by the action of the subgroup $\mathbb{R} \subseteq \Diff(\mathbb{R}^n)$.
If we let $\emptyset \in \psi_\theta(n,k)$ be the
basepoint, then~(\ref{eq:16}) extends uniquely to a continuous map
$S^1 \wedge \psi_\theta(n,k-1) \to \psi_\theta(n,k)$ with adjoint map
\begin{align}\label{eq:18}
\psi_\theta(n,k-1) \to \Omega \psi_\theta(n,k).
\end{align}
\begin{theorem}\label{thm:main-cpt-3}
The map~(\ref{eq:18}) is a homotopy equivalence for $k \geq 2$.
Consequently there is a homotopy equivalence
\begin{align*}
\psi_\theta(n,1) \stackrel{\simeq}{\longrightarrow} \Omega^{n-1} \psi_\theta(n,n).
\end{align*}
\end{theorem}
The homotopy equivalence~(\ref{eq:18}) will be proved in three steps, Propositions \ref{prop:DeloopPsi}, \ref{prop:IdentifyPsi0} and \ref{prop:Psi0IntoPsiDot}. Let $k \geq 2$ be fixed. The idea of the proof is to instead prove a homotopy
equivalence $B\psi_\theta(n,k-1) \simeq \psi_\theta(n,k)$ for a suitable ``monoid
structure'' on $\psi_\theta(n,k-1)$. In fact it is convenient to work with
something which is not quite a monoid structure, but which contains the same
homotopical information. We recall the following well known lemma.
\begin{lemma}\label{lem:monoid}
Let $X_\bullet$ be a simplicial space such that the face maps induce
a homotopy equivalence $X_k \simeq X_1 \times \dots \times X_1$.
(When $k = 0$ this means that $X_0$ is contractible). Then the
natural map
\begin{align*}
X_1 \to \Omega \|X_\bullet\|
\end{align*}
is a homotopy equivalence if and only if $X_\bullet$ is
\emph{group-like} i.e.\ $\pi_0 X_1$ is a group with respect to the
product induced by $d_1: X_2 \to X_1$.
\end{lemma}
\begin{proof}
Here $\Omega \|X_\bullet\|$ should be taken to mean the space of
paths in the fat realisation $\|X_\bullet\|$ which start and end
somewhere in $X_0 \subset \|X_\bullet\|$. This space receives a natural map
from $X_1$. The lemma is a well known variant of the fact that $M
\simeq \Omega BM$ if $M$ is a group-like topological monoid. See \cite[Prop. 1.5]{SegalCatCohom}
Given any pair $a, b \in X_0$, the space $\Omega_{a,b} \|X_\bullet\|$ of paths starting at $a$ and ending at $b$ is homotopy equivalent to $\Omega \|X_\bullet\|$ defined above.
\end{proof}
Instead of defining an actual monoid structure on $\psi_\theta(n,k-1)$ we
can define a simplicial space satisfying the lemma. By abuse of
notation we will call it $N_\bullet \psi_\theta(n,k-1)$ since it plays the
role of the nerve of a monoid. First we have a preliminary
definition.
\begin{definition}
Let $A \subseteq \bR$ be an open subset, and $x_k : \bR^n \to \bR$ denote the projection on the $k$th coordinate. Let
$\psi_\theta^A(n,k)\subseteq \psi_\theta(n,k)$ denote the subset defined by
requiring $M \cap x_k^{-1}(\bR - A) = \emptyset$. In particular
$\psi_\theta(n,k-1) = \psi_\theta^{(-1,1)}(n,k)$.
Let $\psi_\theta^A(n,k)' \subseteq \psi_\theta(n,k)$ be the subset defined by the
condition $M \cap [-1,1]^n \cap x_k^{-1}(\bR - A) = \emptyset$.
\end{definition}
\begin{lemma}\label{lem:monoid-properties}
\
\begin{enumerate}[(i)]
\item\label{item:4:lem:monoid-properties} The inclusion $\psi_\theta^A(n,k) \to \psi_\theta^A(n,k)'$ is a homotopy equivalence.
\item\label{item:1:lem:monoid-properties} If $A_1, \dots, A_l \subseteq \bR$ are disjoint
open sets and $A = \cup A_i$ then union of subsets defines a
homeomorphism
\begin{align*}
\prod \psi_\theta^{A_i}(n,k) \xrightarrow{\cong} \psi_\theta^A(n,k).
\end{align*}
\item\label{item:2:lem:monoid-properties} If $A = (a_0,a_1)$ is an open interval then
$\psi_\theta^A(n,k) \cong \psi_\theta(n,k-1)$.
\item\label{item:3:lem:monoid-properties} If $A = (-\infty,a)$ or $A= (a,\infty)$ for some
$a \in \bR$, then $\psi_\theta^A(n,k)$ and $\psi_\theta^A(n,k)'$ are
contractible.
\end{enumerate}
\end{lemma}
\begin{proof}
For part~(\ref{item:4:lem:monoid-properties}), we prove that the inclusion is a deformation
retract. Pick an isotopy $e_t: \bR \to \bR$, $t \in [0,1]$ of
embeddings, with $e_0$ the identity and $e_1(\bR) = (-1,1)$. Let
\begin{align*}
j_t = (e_t)^{k-1} \times (\mathrm{Id})^{n-k+1}: \bR^n \to \bR^n.
\end{align*}
This gives a path $[0,1] \to \Emb(\bR^n,\bR^n)$, and we define a
homotopy
\begin{align*}
\psi_\theta(n,k) \stackrel{h_t}{\longrightarrow} \psi_\theta(n,k),\quad t \in [0,1]
\end{align*}
by $h_t(M) = e_t^{-1}(M)$. This restricts to a deformation
retraction because the homotopy preserves $\psi_\theta^A(n,k)$ and
$\psi_\theta^A(n,k)'$, starts at the identity, and $h_1$ maps $\psi_\theta^A(n,k)'$
into $\psi_\theta^A(n,k)$.
(\ref{item:1:lem:monoid-properties}) is obvious and for~(\ref{item:2:lem:monoid-properties}), a homeomorphism is
obtained by stretching the $k$th coordinate using the affine map $f: \bR
\to \bR$ with $f(a_0)=-1$ and $f(a_1)=1$. For~(\ref{item:3:lem:monoid-properties}) it suffices to consider $\psi_\theta^A(n,k)$ with $A =
(-\infty,a)$. We define a homotopy by
\begin{align*}
[0,1] \times \psi_\theta^{A}(n,k) &\to \psi_\theta^{A}(n,k)\\
(t,M) & \mapsto M - \tfrac{t}{1-t} \cdot e_k \\
(1,M) & \mapsto \emptyset.
\end{align*}
This contracts $\psi_\theta^A(n,k)$ to the point $\emptyset$.
\end{proof}
\begin{definition}\label{defn:cat-as-poset}
Let $N_\bullet\psi_\theta(n,k-1)$ be the simplicial space defined by
letting $N_l\psi_\theta(n,k-1)\subseteq \bR^{l+1} \times \psi_\theta(n,k)$ be the
set of pairs $(t,M)$ such that $0 < t_0 \leq \dots \leq t_l < 1$ and
such that $M \in \psi_\theta^A(n,k)$ with $A = \bR - \{t_0 , \dots t_l\}$.
\end{definition}
\begin{lemma}
$N_\bullet \psi_\theta(n,k-1)$ satisfies the assumptions of Lemma~\ref{lem:monoid}.
\end{lemma}
\begin{proof}
This follows immediately from Lemma~\ref{lem:monoid-properties}.
\end{proof}
It remains to see that $\pi_0N_1 \psi_\theta(n,k-1)$ is a group, but this follows
from Corollary \ref{cor:grouplike}. We have proved
\begin{proposition}\label{prop:DeloopPsi}
The natural map
\begin{align*}
\psi_\theta(n,k-1) \to \Omega |N_\bullet \psi_\theta(n,k-1)|
\end{align*}
is a homotopy equivalence.\qed
\end{proposition}
\begin{proposition}\label{prop:IdentifyPsi0}
The forgetful map $(t,M) \mapsto M$ defines a homotopy equivalence
\begin{align*}
|N_\bullet\psi_\theta(n,k-1)| \to \psi_\theta^0(n,k),
\end{align*}
where $\psi_\theta^0(n,k)\subseteq \psi_\theta(n,k)$ is the subset where $M$
satisfies $M \cap [-1,1]^n \cap x_k^{-1}(t) = \emptyset$ for some $t
\in (0,1)$.
\end{proposition}
\begin{proof}
There is a simplicial space $N_\bullet \psi_\theta(n,k-1)'$ defined as in
Definition~\ref{defn:cat-as-poset}, but with $\psi_\theta^A(n, k-1)'$ instead of
$\psi_\theta^A(n,k-1)$. By Lemma~\ref{lem:monoid-properties}(\ref{item:4:lem:monoid-properties}), the
inclusion $N_\bullet\psi_\theta(n,k-1) \to N_\bullet\psi_\theta(n,k-1)'$ is a levelwise homotopy equivalence, so it suffices to prove that $|N_\bullet
\psi_\theta(n,k-1)'| \to \psi_\theta^0(n,k)$ is a homotopy equivalence.
The fibre over $M \in \psi_\theta^0(n,k)$ is the classifying space of the
poset of $t \in (0,1)$ such that $M \cap [-1,1]^n \cap x_k^{-1}(t) =
\emptyset$, ordered as usual. This is a totally ordered
non-empty set, so the realisation is a simplex. Therefore the map is a weak homotopy equivalence as in the proof of Theorem \ref{thm:PosetModel}.
\end{proof}
\begin{proposition}\label{prop:Psi0IntoPsiDot}
Let $\psi_\theta^\emptyset(n,k)\subseteq \psi_\theta(n,k)$ denote the path component
of empty set. Then the inclusion
\begin{align*}
\psi_\theta^0(n,k) \to \psi_\theta^\emptyset(n,k)
\end{align*}
is a weak homotopy equivalence.
\end{proposition}
\begin{proof}
For ease of notation we will switch the roles of the coordinates
$x_1$ and $x_k$ in this proof. Thus $\psi_\theta^0(n,k)$
becomes the subspace consisting of manifolds $M$ satisfying $M \cap x_1^{-1}(a) \cap [-1,1]^n = \emptyset$ for some $a$. We prove
that the relative homotopy groups vanish. Let
\begin{align*}
f: (D^m, \partial D^m) \to (\psi_\theta^\emptyset(n,k),
\psi_\theta^0(n,k))
\end{align*}
represent an element of relative $\pi_m$. We may assume $f$ is smooth.
For each $a \in \bR$, let $U_a \subseteq D^m$ the set of points $y
\in D^m$ such that $x_1: f(y) \to \bR$ has no critical points
in $\{a\} \times I^{k-1} \times \bR^{n-k}$. This is an open
condition on $f(y)$, so all $U_a\subseteq D^m$ are open. As in the proof of Lemma \ref{lem:monoid-properties}(\ref{item:4:lem:monoid-properties}), pick an
isotopy of embeddings $e_t: \bR \to \bR$, $t \in [0,1]$ starting at
$e_0 = \mathrm{Id}$ and ending at a diffeomorphism $e_1: \bR \to
(-1,1)$. Let $h_t: \bR^n \to \bR^n$ be given by $h_t = \mathrm{Id}
\times e_t^{k-1} \times \mathrm{Id}$ and define a homotopy $f_t: D^m
\to \psi_\theta^\emptyset(n,k)$, $t \in [0,1]$ by
\begin{align*}
f_t(x) = h_t^{-1}(f(x)).
\end{align*}
This gives a relative homotopy that starts at $f_0 = f$, and ends at a map
$f_1$, where in $f_1(x) = h_1^{-1}(f(x))$ we have ``stretched''
the space
\begin{align*}
\{a\} \times (-1,1)^{k-1} \times \bR^{n-k}
\end{align*}
to be all of $\{a\} \times \bR^{n-1}$. Therefore $x_1: f_1(x) \to
\bR$ now has no critical points in $\{a\} \times \bR^{n-1}$ for all $x
\in U_a$. We now replace our old $f$ by the homotopic $f_1$.
By compactness of $D^m$, we can refine the cover by the $U_a$ to a
cover by finitely many contractible sets $V_1, \dots, V_r \subseteq
D^m$, with corresponding regular values $a_i \in \bR$. After
possibly perturbing the $a_i$, we can assume they are different. We may choose an $\epsilon>0$ such that the intervals $(a_i-2\epsilon, a_i+2\epsilon)$ are disjoint. By
Lemma \ref{Cylindrical} we can suppose, after possibly changing $f$ by
a homotopy concentrated in $(a_i-2\epsilon, a_i+2\epsilon)$, that for all $y \in V_i$, the element $f(y) \in
\psi_\theta(n,k)$ is cylindrical in $x_1^{-1}(a_i-\epsilon, a_i+\epsilon)$, i.e.\ that there is an element $\lambda_i(y)
\in \psi_{\theta_{d-1}}(n-1,k-1)$ such that the two elements
\begin{align*}
f(y) \text{ and } (\bR \times \lambda_i(y)) \in \Psi_\theta(\bR^n)
\end{align*}
become equal in $\Psi_\theta(x_1^{-1}(a_i-\epsilon,a_i+\epsilon))$.
Since $\pi_0 \big(\psi_\theta(n-1,k-1)\big) = \pi_0 \big(\psi_\theta(n,k)\big)$ by Proposition
\ref{prop:IdentifyingComponents}, the element $\lambda_i(y)$ must be in the
basepoint component of $\psi_\theta(n-1,k-1)$ and since the $V_i$ are
contractible, we can pick a smooth homotopy
\begin{align*}
[0,1] \times V_i \overset{\Lambda_i}\lra \psi_{\theta_{d-1}}(n-1,k-1),
\end{align*}
with $\Lambda_i(0,-) = \lambda_i$ and $\Lambda_i(1,-) = \emptyset$.
Pick a $\delta > 0$ with $3 \delta < \epsilon$, and a smooth
function $\rho: \bR \to [0,1]$ which is 1 on $(-\delta,\delta)$ and
has support in $(-2\delta,2\delta)$. Finally pick $\tau_i: V_i \to
[0,1]$ with compact support and with $\cup_i \tau_i^{-1}(1) = D^m$ and define a homotopy by
\begin{align*}
V_i & \overset{h_t^i}\to C^\infty(\bR,\psi_{\theta_{d-1}}(n-1,k-1)), \quad t \in [0,1]\\
x & \mapsto (b \mapsto \Lambda_i(t \tau_i(x)\rho(b - a_i),x)).
\end{align*}
The homotopy starts with the map $h_0^i$ which sends all $x$ to the constant path at
$\lambda_i(x)$. At any time $t$ the map $h_t^i(x) : \mathbb{R} \to \psi_{\theta_{d-1}}(n-1,k-1)$ is constant outside of $(a_i - 2\delta,a_i + 2\delta)$. The homotopy ends at $h_1^i(x)$, which maps $(a_i-\delta, a_i+\delta)$ to the empty set.
By taking graphs (i.e.\
composing with the function $\Gamma$ from Definition \ref{defn:take-theta-graph})
we get a homotopy of maps
\begin{align}\label{eq:27}
\begin{aligned}
V_i & \to \psi_\theta(n,k), \quad t \in [0,1]\\
x & \mapsto \Gamma(h_t^i(x))
\end{aligned}
\end{align}
which at $t=0$ is $x \mapsto \bR \times \lambda_i(x)$, so it agrees
with $f$ on $x_1^{-1}(a_i-3\delta, a_i+3\delta)$, and at $t=1$ maps any $x \in \tau_i^{-1}(1)$ to an
element which is empty inside $x_1^{-1}(a_i-\delta,a_i+\delta)$. At
any time $t$, it agrees with $f$ when restricted to $x_1^{-1}(
(a_i-3\delta,a_i+3\delta) - (a_i-2\delta,a_i+2\delta))$ so by the sheaf
property of $\Psi^\theta$ we can define a homotopy of maps $V_i \to
\psi_\theta(n,k)$ whose restriction to $x_1^{-1}(\bR -
(a-2\delta,a+2\delta))$ is the constant homotopy of $f|_{V_i}$ and
whose restriction to $x_1^{-1}(a_i-3\delta,a_i+3\delta)$ is the
homotopy~(\ref{eq:27}). This homotopy is constant outside a compact
subset of $V_i$, so it extends to a homotopy of the map $f$ which at
time $t=1$ maps $\tau_i^{-1}(1)$ into $\psi_\theta^\emptyset(n,k)$.
We have only changed $f$ inside
$x_1^{-1}(a_i-2\delta,a_i+2\delta)$, so we can carry out this
construction for other $a_j$'s as well. In the end we have
homotoped $f$ into a map to $\psi_\theta^\emptyset(n,k)$ as desired.
It remains to show that this is a relative homotopy. Suppose we have an $x \in \partial D^m$, so $f(x) \in \psi^0_\theta(n,k)$. Then there is a $t \in (0,1)$ that is a regular value of $x_1 : f(x) \to \mathbb{R}$ such that $x_1^{-1}(t) = \emptyset$ (after we have replaced $f$ by $f_1$ as described in the second paragraph). If $t$ is not in $\coprod_i (a_i-\epsilon, a_i+\epsilon)$, then all the homotopies we perform are constant near height $t$, so the level set at $t$ is always empty. If $t \in (a_i-\epsilon, a_i+\epsilon)$ then $\lambda_i = \emptyset$, so we can choose $\Lambda_i$ to be constantly $\emptyset$. The homotopy is then constant near height $t$, so the level set at $t$ is always empty and we remain in $\psi^0_\theta(n,k)$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main-cpt-3}]
We combine the three propositions to get the weak homotopy equivalences
\begin{align*}
\psi_\theta(n,k-1) \to \Omega|N_\bullet \psi_\theta(n,k-1)| \to \Omega
\psi_\theta^0(n,k) \to \Omega \psi_\theta^\emptyset (n,k) = \Omega \psi_\theta(n,k).
\end{align*}
\end{proof}
We will now explain how to identify the homotopy type of $\psi_\theta(n,n) = \Psi_\theta(\mathbb{R}^n)$.
\begin{theorem}\label{thm:Scanning}
There is a homotopy equivalence
$$\Psi_\theta(\mathbb{R}^n) \simeq \Th(\theta_{n}^*(\gamma_{d,n}^\perp) \to \mathscr{X}(\mathbb{R}^n)),$$
where $\gamma_{d,n}^\perp$ is the orthogonal complement to the tautological bundle over $Gr_d(\mathbb{R}^{n})$.
\end{theorem}
\begin{proof}
First let $\Psi_\theta(\mathbb{R}^n)^\circ \subset \Psi_\theta(\mathbb{R}^n)$ be the subspace of those $\theta$-manifolds which contain the origin. Write $L^\theta$ for the subspace of $\Psi_\theta(\mathbb{R}^n)^\circ$ consisting of \textit{linear} $\theta$-manifolds, i.e.\ those where the underlying manifold is a $d$-plane and the $\theta$-structure is constant. There is a map $\mathscr{X}(\mathbb{R}^n) \to L^\theta$ which sends a pair $(V, x)$ of a $d$-dimensional plane $V$ in $\mathbb{R}^d$ and a point $x \in \mathscr{X}$ over $i(V)$ to the pair $(V, \ell)$ of an element of $Gr_d(\mathbb{R}^n)$ and a bundle map $\ell : TV = V \times V \to \theta^*\gamma$ given by $(v, \bar{v}) \mapsto (x, i(\bar{v}))$. This gives a map of fibrations over $Gr_d(\mathbb{R}^n)$, with map on fibres over $V$ given by the inclusion $\mathrm{Fib}(\theta) \to \Bun(V, \theta^*\gamma)$. This is a homotopy equivalence (it is essentially the inclusion of the fibre of $\theta$ into its homotopy fibre, and $\theta$ has been assumed to be a Serre fibration).
Let $F_t : \mathbb{R}^n \to \mathbb{R}^n$ be scalar multiplication by $(1-t)$ and define a homotopy $S: [0,1] \times \Psi_\theta(\mathbb{R}^n)^\circ \lra \Psi_\theta(\mathbb{R}^n)^\circ$ as follows: on underlying manifolds let
$$S(t, W) = \begin{cases}
F_t^{-1}(M) &\text{if} \,\,\, t < 1\\
T_0M &\text{if} \,\,\, t=1.
\end{cases}$$
To define the $\theta$-structure on $S(t,M)$ we use the map $F_t \times \mathrm{Id} : S(t, M) \times \mathbb{R}^n \to M \times \mathbb{R}^n$, which restricts to a fiberwise linear isomorphism
\begin{align*}
T(S(t, M)) \to TM
\end{align*}
over $F_t$ (which is not the same as $DF_t$) and we give $S(t,M)$ the $\theta$-structure obtained by composition. This defines a continuous homotopy such that $S(0,W)=W$, $S(1, W) \in L^\theta$ and $S$ preserves $L^\theta$, so it gives a deformation retraction of $\Psi_\theta(\mathbb{R}^n)^\circ$ to $L^\theta \simeq \mathscr{X}(\mathbb{R}^n)$.
There is a $(n-d)$-dimensional vector bundle $\nu \lra \Psi_\theta(\mathbb{R}^n)^\circ$ which at a point $W$ has fibre $\nu_0 W$ the normal space to the manifold at 0. There is a map $e:\nu \lra \Psi_\theta(\mathbb{R}^n)$ sending $(W, v \in \nu_0 W)$ to the translated manifold $W + v$. Restricted to a neighbourhood of the 0-section in $\nu$, this gives an embedding onto the open subspace $U$ of $\Psi_\theta(\mathbb{R}^n)$ of those manifolds having a unique closest point to the origin. The complement $C$ of this embedding consists of manifolds which do not have a unique closest point to the origin; in particular, they do not contain it. The isotopy $\cdot \frac{1}{1-t} : \mathbb{R}^n \to \mathbb{R}^n$, $t \in [0,1)$ produces a map $H:[0,1) \times C \to C$, as it moves points on a manifold uniformly away from the origin. We can extend it to a continuous map $H:[0,1] \times C \to C$ by $H(1, c) = \emptyset$, which gives a contraction of $C$.
The map $C \to \Psi_\theta(\mathbb{R}^n)$ is a cofibration, as it is a push out of the cofibration $\nu \cap e^{-1}(C) \to \nu$, so collapsing $C$ gives a homotopy equivalent space. On the other hand, collapsing $C$ gives a space homeomorphic to that obtained by collapsing $\nu \cap e^{-1}(C)$ in $\nu$. This is the Thom space of $\nu$, so $\Psi_\theta(\mathbb{R}^n) \simeq \Th(\nu \to \psi_\theta(n,n)^\circ)$.\qedhere
\end{proof}
Combining Theorems \ref{thm:main-cpt-3} and \ref{thm:Scanning} finishes the proof of Theorem \ref{thm:MainThmGMTW}.
\section{Proof of the main theorems}\label{ParametrisedSurgery}
In this section we will prove our main results, Theorems \ref{thm:SubcategoryConnected} and \ref{thm:SubcategoryMonoids}. The inclusion of a full subcategory $\mathcal{D}$ of $\mathcal{C}_\theta^\bullet$ into $\mathcal{C}_\theta$ will be considered in several steps. Recall that by Theorems \ref{thm:CobordismCategoryEquivalence} and \ref{thm:PosetModel}, $B \mathcal{C}_\theta \simeq \psi_\theta(\infty,1)$. We will give a similar model for $B\mathcal{D}$. We first describe some variations on the space $\psi_\theta(\infty, 1)$.
Recall from the introduction that we have chosen a $\theta$-structure on $\mathbb{R}^d$, thought of as a vector bundle over a point, and there is an induced structure on any framed manifold called the \textit{standard $\theta$-structure}.
\begin{definition}
For $1>\epsilon>0$, let $\psi_\theta(n,1)^\epsilon \subseteq \psi_\theta(n,1)$ be the subspace where the manifold satisfies
\begin{align*}
M \subseteq \bR \times (-1,1)^{d-1} \times [0,1)^{n-d}
\end{align*}
and
\begin{align*}
L_\epsilon = \bR \times (-\epsilon,\epsilon)^{d-1} \times \{0\}
\subseteq M,
\end{align*}
and that the tangential structure $l$ is
standard on $L_\epsilon$ with respect to the framing of $L_\epsilon$
given by the vector fields $\partial/\partial x_1,
\dots, \partial/\partial x_d$. If $\epsilon > \epsilon'$, there is an inclusion $\psi_\theta(n,1)^\epsilon \to \psi_\theta(n,1)^{\epsilon'}$. Define $\psi_\theta(n,1)^\bullet = \colim_\epsilon \psi_\theta(n,1)^\epsilon$, with the colimit topology. There is a continuous injection
$\psi_\theta(n,1)^\bullet \to \psi_\theta(n,1)$.
We define $\psi_{\theta_{d-1}}(n-1, 0)^\bullet$ similarly, where $\psi_{\theta_{d-1}}(n-1, 0)^\epsilon \subseteq \psi_{\theta_{d-1}}(n-1, 0)$ is the subspace of those manifolds $M$ that satisfy $M \subseteq (-1,1)^{d-1} \times [0,1)^{n-d}$ and $(-\epsilon, \epsilon)^{d-1} \times \{0\} \subseteq M$.
\end{definition}
In Definition \ref{defn:ThetaDotCategory} we briefly described the objects and morphisms of a category $\mathcal{C}_\theta^\bullet$. A more precise definition is as follows.
\begin{definition}
Let $\mathcal{C}_\theta^\bullet(\mathbb{R}^n)$ have object space the subspace of $\psi_{\theta_{d-1}}(n-1,0)^\bullet$ consisting of connected manifolds. The set of non-identity morphisms from $M_0$ to $M_1$ is the set of $(t, W) \in \mathbb{R} \times \psi_\theta(n, 1)^\bullet$ such that $t>0$, there is an $\epsilon>0$ such that
\begin{align*}
W|_{(-\infty,\epsilon) \times \bR^{n-1}} &= (\bR \times
M_0)|_{(-\infty,\epsilon)\times\bR^{n-1}} \\
W|_{(t-\epsilon,\infty) \times \bR^{n-1}} &= (\bR \times
M_1)|_{(t-\epsilon,\infty)\times\bR^{n-1}},
\end{align*}
and such that $W \cap [0, t] \times \mathbb{R}^{n-1}$ is connected. Composition in the category is as in Definition \ref{defn:CobordismCategory}. The total space
of morphisms is topologised as a subspace of $(\{0\} \amalg
(0,\infty)) \times \psi_\theta(n,1)^\bullet$, where $(0,\infty)$ is given
the usual topology. We shall be mostly concerned with the colimit $\mathcal{C}_\theta^\bullet = \colim_{n \to \infty} \mathcal{C}_\theta^\bullet(\mathbb{R}^n)$
\end{definition}
\begin{definition}
Let the subspace
$$\psi^{nc}_{\theta}(n, 1)^\bullet \subset \psi_{\theta}(n, 1)^\bullet$$
consist of those manifolds $W$ having \textit{no compact path components}.
\end{definition}
\begin{definition}
For $\mathbf{C}$ a collection of elements of $\psi_{\theta_{d-1}}(\infty,0)$, let the subspace
$$\psi_{\theta}(\infty, 1)_{\mathbf{C}} \subset \psi_{\theta}(\infty, 1)$$
consist of those manifolds $W$ for which there exists a regular value $t \in (-1,1)$ of $x_1 : W \to \mathbb{R}$ such that $W_t$ is in $\mathbf{C}$. Write $\mathbf{Conn}$ for the collection of elements of $\psi_{\theta_{d-1}}(\infty,0)$ which are connected manifolds. Thus $\psi_{\theta}(\infty, 1)_{\mathbf{Conn}}$ consists of those $W$ such that some $W_t$ is connected, for $t \in (-1,1)$.
\end{definition}
In this definition we have insisted on regular values in $(-1,1)$ exhibiting an element as a member of the space $\psi_\theta(\infty,1)_\mathbf{C}$. If we merely ask for such regular values in $\mathbb{R}$, we get a weakly homotopy equivalent space, but the current definition will simplify certain constructions later.
\begin{theorem}\label{thm:HtpyTypeCobCat}
Let $\mathbf{C}$ be a collection of objects of $\mathcal{C}_\theta^\bullet$ (so $\mathbf{C} \subseteq \mathbf{Conn}$), and $\mathcal{D}$ be the full subcategory on $\mathbf{C}$. Then there is a weak homotopy equivalence
$$B \mathcal{D} \simeq \psi_{\theta}^{nc}(\infty, 1)^\bullet_{\mathbf{C}}.$$
\end{theorem}
\begin{proof}
This is exactly as the proof of Theorem \ref{thm:CobordismCategoryEquivalence}. Note that if $W \in \psi_\theta^{nc}(\infty,1)^\bullet$ and
$a_0$ and $a_1$ are two regular values of $x_1: W \to \mathbb{R}$ such that
$W_{a_\nu} \in \mathbf{C}$, then the fact that elements of $\mathbf{C}$ are
connected and $W$ is non-compact implies that the manifold
$W \cap x_1^{-1}([a_0,a_1])$ is also connected. The analogous $D_\theta(\mathbb{R}^\infty)_{\mathbf{C}}^\bullet$ is then the topological poset consisting of pairs $(t,W) \in \mathbb{R} \times \psi_\theta^{nc}(\infty,1)^\bullet$ with $t$ regular for $x_1 : W \to \mathbb{R}$ and $W_t \in \mathbf{C}$. This allows one to mimic the proof of Theorem \ref{thm:CobordismCategoryEquivalence}.
However the space $\psi_{\theta}^{nc}(\infty, 1)^\bullet_{\mathbf{C}}$ has such regular values in $(-1,1)$, not merely in $\mathbb{R}$. Thus it does not have a map from the poset $D_\theta(\mathbb{R}^\infty)_{\mathbf{C}}^\bullet$, but only from the full subposet $\mathcal{P} \subseteq D_\theta(\mathbb{R}^\infty)_{\mathbf{C}}^\bullet$ of pairs $(t,W)$ with $t \in (-1,1)$. We must show that the inclusion of this subposet gives an equivalence on classifying-spaces. This is so as it induces a levelwise homotopy equivalence on simplicial nerves: a homotopy inverse to the inclusion $N_k \mathcal{P} \to N_k D_\theta(\mathbb{R}^\infty)_{\mathbf{C}}^\bullet$ is giving by affine scaling in the $\mathbb{R}$ direction until all regular values lie in the interval $(-1,1)$.
As in the proof of Theorem \ref{thm:PosetModel} the fibre of $\mathcal{P} \to \psi_{\theta}^{nc}(\infty, 1)^\bullet_{\mathbf{C}}$ over $W$ becomes the simplex with vertices all possible choices of $a \in (-1,1)$ with $W_a \in \mathbf{C}$, which is contractible.
\end{proof}
To finish the proof of Theorem~\ref{thm:SubcategoryConnected} we need to
prove that the inclusion
\begin{align}
\label{eq:23}
\psi^{nc}_\theta(\infty,1)^\bullet_\mathbf{Conn} \to \psi_\theta(\infty,1)
\end{align}
is a weak equivalence. This will be done
in the rest of this section, and will be broken up into several steps
that we treat separately. The inclusion~(\ref{eq:23}) factors as
\begin{align*}
\psi^{nc}_\theta(\infty,1)^\bullet_\mathbf{Conn} \to
\psi_\theta^{nc}(\infty,1)^\bullet \to \psi_\theta(\infty,1)^\bullet \to
\psi_\theta(\infty,1).
\end{align*}
Starting from the right, we prove in
Lemma~\ref{lem:AddingBasepoint} below that the inclusion
$\psi_\theta(n,1)^\bullet \to \psi_\theta(n,1)$ is a homotopy
equivalence by giving two explicit maps that are homotopy inverse.
Then in Lemma~\ref{lem:ElimCpctComponents} we show that
$\psi_\theta^{nc}(n,1)^\bullet \to \psi_\theta(n,1)^\bullet$ is a weak
homotopy equivalence. These steps are both fairly easy. In
\S\ref{sec:ProofThmA} we prove that $\psi^{nc}_\theta(\infty,1)^\bullet_\mathbf{Conn} \to \psi_\theta^{nc}(\infty,1)^\bullet$ is a weak equivalence. For the proof we use
0-surgery to make objects connected, but in order to get a homotopy
equivalence we need a fairly elaborate procedure for doing 0-surgeries
in families.
\begin{lemma}\label{lem:AddingBasepoint}
The inclusion $i: \psi_{\theta}(n, 1)^\bullet \lra \psi_{\theta}(n, 1)$ is a weak homotopy equivalence.
\end{lemma}
\begin{proof}
For $X$ a compact space, any map $X \to \psi_\theta(n,1)^\bullet$ factors through some $\psi_\theta(n,1)^\epsilon$ by a standard property of the colimit topology. Thus it is enough to show that $\psi_\theta(n,1)^\epsilon \to \psi_\theta(n,1)$ is a weak homotopy equivalence.
We can define a product $\amalg : \psi_\theta(n,1) \times \psi_\theta(n,1) \to \psi_\theta(n,1)$ as follows. The manifold $W_1 \amalg W_2$ is obtained by taking the union of the disjoint manifolds $W_1$ and $W_2 + e_{d+1}$ and scaling the $(d+1)$st coordinate by $\tfrac{1}{2}$.
Now pick a ``cylinder'' $W_0 \in \psi_\theta(n,1)^\epsilon$ and define a map
\begin{align*}
c: \psi_\theta(n,1) & \to \psi_\theta(n,1)^\epsilon\\
M & \mapsto W_0 \amalg M.
\end{align*}
Note that $W_0 \amalg M$ does lie in $\psi_\theta(n,1)^\epsilon$ by construction of the product. Then the composition
\begin{align*}
\psi_\theta(n,1) \xrightarrow{c} \psi_\theta(n,1)^\epsilon \xrightarrow{i}
\psi_\theta(n,1)
\end{align*}
is $M \mapsto i(W_0) \amalg M$, which is homotopic to the
identity map if we pick $W_0$ in the component of the basepoint. The
effect of the reverse composition
\begin{align*}
\psi_\theta(n,1)^\epsilon \xrightarrow{i} \psi_\theta(n,1) \xrightarrow{c}
\psi_\theta(n,1)^\epsilon
\end{align*}
is shown in the following figure
\begin{center}
\includegraphics[bb = 152 623 420 716, scale=1]{figures/join1.eps}
\end{center}
and the homotopy to the identity map is as follows.
\begin{center}
\includegraphics[bb = 156 623 297 716, scale=1]{figures/join2.eps}
\end{center}
\end{proof}
\begin{lemma}\label{lem:ElimCpctComponents}
The inclusion $\psi^{nc}_{\theta}(n, 1)^\bullet \lra \psi_{\theta}(n, 1)^\bullet$ is a weak homotopy equivalence.
\end{lemma}
\begin{proof}
Again we will show that the relative homotopy groups vanish. Let $f: (D^m, \partial D^m) \lra (\psi_{\theta}(n, 1)^\bullet, \psi^{nc}_{\theta}(n, 1)^\bullet)$ be a smooth map. We will show it is relatively homotopic to a map into $\psi^{nc}_{\theta}(n, 1)^\bullet$.
Consider the height function $x_1: \Gamma(f) \lra \mathbb{R}$. After perturbing $f$, we way assume that for all $x \in D^m$, no path component of $f(x)$ is contained in $x_1^{-1}(0)$. For each $x \in X$, we can choose an $\epsilon_x > 0$ such that no path component of $f(x)$ is contained in $f(x) \cap x_1^{-1}[-\epsilon_x, \epsilon_x]$. There is an open neighbourhood $U_x \ni x$ for which this is still true.
The $\{ U_x \}_{x \in X}$ form an open cover of $X$: take a finite subcover $\{ U_i \}$, and let $\epsilon = \min(\epsilon_i)$. Then there is no path component of $f(x)$ in $x_1^{-1}[-\epsilon, \epsilon]$ for any $x \in X$. Choosing an isotopy of embeddings from the identity on $\mathbb{R}$ to a diffeomorphism that takes $\mathbb{R}$ to $(-\epsilon, \epsilon)$ defines a relative homotopy of the map $f$ into the subspace $\psi^{nc}_{\theta}(n, 1)^\bullet$.
\end{proof}
Our aim is to show that the inclusion
$$\psi^{nc}_{\theta}(\infty, 1)^\bullet_{\mathbf{Conn}} \lra \psi^{nc}_{\theta}(\infty, 1)^\bullet$$
is a weak homotopy equivalence, for any dimension $d$ and for all tangential structures $\theta$ such that $\mathscr{X}$ is connected and $S^2$ admits a $\theta$-structure. The idea for doing so is relatively simple, and proceeds by showing the relative homotopy groups $\pi_k(\psi^{nc}_{\theta}(\infty, 1)^\bullet_{\mathbf{Conn}}, \psi^{nc}_{\theta}(\infty, 1)^\bullet)$ are trivial.
It is instructive to first consider the case $k=0$. Suppose we are given a $W \in \psi^{nc}_{\theta}(\infty, 1)^\bullet$. We wish to produce a $\overline{W} \in \psi^{nc}_{\theta}(\infty, 1)^\bullet_{\mathbf{Conn}}$ and a path from $W$ to $\overline{W}$ in $\psi^{nc}_{\theta}(\infty, 1)^\bullet$. The condition to belong to $\psi^{nc}_{\theta}(\infty, 1)^\bullet_{\mathbf{Conn}}$ is the existence of a level set which is connected, so let us choose a regular value $a_0 \in (-1,1)$ of $x_1 : W \to \mathbb{R}$ and write $W_{a_0} = W \cap x_1^{-1}(a_0)$ for the level set. In general $W_{a_0}$ will not be connected, so we intend to obtain $\overline{W}$ by performing 0-surgery on $W$ near $a_0$, so as to make the level set at $a_0$ connected: the result $\overline{W}$ of this surgery will lie in $\psi^{nc}_{\theta}(\infty, 1)^\bullet_{\mathbf{Conn}}$.
To carry out 0-surgery on an element of $\psi_\theta(\infty,1)$ we need to explain how to give a height function and a $\theta$-structure on the handles we attach. We do this in the following section.
\subsection{Parametrised 0-surgery}
\label{sec:surgery}
Recall that if $W$ is an (abstract) smooth $d$-manifold and $e: S^0
\times D^d \to W$ is an embedding, then performing 0-surgery on $W$
along $e$ amounts to removing $e(S^0 \times \mathrm{int}(D^d))$ from $W$ and
gluing in $D^1 \times S^{d-1}$ along $S^0 \times S^{d-1}$. In our
setup, $W \in \psi^\theta(\infty,1)$ has more structure, the essential
parts of which are the height function $x_1: W \to \bR$ and the
$\theta$ structure $l: TW \to \theta^*\gamma$. We explain how to
extend these to the manifold $\overline{W}$ resulting from
performing surgery.
\subsubsection{$\theta$-structures}
\label{sec:structures}
The following depends critically on our assumptions on $\theta: \mathscr{X} \to BO(d)$, namely that $\mathscr{X}$ be path connected and that $S^d$ admit a $\theta$-structure.
\begin{proposition}\label{prop:extend-structures}
Let $W$ be a smooth $d$-manifold equipped with a $\theta$-structure
$l: TW \to \theta^*\gamma$ and let
\begin{align}
\label{eq:36}
e: S^0 \times D^d \to W
\end{align}
be an embedding. Let $\mathcal{K}^e(W)$ denote the result of performing
surgery on $W$ along $e$. After possibly changing $e$ by a
self-diffeomorphism of $D^d$, the $\theta$-structure
on the submanifold
\begin{align*}
W - e(S^0 \times \mathrm{int} D^d) \subseteq \mathcal{K}^e(W)
\end{align*}
extends to a $\theta$-structure on $\mathcal{K}^e(W)$.
\end{proposition}
\begin{proof}
After cutting out $e(S^0 \times \mathrm{int} D^d) \subseteq W$ and gluing in
the $D^1 \times S^{d-1}$, the $\theta$-structure on $W$ gives a
bundle map
\begin{align}\label{eq:37}
T(D^1 \times S^{d-1})|_{\partial D^1 \times S^{d-1}} \to
\theta^*\gamma,
\end{align}
and the problem becomes to extend this to a bundle map over all of
$D^1 \times S^{d-1}$. This obstruction theoretic question only
depends on the homotopy class of the $\theta$ structure on $W$ over
each sphere $e(\{\pm 1\} \times \partial D^d)$.
The space of $\theta$ structures on $D^d$ is homotopy equivalent to
the fibre of the fibration $\theta: \mathscr{X} \to BO(d)$. The assumption
that $\mathscr{X}$ be path connected implies that $\pi_1 BO(d)$ acts
transitively on this fibre, so up to possibly changing the sign of a
coordinate in $D^d$, all $\theta$ structures on it are homotopic. In
particular we can arrange that after possibly composing the embedding
(\ref{eq:36}) with a self-diffeomorphism of $S^0 \times D^d$, the
$\theta$ structures on $D^d$ induced by the embeddings $e(\pm 1, -): D^d
\to W$ are homotopic to the $\theta$ structures on the disks
\begin{align*}
D^d_\pm = \{x \in S^d | \pm x_1 \leq \tfrac{1}{2}\}
\end{align*}
with respect to some $\theta$ structure $l_0: TS^d \to \theta^*\gamma$
on $S^d$, chosen once and for all. (Our assumption on $\theta:X \to
BO(d)$ is precisely that such a choice can be made.) But then the
bundle map~\eqref{eq:37} extends over the cylinder $D^1 \times
S^{d-1}$, since that cylinder can be identified with
\[\{x \in S^d \mid |x_1| \leq \tfrac{1}{2}\}. \qedhere\]
\end{proof}
\subsubsection{Height functions}
\label{sec:level-pres-surg}
Pick once and for all a smooth family of functions
\begin{align*}
\lambda_w: [0,1] \to [w,1], \quad w \in [0,\tfrac{1}{2}]
\end{align*}
such that $\lambda(t) = t$ on $\lambda^{-1}([\tfrac{3}{4},1])$, that
$\lambda^{-1}(w) = [0,\tfrac{1}{2}]$, and such that $\lambda'(t) > 0$ on
$\lambda^{-1}(w,1)$. Define a family of smooth functions $\varphi_w$
by
\begin{align*}
D^1 \times S^{d-1} &\stackrel{\varphi_w}{\to} D^d, \quad w \in [0,\tfrac{1}{2}]\\
(t,x) & \mapsto \lambda_w(|t|)x.
\end{align*}
The domain $D^1 \times S^{d-1}$ is the disjoint union of the open
sets $U^+ = (\tfrac{1}{2},1] \times S^{d-1}$ and $U^- = [-1,-\tfrac{1}{2}) \times
S^{d-1}$, and the closed set $C = [-\tfrac{1}{2}, \tfrac{1}{2}] \times S^{d-1}$. Then the
map $\varphi_w$ restricts to diffeomorphisms
\begin{align}
\label{eq:31}
\begin{aligned}
U^+ &\stackrel{\varphi_w|_{U^+}}{\longrightarrow} D^d - wD^d\\
U^- &\stackrel{\varphi_w|_{U^-}}{\longrightarrow} D^d - wD^d
\end{aligned}
\end{align}
and the restriction to $C = [-\tfrac{1}{2},\tfrac{1}{2}]\times S^{d-1}$ can be factored as
\begin{align}\label{eq:1}
C \xrightarrow{\mathrm{proj}} S^{d-1} \xrightarrow{w} \partial (w
D^d),
\end{align}
where the second map $w: S^{d-1} \to \partial (w D^d)$ denotes
multiplication by $w$ in $\bR^n$.
Let $W$ be a smooth $d$-manifold with a smooth height function $x_1 : W \to \mathbb{R}$. If $e: S^0 \times D^d \to W$ and $w \in [0,\tfrac{1}{2}]$ is such that $x_1\circ e(+1,x) =
x_1\circ e(-1,x)$ for $|x| \leq w$ we define a map $D^1 \times S^{d-1}
\to \bR$ by
\begin{align}\label{eq:38}
(t,x) & \mapsto
\begin{cases}
x_1\circ e(+1,\varphi_w(t,x)) & \text{if $t > -\tfrac{1}{2}$}\\
x_1\circ e(-1,\varphi_w(t,x)) & \text{if $t < \tfrac{1}{2}$}.
\end{cases}
\end{align}
This is well defined because of the factorisation~\eqref{eq:1} (so
$\varphi_w(t,x)$ is independent of $t$ for $t \in [-\tfrac{1}{2},\tfrac{1}{2}]$).
The inverses of the diffeomorphisms~(\ref{eq:31}) give an embedding
\begin{align}
\label{eq:30}
\begin{aligned}
S^0 \times (D^d - wD^d) &\xrightarrow{h_w} D^1 \times S^{d-1}\\
(\pm 1, x) & \mapsto (\varphi_w|_{U^\pm})^{-1}(x)
\end{aligned}
\end{align}
whose image is the complement of $C$. The restriction of $h_w$ to $h:
S^0 \times (D^d - \tfrac{3}{4} D^d) \to D^1 \times S^{d-1}$ is independent of the
parameter $w$.
\begin{definition}\label{defn:surgery}
Let $W \in \psi^\theta(n,1)$, let $w \in [0,\tfrac{1}{2}]$, and let $e: S^0
\times D^d \to W$ be an embedding satisfying
\begin{align*}
x_1 \circ e(+1,x) = x_1 \circ e(-1,x),\quad \text{when $|x| \leq
w$.}
\end{align*}
\begin{enumerate}[(i)]
\item Let $\mathcal{K}^e_w(W)$ be the smooth
manifold obtained by gluing $D^1 \times S^{d-1}$ into $W - e(S^0
\times wD^d)$ along the embedding~\eqref{eq:30}. Let
\begin{align*}
x_1: \mathcal{K}^e_w(W) \to \bR
\end{align*}
be the function which agrees with~\eqref{eq:38} on $D^1 \times
S^{d-1}$ and with the old $x_1$ on $W - e(S^0 \times w D^d)$.
\item Pick functions $x_{2}, x_{3}, \dots: \mathcal{K}^e_w(W)\to (-1,1)$ extending the coordinate functions on $W - e(S^0 \times D^d)$
such that the resulting map $x: \mathcal{K}^e_w(W) \to \bR^\infty$ is an
embedding. Also pick, by
Proposition~\ref{prop:extend-structures}, an extension of the
$\theta$ structure on $W - e(S^0 \times w D^d)$ to a $\theta$
structure on $\mathcal{K}^e_w(W)$ (after possibly changing sign of a
coordinate on $D^d$). We use the same notation $\mathcal{K}^e_w(W)$ for
the resulting element
\begin{align*}
\mathcal{K}^e_w(W) \in \psi^\theta(\infty,1).
\end{align*}
\end{enumerate}
\end{definition}
In the second part of Definition~\ref{defn:surgery}, the
notation $\mathcal{K}^e_w(W) \in \psi^\theta(\infty,1)$ is slightly imprecise,
because the element $\mathcal{K}^e_w(W)$ depends on more data than just $W$,
$w$, and $e$. The remaining data (namely the extension of the embedding and the
$\theta$-structure) will not play any role in the applications of the
construction, so we omit it from the notation. The main properties of
$\mathcal{K}^e_w(W)$ are recorded in the following two lemmas.
\begin{lemma}\label{lemma:thin-surg}
Let $W$, $w$ and $e$ be as in Definition~\ref{defn:surgery} and let
$[a,b] = x_1\circ e(+1,wD^d) = x_1 \circ e(-1,wD^d)$. Then the
level sets $W \cap x_1^{-1}(t)$ and $\mathcal{K}^e_w(W) \cap x_1^{-1}(t)$ are
canonically diffeomorphic, for all $t \in \bR - [a,b]$.
In particular in the extreme case $w=0$, level sets agree except at
the level $a = x_1 \circ e(+1,0) = x_1 \circ e(-1,0)$.
\end{lemma}
\begin{proof}
The manifold $\mathcal{K}^e_w(W)$ is the disjoint union of the open set $W -
e(S^0 \times wD^d)$ and the closed set $C = [-\tfrac{1}{2},\tfrac{1}{2}] \times
S^{d-1}$. The lemma follows because $x_1: W\to \bR$ takes the values
in $[a,b]$ on $e(S^0 \times wD^d)\subseteq W$ and~\eqref{eq:30}
takes values in $[a,b]$ on $C \subseteq \mathcal{K}^e_w(W)$. Outside these
closed sets, $W$ and $\mathcal{K}^e_w(W)$ are canonically diffeomorphic by a
level-preserving diffeomorphism.
\end{proof}
The next lemma is concerned with what happens to the level sets that change, in
some cases. More precisely we will consider $w > 0$ and embeddings
$e: S^0 \times D^d \to W$ that are height preserving, i.e.\ such that
\begin{align}\label{eq:2}
x_1\circ e(\pm 1,y) = a + y_1, \quad\text{for $|y| \leq w$}
\end{align}
where $a = x_1 \circ e(+1,0) = x_1\circ e(-1,0)$ and $y_1$ is the
first coordinate of $y \in D^d \subseteq \bR^d$. In that case $e$
restricts to an embedding
\begin{align*}
S^0 \times (w D^d_0) \to W_a = W \cap x_1^{-1}(a),
\end{align*}
where $D^d_0 = D^d \cap x_1^{-1}(0) \cong D^{d-1}$.
\begin{lemma}\label{lemma:thick-surg}
Let $w > 0$ and $e: S^0 \times D^d \to W$ be as in~\eqref{eq:2}, and
assume $a$ is a regular value of $x_1: W \to \bR$. Then the level
set $\mathcal{K}^e_w(W) \cap x_1^{-1}(a)$ is diffeomorphic to the manifold
obtained by performing 0-surgery on $W_a = W \cap x_1^{-1}(a)$ along
the embedding
\begin{align*}
S^0 \times D^{d-1} \stackrel{\cong}{\longrightarrow} S^0 \times
(wD^d_0) \stackrel{e}{\longrightarrow} W_a.
\end{align*}
\end{lemma}
\begin{proof}
$\mathcal{K}^e_w(W)$ is obtained from $W$ by removing $e(S^0 \times \mathrm{int} (w
D^d))$ and gluing in $C = [-\tfrac{1}{2},\tfrac{1}{2}]\times S^{d-1}$. On the level set
at $a$, this removes the open set $e(S^0 \times \mathrm{int}(w D^d_0)) \cong
S^0 \times D^{d-1}$ and glues in the set
\begin{align*}
C \cap (x_1)^{-1}(a) = [-\tfrac{1}{2},\tfrac{1}{2}] \times S^{d-2},
\end{align*}
where $S^{d-2} \subseteq S^{d-1}$ is the equator.
\end{proof}
Finally, two extended versions of the construction of $\mathcal{K}^e_w(W)$ from
$W,w,e$. First a version with multiple surgeries at once.
\begin{definition}\label{defn:multiple-surg}
Let $W \in \psi^\theta(\infty,1)$ and $w \in [0,\tfrac{1}{2}]$. Let $\Lambda =
\{\lambda_1, \dots, \lambda_r\}$ be a finite set and $e: \Lambda
\times S^0 \times D^d \to W$ an embedding such that
\begin{align}
\label{eq:3}
x_1 \circ e(\lambda_i,+1,x) = x_1 \circ e(\lambda_i,-1,x),\quad \text{when $|x| \leq
w$.}
\end{align}
for all $\lambda_i \in \Lambda$. Then define
\begin{align*}
\mathcal{K}^e_w(W) = \mathcal{K}^{e(\lambda_1,-)}_w \circ \dots \circ
\mathcal{K}^{e(\lambda_r,-)}_w(W) \in \psi^\theta(\infty,1).
\end{align*}
This is well defined because of the disjointness of the embeddings
$e(\lambda_i,-): S^0 \times D^d \to W$.
\end{definition}
Of course, the properties of levelwise surgery from Lemmas
\ref{lemma:thin-surg} and \ref{lemma:thick-surg} imply similar
properties for multiple levelwise surgery. Secondly we need a
parametrised version of this construction.
\begin{proposition}\label{prop:parametrised-surg}
Let $V$ be a contractible manifold, and $f: V \to
\psi^\theta(n,1)$ and $w : V \to [0,\tfrac{1}{2}]$ smooth maps. Let
$\Lambda$ be a finite set and
$e : V \times \Lambda \times S^0\times D^d \to \Gamma(f|_V)$ be an embedding over $V$.
Assume that the triple $(f(x),w(x), e(x))$
satisfies the assumption of Definition~\ref{defn:multiple-surg} for
all $x \in V$. Then there is a smooth map
\begin{align*}
V &\to \psi^\theta(\infty,1)\\
x & \mapsto \mathcal{K}^{e(x)}_{w(x)}(f(x)),
\end{align*}
which for each $x$ is constructed as in
Definition~\ref{defn:multiple-surg}.
\end{proposition}
\begin{proof}[Proof sketch]
The manifold $f(x)$ depends smoothly on $x$. If we pick the
remaining coordinate functions $x_{2}, x_{3}, \dots:
\mathcal{K}^{e(x)}_{w(x)}(f(x)) \to \bR$ in a smooth fashion, the manifold
$\mathcal{K}^{e(x)}_{w(x)}(f(x))$ will also depend smoothly on $x$.
Under the assumption that $V$ is contractible, the obstructions to
extending the $\theta$ structure are the same as in the case where
$V$ is a point.
\end{proof}
Finally, let us remark that the surgery on elements of
$\psi_\theta(\infty,1)$ constructed in this section preserves the
subspace $\psi_\theta^{nc}(\infty,1)$. It also preserves the
subspace $\psi_\theta^{nc}(\infty,1)^\bullet$, provided the
surgery data is disjoint from some strip $L_\epsilon \subseteq W$.
\subsection{Proof of Theorem \ref{thm:SubcategoryConnected}}\label{sec:ProofThmA}
We now prove the main result of this section and finish the proof of
Theorem \ref{thm:SubcategoryConnected}.
\begin{theorem}\label{thm:main-of-4.1}
The relative homotopy groups
\begin{align}\label{eq:35}
\pi_k(\psi_\theta^{nc}(\infty,1)^\bullet,
\psi_\theta^{nc}(\infty,1)^\bullet_\mathbf{Conn})
\end{align}
vanish for all $k$.
\end{theorem}
To prove this theorem we need to construct a null homotopy of an
arbitrary continuous map of pairs
\begin{align*}
(D^k, \partial D^k) \xrightarrow{f}
(\psi_\theta^{nc}(\infty,1)^\bullet,
\psi_\theta^{nc}(\infty,1)^\bullet_\mathbf{Conn}).
\end{align*}
The null homotopy will be constructed using the parametrised surgery
developed in the previous section. We first give the local
construction. It is convenient to have a word for the following
property.
\begin{definition}
Let us say that an element $W \in
\psi_\theta^{nc}(n,1)^\bullet$ is \emph{connected at level
$a$} if $a$ is a regular value of $x_1: W \to \bR$ and if the level
set $W_a = W \cap x_1^{-1}(a)$ is path connected. Thus $W \in
\psi_\theta^{nc}(n,1)^\bullet_\mathbf{Conn}$ if and only if it is
connected at level $a$ for some $a \in (-1,1)$.
\end{definition}
In the following, we will very often use the following construction.
Let $j: \bR \to \bR$ be an embedding which is isotopic through embeddings to the identity,
has image $(-10,10)$, and has $j(t) = t$ for $t \in [-1,1]$. Then let
$s$ be the continuous map
\begin{align}\label{eq:14}
\begin{aligned}
\psi_\theta(n,1) & \xrightarrow{s} \psi_\theta(n,1)\\
M & \mapsto (j \times \mathrm{Id})^{-1}(M).
\end{aligned}
\end{align}
This ``stretching'' map will be used throughout this section.
\begin{proposition}\label{prop:local-surg}
Let $V$ be a contractible space and $f: V \to
\psi_\theta^{nc}(\infty,1)^\bullet$ be a smooth map. Let $a
\in (-1,1)$ be a regular value of $x_1: f(x) \to \bR$ for all $x \in
V$. Finally, let $\Lambda = \{\lambda_1, \dots, \lambda_r\}$ be a
finite set, and
$$p : V \times \Lambda \times [0,1] \to \Gamma(f|_V)$$
be an embedding over $V$, such that
\begin{enumerate}[(a)]
\item\label{item:2} The path $p(x, \lambda,-): [0,1] \to f(x)$ ends
somewhere in $f(x) \cap x_1^{-1}(a)$, outside the path component
of the basepoint; each non-basepoint component contains exactly
one of the ending points $p(x, \lambda,1)$.
\item\label{item:4} The path $p(x, \lambda,-): [0,1] \to f(x)$
starts somewhere in $f(x) - x_1^{-1}([-10,10])$.
\end{enumerate}
Then there is a homotopy $F: [0,1] \times V \to
\psi_\theta^{nc}(\infty,1)^\bullet$, such that
\begin{enumerate}[(i)]
\item\label{item:5} $F(0,-)$ agrees with the composition
\begin{align*}
V \xrightarrow{f} \psi_\theta^{nc}(\infty,1)^\bullet \xrightarrow{s} \psi_\theta^{nc}(\infty,1)^\bullet.
\end{align*}
\item\label{item:6} $F(1,x)$ is connected at level $a$ for all $x \in V$. In
particular,
\begin{align*}
F(\{1\} \times V) \subseteq
\psi_\theta^{nc}(\infty,1)^\bullet_\mathbf{Conn}.
\end{align*}
\item\label{item:9} If $f(x) \in
\psi_\theta^{nc}(\infty,1)^\bullet_\mathbf{Conn}$ then $F(t,x) \in
\psi_\theta^{nc}(\infty,1)^\bullet_\mathbf{Conn}$ for all $t$.
\end{enumerate}
\end{proposition}
\begin{proof}
First construct
$e : [0,1] \times V \times \Lambda \times S^0 \to \Gamma(f|_V)$
an embedding over $V$, in the following way. Recall that we have a trivialised
$L_\epsilon = \bR \times (-\epsilon,\epsilon)^{d-1} \times \{0\}
\subseteq f(x)$ for all $x$. Then pick points $x_\lambda \in
(0,\epsilon)^{d-1} \times \{0\}$ arbitrarily, but with all
$x_\lambda$ different (this is possible because $d-1 > 0$). Then
set
\begin{align*}
e(t, x,\lambda,+1,) &= p(x, \lambda,t)\\
e(t, x,\lambda,-1) &= (x_1\circ p(x, \lambda,t),x_\lambda).
\end{align*}
This has the property that $e(t, x, -)$ embeds $\Lambda \times S^0$ into a single level set of $x_1: f(x) \to \mathbb{R}$.
Thicken each embedding $e(t, x,-): \Lambda \times S^0 \to f(x)$ to
an embedding of $\Lambda \times S^0 \times D^d$. We denote the
resulting map by the same letter
$$e : [0,1] \times V \times \Lambda \times S^0 \times D^d \to \Gamma(f|_V).$$
We can arrange that at $t=1$ we have the analogue of \eqref{eq:2}, namely
\begin{align*}
x_1 \circ e(1, x,\lambda,\pm 1, v) = a + v_1, \quad \text{for
$|v| \leq w_0$}
\end{align*}
for some $w_0 > 0$.
Let $\rho: [0,1] \to [0,1]$ be a smooth function with $[0,\tfrac{1}{2}]
\subseteq \rho^{-1}(0)$ and $\rho(1) = 1$. Set $w(t) = \rho(t)w_0$.
Also pick a function $\tau: [0,1] \to [0,1]$ with $\tau(0) = 0$ and
$[\tfrac{1}{2},1] \subseteq \tau^{-1}(1)$. By
Proposition~\ref{prop:parametrised-surg} we get a smooth
homotopy
\begin{align}\label{eq:5}
\begin{aligned}
{}[0,1] \times V &\to \psi_\theta^{nc}(\infty,1)^\bullet\\
(t,x) & \mapsto \mathcal{K}^{e(\tau(t), x)}_{w(t)}(f(x)).
\end{aligned}
\end{align}
By Lemma~\ref{lemma:thick-surg} this homotopy at time 1 performs
0-surgery on the level set $x_1^{-1}(a)$. This 0-surgery forms the
connected sum of the basepoint component with all the other
components, so the result satisfies~(\ref{item:6}). At time 0, we
have performed 0-surgery on points outside of $x_1^{-1}([-10,10])$,
so we have not changed anything inside that interval. Therefore we
can let $F: [0,1]\times V \to
\psi_\theta^{nc}(\infty,1)^\bullet$ be the composition of
\eqref{eq:5} with the stretching map $s$ from~(\ref{eq:14}), and
then $F$ satisfies (\ref{item:5}) and (\ref{item:6}).
Finally, (\ref{item:9}) follows for any pair $(t,x)$ by cases. Either $\rho(t)=0$, so Lemma \ref{lemma:thin-surg} implies there is at most one level set of $F(t,x)$ in $(-1,1)$ different from its corresponding one in $f(x)$, but if $f(x) \in \psi_\theta^{nc}(\infty,1)^\bullet_{\mathbf{Conn}}$ then there is a small open interval of regular values in $(-1,1)$ having connected level sets, so $F(t,x)$ must still have a connected level set. Otherwise $\tau(t)=1$, so $F(t,x)$ is obtained from $f(x)$ by performing 0-surgery on the level set $x_1^{-1}(a) \cap f(x)$ to make it connected, so $F(t,x)$ has a connected level set.
\end{proof}
Our strategy for proving vanishing of the relative homotopy
groups~(\ref{eq:35}) is to apply Proposition~\ref{prop:local-surg}
locally to construct a null homotopy. More precisely we will use the
following corollary of Proposition~\ref{prop:local-surg} (and its
proof).
\begin{corollary}\label{corollary:homotopy-2}
Let $f: X \to \psi_\theta^{nc}(\infty,1)^\bullet$ be
smooth. Assume that there exists an open cover $X = V_1 \cup
\dots \cup V_r$ by contractible open sets $V_i \subseteq X$ and
different $a_i \in (-1,1)$ such that $a_i$ is a regular value of
$x_1:f(x) \to \bR$ for all $x \in V_i$. Finally, assume there are
finite sets $\Lambda_i$ and an embedding
$$p_i : V_i \times \Lambda_i \times [0,1] \to \Gamma(f|_{V_i})$$
over $V_i$, such that
\begin{enumerate}[(a)]
\item\label{item:10} $(V_i, f|_{V_i}, a_i, \Lambda_i, p_i)$ satisfy
the assumptions of Proposition~\ref{prop:local-surg}.
\item\label{item:11} For $i \neq j$, the images of $p_i(x, -)$ and
$p_j(x, -)$, if both defined, are disjoint.
\end{enumerate}
Then given any collection $\{U_i \subseteq V_i\}$ with $\overline{U}_i \subseteq
V_i$, there is a homotopy $H: [0,1] \times X \to
\psi_\theta^{nc}(\infty,1)^\bullet$ such that
\begin{enumerate}[(i)]
\item $H(0,-)$ agrees with the composition
\begin{align*}
X \xrightarrow{f} \psi_\theta^{nc}(\infty,1)^\bullet \xrightarrow{s} \psi_\theta^{nc}(\infty,1)^\bullet.
\end{align*}
\item $H(1,x)$ is connected at level $a_i$ for all $x \in U_i$. In
particular if the $U_i$ cover $X$, then
\begin{align*}
H(\{1\} \times X) \subseteq
\psi_\theta^{nc}(\infty,1)^\bullet_\mathbf{Conn}.
\end{align*}
\item If $f(x) \in \psi_\theta^{nc}(\infty,1)^\bullet_\mathbf{Conn}$,
then $H(t,x) \in \psi_\theta^{nc}(\infty,1)^\bullet_\mathbf{Conn}$ for
all $t$.
\end{enumerate}
\end{corollary}
\begin{proof}
For $i \in \{1,\dots, r\}$, let $[0,1]^{\{i\}} \subseteq [0,1]^r$ be
the subspace where all coordinates but the $i$th are 0. From the
paths $p_i$, the proof of Proposition~\ref{prop:local-surg}
constructs homotopies $F_i: [0,1]^{\{i\}} \times V_i \to
\psi_\theta(\infty,1)$ in two steps. First we used the path $p_i$ to
construct a map
\begin{align*}
e_i: [0,1]^{\{i\}} \times V_i \times \Lambda_i\times S^0 \times
D^d \to \Gamma(f_{V_i}),
\end{align*}
over $V_i$ that is an embedding for each point of $[0,1]^{\{i\}} \times V_i$, then we let $F_i$ be the composition
\begin{align}\label{eq:9}
\begin{aligned}
\xymatrix @R0mm {
{[0,1]^{\{i\}} \times V_i} \ar[r] & \psi_\theta(\infty,1)\\
(t,x) \ar@{|->}[r] & \mathcal{K}^{e_i(\tau(t_i), x)}_{w(t_i)}(f(x)).&
}
\end{aligned}
\end{align}
The homotopies $F_i$ all start at $s \circ f$ so we can glue them
together to a map
\begin{align}\label{eq:7}
\bigcup_i \big([0,1]^{\{i\}} \times V_i\big) \stackrel{F}{\longrightarrow}
\psi_\theta(\infty,1),
\end{align}
where the union is inside $[0,1]^r \times X$.
Recall that the embedding $e_i(t,x): \Lambda_i \times S^0 \times D^d
\to f(x)$ was constructed the following way. On $\{\lambda\} \times
\{+1\} \times D^d$ it is a disk centered at $p_i(x,\lambda,t)$, and on
$\{\lambda\} \times \{-1\} \times D^d$ it is constructed in the
standard strip $L_\epsilon \subseteq f(x)$. By the disjointness
assumption (\ref{item:11}) we may suppose that the image of
$e_i(t,x)$ is disjoint from the image of $e_j(t',x)$ for $x \in V_i
\cap V_j$ and $t,t' \in [0,1]$, and thus we get an embedding of
$(\Lambda_i \amalg \Lambda_j) \times S^0 \times D^d$. Disjointness
of the $e_i$'s means that we can iterate the
construction~\eqref{eq:9}. Namely if $T = \{i_1, \dots, i_k\}$ and
we set $V_T = V_{i_1} \cap \dots \cap V_{i_k}$, we have the homotopy
\begin{align}\label{eq:6}
\begin{aligned}
{}[0,1]^T \times V_T &\to \psi_\theta(\infty,1)\\
(t,x) & \mapsto \mathcal{K}^{e_{i_1}(\tau(t_{i_1}), x)}_{w(t_{i_1})} \circ
\dots \circ \mathcal{K}^{e_{i_k}(\tau(t_{i_k}),x)}_{w(t_{i_k})}(f(x)),
\end{aligned}
\end{align}
where the iterated surgery is defined because the corresponding
surgery data are disjoint. Composing~\eqref{eq:6} with the
stretching map $s$ gives a homotopy $F_T: [0,1]^T \times V_T \to
\psi_\theta(\infty,1)$. If we regard $[0,1]^T$ as a subset of $[0,1]^r$
in the obvious way, the various $F_T$'s are compatible and we can
glue them to a smooth map extending~\eqref{eq:7}
\begin{align}
\label{eq:8}
\bigcup_{T} \big([0,1]^T \times V_T\big) \xrightarrow{F}
\psi_\theta(\infty,1).
\end{align}
The restriction of $F$ to $\{0\} \times X$ is $s \circ f$, and if
$F(t,x)$ is defined and $t_i = 1$, then $F(t,x)$ is connected at
level $a_i$.
Finally we pick for each $i$ a bump function $\rho_i : X \to [0,1]$
supported in $V_i$, such that $U_i \subseteq \rho_i^{-1}(1)$. Let
$\rho: X \to [0,1]^r$ have the $\rho_i$ as coordinate functions.
Then the map
\begin{align*}
[0,1] \times X &\to [0,1]^r \times X\\
(t,x) & \mapsto (t \rho(x), x)
\end{align*}
has image in the domain of~\eqref{eq:8}, and we let $H$ be the
composition of the two maps.
\end{proof}
We now return to the proof of Theorem~\ref{thm:main-of-4.1}. Given a
map of pairs
\begin{align}\label{eq:MapToNullify}
(D^k, \partial D^k) \xrightarrow{f}
(\psi_\theta^{nc}(\infty,1)^\bullet,\psi_\theta^{nc}(\infty,1)^\bullet_\mathbf{Conn}),
\end{align}
we must produce a relative homotopy, deforming $f$ to a map into
$\psi_\theta^{nc}(\infty,1)^\bullet_\mathbf{Conn}$. The homotopy will be
produced by Corollary~\ref{corollary:homotopy-2} once we construct the
data $(V_i,a_i, p_i)$. We first consider the case $k=0$, in which the
map $p: D^0 \times \Lambda \times [0,1] \to W$
is produced by the following lemma. It is proved by first producing paths $p(\lambda,-)$
which are embedded but not necessarily disjoint, and then making them
disjoint in Lemma~\ref{lemma:vector-flow}.
\begin{lemma}\label{lemma:existence-paths-point}
For any $W \in \psi_\theta^{nc}(n,1)^\bullet$ and any $a
\in (-1,1)$ which is a regular value of $x_1: W \to \bR$, there is a
finite set $\Lambda = \{\lambda_1, \dots, \lambda_r\}$ and an
embedding $p: \Lambda \times [0,1] \to W$ such that
\begin{enumerate}[(a)]
\item\label{item:7} The path $p(\lambda,-): [0,1] \to W$ starts
somewhere in $W \cap x_1^{-1}(a)$, outside the path component of
the basepoint, and each non-basepoint component contains exactly
one of the starting points $p(\lambda,0)$.
\item\label{item:8} The path $p(\lambda,-): [0,1] \to W$ ends
somewhere in $f(x) - x_1^{-1}([-10,10])$.
\end{enumerate}
\end{lemma}
\begin{proof}
First pick points $p(\lambda,0) \in W \cap x_1^{-1}(a)$, one in
each non-basepoint component. The index set $\Lambda$ is then the
set of non-basepoint components. By assumption, no compact
component of $W$ is contained in $x_1^{-1}([-10,10])$, so for each
$\lambda$ there is a path in $W$ from $p(\lambda,0)$ to a point in
$W - x_1^{-1}([-10,10])$. Pick an embedded such path $p(\lambda,-):
[0,1] \to W$. These assemble to a map
\begin{align*}
p: \Lambda \times [0,1] \to W,
\end{align*}
but it need not be an embedding because the images of the paths
$p(\lambda,-)$ need not be disjoint. However, by transversality we may suppose that the paths $p(\lambda,-)$ do not intersect at their endpoints. In
Lemma~\ref{lemma:vector-flow} below we will prove that we can change
each $p(\lambda,-): [0,1] \to W$ by an isotopy of embedded paths
such that the isotoped paths are disjoint.
\end{proof}
\begin{lemma}\label{lemma:vector-flow}
Let $W$ be a smooth manifold and $p_1, \dots, p_r: [0,1] \to W$ a
set of embedded paths, such that
\begin{align*}
p_i([0,1]) \cap p_j(\partial [0,1]) = \emptyset\quad \text{for $i
\neq j$}.
\end{align*}
Then there are paths $q_i: [0,1] \to W$ such that $p_i$ is isotopic
to $q_i$ through an isotopy of embedded paths which is relative to a
neighbourhood of the endpoints, and such that the
\begin{align*}
q_i([0,1]) \cap q_j([0,1]) = \emptyset\quad \text{for $i \neq j$}.
\end{align*}
\end{lemma}
\begin{proof}
By induction on $r$ we can assume that $q_i = p_i$ already have
disjoint images for $i < r$. The derivative $q_i'(t) \in TW$ gives
a vector field on $q_i([0,1]) \subseteq W$. We can extend this to a
vector field $X$ on $W$, with the properties that $X(q_i(t)) =
q_i'(t)$ for $i = 1,\dots, r-1$. Then each $q_i$ is the restriction
of an integral curve $\overline{q}_i$ of $X$. After multiplying $X$
with a function $W \to [0,1]$ which is 1 on a neighbourhood of the
image of the $q_i$'s and vanishes outside a slightly larger
neighbourhood, we may assume that $X$ has compact support and that
$\overline{q}_i$ does not intersect $q_r$ outside the image of
$q_i$.
Finally extend $X$ to a compactly supported vector field on all of $\bR^n$, i.e.\ $X: \bR^n \to \bR^n$. Let $F: \bR \times \bR^n \to \bR^n$ be the flow of $X$. It is defined everywhere because $X$ is compactly supported, it preserves $W$, and it also fixes a neighbourhood of the end points of $p_r$. Then $q_r(s) = F(1,p_r(s))$ will work, since all intersections between $q_r$ and the $\overline{q}_i$'s have been flowed off of $q_i$.
\end{proof}
Combining Lemma~\ref{lemma:existence-paths-point} and Corollary
\ref{corollary:homotopy-2} produces a null homotopy of~(\ref{eq:MapToNullify}),
proving Theorem~\ref{thm:main-of-4.1} for $k=0$. For $k > 0$ we need
a parametrised version of Lemma~\ref{lemma:existence-paths-point}.
\begin{proposition}\label{prop:paths-exist}
For any smooth $f: D^k \to \psi_\theta^{nc}(\infty,1)^\bullet$,
there exists a covering $D^k = V_1 \cup \dots \cup V_k$ by
contractible open sets $V_i$, real numbers $a_i$, finite sets
$\Lambda_i$, and embeddings
$p_i : V_i \times \Lambda_i\times [0,1] \to \Gamma(f|_{V_i})$ over $V_i$
satisfying the assumptions of
Corollary~\ref{corollary:homotopy-2}.
\end{proposition}
\begin{proof}
First pick regular values $a_x \in (-1,1)$ of $x_1: f(x) \to \bR$, one
for each $x \in D^k$. Apply Lemma~\ref{lemma:existence-paths-point}
for $W = f(x)$ and $a = a_x$ to find a finite set $\Lambda_x$ and an
embedding
$$p_x : \{x\} \times \Lambda_x \times [0,1] \to f(x)$$
satisfying (\ref{item:7}) and (\ref{item:8}) of
Lemma~\ref{lemma:existence-paths-point}.
Then extend to a map
$p_x : V_x \times \Lambda_x \times [0,1] \to \Gamma(f|_{V_x})$
defined on a neighbourhood $V_x$ of $x$. After possibly
shrinking the $V_x$, each tuple $(V_x, f|_{V_x} ,\Lambda_x, a_x,
p_x)$ will satisfy the assumptions in
Proposition~\ref{prop:local-surg}. Using compactness of $D^k$ we
get a finite subcover
\begin{align*}
D^k = V_1 \cup \dots \cup V_r,
\end{align*}
with corresponding regular values $a_i \in (-1,1)$ and maps
$$p_i : V_i \times \Lambda_i \times [0,1] \to \Gamma(f|_{V_i})$$
satisfying condition (\ref{item:10}) of
Corollary~\ref{corollary:homotopy-2} but not the disjointness
condition (\ref{item:11}).
\vspace{2ex}
Before achieving full disjointness of all paths, let us explain how
to modify the data $(V_i, a_i, p_i)$ so as to satisfy
\begin{align}\label{eq:11}
p_j(\{x\} \times \Lambda_j \times \partial[0,1]) \text{ disjoint from
$\mathrm{Im}(p_i(x, -))$ for $j < i$.}
\end{align}
To achieve this we proceed by induction on $i$, the case $i=1$ being
vacuous. We can temporarily extend $p_i$ to a map $e_i : V_i \times \Lambda_i \times [0,1] \times D^{d-1} \to \Gamma(f|_{V_i})$ over $V_i$ that is an embedding of $\{x\} \times \Lambda_i \times \{t\} \times D^{d-1}$ into a level set for each $(x, t) \in V_i \times [0,1]$. For each $x \in V_i$ there is a $d_x \in D^{d-1}$ such that $\mathrm{Im}(e_i(x, -, d_x))$ is disjoint from $\bigcup_{j<i}p_j(\{x\} \times \Lambda_j \times \partial[0,1])$. Furthermore, there is a small contractible neighbourhood $x \in V_i^x \subseteq V_i$ where this is still true. Equip $V_i^x$ with the paths
$$p_i^x = e_i(-, d_x) : V_i^x \times \Lambda_i \times [0,1] \to \Gamma(f|_{V_i^x}).$$
We can choose finitely many $\{x_j\}_{j \in J}$ such that $\cup_{j \in J} V_i^{x_j} = V_i$ and the $d_{x_j}$ are distinct points in $D^{d-1}$. Then note that the $\{p_i^{x_j}\}_{j \in J}$ have disjoint images in $\Gamma(f|_{V_i})$.
The $\{p_i^{x_j}\}_{j \in J}$ all start at height $a_i \in (-1,1)$. We wish for the paths to start at different heights: by replacing the paths $p_i^{x_j}$ with their restrictions to a subinterval $[\eta, 1]$, for some small $\eta>0$, we can ensure that they all start at different heights $a_i^{x_j}$. We now replace $(V_i, a_i, p_i)$ by the collection $\{(V_i^{x_j}, a_i^{x_j}, p_i^{x_j})\}_{j \in J}$. This proves the induction step, and hence~(\ref{eq:11}).
\vspace{2ex}
Finally we pass from the ``endpoint disjointness'' expressed
in~(\ref{eq:11}) to the actual disjointness of all paths as required
in (\ref{item:11}) of Corollary~\ref{corollary:homotopy-2}. This is
just an easy parametrised version of Lemma~\ref{lemma:vector-flow}.
Indeed, if we pick an ordering of all the paths involved (i.e.\ of
the disjoint union of the finite sets $\Lambda_i$) and proceed by
induction as in the proof of that lemma. Suppose we have already
made the first $l$ paths disjoint. Then as in the proof of
Lemma~\ref{lemma:vector-flow} we get in the induction step for each
$x \in X$ a compactly supported vector field $X(x): \bR^n \to \bR^n$
preserving $f(x) \subseteq \bR^n$. It is easy to construct this to
depend smoothly on $x$, and then we can flow the $(l+1)$st path
with the flow of $X$ to make it disjoint from the previous paths,
without changing any endpoints.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main-of-4.1}]
Starting with the map of pairs~(\ref{eq:MapToNullify}),
Proposition~\ref{prop:paths-exist} produces a collection $\{(V_i, a_i, p_i)\}$
satisfying (\ref{item:10}) and (\ref{item:11}) of
Corollary~\ref{corollary:homotopy-2}, which then produces a relative
null homotopy.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:SubcategoryConnected}]
Lemma \ref{lem:AddingBasepoint}, Lemma \ref{lem:ElimCpctComponents} and Theorem \ref{thm:main-of-4.1} imply that the map
$$\psi^{nc}_\theta(\infty,1)^\bullet_\mathbf{Conn} \to \psi_\theta(\infty,1)$$
is a weak homotopy equivalence. Theorem \ref{thm:HtpyTypeCobCat} and Theorem \ref{thm:MainThmGMTW} identify these spaces with $B \mathcal{C}_\theta^\bullet$ and $B \mathcal{C}_\theta$ respectively.
\end{proof}
\subsection{Reversing morphisms}\label{sec:ReversingMorphisms}
Let us study the implications of the assumptions on $\theta: \mathscr{X} \to
BO(2)$. We can arrange that $\mathscr{X} = EO(2) \times_{O(2)} F$, for some
space $F$ equipped with an action
\begin{align}
\label{eq:25}
O(2) \times F \to F.
\end{align}
The assumption that $\mathscr{X}$ is path connected says that $\pi_0 O(2)$ acts
transitively on $\pi_0 F$. Therefore $F$ is either path connected, or
has two (homeomorphic) path components which are permuted by the
action of some reflection in $O(2)$. To interpret the assumption that
$S^2$ admits a $\theta$-structure, we consider the induced action of unbased
homotopy classes
\begin{align}
\label{eq:33}
[S^1,O(2)] \times [S^1, F] \to [S^1,F].
\end{align}
We regard $\pi_0 F \subseteq [S^1,F]$ as the homotopy classes of
constant maps.
\begin{proposition}
If $S^2$ admits a $\theta$-structure, then the subset $\pi_0(F) \subseteq
[S^1, F]$ is preserved by elements of $[S^1, O(2)]$ represented by
maps of even degree.
\end{proposition}
That a map $S^1 \to O(2)$ has even degree means that the corresponding
vector bundle over $S^2$ built by clutching has vanishing second Stiefel--Whitney class.
\begin{proof}
The free homotopy classes of maps of even degree form a subgroup
generated by two elements: the element $r$, represented by a
constant map $S^1 \to O(2)$ to some reflection, and the element $2:
S^1 \to SO(2) \subseteq O(2)$ represented by a map of degree 2. It
suffices to see that these preserve $\pi_0(F) \subseteq [S^1, F]$.
This is obvious for $r$ (as it is represented by a constant map), so
we consider 2. Pick a basepoint of $F$ and consider the long exact
sequence in homotopy
\begin{align*}
\pi_2(F) \to \pi_2 X \to \pi_2 BO(2) \to \pi_1 F \to \pi_1 X \to
\dots.
\end{align*}
The assumption says that a classifying map for $T S^2$ gives an
element of $\pi_2 BO(2) = \pi_1 O(2)$ which lifts to $\pi_2 X$.
Equivalently it must vanish in $\pi_1 F$. But $T S^2$ comes from
a map $S^1 \to SO(2) \subseteq O(2)$ which has degree 2, hence this
assumption says that $2 \in [S^1, O(2)]$ acts trivially on $[S^1,F]$.
\end{proof}
We next study $\theta$-structures on $D^2$. More precisely, if we are
given a structure over $\partial D^2$, how do we know whether it
extends to all of $D^2$? Such a structure is called a bounding
structure, and we prove that the set of bounding structures is
preserved by a certain construction that we now define. Let
\begin{align*}
V \in \Gamma(TD^2|_{\partial D^2})
\end{align*}
be a non-zero vector field, and let
\begin{align*}
\phi^V: TD^2|_{\partial D^2} \to TD^2|_{\partial D^2}
\end{align*}
be the bundle map which maps $V \mapsto -V$, and acts as the identity
on $V^\perp$.
\begin{corollary}
Let
\begin{align*}
l: TD^2|_{\partial D^2} \to \theta^* \gamma
\end{align*}
be a bounding $\theta$-structure (i.e.\ one that extends to $D^2$). Then $l
\circ \phi^V$ is also bounding.
\end{corollary}
\begin{proof}
Give $D^2$ the framing coming from $\bR^2$. With respect to this
framing, $l: \partial D^2 \to F$ has homotopy class
\begin{align*}
l \in [S^1, F]
\end{align*}
which is in $\pi_0 F \subseteq [S^1, F]$ because $l$ bounds. With
respect to the framing, the bundle map $\phi^V$ corresponds to the
map $S^1 \to O(2)$ which takes $x \in S^1$ to the reflection taking
$V(x) \mapsto -V(x)$ and preserves $V(x)^\perp$. This map depends
not on the vector $V(x)$, but only on the line $\bR V(x)$, so it
factors through the projection $S^1 \to \R P^1$. It follows that
$\phi^V$ corresponds to an element
\begin{align*}
\phi^V \in [S^1, O(2)]
\end{align*}
which is of even degree. Consequently $l \circ \phi^V \in [S^1, F]$
also bounds, by the previous proposition.
\end{proof}
We now study the extent to which morphisms in $C_\theta$ can be
``turned around''. The following proposition depends crucially on the
assumption that $S^2$ admit a $\theta$-structure. It is false for example for
framed manifolds.
\begin{proposition}\label{prop:ReverseMorphisms}
Let $c_0 = (M_0, l_0) \in C_\theta$ and $c_1 = (M_1,l_1)$ be objects
and let $(W,l): c_0 \to c_1$ be a morphisms in $C_\theta$ whose
underlying manifold is connected. Then there exists a morphism
$(\overline{W},\overline{l}) : c_1 \to c_0$ whose underlying
manifold $\overline{W}$ is diffeomorphic to $W$.
In particular, if $c_0$ and $c_1$ are linked by a zig-zag of morphisms, then there are in fact morphisms between them in both directions.
\end{proposition}
\begin{proof}
Recall that $W \subseteq [0,t] \times \bR^n$ for some large $n$, and
that $W$ is ``cylindrical'' near its boundary. Let $\varphi: [0,t] \to
[0,t]$ be the affine map $s \mapsto t-s$, and let $F = \varphi \times
\mathrm{Id}: [0,t] \times \bR^n \times [0,t] \times \bR^n$.
To construct the morphism $c_1 \to c_0$ we first construct the
underlying manifold as $\overline{W} = F^{-1}(W)$. If there were no
tangential structures to worry about, this would be a morphism from
$c_1$ to $c_0$, but we need to extend the structure given near $\partial
\overline{W} = c_1 \amalg c_0$ to all of $\overline{W}$.
The vector $V = \partial/\partial x_1 \in \bR^{n+1}$ gives a section
of $TW$, defined near $\partial W$ (pointing inwards on the incoming
boundary and outwards on the outgoing). Since $W$ is connected, we
can pick a relative CW structure with only one 2-dimensional cell.
Let $W^1\subseteq W$ be the 1-skeleton in this CW structure. We can
extend $V$ to a non-zero section
\begin{align*}
V \in \Gamma(TW|_{W^1})
\end{align*}
Now let $\phi^V: TW|_{W^1} \to TW|_{W^1}$ be the automorphism constructed
as before: it takes $V \mapsto -V$ and is the identity on $V^\perp$.
The ``reflected'' tangential structure
\begin{align*}
\overline{l} = l \circ \phi^V: TW|_{W^1} \to \theta^*\gamma,
\end{align*}
extends first to a neighbourhood of $W^1 \subseteq W$, and then over
the 2-cell $D^2 \to W$ by the corollary above.
Thus we have a tangential structure $\overline{l} : TW \to
\theta^*\gamma$. If we use the diffeomorphism $F: W \to
\overline{W}$ to transport it to a structure on $\overline{W}$, we
have produced the required morphisms $c_1 \to c_0$.
\end{proof}
\subsection{Proof of Theorem \ref{thm:SubcategoryMonoids}}\label{sec:DimensionTwo}
By Theorem \ref{thm:HtpyTypeCobCat}, we have reduced the first part of Theorem \ref{thm:SubcategoryMonoids} to the following theorem.
\begin{theorem}\label{thm:CToConn}
For tangential structures $\theta : \mathscr{X} \to BO(2)$ such that $\mathscr{X}$ is path connected and $S^2$ admits a $\theta$-structure, the inclusion
$$\psi^{nc}_\theta(\infty,1)^\bullet_\mathbf{C} \to \psi^{nc}_\theta(\infty,1)^\bullet_\mathbf{Conn}$$
is a weak homotopy equivalence of each component onto a component of $\psi^{nc}_\theta(\infty,1)^\bullet_\mathbf{Conn}$.
\end{theorem}
The second part of Theorem \ref{thm:SubcategoryMonoids} will be proved in Proposition \ref{prop:HomotopyCommutative}. Let us first prove Theorem \ref{thm:CToConn} in the case where $\mathbf{C}$ is a disjoint union of path components of $\mathrm{ob}(C_\theta^\bullet)$. We will explain how to remove this assumption in Proposition \ref{prop:PathComponentsOfObjects}.
Suppose we are given a map of pairs
$$f:(D^k, \partial D^k) \lra (\psi^{nc}_{\theta}(\infty, 1)^\bullet_{\mathbf{Conn}}, \psi^{nc}_{\theta}(\infty, 1)^\bullet_{\mathbf{C}}).$$
For each point $x \in D^k$, we can choose a regular value $a_x \in (-1,1)$ of $x_1 :f(x) \to \mathbb{R}$. There is an open neighbourhood $U_x$ of $x$ on which $a_x$ is still regular. We may choose a finite subcover $\{U_i\}_{i \in I}$ of $\{U_x\}_{x \in X}$, and suppose that the $a_i$ are distinct: otherwise we may perturb them so that they are. Choose an $\epsilon >0$ such that the intervals $(a_i-2\epsilon, a_i+2\epsilon)$ are disjoint. By Lemma \ref{Cylindrical}, we may further suppose that for $x \in U_i$ the manifold $f(x)$ is cylindrical in $x_1^{-1}(a_i-\epsilon, a_i+\epsilon)$.
\vspace{2ex}
As we are working in the space of manifolds with basepoint, and the family $D^k$ is compact, there is a $\delta > 0$ such that the strip $L_{3\delta} = \mathbb{R} \times (-3\delta, 3\delta) \times \{0\}$ lies inside every manifold $f(x)$, and furthermore that $f(x) \cap \mathbb{R} \times (-3\delta, 3\delta)^{\infty} = L_{3\delta}$. Recall that inside $L_{3\delta}$ the surface has the standard $\theta$-structure. We can change the manifold inside $\mathbb{R} \times (\delta, 2\delta) \times (-3\delta, 3\delta)^{\infty-1}$ without leaving the space of manifolds with basepoint, as the new manifold will still contain $L_{\delta}$.
The set $\pi_0 \mathrm{ob}( \mathcal{C}_\theta^\bullet )$ is in natural bijection with $\pi_1 \mathrm{Fib}(\theta)$, as an object $C$ has a natural framing of $\epsilon^1 \oplus TC$, extending the given framing on $(-\epsilon,\epsilon) \times \{0\}$, and this framing is unique up to homotopy. Write $*$ for the object having the standard $\theta$-structure with respect to this framing. For each $U_i$, choosing a $u_i \in U_i$ and setting $M_i = f(u_i) \cap \{a_i\}\times \mathbb{R}^\infty$ produces a homotopy class $[M_i] \in \pi_1 \mathrm{Fib}(\theta)$ which is independent of the choice of $u_i$.
\begin{lemma}\label{lem:InverseObjects}
Let $\mathbf{C} \subseteq \mathrm{ob}( \mathcal{C}_\theta^\bullet )$ be a disjoint union of path components: this determines a subset $\pi_0 \mathbf{C} \subseteq \pi_1 \mathrm{Fib}(\theta)$. If an object $M \in \mathcal{C}_\theta^\bullet$ lies in a component of $B \mathcal{C}_\theta^\bullet$ that also contains an object in $\mathbf{C}$, there is another object $N$ such that $[M] \cdot [N] \in \pi_0\mathbf{C} \subseteq \pi_1 \mathrm{Fib}(\theta)$, where $\cdot$ denotes the group product in $\pi_1 \mathrm{Fib}(\theta)$. Furthermore, there are morphisms
$$* \lra N \lra *$$
in $\mathcal{C}_\theta^\bullet$. If $M$ is already in $\mathbf{C}$, we can choose $N$ to be $*$.
\end{lemma}
\begin{proof}
Choose morphisms $E_1 : M \to C \in \mathbf{C}$, and $E_2 : C \to M$ which exist by Proposition \ref{prop:ReverseMorphisms}, and an object $M^{-1}$ such that $[M^{-1}] = [M]^{-1} \in \pi_1 \mathrm{Fib}(\theta)$. Scale the morphisms $E_1$ and $E_2$ to have length 1. Consider the morphism $I_{M^{-1}} = (1, \mathbb{R} \times M^{-1}) : M^{-1} \to M^{-1}$. We may cut this along $\mathbb{R} \times \{0\}$ to obtain a surface with $\theta$-structure and height function, $\tilde{I}_{M^{-1}}$, having two identical boundary components, which we will call $\widetilde{M^{-1}}$ and which are intervals with a $\theta_1$-structure that is standard near their ends.
Let $\epsilon>0$ be small enough that the morphisms $E_1$ and $E_2$ both satisfy $E_i \cap \mathbb{R} \times (-3\epsilon, 3\epsilon)^\infty = L_{3\epsilon}$. In particular, the morphisms come from $\mathcal{C}_\theta^{3\epsilon}$. There is a height-preserving embedding $e: \tilde{I}_{M^{-1}} \to \mathbb{R} \times [\epsilon, 2\epsilon] \times [0, 3\epsilon)^\infty$ sending $\partial \tilde{I}_{M^{-1}}$ to $\mathbb{R} \times \{\epsilon, 2\epsilon\} \times \{0\}$ with image that agrees with $L_{3\epsilon}$ as a $\theta$-manifold near $(\mathbb{R} \times \mathbb{R} - (\epsilon, 2\epsilon) ) \times \mathbb{R}^\infty$.
Let $\tilde{E}_i = (E_i - \mathbb{R} \times (\epsilon, 2\epsilon)\times\{0\}) \cup e(\tilde{I}_{M^{-1}}) \in \psi_\theta^{nc}(\infty, 1)^\bullet_\mathbf{Conn}$. Then $(1, \tilde{E}_1)$ and $(1, \tilde{E}_2)$ are morphisms in $\mathcal{C}_\theta^\bullet$, between
$(M - (\epsilon, 2\epsilon)\times\{0\}) \cup e(\widetilde{M^{-1}})$ and $(C - (\epsilon, 2\epsilon)\times\{0\}) \cup e(\widetilde{M^{-1}})$. The first of these objects is in the path component of $*$, and the second is defined to be $N$.
\end{proof}
We will now construct a $\theta$-manifold $D$ with height function, which we will glue into the manifolds $f(x)$ inside $L_{3\delta}$, similarly to the proof of Lemma \ref{lem:InverseObjects}. Let $C_i$ be the composition $* \to N_i \to *$ obtained from the above Lemma applied to $M_i$, and scaled so that each morphism has length $\epsilon$. Let $\tilde{C}_i$ denote the $\theta$-manifold with height function obtained by cutting $C_i$ along $\mathbb{R} \times \{0\}$, and adding $a_i-\epsilon$ to the height function. The boundary of $\tilde{C}_i$ can be naturally identified with the boundary of $\mathbb{R} \times [0,1]$, where the height function is given by the first coordinate. There is a height-preserving embedding $e_i : \tilde{C}_i \to \mathbb{R} \times [\delta, 2\delta] \times [0, 3\delta)^\infty$ sending $\partial \tilde{C}_i$ to $\mathbb{R} \times \{\delta, 2\delta \}$ with image that agrees with $L_{3\delta}$ as a $\theta$-manifold near
the boundary of $[a_i - \epsilon, a_i+\epsilon] \times [\delta, 2\delta] \times \mathbb{R}^\infty$.
Let $D$ be the $\theta$-manifold obtained from $L_{3\delta}$ by cutting out $[a_i - \epsilon, a_i+\epsilon] \times [\delta, 2\delta] \times \{0\}$ and gluing in $e_i(\tilde{C}_i) \cap [a_i - \epsilon, a_i+\epsilon] \times [\delta, 2\delta] \times \mathbb{R}^\infty$. The following figure shows the part of $D$ near $\mathbb{R} \times (\delta, 2\delta) \times \mathbb{R}^\infty$.
\begin{center}
\includegraphics[bb = 148 512 500 649, scale=1]{figures/D2.eps}
\end{center}
For each $x \in D^k$ we can form $\bar{f}(x) = (f(x) - L_{3\delta}) \cup D$, which defines a continuous map $\bar{f} : D^k \to \psi_\theta(\infty,1)^\bullet_\mathbf{Conn}$. If $x \in U_i$, then $[\bar{f}(x) \cap \{a_i\} \times \mathbb{R}^\infty] = [M_i] \cdot [N_i] \in \pi_1 \mathrm{Fib}(\theta)$ which is in $\pi_0 \mathbf{C}$ by construction. Thus $\bar{f}$ is a map into $\psi_\theta(\infty,1)^\bullet_{\mathbf{C}}$, and we must show it is relatively homotopic to $f$. We will do so by changing the height function on $D$. For ease of notation we will write $h$ instead of $x_1$ for the height function.
\vspace{2ex}
We first deform the height function on $D$ so that $D - L_{3\delta}$ has total height 0. Let $c_1 : \mathbb{R} \to \mathbb{R}$ be a smooth function such that $(t_i -\epsilon, t_i + \epsilon) \subseteq c_1^{-1}(t_i)$, and $c_1' \geq 0$. Let $c_s = s \cdot c_1 + (1-s) \cdot \mathrm{Id}$. Choose a smooth function $\varphi$ on $D$ that is identically 1 on $D-L_{3\delta}$, and identically 0 near $\mathbb{R} \times \mathbb{R} - (\delta,2\delta) \times \mathbb{R}^\infty$. We can modify the height function on $D$ by
$$h_s(d) = \varphi(d) \cdot (c_s \circ h(d) - h(d)) + h(d).$$
Then $h_0 = h$ and if $c \in e_i(\tilde{C}_i) - L_{3\delta} \subseteq D$ then $h_1(c)=t_i$. The level sets of $h_1$ in $D$ at any points except $\{t_1, ...,t_n\}$ are then intervals with the standard $\theta$-structure.
We now define another path of height functions on $D$,
$$h'_s(d) = \varphi(d) \cdot 20 \cdot s + h_1(d).$$
Then $h'_0 = h_1$. If $c \in e_i(\tilde{C}_i) - L_{3\delta} \subseteq D$ then $h'_1(d) = t_i + 20 \in (19, 21)$. With this height function, $D$ has the standard tangential structure between heights $-10$ and $10$.
Applying these paths of height functions to the copy of $D$ inside each $\bar{f}(x)$ gives a path from $\bar{f}(x)$ to a manifold that agrees with $f(x)$ inside $x_1^{-1}(-10,10)$, so composing with the stretching map (\ref{eq:14}) gives a path to $f(x)$.
\vspace{2ex}
We must explain why these are \textit{relative} homotopies. Suppose $f(x) \in \psi^{nc}_\theta(\infty,1)^\bullet_\mathbf{C}$, so there is a $t \in (-1,1)$ and an $\eta>0$ with $f(x) \cap \{t'\} \times \mathbb{R}^\infty \in \mathbf{C}$ for all $t' \in (t-\eta, t+\eta)$. If $t$ is not in $\cup_i (a_i-\epsilon, a_i+\epsilon)$, then $\bar{f}(x)$ agrees with $f(x)$ near $t$, so the level set at $t$ of $\bar{f}(x)$ is in $\mathbf{C}$. The path $h_s$ only changes the level sets in $\cup_i (a_i-\epsilon, a_i+\epsilon)$, so also always has a level set in $\mathbf{C}$. Finally, the path $h'_s$ only has finitely many level sets different from those of $f(x)$ at any time. Thus the final manifold always has a $t' \in (t-\eta, t+\eta)$ with level set in $\mathbf{C}$.
On the other hand, suppose $t \in (a_i-\epsilon, a_i+\epsilon)$. Then $M_i \in \mathbf{C}$ already, and we have thus chosen $N_i$ to be trivial. Thus the level sets of $f(x)$ and $\bar{f}(x)$ at $t$ are the same, and in $\mathbf{C}$. The path $h_s$ does not change this. Finally, the path $h'_s$ only has finitely many level sets in $(-10,10)$ different from $f(x)$ at a time. Thus the final manifold always has a $t'$ with level set in $\mathbf{C}$.
This concludes the proof of Theorem \ref{thm:CToConn}, in the case that $\mathbf{C}$ consists of whole path components of $\psi_{\theta_{d-1}}(\infty,0)$. The general case follows by:
\begin{proposition}\label{prop:PathComponentsOfObjects}
Let $\mathbf{C}$ be a collection of elements of $\psi_{\theta_{d-1}}(\infty,0)$, and $\mathbf{C}'$ be the collection of all elements of $\psi_{\theta_{d-1}}(\infty,0)$ in the same path component as an element in $\mathbf{C}$. Then the inclusion
$$\psi^{nc}_\theta(\infty,1)^\bullet_\mathbf{C} \to \psi^{nc}_\theta(\infty,1)^\bullet_{\mathbf{C}'}$$
is a weak homotopy equivalence.
\end{proposition}
\begin{proof}
We will show that the relative homotopy groups of the inclusion vanish. The argument is almost identical to that of Proposition \ref{prop:Psi0IntoPsiDot}, so we will explain to what extent it differs.
Take a map $f : (D^m, \partial D^m) \to (\psi^{nc}_\theta(\infty,1)^\bullet_{\mathbf{C}'} , \psi^{nc}_\theta(\infty,1)^\bullet_\mathbf{C})$. As usual, there is a finite cover of $D^m$ by contractible open sets $U_i$ and values $a_i \in (-1,1)$ such that $a_i$ is a regular value of the height function on $f(x)$ for $x \in U_i$, and the level sets at $a_i$ are in $\mathbf{C}'$. We may suppose the $a_i$ are distinct, and choose an $\epsilon >0$ so that the intervals $(a_i-2\epsilon, a_i+2\epsilon)$ are disjoint. By Lemma \ref{Cylindrical}, we may further suppose that $f(x \in U_i)$ is cylindrical in $x_1^{-1}(a_i-\epsilon, a_i+\epsilon)$. It is important to note that the homotopy of Lemma \ref{Cylindrical} does not change which level sets occur in $f(x)$, only the heights at which they occur: thus it gives a relative homotopy.
Thus there is a map $\lambda_i : U_i \to \psi_{\theta_{d-1}}(\infty,0)^\bullet$ so that $f(x)$ and $\mathbb{R} \times \lambda_i(x)$ are equal in $\Psi_\theta(x_1^{-1}(a_i-\epsilon, a_i+\epsilon))$. The map $\lambda_i$ takes values in $\mathbf{C}'$. As $U_i$ is contractible we can pick a smooth homotopy $\Lambda_i : [0,1]\times U_i \to \mathbf{C}'$ with $\Lambda_i(0,-)=\lambda_i$ and $\Lambda_i(1,-) \in \mathbf{C}$. We use these functions exactly as in the proof of Proposition \ref{prop:Psi0IntoPsiDot}.
It remains to see why this gives a relative homotopy. Over $U_i$, elements $f(x)$ are cylindrical in $x_1^{-1}(a_i-\epsilon, a_i+\epsilon)$. The homotopy constructed in Proposition \ref{prop:Psi0IntoPsiDot} is constant outside of $x_1^{-1}(a_i-2\delta, a_i+2\delta)$, for some $0 < 3\delta < \epsilon$. In particular, throughout the homotopy there are only new types of level sets added: if $f(x)$ has a level set in $\mathbf{C}$, then either it occurs at a height outside of $(a_i-2\delta, a_i+2\delta)$ and so is unchanged by the homotopy, or it occurs at a height inside $(a_i-2\delta, a_i+2\delta)$, in which case it also occurs at a height inside $(a_i+2\delta, a_i+\epsilon)$, which must be unchanged by the homotopy.
\end{proof}
Theorem \ref{thm:CToConn} implies the first part of Theorem \ref{thm:SubcategoryMonoids}. It remains to see homotopy commutativity of the endomorphism monoids in $\mathcal{C}_\theta^\bullet$; to that end we give yet another model for these spaces.
The monoid of endomorphisms of an object $C \in \mathcal{C}_\theta^\bullet$ is homotopy equivalent to $\coprod_W B\Diff_\theta(W, L\cup \partial W)$, where $W$ ranges over connected compact $\theta$-surfaces with two boundary circles, both identified with $C$. We can cut such surfaces along $L$ to obtain surfaces with one boundary component. More precisely, given a pointed map $\ell : S^1 \to \mathrm{Fib}(\theta)$, define
$$\tilde{\ell} : \mathbb{R}^2 \overset{\pi_2}\to \mathbb{R} \to S^1 \overset{\ell}\to \mathrm{Fib}(\theta)$$
where $\mathbb{R} \to S^1$ collapses the complement of $(0,1)$ to the basepoint. Define an element $B_\ell=(\mathbb{R}^2 \times \{0\}, \tilde{\ell}) \in \Psi_\theta(\mathbb{R}^\infty)$. Let $\mathcal{M}(\ell) \subseteq \Psi_\theta(\mathbb{R}^\infty) \times \mathbb{R}$ be the subspace of pairs $(W, t)$ where $W$ is connected and agrees with $B_\ell$ on an open neighbourhood of the complement of $(0,t)\times (0,1) \times \mathbb{R}^\infty$. This is a monoid via
$$(W_1, t_1) \circ (W_2, t_2) = (W_1 \cup (W_2 + t_1 \cdot e_1), t_1+t_2).$$
An object $C \in \mathcal{C}_\theta^\bullet$ determines a loop $\ell_C : S^1 \to \mathrm{Fib}(\theta)$, after choosing a parametrisation of the underlying circle (of which there are a contractible choice, as it is determined near the basepoint). There is an object $C'$ in the same path component of $\mathrm{ob}( \mathcal{C}_\theta^\bullet )$ as $C$ and a map of monoids $\mathcal{M}(\ell_C) \to \End_{\mathcal{C}_\theta^\bullet}(C')$ given by replacing $B_\ell$ by a cylinder $\mathbb{R} \times C'$ that agrees with $B_\ell$ near $\mathbb{R} \times [0,1] \times \{0\}$. It is a weak homotopy equivalence as they are classifying spaces for equivalent families. Furthermore, the monoids $\End_{\mathcal{C}_\theta^\bullet}(C')$ and $\End_{\mathcal{C}_\theta^\bullet}(C)$ are equivalent in the following sense: choosing a path $C \leadsto C'$ in $\mathrm{ob} (\mathcal{C}_\theta^\bullet)$ gives a weak homotopy equivalence $\End_{\mathcal{C}_\theta^\bullet}(C') \to \End_{\mathcal{C}_\theta^\bullet}(C)$ that is not quite a map of monoids, but is a map of $H$-spaces (and in fact a map of $A_\infty$ spaces).
\begin{proposition}\label{prop:HomotopyCommutative}
Let $\theta : \mathscr{X} \to BO(2)$ be any tangential structure. Then the monoid of endomorphisms of an object $C \in \mathcal{C}_\theta^\bullet$ is homotopy commutative.
\end{proposition}
\begin{proof}
By the above discussion, it is enough to prove the homotopy commutativity of $\mathcal{M}(\ell)$. Consider first the case where $\ell$ is the constant map to $* \in \mathrm{Fib}(\theta)$, the basepoint which determines the standard tangential structure. Then $\tilde{\ell}=*$ also and so $B_\ell = \mathbb{R}^2 \times \{0\}$ with the standard $\theta$-structure induced by the framing given by the coordinate directions. Thus $\mathcal{M}(*)$ has an obvious action of the little 2-cubes operad. Its usual monoid multiplication is homotopic to that induced by the operad, so its usual multiplication is homotopy commutative (in fact, $E_2$).
There is a map $\ell_* : \mathcal{M}(\ell_1) \to \mathcal{M}(\ell \cdot \ell_1)$ given on $(W, t)$ by scaling $W-B_{\ell_1}$ by $\tfrac{1}{2}$ in the $e_2$ direction, and then taking the union with $B_{\ell \cdot \ell_1}$. This is a map of monoids, and composing with $\ell^{-1}_*$ gives a map $\mathcal{M}(\ell_1) \to \mathcal{M}(\ell^{-1} \cdot \ell \cdot \ell_1)$. This is easily seen to have a homotopy inverse, by picking a null homotopy of $\ell^{-1} \cdot \ell$ (though the inverse is not a map of monoids). Thus the maps $\ell_*$ are homotopy equivalences. In particular $\mathcal{M}(\ell) \to \mathcal{M}(\ell^{-1} \cdot \ell) \simeq \mathcal{M}(*)$ is a monoid map and an equivalence, so $\mathcal{M}(\ell)$ is equivalent as a monoid to a homotopy commutative monoid, so is homotopy commutative itself.
\end{proof}
\section{Applications}\label{sec:Applications}
As mentioned in the introduction, taking the tangential structure $\theta : BSO(2) \to BO(2)$ gives a category $\mathcal{C}_{SO(2)}^\bullet$ having a unique path component, and hence cobordism class, of objects. The endomorphism monoid of any object is homotopy equivalent to $\mathcal{M} = \coprod_{g \geq 0} B\Diff^+(F_g, \partial F_g = S^1)$ with the pair of pants product. By \cite{EE}, components of the group $\Diff^+(F_g, \partial F_g = S^1)$ are contractible, so we can replace it by the associated mapping class group $\Gamma_{g,1} = \pi_0 \Diff^+(F_g, \partial F_g = S^1)$. The discussion in \S\ref{sec:harer-stab-mads} proves
\begin{MainCor}[Madsen--Weiss \cite{MW}]\label{cor:MadsenWeiss}
There is a homology equivalence
\begin{align*}
\mathbb{Z} \times B\Gamma_{\infty,1} \to \Omega^\infty \MT{SO}{2},
\end{align*}
where $\Gamma_{\infty,1}$ is the limit of the mapping class groups
$\Gamma_{g,1}$ as $g \to \infty$.
\end{MainCor}
A similar result holds for non-orientable or spin surfaces, though the components of the relevant monoid are slightly more complicated.
\begin{MainCor}[Wahl \cite{Wahl} Theorem B]\label{Cor:Unoriented}
Let $\mathcal{N}_g$ be the mapping class group of a non-orientable surface of genus $g$. Then there is a homology equivalence
$$\mathbb{Z} \times B \mathcal{N}_\infty \to \Omega^\infty \MT{O}{2}.$$
\end{MainCor}
\begin{proof}
We apply Theorem \ref{thm:SubcategoryMonoids} with no tangential structure. The category $\mathcal{C}_{O(2)}^\bullet$ has a unique cobordism class of objects. The relevant homotopy commutative monoid is
$$\mathcal{M} = \coprod_{F} B\Diff(F, \partial F = S^1)$$
where the disjoint union is over all diffeomorphism types of unoriented surfaces with one boundary component, and Theorem \ref{thm:SubcategoryMonoids} identifies its classifying space as $\Omega^{\infty-1}\MT{O}{2}$.
The monoid $\pi_0\mathcal{M}$ is in bijection with the set of pairs $(g_1, g_2) \in \mathbb{N}^2$ where one entry is always 0. The element $(g,0)$ corresponds to $F_{g, 1} = \#^g T - D^2$ and $(0,g)$ corresponds to $N_{g,1} = \#^g \mathbb{R} \mathbb{P}^2 - D^2$. Then addition in this monoid is given by usual addition if both elements are in the same factor, and by the formula $(g, 0) + (0, h > 0) = (0, h+2g)$ otherwise.
By the discussion in \S\ref{sec:harer-stab-mads}, the homology of the group completion of $\mathcal{M}$ is the same as that of the telescope $\mathcal{M}_\infty$ of
$$\mathcal{M} \overset{\cdot(0,3)} \lra \mathcal{M} \overset{\cdot(0,3)} \lra \mathcal{M} \overset{\cdot(0,3)} \lra \cdots$$
as $\pi_0\mathcal{M}$ is generated by $(1,0)$ and $(0,1)$, whose product is $(0,3)$.
There is a submonoid $\mathcal{M}'$ of $\mathcal{M}$ consisting only of the non-orientable surfaces (and the disc), and translation by $(0,1)$ sends $\mathcal{M}$ into $\mathcal{M}'$, so $\mathcal{M}'$ is cofinal. Thus $\mathcal{M}_\infty \simeq \mathcal{M}'_\infty$ and this is equivalent to $\mathbb{Z} \times B \mathcal{N}_\infty$, the classifying-space of the stable non-orientable mapping class group.
\end{proof}
\begin{MainCor}[Harer \cite{HSpin}, Bauer \cite{Bauer}]\label{CorSpin}
Let $\hat{\Gamma}^{\mathfrak{s}}_{g, 1}$ be the (extended) spin mapping class group of an orientable surface of genus $g$ with one boundary circle, and with spin structure $\mathfrak{s}$ that agrees with the trivial spin structure on the boundary circle.
There is a homology equivalence
$$\mathbb{Z} \times \mathbb{Z}/2 \times B\hat{\Gamma}^{\sigma}_\infty \lra \Omega^\infty \MT{Spin}{2}$$
where $\hat{\Gamma}^{\sigma}_{\infty,1}$ is the stable spin mapping class group.
\end{MainCor}
\begin{proof}
We apply Theorem \ref{thm:SubcategoryMonoids} with the tangential structure $\theta : B\mathit{Spin}(2) \to BO(2)$. Write $\gamma^s$ for the universal vector bundle over $B\mathit{Spin}(2)$. Note that $B\mathit{Spin}(2) \simeq BSO(2)$ is connected and $S^2$ admits a spin structure, so the hypotheses of the theorem are fulfilled. The category $\mathcal{C}_{\mathit{Spin}(2)}^\bullet$ has two cobordism classes of objects, corresponding to a trivial and a nontrivial spin structure on the circle. The monoid of endomorphisms of the trivial circle is
$$\mathcal{M} = \coprod_{F} B\Diff^{\mathit{Spin}}(F, \partial F) = \coprod_{F} \Bun^\partial(TF, \gamma^s) /\!\!/ \Diff(F, \partial F).$$
where the disjoint union is over all diffeomorphism types of surfaces with one boundary component. If $F$ is not orientable, $\Bun^\partial(TF, \gamma^s) = \emptyset$ and it plays no role. If $F$ is an oriented surface, the space $\Bun^\partial(TF, \gamma^s)$ is homotopy equivalent to the space of pairs $(\tau, \ell)$ of a map $\tau : F \to BSO(2)$ classifying the tangent bundle and a relative lift $\ell$
\begin{diagram}
& & K(\mathbb{Z}/2\mathbb{Z}, 1) \\
& & \dTo \\
S^1& \rTo^* & B\mathit{Spin}(2) \\
\dTo & \ruTo^\ell[dotted] & \dTo \\
F& \rTo^{\tau} & BSO(2).
\end{diagram}
This space can be (non-canonically, and non-equivariantly, for the obvious action of $\Diff^+(F, \partial F)$ on $H^1(F, \partial F; \mathbb{F}_2)$) identified with $\mathrm{Map}(F, \partial F ; K(\mathbb{Z}/2\mathbb{Z}, 1), *) \simeq H^1(F, \partial F; \mathbb{F}_2) \times K(\mathbb{Z}/2\mathbb{Z}, 1)$ as there is a contractible space of such maps $\tau$. Thus
$$B\Diff^{\mathit{Spin}}(F, \partial F) \simeq H^1(F, \partial F; \mathbb{F}_2) \times K(\mathbb{Z}/2\mathbb{Z}, 1) /\!\!/ \Diff^+(F, \partial F).$$
The group $\Diff^+(F, \partial F)$ acts on $\pi_0(\mathrm{Map}(F, \partial F ; K(\mathbb{Z}/2\mathbb{Z}, 1), *)) \cong H^1(F, \partial F ; \mathbb{F}_2)$ with two orbits (distinguished by their Arf invariant). Pick representatives $\mathfrak{s}_0$ and $\mathfrak{s}_1$ for these orbits. If $\Diff^{\mathfrak{s}_i}(F, \partial F)$ is the stabiliser of $\mathfrak{s}_i \in H^1(F, \partial F ; \mathbb{F}_2)$, then there are fibre sequences
$$K(\mathbb{Z}/2\mathbb{Z}, 1) \lra \Bun^\partial_{\mathfrak{s}_i}(TF, \gamma^s) /\!\!/ \Diff^{\mathfrak{s}_i}(F, \partial F) \lra B\Diff^{\mathfrak{s}_i}(F, \partial F).$$
The contractibility \cite{EE} of the components of $\Diff(F, \partial F)$ means that all three spaces are $K(\pi,1)$'s. Write $\hat{\Gamma}^{\mathfrak{s}_i}_{g,1}$ for the fundamental group of the middle space, when $F$ is an oriented surface of genus $g$. This is by definition the \textit{extended spin mapping class group}.
Now $B\Diff^{\mathit{Spin}}(F, \partial F) = \coprod_{i=0,1} \Bun^\partial_{\mathfrak{s}_i}(TF, \gamma^s) /\!\!/ \Diff^{\mathfrak{s}_i}(F, \partial F)$ and so
$$\mathcal{M} = \coprod_{g \geq 0} \coprod_{i=0,1} B \hat{\Gamma}^{\mathfrak{s}_i}_{g,1}.$$
The components of this monoid are $\mathbb{N} \times \mathbb{Z}/2\mathbb{Z}$, so it can be group-completed by inverting just $(1,0)$: this corresponds to gluing on a torus with a spin structure of Arf invariant 0. The telescope $\mathcal{M}_\infty$ is then $\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \times B \hat{\Gamma}^{\sigma}_{\infty,1}$.
\end{proof}
If $\theta : \mathscr{X} \to BO(2)$ is a tangential structure with $\mathscr{X}$ connected and such that $S^2$ admits a $\theta$-structure, and $Y$ is a path connected space, then $\theta \times Y = \theta \circ \pi_{\mathscr{X}} : \mathscr{X} \times Y \to BO(2)$ again has these properties. This allows us to add maps to a background space $Y$ to any such tangential structure.
Let us develop this in the case of ordinary orientations, where the tangential structure is $\theta : BSO(2) \to BO(2)$. The space $\mathcal{S}_{g,1}(Y; *)$ of Cohen--Madsen \cite{CM} is precisely the space $B\Diff^{\theta \times Y}(F_{g, 1}, \partial F_{g, 1})$, where the map to $Y$ is fixed to be the constant map to the basepoint on the boundary. Cobordism classes of objects in $\mathcal{C}_{\theta \times Y}^\bullet$ are in natural bijection with $H_1(Y, \mathbb{Z})$, via the map that sends a circle in $Y$ to the homology class it represents. Let $C$ be a representative of the trivial homology class.
\begin{comment}
The monoid of endomorphisms of $C$ in $\mathcal{C}_{\theta \times Y}^\bullet$ is $\coprod_{g \geq 0} \mathcal{S}_{g,1}(Y; *)$. Theorem \ref{thm:SubcategoryMonoids} and the discussion in \S\ref{sec:harer-stab-mads} then proves the following generalisation of the theorem of \cite{CM}.
\begin{MainCor}
Let $Y$ be a path connected space. There is a homology equivalence
$$\mathbb{Z} \times \mathcal{S}_{\infty, 1}(Y;*) \lra \Omega^\infty \MT{SO}{2} \wedge Y_+.$$
\end{MainCor}
A similar statement holds for unoriented and spin surfaces with maps to a background space.
\begin{MainCor}
Let $Y$ be a path connected space. There are homology equivalences
\begin{align*}
\hocolim_{g \to \infty} B\Diff^Y(N_{g,1}, \partial N_{g,1}) & \lra \Omega^\infty \MT{O}{2} \wedge Y_+, \\
\hocolim_{g \to \infty} B\Diff^{Spin \times Y}(F_{g,1}, \partial F_{g,1}) & \lra \Omega^\infty \MT{Spin}{2} \wedge Y_+.
\end{align*}
\end{MainCor}
\end{comment}
The monoid of endomorphisms of $C$ in $\mathcal{C}_{\theta \times Y}^\bullet$ is $\mathcal{M} = \coprod_{g \geq 0} \mathcal{S}_{g,1}(Y; *)$. For a general space $Y$ this monoid has tremendously many components, and so it is not clear that one can form the stabilisation $\mathcal{M}_\infty$, as one may not be able to group complete $\pi_0(\mathcal{M})$ by inverting finitely many elements. However, in several interesting one \textit{can} form $\mathcal{M}_\infty$.
Note that there is a homotopy fiber sequence
\begin{equation}\label{MappingSpaceFibration}
\Omega^2 Y \to \mathrm{map}((F_{g,1}, \partial F_{g,1}), (Y, *)) \to (\Omega Y)^{2g}
\end{equation}
onto those components of $(\Omega Y)^{2g}$ represented by tuples $(a_1, b_1, ..., a_g, b_g) \in \pi_1(Y)^{2g}$ such that $\prod_{i=1}^g [a_i, b_i] = 1$.
\subsection{$Y$ simply connected} In this case $\pi_0(\mathcal{S}_{g,1}(Y; *)) \cong \pi_2(Y) \cong H_2(Y ; \mathbb{Z})$, and $\pi_0(\mathcal{M}) = \mathbb{N} \times H_2(Y ; \mathbb{Z})$ as a monoid. This can be group completed by inverting just $(1,0) \in \mathbb{N} \times H_2(Y ; \mathbb{Z})$, and so we can form $\mathcal{M}_\infty$ as the mapping telescope of
$$\mathcal{M} \overset{\cdot(1,0)}\to \mathcal{M} \overset{\cdot(1,0)}\to \mathcal{M} \overset{\cdot(1,0)}\to \cdots.$$
The discussion in \S\ref{sec:harer-stab-mads} and Theorem \ref{thm:SubcategoryMonoids} gives
\begin{MainCor}[Cohen--Madsen \cite{CM}]
Let $Y$ be a simply connected space. Then there is a homology equivalence
$$\mathbb{Z} \times \hocolim_{g \to \infty}\mathcal{S}_{g,1}(Y; *) \to \Omega^\infty \MT{SO}{2} \wedge Y_+$$
where the colimit is formed by the map $\cdot(1,0)$ above.
\end{MainCor}
\subsection{$Y=BG$, $G$ a cyclic group} In this case
$$\pi_0(\mathcal{S}_{g,1}(BG; *)) = \pi_0(\Bun^\partial(TF_{g,1}, \theta^*\gamma)) \times_{\Diff(F_{g,1})} \pi_0(\mathrm{map}^\partial(F_{g,1}, BG))$$
and the homotopy fiber sequence (\ref{MappingSpaceFibration}) gives that $\mathrm{map}^\partial(F_{g,1}, BG) \simeq G^{2g} \simeq H^1(F_{g,1}, \partial F_{g,1} ; G)$ so
\begin{equation}\label{eq:ComponentsOfMonoid}
\pi_0(\mathcal{M}) = \coprod_{g \geq 0} G^{2g} / \Diff^+(F_{g,1})
\end{equation}
where composition is by concatenating (representatives of equivalence classes of) tuples of elements of $G$. Note that this monoid is generated by the $g=1$ elements. This much holds for any abelian group $G$.
\begin{comment}
\begin{lemma}
\
\begin{enumerate}[(i)]
\item The multiplicative subset $\pi_0(M) \subseteq H_0(M)$ is generated by the
symbols $[a,b]$, where $(a,b)$ range over $G^2$.
\item If $t=[x,0]$ for some $x \in G$, then
$$t [a,b] = t [a',b']$$
if $a-a'$ and $b-b'$ are in the subgroup generated by $x$. Therefore in the
ring $H_0(M)[t^{-1}]$, the symbol $[a,b]$ only depends on the image of $a$
and $b$ in the set of cosets $G/(x)$.
\end{enumerate}
\end{lemma}
\begin{proof}
Part i) follows directly from (\ref{eq:ComponentsOfMonoid}). For ii) first note that $\Diff^+(F_{1,1})$ surjects onto $Sp(2, \mathbb{Z})$ and acts on $G^2$ through it. Thus
$$[a,b] = [a \pm b, b] = [a, b \pm a].$$
Let $a'=a+Nx$, $b'=b+Mx$. Consider the element
\[ T(N,M) = \left( \begin{array}{cccc}
1 & 0 & -M & N \\
0 & 1 & 0 & 0 \\
0 & N & 1 & 0 \\
0 & M & 0 & 1 \end{array} \right) \in Sp(4, \mathbb{Z})\]
and recall that $\Diff^+(F_{2, 1})$ surjects onto $Sp(4, \mathbb{Z})$ and acts on $G^{2\cdot 2}$ through it. Then
$$T([0,x][a,b]) = T[0,x,a,b] = [x-b, x, a+b-x, b] = [-b,x,a-x, b]$$
\end{proof}
Therefore, after inverting $[x,0] \in H_0(M)$ for all $x$ running through
a set of generators of $G$, the relation $[a,b] = [0,0] = [x,0]$ holds for
all $a,b \in G$ and hence all such $[a,b]$ are invertible and in turn all
elements in the image from $\pi_0(M)$ are invertible. If the generators
are called $x_1,...,x_n$, then the localization may be formed by
inverting the single element $[x_1,0,x_2,0,...,x_n,0]$.
\end{comment}
\begin{lemma}
Suppose that $G$ is a cyclic group generated by an element $1$. Then in $G^2/\Diff^+(F_{1,1})$ there are relations
$$[a \pm b,b] = [a,b] = [a, b \pm a].$$
In $G^4/\Diff^+(F_{2,1})$ there are relations
$$[0,1]\cdot[a,b] = [0,1]\cdot[a-1, b]$$
and
$$[0,1]\cdot[0,b] = [0,1]\cdot[0,1]$$
in the monoid $\pi_0(\mathcal{M})$.
\end{lemma}
\begin{proof}
$\Diff^+(F_{1,1})$ acts on $G^2$ through the group $Sp(2, \mathbb{Z})$, to which it surjects. The obvious group elements give the relations claimed.
$\Diff^+(F_{2, 1})$ acts on $G^4$ through the group $Sp(4, \mathbb{Z})$, to which it surjects. Define an element
\[ t = \left( \begin{array}{cccc}
1 & 1 & 0 & -1 \\
0 & 1 & 0 & 0 \\
0 & -1 & 1 & 1 \\
0 & 0 & 0 & 1 \end{array} \right)\]
in this group. Then
$$t(0,1,a,b) = (1-b, 1, a-1+b, b) \sim (0,1, a-1, b).$$
We also have
$$t(0,1,0,b) = (1-b, 1, b-1, b) \sim (0,1,b-1, 1) \sim (0,1,0,1).$$
\end{proof}
Thus inverting $[0,1]$ gives $[a,b] = [0,b]$ in the group-completion, and implies $[0,b]$ is invertible. Thus it is enough to just invert $[0,1]$ to group-complete the monoid, so we may form
$$\mathcal{M}_\infty = \hocolim(\mathcal{M} \overset{\cdot [0,1]}\to \mathcal{M} \overset{\cdot [0,1]}\to \mathcal{M} \overset{\cdot [0,1]}\to \cdots)$$
whereby Theorem B gives
\begin{MainCor}
Let $G$ be a cyclic group. Then there is a homology equiavalence
$$\mathbb{Z} \times \hocolim_{g \to \infty} \mathcal{S}_{g,1}(BG; *) \to \Omega^\infty \MT{SO}{2} \wedge BG_+$$
where the colimit is formed by the map $\cdot [0,1]$ above.
\end{MainCor}
Let us spell this result out in certain cases. Recall that $\Diff^+(F_{g,1}) \simeq \pi_0(\Diff^+(F_{g,1}))$ which is the mapping class group $\Gamma_{g,1}$ of $F_{g,1}$. When $G=\mathbb{Z}$, we have $\Gamma_{g,1}$ acting on $\mathbb{Z}^{2g}$, and let us write $\Gamma'_{g,1}$ for the subgroup of $\pi_0(\Diff^+(F_{g,1}))$ which fixes the primitive element $(0,1,0,1,\dots,0,1) \in \mathbb{Z}^{2g}$. The statement of the corollary is then that
$$\mathbb{Z} \times \hocolim(\cdots \to B\Gamma'_{g,1} \overset{\cdot [0,1]}\to B\Gamma'_{g+1,1} \to \cdots) \to \Omega^\infty \MT{SO}{2} \wedge S^1_+$$
is a homology equivalence. This identifies the stable homology of the subgroup $\Gamma'_{g,1} < \Gamma_{g,1}$ which fixes a primitive element of $H^1(F_{g,1}, \partial F_{g,1} ; \mathbb{Z})$.
When $G= \mathbb{Z}/n\mathbb{Z}$, we have $\Gamma_{g,1}$ acting on $(\mathbb{Z}/n\mathbb{Z})^{2g}$, and let us write $\Gamma'_{g,1}(n)$ for the subgroup of $\pi_0(\Diff^+(F_{g,1}))$ which fixes the primitive element $(0,1,0,1,\dots,0,1) \in (\mathbb{Z}/n\mathbb{Z})^{2g}$. The statement of the corollary is then that
$$\mathbb{Z} \times \hocolim(\cdots \to B\Gamma'_{g,1}(n) \overset{\cdot [0,1]}\to B\Gamma'_{g+1,1}(n) \to \cdots) \to \Omega^\infty \MT{SO}{2} \wedge B\mathbb{Z}/n\mathbb{Z}_+$$
is a homology equivalence. On the other hand, $B\Gamma_{g,1}'(n)$ can be identified with the moduli space of Riemann surfaces of genus $g$ with a single framed point, equipped with an $n$-fold cyclic unbranched cover. This identifies the stable homology of these moduli spaces.
A similar analysis holds for any space $Y$ with cyclic fundamental group. The question of whether $\pi_0(\mathcal{M})$ may be group completed by inverting finitely many elements for general groups $G=\pi_1(Y)$ seems to be difficult.
|
1,941,325,220,498 | arxiv | \section{Introduction}
\label{intro}
Galaxies and their supermassive black holes are linked in their evolution, resulting in surprisingly tight observational correlations between parameters such as supermassive black hole mass, stellar velocity dispersion, and host galaxy mass (\citealt{HE14.1} for a review). Active galactic nuclei (AGNs) have emerged as key players in this coevolution, by the primary mechanisms of AGN fueling and AGN feedback. Supermassive black holes build up mass by accreting gas during AGN fueling, while AGN outflows are a crucial regulator of star formation that controls the mass growth of the galaxies (e.g., \citealt{DI05.1,CR06.1,FA12.2}).
In recent years, double-peaked narrow emission lines in AGN host galaxies have been studied as a population (e.g., \citealt{LI10.1,CO11.2,BA12.1,CO12.1,FU12.1,BA13.1,MC15.1}), and have been shown to be signatures of both AGN fueling and AGN outflows. Some of these double-peaked emission lines are produced by dual AGNs, which are a pair of AGNs being fueled during a galaxy merger \citep{FU11.3,LI13.1,CO15.1,MU15.1}, and the majority of double-peaked emission lines are produced by AGN outflows (e.g., \citealt{RO10.1,GR12.2,NE16.1}).
Analogous to the AGNs with double-peaked narrow emission lines, there is also a population of galaxy spectra with single-peaked narrow AGN emission lines that exhibit a statistically significant line-of-sight velocity offset relative to the velocity of the host galaxy's stars; 2\% of AGNs exhibit these velocity offsets \citep{CO09.1,CO13.1}. These objects have been much less well studied than the AGNs with double-peaked narrow emission lines, and numerical simulations of galaxy mergers show that velocity-offset emission lines can be produced by offset AGNs, which are off-nuclear AGNs in ongoing galaxy mergers (e.g., \citealt{BL13.1,ST16.1}). Inflows or outflows of gas could also produce velocity-offset AGN emission lines (e.g., \citealt{AL15.1}).
Here, we investigate the origins of the velocity-offset narrow emission lines observed in the Sloan Digital Sky Survey (SDSS) spectra of seven AGNs at $z<0.12$. We observe each galaxy with the {\it Chandra X-ray Observatory} ACIS ({\it Chandra}/ACIS), to pinpoint the location of the AGN, and the {\it Hubble Space Telescope} Wide Field Camera 3 ({\it HST}/WFC3), to obtain high spatial resolution maps of the stellar continuum and the ionized gas. Our goal is to determine the nature of each galaxy and whether its velocity-offset emission lines are tracers of AGN fueling (via inflows or offset AGNs) or AGN feedback (via outflows).
This paper is organized as follows: In Section 2 we describe the sample selection and characteristics. In Section 3 we describe the observations of the sample (SDSS spectra, Keck/OSIRIS integral-field spectroscopy for three of the seven galaxies, {\it Chandra} observations, and {\it HST}/WFC3 multiband imaging), the astrometry, and our analyses of the data. Section 4 presents our results, including the nature of each velocity-offset AGN. Finally, our conclusions are summarized in Section 5.
We assume a Hubble constant $H_0 =70$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_m=0.3$, and $\Omega_\Lambda=0.7$ throughout, and all distances are given in physical (not comoving) units.
\section{The Sample}
We begin with a parent sample of 18,314 Type 2 AGNs at $z<0.21$ in SDSS, which were identified as AGNs via their optical emission line ratios \citep{BR04.1} and the requirement that the fits to the absorption and emission line systems in the SDSS spectra are robust (by examining the signal, residual noise, and statistical noise; \citealt{OH11.1}). The line-of-sight velocity offsets of the emission lines relative to the stellar absorption lines were then measured. From the parent sample of 18,314 Type 2 AGNs, the velocity-offset AGNs were the systems that fulfilled the following four criteria: 1) the velocity offsets of the forbidden emission lines and the Balmer emission lines are the same to within $1\sigma$; 2) the velocity offsets of the emission lines are greater than $3\sigma$ in significance; 3) the emission line profiles are symmetric; 4) the systems do not have double-peaked emission lines.
The 351 AGNs that meet these criteria are the velocity-offset AGNs \citep{CO14.1}. From these 351 velocity-offset AGNs, we select seven systems with low redshifts ($z<0.12$) and high estimated 2-10 keV fluxes ($>5 \times 10^{-14}$ erg cm$^{-2}$ s$^{-1}$). We estimate the 2-10 keV fluxes from the \mbox{[\ion{O}{3}] $\lambda$5007} $\,$ fluxes of the AGNs (which are $>1.3 \times 10^{-14}$ erg cm$^{-2}$ s$^{-1}$ for this sample of seven systems) and the established Type 2 AGN \mbox{[\ion{O}{3}] $\lambda$5007} $\,$ to X-ray scaling relation \citep{HE05.1}. The low redshifts maximize the physical spatial resolution that we can achieve with {\it Chandra} and {\it HST}, while the high 2-10 keV fluxes minimize the observing time necessary for X-ray detections. The seven systems are listed in Table~\ref{tbl-1}.
\section{Observations and Analysis}
\subsection{Optical SDSS Observations}
\label{sdss}
For each of the seven velocity-offset AGNs, the host galaxy redshift (based on the stellar absorption features), the line-of-sight velocity offset of the emission lines, and the \mbox{[\ion{O}{3}] $\lambda$5007} $\,$ luminosity were determined from the SDSS spectrum \citep{CO14.1}. Three of the AGNs have emission lines with redshifted velocity offsets, and four have emission lines with blueshifted velocity offsets. The absolute values of the velocity offsets range from 50 to 113 km s$^{-1}$ (Table~\ref{tbl-1}).
\begin{deluxetable*}{llll}
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecolumns{4}
\tablecaption{Measurements from SDSS Observations}
\tablehead{
\colhead{SDSS Designation} &
\colhead{$z$} &
\colhead{$\Delta v$} &
\colhead{$L_{\mbox{[\ion{O}{3}] $\lambda$5007}}$} \\
& & (km s$^{-1}$) & ($10^{40}$ erg s$^{-1}$)
}
\startdata
SDSS J013258.92$-$102707.0 & $0.03222 \pm 0.00002$ & \phn \phn \phd $56 \pm 10$ & \phn $5.5 \pm 0.6$ \\
SDSS J083902.97+470756.3 & $0.05236 \pm 0.00006$ & \phn $-50 \pm 10$ & $17.7 \pm 1.2$ \\
SDSS J105553.64+152027.4 & $0.09201 \pm 0.00002$ & $-113 \pm 10$ & $48.1 \pm 5.7$ \\
SDSS J111729.22+614015.2 & $0.11193 \pm 0.00001$ & \phn \phn \phd $85 \pm 12$ & $44.5 \pm 7.2$ \\
SDSS J134640.79+522836.6 & $0.02918 \pm 0.00001$ & \phn $-52 \pm 10$ & \phn $7.5 \pm 0.8$ \\
SDSS J165430.72+194615.5 & $0.05367 \pm 0.00001$ & \phn \phn \phd $66 \pm 11$ & $20.7 \pm 2.8$ \\
SDSS J232328.01+140530.2 & $0.04142 \pm 0.00007$ & \phn $-53 \pm 10$ & $10.2 \pm 1.3$
\enddata
\tablecomments{Column 2: host galaxy redshift, based on stellar absorption features. Column 3: line-of-sight velocity offset of emission lines relative to host galaxy systemic. Column 4: observed \mbox{[\ion{O}{3}] $\lambda$5007} $\,$ luminosity.}
\label{tbl-1}
\end{deluxetable*}
\subsection{Keck/OSIRIS Near-infrared IFU Observations}
\label{osiris}
Three of the velocity-offset AGNs were observed with Keck Laser Guide Star Adaptive Optics with OH-Suppressing Infra-Red Imaging Spectrograph (OSIRIS) integral-field spectroscopy \citep{MU16.1}. In each galaxy (SDSS J1055+1520, SDSS J1117+6140, and SDSS J1346+5228), the peak of the line emission (Pa$\alpha$, Pa$\alpha$, and \feii in each galaxy, respectively) was spatially offset from the galaxy center by $0\farcs1$ (0.2 kpc), $0\farcs2$ (0.5 kpc), and $0\farcs3$ (0.2 kpc), respectively. Based on the kinematics of the gas in the OSIRIS observations, \cite{MU16.1} found that SDSS J1055+1520 and SDSS J1346+5228 host AGN outflows while SDSS J1117+6140 has gas inflow along a bar. They concluded that the spatially-offset peaks in line emission are the result of the outflows or inflows driving shocks into off-nuclear gas.
\subsection{Chandra/ACIS X-ray Observations}
\label{chandra}
The seven velocity-offset AGNs were observed with {\it Chandra}/ACIS for the program GO4-15113X (PI: Comerford). Our exposure times were derived from the observed \mbox{[\ion{O}{3}] $\lambda$5007} $\,$ flux for each system (Table~\ref{tbl-1}) and the scaling relation between \mbox{[\ion{O}{3}] $\lambda$5007} $\,$ flux and hard X-ray (2-10 keV) flux for Type 2 AGNs, which has a scatter of 1.06 dex \citep{HE05.1}. We selected exposure times that would ensure a firm detection of at least 10 counts for each AGN, even in the case of the actual X-ray flux falling in the low end of the 1.06 dex scatter. The galaxies were observed with exposure times of 10 ks to 20 ks (Table~\ref{tbl-2}).
\begin{deluxetable*}{lllllll}
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecolumns{7}
\tablecaption{Summary of Chandra and HST Observations}
\tablehead{
\colhead{SDSS Name} &
\colhead{{\it Chandra}/ACIS} &
\colhead{{\it Chandra}/ACIS} &
\colhead{{\it HST}/WFC3} &
\colhead{{\it HST}/WFC3} &
\colhead{{\it HST/WFC3}} &
\colhead{{\it HST/WFC3}} \\
& exp. time (s) & obs. date (UT) & F160W & F606W & F438W & obs. date (UT) \\
& & & exp. time (s) & exp. time (s) & exp. time (s) &
}
\startdata
J0132$-$1027 & 14871 & 2014-08-23 & 147 & 900 & 1047 & 2014-06-24 \\
J0839+4707 & 9927 & 2014-09-03 & 147 & 945 & 1050 & 2014-09-06 \\
J1055+1520 & 14869 & 2015-02-04 & 147 & 900 & 957 & 2014-10-25 \\
J1117+6140 & 19773 & 2015-02-03 & 147 & 1062 & 1065 & 2014-07-03 \\
J1346+5228 & 9937 & 2014-08-29 & 147 & 996 & 1050 & 2015-02-05 \\
J1654+1946 & 9937 & 2014-07-23 & 147 & 900 & 957 & 2014-07-27 \\
J2323+1405 & 14868 & 2014-08-31 & 147 & 900 & 954 & 2014-06-08
\enddata
\tablecomments{Column 2: exposure time for the {\it Chandra}/ACIS observation. Column 3: UT date of the {\it Chandra}/ACIS observation. Columns 4 -- 6: exposure times for the {\it HST}/WFC3 F160W, F606W, and F438W observations. Column 7: UT date of the {\it HST}/WFC3 observations.}
\label{tbl-2}
\end{deluxetable*}
The galaxies were observed with the telescope aimpoint on the ACIS S3 chip in ``timed exposure'' mode and telemetered to the ground in ``faint'' mode. We reduced the data with the latest {\it Chandra} software (CIAO\,4.6.1) in combination with the most recent set of calibration files (CALDB\,4.6.2).
For each galaxy, we used \texttt{dmcopy} to make a sky image of the field in the rest-frame soft ($0.5-2$ keV), hard ($2-10$ keV) and total ($0.5-10$ keV) energy ranges. Using the modeling facilities in \texttt{Sherpa}, we simultaneously modeled the source as a two-dimensional Lorenztian function (\texttt{beta2d}: $f(r)=A(1+[r/r_{0}]^{2})-\alpha$) and the background as a fixed count rate estimated using a source-free adjacent circular region of $30^{\prime\prime}$ radius. We used the SDSS galaxy coordinates as the initial position of the \texttt{beta2d} component, and then we allowed the model to fit a region of 3 times the PSF size (estimated with \texttt{psfSize}) at that location. We determined the best-fit model parameters with \texttt{Sherpa}'s implementation of the `Simplex' minimization algorithm \citep{LA98.1}, by minimizing the Cash statistic. We also attempted a two-component \texttt{beta2d} model to test for additional sources, but all secondary components were detected with $<1\sigma$ significance. Therefore, none of the systems require a secondary component, and Table~\ref{tbl-3} and Figure~\ref{fig:chandraresults} show the best-fit positions of the X-ray source in each galaxy. Table~\ref{tbl-3} also gives the spatial separations between each X-ray source and the host galaxy's stellar nucleus. The errors on these separations are dominated by the astrometric uncertainties in aligning the {\it Chandra} and {\it HST} images. These astrometric erros are calculated in Section~\ref{astrometry}, and the median astrometric error is $0\farcs5$.
\begin{deluxetable*}{lllllllll}
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecolumns{9}
\tablecaption{Chandra and HST/F160W Positions of Each Source}
\tablehead{
\colhead{SDSS Name} &
\colhead{RA$_{HST/F160W}$} &
\colhead{DEC$_{HST/F160W}$} &
\colhead{{\it Chandra} Energy} &
\colhead{RA$_{Chandra}^a$} &
\colhead{DEC$_{Chandra}^a$} &
\colhead{$\Delta \theta (^{\prime\prime})^b$} &
\colhead{$\Delta x$ (kpc)$^b$} &
\colhead{Sig.} \\
& & & Range (keV) & & & &
}
\startdata
J0132$-$1027 & 01:32:58.927 & $-$10:27:07.05 & $0.5-2$ & 01:32:58.924 & $-$10:27:06.87 & $0.18 \pm 0.33$ & $0.12 \pm 0.21$ & $0.6\sigma$ \\
& & & \phn \phd $2-10$ & 01:32:58.917 & $-$10:27:07.05 & $0.14 \pm 0.46$ & $0.09 \pm 0.29$ & $0.3\sigma$ \\
& & & $0.5-10$ & 01:32:58.922 & $-$10:27:07.02 & $0.08 \pm 0.43$ & $0.05 \pm 0.27$ & $0.2\sigma$ \\
\hline
J0839+4707 & 08:39:02.949 & +47:07:55.88 & $0.5-2$ & 08:39:02.944 & +47:07:55.95 & $0.09 \pm 0.29$ & $0.09 \pm 0.30$ & $0.3\sigma$ \\
& & & \phn \phd $2-10$ & 08:39:02.961 & +47:07:55.84 & $0.13 \pm 0.19$ & $0.13 \pm 0.19$ & $0.7\sigma$ \\
& & & $0.5-10$ & 08:39:02.961 & +47:07:55.88 & $0.12 \pm 0.18$ & $0.12 \pm 0.18$ & $0.7\sigma$ \\
\hline
J1055+1520 & 10:55:53.644 & +15:20:27.87 & $0.5-2$ & 10:55:53.653 & +15:20:27.40 & $0.49 \pm 0.84$ & $0.83 \pm 1.44$ & $0.6\sigma$ \\
& & & \phn \phd $2-10$ & 10:55:53.682 & +15:20:27.16 & $0.90 \pm 0.84$ & $1.54 \pm 1.44$ & $1.1\sigma$ \\
& & & $0.5-10$ & 10:55:53.662 & +15:20:27.30 & $0.62 \pm 0.84$ & $1.07 \pm 1.44$ & $0.7\sigma$ \\
\hline
J1117+6140 & 11:17:29.208 & +61:40:15.38 & $0.5-2$ & 11:17:29.193 & +61:40:16.06 & $0.69 \pm 0.73$ & $1.41 \pm 1.49$ & $0.9\sigma$ \\
& & & \phn \phd $2-10$ & 11:17:29.287 & +61:40:15.63 & $0.62 \pm 0.37$ & $1.26 \pm 0.76$ & $1.7\sigma$ \\
& & & $0.5-10$ & 11:17:29.268 & +61:40:15.56 & $0.46 \pm 0.36$ & $0.94 \pm 0.74$ & $1.3\sigma$ \\
\hline
J1346+5228 & 13:46:40.812 & +52:28:36.22 & $0.5-2$ & 13:46:40.816 & +52:28:36.15 & $0.08 \pm 0.36$ & $0.05 \pm 0.21$ & $0.2\sigma$ \\
& & & \phn \phd $2-10$ & 13:46:40.821 & +52:28:35.76 & $0.48 \pm 0.35$ & $0.28 \pm 0.20$ & $1.4\sigma$ \\
& & & $0.5-10$ & 13:46:40.816 & +52:28:35.76 & $0.47 \pm 0.35$ & $0.27 \pm 0.21$ & $1.3\sigma$ \\
\hline
J1654+1946 & 16:54:30.724 & +19:46:15.56 & $0.5-2$ & 16:54:30.734 & +19:46:15.45 & $0.17 \pm 0.48$ & $0.18 \pm 0.50$ & $0.4\sigma$ \\
& & & \phn \phd $2-10$ & 16:54:30.806 & +19:46:15.78 & $1.17 \pm 0.44$ & $1.22 \pm 0.46$ & $2.6\sigma$ \\
& & & $0.5-10$ & 16:54:30.732 & +19:46:15.42 & $0.18 \pm 0.44$ & $0.18 \pm 0.46$ & $0.4\sigma$ \\
\hline
J2323+1405 & 23:23:28.010 & +14:05:30.08 & $0.5-2$ & 23:23:27.996 & +14:05:30.12 & $0.21 \pm 0.26$ & $0.17 \pm 0.22$ & $0.8\sigma$ \\
& & & \phn \phd $2-10$ & 23:23:28.008 & +14:05:30.12 & $0.05 \pm 0.22$ & $0.04 \pm 0.18$ & $0.2\sigma$ \\
& & & $0.5-10$ & 23:23:28.003 & +14:05:30.12 & $0.11 \pm 0.21$ & $0.09 \pm 0.17$ & $0.5\sigma$
\enddata
\tablecomments{Columns 2 and 3: coordinates of the host galaxy's stellar nucleus, measured from {\it HST}/WFC3/F160W observations. Column 4: rest-frame energy range of {\it Chandra} observations. Columns 5 and 6: coordinates of the X-ray AGN source, measured from {\it Chandra}/ACIS observations in the energy range given in Column 4. Columns 7 and 8: angular and physical separations between the positions of the host galaxy's stellar nucleus and the X-ray AGN source, where the error includes uncertainties in the positions of the {\it HST} and {\it Chandra} sources as well as the astrometric uncertainty. Column 9: significance of the separation between the host galaxy's stellar nucleus and the X-ray AGN source.}
\tablenotetext{a}{The astrometric shifts described in Section~\ref{astrometry} have been applied to the {\it Chandra} source positions.}
\tablenotetext{b}{The errors are dominated by the astrometric uncertainties, which range from $0\farcs2$ to $0\farcs8$.}
\label{tbl-3}
\end{deluxetable*}
\begin{figure*}
\begin{center}
\includegraphics[height=1.5in]{fig1a.pdf}
\includegraphics[height=1.5in]{fig1b.pdf}
\includegraphics[height=1.5in]{fig1c.pdf}
\includegraphics[height=1.5in]{fig1d.pdf}
\includegraphics[height=1.5in]{fig1e.pdf}
\includegraphics[height=1.5in]{fig1f.pdf}
\includegraphics[height=1.5in]{fig1g.pdf}
\includegraphics[height=1.5in]{fig1h.pdf}
\includegraphics[height=1.5in]{fig1i.pdf}
\includegraphics[height=1.5in]{fig1j.pdf}
\includegraphics[height=1.5in]{fig1k.pdf}
\includegraphics[height=1.5in]{fig1l.pdf}
\includegraphics[height=1.5in]{fig1m.pdf}
\includegraphics[height=1.5in]{fig1n.pdf}
\includegraphics[height=1.5in]{fig1o.pdf}
\includegraphics[height=1.5in]{fig1p.pdf}
\includegraphics[height=1.5in]{fig1q.pdf}
\includegraphics[height=1.5in]{fig1r.pdf}
\includegraphics[height=1.5in]{fig1s.pdf}
\includegraphics[height=1.5in]{fig1t.pdf}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[height=1.5in]{fig1u.pdf}
\includegraphics[height=1.5in]{fig1v.pdf}
\includegraphics[height=1.5in]{fig1w.pdf}
\includegraphics[height=1.5in]{fig1x.pdf}
\includegraphics[height=1.5in]{fig1y.pdf}
\includegraphics[height=1.5in]{fig1z.pdf}
\includegraphics[height=1.5in]{fig1aa.pdf}
\includegraphics[height=1.5in]{fig1ab.pdf}
\end{center}
\caption{From left to right: {\it Chandra} restframe $0.5-2$ keV (soft) observations, {\it Chandra} restframe $2-10$ keV (hard) observations, {\it Chandra} restframe $0.5-10$ keV (total) observations, and four-color combined {\it Chandra} and {\it HST} observations for each velocity-offset AGN. The leftmost three panels show $5^{\prime\prime} \times 5^{\prime\prime}$ images centered on the host galaxy (red crosses indicate the centers of the host galaxies, based on {\it HST}/F160W observations). The red curves show linear contours for the {\it HST}/F160W observations. The pixels are one-fourth size {\it Chandra} pixels (soft X-rays shown in blue and hard X-rays shown in magenta), and the blue, magenta, and purple crosses indicate the best-fit locations of the soft, hard, and total X-ray sources, respectively. The rightmost panels show four-color images of {\it HST} F160W (red), F606W (green), F438W (blue), and {\it Chandra} restframe $0.5-10$ keV (purple, one-twelfth size pixels smoothed with a 16 pixel radius Gaussian kernel) observations, where the {\it HST} and {\it Chandra} images have been aligned using the astrometric shifts described in Section~\ref{astrometry}. In all panels, north is up and east is to the left.}
\label{fig:chandraresults}
\end{figure*}
Then, we used the Bayesian Estimation of Hardness Ratios (\texttt{BEHR}) code \citep{PA06.1} to measure the rest-frame soft, hard, and total counts in each X-ray source. We used \texttt{calc$\_$data$\_$sum} to determine the number of observed soft and hard counts from both the source region and a background region, and then \texttt{BEHR} used a Bayesian approach to estimate the expected values and uncertainties of the rest-frame soft counts, hard counts, total counts, and hardness ratio. Table~\ref{tbl-4} shows these values, and we estimated errors on the counts assuming Poisson noise.
\begin{deluxetable*}{llllllll}
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecolumns{8}
\tablecaption{X-ray Counts and Spectral Fits}
\tablehead{
\colhead{SDSS Name} &
\colhead{Soft Counts} &
\colhead{Hard Counts} &
\colhead{Total Counts} &
\colhead{Hardness} &
\colhead{$n_{H,exgal}$} &
\colhead{$\Gamma$} &
\colhead{Reduced} \\
& ($0.5-2$ keV) & ($2-10$ keV) & ($0.5-10$ keV) & Ratio & ($10^{22}$ cm$^{-2}$) & & C-stat
}
\startdata
J0132$-$1027 & $14.9^{+3.1}_{-4.2}$ & $8.2^{+2.3}_{-3.4}$ & $23.1^{+4.2}_{-5.2}$ & $-0.30^{+0.19}_{-0.22}$ & $<0.02$ & $1.73^{+0.41}_{-0.38}$ & 0.24 \\
J0839+4707 & $8.7^{+2.1}_{-3.4}$ & $66.1^{+7.4}_{-8.6}$ & $74.8^{+7.9}_{-9.1}$ & $0.77^{+0.09}_{-0.06}$ & $8.18^{+2.21}_{-0.29}$ & $1.70$ (fixed)$^a$ &0.59 \\
J1055+1520 & $15.8^{+3.2}_{-4.5}$ & $6.8^{+2.0}_{-3.3}$ & $22.7^{+4.1}_{-5.3}$ & $-0.40^{+0.19}_{-0.22}$ & $<0.03$ & $1.71^{+0.39}_{-0.47}$ & 0.22 \\
J1117+6140 & $7.6^{+2.0}_{-3.3}$ & $3.1^{+1.1}_{-2.5}$ & $10.7^{+2.6}_{-4.0}$ & $-0.43^{+0.25}_{-0.35}$ & $<0.10$ & $1.87^{+0.78}_{-0.76}$ & 0.12 \\
J1346+5228 & $10.8^{+2.4}_{-3.9}$ & $7.2^{+1.9}_{-3.3}$ & $18.0^{+3.8}_{-4.7}$ & $-0.21^{+0.23}_{-0.25}$ & $<0.12$ & $1.35^{+0.54}_{-0.85}$ & 0.19 \\
J1654+1946 & $23.9^{+4.2}_{-5.3}$ & $4.2^{+1.4}_{-2.7}$ & $28.1^{+4.4}_{-6.0}$ & $-0.71^{+0.11}_{-0.17}$ & $<0.02$ & $1.70$ (fixed)$^a$ & 0.20 \\
J2323+1405 & $17.4^{+3.5}_{-4.7}$ & $32.5^{+5.2}_{-6.2}$ & $49.9^{+6.3}_{-7.6}$ & $0.30^{+0.15}_{-0.13}$ & $0.41^{+0.18}_{-0.16}$ & $1.70$ (fixed)$^a$ & 0.52
\enddata
\tablecomments{Column 2: soft X-ray (restframe 0.5-2 keV) counts (S). Column 3: hard X-ray (restframe 2-10 keV) counts (H). Column 4: total X-ray (restframe 0.5-10 keV) counts. Column 5: hardness ratio HR = (H-S)/(H+S). Column 6: extragalactic column density. Column 7: best-fit spectral index. Column 8: reduced Cash statistic of the fit.}
\tablenotetext{a}{The best-fit spectrum had a spectral index of $\Gamma <1$ or $\Gamma>3$, so we redid the fit by freezing the spectral index to $\Gamma=1.70$.}
\label{tbl-4}
\end{deluxetable*}
To model the energy spectra of the extracted regions over the observed energy range 2-8 keV, we used \texttt{Sherpa}. We fit each unbinned spectrum with a redshifted power law, $F\sim E^{-\Gamma}$ (which represents the intrinsic AGN X-ray emission at the SDSS spectroscopic redshift $z$). This spectrum is attenuated by passing through two absorbing column densities of neutral Hydrogen. One of these is fixed to the Galactic value, $n_{H,Gal}$, and the other is assumed to be intrinsic to the source, $n_{H,exgal}$, at the redshift $z$. We determined $n_{H,Gal}$ using an all-sky interpolation of the \mbox{\ion{H}{1}} $\,$ in the Galaxy \citep{DI90.1}.
For our first fit to each spectrum, we allowed $\Gamma$ and $n_{H,exgal}$ to vary freely. If the best-fit value of $\Gamma$ was not within the typical range of observed power-law indices, i.e. $1\le \Gamma \le 3$ \citep{IS10.1,PI05.1,NA94.2,RE00.1}, then we fixed $\Gamma$ at a value of 1.7, which is a typical value for the continuum of Seyfert galaxies, and ran the fit again.
To determine the best-fit model parameters for each spectrum, we used \texttt{Sherpa}'s implementation of the Levenberg-Marquardt optimization method \citep{BE69.1} to minimize the Cash statistic. Table~\ref{tbl-4} shows the results of these spectral fits. All fluxes are $k$-corrected, and we calculated the observed flux values from the model sum (including the absorbing components) and the intrinsic flux values from the unabsorbed power law component. Finally, we used the redshift to determine the distance to each system and convert the X-ray fluxes to X-ray luminosities (Table~\ref{tbl-5}).
\begin{deluxetable*}{llll}
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecolumns{4}
\tablecaption{X-ray Luminosities}
\tablehead{
\colhead{SDSS Name} &
\colhead{$L_{X, 0.5-2 \mathrm{keV}}$ abs (unabs)} &
\colhead{$L_{X, 2-10 \mathrm{keV}}$ abs (unabs)} &
\colhead{$L_{X, 0.5-10 \mathrm{keV}}$ abs (unabs)} \\
& ($10^{40}$ erg s$^{-1}$) & ($10^{40}$ erg s$^{-1}$) & ($10^{40}$ erg s$^{-1}$)
}
\startdata
J0132-1027 & $1.2^{+0.4}_{-0.3} (1.3^{+0.4}_{-0.4})$ & $2.3^{+2.1}_{-1.2} (2.3^{+2.1}_{-1.2})$ & $3.3^{+2.5}_{-1.2} (3.5^{+2.6}_{-1.3})$ \\
J0839+4707 & $0.2^{+0.5}_{-0.1} (89.7^{+29.6}_{-29.0})$ & $165.0^{+44.7}_{-37.6} (245.0^{+61.2}_{-59.8})$ & $170.0^{+39.6}_{-40.1} (338.0^{+83.0}_{-84.3})$ \\
J1055+1520 & $9.7^{+3.2}_{-3.5} (11.6^{+3.3}_{-3.8})$ & $18.4^{+19.2}_{-8.8} (18.5^{+19.2}_{-8.8})$ & $30.1^{+22.5}_{-12.7} (32.3^{+21.1}_{-13.3})$ \\
J1117+6140 & $3.9^{+2.4}_{-1.9} (6.4^{+2.5}_{-2.9})$ & $7.7^{+14.6}_{-5.1} (7.8^{+14.6}_{-5.1})$ & $12.3^{+15.1}_{-7.1} (14.9^{+14.0}_{-8.2})$ \\
J1346+5228 & $0.8^{+0.4}_{-0.3} (1.0^{+0.4}_{-0.4})$ & $2.7^{+3.1}_{-1.4} (2.7^{+3.1}_{-1.5})$ & $3.7^{+3.5}_{-1.7} (3.8^{+3.6}_{-1.7})$ \\
J1654+1946 & $5.5^{+1.2}_{-1.2} (6.6^{+1.2}_{-1.4})$ & $11.9^{+2.1}_{-2.6} (12.0^{+2.1}_{-2.7})$ & $17.2^{+3.5}_{-3.2} (18.2^{+3.6}_{-3.5})$ \\
J2323+1405 & $2.9^{+1.2}_{-0.8} (7.0^{+1.3}_{-1.4})$ & $12.3^{+2.4}_{-2.6} (12.7^{+2.4}_{-2.6})$ & $15.4^{+3.3}_{-3.0} (19.8^{+3.9}_{-3.8})$
\enddata
\tablecomments{Column 2: absorbed (and unabsorbed) soft X-ray 0.5-2 keV luminosity. Column 3: absorbed (and unabsorbed) hard X-ray 2-10 keV luminosity. Column 4: absorbed (and unabsorbed) total X-ray 0.5-10 keV luminosity.}
\label{tbl-5}
\end{deluxetable*}
\subsection{HST/WFC3 F438W, F606W, and F160W Observations}
\label{hst}
The seven velocity-offset AGNs were also observed with {\it HST}/WFC3 (GO 13513, PI: Comerford), and the observations covered three bands: UVIS/F438W ($B$ band), UVIS/F606W ($V$ band), and IR/F160W ($H$ band). The exposure times are summarized in Table~\ref{tbl-2}.
Each band revealed different properties of the galaxies. The F438W observations covered $H\delta$, $H\gamma$, and \mbox{[\ion{O}{3}] $\lambda$4363} $\,$ for the $0.02 < z < 0.06$ galaxies and \oiidouble and $H\delta$ for the $0.09 < z < 0.12$ galaxies. The F606W observations covered \mbox{H$\beta$}, \mbox{[\ion{O}{3}] $\lambda\lambda$4959, 5007}; \mbox{[\ion{O}{1}] $\lambda\lambda$6300, 6363}; \mbox{[\ion{N}{2}] $\lambda\lambda$6548, 6583}; \mbox{H$\alpha$}, and \mbox{[\ion{S}{2}] $\lambda\lambda$6716, 6730} $\,$ for the $0.02 < z < 0.06$ galaxies; \mbox{H$\beta$}, \mbox{[\ion{O}{3}] $\lambda\lambda$4959, 5007}; and \mbox{[\ion{O}{1}] $\lambda\lambda$6300, 6363} $\,$ for the $z=0.09$ galaxy; and $H\gamma$, \mbox{[\ion{O}{3}] $\lambda$4363}, \mbox{H$\beta$}, \mbox{[\ion{O}{3}] $\lambda\lambda$4959, 5007}; and \mbox{[\ion{O}{1}] $\lambda\lambda$6300, 6363} $\,$ for the $z=0.11$ galaxy. The F160W observations primarily traced the stellar continuum, although they may also have included \feii 1.6436 $\mu$m emission for the $0.02 < z < 0.04$ galaxies and Pa$\beta$ emission for the $0.09 < z < 0.12$ galaxies.
To locate the stellar centroid of each galaxy, we fit a S\'ersic profile (plus a fixed, uniform sky component) to each galaxy's F160W image using GALFIT V3.0 \citep{PE10.1}. We ran each fit on a square region of projected physical size 40 kpc on each side, with the angular size scale calculated from $z$ and assuming the cosmology stated in Section 1.
The errors returned by GALFIT are purely statistical in that they are computed directly from the variance of the input images. We note that in reality, the true radial profiles may deviate from the parametric model components used in GALFIT, particularly at large radii. We previously examined this in \cite{CO15.1} by creating radial profiles of the S\'ersic fits to merger-remnant galaxies, where we found that, even with significant residuals at large radii, the S\'ersic component peaks are excellent tracers of the photometric peaks.
In our fitting procedure, we also attempted a two-S\'ersic component fit (over the same fitting region) to test for the presence of secondary nuclei and/or close interacting neighbors. In these cases, we adopted the two-component model if the secondary component is detected at $>3\sigma$ significance above the background. We found one system, SDSS J0839+4707, with a nearby neighbor galaxy. GALFIT returned the positions of the sources and their integrated magnitudes, which we used to determine the spatial separation on the sky between the two galaxies and their merger mass ratio. We approximated the merger mass ratio as the luminosity ratio of the two stellar bulges.
We also measured the centroid of emission for each galaxy, using Source Extractor \citep{BE96.1} on the F606W images. According to the SDSS spectra, the \mbox{[\ion{O}{3}] $\lambda$5007} $\,$ emission line is the dominant line in the F606W image for each galaxy, within the central $3^{\prime\prime}$. Therefore, the centroid of F606W emission within the central $3^{\prime\prime}$ is a proxy for the centroid of \mbox{[\ion{O}{3}] $\lambda$5007} $\,$ emission. We ran Source Extractor with a detection threshold of $5\sigma$ above the background, and the errors on the positions are statistical.
The positions of the emission centroids, as well as their separations from the stellar centroids, are shown in Table~\ref{tbl-6} and Figure~\ref{fig:hstresults}. We determined the spatial separation errors by combining the errors on the GALFIT positions in the F160W data, the Source Extractor positions in the F606W data, and the relative astrometric uncertainties in the F160W (10 mas) and F606W observations (4 mas; \citealt{DE16.1}). The relative astrometric uncertainties dominate the errors, so that the spatial separation errors are all $0\farcs01$. We found that all of the spatial separations between the emission centroids and the stellar centroids are greater than $3\sigma$ in significance.
\begin{deluxetable*}{lllll}
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecolumns{5}
\tablecaption{HST/F606W Positions of Each Source}
\tablehead{
\colhead{SDSS Name} &
\colhead{RA$_{HST/F606W}$} &
\colhead{DEC$_{HST/F606W}$} &
\colhead{$\Delta \theta (^{\prime\prime})^a$} &
\colhead{$\Delta x$ (kpc)$^a$}
}
\startdata
J0132$-$1027 & 01:32:58.922 & $-$10:27:07.01 & $0.078 \pm 0.011$ & $0.050 \pm 0.007$ \\
J0839+4707 & 08:39:02.937 & +47:07:56.02 & $0.193 \pm 0.011$ & $0.197 \pm 0.011$ \\
J1055+1520 & 10:55:53.638 & +15:20:27.96 & $0.124 \pm 0.012$ & $0.212 \pm 0.020$ \\
J1117+6140 & 11:17:29.218 & +61:40:15.31 & $0.179 \pm 0.011$ & $0.365 \pm 0.023$ \\
J1346+5228 & 13:46:40.802 & +52:28:36.23 & $0.252 \pm 0.011$ & $0.147 \pm 0.006$ \\
J1654+1946 & 16:54:30.734 & +19:46:15.48 & $0.152 \pm 0.011$ & $0.159 \pm 0.011$ \\
J2323+1405 & 23:23:28.004 & +14:05:30.03 & $0.111 \pm 0.011$ & $0.091 \pm 0.009$
\enddata
\tablecomments{Columns 2 and 3: coordinates of the peak of the emission, measured from {\it HST}/WFC3/F606W observations. Columns 4 and 5: angular and physical separations between the positions of the peak of the emission and the host galaxy's stellar nucleus. All separations are $>3\sigma$ in significance.}
\tablenotetext{a}{The errors are dominated by the astrometric uncertainties, which are $0\farcs01$.}
\label{tbl-6}
\end{deluxetable*}
\begin{figure}
\begin{center}
\includegraphics[height=1.5in]{fig2a.pdf}
\includegraphics[height=1.5in]{fig2b.pdf}
\includegraphics[height=1.5in]{fig2c.pdf}
\includegraphics[height=1.5in]{fig2d.pdf}
\includegraphics[height=1.5in]{fig2e.pdf}
\includegraphics[height=1.5in]{fig2f.pdf}
\includegraphics[height=1.5in]{fig2g.pdf}
\includegraphics[height=1.5in]{fig2h.pdf}
\includegraphics[height=1.5in]{fig2i.pdf}
\includegraphics[height=1.5in]{fig2j.pdf}
\includegraphics[height=1.5in]{fig2k.pdf}
\includegraphics[height=1.5in]{fig2l.pdf}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[height=1.5in]{fig2m.pdf}
\includegraphics[height=1.5in]{fig2n.pdf}
\end{center}
\caption{{\it HST}/F160W (left) and {\it HST}/F606W (right) images of each galaxy. The white circle illustrates the SDSS fiber, the red cross shows the galaxy's stellar centroid based on {\it HST}/F160W observations, and the green cross shows the centroid of the emission source detected in the {\it HST}/F606W observations. All images are $5^{\prime\prime} \times 5^{\prime\prime}$, except for the J2323+1405 images, which are $8^{\prime\prime} \times 8^{\prime\prime}$. In all panels, north is up and east is to the left.}
\label{fig:hstresults}
\end{figure}
Finally, we measured the spatial separation between the X-ray AGN source and the center of the stellar bulge (Table~\ref{tbl-3}). The error on each spatial separation incorporate the errors from the \texttt{beta2d} model fit to the {\it Chandra} data (Section~\ref{chandra}), the GALFIT fit to the {\it HST}/F160W data, and the astrometric uncertainty (Section~\ref{astrometry}). The error budget is dominated by the uncertainty in aligning the {\it Chandra} and {\it HST} images, where the median astrometric uncertainty is $0\farcs5$. Due in part to these large astrometric uncertainties, all of the spatial separations are less than $3\sigma$ in significance.
\subsection{Astrometry}
\label{astrometry}
To determine if any {\it Chandra} sources are significantly spatially offset from the stellar bulges seen in the {\it HST}/F160W data, we registered each pair of {\it HST}/F160W and $Chandra$ images and estimated their relative astrometric uncertainties. Due to the small number of {\it Chandra}/ACIS sources and the relatively small {\it HST}/F160W field of view, we registered each image separately to SDSS ($u$, $g$, $r$, $i$, and $z$) and the 2MASS point source catalog \citep{CU03.1}. Then, we combined the two transformations to register the {\it Chandra} and {\it HST} images.
We used \texttt{wavdetect} with a threshold of \texttt{sigthresh}=$10^{-8}$ to detect sources in {\it Chandra}, and Source Extractor with a threshold of $3\sigma$ to detect sources in SDSS, 2MASS, and {\it HST}. Then, we matched sources in each pair of images using the \texttt{xyxymatch} task in IRAF. Next, we used the \texttt{geomap} task in IRAF to calculate $X$ and $Y$ linear transformations for each matched pair ($X_{\mathrm{shift},j}$, $Y_{\mathrm{shift},j}$). We took the final linear transformations in $X$ and $Y$ to be the error-weighted averages, $\overline{X_{\mathrm{shift}}}={\sum_{j=1}^{n} X_{\mathrm{shift},j}\times w_{j,X}}$ and $\overline{Y_{\mathrm{shift}}}={\sum_{j=1}^{n} Y_{\mathrm{shift},j}\times w_{j,Y}}$, where $n$ is the number of sources matched between two images and $w$ is the error weighting.
For each dimension, $X$ and $Y$, we combined in quadrature the errors on the {\it Chandra} and SDSS/2MASS source positions in each band. We repeated this procedure to determine the uncertainty of the relative astrometry for the {\it HST} and SDSS/2MASS images. Then, we added the relative astrometric errors between {\it Chandra} and SDSS/2MASS and between {\it HST} and SDSS/2MASS in quadrature to determine the relative astrometric errors between the {\it Chandra} and {\it HST} images. The final astrometric errors ($\overline{\Delta X}$, $\overline{\Delta Y}$) are then the error-weighted averages of these bands, shown in Table~\ref{tbl-7}. These uncertainties range from $0\farcs2$ to $0\farcs8$, and they dominate the errors when we measure the spatial separations between sources in {\it Chandra} and {\it HST}.
\begin{deluxetable}{lllll}
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecolumns{5}
\tablecaption{Astrometry Measurements}
\tablehead{
\colhead{SDSS Name} &
\colhead{$n_{CS}$} &
\colhead{$n_{HS}$} &
\colhead{$\overline{\Delta X}$ ($\prime\prime$)} &
\colhead{$\overline{\Delta Y}$ ($\prime\prime$)}
}
\startdata
J0132$-$1027 & 0,1,1,0,0,1 & 0,0,1,0,0,0 & 0.4656 & 0.2622 \\
J0839+4707 & 2,2,2,2,1,1 & 0,2,2,2,2,1 & 0.2612 & 0.2607 \\
J1055+1520 & 0,0,0,0,0,0 & 0,0,0,1,0,0 & 0.8382 & 0.8382 \\
J1117+6140 & 0,1,1,1,0,0 & 0,1,1,1,0,0 & 0.5299 & 0.7136 \\
J1346+5228 & 1,2,1,1,1,0 & 0,1,1,1,1,0 & 0.2837 & 0.3517 \\
J1654+1946 & 0,0,1,1,0,0 & 1,2,3,3,2,1 & 0.4721 & 0.4375 \\
J2323+1405 & 1,1,2,1,1,1 & 0,0,1,0,0,0 & 0.2035 & 0.2388
\enddata
\tablecomments{Column 2: number of sources matched between {\it Chandra} and SDSS $u$, $g$, $r$, $i$, $z$ and 2MASS images. Column 3: number of sources matched between {\it HST}/F160W and SDSS $u$, $g$, $r$, $i$, $z$ and 2MASS images. Columns 4 and 5: astrometric accuracy measurements based on matching these sources, in native $X$ and $Y$ coordinates of the {\it HST}/F160W image.}
\label{tbl-7}
\end{deluxetable}
\section{Results}
\subsection{The Galaxies Host Central AGNs, Where Shocks Produce Off-nuclear Peaks in Emission}
\label{shocks}
We use the {\it Chandra} observations to pinpoint the location of the AGN in each galaxy, and we find that each AGN's position is consistent with the host galaxy center to within $3\sigma$ (Table~\ref{tbl-3}). Some of the AGNs may have small, but real, spatial offsets from the galaxy center, but the {\it HST}/F160W images do not show evidence of secondary stellar cores that would accompany these offset AGNs. This leads us to conclude that each galaxy in our sample most likely hosts a central AGN, and not an offset AGN.
The emission line maps for each galaxy are probed by the {\it HST}/F606W observations, which are dominated by \mbox{[\ion{O}{3}] $\lambda$5007}. We find that the emission line centroids are spatially offset from the host galaxy centers by 0.05 to 0.4 kpc, and that all of the spatial separations are greater than $3\sigma$ in significance (Table~\ref{tbl-6}). For the three galaxies that were also observed with Keck/OSIRIS, in all three galaxies the spatial offsets of the emission in the OSIRIS data are consistent with those measured in the F606W data.
Such spatially-offset peaks in emission could be produced by photoionization of an off-nuclear cloud of gas. Outflows and inflows can drive gas into off-nuclear dense regions, but this gas need not necessarily be excited by shocks (e.g., \citealt{RO04.1,RO10.3}). Spatially-offset peaks in emission can also be a signature of shocks. Interacting gas clouds shock the gas, enhancing the ionized gas emission and producing an off-nuclear peak of emission within the narrow line region (e.g., \citealt{MA13.3}).
To search for further evidence of shocks, we examine the optical line flux ratios $\mbox{[\ion{O}{3}] $\lambda$4363}/ \mbox{[\ion{O}{3}] $\lambda$5007}$, $\mbox{[\ion{O}{3}] $\lambda$5007} / \mbox{H$\beta$}$, and $\oin /\mbox{[\ion{O}{3}] $\lambda$5007}$ measured from the SDSS spectrum of each galaxy. Shocks driven into the surrounding gas clouds compress the gas, increasing its density and temperature. The \mbox{[\ion{O}{3}] $\lambda$4363} $\,$ emission line indicates a very high kinetic temperature, which is produced by shock wave excitation and is inconsistent with photoionized low-density clouds. Consequently, the $\mbox{[\ion{O}{3}] $\lambda$4363} $\,$/ \mbox{[\ion{O}{3}] $\lambda$5007}$ line ratio is temperature sensitive and a good indicator of shock activity. Shock heating can also be probed by the $\mbox{[\ion{O}{3}] $\lambda$5007} / \mbox{H$\beta$}$ line flux ratio (e.g., \citealt{SH79.1}). The \oi emission line is another indicator of shocks (e.g., \citealt{DO76.1}), and $\oin /\mbox{[\ion{O}{3}] $\lambda$5007}$ is an ionization level-sensitive line flux ratio.
We compare the $\mbox{[\ion{O}{3}] $\lambda$5007} / \mbox{H$\beta$}$ vs. $\mbox{[\ion{O}{3}] $\lambda$4363}/ \mbox{[\ion{O}{3}] $\lambda$5007}$ line flux ratios, as well as the $\oin /\mbox{[\ion{O}{3}] $\lambda$5007}$ vs. $\mbox{[\ion{O}{3}] $\lambda$4363}/ \mbox{[\ion{O}{3}] $\lambda$5007}$ line flux ratios, to models of pure AGN photoionization and combined AGN photoionization and shocks \citep{MO02.1}. The pure photoionization models are computed with CLOUDY \citep{FE96.1} and use a spectral index $\alpha=-1$ of the ionizing continuum and an ionization parameter ranging from $-4 \leq \log U \leq -1$. The hydrogen density is 100 cm$^{-3}$, which is typical for extended emission line regions \citep{MC90.1}, and the metallicity is solar. The shock models are computed with MAPPINGSIII \citep{DO96.1}, and have a range of shock velocities $100 < v_s ($km s$^{-1}) < 1000$. We find that none of the velocity-offset AGNs have line flux ratios consistent with pure photoionization, and that instead their spectra are explained by a combination of photoionization and shocks (Figure~\ref{fig:shocks}).
To further explore the role of photoionization and shocks in these galaxies, we compare our data to the radiative shock models of \cite{AL08.1}. They assume solar abundance, a preshock density 1 cm$^{-3}$, magnetic parameters ranging from $10^{-4}$ to 10 $\mu$G cm$^{3/2}$, and shock velocities ranging from 200 to 1000 km s$^{-1}$, and they use MAPPINGSIII to model both the shock and its photoionized precursor. For shocks with velocities $\gtrsim170$ km s$^{-1}$, the ionizing front is moving faster than the shock itself, and the ionizing front dissociates and spreads out to form a precursor \mbox{\ion{H}{2}} $\,$ region in front of the shock. Hence, a shocked region can have both shocked gas and photoionized gas. We find that the line flux ratios of our seven velocity-offset AGNs are consistent with the shock plus precursor models of \cite{AL08.1}.
\begin{figure*}
\begin{center}
\includegraphics[width=3.5in]{fig3a.pdf}
\includegraphics[width=3.5in]{fig3b.pdf}
\end{center}
\caption{Optical line flux ratios for the seven velocity-offset AGNs in this paper (filled colored circles), and a velocity-offset AGN that is an offset AGN (open black star). Left: $\mbox{[\ion{O}{3}] $\lambda$5007} / \mbox{H$\beta$}$ vs. $\mbox{[\ion{O}{3}] $\lambda$4363}/ \mbox{[\ion{O}{3}] $\lambda$5007}$, right: $\oin /\mbox{[\ion{O}{3}] $\lambda$5007}$ vs. $\mbox{[\ion{O}{3}] $\lambda$4363}/ \mbox{[\ion{O}{3}] $\lambda$5007}$. The lines illustrate photoionization and shock models from \cite{MO02.1}: pure photoionization models (solid line, where the ionization parameter varies from $\log U=-4$ at the bottom to $\log U=-1$ at the top), and photoionization plus shock models with shock velocities $v_s=1000$ km s$^{-1}$ (dotted line), $v_s=300$ km s$^{-1}$ (dashed line), and $v_s=100$ km s$^{-1}$ (dot-dashed line). The offset AGN's emission lines are consistent with pure photoionization models, whereas the other velocity-offset AGNs (which do not have offset AGNs) have contributions from shocks.}
\label{fig:shocks}
\end{figure*}
We conclude that all seven of the galaxies host both shocked gas and photoionized gas. In the three galaxies observed with Keck/OSIRIS, the OSIRIS data show that the velocity-offset emission lines in the SDSS integrated spectra originate from the shocked off-nuclear emission peak in the gas \citep{MU16.1}. The same is most likely true for the other four galaxies in our sample, and spatially resolved spectra would show it definitively.
\subsection{Sources of the Shocks in the Galaxies}
Here we explore the nature of the shocks in each of the seven galaxies individually.
\subsubsection{Four AGN Outflows}
\label{outflows}
{\bf SDSS J0132$-$1027.} This galaxy displays several colinear knots of emission (Figure~\ref{fig:hstresults}), which are often seen in radio jets driving collimated AGN outflows (e.g., \citealt{MI04.1, RO10.1,TO12.1}). Indeed, SDSS J0132$-$1027 is detected in the FIRST radio survey \citep{BE95.1} with a 20 cm flux density of 1.6 mJy, and higher resolution radio observations would reveal whether it hosts a radio jet. We also note that the southwestern most knot is also detected in the F160W observations. While it is possible that this the stellar bulge of a minor merger, it seems an unlikely coincidence that the minor merger would be colinear with the other knots of emission. Instead, the F160W observations may be tracing \feii 1.6436 $\mu$m emission, which is a common indicator of shocks (e.g., \citealt{AL97.1}) and could be produced as the jet drives into the interstellar medium.
{\bf SDSS J1055+1520.} The outflow in this galaxy has been modeled as a bicone using the Keck/OSIRIS observations of Pa$\alpha$, and the ratio of the outflow energy to the AGN bolometric luminosity was found to be $\dot{E}_{out}/L_{bol} = 0.06 \pm 0.015$ \citep{MU16.1}. This exceeds the energy threshold for an outflow to drive a powerful two-stage feedback process that removes cold molecular gas from the inner parts of a galaxy and suppresses star formation, as found by theoretical studies ($\dot{E}_{out}/L_{bol} > 0.005$; \citealt{HO10.2}). Further, the bicone is oriented with a position angle $138^\circ \pm 6^\circ$ east of north, which is consistent with the spatial orientation of the X-rays (Figure~\ref{fig:chandraresults}). This hints that there may be spatially extended X-ray emission associated with the spatially extended ionized gas, and deeper X-ray observations would be required to confirm this. SDSS J1055+1520 is not detected in FIRST, so its outflow is not radio jet driven. The {\it HST} image also shows that the galaxy itself is asymmetric, which suggests that it may be a merger-remnant galaxy (Figure~\ref{fig:chandraresults}).
{\bf SDSS J1346+5228.} This galaxy's outflow was modeled as a bicone with the Keck/OSIRIS observations, and it is energetic enough ($\dot{E}_{out}/L_{bol}=0.01 \pm 0.002$; \citealt{MU16.1}) to suppress star formation in the galaxy (as was also the case for
SDSS J1055+1520, above). SDSS J1346+5228 is also detected in FIRST with a 20 cm flux density of 1.1 mJy, indicating that there may be a radio jet powering the outflow.
{\bf SDSS J2323+1405.} This galaxy has symmetric emission line gas extending north and south of the galaxy center, out of the plane of the galaxy (Figure~\ref{fig:hstresults}). This morphology is typical of AGN outflows (e.g., \citealt{MU96.1,SC03.3}), and we conclude that SDSS J2323+1405 most likely hosts an AGN outflow.
\subsubsection{Two Inflows of Gas along a Bar}
\label{inflow}
{\bf SDSS J0839+4707.} This galaxy has a stellar bar that is visible in Figure~\ref{fig:chandraresults}, and the peak of emission is spatially offset along the bar (Figure~\ref{fig:hstresults}). SDSS J0839+4707 is also the only galaxy in our sample that has a close companion. The companion galaxy, SDSS J083902.50+470813.9, is located 18.8 kpc ($18\farcs4$) to the northwest and has a redshift of $z=0.053454 \pm 0.000045$ (Figure~\ref{fig:0839companions}). This corresponds to a velocity difference of $311.9 \pm 21.4$ km s$^{-1}$ redshifted away from the primary galaxy. Emission line diagnostics of the companion's SDSS spectrum show that it is a star-forming galaxy \citep{BR04.1}. Using the ratio of the stellar bulge luminosities as a proxy for the merger mass ratio, the merger ratio is 3.59:1 (SDSS J0839+4707 is the more massive galaxy). There is no morphological evidence that SDSS J0839+4707 and its companion are interacting, though a future interaction may trigger new accretion onto the central AGN.
\begin{figure*}
\begin{center}
\includegraphics[width=3in]{fig4a.pdf}
\includegraphics[width=3in]{fig4b.pdf}
\end{center}
\caption{Images of the galactic environment of SDSS J0839+4707. Left: $40^{\prime\prime} \times 40^{\prime\prime}$ four-color image of {\it HST} F160W (red), F606W (green), F438W (blue), and {\it Chandra} restframe $0.5-10$ keV (purple, one-twelfth size pixels smoothed with a 16 pixel radius Gaussian kernel) observations. The {\it HST} and {\it Chandra} images have been aligned using the astrometric shifts described in Section~\ref{astrometry}. Right: $100^{\prime\prime} \times 100^{\prime\prime}$ SDSS $gri$ color composite image. In both panels, north is up and east is to the left.}
\label{fig:0839companions}
\end{figure*}
{\bf SDSS J1117+6140.} The OSIRIS observations of this galaxy reveal two kinematic components: a disturbed rotating disk on large scales and a counterrotating nuclear disk on the small scales of the central kpc \citep{MU16.1}. The galaxy's stellar bar is apparent in Figure~\ref{fig:chandraresults}, and the peak of emission is spatially offset along the bar (Figure~\ref{fig:hstresults}). Based on the model of the counterrotating disk \citep{MU16.1}, the emission peak is located where the nuclear disk and the bar intersect.
\subsubsection{One Ambiguous System}
{\bf SDSS J1654+1946.} The {\it HST} observations of this galaxy show no obvious signatures of an outflow, a bar, or a merger. There is a knot of emission northwest of the galaxy center (Figure~\ref{fig:hstresults}), which could be a nuclear star cluster (e.g., \citealt{GE14.1}). Since SDSS J1654+1946 is highly inclined (almost edge-on), we hypothesize that there could be a small nuclear bar that is too inclined to clearly see in the {\it HST} data. Gas inflowing along this bar could be the cause of the off-nuclear peak in emission, though without evidence of this bar we classify this system as ambiguous.
\subsection{Distinguishing between Velocity Offsets Produced by Shocks and by Offset AGNs}
We have determined that the velocity offsets in our seven targets are produced by shocks and not offset AGNs. Now, for comparison, we consider a velocity-offset AGN that has been confirmed as an offset AGN: SDSS J111519.98+542316.65. SDSS J1115+5423 is $z=0.07$ galaxy that is in the velocity-offset AGN catalog \citep{CO14.1} from which we selected the seven targets in this paper, and it is the only galaxy in that catalog that has been shown to be an offset AGN so far. The emission lines in SDSS J1115+5423 are offset $-68.5 \pm 11.9$ km s$^{-1}$ from systemic. By analyzing archival {\it Chandra} observations of this galaxy, \cite{BA16.1} found that it has a hard X-ray source with $L_{2-10 \mathrm{keV}}=4 \times 10^{43}$ erg s$^{-1}$ that is located $0.8 \pm 0.1$ kpc ($0\farcs64 \pm 0\farcs05$) from the host galaxy center. This offset AGN is located within the $3^{\prime\prime}$ SDSS fiber and presumably is the source of the velocity-offset emission lines in the SDSS spectrum, which could be confirmed with a spatially resolved spectrum of the system.
Interestingly, SDSS J1115+5423's $\mbox{[\ion{O}{3}] $\lambda$5007} / \mbox{H$\beta$}$ vs. $\mbox{[\ion{O}{3}] $\lambda$4363}/ \mbox{[\ion{O}{3}] $\lambda$5007}$ line flux ratios, as well as its $\oin /\mbox{[\ion{O}{3}] $\lambda$5007}$ vs. $\mbox{[\ion{O}{3}] $\lambda$4363}/ \mbox{[\ion{O}{3}] $\lambda$5007}$ line flux ratios, are consistent with models of pure photoionization, in contrast to the seven velocity-offset AGNs studied here (Figure~\ref{fig:shocks}). In the case of the offset AGN (SDSS J1115+5423), the emission lines are produced by photoionization from an AGN that is off-nuclear from the galaxy center but still within the SDSS fiber; this explains the velocity-offset emission lines observed in the SDSS spectrum. On the other hand, in each of the seven velocity-offset AGNs studied here, the emission lines originate from a {\it central} (not offset) AGN. Inflowing or outflowing gas is shocked, producing off-nuclear peaks in emission (still within the SDSS fiber) that result in the velocity-offset emission lines in the SDSS spectrum.
Consequently, we suggest that it is possible to separate a sample of velocity-offset AGNs into offset AGNs and central AGNs (which have shocks resulting from inflows or outflows of gas) using the shocks vs. photoionization diagnostic line flux ratios $\mbox{[\ion{O}{3}] $\lambda$5007} / \mbox{H$\beta$}$ vs. $\mbox{[\ion{O}{3}] $\lambda$4363} / \mbox{[\ion{O}{3}] $\lambda$5007}$, or $\oin /\mbox{[\ion{O}{3}] $\lambda$5007}$ vs. $\mbox{[\ion{O}{3}] $\lambda$4363} / \mbox{[\ion{O}{3}] $\lambda$5007}$. These line flux ratios are measurable with the SDSS spectrum alone; no follow-up observations are required.
\section{Conclusions}
We have presented {\it Chandra} and multiband {\it HST} observations of seven velocity-offset AGNs. The seven AGNs are at $z<0.12$ and have SDSS spectra that show emission lines that are offset in line-of-sight velocity from systemic by 50 to 113 km s$^{-1}$. To determine the nature of the velocity offset in each galaxy, we use the {\it Chandra} observations to determine the location of the AGN and the {\it HST} observations to identify the galaxy's stellar centroid and the location of the peak of the ionized gas emission. Our main results are summarized as follows.
1. All seven velocity-offset AGNs have central AGNs, yet each galaxy's peak in emission is spatially offset from the stellar centroid. The spatial offsets range from 0.05 to 0.4 kpc, and they are all $>3\sigma$ in significance. The spatially offset emission is produced by shocks, and the velocity offsets of the emission lines observed in the SDSS spectra originate from the spatially offset, shocked emission.
2. The shocks are produced by gas falling onto the AGN along a bar, or by AGN outflows propelling outward into the interstellar medium. The seven velocity-offset AGNs are classified as follows: four outflows, two inflows of gas along a bar, and one ambiguous case (since this galaxy is nearly edge-on, it may have a bar that is difficult to see).
3. All of the velocity-offset AGNs studied here fall in the regions of the $\mbox{[\ion{O}{3}] $\lambda$5007} / \mbox{H$\beta$}$ vs. $\mbox{[\ion{O}{3}] $\lambda$4363} / \mbox{[\ion{O}{3}] $\lambda$5007}$ and $\oin /\mbox{[\ion{O}{3}] $\lambda$5007}$ vs. $\mbox{[\ion{O}{3}] $\lambda$4363} / \mbox{[\ion{O}{3}] $\lambda$5007}$ diagrams that are consistent with a combination of photoionization and shock contributors. However, a comparison velocity-offset AGN (where the velocity offset is caused by an offset AGN in a galaxy merger) is consistent with models of pure photoionization and no shocks. We suggest that these emission lines, measured from the SDSS spectrum alone, may efficiently separate the velocity-offset AGNs produced by offset AGNs (photoionization only) from those produced by central AGNs with shocked gas in inflows or outflows (photoionization plus shocks).
Additional follow-up observations, including spatially resolved spectroscopy, X-ray observations, and radio observations, of a large sample of velocity-offset AGNs could test the hypothesis that the $\mbox{[\ion{O}{3}] $\lambda$5007} / \mbox{H$\beta$}$, $\mbox{[\ion{O}{3}] $\lambda$4363} / \mbox{[\ion{O}{3}] $\lambda$5007}$, and $\oin /\mbox{[\ion{O}{3}] $\lambda$5007}$ line flux ratios distinguish the offset AGNs from the central AGNs with shocks. The offset AGNs could then be used for studies of AGN fueling during galaxy mergers (e.g., \citealt{BA17.1}), while the central AGNs with outflows may be particularly effective drivers of feedback. Since the outflows selected from velocity-offset AGNs are outflows {\it with shocks}, these outflows have already been pre-selected to be interacting with their host galaxies. In fact, we found that the two outflows in our sample that were modeled as bicones are energetic enough to drive cold molecular gas out of the galaxy's inner regions and regulate star formation. Thus, AGN outflows with velocity offsets may be a rich source of examples of feedback.
\acknowledgements We thank the anonymous referee for comments that have improved the clarity of this paper. Support for this work was provided by NASA through Chandra Award Number GO4-15113X issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA under contract NAS8-03060. Support for HST program number GO-13513 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 NAS5-26555.
The scientific results reported in this article are based in part on observations made by the Chandra X-ray Observatory, and this research has made use of software provided by the Chandra X-ray Center in the application packages CIAO, ChIPS, and Sherpa. The results reported here are also based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with program number GO-13513.
{\it Facilities:} \facility{{\it CXO}}, \facility{{\it HST}}
\bibliographystyle{apj}
|
1,941,325,220,499 | arxiv | \section{Introduction}
In 1927, S. N. Bernstein proved that the only entire minimal graphs in $\mathbb{R}^3$ are planes. The analogous problem in higher dimension is known as Bernstein problem. Namely, given $u: \mathbb{R}^n\to \mathbb{R}$ a minimal graph, is the graph of $u$ a flat hyperplane? It turns out that the answer is yes for $n\leq7$ and that has been settled down by a series of very significative papers \cite{almgren1966some,de1965estensione,fleming1962oriented,simons1968minimal}.
On the contrary, for $n\geq8$, one has the famous counterexamples by E. Bombieri, E. De Giorgi and E. Giusti \cite{BDG}. Afterwards, many generalizations of Bernstein theorem have arised. As an example, we mention
\cite{schoen1975curvature}
where R. Schoen, L. Simon and S. T. Yau studied a Bernstein type theorem for stable minimal hypersurfaces.
Later on, Bernstein type theorems for constant mean curvature hypersurfaces in Euclidean and in hyperbolic space
$\mathbb{H}^{n+1}$ have been studied. Let us give a very simple example in the Euclidean space.
Does it exist an entire graph $M$ in $\mathbb{R}^{n+1}$ with constant mean curvature $H\not=0$?
It is well known that the answer is no and here is the proof. Assume $M$ exists. Without loss of generality, we can assume that the mean curvature vector of $M$ points upward. Consider a sphere $\mathcal S$ in $\mathbb{R}^{n+1}$ of mean curvature $H$.
As $\mathcal S$ is compact, up to an ambient isometry, we can assume that $\mathcal S$ is above $M.$ Then, translate down
$\mathcal S.$ Clearly, there will be a first contact point $p$ between $M$ and $\mathcal S.$ At $p,$ $\mathcal S$ and $M$ are tangent and applying the maximum principle, one gets that $\mathcal S$ and $M$ should coincide, that is a contradiction.
In the hyperbolic space, there is more variety of constant mean curvature hypersurfaces.
We point out three important results that have been proved throughout history.
{\em
A complete hypersurface $\Sigma$ of ${\mathbb H}^{n+1}$ with constant mean curvature is a horosphere, provided:
\begin{enumerate}
\item \label{DL} $\Sigma$ is properly embedded and has exactly one point in its asymptotic boundary \cite[Theorem A]{do1983alexandrov}.
\item \label{AD} $n=2$ and $\Sigma$ is properly immersed between two horospheres in ${\mathbb H}^3$ with the same asymptotic point \cite[Theorem 1]{alias2006uniqueness}.
\item \label{BQJ} $\Sigma$ is immersed, has all the principal curvatures uniformly are larger than $-1$ and has exactly one point in its asymptotic boundary \cite[Theorem 1.5]{bonini2017weakly}.
\end{enumerate}
}
We will study analogous problems for hypersurfaces with constant higher order mean curvature functions
($H_r$-hypersurface in the following). Our results are generalizations of the three statements above.
Moreover, motivated by the recent work by R. Souam \cite{souam2019mean}, we show the $r$-mean curvature rigidity of horospheres and equidistant spheres (notice that $H_r$ may be zero).
The article is organized as follows. In Section 2, we fix notations and collect some preliminary results.
The result of Section 3 is the following uniqueness theorem for horospheres and equidistant spheres, which is a generalization of \eqref{DL}.
\begin{thmv}{DL r-mean}
Let $\Sigma$ be a complete $H_r$-hypersurface properly embedded in the hyperbolic space $\mathbb{H}^{n+1},$ $r\geq 2.$ Denote the asymptotic boundary of $\Sigma$ by $\partial_\infty\Sigma.$ Then we have:
(1) if $\partial_\infty\Sigma$ is a point, then $\Sigma$ is a horosphere;
(2) if $\partial_\infty\Sigma$ is a sphere and $\Sigma$ separates poles, then $\Sigma$ is a equidistant sphere.
\end{thmv}
For the definition of hypersurface separating poles, see Section 3.
For $r=1$ the result is contained in \cite[Theorem B]{do1983alexandrov}.
In Section 4, we consider the Bernstein problem for immersed hypersurfaces, either with constant $r$-mean curvature
or satisfying a Weingarten equation. In particular, we generalize results by L. Al{\'\i}as and M. Dajczer, \cite{alias2006uniqueness}, which concerns a more general problem in warped products, by L. Al{\'\i}as, D. Impera and M. Rigoli \cite{alias2013hypersurfaces} and by Bonini, Qing and the second author \cite{bonini2017weakly}. Let us mention our main results in Section 4.
\begin{thmv}{theorem-parabolic}
Let $\Sigma$ be a $r$-admissible, $L_{r-1}$-parabolic $H_r$-hypersurface properly immersed in ${\mathbb H}^{n+1}$. If $\Sigma$ is contained in a slab and the angle function does not change sign, then $\Sigma$ is a horosphere.
\end{thmv}
\begin{thmv}{weakly horospherically convex}
Let $\Sigma$ be an immersed, complete, uniformly admissible Weingarten hypersurface in $\mathbb{H}^{n+1}.$ Then $\Sigma$ is a horosphere provided its asymptotic boundary is a single point.
\end{thmv}
A slab is the space between two horospheres that share the same asymptotic point. For the definition of $r$-admissibility, $L_{r-1}$-parabolicity, uniformly admissible Weingarten hypersurface, see Section 2.
Finally, in Section 5, we show the $r$-mean curvature rigidity of horospheres and equidistant spheres, which generalizes \cite{souam2019mean}.
\begin{thmv}{rigidity}
Let $M$ be a horosphere or a equidistant sphere in hyperbolic space $\mathbb{H}^{n+1}$, $n\geq2$ and $H_M>0$ denote its $r$-mean curvature, $r\geq1$, with respect to the orientation given by the mean curvature vector. Let $\Sigma$ be a connected properly embedded $C^2$ hypersurface in $\mathbb{H}^{n+1}$ which coincides with $M$ outside a compact subset $B$ in $\mathbb{H}^{n+1}$. Choose the orientation on $\Sigma$ such that the $r$-mean curvature $H_r$ of $\Sigma$ is equal to $H_M$ outside the compact set $B$. With respect to this orientation, if either $H_r\geq H_M$ or $|H_r|\leq H_M$, then $\Sigma=M$.
\end{thmv}
{\bf Acknowledgements}
The first author was partially supported by INdAM-GNSAGA.
Part of the present work was done during the visit of the second author to Scuola Normale Superiore. The second author would like to thank Professor J\"urgen Jost and Professor Andrea Malchiodi for their support to his visit.
\section{Preliminaries}
\subsection{Models for the hyperbolic space ${\mathbb H}^{n+1}$}
\label{models}
We will work in different well-known models for the hyperbolic space. For the sake of completeness we briefly describe them.
\
{\em The half-space model.} Consider the upper half-space
\begin{align*}
\mathbb{R}^{n+1}_+=\{(x_1,\cdots,x_{n+1})\in\mathbb{R}^{n+1}|x_{n+1}>0\}
\end{align*}
with the metric $\frac{dx_1^2+\dots+dx_{n+1}^2}{x_{n+1}^2}.$ In this model, horospheres are either horizontal hyperplanes
or Euclidean spheres tangent at some point to the hyperplane $\{x_{n+1}=0\}$. Moreover, the intersections of the upper half-space with Euclidean spheres not contained in the upper half-space are totally umbilical hypersurfaces, whose absolute values of the principal curvatures are strictly less than $1$. Such hypersurfaces are usually called {\em equidistant spheres} when the principal curvatures are not zero. The ones with centers on the hyperplane $\{x_{n+1}=0\}$ are {\em (totally geodesic) hyperplanes}.
\
{\em The warped product model.} $\mathbb{H}^{n+1}$ can be viewed as the warped product $\mathbb{R}\times_{e^t}\mathbb{R}^n$, that is the product manifold ${\mathbb R}\times\mathbb{R}^n$ endowed with the following metric
\begin{equation}
\langle\cdot,\cdot\rangle=\pi_1^*(dt^2)+e^{2t}\pi_{2}^*(\langle\cdot,\cdot\rangle_{\mathbb{R}^{n}}),
\end{equation}
where $\pi_1$ and $\pi_2$ denote the projections onto the two factors and $\langle\cdot,\cdot\rangle_{\mathbb{R}^{n}}$ is the Euclidean metric. Notice that the leafs $\mathbb{R}_t=\{t\}\times\mathbb{R}^n$ are horospheres with r-mean curvature one, for every
$r=1,\dots,n$ with respect to $-T$, where $T$ is the lift of $\frac{\partial}{\partial t}$. All the $\mathbb{R}_t$'s share the same point at infinity.
For an immersed hypersurface $\Sigma$ of $\mathbb{R}\times_{e^t}\mathbb{R}^n$, oriented by $\nu$, we define the height function $h\in C^\infty(\Sigma)$ to be the restriction of $\pi_{1}$ to $\Sigma$ and the angle function by $\Theta=\langle\nu,T\rangle$.
\
{\em The hyperboloid model.} For $n\geq2$, The Minkowski space $\mathbb{L}^{n+2}$, is the vector space $\mathbb{R}^{n+2}$ endowed with the Lorentzian metric $\langle, \rangle$ given by\begin{equation*}
\langle \bar{x},\bar{x}\rangle=-x_0^2+\sum_{i=1}^{n+1}x_i^2,
\end{equation*}
where $\bar{x}=(x_0,x_1,\dots,x_{n+1})\in\mathbb{R}^{n+2}$. Then hyperbolic space, de Sitter spacetime and the positive null cone are given by
\begin{equation*}
\begin{split}
& \mathbb{H}^{n+1}=\{\bar{x}\in\mathbb{L}^{n+2}|\langle \bar{x},\bar{x}\rangle=-1, x_0>0\},\\
&\mathbb{S}^{1,n}=\{\bar{x}\in\mathbb{L}^{n+2}|\langle \bar{x},\bar{x}\rangle=1\},\\
&\mathbb{N}^{n+1}_+=\{\bar{x}\in\mathbb{L}^{n+2}|\langle \bar{x},\bar{x}\rangle=0, x_0>0\},
\end{split}
\end{equation*}
respectively. We identify the ideal boundary at infinity of hyperbolic space $\mathbb{H}^{n+1}$ with the unit round sphere $\mathbb{S}^n$ sitting at height $x_0=1$ in the null cone $\mathbb{N}^{n+1}_+$ of Minkowski space $\mathbb{L}^{n+2}$.
Here, horospheres are the intersections of affine {\em null hyperplanes} of $\mathbb{L}^{n+2}$ with $\mathbb{H}^{n+1}.$
A {\em null hyperplane} is such that its normal vector field belongs to $\mathbb{N}^{n+1}_+.$
\subsection{The $k$-mean curvatures $H_k$}
Let $\Sigma$ be an orientable, connected, immersed hypersurface in hyperbolic space $\mathbb{H}^{n+1}$. Let
$\nu$ be an orientation on $\Sigma$ and denote by $A$ the second fundamental form of the immersion with respect to $\nu$. Denote by $\kappa_1,\cdots,\kappa_n$ the principal curvature of $\Sigma$, that is the eigenvalues of $A.$
The $k$-mean curvatures $H_k$ of $\Sigma$, $1\leq k\leq n$, is defined by
\begin{equation}
\binom{n}{k}H_k(x)=\sigma_k(\kappa(x)).
\end{equation}
where $\sigma_k: \mathbb{R}^n\to\mathbb{R}$ is the $k$-elementary symmetric function defined by
\begin{equation}
\sigma_k(\lambda_1,\dots,\lambda_n)=\sum_{i_1<i_2<\cdots<i_k}\lambda_{i_1}\cdots \lambda_{i_k}
\end{equation}
Thus, $H_1$ is the mean curvature, $H_n$ is the Gauss-Kronecker curvature and $H_2$ is a multiple of the scalar curvature, when the ambient space is Einstein.
Functions like $\sigma_k$ are a particular case of hyperbolic polynomials (see \cite{Garding1959}).
It was proved by R. Reilly in \cite{reilly1973} that the study of the $k$-mean curvatures is related to the study of the classical Newton transformations $P_{k}$, that are defined inductively as follows.
$$\begin{array}{l}
\label{Pk}
P_{0}= I,\\
P_{k}=\sigma_{r}I-AP_{k-1},\\
\end{array}$$
where $I$ is the identity matrix and $A$ is a symmetric matrix. Each $P_{k}$ is a self-adjoint operator
that has the same eigenvectors of $A$.
Before establishing the relation between $P_k$ and $H_k$, let us recall that
J. L. Barbosa and G. Colares extended the relation to space forms \cite{barbosacolares1997} and M. F. Elbert to any Riemannian manifold \cite{elbert2002} (see also \cite{elbertnelli2019}).
Let $f:\Sigma\longrightarrow {\mathbb H}^{n+1}$ be an isometric immersion of a
connected oriented Riemannian $n$-manifold into the hyperbolic space and let $A$ its second fundamental form with respect to
an orientation $\nu.$
Let $D\subset \Sigma$ be a domain. A variation of $D$ is a differentiable
map $F:(-\varepsilon,\varepsilon)\times \bar{D}\longrightarrow
{\mathbb H}^{n+1}$, $\varepsilon >0$, such that for each $t\in
(-\varepsilon,\varepsilon)$ the map $F_{t}:\{t\}\times \bar{D}
\longrightarrow{\mathbb H}^{n+1}$ defined by $F_{t}(p)=F(t,p)$ is an immersion
and $F_{0}=f|_{\bar{D}}$. Define $V_{t}(p)=\frac{\partial F}{\partial t}(t,p)$ and $u(t)=\left<V_{t},\nu_{t}\right >,$ where $\nu_{t}$ is the unit normal vector field in $F_{t}(D)$ such that $\nu_0=\nu.$ We say that a variation $F$ of $D$ has compact support if ${\rm supp}(F_{t})\subset K$, for all $t\in (-\varepsilon,\varepsilon)$,
where $K\subset D$ is a compact domain.
Let $H_k^t$ the $k$-mean curvature of $F_{t}$, and $\sigma_k^t= {\binom{n}{k}} H_k^t.$ Then one has
\begin{equation*}\frac{\partial}{\partial t}(\sigma_{k+1}^t)|_{t=0}=L_{k}(u)+u(\sigma_{1}\sigma_{k+1}-(k+2)\sigma_{k+2}-(n-k)\sigma_k)
+V^T(\sigma_{k+1})
\end{equation*}
where $L_k(u)=tr(P_k({\rm Hess}(u)))$ and $V^T$ is the projection of $V$ on $T\Sigma.$ Notice that, in the case $\sigma_{k+1}$ is constant, then the left-hand side and the last term in the previous equality are zero.
\subsection{Ellipticity of $L_k$ and $L_k$-parabolicity}
As $L_k(u)=tr(P_k({\rm Hess}(u)))$, $L_k$ is an elliptic operator if and only if $P_k$ is a positive definite matrix. In particular, $L_0$ is the Laplace-Beltrami operator $\Delta$.
Let us establish a geometric condition that guarantees the ellipticity of $L_k.$
Denote by $\Gamma_k$ the connected component in $\mathbb{R}^n$ of the set $\{H_k>0\}$ that
contains the vector $(1,\dots,1)$. As it is proved in \cite[Section 2]{fontenele2001tangency}, for any $k=1,\dots,n-1,$
\begin{equation}\label{garding ineq}
\Gamma_{k+1}\subset \Gamma_k
\end{equation}
Notice that $\Gamma_n$ is the positive cone in ${\mathbb R}^n.$ Moreover, since $\Gamma_1$ is the largest cone, the mean curvature is positive at any point where the principal curvatures vector stays in the cone $\Gamma_k$.
Moreover, we recall the classical G$\mathring{\rm a}$rding inequality \cite{Garding1959}:
\begin{equation}\label{garding-ineq2}
H_1\geq H_2^{1/2}\geq\cdots\geq H_{k}^{1/k}\geq H_{k+1}^{1/{k+1}}>0,
\end{equation}
providing all the $r$-mean curvature involved are positive.
\begin{defn}\label{k admissible}
A hypersurface $\Sigma$ of $\mathbb{H}^{n+1}$ is called $k$-admissible if the principal curvatures vector at any point of $\Sigma$ stays in the cone $\Gamma_k$, that is,
\begin{equation}
\lambda(x)=(\kappa_1(x),\dots, \kappa_n(x))\in\Gamma_k
\end{equation}
for all $x\in \Sigma$.
\end{defn}
It is well known that the existence of an elliptic point on a $H_k$-hypersurface with $H_k>0$ yields that the hypersurface is $k$-admissible \cite{Garding1959, fontenele2001tangency}. Moreover, $k$-admissibility yields that $L_{k-1}$ is elliptic.
In \cite{alias2013hypersurfaces}, L. Al{\'i}as, D. Impera and M. Rigoli assumed $L_{k}$-parabolicity to study the Bernstein type theorems for hypersurfaces with constant $k$-mean curvature in warped product spaces.
\begin{defn}\label{def-parabolic}
A hypersurface $\Sigma$ in $\mathbb{H}^{n+1}$ is $L_{k}$-parabolic if the only bounded above $C^1$ solutions $u:\Sigma\longrightarrow {\mathbb R},$ of the inequality
\begin{equation}
L_k u\geq0
\end{equation}
are constants.
\end{defn}
\subsection{Weakly horospherically convexity}
Intuitively, a hypersurface is {\em weakly horospherically convex} at $p$ if and only if all the principal curvatures of the hypersurface at $p$ are simultaneously $<-1$ or $>-1$.
For later use, we recall some basic definitions related to the normal geodesic flow in \cite{bonini2015hypersurfaces, bonini2017weakly}.
Let $f:M\longrightarrow \mathbb{H}^{n+1}$ be an isometric immersion of an orientable connected Riemannian manifold of dimension $n$,
and $\eta$ a unit normal vector field orienting $M.$
The {\em hyperbolic Gauss map } $G$ of $M$ is
defined as follows: for every $p\in M$, $G(p)$ is the point at infinity of the unique geodesic starting at $f(p)$ with tangent vector $-\eta(p).$
Notice that $G(p)$ coincides with the point at infinity of the unique horosphere in $\mathbb{H}^{n+1}$ passing through $f(p)$ whose mean curvature vector coincides with $-\eta(p)$ at $f(p)$.
Moreover, with our notion of Gauss map, an horosphere oriented by the mean curvature vector ($\kappa_i=1$) has injective Gauss map.
Now, we give a notion of weak horospherical convexity, using the definition in \cite{bonini2015hypersurfaces, bonini2017weakly} (notice that the orientation is different from that in \cite{espinar2009hypersurfaces}).
\begin{defn}\cite{bonini2017weakly}
Let $f : M^n\to\mathbb{H}^{n+1}$ be an immersed, oriented hypersurface in $\mathbb{H}^{n+1}$ with unit normal vector field $\eta.$
Let $\mathcal{H}_p$ denote the horosphere in $\mathbb{H}^{n+1}$ that is tangent to the hypersurface at $f(p)$ and whose mean curvature vector at $f(p)$ coincides with $-\eta(p)$. We will say that $f : M^n\to\mathbb{H}^{n+1}$ is weakly horospherically convex at $p$ if there exists a neighborhood $V\subset M^n$ of $p$ so that $f(V\setminus\{p\})$ does not intersect with $\mathcal{H}_p$. Moreover, the distance function of the hypersurface $f : M^n\to\mathbb{H}^{n+1}$ to the horosphere $\mathcal{H}_p$ does not vanish up to the second order at $f(p)$ in any direction.
\end{defn}
As we say at the beginning, the formal definition of weakly horospherically convex at a point $p$ implies that all the principal curvatures of the hypersurface at $p$ are simultaneously $<-1$ or $>-1$. By choosing the orientation, we may assume that all the principal curvatures of a weakly horospherically convex hypersurface are $>-1$.
We say that a hypersurface is {\em uniformly} weakly horospherically convex if all the principal curvatures $\kappa_i$ are uniformly larger than $-1$, i.e. $\kappa_i\geq c_0>-1$.
It is clear that the Gauss map of a weakly horospherically convex hypersurface is a local diffeomorphism, therefore, such hypersurface can be parametrized by a subset of $\Omega\subset\mathbb{S}^n$. Now,
let $f: \Omega\subset{\mathbb S}^n\to\mathbb{H}^{n+1}$ be a properly immersed, complete, and uniformly weakly horospherically convex hypersurface.
The (past) normal geodesic flow $\{f^t\}_{t \in \mathbb{R}}$ in $\mathbb{H}^{n+1}$ of $f$ is given by
\begin{equation}\label{Eq:NormalFlow}
f^t(x) := \exp_{f(x)}(-t\eta(x)) = f(x) \cosh{t} - \eta(x)\sinh{t}:\Omega \to \mathbb{H}^{n+1} \subset \mathbb{R}^{1,n+1}
\end{equation}
It is well known \cite{bonini2015hypersurfaces, bonini2017weakly, espinar2009hypersurfaces} that the principal curvatures $\kappa_i^t$ of $f^t$ are given by
\begin{equation}
\label{kappat}
\kappa_i^t = \frac{\kappa_i + \tanh{t}}{1+\kappa_i \tanh{t}}
\end{equation}
Moreover, it is easily seen that the hyperbolic Gauss map $G^t$ is invariant under the normal geodesic flow.
\subsection{ Admissible Weingarten hypersurfaces.}
We will briefly introduce the elliptic problem of Weingarten hypersurfaces \cite{bonini2015hypersurfaces} in our context, that is, restricted to weakly horospherically convex hypersurfaces with the orientation under which all the principal curvatures are simultaneously larger than $-1$.
Let $\mathcal{W}(x_1,\cdots,x_n)$ be a symmetric function of $n$-variables, such that $\mathcal{W}(\kappa_0,\cdots,\kappa_0)=0$ for some number $\kappa_0>-1$. Moreover, let
\begin{equation*}
\mathcal{K}:=\{(x_1,\cdots,x_n)\in\mathbb{R}^n: x_i>-1, i=1,\cdots,n\},
\end{equation*}
Let $\Gamma_+=\{(x_1,\cdots,x_n) : x_1>0,\cdots,x_n>0\}$ and $\Gamma^*$ be an open connected component of
$\{(x_1,\cdots,x_n) : \mathcal{W}(x_1,\cdots,x_n)>0\}$
satisfying
\begin{enumerate}[leftmargin=*]
\item \label{positive}
$(\kappa,\cdots,\kappa)\in\Gamma^*\cap\mathcal{K},$ for every $\kappa\in (\kappa_0,\infty), $
\item \label{connected}
For every $(x_1,\cdots,x_n)\in\Gamma^*\cap\mathcal{K},$ and $(y_1,\cdots,y_n)\in \Gamma^*\cap\mathcal{K}\cap((x_1,\cdots,x_n)+\Gamma_+),$
there exists a curve
$ \gamma $ connecting $(x_1,\cdots,x_n)$ to $(y_1,\cdots,y_n)$ inside $\Gamma^*\cap\mathcal{K}$
such that $\gamma'\in\Gamma_+ $ along $\gamma$
\item\label{elliptic}
$\mathcal{W}\in C^1(\Gamma^*)$ and $\frac{\partial \mathcal{W}}{\partial x_i}>0$ in $ \Gamma^*.$
\end{enumerate}
Suppose $\Sigma$ is a hypersurface of $\mathbb{H}^{n+1}$ satisfying the following {\em general Weingarten equation}
\begin{equation}\label{weingarten}
\mathcal{W}(\kappa_1,\cdots,\kappa_n)=K \ \text{and} \ (\kappa_1,\cdots,\kappa_n)\in \Gamma^*\cap\mathcal{K} \ \text{on} \ \Sigma,
\end{equation}
for some positive constant $K$, where $(\kappa_1,\cdots,\kappa_n)$ are the principal curvatures of the $\Sigma$.
\begin{defn}
In \eqref{weingarten}, a positive number $K$ is admissible for a given curvature function $\mathcal{W}$ if $\mathcal{W}(\bar\kappa_0,\cdots,\bar\kappa_0)=K$, $\frac{\partial\mathcal{W}}{\partial x_i}(\bar\kappa_0,\cdots,\bar\kappa_0)>0$, and $\bar\kappa_0>\kappa_0$.
\end{defn}
A hypersurface $\Sigma$ such that \eqref{weingarten} is satisfied for an admissible constant is called {\em admissible Weingarten hypersurface}. If the principal curvatures have a uniform lower bound which is strictly bigger than $-1$, then we call it uniformly admissible Weingarten hypersurface.
In particular if $ \mathcal{W}$ is an elementary symmetric function of the principal curvatures, all the assumptions are satisfied.
Hence a weakly horospherical convex, $r$-admissible $H_r$-hypersurface is an admissible Weingarten hypersurface.
We will always chose $K$ such that
\begin{equation}
\label{K-adm}
\mathcal{W}(\bar\kappa_0,\cdots,\bar\kappa_0)=K, \ \frac{\partial\mathcal{W}}{\partial x_i}(\bar\kappa_0,\cdots,\bar\kappa_0)>0
\end{equation}
for some $\bar\kappa_0>\kappa_0$.
\section{Bernstein theorem for embedded hypersurfaces}
In this section, we extend the Bernstein type theorem by Do Carmo and Lawson in \cite{do1983alexandrov} to
hypersurfaces with constant $r$-mean curvature in the hyperbolic space.
Recall that $\partial_{\infty} {\mathbb H}^{n+1}$ has a natural conformal structure of a sphere ${\mathbb S}^n(\infty).$ When the
asymptotic boundary of a hypersurface $\Sigma$ is a sphere in ${\mathbb S}^n(\infty),$ we can assume that it is an equator. We say that
$\Sigma$ separates poles if the north and the south poles with respect to such equator are in distinct connected components of
${\mathbb H}^{n+1}\cup{\mathbb S}^n(\infty)\setminus(\Sigma\cup\partial_{\infty}\Sigma).$
\begin{thm}\label{DL r-mean}
Let $\Sigma$ be a complete hypersurface properly embedded in hyperbolic space $\mathbb{H}^{n+1}$ with constant $r$-mean curvature $(r\geq2)$. Denote by $\partial_\infty \Sigma\subset {\mathbb S}^n(\infty)$ the asymptotic boundary of $\Sigma$. Then we have the following:
\begin{enumerate}[leftmargin=*]
\item if $\partial_\infty \Sigma$ is a point, then $\Sigma$ is a horosphere;
\item if $\partial_\infty \Sigma$ is a sphere and $\Sigma$ separates poles, then $\Sigma$ is an equidistant sphere.
\end{enumerate}
\end{thm}
\begin{proof} {\em (1)} Suppose the asymptotic boundary of $\Sigma$ is only one point $q_\infty\in{\mathbb S}^n(\infty)$. First, inspired by \cite{nelli1995some}, we prove that $\Sigma$ has a strictly convex point. We consider the half-space model for $\mathbb{H}^{n+1}$ so that $q_\infty$ corresponds to the infinity point. In this model, the horospheres whose asymptotic boundary is $q_\infty$ are given by the equations $x_{n+1}=\text{constant}$. We write the coordinates as $(\bar{x}, x_{n+1})$, where $\bar{x}=\{x_1,\cdots,x_n\}$. Then the geodesics orthogonal to the horospheres with $q_{\infty}$ as asymptotic point, are one-to-one correspondence with the points $\bar{x}\in\mathbb{R}^{n}$ and can be written as $\gamma_{\bar{x}}(s)=(\bar{x},s)$ for $s>0$. Each such geodesic $\gamma=\gamma_{\bar{x}}$ determines a family of hyperplanes orthogonal to $\gamma,$ $h_t(\bar{x})=\{x\in\mathbb{R}^{n+1}_+:\|x-(\bar{x},0)\|=t\}$, where $\|\cdot\|$ denotes the standard Euclidean norm.
Let ${\mathcal E}_t$ be a family of equidistant spheres such that for any $t$, the asymptotic boundary is $\partial_{\infty}{\mathcal E}_t=\{(\bar x,0)\in{\mathbb R}^{n+1}\ |\ x_1^2+\dots+x_n^2=t^2\}$ and that the mean curvature vector at the highest point, points upward.
Notice that every principal curvature of ${\mathcal E}_t$ is equal to a constant $0<H_0<1$ at any point.
For $t$ small, ${\mathcal E}_t\cap \Sigma=\emptyset.$ Then increase $t$ till the first $\bar t$ such that
${\mathcal E}_{\bar t}\cap \Sigma$ contains a point $p.$ Notice that, in a neighborhood of $p$, the hypersurface $\Sigma$ lies above
${\mathcal E}_{\bar t}.$ For any tangent vector $X$ at $p$ (tangent to $\Sigma$ and ${\mathcal E}_{\bar t}$), consider the 2-plane $P_X$ generated by $X$ and the $x_{n+1}$-axis. In a neighborhood of $p$, $P_X\cap\Sigma$ is a regular curve, that lies above the regular curve$P_X\cap{\mathcal E}_{\bar t}$ that has curvature $H_0.$ Hence the curvature of $P_X\cap\Sigma$ is larger or equal to $H_0$.
Then, $p$ is a strictly convex point of $\Sigma.$
Since $\partial_\infty\Sigma=\{q_{\infty}\}$, if $t$ small enough, we have
\begin{equation*}
h_t\cap\Sigma=\emptyset
\end{equation*}
For any $t>0$, we denote by $\mathcal{H}^{n+1}_+(t)$ and $\mathcal{H}^{n+1}_-(t)$ the half-spaces determined by $h_t=h_t(\bar x)$. We set
\begin{equation*}
\Sigma_{\pm}(t)=\Sigma\cap\mathcal{H}^{n+1}(t)
\end{equation*}
and $\Sigma_-(t)=\emptyset$ for any $\bar x$ and $t$ sufficiently small.
Note also that $\Sigma$ separates $\mathbb{H}^{n+1}$ into two connected components $\Omega_+$ and $\Omega_-$ where $\partial_{\infty}\Omega_+=\{q_{\infty}\}$ and $\partial_\infty\Omega_-\cong\{\mathbb{R}^n\times\{0\}\}$.
Let $t_0$ be the smallest $t$ for which $h_t(\bar x)\cap\Sigma\neq\emptyset$. Then for all $t>t_0$ such that $t-t_0$ sufficiently small, consider the reflected hypersurfaces $\Sigma^{'}_{-}(t)=r_{h_t(\bar x)}(\Sigma_-(t))$, where $r_{h_t}$ is the hyperbolic isometry fixing $h_t(\bar x).$
The hypersurfaces $\Sigma^{'}_{-}(t)$ have the following properties:
\begin{equation}\label{reflect}
\Sigma_{-}^{'}(t)\subset\Omega_+,
\end{equation}
\begin{equation}
\Sigma_{-}^{'}(t)-\partial\Sigma_{-}^{'}(t)\subset\text{int}(\Omega_+).
\end{equation}
Let us now suppose that condition \eqref{reflect} does not hold for some large $t$. Then there must be some $\bar{t}>t_0$ such that the surfaces $\Sigma_{-}^{'}(\bar{t})$ and $\Sigma_{+}(\bar{t})$
have a point $p$ of common tangency (possibly at the boundary), and that $\Sigma_{-}^{'}(\bar{t})$
lies above $\Sigma_{+}(\bar{t})$ in a neighborhood of $p$. By the tangency principle in \cite[Theorem 1.1]{fontenele2001tangency}, we conclude that these hypersurfaces coincide. From this, it easily follows that $\Sigma$ is compact, which is a contradiction. Thus we conclude that \eqref{reflect} holds for all $t$ and all $\bar{x}\in\mathbb{R}^n$.
Relation \eqref{reflect} yields that at any of its points, $\Sigma$ is tangent to the (horizontal) horosphere passing through the point.
If this was not the case, then it is easily seen that there is a geodesic hyperplane $h_t(\bar x)$ (for an appropriate choice of $\bar{x}$ and $t$) such that $\Sigma_{-}^{'}(t)\not\subset\Omega_+$ and this contradicts \eqref{reflect}. It follows that the function $x_{n+1}$ must be constant on $\Sigma$ and $\Sigma$ is a horosphere.
{\em (2)} First we prove that $\Sigma$ has a strictly convex point. By the embeddedness, $\Sigma$ divides the hyperbolic space into two connected components. Also, $\partial_\infty\Sigma$ separates $S^n(\infty)$ into two components: $S^n_+\cup S^n_-=S^n(\infty)-\partial_\infty\Sigma$. Denote by $\nu$ the unit normal orienting $\Sigma$.
We first prove that $\Sigma$ has a strictly convex point, unless $\Sigma$ is a hyperplane.
Let $N_{\varepsilon}(t)$ $t\in[0,1],$ be a family of equidistant spheres with the following properties.
\begin{enumerate}[leftmargin=*]
\item The mean curvature vector of $N_{\varepsilon}(t)$ points upward for any $t\in[0,1].$
\item The angle between $N_{\varepsilon}(0)$ and $\{x_{n+1}=0\},$ is $\frac{\pi}{2}+\varepsilon$
with $\varepsilon>0$ small and $\Sigma$ is contained in the mean-convex side of $N_{\varepsilon}(0).$
\item $N_{\varepsilon}(t)$ is obtained from $N_{\varepsilon}(0)$ by a homothety from the euclidean center of $N_{\varepsilon}(0).$
By construction, the angle $\theta_{\varepsilon}(t)$ between $N_{\varepsilon}(t)$ and $\{x_{n+1}=0\}$ satisfies
$\theta_{\varepsilon}(t)>\frac{\pi}{2}$ hence the mean curvature vector of $N_{\varepsilon}(t)$ points towards $\Sigma.$
\item $\partial_{\infty} N_{\varepsilon}(1)=\partial_{\infty}\Sigma.$
\end{enumerate}
Increasing $t$, there exists a first $\bar t<1$ such that $N_{\varepsilon}(\bar t)$ and $\Sigma$ has a contact point $p$, then $p$ is a strictly convex point. If such point does not exists, then $\Sigma$ lies above $N_{\varepsilon}(1).$ Notice that, letting
$\varepsilon\longrightarrow 0,$ $N_{\varepsilon}(1)$ tends to the hyperplane whose asymptotic boundary coincides with $\partial\Sigma.$
Hence $\Sigma$ lies above such hyperplane.
Let $S_{\varepsilon}(t)$ $t\in[0,1],$ be a family of equidistant spheres with the following properties.
\begin{enumerate}[leftmargin=*]
\item The mean curvature vector of $S_{\varepsilon}(t)$, at the highest point, points downward for any $t\in[0,1].$
\item The angle between $\partial_{\infty}S_{\varepsilon}(0)$ and $\{x_{n+1}=0\},$ is $\frac{\pi}{2}-\varepsilon$
with $\varepsilon>0$ small and $\Sigma$ is contained in the mean-convex side of $S_{\varepsilon}(0).$
\item $S_{\varepsilon}(t)$ is obtained from $S_{\varepsilon}(0)$ by a homothety from the euclidean center of $S_{\varepsilon}(0).$
By construction, the angle $\theta_{\varepsilon}(t)$ between $S_{\varepsilon}(t)$ and $\{x_{n+1}=0\}$ satisfies
$\theta_{\varepsilon}(t)<\frac{\pi}{2}$ hence the mean curvature vector of $S_{\varepsilon}(t)$ points towards $\Sigma$.
\item $\partial_{\infty} S_{\varepsilon}(1)=\partial\Sigma.$
\end{enumerate}
Increasing $t$, there exists a first $\bar t<1$ such that $S_{\varepsilon}(\bar t)$ and $\Sigma$ has a contact point $p$, then $p$ is a strictly convex point. If such point does not exists, then $\Sigma$ lies below $S_{\varepsilon}(1).$ Notice that, letting
$\varepsilon\longrightarrow 0,$ $S_{\varepsilon}(1)$ tends to the hyperplane whose asymptotic boundary coincides with $\partial\Sigma.$
Hence $\Sigma$ lies below such hyperplane.
We conclude that either there is a strictly convex point on $\Sigma$ or $\Sigma$ is a hyperplane.
Hence we may assume that there is a strictly convex point. By the argument after \eqref{garding ineq}, $H_1$ is positive on $\Sigma$.
Moreover, since $\Sigma$ separates poles, we can select $S^n_+$ as asymptotic boundary of the region into which the mean curvature vector of $\Sigma$ points.
Now let us use the half-space model. We take the center of $S^-_n$ to be the origin of the half-space model and then $S^n_+$ is the component which is unbounded in the Euclidean topology.
First we prove that the $r$-mean curvature of $\Sigma$ satisfies $H_r< 1.$
Fix a point $x\in S^n_+$ and consider the family of horospheres having $x$ as asymptotic point. There is a horopshere first touches
$\Sigma.$ At this contact point, the horosphere and $\Sigma$ are tangent. Moreover, with respect to that normal vector, $\Sigma$ is below the horosphere. Therefore $r$-mean curvature $H_r$ of $\Sigma$ is strictly less than $1$ by the tangency principle in \cite{fontenele2001tangency}.
Let ${\mathcal E}$ be the equidistant sphere of $r$-mean curvature $H_r,$ such that
$\partial_{\infty}{\mathcal E}=\partial_\infty\Sigma$ and the mean curvature vector of ${\mathcal E}$ also points to $S^n_+$. By applying to ${\mathcal E}$ the isometries of ${\mathbb H}^{n+1}$ given by homotheties with respect to the center of $S^n_-$, we get a foliation of $\mathbb{H}^{n+1}$ consisting of equidistant spheres, denoted by ${\mathcal E}_t,$ $t\in {\mathbb R}.$ Choose the parameter $t$ such that
${\mathcal E}_0={\mathcal E}$ and ${\mathcal E}_t$ goes to the origin as $t\to-\infty$ (to the infinity point as $t\to+\infty$). Since $\partial_{\infty}{\mathcal E}=\partial_\infty\Sigma$, we have that $\Sigma\cap {\mathcal E}_t$ is compact for all $t\neq0$ and $\Sigma\cap {\mathcal E}_t=\emptyset$ for all $|t|$ sufficiently large. If $\Sigma\neq {\mathcal E}$, then $\Sigma\cap {\mathcal E}_{\bar t}\neq\emptyset$ for some $\bar{t}\neq0$. Suppose $\bar{t}>0$, let $t_1=\sup\{t: \Sigma\cap {\mathcal E}_{\bar t}\neq\emptyset\}$. Then $\Sigma$ is below ${\mathcal E}_{t_1}$ with respect to $\nu$ near the contact point. Thus, by the tangency principle in \cite{fontenele2001tangency}, $\Sigma={\mathcal E}_{t_1}$ which contradicts $\partial_{\infty}{\mathcal E}=\partial_\infty\Sigma$. The case of $\bar{t}<0$ is similar. Hence, $\Sigma={\mathcal E}$.
\end{proof}
\section{Bernstein theorem for immersed hypersurface}
In this section, we consider the Bernstein theorem for immersed hypersurfaces, either with constant $r$-mean curvature, or satisfying a general elliptic Weingarten equation.
\subsection{The case of constant $r$-mean curvature hypersurface contained in a slab}
In this section, we consider the Bernstein theorem for immersed hypersurfaces with constant $r$-mean curvature contained in a slab.
The starting point is the following non-existence theorem.
\begin{thm}\label{nonexistence}
In any slab of hyperbolic space $\mathbb{H}^{n+1}$, there is no complete properly immersed hypersurface $\Sigma$ with $r$-mean curvature satisfying $\Bar{H}:=\sup_{\Sigma}|H_r|<1$ for any $r\geq1$.
\end{thm}
\begin{proof}
In the upper half-space model of hyperbolic space $\mathbb{H}^{n+1}$, we assume that the slab is between two horospheres given by two horizontal Euclidean hyperplanes. We can foliate the whole space by a family of equidistant spheres ${\mathcal E}(t)$ with mean curvature being $\bar{H}^{\frac1r}$ with respect to the normal vector field that points upward at the highest point, for $t\in[0,\infty)$. When $t$ is small, ${\mathcal E}(t)$ and $\Sigma$ are disjoint. Then, consider $t_0$ such that ${\mathcal E}(t_0)$ and the hypersurface $\Sigma$ first touch at some point $p$. In a neighborhood of $p$, $\Sigma$ is above ${\mathcal E}(t_0)$ with respect the upward normal vector of ${\mathcal E}(t_0)$ at $p$. We also have $H_r({\mathcal E}(t_0))\geq H_r(\Sigma)$ in that neighborhood and ${\mathcal E}(t_0)$ is $r$-admissible. Therefore, by the tangency principle \cite[Theorem 1.1]{fontenele2001tangency}, $\Sigma={\mathcal E}(t_0)$, which is a contradiction, because ${\mathcal E}(t_0)$ is not contained in any slab.
\end{proof}
As a consequence of the previous Theorem, we are able to prove the analogous of \cite[Theorem 1]{alias2006uniqueness}
for surfaces of constant Gaussian curvature.
\begin{cor}
If $\Sigma$ is a properly immersed complete surface in ${\mathbb H}^3$ with constant $2$-mean curvature $0<H_2\leq1$ contained in a slab then $\Sigma$ is a horosphere.
\end{cor}
\begin{proof}
By Theorem \ref{nonexistence}, we have $H_2=1$, which implies that $\Sigma$ is a complete flat immersion. Then the result follows from \cite[Theorem 5]{galvez2009surfaces}.
\end{proof}
In higher dimension, we need to add $L_{k}$-parabolicity (see Definition \ref{def-parabolic}) and a geometric assumption, in order to get a Bernstein type Theorem.
\begin{thm}\label{theorem-parabolic}
Let $\Sigma$ be a complete, $r$-admissible, $L_{r-1}$-parabolic properly immersed hypersurface with constant $r$-mean curvature. If $\Sigma$ is contained in a slab and the angle function does not change sign, then $\Sigma$ is a horosphere.
\end{thm}
\begin{proof}
Define $\phi=e^hH_r^{1/r}+e^h\Theta$ where $\Theta$ is the angle function and $h$ is the height, as we defined in Section 2.1. It follows from the proof of Theorem 32 in \cite{alias2013hypersurfaces} that
\begin{equation}
\begin{split}
L_{r-1}\phi\geq&c_{k-1}e^hH_r^{1/r}(H_{r-1}-H_r^{\frac{r-1}{r}})\\
&-\binom{n}{k}e^h\Theta(nH_1H_r-(n-r)H_{r+1}-rH_r^{\frac{r+1}{r}})\geq0,
\end{split}
\end{equation}
where we have used the Garding inequality \eqref{garding-ineq2} in the last inequality. Since $\Sigma$ is $L_{r-1}$-parabolic, $\phi$ is a constant. In particular, $\Delta\phi=0.$ Then, equation (3.8) in \cite{alias2007constant} gives that
\begin{equation*}
0=\Delta \phi=e^h\Theta(||A||^2-(n-1)H^2)
\end{equation*}
Notice that, in the notation of (3.8) in \cite{alias2007constant}, $\rho(t)=e^t,$ ${\mathcal H}(t)=1$ and $Ric_{\mathbb P} (\hat N)=0$ as ${\mathbb P}={\mathbb R}^n.$
We conclude that $\|A\|^2=(n-1)H^2,$ that yields $\Sigma$ is a totally umbilical hypersurface. Thus, all the principal curvatures are equal to a constant. Moreover, since the hypersurface is contained in a slab, then all the principal curvatures are larger or equal than $1$ by Theorem \ref{nonexistence}. Then it is either a horopshere or sphere by \cite[Theorems A,B]{currier1989hypersurfaces}. However, in the latter case, the angle function changes sign. Therefore, it has to be a horosphere.
\end{proof}
\subsection{The case of admissible Weingarten hypersurfaces}
In this section, we get a Bernstein type theorem for admissible Weingarten hypersurfaces.
Note that uniformly weakly horospherically convex hypersurfaces with injective hyperbolic Gauss map become embedded under the (past) normal geodesic flow (see Theorem 1.3 in \cite{bonini2017weakly}).
\begin{thm}\label{weakly horospherically convex}
Suppose that $ \Sigma$ is an immersed, complete, uniformly weakly horospherically convex admissible Weingarten hypersurface in $\mathbb{H}^{n+1}$. Then $\Sigma$ is a horosphere provided its asymptotic boundary is a single point.
\end{thm}
\begin{proof}
As the Gauss map of $\Sigma$ is locally injective, we can apply Theorem 4.2 in \cite{bonini2017weakly}. Then, for $t$ large enough, the past normal geodesic flow defined in \eqref{Eq:NormalFlow}, deforms $\Sigma$ into a properly embedded, uniformly weakly horospherically convex hypersurface $\Sigma_t$ with single point boundary at infinity. Moreover, the principal curvatures of $\Sigma_t$ are given by (see \eqref{kappat}):
\begin{equation}
\kappa_i^t=\frac{\kappa_i + \tanh(t)}{1+\kappa_i \tanh(t)} \ \text{and} \ \kappa_i=\frac{\kappa_i^t-\tanh(t)}{1-\kappa_i^t\tanh(t)}
\end{equation}
Let
\begin{equation}
\mathcal{W}^t(x_1,\cdots,x_n):=\mathcal{W}\bigg(\frac{x_1-\tanh(t)}{1-x_1\tanh(t)},\cdots,\frac{x_n-\tanh(t)}{1-x_n\tanh(t)}\bigg).
\end{equation}
Then it follows from the definition of $\mathcal{W}$ that $\mathcal{W}^t$ is a symmetric function of $n$-variables with
\begin{equation}
\mathcal{W}^t\left(\frac{\kappa_0 + \tanh(t)}{1+\kappa_0 \tanh(t)},\dots,\frac{\kappa_0 + \tanh(t)}{1+\kappa_0 \tanh(t)}\right)= \mathcal{W}(\kappa_0,\dots,\kappa_0)=0
\end{equation}
and $\frac{\kappa_0 + \tanh(t)}{1+\kappa_0 \tanh(t)}>-1$.
Define
\begin{equation}
\mathcal{T}(x_1,\cdots,x_n)=\bigg(\frac{x_1+\tanh(t)}{1+x_1\tanh(t)},\cdots,\frac{x_n+\tanh(t)}{1+x_n\tanh(t)}\bigg).
\end{equation}
We then have
\begin{equation}
\Gamma_t^*\cap\mathcal{K}=\mathcal{T}(\Gamma^*\cap\mathcal{K})
\end{equation}
and
\begin{equation}
\mathcal{T}((x_1,\cdots,x_n)+\Gamma_n)=\mathcal{T}(x_1,\cdots,x_n)+\Gamma_n.
\end{equation}
For ellipticity, one can easily compute
\begin{equation}
\frac{\partial \mathcal{W}}{\partial x_i}=\frac{1-\tanh^2(t)}{(1-x_i\tanh(t))^2}\frac{\partial\mathcal{W}}{\partial y_i}.
\end{equation}
Therefore, $(\mathcal{W}_t,\Gamma_t^*)$ satisfies \eqref{positive}-\eqref{elliptic}. Thus, \cite[Theorem 4.4]{bonini2015hypersurfaces}
yields that $\Sigma_t$ is a horosphere and so is $\Sigma$, since $\Sigma$ is a time-slice of the foliation formed by a horosphere under the normal geodesic flow.
\end{proof}
\begin{cor}\label{cor-weingarten}
Suppose that $ \Sigma$ is an immersed, complete, uniformly weakly horospherically convex, $r$-admissible $H_r$-hypersurface in $\mathbb{H}^{n+1}$. Then $\Sigma$ is a horosphere provided its asymptotic boundary is a single point.
\end{cor}
\section{$r$-mean curvature rigidity of horospheres and equidistan spheres}
Motivated by the following result of M. Gromov \cite{gromov2018}:
\
{\em A hyperplane in a Euclidean space $\mathbb{R}^n$ cannot be perturbed on a compact set so
that its mean curvature satisfies $H\geq0$.}
\
R. Souam proved the following extension to hyperbolic space \cite{souam2019mean}:
\
{\em Let $M$ denote a horosphere, an equidistant sphere or a hyperplane in a hyperbolic
space $\mathbb{H}^{n+1}$, $n\geq2$ and $H_M\geq 0$ its constant mean curvature. Let $\Sigma$ be a connected properly embedded $C^2$-hypersurface in $\mathbb{H}^{n+1}$ which coincides with $M$ outside a compact subset of $\mathbb{H}^{n+1}$. If the mean curvature of $\Sigma$ is $\geq H_M$, then $\Sigma=M$. }
\
The proof in \cite{souam2019mean} is based on the tangency principle for mean curvature. We are able to extend Souam's result to the case of $r$-mean curvature by the tangency principle in \cite{fontenele2001tangency}.
\begin{thm}\label{rigidity}
Let $M$ be a horosphere or an equidistant sphere in hyperbolic space $\mathbb{H}^{n+1}$, $n\geq2$ and denote by $H_M>0$ its $r$-mean curvature, $r\geq1$, with respect to the orientation given by the mean curvature vector. Let $\Sigma$ be a connected properly embedded $C^2$ hypersurface in $\mathbb{H}^{n+1}$ which coincides with $M$ outside a compact subset $B$ in $\mathbb{H}^{n+1}$. Choose the orientation on $\Sigma$ such that the $r$-mean curvature $H_r$ of $\Sigma$ is equal to $H_M$ outside the compact set $B.$ With respect to this orientation, if either $H_r\geq H_M$ or $|H_r|\leq H_M$, then $\Sigma\equiv M$.
\end{thm}
\begin{proof}
We take the upper half-space model of hyperbolic space.
(1) The case of horospheres.
Consider the family of horospheres ${\mathcal O}_t=\{x\in\mathbb{R}^{n+1}|x^{n+1}=t\},$ $t>0$ and assume that $M={\mathcal O}_1$. Notice that, $H_M=1.$
Let $\Sigma$ be a connected properly embedded $C^2$ hypersurface in $\mathbb{H}^{n+1}$ with $r$-mean curvature either $H_r\geq 1$ or $H_r\leq 1$.
Now, assume that $H_r\geq 1$.
Since $\Sigma$ coincides with $M$ outside a compact subset $B$ of $\mathbb{H}^{n+1}$, the mean convex side of $\Sigma,$ that is the component where the mean curvature vector points towards, coincides with the domain $\{x\in\mathbb{R}^{n+1}|x^{n+1}>1\},$ outside a compact set.
We consider the largest $T\geq 1$ such that $\Sigma\cap{\mathcal O}_T\not=\emptyset$ and let $p\in \Sigma\cap{\mathcal O}_T$.
At the point $p$, $\Sigma$ and ${\mathcal O}_T$ are tangent, in a neighborhood of $p$, the horosphere ${\mathcal O}_T$ lies above $\Sigma,$ while the $r$-mean curvature of $\Sigma$ is larger or equal than the $r$-mean curvature of ${\mathcal O}_T$ (with respect to the upward normal vector). By the tangency principle \cite[Theorem 1.1]{fontenele2001tangency}, $\Sigma$ coincide with ${\mathcal O}_T$ in a open neighborhood of $p$.
Hence the subset $\Sigma\cap{\mathcal O}_T$ is open. As it is also closed, we get that $\Sigma$ coincides with ${\mathcal O}_T$ and $T=1$.
Now, assume that $H_r\leq 1$.
We consider the smallest $\tau\leq 1$ such that $\Sigma\cap{\mathcal O}_{\tau}\not=\emptyset$ and let $p\in \Sigma\cap{\mathcal O}_{\tau}$.
At the point $p$, $\Sigma$ and ${\mathcal O}_{\tau}$ are tangent, in a neighborhood of $p$, the horosphere ${\mathcal O}_{\tau}$ lies below $\Sigma$, while the $r$-mean curvature of $\Sigma$ is less than or equal to the $r$-mean curvature of ${\mathcal O}_T$ (with respect to the upward normal vector). Notice that, we do not know in advance whether the normal vector to $\Sigma$ at $p$ points upward or downward. However, by the assumption $|H_r|\leq H_M$, one can check that the $r$-mean curvature of $\Sigma$ with respect to the upward normal is always less than or equal to $H_M$. Then it follows from the tangency principle \cite[Theorem 1.1]{fontenele2001tangency} that $\Sigma$ coincide with ${\mathcal O}_{\tau}$ in a open neighborhood of $p$. Hence the subset $\Sigma\cap{\mathcal O}_{\tau}$ is open. As it is also closed, we get that $\Sigma$ coincides with ${\mathcal O}_{\tau}$ and $\tau=1$. Notice that, in spite of the fact that $\Sigma$ may have non positive $r$-mean curvature, we can apply \cite[Theorem 1.1]{fontenele2001tangency} because the principal curvature vector of ${\mathcal O}_{\tau}$ lies in the positive cone.
(2) The case of equidistant spheres.
We may assume that the mean curvature vector of $M$ points upward and that the asymptotic boundary of $M$ is a $(n-1)$-sphere of $\partial_{\infty}{\mathbb H}^{n+1}$ centered at the origin of ${\mathbb R}^{n+1}.$ As in the previous case, since $\Sigma$ coincides with $M$ outside a compact subset $B$ of $\mathbb{H}^{n+1}$, the mean convex side of $\Sigma,$ coincides with the mean convex side of $M$ outside a compact set.
Let ${\mathcal E}(t)$ be a foliation in equidistant spheres, obtained rescaling $M$, with respect to the origin, such that ${\mathcal E}(0)=M$ and ${\mathcal E}(t)$ is above $M$ for positive $t$ and below $M$ for negative $t.$ Notice that, all the ${\mathcal E}(t)$ has the same $r$-mean curvature.
Similar as in (1), when $H_r\geq H_M$, $\Sigma$ touches ${\mathcal E}(t)$, $t\geq 0$ from below. When $|H_r|\leq H_M$, $\Sigma$ touches ${\mathcal E}(t)$, $t\leq 0$ from above. In both cases, by tangency principle, $\Sigma=M$.
\end{proof}
\nocite{*}
\bibliographystyle{amsplain}
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1,941,325,220,500 | arxiv | \section{Introduction}
Planets form in gaseous and dusty protoplanetary disks around young stars a few million years old. Forming planets gravitationally interact with the host disk, producing structures such as gaps, spiral arms, and vortices \citep{kley12}. By comparing observations with theoretical models, spatially resolved disk structures may yield rich information about the properties of embedded planets, such as their orbits, and dynamical constraints on their masses.
In the past decade, near-infrared imaging of disks with high spatial resolution has discovered spiral arms at tens of au in a few systems (e.g., SAO~206462: \citealp{muto12, garufi13, stolker16}; LkH$\alpha$~330: \citealp{akiyama16}; MWC~758: \citealp{grady13, benisty15}; HD~100453: \citealp{wagner15, benisty17}; {and HD~141569~A: \citealp{mouillet01, clampin03, konishi16}}). Hydrodynamical and radiative transfer simulations have suggested two mechanisms for reproducing such structure: gravitational instability \citep{lodatorice15, dong15b} which occurs in disks with sufficient mass \citep{kratter16}, and companion-disk interaction \citep{dong15, zhu15, bae16}. Because the host disks in these few systems are probably not massive enough to trigger the gravitational instability \citep[e.g., ][]{andrews11}, the latter scenario is more likely.
Detailed numerical simulations have quantified the dependencies of arm separation and contrast on the companion mass and disk properties \citep{fung15, dong17}. These relations have been used to infer the parameters of hypothesized arm-driving companions. A proof of concept of this mechanism is recently provided by the HD~100453 system, where both the arms and the companion have been found, with their physical connections numerically supported \citep{dong16, wagner18}. Extensive direct imaging observations have been carried out to look for the predicted arm-driving companions in a few other systems. Assuming hot start planet formation models \citep[e.g.,][]{baraffee15}, they have generally ruled out all but planetary mass objects of a few Jupiter masses or less \citep[e.g., ][]{maire17}.
Companion-driven arms co-rotate with their driver. Therefore, by measuring their pattern speed, the orbital period, thus semi-major axis, of their companion can be constrained \citep[e.g.,][]{lomax16}. We perform such an exercise for the spiral arm system MWC~758, taking advantage of observations of the arms over a decade-long baseline established by a 2005 {\it HST}/NICMOS observation and 2015/2017 {\it VLT}/SPHERE and {\it Keck}/NIRC2 observations.
MWC~758 is a Herbig Ae star located at $151_{-9}^{+8}$ pc \citep{gaia16} with an age of $3.5\pm2.0$ Myr \citep{meeus12}, and mass of $\sim2.0 M_{\odot}${\footnote{We derive the stellar mass from the \citet{siess00} pre-main sequence evolutionary tracks, assuming stellar effective temperature and luminosity of 7580K and 19.6 $L_\odot$ (\citealp{vanderMarel16}; after scaling the stellar luminosity using the new {\it Gaia} distance).}} . The disk has a low inclination of ${\sim}20^\circ$ \citep{isella10}. Its two prominent, roughly $180^\circ$ rotationally symmetric arms were first discovered with {\it Subaru}/HiCIAO \citep{grady13}, and subsequently characterized in detail with {\it VLT}/SPHERE \citep{benisty15}, with a third arm and point-source candidate {at ${\sim}0\farcs11$ (17 au)} recently reported in \citet{reggiani18} using {\it Keck}/NIRC2. Numerical simulations by \citet{dong15} suggested that both arms can be produced by a multi-Jupiter-mass planet at $\sim0\farcs6$ from the star.
\section{Data Acquisition \& Reduction}
In this Section, we describe the observations and data reduction for our 2005 {\it HST} and 2017 {\it Keck} programs.
\subsection{NICMOS}
The {\it HST}/NICMOS coronagraphic instrument observed the MWC~758 system in total intensity with the F110W filter ($\lambda_{\text{cen}}=1.1\ \micron$) on 2005 January 7 (Proposal ID: 10177, PI: G.~Schneider), and the unresolved disk morphology was presented in \citet{grady13}. To retrieve the morphology of the spiral arms, we obtain calibrated NICMOS images of MWC~758 and another 814 reference star exposures, i.e., point-spread functions (PSFs), from the Archival Legacy Investigations for Circumstellar Environments (ALICE) project (PI: R.~Soummer; \citealp{choquet14,hagan18}). We align the observations for better astrometry by employing a Radon-transform-based technique (\citealp{pueyo15}; C.~Poteet et al., ApJ submitted), which focuses on the diffraction spikes in each exposure. To minimize color mismatch, telescope breathing and cold mask alignment, we select 81 closest PSFs in the $L^2$-norm sense, and perform PSF subtraction with the Non-negative Matrix Factorization (NMF) method \citep{ren18}, which is shown to preserve the morphology of circumstellar disks better, especially in reference differential imaging scenarios.
In Fig.~\ref{fig:raw}, we present the reduction results of the NICMOS observations at two telescope orientations (three exposures each) and their signal-to-noise (S/N) maps. We argue the physical existence of the detection since the spiral pattern is (1) consistent within the same telescope orientation, as shown from the S/N maps which are calculated from dividing the combined result by the pixel-wise standard deviation of the ones constituing them; (2) independent of telescope orientation ($30^\circ$ separation), which excludes the scenario of unsuccessful noise removal; (3) not depending on the number of NMF components, reducing the possibility of underfitting and overfitting; (4) not resembling the null detections in the ALICE archive, as well as a reduction consistency using a principal-component-analysis-based reduction method \citep{soummer12}.
The integrated flux for $0\farcs3<r<0\farcs5$ is $2.0\pm0.8$ mJy at $1\sigma$ level, consistent with the upper limit reported in \citet{grady13}. We notice flux variations between the two telescope orientations, however we do not address the origin of this difference in this letter, but focus on the morphology only.
\begin{figure}[htb!]
\center
\includegraphics[width=.49\textwidth]{fig-raw.pdf}
\caption{{\it Left}: The NICMOS images of MWC~758, including the combined ({\bf a}) and two different rolls in ({\bf b}, {\bf c}). {\it Right}: The S/N maps, calculated from dividing the final images by the pixel-wise standard deviation of their constituting ones. The inner working angles are marked with gray circles, and stellar locations with white crosses.}
\label{fig:raw}
\end{figure}
\begin{figure*}[htb!]
\center
\includegraphics[width=.95\textwidth]{fig-r2-pol.pdf}
\caption{The $r^2$-scaled NICMOS, SPHERE, and NIRC2 (from left to right) observations of MWC~758 in Cartesian (top) and polar (bottom) coordinates with total flux normalized to unity. The gray dotted circles and lines mark the inner working angles.}
\label{fig:carpol}
\end{figure*}
\subsection{NIRC2}\label{nirc2reduction}
We observed MWC~758 with {\it Keck}/NIRC2 in $L'$-band total intensity ($\lambda_{\text{cen}}=3.8\ \micron$) on 2017 February 2 (PI: E.~Chiang). The data were obtained with the narrow camera (9.971 mas pixel$^{-1}$; \citealp{service16}) in ``vertical angle mode'' to allow for angular differential imaging (ADI; \citealp{marois06}). Our observations totaled 262 images, each consisting of 30 coadds of 1.0-s exposures, covering 161$^\circ$ of field rotation. Airmass varied from 1.01 to 1.39 and precipitable water vapor was approximately 2.5 mm.
The vector vortex coronagraph \citep{serabyn17} was used in combination with the existing Keck II adaptive optics system to suppress host star light. The QACITS control system \citep{huby17} maintained alignment of the vortex mask with the star during observations, and images are aligned with each other to sub-pixel precision using a downhill simplex algorithm to minimize residuals of the stellar PSF in frames differenced with a selected reference frame. Calibrated images are produced from raw images by performing dark subtraction, flat-fielding, thermal background subtraction with dedicated sky frames, and distortion correction \citep{service16}. The absolute star positions are then determined to 0.5 pixel precision in both spatial dimensions by a Radon transform of the averaged frames \citep{pueyo15}.
We subtract the stellar PSF from the calibrated images using the NMF method. PSF-subtraction algorithms with ADI are known to distort the morphology of extended objects from self-subtraction \citep[e.g.,][]{follette17}; therefore, for each image, this bias is avoided by a minimum rotation threshold of $45^\circ$ for the selection of its reference images; the final result is then the median of the PSF-subtracted images.
\subsection{SPHERE}
We obtain the {\it VLT}/SPHERE polarized intensity result in $Y$-band ($\lambda_{\text{cen}}=1.04\ \micron$) on 2015 March 3 from \citet{benisty15}.
\section{Data Analysis}\label{analysis}
To measure the pattern speed of the arms, we first scale the surface brightnesses of the reduced NICMOS, NIRC2, and SPHERE images by the distance-dependent factor $r^2$. The results in Cartesian and polar coordinates are presented in Fig.~\ref{fig:carpol}.
There are three main differences among our observational datasets: (1) the pixel size of the NICMOS instrument is ${\sim}8$ times larger than the other two; (2) the NICMOS and NIRC2 observations measure the total intensity while SPHERE traces the polarized light, and (3) the NICMOS and SPHERE observations are at ${\sim}1\ \micron$ while the NIRC2 observation is at ${\sim}3.8\ \micron$. For (1), we interpolate the NICMOS image to match the pixel scales of the others. For (2), we reduced the SPHERE ADI total intensity observation, compared it with the polarized map, and found no discernible discrepancy; this agreement is also endorsed by simulation in \citet{dong16b}. For (3), we compare observations at roughly the same central wavelength ($\lambda_{\text{cen}}$).
\subsection{Measurement of Rotation of Spirals}\label{rotation_measured}
\subsubsection{NIRC2: 2015 vs 2017 (1.28 yr apart)}\label{consistency}
To mitigate any systematic offset between instruments, and provide an initial constraint on the rotation, we compare two observations from the same {\it Keck}/NIRC2 instrument: our 2017 February 02 observation, and the 2015 October 24 observation (PI: E.~Serabyn, \citealp{reggiani18}) which is aligned and reduced with identical procedure.
We quantify the rotation of the spiral arms as follows: in polar coordinates, we first fit Gaussian profiles to the brightness of the spiral arms at a fixed radial separation; then for each arm, we perform the weighted Least Square Dummy Variable (LSDV, Appendix~\ref{lsdv}) analysis to fit the same morphological profile in both epochs to obtain their relative rotation. For the Southern and Northern primary arms (marked with ``1'' and ``3'' in Fig.~\ref{fig:nirc2}), we obtain a rotation of $\Delta\theta_S^{(1.28\,\mathrm{yr})}=0\fdg77\pm10\fdg65$\footnote{The calculated uncertainty in this letter is $3\sigma$ unless otherwise specified.}, and $\Delta\theta_N^{(1.28\,\mathrm{yr})}=-0\fdg70\pm6\fdg78$, respectively. Since spiral arms in disks are trailing patterns, the MWC~758 arms are expected to rotate in a clockwise direction, i.e., $\Delta\theta\geq0$, we therefore adopt the constraints from the Southern primary arm, $\dot\theta=\Delta\theta_S^{(1.28\,\mathrm{yr})}\big/1.28\mathrm{yr}=0\fdg6\pm8\fdg3$ yr$^{-1}$, as the rotation of the two.
For consistency check, we measure the rotation with another method: in polar coordinates, we obtain the cross-correlate maps \citep{tonry79}, and measure $\dot\theta=0\fdg7\pm56\fdg2$ yr$^{-1}$. We adopt the value from the LSDV method, since it is less biased by the non-spiral structures in the entire field of view as the cross-correlation method, and the best-fit values agree within $0\fdg1$.
\begin{figure*}[htb!]
\center
\includegraphics[width=\textwidth]{fig-nirc2-comparison.pdf}
\caption{MWC~758 spiral arms observed by {\it Keck}/NIRC2 in polar coordinates (total flux normalized to 1 for comparison) in 2015 ({\bf a}) and 2017 ({\bf b}), and contour of 2015 observation overplotted on the 2017 one ({\bf c}).}
\label{fig:nirc2}
\end{figure*}
\subsubsection{2005 NICMOS vs 2015 SPHERE (10.17 yr apart)}\label{quantify}
By analyzing the rotation between the NICMOS and SPHERE images, we narrow down the uncertainty determined from the NIRC2 observations here. We adopt the NIRC2 best-fit and uncertainty values, rotate the SPHERE result back to the NICMOS epoch, then the two images are expected to have no azimuthal shift. By fitting identical profiles and offsets for the rotated SPHERE and original NICMOS observations, and given the Northern arm is blended with its secondary arm (marked by ``4'' in Fig.~\ref{fig:nirc2}) but with a smaller uncertainty than the Southern arm, we adopt the results from the Southern arm, obtaining a conservative measurement of $\Delta\theta^{(10.17\,\mathrm{yr})}=6\fdg1\pm29\fdg4$. This corresponds to a statistical uncertainty for the angular speed: $(\delta\dot\theta)_{\mathrm{statistical}} = 29\fdg4\big/10.17\mathrm{yr}= 2\fdg89$ yr$^{-1}$.
\subsection{Additional Systematics}\label{rotation_systematics}
We identify and study the impact of two possible systematics associated with the NICMOS results: the alignment uncertainty of stellar center determination (CD), and the misalignment uncertainty of the star behind the focal plane mask (FPM).
To quantify the stellar center determination uncertainty, we cross-correlate the raw MWC~758 exposures with the $814$ ALICE references, and determine the $3\sigma$ uncertainty to be $0.5$ pixel along both horizontal and vertical directions. We then draw $1,000$ possible centers within $\pm 0.5$ pixel from the center determined by our Radon Transform method, and cross-correlate the arm images in polar coordinates with the SPHERE result, and obtain a $3\sigma$ quantile of $\Delta\theta_{\mathrm{CD}}=0.6^\circ$. We therefore adopt a $3\sigma$ upper limit of {$\delta\dot\theta_{\text{CD}}=0.6^\circ\big/10.17\mathrm{yr}=0\fdg06$ yr$^{-1}$}.
Since the arms lie near the edge of the NICMOS FPM, if they do have rotated in this $\sim$10-yr span, with the star not well-centered on the FPM during the time of the observation, this may still yield nonsignificant moving spirals. To account for this, we simulate $1,000$ SPHERE images with the following two parameters: (1) rotations within $\pm60^\circ$ (denoted as $\alpha$): a range that the arms would rotate in ${\sim}10$ years if they are driven by the protoplanet candidate reported by \citet{reggiani18}, and (2) shifted centers within $\pm 0\farcs04$ ($0.5$ NICMOS pixel) along both horizontal and vertical directions. We mask the resampled SPHERE data with a circle of the NICMOS FPM size, then cross-correlate them with the original SPHERE image, and obtain their relative azimuthal shift ($\Delta\theta'$), which is then subtracted by introduced shift ($\alpha$). We obtain a $3\sigma$ upper limit $\delta(\Delta\theta)_{\text{FPM}} = \Delta\theta' - \alpha = 11^\circ$. {Therefore, $(\delta\dot\theta)_{\mathrm{FPM}}=11^\circ\big/10.17\mathrm{yr}=1\fdg08$ yr$^{-1}$.}\\
\section{Result}\label{result}
From the previous analyses of statistical and systematical uncertainties, we obtain a total uncertainty in the rotation of the arms at $3\sigma$:
\begin{align*}
&\delta\dot\theta= \sqrt{(\delta\dot\theta)^2_{\text{statistical}}+(\delta\dot\theta)^2_{\text{systematic}}}\\
&= \sqrt{(\delta\dot\theta)^2_{\text{statistical}}+\left[(\delta\dot\theta)^2_{\text{CD}}+(\delta\dot\theta)^2_{\text{FPM}}+(\delta\dot\theta)^2_{\text{pixel}}\right]}\\
&= \sqrt{2.89^2+(0.06^2+1.08^2+1.18^2)} = 3\fdg31\, \mathrm{yr}^{-1},
\end{align*}where the pixel uncertainty is accounted for the NICMOS pixel size of $12^\circ$ at $r{\sim}0\farcs6$ (i.e., $\delta\dot\theta_{\text{pixel}}=12^\circ\big/10.17\mathrm{yr}=1\fdg18$ yr$^{-1}$). Together with the best-fit value, we obtain \begin{equation}
\dot\theta=0\fdg6^{+3\fdg3}_{-0\fdg6}\, \mathrm{yr}^{-1},
\end{equation}
where the lower limit is physically constrained from the clock-wise rotation of the MWC~758 arms.
For hypothesized arm-driving planet(s) on a circular orbit (eccentricity $e=0$), the best-fit pattern speed corresponds to a period of $T=598$ yr, or a radial separation of $r_{\text{best}} = 89$ au; and the $3\sigma$ upper limit leads to $T=92$ yr and $r_{3\sigma}=26$ au; see Fig.~\ref{fig:planet} for graphical representations
For $e>0$, if the planet reaches its apogee in the current epoch, the limit on the arm motion translates into a stellocentric separation $r_{\text{min}}(e) = (1-e)^{1/3}r$ au. For giant planets with several Jupiter mass forming in and interacting with a gaseous disk, their eccentricities are not expected to grow beyond $\sim$0.1 \citep{dunhill13, duffell15}. Furthermore, density waves excited by Jovian planets with $e\gtrsim0.2$ start to deviate from their usual morphology, as the waves launched at different epochs interact with each other (Hui Li \& Shengtai Li, private commnication), which provide poor fits to the arms around MWC~758. In this case, $e=0.2$ leads to a $7\%$ decrease in the minimum stellocentric separation, and the hypothetical arm-driving planet(s) should be located at least $25$ au from the star in 2017 at $3\sigma$.
One might consider another scenario in which the two spiral arms exchanged positions between 2005 and 2015, i.e., rotated ${\sim}180^\circ$. However, this means the major arms should have rotated an additional $22\fdg6$ between the 2015 and 2017 NIRC2 observations, this is ruled out at $6\sigma$ with our constraints.
\begin{figure}[htb!]
\center
\includegraphics[width=.45\textwidth]{fig-planet-limit.pdf}
\caption{{The best-fit and $3\sigma$ lower limit (solid lines) of where the major-arm-driving planets should be based on the measured pattern speed, plotted over the SPHERE data. The $1\sigma$ limit (dashed line), which is interpolated from the best-fit and the $3\sigma$ limit, is presented for illustration purpose and it may not represent a true 68\% likelihood.}}
\label{fig:planet}
\end{figure}
\section{Summary and Discussion}
We present reprocessed 2005 {\it HST}/NICMOS observations of the MWC 758 disk, and successfully retrieve the two spiral arms in the system revealed by ground-based high-contrast imaging facilities. Thanks to the {\it HST} image, we are able to establish a 10-yr baseline in observations to constrain the pattern speed of the major arms. Together with a 2015 {\it VLT}/SPHERE dataset, and two {\it Keck}/NIRC2 observations in 2015 and 2017, we obtain a rotation speed of {$0\fdg6^{+3\fdg3}_{-0\fdg6}\, \mathrm{yr}^{-1}$ at $3\sigma$ for the two major spiral arms. The results correspond to a best-fit value of $89$ au ($0\farcs59)$, and a $3\sigma$ lower limit of $26$ au ($0\farcs17$), for the orbital distance of the hypothesized arm-driving perturber on a circular orbit.}
Our constraint is consistent with the \citet{dong15} prediction of the arm-driving planet at ${\sim}90$ au (${\sim}0\farcs6$). In addition, we rule out at a $6\sigma$ level the scenario that the companion candidate at $0\farcs11$ (17 au) reported by \citet{reggiani18} is driving the major two spiral arms, assuming the candidate is on a circular orbit coplanar with the arms. This is further supported by \citet{bae18}, that a Jovian planet can drive only one external arm assuming a reasonable disk scale height. For new arm(s) and planet candidate(s) \citep[e.g.,][]{reggiani18}, more observations are needed to confirm their existence and dynamical connections.
The possible arm-driving planets in the MWC~758 system are excellent targets for future observations in direct imaging observations both from the ground and with the {\it James Webb Space Telescope}, and in millimeter observations using {\it ALMA} to search for evidence of a circumplanetary disk \citep[e.g.,][]{zhu15a, eisner15, perez15, szulagyi17}.
\acknowledgments
We are grateful to an anonymous referee for constructive suggestions that improved our letter, and Sean Brittain for insightful discussions. B.R.~acknowledges the computational resources from the Maryland Advanced Research Computing Center (MARCC), which is funded by a State of Maryland grant to Johns Hopkins University through the Institute for Data Intensive Engineering and Science (IDIES). E.C.~acknowledges support from NASA through Hubble Fellowship grant HST-HF2-51355 awarded by STScI, operated by AURA, Inc.~under contract NAS5-26555, and support from HST-AR-12652, for research carried out at the Jet Propulsion Laboratory, California Institute of Technology. T.E.~was supported in part by NASA Grants NNX15AD95G/NEXSS, NNX15AC89G, and NSF AST-1518332. This research has made use of data reprocessed as part of the ALICE program, which was supported by NASA through grants HST-AR-12652 (PI: R. Soummer), HST-GO-11136 (PI: D.~Golimowski), HST-GO-13855 (PI: E.~Choquet), HST-GO-13331 (PI: L.~Pueyo), and STScI Director's Discretionary Research funds, and was conducted at STScI which is operated by AURA under NASA contract NAS5-26555. The input images to ALICE processing are from the recalibrated NICMOS data products produced by the Legacy Archive project, ``A Legacy Archive PSF Library And Circumstellar Environments (LAPLACE) Investigation,'' (HST-AR-11279, PI: G.~Schneider). This work benefited from NASA's Nexus for Exoplanet System Science (NExSS) research coordination network sponsored by NASA's Science Mission Directorate. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea, host to the W.~M.~Keck Observatory, has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.
|
1,941,325,220,501 | arxiv | \section{Compare two lowest states with $\Omega$ value of 5/2 and 7/2 on RASSI level of theory}
Recently, the highly resolution spectra of IrO has been studied by Pang \textit{et al.} (J. Phys. Chem. A, 116, 9379, 2013) and Adam \textit{et al.} (J. Mol. Spectrosc., 286,46,2013). Both of these studies determined that the ground state of IrO had the $\Omega$ value of 5/2. These inspired Suo and Han to reconsider previous calculation (Chem. Phys. Lett., 548, 12, 2012). As pointed out by Adam and coworkers (J. Mol. Spectrosc., 286, 46, 2013), the $^4\Delta_{5/2}$ state may drop below the $^4\Delta_{7/2}$ state when the influence of spin-orbital coupling between the $^4\Delta$ and $^2\Delta$ is included. In previous work by Suo \textit{et. al}, the interaction between the $^4\Delta_{5/2}$ and $^2\Delta_{5/2}$ states is not be considered. Therefore, Suo and Han performed the SOC calculation for $^4\Delta$, $A^2\Delta$ and $B^2\Delta$ states. The results of the $^4\Delta_{7/2}$ and $^4\Delta_{5/2}$ substates are shown in the Table \ref{RASSIRE}. In the new calculation, the energy separation of the 7/2 and 5/2 is reduced to 1108 cm$^{-1}$, which is smaller than 2425 cm$^{-1}$ in previous calculation (Chem. Phys. Lett., 548, 12, 2012). However, we still could not assign a 5/2 ground state due to larger energy gap of the 7/2 and 5/2 states. The discrepancy between theoretical and experimental results may come from insufficient treatment the relativistic effect. The large SOC effect in IrO seems to be calculated by the variational method rather than the perturbation theory. Therefore, multi-configurational electron correlation calculation in the frame of fully relativistic theory is highly desired to give the more accurate description to the ground state of IrO. This is our main motivation of the present work.
\begin{table*}[h]
\caption{Comparing the $^4\Delta_{7/2}$ and $^4\Delta_{5/2}$ substates of IrO.}\label{RASSIRE}
\begin{tabular}{ccccc}
\hline\hline
$\Omega$ & R$_e$ \AA & $\Delta$E & $\omega$ (cm$^{-1}$) & Configurations \\
\hline
7/2 & 1.700 & 0 & 950 & $^4\Delta_{7/2}(100\%)$ \\
5/2 & 1.704 & 1108 & 937 & $^4\Delta_{5/2}(89.6\%),A^2\Delta_{5/2}(9.6\%), B^2\Delta_{5/2}(6.4\%)$ \\
\hline\hline
\end{tabular}
\end{table*}
\section{Excited state of IrO}
More roots are required in the KRCI implementation in order to get the low-lying excited state of IrO, which increases the computation demand greatly. Therefore, the calculation for the excited states is mainly performed on the small basis set, then a composite scheme is adapted to give the estimated value $T^{Com}$ of the excited energy at larger basis set. The correction due to the finite basis set is determined as $\Delta T^{basis}=T^{Ver}_{cv3z}-T^{Ver}_{cv2z}$, where $T^{Ver}_{cv3z}$ and $T^{Ver}_{cv2z}$ are the vertical excitation energies that are calculated at the basis sets [dyall.cv3z (Ir) + aug-cc-pCVTZ (O)] and [dyall.cv2z (Ir) + aug-cc-pCVDZ (O)], respectively. Then, $T^{Com}$ is determined as $T^{Com}=T^{Adi}_{cv2z}+\Delta T^{basis}$, where $T^{Adi}_{cv2z}$ is the adiabatic excited energy obtained at the basis set [dyall.cv2z (Ir) and aug-cc-pCVDZ (O)]. The correction $\Delta T^{basis}$ is regarded as the corresponding uncertainty.
\begin{table*}[btp]
\caption{The excited energy of IrO with $\Omega$=5/2 and 7/2 deduced by the composite technique at the KRCI SD8(13in9)SD level in the MCSCF+CI implementations.}
\begin{tabular}{p{1.2cm} p{1.2cm} p{1.2cm} p{1.2cm} p{1.2cm} p{1.2cm} }\hline\hline
state &$T^{Adi}_{cv2z}$ &$T^{Ver}_{cv2z}$ &$T^{Ver}_{cv3z}$ & $\Delta T^{basis}$ & $T^{Com}$ \\ \hline
\multicolumn{6}{c}{$\Omega$=5/2}\\
1 &8481 &8481 &8373 &-108 &8373 \\
2 &10350 &10994 &12256 &1262 &11612 \\
3 &12691 &13369 &14140 &771 &13462 \\
4 &13893 &14260 &14752 &492 &14385 \\
5 &14723 &14901 &15673 &772 &15495 \\
6 &15448 &15866 &16900 &1035 &16483 \\
7 &17231 &17249 &17712 &462 &17694 \\
8 &18316 &18762 &19750 &988 &19304 \\
9 &20043 &20519 &21511 &992 &21035 \\
10 &22022 &22807 &23360 &553 &22575 \\
11 &22320 &23086 &23699 &613 &22933 \\
12 &22914 &23735 &25247 &1513 &24426 \\
13 &23518 &24470 &25660 &1190 &24708 \\
14 &24796 &25935 &27176 &1241 &26037 \\
15 &25324 &27778 &29215 &1437 &26761 \\ \hline
\multicolumn{6}{c}{$\Omega$=7/2}\\
1 &8662 &9165 &10063 &898 &9560 \\
2 &12061 &12815 &13880 &1065 &13126 \\
3 &15684 &15684 &15428 &-256 &15428 \\
4 &16209 &16771 &17110 &338 &16547 \\
5 &17661 &17963 &18887 &924 &18585 \\
6 &20028 &21021 &22006 &985 &21013 \\
7 &22319 &23674 &24833 &1159 &23478 \\
8 &24497 &24965 &26014 &1049 &25546 \\
9 &26374 &28992 &29288 &296 &26671 \\ \hline\hline
\end{tabular}
\end{table*}
\end{document}
\section{Introduction}
Though Iridium Monoxide is simple diatomic molecule, it is a difficult system both on experimental and theoretical studies due to the existence of nearly degenerated open 5d shells and also the notably relativistic effect of the Ir atom. The multiconfigurational character of electron structure of IrO could give rise to strongly perturbed electronic transitions, which would produce complicated spectra in the experimental study. In particular, as a heavy-element with open d shell, Ir is an typically interesting system that requires accurate description of electron correlation, both static and dynamic, and also the relativistic effect contributions in theoretical study. This poses a great challenge to the modern electron structure theory.
The emission spectra of IrO was first recorded by Raziunas \textit{et al.}, who observed four bands situated at 5856, 5990, 6972 and 6899{\AA} \cite{Raziunas-JCP-1965}, respectively, but no rotational and vibrational analysis is made because of low resolution of the spectra. In 1972, Jansson and Scullman \cite{Jansson-JMS-1972} analyzed fifteen emission bands in the region of 4200-6400 {\AA}, and one of them is assigned as the band of 5950 {\AA} observed in Raziunas's study \cite{Raziunas-JCP-1965}. Three subsystems with $\Delta \Omega=0$ are resolved and assigned as sub-state transitions between two $^2\Delta$ states \cite{Jansson-JMS-1972}. After that, several theoretical studies have been carried out for the IrO molecule \cite{Citra-JPCA-1999,Song-TCA-2007,Suo-JPB-2012}. The first theoretical work is performed by Citra and Andrews on density function theory (DFT) \cite{Citra-JPCA-1999}. They assigned $^4\Sigma^-$ as the ground state of IrO. Later, a systemic study of 5d metal oxide by Yao \textit{et al.} also on DFT level of theory supported this conclusion \cite{Song-TCA-2007}. In general, DFT is insufficient to describe electron structure of such complicate molecule as IrO because the static correlation is difficult to be caught by a single determinate approach.
Therefore, two authors in present study, Suo and Han have studied 25 low-lying electronic states of IrO by using the multi-state complete active space second order perturbation theory (MS-CASPT2) method. Some important states are calculated by the restrict active space state interaction (RASSI) method to take into account the spin-orbit coupling (SOC) effect \cite{Suo-JPB-2012}. Different with previous theoretical prediction, the $^4\Delta_{7/2}$ is assigned as the ground state of IrO. If the spin-orbital coupling effect is omitted, the $^2\Pi$ state is lowest in energy. Almost at same time with Suo \textit{et al.}'s theoretical work, Pang, \textit{et al.} have recorded five electronic transition spectra of IrO in the region between 448 and 650 nm, which are identified as [17.6] 2.5-X$^2\Delta_{5/2}$, [17.8] 2.5-X$^2\Delta_{5/2}$, [21.5] 2.5-X$^2\Delta_{5/2}$, [22.0] 2.5-X$^2\Delta_{5/2}$, and [21.9] 3.5-3.5 systems \cite{Pang-JPCA-2010}. According to the observed transitions, they concluded that IrO has a X$^2\Delta_{5/2}$ ground state. Obviously, the ground state $^4\Delta_{7/2}$ obtained from high level theoretical calculation is inconsistent with Pang \textit{et al.}'s assignment of the $\Omega=5/2$ ground state. In Suo \textit{et al.}'s study, a state with $\Omega=5/2$ lies 2425 $cm^{-1}$ above the ground state and is dominated by $^4\Delta_{5/2}$. The $^2\Delta_{5/2}$ is higher in energy and locates at 8310 $cm^{-1}$ above the $^4\Delta_{7/2}$ (In Suo \textit{et al.}'s work \cite{Suo-JPB-2012}, the $^2\Delta_{5/2}$ is mislabeled as $^2\Delta_{3/2}$ and the $^2\Delta_{3/2}$ is mislabeled as $^2\Delta_{1/2}$). However, Pang \textit{et al.}'s experiment is performed on cold molecular beam, the energy separation 2425 $cm^{-1}$ between $^4\Delta_{7/2}$ and $^4\Delta_{5/2}$ is too large to be overcome by thermal fluctuation. Therefore, the low-lying state in the experimental spectra should almost be the ground state.
Inspired by the serious discrepancy between the theoretical and experimental studies, Adam \textit{et al.} have re-examined spectra of IrO via high resolution LiF experiments \cite{Adam-JMS-2013}. Two transitions $[17.5]2.5-X2.5$ and $[23.3]2.5-X2.5$ are fully resolved and the ground state is verified to have spin-orbit component $\Omega$ value of $5/2$ \cite{Adam-JMS-2013}. These researchers proposed that the $^2\Delta_{5/2}$ state may interact with $^4\Delta_{5/2}$ strongly and push the later below the $^4\Delta_{7/2}$ state \cite{Adam-JMS-2013}. According to Adam \textit{et al.}'s suggestion, Suo and Han include $A^2\Delta_{5/2}$ into SOC calculation and find that the first state with $\Omega=5/2$ is highly mixed with $^4\Delta_{5/2}$, $A^2\Delta_{5/2}$ and $B^2\Delta_{5/2}$ \cite{Suo-supplement}. The energy gap between the $\Omega=5/2$ and $\Omega=7/2$ is reduced to 1108 cm$^{-1}$. However, such energy gap is still too large to identify the $\Omega=5/2$ ground state doubtlessly.
With increase of the nuclear charge, the relativistic effect becomes crucial in theoretical description of the chemistry of heavy elements. Therefore, j-j coupling is more appropriate than L-S coupling for interpretation of the experimental spectra. However, Suo et al.'s calculation reckons the scalar relativistic effect via spin-free second order Douglas-Kroll Hamiltonian (DKH) Hamiltonian \cite{DKH1974,DKH1985,DKH1986}. Then, SOC was treated as the perturbation and evaluated via RASSI \cite{RASSI}. This calculation may not sufficient for IrO due to the strong spin-orbital coupling in heavy atom leaves only total angular moment J as a good quantum number. Therefore, in this work, the electron states of IrO are calculated for individual J components by using fully relativistic multi-reference configuration interaction method. The relativistic effects and the electronic correlations are considered on the same footing in Kramers-restricted configuration interaction (KRCI) implementation based on the exact 2-Component (X2C) Hamiltonian \cite{X2C_Liu, X2C_Ilias}. We present and discuss the electronic ground and excited states of $\Omega$=1/2, 3/2, 5/2, and 7/2 of IrO. Our calculated results show the strong degeneracy of the lowest state with $\Omega$=1/2, 5/2, and 7/2, which can explain the discrepancy between the calculated and observed ground states. In addition, the excited stated obtained for $\Omega$=5/2 and 7/2 are used to assign the observed six transition bands of LIF spectrum in the visible region.
\section{Computational method}
All our calculations utilize the exact two-component (X2c) Hamiltonian that includes by default atomic-mean-field two-electron spin-same-orbit corrections \cite{X2C_Liu, X2C_Ilias}. Firstly, average-of-configuration Dirac-Hatree-Fock (AC-DHF) \cite{Tyssen-DHF} or Kramers restricted multi-configuration self consistent field (KR-MCSCF) \cite{KR-MCSCF} are performed to optimize molecular spinors in which static correlation has been evaluated. In AC-DHF and KR-MCSCF calculations, 13 electrons are allowed to distribute in 9 Kramers pairs (13in9) consisting of the Ir 5d6s and O 2p orbitals. Then, the Kramers-restricted multi-reference configuration interaction (KR-MRCI) are carried out by using molecular spinors from AC-DHF or KR-MCSCF calculations \cite{KRCI2010}. It takes advantage of the concept of generalized active spaces (GAS) to define suitable correlation spaces thereby allowing for arbitrary occupation constraints. In our calculation, two sets of GAS are applied. The first GAS, namely, ``SD8(13in9)SD" includes the frozen core [$1s^2$ (O) and $1s^2 2s^2 2p^6 3s^2 3p^6 3d^{10} 4s^2 4p^6 4d^{10}4f^{14}5s^2$ (Ir)], the outer core [$2s^2$ (O) and $5p^6$ (Ir)], the valence [$2p^4$ (O) and $5d^7 6s^2$ (Ir)], and the virtual shells less than 2 a.u. The notation ``SD8(13in9)SD" means that at least two holes in the 8 electron of the outer core is allowed, 13 electrons are distributed in the valence shell with plus of excited electrons from the outer core to valence shells, and finally that all single and double excitations into virtual orbitals are taken into account. Therefore, the first GAS will bring 21 electrons at most into correlation. The second GAS ``SD20(15in10)SD" has a larger outer core [$4f^{14} 5p^6$ (Ir)], the valence [$2s^2 2p^4$ (O) and $5d^7 6s^2$ (Ir)], and the virtual orbitals less than 2 a.u, which brings 35 electrons at most into correlation. The quantum number for each individual electronic state is assigned through calculating the expectation value for the one-electron operator $j_z$=$l_z$+$s_z$. All our calculations are implemented with the relativistic quantum chemistry calculation package DIRAC \cite{DIRAC}.
In order to investigate the ground state of IrO, the four lowest-lying states for $\Omega$=1/2, 3/2, 5/2, and 7/2 are calculated by employing the Dunning's aug-cc-pCVTZ basis set \cite{Dunning} for O and Dyall's cv3z basis set for Ir \cite{Dyall}. For understanding the ground electronic structure of IrO, we also conduct the comparative calculation of CoO and RhO by using Dunning's aug-cc-pCVDZ for O \cite{Dunning} and dyall's cv2z basis sets for Co and Rh \cite{Dyall}. The excited states of IrO are calculated by the additional KRCI implementation. The adiabatic energy is obtained by using Dunning's aug-cc-pCVDZ (O) and dyall's cv2z basis sets (Ir), which is modified with the vertical excited energy calculated by using Dunning's aug-cc-pCVTZ (O) and dyall's cv3z basis sets (Ir) (See supplemental material for computational details). All calculations were performed using uncontracted basis sets. The expectation values $\langle s_z\rangle=\Sigma_{i}m_s$ and $\langle l_z\rangle=\Sigma_{i}m_l$ are given that suggest the major component in the $\Lambda-S$ notation. The KRCI prosperity module gives the transition dipole moment and therefore the oscillation strength is obtained in terms of
\begin{equation}
f=\frac{2m}{e^2\hbar\omega}|D|^2
\end{equation}
, where $\hbar\omega$ and $D$ is the excited energy and transition dipole moment, respectively, $e$ and $m$ is electronic charge and mass. The oscillation strength helps us to assign the excited states to the observed visible transition bands.
\section{Results and discussion}
\subsection{The ground state of IrO}
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{fig_1.eps}
\end{center}\vspace{-20mm}
\caption{PECS of the lowest states with $\Omega$=1/2, 3/2, 5/2, and 7/2, as calculated at the KRCI SD8(13in9)SD level implemented by the DHF+CI and MCSCF+CI ways.}
\label{fig1}
\end{figure}
We first compare the ground state of IrO on the different level of calculations. In Table \ref{tbl:groundstate}, we illustrate the equilibrium bond lengths, energy separations, harmonic vibrational frequencies and major configurations of each lowest $\Omega$ states on three different KRCI calculations. The first two calculations employ the same GAS SD8(13in9)SD but with different molecular spinors in which one is obtained by AC-DHF and the other is from KR-MCSCF. It is found that the two different implementations, the DHF+CI and MCSCF+CI, yield the almost same results for the spectroscopy constant and electronic configuration. This suggests that the AC-DHF has captured sufficient static correlation, even without more expensive KR-MCSCF calculations. Both of two calculations prefer an $\Omega=1/2$ ground state. The 5/2 and 7/2 states are slightly higher in energy and locates at 58 and 91 cm$^{-1}$ above the ground state on the MCSCF+CI calculation. The 3/2 state is well separated with three low-lying states and locates 2632 cm$^{-1}$ above the 1/2 state.
It is noteworthy that the energy separations of 1/2, 5/2 and 7/2 are so small that the energy gap of these three states may sensitive to different level of theories. Therefore, we have performed an extra KRCI calculation with a larger GAS space, in which the 4f shell is included in correlation calculation and is labeled as GAS SD20(15in10)SD. This calculation sufficiently considered correlations of the 4f electrons of Ir atom, and was our highest level of calculation in present work. As can be seen from Table \ref{tbl:groundstate}, the energy order of four lowest states does not change when the GAS space is enlarged from SD8(13in9)SD to SD20(15in10)SD. However, the energy gap of the 1/2 and 5/2 states is increased to 234 cm$^{-1}$, which is larger than the corresponding value 58 on SD8(13in9)SD calculation. Interestingly, the energy separation between the 5/2 and 7/2 states is still small with the value of 26 cm$^{-1}$, and is close to 23 cm$^{-1}$ obtained by the small-core calculation without the 4f electrons. Including 4f electrons in correlation calculation introduces more core-core and core-active correlations than small-core calculation. If assuming core-core correlation is similar in each states, it is correlation between the 4f and active electrons that stabilizes the 1/2 state and pushes this state lower in energy than SD8(13in9)SD result. The energy variation between the small-core and large-core calculations is minor, for example the energy separation of the 1/2 and 5/2 states is changed less than 200 cm$^{-1}$. Such small variation is not supposed change the order of the high lying excited state for a given $\Omega$ value. Therefore, we still use the small-core calculation in the following calculation for the excited states because of the economical computational cost. We intend to give a description of the excited states of IrO at a qualitative level that is sufficient for assignment of the experimental spectra.
\begin{spacing}{1}
\begin{table*}\small
\caption{The lowest-lying electronic states of IrO for $\Omega$=1/2, 3/2, 5/2, and 7/2, as calculated at the KRCI SD8(13in9)SD level in the DHF+CI and MCSCF+CI implementations and KRCI SD20(15in10)SD level in the MCSCF+CI, where $\langle s_z \rangle$ and $\langle l_z \rangle$ are expectation value of $s_z$ and $l_z$ operators, and $R$, $T$, $\omega$ are the equilibrium bond length ({\AA}), excited energy (cm$^{-1}$), and harmonic frequency (cm$^{-1}$). The major configurations have a common part ($14\sigma_{1/2}^2 15\sigma_{1/2}^2 9\pi_{1/2}^2 9\pi_{3/2}^2$).}\label{tbl:groundstate}
\begin{tabular}{p{0.8cm} p{0.8cm} p{0.8cm} p{7cm} p{1cm} p{0.8 cm} p{0.8 cm} } \hline\hline
$\Omega$ &$\langle s_z \rangle$ &$\langle l_z \rangle$ &Major configuration & $R$ & $T$&$\omega$ \\\hline
\multicolumn{7}{c}{SD8(13in9)SD, DHF-CI }\\
1/2 &-0.39&0.89&$4\delta_{3/2}^2 4\delta_{5/2}^2 10\pi_{1/2}^2 16(17)^{a}\sigma_{1/2}^1$ ($^{4}\Sigma^-$) 46\% & 1.674&0 &858 \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^2 10\pi_{1/2(3/2)}^1 16\sigma_{1/2}^2$ ($^{2}\Pi$) 15\% & & & \\
& & &$4\delta_{3/2}^1 4\delta_{5/2}^2 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^2$ ($^{2}\Delta$) 1\% & & & \\ [+1 ex]
5/2 &0.50&2.00 &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2(3/2)}^2 16\sigma_{1/2}^2$ ($^{2}\Delta$) 42\% &1.712&79&974 \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2(3/2)}^2 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ ($^{2}\Delta$)14\% & & & \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^2 16\sigma_{1/2}^1$ ($^{4}\Pi$) 2\% & & & \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^2 16\sigma_{1/2}^1$ ($^{2}\Phi$) 1\% & & & \\ [+1 ex]
7/2 &1.49&2.01 &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^2$ ($^{4}\Delta$) 38\%&1.712&125 &884 \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^1 17\sigma_{1/2}^1$($^4\Delta$)13\%& & & \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^2 10\pi_{3/2}^1 16(17)\sigma_{1/2}^1$ ($^{2}\Phi$) 17\% & & & \\ [+1 ex]
3/2 &-0.46&1.96&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^2$ ($^{2}\Delta$) 15\% & 1.702&2629 &904 \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^1 17\sigma_{1/2}^1$($^{2}\Delta$)5\% & & & \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^2 10\pi_{3/2}^1 16(17)\sigma_{1/2}^1$ ($^{2}\Pi$) 10\% & & & \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^2 10\pi_{1/2}^1 10\pi_{3/2}^1 16(17)\sigma_{1/2}^1$($^{4}\Sigma$)18\% & & & \\
& & &$4\delta_{3/2}^1 4\delta_{5/2}^2 10\pi_{1/2}^2 16\sigma_{1/2}^2$($^{2}\Delta$)8\% & & & \\ \hline
\multicolumn{7}{c}{SD8(13in9)SD, MCSCF-CI }\\
1/2 &-0.55&0.94&$4\delta_{3/2}^2 4\delta_{5/2}^2 10\pi_{1/2}^2 16(17)\sigma_{1/2}^1$ ($^{4}\Sigma^-$) 48\% & 1.670& 0 &983 \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^2 10\pi_{1/2}^1 16\sigma_{1/2}^2$ ($^{2}\Pi$) 13\% & & & \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^2 10\pi_{1/2}^1 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ ($^{2}\Pi$)4.7\%& & & \\ [+1 ex]
5/2 &0.50 &2.00&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2(3/2)}^2 16\sigma_{1/2}^2$ ($^{2}\Delta$) 41\% &1.691& 58&1022\\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2(3/2)}^2 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ ($^{2}\Delta$)8\% & & & \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^2 16\sigma_{1/2}^1$ ($^{4}\Pi$) 2\% & & & \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^2 16\sigma_{1/2}^1$ ($^{2}\Phi$) 1\% & & & \\ [+1 ex]
7/2 &1.47 &2.03&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^2$ ($^{4}\Delta$) 39\%&1.702& 91 &1001\\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^1 17\sigma_{1/2}^1$($^4\Delta$)13\% & & & \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^2 10\pi_{3/2}^1 16(17)\sigma_{1/2}^1$ ($^{2}\Phi$) 18\% & & & \\ [+1 ex]
3/2 &1.39 &0.11&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^2$ ($^{2}\Delta$) 18\% & 1.681&2632&954 \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^1 17\sigma_{1/2}^1$($^{2}\Delta$)7\% & & & \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^2 10\pi_{3/2}^1 16(17)\sigma_{1/2}^1$ ($^{2}\Pi$) 12\% & & & \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^2 10\pi_{1/2}^1 10\pi_{3/2}^1 16(17)\sigma_{1/2}^1$($^{4}\Sigma$)11\% & & & \\
& & &$4\delta_{3/2}^1 4\delta_{5/2}^2 10\pi_{1/2}^2 16\sigma_{1/2}^2$($^{2}\Delta$)8\% & & & \\ \hline
\multicolumn{7}{c}{SD20(15in10)SD, MCSCF-CI }\\
1/2 &-0.49&0.99&$4\delta_{3/2}^2 4\delta_{5/2}^2 10\pi_{1/2}^2 16(17)\sigma_{1/2}^1$ ($^{4}\Sigma^-$) 52\%& 1.681&0 &1028 \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^2 10\pi_{1/2(3/2)}^1 16\sigma_{1/2}^2$ ($^{2}\Pi$) 12\% & & & \\[+1 ex]
5/2 &0.55&1.95&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^2 16\sigma_{1/2}^2$ ($^{2}\Delta$)41\%&1.712 &234 &903 \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2(3/2)}^2 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ ($^{2}\Delta$)17\% & & & \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^2 16\sigma_{1/2}^1$ ($^{4}\Pi$) 2\% & & & \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^2 16\sigma_{1/2}^1$ ($^{2}\Phi$) 1\% & & & \\[+1 ex]
7/2 &1.46 &2.04&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^2$ ($^{4}\Delta$) 43\%&1.712 &260 &954 \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^1 17\sigma_{1/2}^1$($^4\Delta$)14\%& & & \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^2 10\pi_{3/2}^1 16(17)\sigma_{1/2}^1$ ($^{2}\Phi$) 15\% & & & \\ [+1 ex]
3/2 &-0.38&1.88&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^2$ ($^{2}\Delta$) 13\% &1.702 &3300 &934\\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^1 17\sigma_{1/2}^1$($^{2}\Delta$)5\% & & & \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^2 10\pi_{3/2}^1 16(17)\sigma_{1/2}^1$ ($^{2}\Pi$) 10\% & & & \\
& & &$4\delta_{3/2}^2 4\delta_{5/2}^2 10\pi_{1/2}^1 10\pi_{3/2}^1 16(17)\sigma_{1/2}^1$($^{4}\Sigma$)22\% & & & \\
& & &$4\delta_{3/2}^1 4\delta_{5/2}^2 10\pi_{1/2}^2 16\sigma_{1/2}^2$($^{2}\Delta$)8\% & & \\ \hline\hline
\multicolumn{6}{l}{$^{a}$ The spinor in the bracelet also exists but with less than 5\% composition. }
\end{tabular}
\end{table*}
\end{spacing}
As mentioned above, the previously theoretical predictions of the ground state disagree with the experimental results. The DFT \cite{Citra-JPCA-1999} and RASSI \cite{Suo-JPB-2012} calculations predicted the $^4\Sigma^-$ and $^4\Delta_{7/2}$ ground states, respectively, whereas the experiment results present the $^2\Delta_{5/2}$ ground state. Our calculation gives the ground state of $\Omega$=1/2, which is inconsistent with the RASSI results and also does not support experimental assignment. However, the first excited state is predicted as $\Omega$=5/2, which only locates 234 cm$^{-1}$ above the ground state on our highest level of calculation. In addition, our predicted energy separation between $\Omega$=5/2 and 7/2 is only several tens cm$^{-1}$, which is far less than the RASSI value of 1327 cm$^{-1}$ \cite{Suo-supplement}. The 1/2, 5/2 and 7/2 are so close that the ground state can be easily polluted by two other low-lying excited states. As can be seen in Table \ref{tbl:groundstate}, the equilibrium bond length of 5/2 state is 1.712 \AA \ that is only 0.012 \AA \ shorter than Pang \textit{et al.}'s result \cite{Pang-JPCA-2010}. The harmonic vibrational frequency 903 $cm^{-1}$ agrees well with the experimental value of 909 $cm^{-1}$ obtained by Pang and co-workers \cite{Pang-JPCA-2010}. Therefore, the 5/2 state in our calculation should be the low-lying state that contributes to the experimental spectra observed by Pang \textit{et al.} and Adam \textit{et al.} \cite{Adam-JMS-2013}.
\begin{spacing}{1}
\begin{table*}\small
\caption{The lowest-lying electronic states of CoO and RhO for $\Omega$=1/2, 3/2, 5/2, and 7/2 symmetries, as calculated at the KRCI SD8(13in9)SD level in the MCSCF+CI implementations, where $\langle s_z \rangle$ and $\langle l_z \rangle$ are expectation values of $s_z$ and $l_z$ operators, and $R$, $T$ are the equilibrium bond length ({\AA}) and excited energy (cm$^{-1}$). The major configuration of CoO and RhO have the common parts ($7\sigma_{1/2}^2 8\sigma_{1/2}^2 3\pi_{1/2}^2 3\pi_{3/2}^2$) and ($10\sigma_{1/2}^2 11\sigma_{1/2}^2 5\pi_{1/2}^2 5\pi_{3/2}^2$), respectively.}\label{tbl:CoORhO}
\begin{tabular}{p{0.8cm} p{0.8cm} p{0.8cm} p{7cm} p{1cm} p{0.8 cm} }\hline\hline
$\Omega$ &$\langle s_z \rangle$&$\langle l_z \rangle$ &Major configuration & $R$ & $T$ \\ \hline
\multicolumn{6}{c}{ CoO }\\
7/2 &1.50&2.00&$1\delta_{3/2}^2 1\delta_{5/2}^1 4\pi_{1/2}^1 4\pi_{3/2}^1 9(10)^{a}\sigma_{1/2}^2$ ($^{4}\Delta$) 65\%&1.620 &0 \\ [+2 ex]
5/2 &0.50&2.00&$1\delta_{3/2}^2 1\delta_{5/2}^1 4\pi_{1/2(3/2)}^2 9\sigma_{1/2}^2$ ($^{2}\Delta$)31\%&1.620 &290 \\
& & &$1\delta_{3/2}^2 1\delta_{5/2}^1 4\pi_{1/2(3/2)}^2 9\sigma_{1/2}^1 10\sigma_{1/2}^1$ ($^{2}\Delta$)23\% & & \\[+2 ex]
3/2 &-0.50&2.00&$1\delta_{3/2}^2 1\delta_{5/2}^1 4\pi_{1/2}^1 4\pi_{3/2}^1 9\sigma_{1/2}^2$ ($^{2}\Delta$) 16\% &1.620 &670 \\
& & &$1\delta_{3/2}^2 1\delta_{5/2}^1 4\pi_{1/2}^1 4\pi_{3/2}^1 9\sigma_{1/2}^1$($^{2}\Delta)$13\% & & \\[+2 ex]
1/2 &-1.50&1.98&$1\delta_{3/2}^1 1\delta_{5/2}^2 4\pi_{1/2}^1 4\pi_{3/2}^1 9\sigma_{1/2}^2$ ($^{4}\Delta$) 42\%& 1.610&786 \\
& & &$1\delta_{3/2}^1 1\delta_{5/2}^2 4\pi_{1/2}^1 4\pi_{3/2}^1 9\sigma_{1/2}^1 10\sigma_{1/2}^1$ ($^{4}\Delta$) 35\%& & \\\hline
\multicolumn{6}{c}{RhO}\\
3/2 &1.49&0.008&$2\delta_{3/2}^2 2\delta_{5/2}^2 6\pi_{1/2}^1 6\pi_{3/2}^1 12(13)\sigma_{1/2}^1$ ($^{4}\Sigma^-$) 76\%&1.712 &0 \\
& & &$2\delta_{3/2}^2 2\delta_{5/2}^2 6\pi_{1/2}^2 6\pi_{3/2}^1 $ ($^{2}\Pi$) 2\% & & \\ [+2 ex]
1/2 &0.49&0.01 &$2\delta_{3/2}^2 2\delta_{5/2}^2 6\pi_{1/2}^2 12(13)\sigma_{1/2}^1$ ($^{2}\Sigma^-$) 32\%&1.712 &223 \\
& & &$2\delta_{3/2}^2 2\delta_{5/2}^2 6\pi_{3/2}^2 12(13)\sigma_{1/2}^1$ ($^{2}\Sigma^-$) 21\%& & \\ [+2 ex]
7/2 &1.49&2.01&$2\delta_{3/2}^2 2\delta_{5/2}^1 6\pi_{1/2}^1 6\pi_{3/2}^1 12\sigma_{1/2}^2$ ($^{4}\Delta$) 66\% &1.778 &3793 \\
& & &$2\delta_{3/2}^2 2\delta_{5/2}^1 6\pi_{1/2}^1 6\pi_{3/2}^1 12\sigma_{1/2}^1 13\sigma_{1/2}^1$($^4\Delta$)5\%& & \\
& & &$2\delta_{3/2}^2 2\delta_{5/2}^1 6\pi_{1/2}^2 6\pi_{3/2}^1 12\sigma_{1/2}^1$ ($^{2}\Phi$) 2\% & & \\[+2 ex]
5/2 &0.50&2.00&$2\delta_{3/2}^2 2\delta_{5/2}^1 6\pi_{1/2(3/2)}^2 12\sigma_{1/2}^2$ ($^{2}\Delta$)35\% &1.740 &3976 \\
& & &$2\delta_{3/2}^2 2\delta_{5/2}^1 6\pi_{1/2(3/2)}^2 12\sigma_{1/2}^1 13\sigma_{1/2}^1$ ($^{2}\Delta$)15\% & & \\
& & &$2\delta_{3/2}^2 2\delta_{5/2}^1 6\pi_{1/2}^1 6\pi_{3/2}^2 12\sigma_{1/2}^1$ ($^{4}\Pi$) 2\% & & \\
& & &$2\delta_{3/2}^2 2\delta_{5/2}^1 6\pi_{1/2}^1 6\pi_{3/2}^2 12\sigma_{1/2}^1$ ($^{2}\Phi$) 1\% & & \\ [+2 ex]\hline\hline
\multicolumn{6}{l}{$^{a}$ The spinor in the bracelet also exists but with less than 5\% composition. }
\end{tabular}
\end{table*}
\end{spacing}
The strongly relativistic effect leads the electronic states of IrO could only be identified by their $\Omega$ values, not by $\lambda$ value. However, the expectation values $\langle s_z\rangle=\Sigma_{i}m_s$ and $\langle l_z\rangle=\Sigma_{i}m_l$ suggest the major component in the $\Lambda-S$ notation. Here, we also provide the corresponding $\Lambda-S$ notation for each configuration in Table \ref{tbl:groundstate}. As illustrated by Table \ref{tbl:groundstate}, the lowest lying states of $\Omega$=1/2, 3/2, 5/2, and 7/2 have a common electronic occupation of $14\sigma_{1/2}^2 15\sigma_{1/2}^2 9\pi_{3/2}^2 9\pi_{1/2}^2$ in valance orbitals. The $14\sigma_{1/2}$ is mainly $2s$ atomic orbital of O. The $15\sigma_{1/2}$ is bonding orbital of Ir$(5d_{\sigma})$ + O$(2p_{\sigma})$. The $9\pi_{1/2}$ and $9\pi_{3/2}$ are two components of bonding $\pi$ orbital (Ir$(5d_{\pi})$+O$(2p_{\pi})$). These orbitals are low in energy and are fully occupied in all states in present study. Remaining six spinors in valence space, namely $16\sigma_{1/2}$, $17\sigma_{1/2}$, $4\delta_{5/2}$, $4\delta_{3/2}$ and $10\pi_{1/2}$ and $10\pi_{3/2}$ are occupied by 7 electrons in flexible ways that could generate many electronic states with relatively similar energies. Two components of the $\delta$ spinor are mainly composed of Ir $5d_{\delta}$ atomic orbital that is non-bonding. Two $10\pi$ spinors are anti-bonding combination of Ir $5d_{\pi}$ and O $2p_{\pi}$. The $16\sigma$ is mainly composed of Ir $6s^2$ and $17\sigma$ is anti-bonding of Ir $5d_{\sigma}$ and O $2p_{\sigma}$.
The ground state $\Omega$=1/2 has the leading configuration of $4\delta_{3/2}^2 4\delta_{5/2}^2 10\pi_{1/2}^2 16\sigma_{1/2}^1$ (or $17\sigma_{1/2}^1$ with composition less than 5\%), which corresponds to $^{4}\Sigma^-$ with $\delta^4\pi^2\sigma^1$ molecular orbital occupation. The configuration $4\delta_{3/2}^2 4\delta_{5/2}^2 10\pi_{1/2}^1 16\sigma_{1/2}^2$ corresponding to $^{2}\Pi$ ($\delta^4\pi^1\sigma^2$) also contributes to the lowest $\Omega$=1/2 state but with smaller component. Comparing with predictions of RASSI calculation, it suggests that the 1/2 state is mixture of $^2\Pi, ^4\Delta, ^4\Sigma^-$ components. The lowest lying state of $\Omega$=5/2 has the leading configuration $4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^2 16\sigma_{1/2}^2$, corresponding to $^2\Delta$. Therefore, the present result supports the previous assignment that the $\Omega$=5/2 state is arising from the $^2\Delta$ state in Pang's work \cite{Pang-JPCA-2010}. Nevertheless, our results show that the lowest $\Omega$=5/2 state has $\delta^3\pi^2\sigma^2$ molecular orbital occupation, not $\delta^3\pi^4$ as supposed by Pang and co-workers \cite{Pang-JPCA-2010}. The lowest lying state of $\Omega$=7/2 has a major configuration $4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^2$, which corresponds to $^{4}\Delta$ if described in the $\Lambda-S$ notation. Therefore, the lowest lying state of $\Omega$=7/2 comes mainly from the $\delta^3\pi^2\sigma^2$ occupation, which is consistent with the RASSI results for the $^4\Delta_{7/2}$. By analyzing the energy separations and leading configurations for the lowest $\Omega$=1/2, 5/2, and 7/2 states, we suggest that the none-bonding $16\sigma$, $4\delta$ and anti-bonding $10\pi$ orbitals are close in energy and $17\sigma$ is slightly higher. This is consistent with previous CASPT2 result that has included scaler relativistic effect in calculation.
It is interesting to compare IrO with its iso-valent molecules CoO and RhO. It already known that CoO and RhO have the $^4\Delta$ and $^4\Sigma^-$ ground states, respectively. Here, we have calculated these two molecules at KRCI level and the results of the low-lying states are summarized in Table. \ref{tbl:CoORhO}. For CoO, the $\Omega$=7/2 is assigned as the ground state, which has the leading configuration $1\delta_{3/2}^2 1\delta_{5/2}^1 4\pi_{1/2}^1 4\pi_{3/2}^1 9(10)\sigma_{1/2}^2$ ($^{4}\Delta$). The first excited state is $\Omega$=5/2, which corresponds to a $^2\Delta$ state. The adiabatic excitation energy is 290 cm$^{-1}$, being closed to the previous experimental value of $244$ cm$^{-1}$ \cite{CoO1987}. For RhO, our calculations show that the lowest-lying state has the $\Omega$ value of 3/2, which is mainly contributed by the leading configuration $2\delta_{3/2}^2 2\delta_{5/2}^2 6\pi_{1/2}^1 6\pi_{3/2}^1 12(13)\sigma_{1/2}^1$ ($^{4}\Sigma^-$). Our results of the ground state of CoO and RhO are in agreement with the previous experimental and theoretical studies \cite{CoO1979,CoO1987,CoO1993,CoO1997,CoO2005,CoO2006, RhO1998, RhO1999, RhO2002, RhO2003,RhO2005,RhO2007,RhO2009}. The low-lying states of three molecules present different characters. For instance, both of the CoO and IrO molecules have very dense states in ground state region. The energy gaps of four lowest states of CoO are only several hundreds cm$^{-1}$ and are much closer than corresponding values in RhO. For RhO, four lowest states are divided into two groups. As illustrated in Table \ref{tbl:CoORhO}, the 3/2 and 1/2 states are closed in energy and the 7/2 and 5/2 are nearly degenerate. However, the 7/2 and 5/2 states are well separated with the 1/2 and 3/2 states and locate more than 3500 $cm^{-1}$ higher. Thus, it is easier to assign the ground state of RhO than CoO. For the IrO molecule, it is much trouble because all of the three lowing states 1/2, 5/2 and 7/2 locate within 250 cm$^{-1}$, as shown in our calculation. Such near-degeneracy causes great difficulty in identification of the ground state of IrO.
Tracing back to the ground states of the group-9 elements Co, Rh and Ir, both the ground states of Co and Ir have a $d^7s^2$ configuration, whereas the Rh atom prefers a $d^8s^1$ configuration, which suggests the smaller electronic repulsion of d shells of the Rh atom. Provided that the group-9 metals bonded with oxygen atom somehow do not change the occupation numbers of those none-bonding $\sigma$ and $\delta$ orbitals, one can expect that IrO should be similar to CoO instead of RhO. However, our results show that the strongly relativistic effect of the Ir atom should be taken into account during such comparison. First of all, the relativistic effect causes the contraction of the $s$ and $p$ orbitals and then the increasing screening effect, which yields the expansion of the $d$ orbital. Thus, the repulsion of d electrons in the Ir atom should smaller than that in the Co atom. Also, the non-bonding $\delta$, $\sigma$ and anti-bonding $\pi$ orbitals are much closer in IrO than CoO, which leads to denser low-lying states of IrO than CoO. When the multiply-degeneracy of $d$ open shell of Ir element prevails over the $d$-shell electron repulsion effect, it will give rise to a low spin state, i.e., $\Omega$=1/2, of the IrO molecule in which $\delta$ orbital is fully occupied. In addition, these non-bonding orbitals are so close that it is easily to put electron in none-bonding $\sigma$(Ir 6s) orbital due to the electron repulsion in $\delta$ (Ir $5d_{\delta}$) orbital \cite{Eugen-JCE-2010}, which results in the high spin state, $\Omega=$7/2 and 5/2, of IrO close to the ground-state just like their counter part component of CoO.
\subsection{Excited states and assignment of the experimental spectra}
\begin{spacing}{1}
\begin{table*}\small
\caption{The low-lying states of IrO with $\Omega=5/2$, as calculated at the KRCI SD8(13in9)SD level with spinor obtained by KR-MCSCF implementations, where $\langle s_z \rangle$ and $\langle l_z \rangle$ are expectation values of $s_z$ and $l_z$ operators, $R$ is equilibrium bond length (\AA), $T^{Adi}_{cv2z}$ and $T^{Com}$ are excited energy (cm$^{-1}$), and $f$ is oscillation strength (a.u.). The major configurations have a common part ($14\sigma_{1/2}^2 15\sigma_{1/2}^2 9\pi_{1/2}^2 9\pi_{3/2}^2$).}\label{grid_mlmmh}
\begin{tabular}{p{0.9cm} p{0.9cm} p{0.9cm} p{0.9cm} p{6cm} p{1cm} p{1cm} p{1.1cm} } \hline\hline
State&$\langle s_z \rangle$ &$\langle l_z \rangle$&$R$&Major configuration &$T^{Adi}_{cv2z}$ & $T^{Com}$ & $f$\\\hline
0 &0.50 &2.00&1.712&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^2 16\sigma_{1/2}^2$ 41\% &0 & 0 & \\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^2 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ 15\%& & & \\[+2 ex]
1 &0.50 &2.00&1.712&$4\delta_{3/2}^2 4\delta_{5/2}^2 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^1$ 19\% &8481 &8373 &0.0003 \\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^2 10\pi_{1/2}^1 10\pi_{3/2}^1 17\sigma_{1/2}^1$ 8\% & & & \\[+1 ex]
2 &0.52 &1.98&1.774&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^2 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ 19\% &10350&11612&0.0003\\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^2 16\sigma_{1/2}^1 $ 12\% & & & \\[+1 ex]
3 &1.47 &1.03&1.787&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^2 16\sigma_{1/2}^1$ 12\% &12691&13462&0.0003\\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 16\sigma_{1/2}^2 17\sigma_{1/2}^1$ 7\% & & & \\[+2 ex]
4 &1.72 &0.78&1.774&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 16\sigma_{1/2}^2 17\sigma_{1/2}^1$ 18\% &13893&14385&0.0021 \\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^2 16\sigma_{1/2}^1$ 12\% & & & \\[+2 ex]
5 &0.46 &2.04&1.787&$4\delta_{3/2}^2 4\delta_{5/2}^2 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^1$ 12\% &14723&15495&0.0001\\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^2 10\pi_{1/2}^1 10\pi_{3/2}^1 17\sigma_{1/2}^1$ 10\% & & & \\[+2 ex]
6 &-0.46&2.96&1.787&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 16\sigma_{1/2}^2 17\sigma_{1/2}^1$ 10\% &15448&16483&0.0054\\
& & & &$4\delta_{3/2}^1 4\delta_{5/2}^2 10\pi_{1/2}^1 10\pi_{3/2}^2 16\sigma_{1/2}^1$ 11\% & & & \\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^2 16\sigma_{1/2}^1$ 7\% & & & \\[+2 ex]
7 &-0.48&2.98&1.787&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 16\sigma_{1/2}^2 17\sigma_{1/2}^1$ 22\% &17231&17694&0.0019\\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{3/2}^2 16\sigma_{1/2}^2$ 6\% & & & \\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^2 10\pi_{3/2}^1 16\sigma_{1/2}^1$ 5\% & & & \\[+2 ex]
8 &-0.48&2.98&1.787&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 16\sigma_{1/2}^2 17\sigma_{1/2}^1$ 13\% &18316&19304&0.0145\\
& & & &$4\delta_{3/2}^1 4\delta_{5/2}^1 10\pi_{1/2}^2 10\pi_{3/2}^1 16\sigma_{1/2}^2$ 11\% & & & \\[+2 ex]
9 &0.48 &2.02&1.787&$4\delta_{3/2}^1 4\delta_{5/2}^1 10\pi_{1/2}^2 10\pi_{3/2}^1 16\sigma_{1/2}^2$ 16\% &20043&21035&0.0047\\
& & & &$4\delta_{3/2}^1 4\delta_{5/2}^1 10\pi_{1/2}^2 10\pi_{3/2}^1 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ 5\% & & & \\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{3/2}^2 16\sigma_{1/2}^2$ 5\% & & & \\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^2 16\sigma_{1/2}^1$ 5\% & & & \\
10&0.52 &1.98&1.794&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{3/2}^2 16\sigma_{1/2}^2$ 7\% &22022&22575&0.0217\\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^2 17\sigma_{1/2}^1$ 6\% & & & \\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^2 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ 3\% & & & \\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^2 10\pi_{3/2}^1 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ 7\% & & & \\ [+2 ex]
11&-0.50&3.00&1.787&$4\delta_{3/2}^1 4\delta_{5/2}^2 10\pi_{1/2}^1 10\pi_{3/2}^2 16\sigma_{1/2}^1$ 10\% &22320&22933&0.0069\\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ 8\% & & & \\[+2 ex]
12&1.41 &1.09&1.787&$4\delta_{3/2}^2 4\delta_{5/2}^2 10\pi_{3/2}^1 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ 29\% &22913&24426&$<10^{-4}$ \\[+2 ex]
13&0.52 &1.98&1.787&$4\delta_{3/2}^1 4\delta_{5/2}^2 10\pi_{1/2}^2 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ 12\% &23518&24708&0.034\\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^2 10\pi_{1/2}^1 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ 6\% & & & \\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^1 17\sigma_{1/2}^1$11\% & & & \\[+2 ex]
14&2.38 &0.12&1.808&$4\delta_{3/2}^1 4\delta_{5/2}^1 10\pi_{1/2}^1 16\sigma_{1/2}^2 17\sigma_{1/2}^1$ 30\% &24796&26037&0.0005\\[+2 ex]
15&-0.21&2.71&1.808&$4\delta_{3/2}^1 4\delta_{5/2}^2 10\pi_{1/2}^2 10\pi_{3/2}^1 16\sigma_{1/2}^2$ 9\% &25324&26761&0.0106\\[+2 ex] \hline\hline
\end{tabular}
\end{table*}
\end{spacing}
\begin{spacing}{1}
\begin{table*}\small
\caption{The low-lying states of IrO with $\Omega=7/2$, as calculated at the KRCI SD8(13in9)SD level with spinors obtained by KR-MCSCF, where $\langle s_z \rangle$ and $\langle l_z \rangle$ are expectation values of $s_z$ and $l_z$ operators, $R$ is equilibrium bond length (\AA), $T^{Adi}_{cv2z}$ and $T^{Com}$ are excited energy (cm$^{-1}$), and $f$ is oscillation strength (a.u.). The major configurations have a common part ($14\sigma_{1/2}^2 15\sigma_{1/2}^2 9\pi_{1/2}^2 9\pi_{3/2}^2$).}\label{grid_mlmmh}
\begin{tabular}{p{0.8cm} p{0.8cm} p{0.8cm} p{0.8cm} p{6cm} p{1cm} p{1cm} p{1.1cm} } \\ \hline\hline
State&$\langle s_z \rangle$ &$\langle l_z \rangle$&$R$({\AA})&Major configuration &$T^{Adi}_{cv2z}$ & $T^{Com}$ & $f$ \\
\hline
0&1.48 &2.02&1.723&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^2$ 36\% & 0 &0 & \\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^2 10\pi_{3/2}^1 16\sigma_{1/2}^1$ 15\% & & & \\[+2 ex]
1&1.49 &2.01&1.776&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^2 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ 11\% &8662 &9560 &$<10^{-4}$\\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^2 16\sigma_{1/2}^1$ 7\% & & & \\[+2 ex]
2&0.49 &3.01&1.776&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^2 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ 18\% &12060&13126 &0.005\\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^2 16\sigma_{1/2}^1$ 5\% & & & \\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{3/2}^2 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ 4\% & & & \\[+2 ex]
3&-0.43&3.93&1.723&$4\delta_{3/2}^1 4\delta_{5/2}^2 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^2$ 35\% &15684&15428 &0.0001\\
& & & &$4\delta_{3/2}^1 4\delta_{5/2}^2 10\pi_{1/2}^2 10\pi_{3/2}^1 16\sigma_{1/2}^1$ 9\% & & & \\[+2 ex]
4&-0.46&3.96&1.776&$4\delta_{3/2}^1 4\delta_{5/2}^2 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^2$ 32\% &16208&16547 &$<10^{-4}$\\[+2 ex]
5&0.40 &3.10&1.776&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 16\sigma_{1/2}^2 17\sigma_{1/2}^1$ 21\% &17661&18585 &0.0019 \\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^2 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ 11\% & & & \\[+2 ex]
6&1.47&2.03&1.820&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ 14\%&20027&21013 &0.0046\\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^2 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ 14\% & & & \\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 16\sigma_{1/2}^2 17\sigma_{1/2}^1$ 8\% & & & \\[+2 ex]
7&1.48&2.02&1.820&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^1 17\sigma_{1/2}^1$ 11\%&22319&23478 &0.099\\
& & & &$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^2$ 11\% & & & \\[+2 ex]
8&0.52&2.98&1.820&$4\delta_{3/2}^2 4\delta_{5/2}^1 10\pi_{1/2}^1 16\sigma_{1/2}^2 17\sigma_{1/2}^1$ 16\% &24497&25546 &0.0035 \\[+2 ex]
9&0.58&2.92&1.820&$4\delta_{3/2}^1 4\delta_{5/2}^2 10\pi_{1/2}^1 10\pi_{3/2}^1 16\sigma_{1/2}^2$ 10\% &26374&26671 &0.0162 \\
& & & &$4\delta_{3/2}^1 4\delta_{5/2}^2 10\pi_{1/2}^2 10\pi_{3/2}^1 16\sigma_{1/2}^1$ 12\% & & & \\ \hline\hline
\end{tabular}
\end{table*}
\end{spacing}
The calculated excited states for $\Omega$=5/2 and 7/2 are summarized in Table 3 and 4 in order to explain the observed visible transition bands. The adiabatic excited states are calculated at the [dyall.cv2z (Ir) and aug-cc-pCVDZ (O)] basis sets, which yields the equilibrium bond length $R$ and the adiabatic excited energy $T^{Adi}_{cv2z}$. Besides, we also give a composite data of the excited energy $T^{Com}$, as evaluated for larger basis set (See supplemental material for the computational details and the complete data tabular). The oscillator strength $f$ associated to the transition from the lowest state in each $\Omega$ component to the upper states are given along with the excitation energy, which provides a reference to determinate which excited state can be assigned to the observed transition bands. The excited states of IrO exhibit stronger multiconfigurational features and therefore are more difficult to analyze the molecular orbital picture than the ground state. Moreover, our calculation for excited states is mainly based on the small basis set, limited by the huge computational demanding, which is supposed to bring uncertainty of 100-2000 $cm^{-1}$ due to the basis set incompleteness. Therefore, our analysis for the experimental transition band is conducted in a qualitative way.
As shown in Table 3, the 6th to 8th excited states have the excited energies $T$=15448 and 18316 cm$^{-1}$, respectively, that are close to the experimental values [17.6] and [17.8], and also have significant oscillator strengths. As compared with the electronic configurations of the ground state, the 6th to 8th excited states can be traced back to the electron excitations from $10\pi_{1/2}$ to $17\sigma_{1/2}$. The obtained equilibrium bond length for the 6th to 8th excited states is 1.787 {\AA}, which differs from the experimental value $R$=1.7969 {\AA} by about 0.1 {\AA}. This deviation in the bond length is not surprised for KRCI calculation of excited states. An assessment of accuracy of KRCI is conducted by Stefan, et al., \cite {Stefan-Thesis-2013} which shows that the KRCI calculation performs well for prediction of excited energy but with apparently large deviations in bond length. Therefore, consider the good agreement of the excited energy, the strong oscillation strength, and the clear electronic excitation path, we can suggest that the 6th to 8th excited states could contribute to the observed visible bands [17.6] 2.5-X$^2\Delta_{5/2}$ and [17.8] 2.5-X$^2\Delta_{5/2}$.
The 9th to 11th excited states give the excitation energies of 22043 and 22320 cm$^{-1}$ and the significant oscillator strengths of around 0.0047 - 0.0217. Such excited energies are close to the experimental values of transitions [21.5]-[22.0] 2.5-X2.5. The corresponding bond length is around 1.787 and 1.794 \AA, which is consistent with the experimental value 1.7874 \AA. Be compared with the electronic configurations of the ground state, the 9th to 11th excited states are mainly arising from electron excitation from $4\delta_{3/2}$ to $10\pi_{3/2}$, $4\delta_{3/2}$ to $4\delta_{5/2}$, and $10\pi_{1/2}$ to $10\pi_{3/2}$. The good agreement with the experimental values suggests that the 9th to 11th excited states can be assigned to the observed transition bands [21.5] 2.5-X$^2\Delta_{5/2}$ and [22.0] 2.5-X$^2\Delta_{5/2}$.
The 13th excited state is arising from electron excitation from $4\delta_{3/2}$ to $4\delta_{5/2}$ and $10\pi_{1/2}$ to $10\pi_{3/2}$, which gives the excited energy $T$=23518 cm$^{-1}$ that is close to the experimental value [23.3]. The corresponding oscillator strength is strong, be around 0.034, which indicates that this state is highly probably observed in experiment. Therefore, we can assign the 13th state to the observed band [23.3] 2.5-2.5.
Compared to the wealth of the experimental spectra of $\Omega$=5/2, only one transition band [21.9] 3.5-3.5 is observed for $\Omega$=7/2 \cite{Pang-JPCA-2010}. As shown in Table. 4, the 6th and 7th excited states give the excited energies around 20027-22318 cm$^{-1}$ that are close to the experimental value [21.9]. Their oscillation lengths are around 0.0046-0.099, indicating that these states are highly probable to be observed in experiments. The corresponding electronic excitations are also clear, mainly arising from singly excitation from $16\sigma_{1/2}$ and $10\pi_{3/2}$ to $17\sigma_{1/2}$. Therefore, we suggest that one of the 6th and 7th excited states with $\Omega$=7/2 may contribute to the experimental visible transition band [21.9] 3.5-3.5.
\section{Conclusions}
In summary, the electronic structure of IrO is calculated by using fully relativistic multi-reference configuration interaction method. Four lowest-lying states for $\Omega$=1/2, 3/2, 5/2, and 7/2 are calculated with the aim to determinate the ground state of IrO. Our results indicate that the ground state of IrO is $\Omega$=1/2, which is highly mixed by component of the $^4\Sigma^-$ and $^2\Pi$ states. Two low-lying states, namely, 5/2 and 7/2, are nearly degenerated with the 1/2 state and locate only 234 and 260 cm$^{-1}$ above. The quite small energy separations among the three $\Omega$ states bring the great difficulty for the identification of the ground state, which causes the discrepancy between the experiments and the theoretical calculations. However, our calculation supports that the low-lying state $5/2$ should contribute to most of the experimental observed spectra. This state has the equilibrium bond length of 1.712 \AA \ and vibrational frequency of 903 cm$^{-1}$, which agrees with the experimental value 1.724 \AA \ and 909 cm$^{-1}$ \cite{Pang-JPCA-2010}. Furthermore, the excited states of IrO are investigated for $\Omega$=5/2 and 7/2 that can be used to interpret the experimental spectra. Six excited states are assigned to the observed six transition bands of LIF spectrum in the visible region, i.e., [17.6] 2.5-$^2\Delta_{5/2}$, [17.8] 2.5-$^2\Delta_{5/2}$, [21.5] 2.5-$^2\Delta_{5/2}$, [22.0] 2.5-$^2\Delta_{5/2}$, [21.9] 3.5-$\Omega$=3.5, and [23.3] 2.5-2.5, through comparing the excited energy values, the oscillator strengths and the possible excitation paths.
Our calculation is implemented at the high level that the relativistic effect and spin-orbit coupling are taken into account at the same foot. The multi-reference properties of 5d shell of Ir is sufficiently considered through KR-MCSCF implementation. The electronic correlation is considered up to the 4f electrons of Ir. The strong multiconfigurational features of the electronic structure of IrO have been clearly demonstrated in our calculations. Our calculation still cannot provide the direct theoretical proof that the ground state is the $\Omega$=5/2 state. This urges more systematic theoretical and experimental work, for example, to give the relative position of the $\Omega$= 1/2, 5/2 and 7/2 states. More accurate electronic structure computation technique, for example, multi-reference couple cluster (MRCC) method that is considered as the most accurate method to treat the electronic correlation, is also strongly urged to be adopted for this challenge question.
\begin{acknowledgments}
This work is supported by NSFC 61275129, NFSC 21033001, 2012CB821305, NFSC 21203147, and CAS KJZD-EW-W02. Suo and Han would like to thank Professor C. Linton of University of New Brunswick for some beneficial suggestions and bringing new experimental studies of IrO to our attention.
\end{acknowledgments}
\nocite{*}
|
1,941,325,220,502 | arxiv |
\section{Introduction and Overview}
\label{Sec:intro}
This paper presents measurements of jet properties in
proton-proton ($pp$)
collisions at a center of mass energy of $7 \ {\rm TeV}$ at the CERN
LHC using the ATLAS detector. Jets are identified and their
momenta measured using the calorimeters. Charged particles
measured by the tracking system are then associated with these
jets using a geometric definition. The structure of the jets is
studied using these associated particles.
Jets produced at large transverse momentum in proton-proton collisions
arise from the scattering of proton constituents leading to outgoing partons
(quarks and gluons)
with large transverse momenta.
These manifest themselves as jets of hadrons
via a ``fragmentation process''. While the scattering of the proton
constituents
is well described by perturbative QCD and leads, at lowest order,
to final states of $gg$, $gq$, and $qq$, the fragmentation process is
more complex.
First, fragmentation must connect the outgoing partons with
the rest of the event as the jet consists of colourless hadrons while the
initiating parton carries colour. Second, the process involves the production
of hadrons and takes place at an energy scale where the QCD coupling
constant is large and perturbation theory cannot be used.
Fragmentation is therefore described using a QCD-motivated model with
parameters that must be determined from experiment.
The fragmentation function $D^h_i(z,Q)$ is defined as the
probability that a hadron of type $h$ carries longitudinal
momentum fraction $z$ of the momentum $\vec{p}_i$ of a parton of type $i$
\begin{equation}
z \equiv \frac{\vec{p}_i\cdot\vec{p_{h}}}{\abs{\vec{p}_i}^2}.
\label{eq:1}
\end{equation}
$D(z,Q)$ depends on $z$ and on the scale $Q$ of the hard scattering
process which produced the parton.
While the value of $D^h_i(z,Q)$ cannot be calculated in perturbative
QCD, the variation with $Q$ can be predicted provided $Q$ is sufficiently
large\cite{Gribov:1972rt,Altarelli:1977zs,Dokshitzer:1977sg,Georgi:1977mg,Owens:1978qz,Uematsu:1978yw}.
In this paper a quantity related to $D^h_i(z,Q)$ is
measured. After jets have been
reconstructed, the data are binned for fixed ranges of jet transverse
momenta (\ensuremath{p_{\mathrm{T\,jet}}}),
each bin containing \ensuremath{N_{\mathrm jet}}\ jets; $z$ is then
determined for each charged particle associated with the jet
\begin{equation}
z = \frac{\vec{p}_{\mathrm jet}\cdot\vec{p}_{ch}}{\abs{\vec{p}_{\mathrm jet}}^2},
\end{equation}
where $\vec{p}_{\mathrm jet}$ is the momentum of the reconstructed jet and
$\vec{p}_{ch}$ the momentum of the charged particle.
The following quantity is measured
\begin{equation}
F(z,\ensuremath{p_{\mathrm{T\,jet}}}) \equiv \frac{1}{\ensuremath{N_{\mathrm jet}}} \frac{d\ensuremath{N_{ch}}}{dz},
\label{eq:2}
\end{equation}
where \ensuremath{N_{ch}}\ is the number of charged particles in the jet.
$F(z,\ensuremath{p_{\mathrm{T\,jet}}})$ is a sum over $D^h_i(z,Q)$ weighted
by the rate at which each parton species ($i$) is produced from the hard scattering process.
As particle identification is not used, $h$ is summed over all charged hadrons.
The hard scattering scale $Q$ is of the same order of magnitude
as \ensuremath{p_{\mathrm{T\,jet}}}.
At small \ensuremath{p_{\mathrm{T\,jet}}}, gluon jets dominate due to the larger gluon
parton densities in the proton and larger scattering rates
for $gg\to gg$. In the pseudorapidity range used for jets in this
analysis ($\abs{\ensuremath{\eta_{\mathrm jet}}} < 1.2$)\footnote{ATLAS uses a
right-handed coordinate system with its origin
at the nominal interaction point (IP) in the centre of the
detector and the $Z$-axis coinciding with the axis of the beam
pipe. The $X$-axis points from the IP to the centre of the LHC
ring, and the $Y$-axis points upward. Cylindrical coordinates
($r$,$\phi$) are used in the transverse plane, $\phi$ being the azimuthal
angle around the beam pipe. The pseudorapidity is defined in
terms of the polar angle $\theta$ as $\eta=-\ln \tan(\theta/2)$.
The rapidity $y$ for a track or jet is defined by
$y=0.5\ln \left [ (E+p_Z)/(E-p_Z) \right ] $
where $E$ denotes the energy and $p_Z$ is
the momentum along the beam direction. For tracks, the energy is
calculated assuming the particle is a pion.}
the fraction of jets originating from
a hard scattering that produces a gluon falls from 80\%\ for
$\ensuremath{p_{\mathrm{T\,jet}}} \sim 25 \ {\rm GeV}$ to 50\%\ for $\ensuremath{p_{\mathrm{T\,jet}}}\sim 300 \ {\rm GeV}$ according
to the {\sc Pythia}~\cite{Sjostrand:2006za} event generator.
The jets measured experimentally also contain particles produced from
the hadronization of the beam remnants (the ``underlying event'').
It should be emphasized that because colour fields connect all the strongly
interacting partons in the $pp$ event, no unambiguous assignment of
particles
to the hard scattering parton or underlying event is possible.
The integral of
$F(z,\ensuremath{p_{\mathrm{T\,jet}}})$ with respect to $z$ corresponds to the multiplicity of
charged particles within the jet. A clear summary
of fragmentation phenomenology
is provided in~\cite{Nakamura:2010zzi}
(Section 17) whose
notation
is followed here.
The derivation of $D^h_i(z,Q)$ from $F(z,\ensuremath{p_{\mathrm{T\,jet}}})$ is
beyond the scope
of this paper, but comparisons of $F(z,\ensuremath{p_{\mathrm{T\,jet}}})$ with the
predictions of several Monte Carlo (MC) generators will be made. Different
features of the Monte Carlo models are probed by these studies.
At low values of \ensuremath{p_{\mathrm{T\,jet}}}, the comparisons are most sensitive to the
non-perturbative models of fragmentation, the connection of the partons
to the remainder of the event and to the accretion of
particles from the underlying event into the jet.
As \ensuremath{p_{\mathrm{T\,jet}}}\ rises, the impact of these effects is diluted and,
if all the Monte Carlo models implemented perturbative QCD in the
same way, $F(z,\ensuremath{p_{\mathrm{T\,jet}}})$ would become similar.
In particular the increase
of the total particle multiplicity with the hard scattering energy, here
\ensuremath{p_{\mathrm{T\,jet}}}, is predicted by
perturbative QCD\cite{Mueller:1981ex}.
Two other related quantities that describe the transverse shape of the
jets are also studied here.
The variable \ensuremath{p_{\mathrm{T}}^{rel}}\ is the momentum of charged
particles in a jet transverse to that jet's axis:
\begin{equation} \ensuremath{p_{\mathrm{T}}^{rel}} = \frac{|\vec{p}_{ch} \times
\vec{p}_{\mathrm jet}|}{|\vec{p}_{\mathrm jet}|}. \end{equation}
\noindent The following distribution is measured
\begin{equation}
f(\ensuremath{p_{\mathrm{T}}^{rel}},\ensuremath{p_{\mathrm{T\,jet}}}) \equiv \frac{1}{N_{\mathrm jet}} \frac{d\ensuremath{N_{ch}} }{d\ensuremath{p_{\mathrm{T}}^{rel}}}.
\label{eq:3}
\end{equation}
\noindent Finally, the density of charged particles in $y$-$\phi$ space,\\
$\rho_{ch}(r,\ensuremath{p_{\mathrm{T\,jet}}})$, is
measured as a function of the angular distance $r$ of charged particles
from the axis of the jet that contains them, where $r$ is given by:
\begin{equation} \label{eqn:rdef} r = \Delta R(ch,jet) =
\sqrt{(\phi_{ch} - \phi_{\mathrm jet})^2+(y_{ch} - y_{\mathrm jet})^2}
\end{equation}
\noindent Thus $\rho_{ch}(r,\ensuremath{p_{\mathrm{T\,jet}}})$ is given by:
\begin{equation} \rho_{ch}(r,\ensuremath{p_{\mathrm{T\,jet}}}) \equiv \frac{1}{\ensuremath{N_{\mathrm jet}}}
\frac{dN_{ch}}{2\pi rdr}.
\label{eq:5}
\end{equation}
As in the case of the longitudinal variables, a comparison of these
transverse quantities with Monte Carlo
generators is sensitive to many of their features. The non-perturbative
hadronization processes produce particles that
have limited transverse momentum with respect to the parton direction.
The mean value of this transverse momentum is of order
a few hundred {\rm MeV}, the scale where the QCD coupling constant becomes
non-perturbative.
At low \ensuremath{p_{\mathrm{T\,jet}}}\ this effect dominates.
If there were no other contributions,
\ensuremath{p_{\mathrm{T}}^{rel}}\ would remain constant
with increasing \ensuremath{p_{\mathrm{T\,jet}}}. Therefore more of the energy would be
concentrated in the core of the jet as \ensuremath{p_{\mathrm{T\,jet}}}\ increases
and the jets would become narrower.
However, as \ensuremath{p_{\mathrm{T\,jet}}}\ increases
contributions from processes controlled by perturbative QCD radiation
become more
important, contributing to
jet broadening and causing the
mean value of \ensuremath{p_{\mathrm{T}}^{rel}}\ to rise slowly (approximately logarithmically).
The phenomena described above are incorporated in all the Monte Carlo
generators used to describe jet production in $pp$ collisions, although
there are significant differences in how these effects are implemented.
For example, {\sc Pythia}\
describes non-perturbative hadronization using a
string model while {\sc Herwig}~\cite{Corcella:2000bw} uses a cluster model.
In {\sc Pythia}, coherent
colour effects are described partly by string fragmentation.
These effects are also produced in {\sc Herwig}\ and {\sc Pythia}\
from gluon radiation. Treatments of the
proton remnants are also described using different phenomenological
approaches. For both generators, the implementations require that
a number of input parameters be tuned to the data. The results presented
in this paper will test whether these Monte Carlo models and their current
input parameters adequately describe jets produced at the LHC.
As the results are presented in bins of \ensuremath{p_{\mathrm{T\,jet}}}, the explicit dependence on
\ensuremath{p_{\mathrm{T\,jet}}} in the variables defined in
Equations~\ref{eq:2}, \ref{eq:3} and~\ref{eq:5} is
often suppressed in the following.
The measurement is performed using
data with an integrated luminosity of $ 36\ {\rm pb}^{-1} $
recorded in 2010 with the ATLAS
detector at the LHC at a center-of-mass energy of 7~${\rm TeV}$.
The measurement covers a
kinematic range of $25\ {\rm GeV} < \ensuremath{p_{\mathrm{T\,jet}}} < 500\ {\rm GeV}$ and $|\ensuremath{\eta_{\mathrm jet}}| < 1.2$.
Events are triggered using a minimum bias trigger and a
combination of calorimeter jet triggers. A complementary ATLAS
analysis\cite{Aad:2011gn} studying the jet fragmentation
function and transverse profile
of jets reconstructed from charged particle
tracks using a total integrated luminosity of ~800 $\mu {\rm b}^{-1}$ has
been completed.
It explores the properties of jets at lower transverse momentum
than those typically studied in this paper.
Previous measurement of jet fragmentation functions have been made in $e^+e^-$ collisions
\cite{Akers:1995wt,Abreu:1996na,Buskulic:1995aw,Mattig:1988sv}, in $p\overline{p}$ collisions
\cite{Arnison:1983gv,Abe:1990ii} and in $ep$ collisions~\cite{Breitweg:1997ra,Aaron:2007ds}.
This paper is organized as follows.
The ATLAS detector is described briefly in Section~\ref{sec:detector}.
The Monte Carlo generator samples are discussed in Section~\ref{sec:samples}.
The event and object selections are described in Section~\ref{sec:selection}.
Section \ref{sec:analysis} contains a description of the
analysis. In Section~\ref{sec:systematics} the treatment of systematic
uncertainties is presented. Results and conclusions are shown in Sections~\ref{sec:results} and~\ref{sec:conclusions}.
\section{The ATLAS Detector}
\label{sec:detector}
The ATLAS detector is described in detail in
Ref.~\cite{ATLAS}. The subsystems
relevant for this analysis are the inner detector (ID),
the calorimeter and the trigger.
The ID is used to measure the momentum of charged particles.
It consists of three subsystems: a pixel detector, a silicon strip tracker
(SCT) and a transition radiation straw tube tracker (TRT). These
detectors are located
inside a solenoid that provides a 2~T axial field.
The ID has full coverage in the azimuthal angle
$\phi$ and over the pseudorapidity range
$0<|\eta^{\mathrm track}|<2.5$.
The electromagnetic calorimeters use liquid argon as the
active detector medium. They consist of accordion-shaped
electrodes and lead absorbers
and cover the pseudorapidity range $|\eta|<3.2$.
The technology used for the hadronic calorimeters varies with $\eta$.
In the barrel region ($|\eta|<1.7$) the detector is made of
scintillating tiles with steel radiator. In the endcap region
($1.5<|\eta|<3.2$) the detector uses liquid argon and copper.
A forward calorimeter consisting of liquid argon and tungsten/copper absorbers
serves as both electromagnetic and hadronic calorimeter at large
pseudorapidity and extends the coverage to $|\eta| < 4.9$.
The calorimeters are calibrated at the electromagnetic scale
which correctly reconstructs the energy
deposited by electrons and photons.
The calorimeters are not compensating and the
response of hadrons is lower than that of electrons ($e/h>1$).
Some fraction of the
hadronic energy can also be deposited in the
material in front of and in-between calorimeters.
The response for hadronic jets~\cite{ATLAS-CONF-2011-032}
is $\sim 50\%$ of the true energy
for $\ensuremath{p_{\mathrm{T\,jet}}}=20\,{\rm GeV}$ and $|\ensuremath{\eta_{\mathrm jet}}|<0.8$
and rises both with \ensuremath{p_{\mathrm{T\,jet}}}\ and \ensuremath{\eta_{\mathrm jet}}. For $|\ensuremath{\eta_{\mathrm jet}}|<0.8$, the
response at $\ensuremath{p_{\mathrm{T\,jet}}}=1\,{\rm TeV}$ is $\sim 80\% $.
The ATLAS trigger consists of three levels of event selection:
Level-1 (L1), Level-2 (L2), and Event Filter. The L2 and event filter
together form the High-Level Trigger (HLT).
The L1 trigger is implemented using custom-made electronics, while the
HLT is based on fast data reconstruction online algorithms running
on commercially available computers and networking systems.
The triggers relevant for this analysis are
the L1 minimum bias triggers (MBTS) and the L1 and HLT calorimeter triggers.
The minimum bias trigger is based on signals from 32 scintillation
counters located at pseudorapidities $2.09 <|\eta| < 3.84$.
Because non-diffractive events fire the MBTS with
high efficiency and negligible bias, this trigger can be used to study
jets with low \ensuremath{p_{\mathrm{T\,jet}}}. However, MBTS triggers were highly prescaled at large
instantaneous luminosities, making them unsuitable for studies
of high \pt\ jets that are produced at low rate.
A series of single jet inclusive
triggers with different jet $E_T$ thresholds and prescales were
deployed to ensure that significant data samples were taken over the full
range of \ensuremath{p_{\mathrm{T\,jet}}}~\cite{bib:jet}.
\section{Monte Carlo Samples}
\label{sec:samples}
Several Monte Carlo samples are used in this analysis. Some
samples were processed
with the
ATLAS full detector simulation~\cite{bib:atlas-sim} which is based on the
{\sc GEANT4} toolkit~\cite{bib:g4}. The simulated events are then
passed through the same reconstruction software as the data.
These are used to model the response of the
detector and to correct the data for experimental effects.
The baseline Monte Carlo sample used to determine these
corrections is produced using {\sc Pythia}\cite{Sjostrand:2006za} 6.421 with the
ATLAS tune~{\sc AMBT1} which uses the MRST2007LO* PDFs~\cite{Sherstnev:2007nd}
and was derived using the measured properties of
minimum bias events~\cite{Aad:2010ir}. Several other
fully simulated samples are used
to assess systematic uncertainties: {\sc Pythia}\ using the {\sc Perugia2010}\
tune~\cite{Skands:2010ak} (CTEQ5L PDFs~\cite{Pumplin:2002vw});
Herwig 6.5~\cite{Corcella:2000bw} using Jimmy 3.41~\cite{Butterworth:1996zw}
and {Herwig\raise0.2ex\hbox{\scriptsize\kern-0.5pt+\kern-0.5pt+}}\ 2.4.2~\cite{Bahr:2008pv} (MRST2007LO* PDFs).
Additional Monte Carlo generator samples
are used to
compare with the final corrected data: {\sc Pythia} 6.421 with the ATLAS MC09
tune~\cite{ATL-PHYS-PUB-2010-002} (MRST2007LO* PDFs),
{Herwig\raise0.2ex\hbox{\scriptsize\kern-0.5pt+\kern-0.5pt+}}\ 2.5.1~\cite{Gieseke:2011na} (MRST2007LO* PDFs) ,
Sherpa~\cite{Gleisberg:2008ta} (CTEQ6L\cite{Pumplin:2005rh} PDFs) and
\Pythia8 (8.105)\cite{Sjostrand:2007gs}\\
(MRST2007LO* PDFs).
\section{Reconstruction and Event Selection}
\label{sec:selection}
\begin{figure*}
\begin{center}
\includegraphics[width=0.49\textwidth]{zProjectionSyst_25_40.eps}
\includegraphics[width=0.49\textwidth]{zProjectionSyst_400_500.eps}
\end{center}
\caption{Systematic uncertainty in $F(z,\ensuremath{p_{\mathrm{T\,jet}}})$ from uncertainties
in the jet energy scale and resolution, the track reconstruction efficiency and momentum
resolution and the response matrix for $25\ {\rm GeV} <\ensuremath{p_{\mathrm{T\,jet}}} < 40\ {\rm GeV}$ (\emph{left}) and $400\ {\rm GeV} <
\ensuremath{p_{\mathrm{T\,jet}}} < 500\ {\rm GeV}$ (\emph{right}). The total uncertainty from the combination is also shown.}
\label{Fig:UnfoldingSystProjections}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=0.49\textwidth]{ptRelProjectionSyst_25_40.eps}
\includegraphics[width=0.49\textwidth]{ptRelProjectionSyst_400_500.eps}
\end{center}
\caption{Systematic uncertainty in $f(\ensuremath{p_{\mathrm{T}}^{rel}},\ensuremath{p_{\mathrm{T\,jet}}})$ from
uncertainties in the jet energy scale and resolution, the track reconstruction efficiency and momentum
resolution and the response matrix for $25\ {\rm GeV} <\ensuremath{p_{\mathrm{T\,jet}}} < 40\ {\rm GeV}$ (\emph{left}) and $400\ {\rm GeV} <
\ensuremath{p_{\mathrm{T\,jet}}} < 500\ {\rm GeV}$ (\emph{right}). The total uncertainty from the combination is also shown. }
\label{Fig:UnfoldingPtRelSystProjections}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=0.49\textwidth]{rhoProjectionSyst_25_40.eps}
\includegraphics[width=0.49\textwidth]{rhoProjectionSyst_400_500.eps}
\end{center}
\caption{Systematic uncertainty in $\rho_{ch}(r,\ensuremath{p_{\mathrm{T\,jet}}})$ from
uncertainties in the jet energy scale and resolution, the track reconstruction efficiency and momentum
resolution and the response matrix for $25\ {\rm GeV} <\ensuremath{p_{\mathrm{T\,jet}}} < 40\ {\rm GeV}$ (\emph{left}) and $400\ {\rm GeV} <
\ensuremath{p_{\mathrm{T\,jet}}} < 500\ {\rm GeV}$ (\emph{right}). The total uncertainty from the combination is also shown.}
\label{Fig:UnfoldingRhoSystProjections}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=0.49\textwidth]{zProjectionData_25_40.eps}
\includegraphics[width=0.49\textwidth]{zProjectionData_400_500.eps}
\end{center}
\caption{Distributions of $F(z)$ for $25\ {\rm GeV} < \ensuremath{p_{\mathrm{T\,jet}}} < 40\ {\rm GeV}$
(\emph{left}) and $400\ {\rm GeV} < \ensuremath{p_{\mathrm{T\,jet}}} < 500\ {\rm GeV}$ (\emph{right}). The gray band indicates the total
uncertainty.}
\label{Fig:UnfoldingDataProjections}
\end{figure*}
Events are required to have at least one primary vertex reconstructed
using ID tracks. If the event has multiple primary vertices, the vertex
with the largest $\sum (\ensuremath{p_{\mathrm{T\,track}}})^2$
is tagged as the
hard-scattering vertex.
Jets are reconstructed using the
infrared- and collinear-safe anti-$k_t$\ algorithm~\cite{bib:antikt2}
with radius parameter $R_c=0.6$
using the FastJet package~\cite{bib:antikt3}.
The detector input is based on topological clusters~\cite{bib:topo}.
A topological cluster is defined to have an energy equal to the
energy sum of all the included calorimeter cells, zero mass and a
reconstructed direction calculated from the weighted averages
of the pseudo-rapidities and azimuthal angles of the constituent
cells. The weight used is the absolute cell energy and the
positions of the cells are relative to the nominal ATLAS coordinate
system.
The energy of these clusters is measured at the electromagnetic scale,
which provides the appropriate calibration for electrons and photons.
A \ensuremath{p_{\mathrm{T\,jet}}}\ and \ensuremath{\eta_{\mathrm jet}}\ dependent
calibration is then applied to each jet\cite{ATLAS-CONF-2011-032}.
These calibrations are
based on comparing the
response from simulated calorimeter jets to that of
jets reconstructed using generator particles and matched to the
reconstructed jets in $\eta$-$\phi$ space.
The $\eta$-$\phi$ position of the jet (and hence its momentum)
is corrected to account for the fact that the primary vertex of the interaction
is not at the geometric centre of the detector.
Quality criteria are applied to ensure that
jets are not produced by noisy calorimeter cells, and to
avoid problematic detector regions. The jet energy is corrected
for the presence of additional $pp$ interactions in the
same bunch crossing using correction constants measured {\it in-situ}
that depend on the number of reconstructed primary
vertices.
Jets are required to have $\abs{\ensuremath{\eta_{\mathrm jet}}}<1.2$.
For events selected with the MBTS trigger, jets are required to pass
a minimum cut of $\ensuremath{p_{\mathrm{T\,jet}}} > 20$~GeV.
For events selected using jet triggers,
a trigger-dependent minimum \ensuremath{p_{\mathrm{T\,jet}}}\ threshold is imposed
on jets used in the final measurements to ensure a jet trigger efficiency larger than 99\%.
Tracks are selected using the following cuts:
\begin{eqnarray*}
&\ensuremath{p_{\mathrm{T\,track}}} >0.5\,{\rm GeV}, \ N_{\rm pixel} \ge 1, \ N_{\rm SCT} \ge 6, & \\
&|d_0| < 1.5\,{\rm mm}, \ |z_0\sin\theta| < 1.5\,{\rm mm},&
\end{eqnarray*}
where $N_{\rm pixel}$ and $N_{\rm SCT}$ are the number of hits from the
pixel and SCT detectors, respectively, that are associated with the
track and
$d_0$ and $z_0$ are the transverse and longitudinal impact parameters
measured with
respect to the hard-scattering vertex.
Tracks are associated with jets using a simple geometric algorithm. If
the distance in $\eta$-$\phi$ between the track and the jet
is less than the radius parameter used in the jet reconstruction ($R_c=0.6$),
the tracks are considered to belong to the jet. Track parameters
are evaluated at the perigee to the primary vertex
and are {\it not} extrapolated to the
calorimeter.
This simple association
algorithm facilitates comparison with particles from the event generator
whose parameters correspond
to those measured at the primary vertex..
\section{Analysis}
\label{sec:analysis}
The results presented here are obtained using four measured distributions:
the jet transverse momentum spectrum,
${dN_{\mathrm jet}(\ensuremath{p_{\mathrm{T\,jet}}})}/{d\ensuremath{p_{\mathrm{T\,jet}}}}$, and
three differential distributions of the number of charged tracks,
${dN_{\mathrm tracks}(z,\ensuremath{p_{\mathrm{T\,jet}}})}/{dz}$,
${dN_{\mathrm tracks}(\ensuremath{p_{\mathrm{T}}^{rel}},\ensuremath{p_{\mathrm{T\,jet}}})}/{d\ensuremath{p_{\mathrm{T}}^{rel}}}$ and
${dN_{\mathrm tracks}(r,\ensuremath{p_{\mathrm{T\,jet}}})}/{dr}$.
To facilitate comparison with the predictions of Monte Carlo event generators,
these distributions are corrected for detector acceptance, reconstruction
efficiency and migration due to track and jet momentum resolution effects.
This correction procedure is called unfolding.
The distributions $F(z,\ensuremath{p_{\mathrm{T\,jet}}})$, $f(\ensuremath{p_{\mathrm{T}}^{rel}},\ensuremath{p_{\mathrm{T\,jet}}})$ and $\rho_{ch}(r,\ensuremath{p_{\mathrm{T\,jet}}})$
are obtained from the charged particle differential distributions
by normalizing the distribution for each \ensuremath{p_{\mathrm{T\,jet}}}\ range to the value
of $N_{\mathrm jet}(\ensuremath{p_{\mathrm{T\,jet}}})$ obtained from the unfolding of the jet
transverse momentum
spectrum.
This paper presents results for $\ensuremath{p_{\mathrm{T\,jet}}} > 25$~GeV; however,
to decrease the systematic uncertainty associated with the modeling
of the \ensuremath{p_{\mathrm{T\,jet}}}\ spectrum,
jets with $20\ {\rm GeV} < \ensuremath{p_{\mathrm{T\,jet}}} < 25\ {\rm GeV}$ are also used in the unfolding.
A Bayesian iterative unfolding method \cite{D'Agostini:1994zf}
implemented in the RooUnfold \cite{RooUnfold} software package is used.
This procedure takes as its input the measured distributions and a
response matrix obtained from simulated data that provides a mapping
between reconstructed objects and those obtained directly from the
event generator.
This response matrix is not unitary because in mapping from generator to
reconstruction some events and objects are lost due to inefficiencies
and some are gained due to misreconstruction or migration of truth
objects from outside the fiducial acceptance into the reconstructed
observables. It is therefore not possible to obtain the unfolded distributions
by inverting the response matrix and applying it to the measured data.
Instead, an
assumed truth distribution (the ``prior'')
is selected, the response matrix is applied and
the resulting trial reconstruction set is compared to the observed
reconstruction set.
A new prior is then constructed from the old prior
and the difference between the trial and the observed distributions.
The procedure can iterated until this difference becomes small.
Monte Carlo based studies of the performance of the procedure demonstrate
that in this analysis no iteration is necessary.
The initial truth prior is taken to be the prediction of the baseline
Monte Carlo generator. Systematic uncertainties associated with this
choice and with the modeling of the response matrix are discussed in
Section~\ref{sec:systematics}.
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{zProjectionsWithMC.eps}
\end{center}
\caption{Distributions of $F(z)$ in bins of $\ensuremath{p_{\mathrm{T\,jet}}}$. The circles show unfolded data and the
lines are the predictions from AMBT1 {\sc Pythia}. }
\label{Fig:zProjections}
\end{figure}
\section{Systematic Uncertainties}
\label{sec:systematics}
The following sources of systematic uncertainties are considered:
\begin{enumerate}
\item The jet energy scale (JES) and resolution (JER) uncertainties which
affect the measurement of the number of jets in a given \ensuremath{p_{\mathrm{T\,jet}}}\ bin and consequently the measured value of
$z$.
\item The track reconstruction efficiency and momentum reconstruction uncertainties which affect the number of tracks in each
$z$, \ensuremath{p_{\mathrm{T}}^{rel}}\ and $N_{ch}(r)$ bin.
\item The uncertainty in the response matrix which is derived using a
particular Monte Carlo sample and
depends on the details of the event generator.
\item Potential bias due to the failure of the unfolding procedure to converge
to the correct value.
\end{enumerate}
These systematic uncertainties are addressed using Monte Carlo methods.
The first two systematic uncertainties, potential bias due to incorrect
Monte Carlo modeling of the JES and/or JER
and potential bias due to mismodeling by the simulation
of the track reconstruction efficiency and/or resolution,
are studied by modifying the detector response in simulated
data.
These modified Monte Carlo events are then unfolded and
compared to the baseline.
The systematic uncertainty on the JES is studied by varying
the jet energy response by its uncertainty. The JES uncertainty
varies from 4.6\%\ at $\ensuremath{p_{\mathrm{T\,jet}}}=20 $~GeV to ~2.5\%\ at
$\ensuremath{p_{\mathrm{T\,jet}}}=500 $~GeV\cite{ATLAS-CONF-2011-032}.
Systematic uncertainties on the JER are studied by broadening
the jet energy resolution with an additional \ensuremath{\eta_{\mathrm jet}}\ and \ensuremath{p_{\mathrm{T\,jet}}}\
dependent Gaussian term.
The uncertainty on the JER is below 14\%\ for the full
\ensuremath{p_{\mathrm{T\,jet}}}\ and \ensuremath{\eta_{\mathrm jet}}\ range used in this analysis\cite{ATLAS-CONF-2010-054}.
The uncertainty on the tracking efficiency is studied by
randomly removing a fraction of the tracks in the simulated data.
Uncertainties on the tracking efficiency are $\eta$-dependent and vary
between 2\%\ and 3\%\ for the relevant range of $\eta^{\mathrm track}$\cite{mb2paper}, dominated by the
accuracy of the
description of the detector material in the simulation.
In addition, there can be a loss of tracking efficiency in the core
of jets at high \ensuremath{p_{\mathrm{T\,jet}}}\ due to a single pixel hit receiving contributions from
more than track. Studies of such hit sharing show that
the simulation and data agree well and that
the resulting systematic uncertainty is negligible for \ensuremath{p_{\mathrm{T\,jet}}}$<500\ {\rm GeV}$.
Uncertainties on the track momentum resolution are parametrized
as an additional $\eta$-dependent broadening of the resolution in curvature
with values that vary from $0.0004$~GeV$^{-1}$ to
$0.0009$~GeV$^{-1}$\cite{ATLAS-CONF-2011-003}.
While the studies described above account for systematic uncertainties
associated with the accuracy of the detector simulation, they do not
account for the fact that the response matrix itself depends on the
fragmentation properties of the jets and hence on the physics description in
the event generator. Because the response of the calorimeter to hadrons
depends on the hadron momentum\cite{ATLAS-CONF-2011-028}, the JES
depends at the few per cent level on the momentum spectrum of
particles within the jet. Because the probability that a track will
share hits in the ID with another track is dependent upon the local
density of particles within the jet, the tracking resolution depends weakly
on the transverse profile of particles within the jet.
These effects have been studied by unfolding fully simulated
Monte Carlo samples
created from {\sc Perugia2010}, Herwig 6.5 (with Jimmy 3.41) and {Herwig\raise0.2ex\hbox{\scriptsize\kern-0.5pt+\kern-0.5pt+}}\
using the baseline response matrix obtained with {\sc Pythia}\ AMBT1.
Differences between the unfolded results for each tune
and the true distributions obtained from that same tune are studied
as a function of $z$, \ensuremath{p_{\mathrm{T}}^{rel}}\ and $N_{ch}(r)$ for each bin in true \ensuremath{p_{\mathrm{T\,jet}}}\
and used to assess the systematic uncertainty.
Potential bias in the unfolding procedure itself is studied by creating
1000 pseudo-experiments where the ``data'' are drawn from the baseline
fully simulated Monte Carlo samples
via a bootstrap method\cite{erfon:bootstrap}
and unfolding these "data" using
the standard procedure. The mean results obtained from these samples show
negligible bias and have a spread that is consistent with the reported
statistical uncertainties.
The systematic uncertainty due to the unfolding
procedure is thus deemed to be negligible in comparison to the other
uncertainties.
The resulting systematic uncertainties on
$F(z,\ensuremath{p_{\mathrm{T\,jet}}})$, $f(\ensuremath{p_{\mathrm{T}}^{rel}},\ensuremath{p_{\mathrm{T\,jet}}})$ and
$\rho_{ch}(r,\ensuremath{p_{\mathrm{T\,jet}}})$ for the $25\ {\rm GeV} < \ensuremath{p_{\mathrm{T\,jet}}} < 40\ {\rm GeV}$
(\emph{left}) and $400\ {\rm GeV} < \ensuremath{p_{\mathrm{T\,jet}}} < 500\ {\rm GeV}$ (\emph{right})
are shown in Figures~\ref{Fig:UnfoldingSystProjections}
through~\ref{Fig:UnfoldingRhoSystProjections}. For
$F(z,\ensuremath{p_{\mathrm{T\,jet}}})$, uncertainties on the tracking efficiency and response
matrix dominate at low $z$ while the jet energy scale dominates at
high $z$. For $f(\ensuremath{p_{\mathrm{T}}^{rel}},\ensuremath{p_{\mathrm{T\,jet}}})$ the jet energy scale, response matrix
and tracking efficiency uncertainties are all significant and the overall
uncertainty rises with \ensuremath{p_{\mathrm{T}}^{rel}}.
For $\rho_{ch}(r,\ensuremath{p_{\mathrm{T\,jet}}})$, the response matrix and tracking efficiency
uncertainties are significant for all \ensuremath{p_{\mathrm{T\,jet}}}\ and $r$ while the jet energy
scale contribution is most important for small \ensuremath{p_{\mathrm{T\,jet}}}.
\section{Results}
\label{sec:results}
This section presents comparisons of acceptance-corrected, unfolded data
to the predictions of several Monte Carlo generators. The gray
band on all the figures indicates the total uncertainty which is
dominated by the systematic uncertainty.
Figure~\ref{Fig:UnfoldingDataProjections} shows distributions of $F(z)$ in
two bins of \ensuremath{p_{\mathrm{T\,jet}}}.
Figure~\ref{Fig:zProjections} shows distributions
of $F(z)$ in all bins of \ensuremath{p_{\mathrm{T\,jet}}}\ compared
to AMBT1 Monte Carlo. Comparisons of the data and the Monte Carlo samples
are shown in Figure~\ref{Fig:zTruthOver0}.
All the {\sc Pythia}~6 tunings show
good agreement with the data. Herwig+Jimmy disagrees with the data
at large $z$ for
$\ensuremath{p_{\mathrm{T\,jet}}}> 200\ {\rm GeV} $. {Herwig\raise0.2ex\hbox{\scriptsize\kern-0.5pt+\kern-0.5pt+}}~2.5.1 is below the data at low $z$ for
$\ensuremath{p_{\mathrm{T\,jet}}}> 100\ {\rm GeV} $ while {Herwig\raise0.2ex\hbox{\scriptsize\kern-0.5pt+\kern-0.5pt+}}~2.4.2 has too many particles at low $z$
for $\ensuremath{p_{\mathrm{T\,jet}}}< 100\ {\rm GeV}$. {\sc Pythia} 8 and Sherpa provide a poor
description of the data.
Figure~\ref{Fig:UnfoldingData} (\emph{left}) shows the
distribution of $\langle z \rangle$ for the data and for a selection of
Monte Carlo samples as a function of \ensuremath{p_{\mathrm{T\,jet}}}. A comparison with
the Monte Carlo generators shows that the AMBT1 and
MC09 {\sc Pythia}\
and {\sc Perugia2010}\
datasets show good agreement with the data
over the entire
\ensuremath{p_{\mathrm{T\,jet}}}\ range. The agreement with Herwig+Jimmy is satisfactory. {Herwig\raise0.2ex\hbox{\scriptsize\kern-0.5pt+\kern-0.5pt+}}\ 2.5.1
is inconsistent with the data for $\ensuremath{p_{\mathrm{T\,jet}}}> 40\ {\rm GeV}$ and 2.4.2 is inconsistent for
$\ensuremath{p_{\mathrm{T\,jet}}} < 100\ {\rm GeV}$. \Pythia8 \ is $\sim 8$\%\ below the data
at all \ensuremath{p_{\mathrm{T\,jet}}}.
{\sc Sherpa} \ agrees well.
The charged particle multiplicity as a function of $\ensuremath{p_{\mathrm{T\,jet}}}$ is shown in Figure~\ref{Fig:meanmult} (\emph{right}). The {\sc Pythia}~6
tunes show reasonable agreement, with AMBT1 being higher than the others. Herwig+Jimmy has slightly
too few particles for $\ensuremath{p_{\mathrm{T\,jet}}}> 200\ {\rm GeV}$. {Herwig\raise0.2ex\hbox{\scriptsize\kern-0.5pt+\kern-0.5pt+}}\ 2.4.2 (2.5.1) has too many (few) particles
for $\ensuremath{p_{\mathrm{T\,jet}}}< 200\ (>300)\ {\rm GeV}$. Sherpa describes the data well
while \Pythia8 has $\sim 8$\%\ too many particles at all \ensuremath{p_{\mathrm{T\,jet}}}.
The transverse profile of the jets is described by the
\ensuremath{\rho_{ch}(r)}\ and $f(\ensuremath{p_{\mathrm{T}}^{rel}})$ distributions.
Figure~\ref{Fig:UnfoldingDataProjectionsrho}
shows the distribution of \ensuremath{\rho_{ch}(r)}\ in two bins of \ensuremath{p_{\mathrm{T\,jet}}}.
The sharp decrease in population in the last bin
is a feature of the jet algorithm, which tends to
incorporate particles close to the radius parameter into the jet.
The effect is also seen
in \cite{Aad:2011gn} (Figure 6) where distributions for two radius
parameters are shown.
Figure~\ref{Fig:UnfoldingDataProjectionsptrel} shows the distribution
of $f(\ensuremath{p_{\mathrm{T}}^{rel}})$
in the same two \ensuremath{p_{\mathrm{T\,jet}}}\ bins.
Figures~\ref{Fig:rhoProjections} and~\ref{Fig:ptRelProjections}
show distributions
of $\rho_{ch}(r)$ and $f(\ensuremath{p_{\mathrm{T}}^{rel}})$, respectively,
in all \ensuremath{p_{\mathrm{T\,jet}}}\ bins together with the predictions of the
AMBT1 Monte Carlo.
Comparisons of \ensuremath{\rho_{ch}(r)}\ for all data and Monte Carlo
are shown in Figure~\ref{Fig:rhoTruthOver0}.
Sherpa, {Herwig\raise0.2ex\hbox{\scriptsize\kern-0.5pt+\kern-0.5pt+}}\ 2.4.2 and \Pythia8 disagree
significantly with the data over the full range of the measurement.
\Pythia8 \ is consistent with the data only over a very restricted range of
\ensuremath{p_{\mathrm{T\,jet}}}\ around $80\ {\rm GeV}$. {Herwig\raise0.2ex\hbox{\scriptsize\kern-0.5pt+\kern-0.5pt+}}\ 2.5.1
shows good agreement except at small $r$ and for
$\ensuremath{p_{\mathrm{T\,jet}}}> 200\ {\rm GeV}$.
Herwig+Jimmy is consistent with the data only for $\ensuremath{p_{\mathrm{T\,jet}}}> 160\ {\rm GeV}$.
All the
{\sc Pythia}~6 tunings except AMBT1 agree; AMBT1 shows disagreement for
$\ensuremath{p_{\mathrm{T\,jet}}} > 200\ {\rm GeV}$.
Comparison of $f(\ensuremath{p_{\mathrm{T}}^{rel}})$ for all data and Monte Carlos
are shown in Figure~\ref{Fig:ptrelTruthOver0}.
None of the generators agree with the data within the systematic uncertainties.
The mean value of \ensuremath{p_{\mathrm{T}}^{rel}}\ as a function of $\ensuremath{p_{\mathrm{T\,jet}}}$ is shown in Figure~\ref{Fig:ptrelaverage}.
{Herwig\raise0.2ex\hbox{\scriptsize\kern-0.5pt+\kern-0.5pt+}}\ 2.5.1 has much too large a value of $\langle \ensuremath{p_{\mathrm{T}}^{rel}} \rangle$ for $\ensuremath{p_{\mathrm{T\,jet}}}> 100\ {\rm GeV}$ and 2.4.2 has too small
a value for $\ensuremath{p_{\mathrm{T\,jet}}}< 80\ {\rm GeV}$. AMBT1 has too small a value at all
\ensuremath{p_{\mathrm{T\,jet}}}. Herwig+Jimmy has too large a value for
$\ensuremath{p_{\mathrm{T\,jet}}}> 200\ {\rm GeV}$. Agreement of the remaining Monte Carlos is quite good.
\begin{figure*}
\begin{center}
\includegraphics[width=0.48\textwidth]{c_zProjectionsTruthOverData_0.eps}
\includegraphics[width=0.48\textwidth]{c_zProjectionsTruthOverData_1.eps}
\end{center}
\caption{The ratio of $F(z)$ predicted by various Monte Carlo generators
to that measured.
The gray band indicates the combined statistical and systematic
uncertainties.}
\label{Fig:zTruthOver0}
\end{figure*}
\begin{figure*}
\subfigure{
\includegraphics[width=0.48\textwidth]{meanZData.eps}
}
\subfigure{
\includegraphics[width=0.48\textwidth]{meanNTrackData.eps}
}
\caption{Distributions of $<z>$ (left) and of the mean number of charged particles selected with
the requirement \ensuremath{p_{\mathrm{T\,track}}}\ > 500 MeV (right) as a function of \ensuremath{p_{\mathrm{T\,jet}}}\ for data and various
Monte Carlos. The gray band indicated the total uncertainty.}
\label{Fig:UnfoldingData}
\label{Fig:meanmult}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=0.49\textwidth]{rhoProjectionData_25_40.eps}
\includegraphics[width=0.49\textwidth]{rhoProjectionData_400_500.eps}
\end{center}
\caption{Distributions of \ensuremath{\rho_{ch}(r)}\ for $25\ {\rm GeV} < \ensuremath{p_{\mathrm{T\,jet}}} < 40\ {\rm GeV}$
(\emph{left}) and $400\ {\rm GeV} < \ensuremath{p_{\mathrm{T\,jet}}} < 500\ {\rm GeV}$ (\emph{right}). The gray band indicates the total
uncertainty.}
\label{Fig:UnfoldingDataProjectionsrho}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=0.49\textwidth]{ptRelProjectionData_25_40.eps}
\includegraphics[width=0.49\textwidth]{ptRelProjectionData_400_500.eps}
\end{center}
\caption{Distributions of $f(\ensuremath{p_{\mathrm{T}}^{rel}})$ for $25\ {\rm GeV} < \ensuremath{p_{\mathrm{T\,jet}}} < 40\ {\rm GeV}$
(\emph{left}) and $400\ {\rm GeV} < \ensuremath{p_{\mathrm{T\,jet}}} < 500\ {\rm GeV}$ (\emph{right}). The gray band indicates the total
uncertainty.}
\label{Fig:UnfoldingDataProjectionsptrel}
\end{figure*}
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{rhoProjectionsWithMC.eps}
\end{center}
\caption{Distributions of \ensuremath{\rho_{ch}(r)}.
The circles show unfolded data. The
lines are the predictions from AMBT1 {\sc Pythia}.
}
\label{Fig:rhoProjections}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{ptRelProjectionsWithMC.eps}
\end{center}
\caption{Distributions of $f(\ensuremath{p_{\mathrm{T}}^{rel}})$. The circles show unfolded data. The
lines are the predictions from AMBT1 {\sc Pythia}.
}
\label{Fig:ptRelProjections}
\end{figure}
\clearpage
\begin{figure*}
\begin{center}
\includegraphics[width=0.48\textwidth]{c_rhoProjectionsTruthOverData_0.eps}
\includegraphics[width=0.48\textwidth]{c_rhoProjectionsTruthOverData_1.eps}
\end{center}
\caption{The ratio of \ensuremath{\rho_{ch}(r)}\ predicted by various Monte Carlo generators
to that measured.
The gray band indicates the combined statistical and systematic
uncertainties.}
\label{Fig:rhoTruthOver0}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=0.48\textwidth]{c_ptRelProjectionsTruthOverData_0.eps}
\includegraphics[width=0.48\textwidth]{c_ptRelProjectionsTruthOverData_1.eps}
\end{center}
\caption{The ratio of $f(\ensuremath{p_{\mathrm{T}}^{rel}} )$ predicted by various Monte Carlo generators
to that measured.
The gray band indicates the combined statistical and systematic
uncertainties.}
\label{Fig:ptrelTruthOver0}
\end{figure*}
\clearpage
\begin{figure}
\begin{center}
\includegraphics[width=0.48\textwidth]{meanPtRelData.eps}
\end{center}
\caption{Comparison of the measured value of the average value of
\ensuremath{p_{\mathrm{T}}^{rel}}\ as a function of $\ensuremath{p_{\mathrm{T\,jet}}}$ with various Monte
Carlo expectations.}
\label{Fig:ptrelaverage}
\end{figure}
\section{Conclusion}
\label{sec:conclusions}
A measurement of the jet fragmentation properties for charged particles in proton-proton collisions at a
center-of-mass energy of 7~${\rm TeV}$ is presented.
The dataset recorded with the ATLAS detector at the LHC in 2010 with an integrated luminosity of $36\,$pb$^{-1}$ is used.
Systematic uncertainties for the
fragmentation function which describes how the jet momentum is distributed amongst its constituents
vary between approximately 4\%\ and 40\%\
depending on $z$ and \ensuremath{p_{\mathrm{T\,jet}}}. The uncertainties increase strongly with $z$ and are largest at small
\ensuremath{p_{\mathrm{T\,jet}}}.
The measurements of the distributions
$\rho_{ch}(r,\ensuremath{p_{\mathrm{T\,jet}}})$ and $f(\ensuremath{p_{\mathrm{T}}^{rel}},\ensuremath{p_{\mathrm{T\,jet}}})$ which describe the shape of jets transverse to the jet direction
have uncertainties that fall as \ensuremath{p_{\mathrm{T\,jet}}}\ increases, increase at large values
of
\ensuremath{p_{\mathrm{T}}^{rel}}\ and are almost independent of $r$. They are less than 5\%\ except in the lowest \ensuremath{p_{\mathrm{T\,jet}}}\ range and for
$\ensuremath{p_{\mathrm{T}}^{rel}} > 1 \;{\rm GeV}$.
The measurements are sensitive to several properties of QCD as implemented in and modeled by
Monte Carlo event generators. The additional QCD radiation present as \ensuremath{p_{\mathrm{T\,jet}}}\ increases is modeled by
perturbative QCD and results in a growth of the particle multiplicity. This growth is very well modeled
by all the Monte Carlo generators used here.
Two other effects that
cannot be described by perturbative QCD impact the measured distributions.
The hadronization of partons produced in a QCD radiative shower into the observed hadrons must be modeled in
the Monte Carlo generators and is described by a large number of parameters which are tuned to agree with data.
Particles produced from remnants of the initial protons (underlying event) can be incorporated into jets whose constituents mainly come from the hard
scattering,
so the measured jet properties can be sensitive to this modeling.
The measured fragmentation functions agree well
with the AMBT1 {\sc Pythia}\ and {\sc Perugia2010}\
Monte Carlo predictions within statistical and systematic uncertainties.
Other tunes and generators show less good agreement indicating that
the non-perturbative physics is not adequately modeled in these cases.
Measurements of the transverse distributions
$f(\ensuremath{p_{\mathrm{T}}^{rel}},\ensuremath{p_{\mathrm{T\,jet}}})$ and $\rho_{ch}(r,\ensuremath{p_{\mathrm{T\,jet}}})$ are also presented.
For the \ensuremath{p_{\mathrm{T}}^{rel}}\ distribution, none of
the generators agree with data within systematic uncertainties
over the full kinematic range. For the $\rho_{ch}(r,\ensuremath{p_{\mathrm{T\,jet}}})$
distribution, Herwig+Jimmy, {\sc Pythia}~MC09 and {\sc Perugia2010}\ are
in reasonable agreement with the data.
In summary, none of the Monte Carlo
generators studied provide a good description of all the data.
The measurements presented here provide valuable inputs to constrain future
improvements in Monte Carlo modeling of fragmentation.
The full results are available in the
HEPDATA database\cite{Buckley:2010jn}, and a Rivet\cite{Buckley:2010ar} module for the analysis is also available.
\section{Acknowledgments}
We honour the memory of our young colleague Christoph Ruwiedel, who was
closely involved in the work described here and died shortly before its
completion.
We thank CERN for the very successful operation of the LHC, as well as the
support staff from our institutions without whom ATLAS could not be
operated efficiently.
We acknowledge the support of ANPCyT, Argentina; YerPhI, Armenia; ARC,
Australia; BMWF, Austria; ANAS, Azerbaijan; SSTC, Belarus; CNPq and FAPESP,
Brazil; NSERC, NRC and CFI, Canada; CERN; CONICYT, Chile; CAS, MOST and
NSFC, China; COLCIENCIAS, Colombia; MSMT CR, MPO CR and VSC CR, Czech
Republic; DNRF, DNSRC and Lundbeck Foundation, Denmark; ARTEMIS, European
Union; IN2P3-CNRS, CEA-DSM/IRFU, France; GNAS, Georgia; BMBF, DFG, HGF, MPG
and AvH Foundation, Germany; GSRT, Greece; ISF, MINERVA, GIF, DIP and
Benoziyo Center, Israel; INFN, Italy; MEXT and JSPS, Japan; CNRST, Morocco;
FOM and NWO, Netherlands; RCN, Norway; MNiSW, Poland; GRICES and FCT,
Portugal; MERYS (MECTS), Romania; MES of Russia and ROSATOM, Russian
Federation; JINR; MSTD, Serbia; MSSR, Slovakia; ARRS and MVZT, Slovenia;
DST/NRF, South Africa; MICINN, Spain; SRC and Wallenberg Foundation,
Sweden; SER, SNSF and Cantons of Bern and Geneva, Switzerland; NSC, Taiwan;
TAEK, Turkey; STFC, the Royal Society and Leverhulme Trust, United Kingdom;
DOE and NSF, United States of America.
The crucial computing support from all WLCG partners is acknowledged
gratefully, in particular from CERN and the ATLAS Tier-1 facilities at
TRIUMF (Canada), NDGF (Denmark, Norway, Sweden), CC-IN2P3 (France),
KIT/GridKA (Germany), INFN-CNAF (Italy), NL-T1 (Netherlands), PIC (Spain),
ASGC (Taiwan), RAL (UK) and BNL (USA) and in the Tier-2 facilities
worldwide.
\bibliographystyle{atlasstylem}
|
1,941,325,220,503 | arxiv | \section{Introduction}\label{introduction}
Orbital period variation (OPV) of post-common envelope binaries has become a powerful tool to search for circumbinary planets \citep[see, e.g.][]{2009AJ.137.3181L,Beuermann+2010,2010MNRAS.401L.34Q,Almeida+2013,Almeida+2019}, as well as to investigate intrinsic phenomena of the binary, e.g., the magnetic cycle of one active component, apsidal motion, mass transfer events, angular momentum loss via magnetic braking and gravitation wave emission \citep[see, e.g.,][]{Claret+2010,Parsons+2010,Schreiber+2010,Zorotovic+2013,Bours+2016,Almeida+2019,Burdge+2019}. While some of these phenomena induce a decrease or an increase in the orbital period of the binary, the magnetic cycle, the apsidal motion, and the gravitational influence of a third body produce cyclic and periodic variations. These three last effects have similar features in time-scales from months to decades and therefore difficult to be distinguished. Despite of some important clues to solve this issue, in both observational and theoretical sides, have been reported in the literature, e.g. \citet[][]{Applegate1992, Brinkworth2006, Parsons+2010, Parsons+2014, Bours+2016, Volschow+2016, Volschow+2018, Almeida+2019}, it is still an open question and more post-common envelope systems with long-baseline of eclipse time monitoring are needed.
GK\,Vir is a detached eclipsing binary system consisting of a white dwarf (WD -- primary) and a low-mass main-sequence star (secondary) with an orbital period of 8.27~h \citep{Green+1978}. This system was discovered in a survey for blue stars at high Galactic latitude \citep[][]{Green+1976} and it is located at $\sim$475~pc from us \citep{Gaia+2018}. \citet[][]{Fulbright+1993} collected spectroscopic data of GK\,Vir and together with photometric information reported by \citet{Green+1978} improved the physical property measurements of the system. With high-resolution spectroscopic data and optical and infrared photometric data, \citet[][]{Parsons+2012} characterised both components as well as the geometrical parameters of GK\,Vir. The authors derived the binary's inclination (i = 89$\fdg$5), masses and radii for both, primary and secondary components (M$_{\rm WD} = 0.564~\rm M_{\sun}$, M$_{\rm sec} = 0.116~\rm M_{\sun}$ and R$_{\rm WD} = 0.0170~\rm R_{\sun}$, R$_{\rm sec} = 0.155~\rm R_{\sun}$). Using evolutionary models, \citet[][]{Parsons+2012} obtained an effective temperature T$_{\rm eff} = 50000$~K and a carbon-oxygen core for the WD.
The eclipse times of GK\,Vir have a long history. \citet[][]{Green+1978} presented the first nine measurements from 1975 to 1978 with uncertainties varying from 1 to 10~s. \citet[][]{Parsons+2010} reported seven new eclipsing time measurements derived from the high-speed ULTRACAM photometric data collected between 2002 and 2007, which showed a slight decrease in the O-C~diagram. Another study by \citet[][]{Parsons+2012} listed out one new eclipse time measure obtained in 2010, which showed an increase in the O-C~diagram. The same trend in this diagram was found by \citet{Bours+2016} with 10 new measurements from 2012 to 2015. As pointed out by these last authors, GK\,Vir is the system with the second-largest observational baseline among the 58 eclipsing detached binaries, which are composed of a WD plus a main-sequence star or a brown dwarf. Also, in this sample, GK\,Vir has one of the smallest OPV amplitude. Therefore, this system is an important target to perform an OPV analysis.
Here we present 10 new mid-eclipse times of GK\,Vir obtained between 2013 August and 2020 April. We combine these data with previous measurements from the literature and performed a new OPV analysis. In Section~\ref{observation}, we describe our data and the reduction procedure. The methodology used to obtain the mid-eclipse times, the procedure to examine the OPV, and their possible related physical effects are presented in Section~\ref{analysis}. In Section~\ref{discussion}, we discuss our results.
\section{Observations and data reduction}\label{observation}
The photometric data of GK\,Vir were collected in an observational program to search for OPVs in compact binaries. This project is carried out using the facilities of the \textit{Observat\'orio do Pico dos Dias} that is operated by the \textit{Laborat\'orio Nacional de Astrof\'isica} (LNA/MCTI) in Brazil. Photometric observations were performed with CCD cameras attached to the 0.6-m, and 1.6-m telescopes. The procedure to remove undesired effects from the CCD data includes typically obtaining 100 bias frames and $30$ dome flat-field images for each night of observations. The characteristics of the GK\,Vir photometric data are summarised in Table~\ref{table:1}. In this table, $n$ and $t_{\rm exp}$ are the number and the time of exposure, respectively.
\begin{table}
\caption{Log of the photometric observations.}
\footnotesize
\label{table:1}
\centering
\begin{tabular}{l c c c c}
\hline\hline
Date~~~~~ & $n$ & \ t$_{\rm exp}$(s) & Telescope & Filter \\
\hline
2013 Aug 12 & 280 & 4 & 1.6-m & Unfiltered \\
2017 Apr 03 & 310 & 10 & 0.6-m & Unfiltered \\
2017 Apr 05 & 120 & 10 & 0.6-m & Unfiltered \\
2019 Apr 03 & 48 & 30 & 0.6-m & Unfiltered \\
2019 Jun 02 & 216 & 10 & 0.6-m & Unfiltered \\
2019 Jul 14 & 377 & 10 & 1.6-m & Unfiltered \\
2019 Jul 24 & 111 & 15 & 0.6-m & Unfiltered \\
2020 Mar 04 & 289 & 15 & 0.6-m & Unfiltered \\
2020 Mar 05 & 352 & 10 & 0.6-m & Unfiltered \\
2020 Apr 01 & 151 & 20 & 0.6-m & Unfiltered \\
\hline
\end{tabular}
\end{table}
The data reduction was done using the standard {\small IRAF}\footnote{IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation.} tasks. We have an automatic procedure to obtain the relative flux of GK\,Vir. The procedure consists of subtracting a master median bias image from each program image and dividing the result by a normalized flat-field. Then, differential photometry is used to obtain the relative flux between our target and a star of the field with constant flux. The extraction of the fluxes was done using aperture photometry as the GK\,Vir field is not crowded. Figure~\ref{mid-eclipse} shows the result of this procedure for the data collected on 2019 July 14.
\section{Analysis and Results}\label{analysis}
\subsection{Eclipse fitting}
\label{eclipe_fit}
In order to obtain the mid-eclipse times of GK\,Vir, we performed a model fit for each observed event. The Wilson-Devinney code \citep[WDC;][]{1971ApJ.166.605W} was used to generate the synthetic light curves. As GK\,Vir is a detached binary, we used mode 2 of the WDC which is appropriate for such type of system. The luminosity of both components was computed assuming stellar atmosphere radiation. For the fitting procedure, we adopted as search intervals the range of the geometrical and physical parameters, e.g., mass ratio, inclination, radii, temperatures, and masses obtained by \citet{Parsons+2012} for GK\,Vir.
A procedure similar to that described in \citet{2012MNRAS.423.478A} was used to search for the mid-eclipse times of GK\,Vir. The light curves generated by the WDC were used as a `function'~to be optimized by the genetic algorithm \textsc{PIKAIA} \citep{char1995}. To measure the goodness of fit, we use the reduced chi-square defined as
\begin{equation}
\centering
\chi_{\rm red}^2= \frac{1}{(n-m)}\sum_1^n\left(\frac{O_j-C_j}{\sigma_j}\right)^2,
\end{equation}
where $O_j$ is the observed points, $C_j$ is the corresponding model, $\sigma_j$ is the uncertainties at each point, $n$ is the number of measurements, and $m$ is the number of fitted parameters. Figure~\ref{mid-eclipse} shows an eclipse of GK\,Vir with the best solution superimposed. To establish realistic uncertainties, we used the solution obtained by \textsc{PIKAIA} as input to a Markov chain Monte Carlo (MCMC) procedure \citep[][]{Foreman-Mackey2013} and examine the marginal posterior distribution of the probability of the parameters. The median of the distribution gives the time of mid-eclipse and the area corresponding to the $\sim$68.3\% in a normal distribution gives the corresponding standard uncertainty. The results are presented in Table~\ref{timing}.
\begin{figure}
\includegraphics[scale=0.46,angle=270,bb=80 100 420 340]{Figure01.pdf}
\caption{Light curve of the primary eclipse of GK\,Vir observed on 2019 July 14. The red line represents the best fit obtained using the procedure described in Section~\ref{eclipe_fit}.}
\label{mid-eclipse}
\end{figure}
\subsection{Orbital period variations}
To analyse the OPVs of GK\,Vir, we collected the available mid-eclipse times in the literature and added the 10 new measurements. We first fit a linear ephemeris,
\begin{equation}
T_{\rm min} = T_0 + E\times P_{\rm bin},
\label{lin_efem}
\end{equation}
to all mid-eclipse times. In the last equation, $T_{\rm min}$ is the mid-eclipse time, $T_0$, $E$, and $P_{\rm bin}$ are the initial epoch, the cycle, and the orbital period of the binary, respectively. The residuals obtained from the fit show a cyclic variation (see Figure~\ref{figo-c}). This kind of variation can be explained by the light travel time (LTT) effect, the apsidal motion, or the Applegate mechanism. In the next sections, these three possible scenarios are discussed.
\begin{table}
\caption{Mid-eclipse times of GK\,Vir.}
\label{timing}
\centering \scriptsize
\begin{tabular}{l c c r}
\hline\hline
Cycle &Eclipse timing & $\sigma$ & Ref. \\
&BJD(TDB) 2400000+ & (d) & \\
\hline
-67 & 42520.76747 & $1.0\times10^{-5}$ & \footnotemark[1] \\
-32 & 42532.81905 & $2.0\times10^{-5}$ & \footnotemark[1] \\
-29 & 42533.85204 & $9.0\times10^{-5}$ & \footnotemark[1] \\
0 & 42543.83769 & $1.0\times10^{-5}$ & \footnotemark[1] \\
3 & 42544.87068 & $1.0\times10^{-5}$ & \footnotemark[1] \\
851 & 42836.86314 & $6.0\times10^{-5}$ & \footnotemark[1] \\
1966 & 43220.79202 & $1.2\times10^{-4}$ & \footnotemark[1] \\
2132 & 43277.95101 & $6.0\times10^{-5}$ & \footnotemark[1] \\
2896 & 43541.01972 & $1.2\times10^{-4}$ & \footnotemark[1] \\
28666 & 52414.425572 & $1.0\times10^{-6}$ & \footnotemark[2] \\
29735 & 52782.515227 & $1.0\times10^{-6}$ & \footnotemark[2] \\
29738 & 52783.548219 & $1.0\times10^{-6}$ & \footnotemark[2] \\
30746 & 53130.633688 & $3.0\times10^{-6}$ & \footnotemark[2] \\
32706 & 53805.522115 & $2.0\times10^{-6}$ & \footnotemark[2] \\
32709 & 53806.555113 & $1.0\times10^{-6}$ & \footnotemark[2] \\
34054 & 54269.680087 & $1.0\times10^{-6}$ & \footnotemark[2] \\
37069 & 55307.837585 & $1.0\times10^{-6}$ & \footnotemark[3] \\
38913 & 55942.783670 & $1.0\times10^{-6}$ & \footnotemark[4] \\
37963 & 55615.669378 & $9.0\times10^{-6}$ & \footnotemark[4] \\
38076 & 55654.578751 & $7.0\times10^{-6}$ & \footnotemark[4] \\
38250 & 55714.492323 & $4.0\times10^{-6}$ & \footnotemark[4] \\
39023 & 55980.660063 & $5.0\times10^{-6}$ & \footnotemark[4] \\
40121 & 56358.735346 & $4.0\times10^{-6}$ & \footnotemark[4] \\
40211 & 56389.725126 & $7.0\times10^{-6}$ & \footnotemark[4] \\
40234 & 56397.644731 & $4.0\times10^{-6}$ & \footnotemark[4] \\
40582 & 56517.47188 & $1.0\times10^{-5}$ & \footnotemark[5] \\
41084 & 56690.325955 & $3.0\times10^{-6}$ & \footnotemark[4] \\
41404 & 56800.511828 & $9.0\times10^{-6}$ & \footnotemark[4] \\
42214 & 57079.419823 & $2.0\times10^{-6}$ & \footnotemark[4] \\
44445 & 57847.621959 & $3.0\times10^{-6}$ & \footnotemark[5] \\
44451 & 57849.68797 & $2.5\times10^{-5}$ & \footnotemark[5] \\
46565 & 58577.60336 & $1.0\times10^{-5}$ & \footnotemark[5] \\
46739 & 58637.51692 & $1.0\times10^{-5}$ & \footnotemark[5] \\
46861 & 58679.525310 & $9.0\times10^{-6}$ & \footnotemark[5] \\
46890 & 58689.510895 & $7.0\times10^{-6}$ & \footnotemark[5] \\
47541 & 58913.670275 & $7.0\times10^{-6}$ & \footnotemark[5] \\
47544 & 58914.703256 & $5.0\times10^{-6}$ & \footnotemark[5] \\
47622 & 58941.561085 & $1.4\times10^{-5}$ & \footnotemark[5] \\
\hline
\end{tabular}\\
\footnotemark[1]\citet{Green+1978};~
\footnotemark[2]\citet{Parsons+2010};~
\footnotemark[3]\citet{Parsons+2012};~
\footnotemark[4]\citet{Bours+2015};~
\footnotemark[5]This study
\end{table}
\begin{figure*}
\begin{minipage}{175mm}
\resizebox{\hsize}{!}{\includegraphics[angle=0]{Figure02.pdf}}
\caption{(O$-$C) diagram of the mid-eclipse times of GK\,Vir built with respect to the linear part of the ephemeris in Eqs.~\ref{ephem_ltt} and \ref{eqam}. Our measurements are presented with blue circles and the red and black lines represent the best fit including one LTT effect and the apsidal motion, respectively. The green and black points are measurements from \citet[][]{Green+1978}, \citet[][]{Parsons+2010,Parsons+2012}, and \citet[][]{Bours+2015}. Four measurements from \citet[][]{Green+1978} which have smaller error bars and are overlapped in this plot (see Table~\ref{timing}), were highlighted with green colour.}
\label{figo-c}
\end{minipage}{}
\end{figure*}
\subsubsection{Light travel time effect}\label{lttsec}
The LTT effect is observed when additional components gravitationally interact with an object that has a stable clock, in our case, the primary eclipses of GK Vir, forcing it to rotate around the common mass centre of the entire system. Thus, the binary moves away from and closer to an external observer at rest. Because the speed of light is constant, the observer will see the period of the binary become larger or smaller when it is moving away or approaching, respectively. Adding the LTT mathematical relation ($\tau_j$) obtained by \citet{irwin1952} to Equation~\ref{lin_efem},
\begin{equation}
T_{\rm min} = T_0 + E\times P_{\rm bin} + \sum_1^n \tau_j,
\label{ephem_ltt}
\end{equation}
where,
\begin{equation}
\tau_{\rm j} = \frac{a_{\rm bin;j}\sin i_j}{c}\left[\frac{1-e^2_j}{1+e_j \cos f_j}\sin (f_j+\omega_j)
+ e_j\sin (\omega_j)\right].
\label{ltt}
\end{equation}
In the last equation, $a_{\rm bin;j}$ is the semimajor axis, $c$ is the speed of light, $i_j$ is the inclination, $e_j$ is the eccentricity, $\omega_j$ is the periastron argument, and $f_j$ is the true anomaly. These parameters are relative to the orbit of the centre of mass of the inner binary around the common mass centre consisting of the inner binary and of the $j-$th body.
We fitted Equation~\ref{ephem_ltt} with one LTT effect to the mid-eclipse times. The resulting $\chi_{\rm red}^2$ was 2.1, and the residuals have no indication of another cyclic variation. To search for the best solution and to sample the parameters of Equation~\ref{ephem_ltt}, we used the \textsc{PIKAIA} algorithm \citep[][]{char1995} and an MCMC procedure \citep[][]{Foreman-Mackey2013}, respectively. The best solution is shown with red line in Figure~\ref{figo-c}, the posterior distributions of the fitted parameters are displayed in Figure~\ref{histo}, and the numerical values with their corres\-ponding standard uncertainties are presented in Table~\ref{parameters}.
\begin{table}
\caption{Parameters of the linear ephemeris, LTT effect, and apsidal motion (Eqs.~\ref{ephem_ltt} and \ref{eqam}) adjusted to the mid-eclipse times of GK Vir.}
\label{parameters}
\begin{center}
\begin{tabular}{l c r }
\hline
\small Linear ephemeris & LTT and apsidal motion \\
\hline
Parameter & Value & Unity \\
\hline
$P_{\rm bin}$ & $0.3443308426(3)$ & d \\
$T_0$ & $2442543.83763(5)$ & BJD \\
\hline
& \small LTT $\tau_1$ term & \\
\hline
Parameter & Value & Unity \\
\hline
$P$ & $24.34^{+2.15}_{-1.64}$ & yr \\
$T$ & $2453028^{+443}_{-370}$ & BJD \\
$a_{\rm bin}\sin i$& $0.0109^{+0.0014}_{-0.0007}$ & au \\
$e$ & $0.14\pm0.04$ & \\
$\omega$ & $198^{+22}_{-18}$ & $\degree$ \\
$f(m)$ & $(1.6^{+1.3}_{-0.7}) \times 10^{-9}$ & M$_{\odot}$ \\ %
$(a_{\rm min})^{a}$ & $7.38^{+1.26}_{-0.72}$ & au \\
$(m_{\rm min})^{b}$ & $0.95^{+0.22}_{-0.13}$ & M$_{\rm Jup}$ \\
$\chi_{\rm red}^2$ & 2.1 & \\
\hline
& \small Apsidal motion & \\
\hline
Parameter & Value & Unity \\
\hline
$P_{\rm AM}$ & $24.0\pm0.3$ & yr \\
$e$ & $(5.35\pm0.02) \times 10^{-5}$ & \\
$\chi_{\rm red}^2$ & 3.4 & \\
\hline
\end{tabular} \\
\footnotesize{$^{\rm a}$Minimum semimajor axis of the outer body} \\
\footnotesize{$^{\rm b}$Minimum mass of the outer body}
\end{center}
\end{table}
In the LTT scenario, one important test is to check if the third-body orbital solution is long-term stable. To do this, we used the solution shown in Table~\ref{parameters} and performed numerical integration using an $N$-body code with the usual Bulirch-Stoer integrator \citep[][]{Cincotta+2016}.
Initially, to verify if the LTT solution shown in Figure~\ref{figo-c} agrees with the numerical one, we performed an orbital integration with a time-step of 6~d and the total time of 45~yr, which is approximately the observational coverage of the GK\,Vir eclipse times. The O-C diagram obtained from the numerical simulation is presented with red line in panel (c) of Figure~\ref{nbody}, which agrees with the LTT analytical solution shown in this same panel with green line.
In the second step, to test the third-body long-term stability, we performed numerical integrations considering the inclination between the external body and our line of sight equal to 15, 30, 45, 60, 75, and 90$\degree$. The time-step was set to 1.3~yr and the total integration time of $10^5$~yr. For each inclination, the mass of the outer body was obtained using a Newton-Raphson iteration in the following mass function:
\begin{equation}
f(m_j) = \frac{4\pi^2(a_{\rm bin,j}\sin i_j)^3}{GP^2_j} = \frac{(m_j\sin i_j)^3}
{(M_{\rm bin}+m_j)^2},
\label{mass_function}
\end{equation}
where $G$ is the gravitational constant, $M_{\rm bin}$ is the mass of the inner binary, $P_j$ and $m_j$ are the orbital period and mass of the external body, respectively. In panels (a) and (b) of Figure~\ref{nbody} are displayed the mass and the semimajor axis of the third body as a function of the inclination. In the same figure, panels (d), (e), and (f) show, respectively, the orbit of the third body in the inner-binary mass-centre reference system, the temporal evolution of the eccentricity and of the semi-major axis of the outer body for the inclination equal to 90$\degree$, which is approximately the same inclination of the inner binary. Inside of panels (e) and (f), two regions were zoomed in to better visualize the effect in the eccentricity and semimajor axis due to the orbital movement of the third body around the central binary. As one can see in Figure~\ref{nbody}, for the inclination equal to 90$\degree$, both eccentricity and semimajor axis of the third body are almost constant during $10^5$~yr, only varying with a small amplitude ($\sim$10$^{-7}$) in the same frequency of the outer body period, indicating that the third body has a stable orbit. The same result was found for the other inclinations.
\begin{figure*}
\begin{minipage}{175mm}
\resizebox{\hsize}{!}{\includegraphics[angle=0]{Figure03.pdf}}
\caption{Numerical integrations of the outer body orbit around GK\,Vir: (a) the mass and (b) semimajor axis as a function of the inclination, (c) the O-C diagram, (d) the outer body orbit in the inner-binary mass centre reference system, and (e) the evolution of the eccentricity and (f) semimajor axis for $10^5$~yr. In panels from (c) to (f) are shown the case for inclination equal to 90$\degree$. In panels (e) and (f) two regions are zoomed in with their amplitudes of variation expanded by a factor of $10^4$ to better visualize the short-term effects.}
\label{nbody}
\end{minipage}{}
\end{figure*}
\subsubsection{Apsidal motion}
The second possible scenario to explain the OPV of GK Vir is the apsidal motion. This effect consists of the rotation of the apsidal line due to tidal interactions in the close binaries and it can be observed when the orbital eccentricity of the binary is non-zero. One direct way to verify if the OPV of an eclipsing binary is due to the apsidal motion is by measuring the time between the primary and secondary eclipses throughout the cycle \citep[see e.g.,][]{Parsons+2014}. However, as the secondary eclipse of GK Vir was not measured yet, we adopted the following approaches to check if the apsidal motion is the one responsible for the OPV of this binary:
(i) Check if the period generated by the apsidal motion is consistent with the period found in the O-C diagram.
The rotational variation rate of the apsidal motion has three contributions: tidal distortions generated by the non-spherical mass distribution of the stars ($\dot \omega_{\rm tide}$), rotation ($\dot \omega_{\rm rot}$), and effect due to the general relativity ($\dot \omega_{\rm GR}$). These three contributions, in degrees per year, can be calculated by
\begin{equation}
\dot \omega_{\rm tide} = 15\frac{360}{P_{\rm bin}}\left(\frac{R_2}{a_{\rm bin}}\right)^5\frac{m_1}{m_2}\frac{1 + 1.5e^2 + 0.125e^4}{(1-e^2)^5} {\rm k_2},
\label{omegat}
\end{equation}
\begin{equation}
\dot \omega_{\rm rot} = \frac{360}{P_{\rm bin}}\left(\frac{R_2}{a_{\rm bin}}\right)^5\frac{m_1+m_2}{m_2}\frac{1}{(1-e^2)^2}{\rm k_2},
\label{omegar}
\end{equation}
and
\begin{equation}
\dot \omega_{\rm GR} = \frac{360}{P_{\rm bin}}\left(\frac{3G}{c^2}\right)\frac{m_1+m_2}{a_{\rm bin}(1-e^2)},
\label{omegagr}
\end{equation}
where $R_2$ is the radius of the secondary star, $m_1$ and $m_2$ are the masses of the primary and secondary components, $G$ is the gravitational constant, $c$ is the speed of light, $k_2$ is the apsidal constant, and the other parameters are the same as defined in the previous sections. The apsidal constant, which is related to the concentration of mass of the tidally distorted star, has been the subject of several theoretical studies \citep[e.g.,][]{Sirotkin+2009, Claret+2010}. Following the work done by \citet[][]{Feiden+2011}, and extrapolating the values of mass presented in its Table 1 to the secondary mass of GK Vir ($m_2 = 0.116$~\rm M$_\odot$), we obtained $k_2 \sim 0.156$. Using the parameters derived by \citet[][see section~\ref{introduction}]{Parsons+2012}, $k_2 = 0.156$, and $e = 0$ in Eqs.~\ref{omegat}, \ref{omegar}, and \ref{omegagr}, the upper limit for the period of the apsidal motion would be $\sim$16.4 yr, which is smaller than the period obtained in the O-C diagram of GK\,Vir. For this calculation, we only consider the $\dot \omega_{\rm tide}$ and $\dot \omega_{\rm rot}$ generated by $m_2$, as the WD contribution is much smaller than the secondary star one.
(ii) Verify if the circularization time for GK Vir's orbit is smaller or higher than the WD cooling time.
Theoretically, it is possible to check if the GK Vir system had enough time to completely circularize its orbit. To do so, we need to compute the cooling time of the WD, which can be considered as the age that the system has in the current configuration, and then, compare it with the circularization time of its orbit.
Following \citet[][]{Althaus+2010}, the WD cooling time can be estimated using the Mastel's law \citep[][]{Mestel1952} through the approximation
\begin{equation}
\tau_{\rm cool} \approx \left(\frac{10^8}{A}\right)\left(\frac{m_1}{M_{\rm \odot}}\right)^{5/7}\left(\frac{L_1}{L_{\rm \odot}}\right)^{-5/7},
\end{equation}
where $A$ is the mean atomic number, and $L_1$ is the luminosity of the WD. Replacing the values obtained by \citet[][]{Parsons+2012} for the WD of GK Vir and assuming for $A$ the atomic number of Carbon, yields $\tau_{\rm cool} \sim 5.7$ Myr.
The orbital circularization of a binary system is an effect caused due to the interaction of the tides between its components \citep[see e.g.,][]{Zahn1984}. The time to circularize the binary's orbit can be estimated by
\begin{equation}
\tau_{\rm cir} = \left[21k_2q(q+1) \left(\frac{L_2}{m_2R_2^2}\right)^{1/3}\left(\frac{R_2}{a_{\rm bin}}\right)^{6}\right]^{-1},
\end{equation}
where $k_2$ is the apsidal constant, $q = m_2/m_1$ is the mass ratio, and $L_2$, $m_2$, and $R_2$ are the luminosity, mass, and radius of the secondary star, respectively. Using the parameters derived by \citet[][]{Parsons+2012} for GK Vir, we obtained the circularization time $\sim$0.7 Myr. Thus, as $\tau_{\rm cool}$ is $\sim$8.1 times larger than $\tau_{\rm cir}$ and considering only the tidal interaction acting on the orbital parameters of the binary, we conclude that GK Vir had enough time to circularize its orbit.
(iii) Verify if the equation of the apsidal motion fits well to the mid-eclipse times of GK Vir.
Following the study done by \citet[][]{Todoran1972}, the equation that describes the linear ephemeris plus the apsidal motion using the first-order approximation for the orbital eccentricity ($e$) is
\begin{equation}
T_{\rm min} = T_0 + E\times P_{\rm bin} + \frac{eP_{\rm bin}}{2\pi}({\rm cosec}^2i_{\rm bin}+1)\sin\left(\frac{2\pi t}{P_{\rm am}}\right),
\label{eqam}
\end{equation}
where, $i_{\rm bin}$ is the binary orbital inclination, $t$ is mid-eclipse time obtained from the linear ephemeris, and $P_{\rm am}$ is the period of the apsidal motion. We fit this equation to the mid-eclipse times of GK Vir adopting the same procedure used for the LTT analysis. The fitted parameters with their uncertainties are presented in Table~\ref{timing} and the best solution is shown with the black line in Figure~\ref{figo-c}. The best solution provides $\chi^2_{\rm red} = 3.4$, which is larger than the one obtained for the LTT best solution (see Section~\ref{lttsec}).
\subsubsection{Applegate mechanism}
The third possible scenario for the OPV of GK Vir is associated with the magnetic cycle of active stars. This effect proposed by \citet{Applegate1992}, called Applegate mechanism, consists of the OPV of the system due to the changes in the form of a magnetically active component. The shape of the star may change due to the variation of the quadrupole moment, which in turn leads to changes in the orbital period of the binary. These changes must occur at the same time-scale as the magnetic activity cycle (MAC) of the star.
In a series of papers \citep{1998MNRAS.296.893L,1999AA.349.887L,2006MNRAS.369.1773L}, Lanza and collaborators refined the treatment done by \citet{Applegate1992}. \citet{2006MNRAS.369.1773L} discarded the Applegate mechanism for RS~CVn systems. Moreover, the author commented that the Applegate hypothesis cannot explain the orbital period modulation of close binary systems composed by a late-type secondary star.
In the same direction, \citet[][]{Brinkworth2006} included a stellar thick outer shell to the Applegate theory. More recently, \citet{Volschow+2016} developed a new formulation to add the quadrupole moment changes in two finite regions, core and external shell. With this new model, the authors using 16 compact binaries concluded that the Applegate mechanism can explain the eclipse time variation for four systems.
In this context, the way to verify if this mechanism can explain the modulation in the binary orbital period is checking if the observed variation amplitude in the O-C diagram can be produced by the energy of the secondary star. Following the work done by \citet{Volschow+2016} and using their online calculator\footnote{http://theory-starformation-group.cl/applegate/index.php}, we obtained that the required energy for the finite-shell constant density model is $\sim$10$^5$ times larger than the energy of the secondary star.
Recently, a new model based on the exchange of angular momentum between the active component spin and the orbital motion was proposed by \citet[][]{Lanza2020}. This author found that the systems with energy $\sim$10$^2$ to 10$^3$ times smaller than the required energy to explain the OPV, reported by previous models, can be explained using this new approach.
\section{Discussion and conclusions}\label{discussion}
We present 10 new mid-eclipse times of GK\,Vir from 2013 August to 2020 April. We combined these measurements with all mid-eclipse times available in the literature and performed an orbital period analysis. One cyclic modu\-lation is seen in the O-C diagram (see Figure~\ref{figo-c}). Based on the modulation period, which is $\sim$24~yr, we investigate if this variation could be explained by the Applegate mecha\-nism, the apsidal motion, or the LTT effect.
For the Applegate mechanism, following \citet{Volschow+2016} we showed that the amount of required energy to explain the O-C diagram of GK\,Vir is $\sim$10$^5$ times larger than the energy of the secondary star. Based on this amount of required energy, even considering the new model proposed by \citet[][]{Lanza2020}, this mechanism would hardly explain the O-C diagram of GK Vir.
Besides the energy test, we can verify if the period found in O-C diagram could be explained by one hypothetical MAC of the secondary star, which is directly correlated to the Applegate mechanism. To do so, we consider the period found in the O-C diagram ($\sim$24~yr) as the MAC of the secondary star. Furthermore, we assume that the secondary star is synchronized with the orbital period of GK\,Vir and therefore its rotational period would be $\sim$8.3~h. Adding these values in the MAC versus rotation period diagram (see the magenta point at the top right hand corner in Figure~\ref{figorbit}), we conclude that it does not agree with the empirical trends. Also, the first evolved system, similar to GK\,Vir, with measured MAC reported by \citet[][see the black square in Figure~\ref{figorbit}]{Almeida+2019} has the same trend than the other measures for single stars in this diagram. Therefore, it is one additional evidence against the possibility of the O-C diagram of GK Vir being explained by the Applegate mechanism. However, we emphasize that this is a particular case, and thus, this result does not rule out this mechanism as a possible cause of the orbital variation of other eclipsing post-common envelope binaries, as for example, it is the most likely cause of V471 Tau \citep[][]{Hardy+2015}.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[angle=0]{Figure04.pdf}}
\caption{MAC versus rotational period diagram as shown in \citet[][]{Vida+2014} and \citet[][]{Almeida+2019}. The black dots, blue squares, green triangles, and red squares represent measurements from \citet[][]{Vida+2013,Vida+2014,Olah+2009}, and \citet[][]{Savanov2012}, respectively. The gray dots show the data from different surveys presented in \citet[][]{Olah+2009}. The gray line represents the fit (using linear regression) to all the data from \citet[][]{Olah+2009} and \citet[][]{Vida+2013,Vida+2014} excluding M stars, while the black and red lines show the fit to the shortest cycles of that data set. The Sun is shown with its standard symbol and the measurement derived by \citet[][]{Almeida+2019} is presented with a black square. The magenta point (see top right) is the hypothetical measurement for GK\,Vir (see Section~\ref{discussion} for more details). As this point does not agree with the trends of the experimental measurements, this hypothetical scenario can be ruled out.}
\label{figorbit}
\end{figure}
In the apsidal motion context, we used three approaches to analyse if the OPV of GK Vir could be explained by this effect. In the first two cases, we showed that the predicted theoretical period by the apsidal motion is $\sim$1.5 times smaller than the period found in the O-C diagram, and the cooling time of the WD is $\sim$8.1 times larger than the circularization time, which are pieces of evidence against the explanation via the apsidal motion. In the third analysis, we fitted the equation of the apsidal motion to the mid-eclipse times of GK Vir and obtained $\chi^2_{\rm red} = 3.4$ (see Figure~\ref{figo-c}). Despite this relatively low $\chi^2_{\rm red}$, which would indicate a good fit, it is larger than the one obtained for the LTT effect (see below).
Finally in the LTT scenario, we showed that the equation that represents a circumbinary body fits well to the mid-eclipse times of GK Vir, see Figure~\ref{figo-c}. The best solution, which provides $\chi^2_{\rm red} = 2.1$, yields orbital period $P = 24.34^{+2.15}_{-1.64}$~yr, and eccentricity $e = 0.14\pm0.04$, for the outer body. Adopting the mass and the inclination of the inner binary, $0.68~\rm M_{\odot}$ and 89$\fdg$5 \citep{Parsons+2012}, and under the assumption of coplanarity between the outer body and the inner binary, the mass of the circumbinary body is $m_3\sim0.95~ M_{\rm Jup}$. Therefore, in this scenario, GK\,Vir would be composed of an inner binary and a Jupiter-like planet. However, as the observational baseline of GK\,Vir is smaller than twice the period found in the O-C diagram and the first eclipse time measurements from \citet[][]{Green+1978} have large error bars, we must take this solution as preliminary.
As some studies have suggested additional circumbinary bodies as a possible explanation for the OPVs of post-common envelope binaries \citep[e.g.,][]{2009AJ.137.3181L, Almeida+2011IAUS} and further works have shown that their orbits are in an unstable configuration \citep[e.g.,][]{Horner+2012,Horner+2013MNRAS}, an important test for the LTT scenario is to verify if the third-body orbital solution shows long-term stability. To do that, we performed a dynamical analysis by using $N$-body numerical integrations. The results for six different inclinations (15, 30, 45, 60, 75, and 90$\degree$) between the third-body orbital plane and our line of sight showed that the outer body around GK\,Vir has a stable orbital configuration over, at least, $10^5$ yr (see Figure~\ref{nbody}). Therefore, this reinforces the LTT effect as the most likely explanation for the OPV behavior of GK\,Vir.
If the LTT effect is confirmed with future data as the true cause of the orbital modulation of GK\,Vir, the third body would be the planet with one of the longest orbital periods, with a full observational baseline, discovered so far. Considering the possible formation scenarios, according to \citet{Perets2011} this circumbinary body could have been formed either at the same time as the inner binary formation (called as the first generation of planets) or after the common envelope phase of the inner binary (known as the second generation of planets). Although our data are not conclusive on these two possible formation scenarios, our results place GK\,Vir as a promising target for further study on this subject with the new generation of large telescopes, e.g., Giant Magellan Telescope, Thirty Meter Telescope, and Extremely Large Telescope.
\section*{Acknowledgements}
We thank the anonymous referee for helpful suggestions which were important to improve the paper. This study was partially supported by Coordena\c{c}\~ao de Aperfei\c{c}oamento de Pessoal de N\'ivel Superior (CAPES). LAA, AD and TAM thank Funda\c{c}\~ao de Amparo \`a Pesquisa do Estado de S\~ao Paulo (FAPESP) through the projects (LAA and AD: 2011/51680-6, LAA: 2012/09716-6, 2013/18245-0, and TAM: 2016/13750-6). GMV thanks Conselho Nacional de Desenvolvimento Cient\'ifico e Tecnol\'ogico (CNPq) for funding. This study was based on observations carried out at the Observatorio do Pico dos Dias (OPD/LNA) in Brazil. We thank the OPD staff for support and help during the observations.
\section*{Data availability}
The data underlying this article will be shared on reasonable request to the corresponding author.
|
1,941,325,220,504 | arxiv | \section{Introduction}\label{}
The problem of convergence of discrete-time financial models to the
models with continuous time is well developed; see, e.g., \cite{Broad,Ch,Cutl,delbaen,duff,HZ,Hu-Sc}. The reason for such an interest
can be explained as follows: from the analytical point of view, it is
much simpler to deal with continuous-time models although all
real-world models operate in the discrete time. In what concerns the
rate of convergence, there can be different approaches to its
estimation. Some of this approaches are established in \cite{Broad,Ch,Mishura3,Mishura1,Mishura,Mish_Mun_2,Mish_Mun_3}. In this paper, we
consider the Cox--Ingersoll--Ross process and its approximation on a
finite time interval. The CIR process was originally proposed by Cox,
Ingersoll, and Ross \cite{Cox_Ing_Ross} as a model for short-term
interest rates. Nowadays, this model is widely used in financial
modeling, for example, as the volatility process in the Heston model
\cite{Heston}. The strong global approximation of CIR process is
studied in several articles. Strong convergence (without a rate or with
a logarithmic rate) of several discretization schemes is shown by \cite
{Alfonsi,Bossy,delbaen1,Gyongy,Higham}. In \cite{Alfonsi},
a~general framework for the analysis of strong approximation of the CIR
process is presented along with extensive simulation studies.
Nonlogarithmic convergence rates are obtained in \cite{Berkaoui}. In
\cite{Deelstra}, the author extends the CIR model of the short interest
rate by assuming a stochastic reversion level, which better reflects
the time
dependence caused by the cyclical nature of the economy or by
expectations concerning the future impact of monetary policies. In this
framework, the convergence of the long-term return by using the theory
of generalized Bessel-square processes is studied. In \cite{Pearson},
the authors propose an empirical method that utilizes the conditional
density of the state variables to estimate and test a term structure
model with known price formula using data on both discount and coupon
bonds. The method is applied to an extension of a two-factor model due
to Cox, Ingersoll, and Ross. Their results show that estimates based
solely on bills imply unreasonably large price errors for longer
maturities. The process is also discussed in \cite{Brigo}.
In this article, we focus on the regime where the CIR process does not
hit zero and study weak approximation of this process. In the first
case, the sequence of prelimit markets is modeled as the sequence of
the discrete-time additive stochastic processes, whereas in the second
case, the sequence of multiplicative stochastic processes is modeled.
The additive scheme is widely used, for example, in the papers \cite
{Alfonsi,Bossy,Andreas}. The papers \cite{Deelstra,Pearson} are
recent examples of modeling a stochastic interest rate by the
multiplicative model of CIR process. In \cite{Deelstra}, the authors
say that the model has the ``strong convergence property,'' whereas
they refer to models as having the ``weak convergence property'' when
the returns converge to a constant, which generally depends upon the
current economic environment and that may change in a stochastic
fashion over time. We construct a discrete approximation scheme for the
price of asset that is modeled by the Cox--Ingersoll--Ross process. In
order to construct these additive and multiplicative processes, we take
the Euler approximations of the CIR process itself but replace the
increments of the Wiener process with iid bounded vanishing symmetric
random variables. We introduce a ``truncated'' CIR process and use it
to prove the weak convergence of asset prices.
The paper is organized as follows. In Section~\ref{section2}, we
present a complete and ``truncated'' CIR process and establish that the
``truncated'' CIR process can be described as the unique strong
solution to the corresponding stochastic differential equation. We
establish that this ``truncated'' process does not hit zero under the
same condition as for the original nontruncated process. In Section~\ref{section3}, we present discrete approximation schemes for both these
processes and prove the weak convergence of asset prices for the
additive model. In the next section, we prove the weak convergence of
asset prices for the multiplicative model. Appendix contains additional
and technical results.\looseness=1
\section{Original and ``truncated'' Cox--Ingersoll--Ross processes and
some of their\hfill\break properties}\label{section2}
Let $\Om_{\F}= (\Om,\F,(\F_t,t\ge0),\pr )$ be a complete
filtered probability space, and $W= \{W_t,\F_t,t\ge0 \}$ be an
adapted Wiener process. Consider a Cox--Ingersoll--Ross process with
constant parameters on this space. This process is described as the
unique strong solution of the following stochastic differential equation:
\begin{equation}
\label{CIR} dX_t=(b-X_t)dt+\si\sqrt{X_t}dW_t, \quad X_0=x_0>0, \; t \ge0,
\end{equation}
where $b>0$, $\si>0$.
The integral form of the process $X$ has the following form:
\begin{equation*}
\label{explicit_CIR} X_t=x_0+\int\limits
_0^t
(b-X_s )ds+\si\int\limits
_0^t\sqrt{X_s}dW_s.
\end{equation*}
According to the paper \cite{Cox_Ing_Ross}, the condition $\si^2\le2b$
is necessary and sufficient for the process $ X $ to get positive
values and not to hit zero. Further, we will assume that this condition
is satisfied.
For the proof of functional limit theorems, we will need a modification
of the Cox--Ingersoll--Ross process with bounded coefficients. This
process is called a truncated Cox--Ingerssol--Ross process.
Let $C>0$. Consider the following stochastic differential equation with
the same coefficients $b$ and $\si$ \xch{as in}{as in Eq.}~\eqref{CIR}:
\begin{equation}
\label{procc} dX_t^C=\bigl(b-X_t^C \wedge C\bigr)dt+\si\sqrt{\bigl(X_t^C\vee0\bigr) \wedge C}dW_t, \quad X_0=x_0 >0, \; t \ge0.
\end{equation}
\begin{lem}\label{lem1}
For any \xch{$C>0$,}{$C>0$, Eq.}~\eqref{procc} has a unique strong solution.
\end{lem}
\begin{proof}
Since the coefficients $\si(x)=\si\sqrt{(x\vee0)\wedge C}$ and
$b(x)=b-(x\wedge C)$ satisfy the conditions of Theorem~\ref{watanabe2}
and also the growth condition \eqref{gr_cond}, a~global strong solution
$X_t^C$ exists uniquely for every given initial value $x_0$.
\end{proof}
\begin{zau}
Denote $\si_{-\epsilon}=\inf{ \{t: X_t^C=-\epsilon \}}$ with
$\epsilon>0$ such that $-\epsilon+b>0$. Suppose that $\pr(\si_{-\epsilon
}<\infty)>0$. Then for any $r<\si_{-\epsilon}$ such that $X_t^C<0$ for
$t\in(r,\si_{-\epsilon})$, we would have, with positive probability,
\[
dX_t^C=\bigl(b-X_t^C\wedge C
\bigr)dt>0
\]
on the interval $(r,\si_{-\epsilon})$, and hence $t\rightarrow X_t^C$
would increase in this interval. This is obviously impossible.
Therefore, $X_t^C$ is nonnegative and can be written as
\begin{equation}
\label{procc1}
dX_t^C=\bigl(b-X_t^C \wedge C\bigr)dt+\si\sqrt{X_t^C\wedge C}dW_t, \quad X_0^C=x_0>0, \ t \ge0.
\end{equation}
\end{zau}\eject
The integral form of the process $X^C$ is as follows:
\begin{equation*}
\label{truncated_explicit_CIR} X_t^C=x_0+\int\limits
_0^t
\bigl(b-X_s^C\wedge C\bigr)ds+\si\int\limits
_0^t
\sqrt {X_s^C\wedge C}dW_s.
\end{equation*}
\begin{lem}\label{lem2}
Let $2b\ge\si^2$ and $C>b\vee1$. Then the trajectories of the process
$X^C$ are positive with probability 1.
\end{lem}
\begin{proof}
In order to prove that the process $X^C$ is positive, we will use the
proof similar to that given in \cite[p. 308]{Mao} for the complete
Cox--Ingersoll--Ross process with corresponding modifications. Note
that the coefficients $g(x):=\si\sqrt{x\wedge C}$ and $f(x):=b-x\wedge
C$ \xch{of}{of Eq.}~\eqref{procc1} are continuous and $g^2(x)>0$ on $x\in(0,\infty
)$. Fix $\alpha$ and $\beta$ such that $0<\alpha<x_0<\beta$. Due to the
nonsingularity of $g $ on $[\alpha,\beta]$, there exists a unique
solution $F(x)$ of the ordinary differential equation
\[
f(x)F'(x)+\frac{1}{2}g^2(x)F''(x)=-1, \quad \alpha<x<\beta,
\]
with boundary conditions $F(\alpha)=F(\beta)=0$, and this solution is
nonnegative, which follows from its representation through a
nonnegative Green function given in \cite[p.~343]{Karatzas}. Define the
stopping times
\[
\tau_\alpha=\inf \bigl\{t\ge0: X_t^C\le\alpha
\bigr\} \quad \text{and} \quad \tau_\beta=\inf \bigl\{t\ge0:
X_t^C\ge\beta \bigr\}.
\]
By the It\^o formula, for any $t>0$,
\begin{equation}
\label{equation1} \M F \bigl(X^C(t\wedge\tau_\alpha\wedge
\tau_\beta) \bigr)=F(x_0)-\M (t\wedge\tau_\alpha
\wedge\tau_\beta).
\end{equation}
This formula and nonnegativity of $ F $ imply that
\[
\M(t\wedge\tau_\alpha\wedge\tau_\beta)\le F(x_0)
\]
and, as $t\rightarrow\infty$,
\[
\M(\tau_\alpha\wedge\tau_\beta)\le F(x_0)<\infty.
\]
This means that $X^C$ exits from every compact subinterval of $[\alpha,
\beta]\subset(0,\infty)$ in finite time. It follows from the boundary
conditions and equality $\pr(\tau_\alpha\wedge\tau_\beta<\infty)=1$
that $\lim_{t\rightarrow\infty}\M F (X^C(t\wedge\tau_\alpha
\wedge\tau_\beta) )=0$, and then from \eqref{equation1} we have
\begin{equation*}
\M(\tau_\alpha\wedge\tau_\beta)=F(x_0).
\end{equation*}
Let us now define the function
\[
V(x)=\int\limits
_{1}^{x}\exp \biggl\{-\int\limits
_{1}^{y}
\frac
{2f(z)}{g^2(z)}dz \biggr\}dy, \quad x\in(0,\infty),
\]
which has a continuous strictly positive derivative $V'(x)$, and the
second derivative $V''(x)$ exists and satisfies
$V''(x)=-\frac{2f(x)}{g^2(x)}V'(x)$.
The It\^o formula shows that, for any $t>0$,\vadjust{\eject}
\[
V \bigl(X^C(t\wedge\tau_\alpha\wedge\tau_\beta)
\bigr) =V(x_0)+\int\limits
_{0}^{t\wedge\tau_\alpha\wedge\tau_\beta}V'
\bigl(X^C_u \bigr)g \bigl(X^C_u
\bigr)dW(u)
\]
and
\[
\M V \bigl(X^C(t\wedge\tau_\alpha\wedge\tau_\beta)
\bigr) =V(x_0).
\]
Taking the limit as $t\rightarrow\infty$, we get\begingroup\abovedisplayskip=12pt\belowdisplayskip=12pt
\[
V(x_0)=\M V\bigl(X^C(\tau_\alpha\wedge
\tau_\beta)\bigr)=V(\alpha)\pr(\tau_\alpha <\tau_\beta)+V(
\beta)\pr(\tau_\beta<\tau_\alpha),
\]
and hence
\begin{equation}
\label{equation2} \pr(\tau_\alpha<\tau_\beta)=\frac{V(\beta)-V(x_0)}{V(\beta)-V(\alpha)}
\quad \text{and} \quad \pr(\tau_\beta<\tau_\alpha)=\frac{V(x_0)-V(\alpha
)}{V(\beta)-V(\alpha)}.
\end{equation}
Consider the integral
\begin{equation*}
\begin{split} \label{compute} V(x)&=\int\limits
_{1}^{x}\exp
\biggl\{-\int\limits
_{1}^{y}\frac
{2(b-z\wedge C)}{\si^2(z\wedge C)}dz \biggr\}dy. \end{split}
\end{equation*}
First, consider the case $x<1 $. Then
\begin{equation*}
\begin{split} \label{compute1} V(x)&=\int\limits
_{1}^{x}\exp
\biggl\{-\int\limits
_{1}^{y}\frac{2(b-z)}{
\si^2z}dz \biggr\}dy=\int\limits
_{1}^{x}y^{-\frac{2b}{\si^2}}
\exp \biggl\{ \frac{2(y-1)}{\si^2} \biggr\}dy, \end{split}
\end{equation*}
and if $\si^2\le2b$, then
\[
\lim\limits
_{x\downarrow0}V(x)=-\infty.
\]
Now let $x$ increase and tend to infinity. Denote
$C_1=\int_{1}^{C}\exp \{\frac{2(y-1)}{\si^2} \}
y^{-\frac{2b}{\si^2}}dy$. Then, for $x>C$,\endgroup
\begin{align*}
\label{compute2}
V(x)&=\int\limits_{1}^{C}\exp \biggl\{-\int\limits_{1}^{y}\frac{2(b-z)}{\si^2z}dz \biggr\}dy\\
&\quad +\int\limits_{C}^{x}\exp \biggl\{-\int\limits_{1}^{C}\frac{2(b-z)}{\si^2z}dz-\int\limits_{C}^{y}\frac{2(b-C)}{\si^2C}dz \biggr\}dy\\
&=\int\limits_{1}^{C}\exp \biggl\{\frac{2(y-1)}{\si^2} \biggr\}y^{-\frac{2b}{\si^2}}dy +C^{-\frac{2b}{\si^2}}\exp{ \biggl\{\frac{2(C-1)}{\si^2} \biggr\}}\\
&\quad \times\int\limits_{C}^{x}\exp \biggl\{-\frac{2(b-C)}{\si^2C}(y-C)\biggr\} dy\\
&=C_1+C^{-\frac{2b}{\si^2}+1}\frac{\si^2}{2(C-b)} \exp \biggl\{\frac{2(C-1)}{\si^2} \biggr\}\\
&\quad \times \biggl(\exp \biggl\{\frac{2(C-b)}{\si^2C}(x-C) \biggr\}-1 \biggr),
\end{align*}
and thus $\lim_{x\uparrow\infty}V(x)=\infty$.
Define
\[
\tau_0=\lim\limits_{\alpha\downarrow0}\tau_\alpha \quad \text{and} \quad \tau_\infty=\lim\limits_{\beta\uparrow\infty}\tau_\beta
\]
and put $\tau=\tau_0\wedge\tau_\infty$. From \eqref{equation2} we get
\[
\pr \Bigl(\inf\limits
_{0\le t <\tau}X_t^C\le\alpha \Bigr)\ge\pr(\tau
_\alpha<\tau_\beta)=\frac{1-V(x_0)/V(\beta)}{1-V(\alpha)/V(\beta)},
\]
and, as $\beta\uparrow\infty$, we get that, for any $\alpha>0$,
$\pr (\inf_{0\le t <\tau}X_t^C\le\alpha )=1$,
whence, finally,
$\pr (\inf_{0\le t <\tau}X_t^C=0 )=1$.
Similarly,
$\pr (\sup_{0\le t <\tau}X_t^C=\infty )=1$.
Assume now that $\pr(\tau<\infty)>0$. Then
\[
\pr \Bigl(\lim\limits_{t\rightarrow\tau}X_t^C \ \text{exists and equals}\ 0 \ \text{or}\ \infty \Bigr)>0.
\]
So the events $ \{\inf_{0\le t <\tau}X_t^C=0 \}$ and
$ \{\sup_{0\le t <\tau}X_t^C=\infty \}$ cannot both
have probability 1. This contradiction shows that $\pr(\tau<\infty)=0$, whence
\[
\pr(\tau=\infty)=\pr \Bigl(\inf\limits
_{0\le t <\tau}X_t^C=0 \Bigr)=
\pr \Bigl(\sup\limits
_{0\le t <\tau}X_t^C=\infty \Bigr)=1
\]
if $2b\ge\si^2$.
\end{proof}
Now, let $T>0$ be fixed.
\begin{lem}
\[
\pr\bigl\{\exists\, t\in[0,T]: X_t\neq X_t^C
\bigr\}\rightarrow0
\]
as $C\rightarrow\infty$.
\end{lem}
\begin{proof}
Obviously, it suffices to show
that
\[
\pr \Bigl\{\sup\limits
_{t\in[0,T]}|X_t|\ge C \Bigr\}\rightarrow0 \quad \text{as} \; C\rightarrow\infty.
\]
It is well known (see, e.g., \cite{Zhu}) that $\frac{4}{\si
^2(1-e^{-t})}X_t$ follows a noncentral $\chi^2$ distribution with (in
general) noninteger degree of freedom $\frac{4b}{\si^2}$ and
noncentrality parameter $\frac{4}{\si^2(1-e^{-t})}x_0 e^{-t}$. The
first and second moments for any $t\ge0$ are given by
\[
\M X_t=x_0e^{-t}+b\bigl(1-e^{-t}
\bigr),
\]
\[
\M(X_t)^2=x_0 \bigl(2b+\si^2
\bigr)e^{-t}+ \bigl(x_0^2-x_0\si
^2-2x_0b \bigr)e^{-2t}+ \biggl(
\frac{b\si^2}{2}+b^2 \biggr) \bigl(1-e^{-t}
\bigr)^2.
\]\eject
\noindent Therefore, there exists a constant $B>0$ such that $\M X_t^2\le B$,
whence $\M X_t\le B^{1/2}$, $0\le t\le T$.
Using the Doob inequality, we estimate
\begin{align*}
&\pr \Bigl\{\sup\limits_{t\in[0,T]}|X_t|\ge C \Bigr\}\le\frac{1}{C^2}\M \sup\limits_{t\in[0,T]}X_t^2\\
&\quad =\frac{1}{C^2}\M\sup\limits_{t\in[0,T]} \biggl\{ \biggl(X_0+\int\limits_0^t (b-X_s )ds+\si\int\limits_0^t\sqrt{X_s}dW_s \biggr)^2 \biggr\}\\
&\quad \le\frac{3}{C^2} \biggl\{X_0^2+T\M \biggl(\int\limits_0^T\llvert b-X_s\rrvert ds \biggr)^2+\si^2\M\sup\limits_{t\in[0,T]} \biggl(\int\limits_0^t\sqrt {X_s}dW_s \biggr)^2 \biggr\}\\
&\quad \le\frac{3}{C^2} \biggl\{X_0^2+T\M\int\limits_0^T (b-X_s )^2ds+4\si^2\M\int\limits_0^TX_sds \biggr\}\le\frac{B_1}{C^2}
\end{align*}
for some constant $B_1>0$.
The lemma is proved.
\end{proof}
\section{Discrete approximation schemes for complete and ``truncated''\hfill\break Cox--Ingersoll--Ross processes}\label{section3}
Consider the following discrete approximation scheme for the process
$X$. Assume that we have a sequence of the probability spaces $(\Om
^{(n)},\F^{(n)},\pr^{(n)})$, $n\ge1$. Let $\{q_k^{(n)}, n\ge1$, $0\le
k \le n\}$ be the sequence of symmetric iid random variables defined on
the corresponding probability space and taking values~ $\pm\sqrt{\frac
{T}{n}}$, that is, $\pr^n (q_k^{(n)}=\pm\sqrt{\frac{T}{n}}
)=\frac{1}{2}$. Let further $n>T$.
We construct discrete approximation schemes for the stochastic
processes $X$ and $X^C$ as follows. Consider the following
approximation for the complete process:
\begin{align}
\label{rec_sch_CIR_E}
&X_0^{(n)}=x_0>0, \qquad X_{k}^{(n)} = X_{k-1}^{(n)}+\frac{(b-X_{k-1}^{(n)})T}{n}+\si q_k^{(n)}\sqrt{X^{(n)}_{k-1}},\nonumber\\
&Q_k^{(n)}:=X_k^{(n)}-X_{k-1}^{(n)} = \frac{(b-X_{k-1}^{(n)})T}{n}+\si q_k^{(n)}\sqrt{X^{(n)}_{k-1}}, \quad 1\le k \le n,
\end{align}
and the corresponding approximations for $X^C$ given by
\begin{align}
X_0^{(n,C)}&=x_{0}>0,\nonumber\\
X_k^{(n,C)}&=X_{k-1}^{(n,C)}+\frac{(b-(X_{k-1}^{(n,C)}\wedge C))T}{n}+\si q_k^{(n)}\sqrt{X^{(n,C)}_{k-1}\wedge C},\nonumber\\
Q_k^{(n,C)}:&=X_k^{(n,C)}-X_{k-1}^{(n,C)}\nonumber\\
&=\frac{(b-(X_{k-1}^{(n,C)}\wedge C))T}{n}+\si q_k^{(n)}\sqrt{X^{(n,C)}_{k-1}\wedge C}, 1\le k \le n.\label{rec_sch_E}
\end{align}
The following lemma confirms the correctness of the construction of
these approximations.
\begin{lem}
Let, $n>2T$.
\begin{itemize}
\item[$1)$] If $2b\geq\sigma^2$, then all values given \xch{by}{by Eqs.}~\eqref
{rec_sch_CIR_E} and \eqref{rec_sch_E} are positive.
\item[$2)$] We have
\begin{equation}
\label{eq11} \pr \bigl\{\exists k, 0\le k\le n: \; X_k^{(n)}
\neq X_k^{(n,C)} \bigr\} \rightarrow0
\end{equation}
as $C\rightarrow\infty$.
\end{itemize}
\end{lem}
\begin{proof}
$1)$ We apply the method of mathematical induction. When $k=1$,
\[
X_1^{(n)}=x_{0}+\frac{(b-x_{0})T}{n}+\si
q_1^{(n)}\sqrt{x_{0}}.
\]
Let us show that
\begin{equation}
\label{rec_sch_E11}x_{0}+\frac{(b-x_{0})T}{n}+\si q_1^{(n)}
\sqrt{x_{0}}>0.
\end{equation}
We denote $\alpha:=\sqrt{x_0}$ and reduce \eqref{rec_sch_E11} to the
quadratic inequality
\[
\alpha^2 \biggl(1-\frac{T}{n} \biggr)\pm\si\sqrt{
\frac{T}{n}}\alpha+\frac
{bT}{n}>0,
\]
which obviously holds because the discriminant $D=\frac{\si^2T}{n}-\frac
{4bT}{n} (1-\frac{T}{n} )<0$ when $\si^2\le2b$ and $n>2T $.
So, $X_1^{(n)}>0$. Assume now that $X_{k}^{(n)}>0$. It can be shown by
applying the same transformation that when $\si^2\le2b$ and $n>2T $,
the values $X_{k+1}^{(n)}>0$.
It can be proved similarly that the values given by \eqref{rec_sch_E}
are positive.
2) $X_k^{(n)}$ can be represented as
\begin{align}
X_k^{(n)}&=x_0+\sum\limits_{i=1}^k\bigl(b-X_{i-1}^{(n)}\bigr)\frac{T}{n}+\si\sum\limits_{i=1}^k q_i^{(n)}\sqrt{X_{i-1}^{(n)}}\nonumber\\
&=X_{k-1}^{(n)}+\frac{(b-X_{k-1}^{(n)})T}{n}+\si q_k^{(n)}\sqrt{X^{(n)}_{k-1}}. \label{eq_X_k^n}
\end{align}
Compute
\begin{align}
\M\bigl(X_{i}^{(n)} \bigr)^2 & =\M \biggl(X_{i-1}^{(n)} \biggl(1- \frac{T}{n} \biggr)+\frac {bT}{n}+\si\sqrt{X_{i-1}^{(n)}}q_i^{(n)} \biggr)^2\nonumber\\
& =\M \biggl(X_{i-1}^{(n)} \biggl(1-\frac{T}{n} \biggr)+ \frac{bT}{n} \biggr)^2+\frac{\si^2T}{n}\M X_{i-1}^{(n)} \nonumber\\
& = \biggl(\frac{bT}{n} \biggr)^2+ \biggl[\frac{\si^2T}{n}+ \frac{2bT}{n} \biggl(1-\frac{T}{n} \biggr) \biggr]\M X_{i-1}^{(n)} \nonumber\\
& \quad + \biggl(1-\frac{T}{n} \biggr)^2\M \bigl(X_{i-1}^{(n)} \bigr)^2.\label{EXkn2}
\end{align}
Assume that $\M (X_{j}^{(n)} )^2\le\beta^2$, $1\le j \le i-1$,
for some $\beta>0$. Then $\M X_{j}^{(n)}\le\beta$, $1\le j \le i-1$. We
get the quadratic inequality of the form
\[
\biggl(1-\frac{T}{n} \biggr)^2\beta^2+ \biggl[
\frac{\si^2T}{n}+\frac
{2bT}{n} \biggl(1-\frac{T}{n} \biggr) \biggr]
\beta+ \biggl(\frac{bT}{n} \biggr)^2<\beta^2
\]
or, equivalently,
\[
\biggl( \biggl(1-\frac{T}{n} \biggr)^2-1 \biggr)
\beta^2+ \biggl[\frac{\si
^2T}{n}+\frac{2bT}{n} \biggl(1-
\frac{T}{n} \biggr) \biggr]\beta+ \biggl(\frac
{bT}{n}
\biggr)^2<0,
\]
which obviously holds when $\beta>\frac{\si^2+2b+\sqrt{\si^4+4b\si
^2+8b^2}}{\frac{3}{2}}$. So, for all $1\le i \le n$,
$\M X_{i}^{(n)}\le\frac{\si^2+2b+\sqrt{\si^4+4b\si^2+8b^2}}{\frac
{3}{2}} \vee x_0=:\gamma$.
Using the Burkholder inequality, we estimate
\begin{align*}
0\le\M\sup\limits_{0\le k \le n} \bigl(X_k^{(n)} \bigr)^2&\le2 (x_0+bT )^2+ 2\si^2\M\sup\limits_{0\le k \le n}\Biggl(\sum\limits_{i=1}^{n} q_{i}^{(n)}\sqrt{X_{i-1}^{(n)}} \Biggr)^2\\
&\le2 (x_0+bT )^2+ 8\si^2\M \Biggl(\sum\limits_{i=1}^{n} q_{i}^{(n)}\sqrt{X_{i-1}^{(n)}} \Biggr)^2\\
&\le2 (x_0+bT )^2+8\sigma^2\gamma T.
\end{align*}
Therefore,
\begin{align*}
\pr \bigl\{\exists k, 0\le k \le n: X_k^{(n)}\neq X_k^{(n,C)} \bigr\}&=\pr \Bigl\{\sup_{0\le k \le n}X_k^{(n)}\geq C\Bigr\}\leq C^{-2}\M\sup_{0\le k \le n}\bigl(X^{(n)}_k\bigr)^2\\
&\leq 2C^{-2}(x_0+bT)^2+8\sigma^2C^{-2}\gamma T,
\end{align*}
whence the proof follows.
\end{proof}
Consider the sequences of step processes corresponding to these schemes:
\begin{equation*}
\label{eq_disproc} X^{(n)}_t=X^{(n)}_{k} \;
\text{for } \frac{kT}{n} \le t < \frac{(k+1)T}{n}
\end{equation*}
and
\begin{equation*}
\label{eq_disproc1} X^{ (n,C) }_t=X^{(n,C)}_{k} \;
\text{for } \frac{kT}{n} \le t < \frac
{(k+1)T}{n}.
\end{equation*}
Thus, the trajectories of the processes $X^{(n)}$ and $X^{(n,C)}$ have
jumps at the points
$kT/n\;,k=0,\ldots, n$, and are constant on the interior intervals.
Consider the filtrations $\F_k^{n}=\sigma(X^{(n)}_t,\, t\le\frac
{kT}{n} )$. The processes $X^{(n,C)}$ are adapted with respect to them.
Therefore, we can consider the same filtrations for all discrete
approximation schemes. So, we can identify $\F^n_t$ with $\F_k^{n}$ for
$\frac{kT}{n} \le t < \frac{(k+1)T}{n}$.
\begin{zau}\label{zau3.1}
Now we can rewrite relation \eqref{eq11} as follows:
\[
\pr \bigl\{\exists t, t\in[0,T]: X_t^{(n)}\neq
X_t^{(n,C)} \bigr\} \rightarrow0
\]
as $C\rightarrow\infty$.
\end{zau}
Denote by $\mathbb{Q}$ and $\mathbb{Q}^n, n\ge1$, the measures
corresponding to the processes
$X$ and $X^{(n)}, n\ge1$, respectively, and by $\mathbb{Q}^C$ and
$\mathbb{Q}^{n,C}, n\ge1$, the measures corresponding to the processes
$X^C$ and $X^{(n,C)}, n\ge1$, respectively. Denote by $\stackrel
{W}{\longrightarrow}$ the weak convergence of measures corresponding to
stochastic processes. We apply Theorem~3.2 from \cite{Mishura3} to
prove the weak convergence of measures $\mathbb{Q}^{n,C}$ to the
measure $\mathbb{Q}^C$. This theorem can be formulated as follows.
\begin{thm}
\label{conditions_E}
\begingroup
\abovedisplayskip=6pt
\belowdisplayskip=6pt
Assume that the following conditions are satisfied:
\begin{itemize}
\item[\rm(i)] For any $\epsilon>0$,
\[
\lim\limits
_{n}\pr \Bigl(\sup\limits
_{1\le k \le n}\big|Q_k^{(n,C)}\big|
\ge \epsilon \Bigr)=0;
\]
\item[\rm(ii)] For any $\epsilon>0$ and $a\in (0,1] $,
\begin{align*}
&\lim\limits_{n}\pr^n \biggl(\sup\limits_{t\in\T}\biggl\llvert\sum\limits_{1\le k\le [\frac{nt}{T} ]}\M \bigl(Q_k^{(n,C)}\I_{|Q_k^{(n,C)}|\le a} \big\rrvert \F_{k-1}^n \bigr)\\
&\qquad -\int\limits_{0}^{t}\bigl(b-X_s^{(n,C)}\wedge C\bigr)ds\biggr\rrvert \ge \epsilon \biggr)=0;
\end{align*}
\item[\rm(iii)] For any $\epsilon>0$ and $a\in (0,1] $,
\begin{align*}
&\lim\limits_{n}\pr^n \biggl(\sup\limits_{t\in\T}\biggl\llvert\sum\limits_{1\le k\le [ \frac{nt}{T} ]} \Bigl(\M \bigl( \bigl(Q_k^{(n,C)}\bigr)^2\I_{|Q_k^{(n,C)}|\le a} \big\rrvert \F_{k-1}^n\bigr) \\
&\qquad - \bigl(\M \bigl(Q_k^{(n,C)}\I_{|Q_k^{(n,C)}|\le a} \big\rrvert\F_{k-1}^n \bigr) \bigr)^2 \Bigr)-\si^2\int\limits_{0}^{t}\bigl(X_s^{(n,C)}\wedge C\bigr) ds\biggr\rrvert \ge\epsilon \biggr)=0;
\end{align*}
\end{itemize}
\endgroup
Then
$\mathbb{Q}^{n,C}\stackrel{W}{\longrightarrow}\mathbb{Q}^C$.
\end{thm}
Using Theorem~\ref{conditions_E}, we prove the following result.
\begin{thm}
\label{weak_conv_E}
$\mathbb{Q}^{n,C}\stackrel{W}{\longrightarrow}\mathbb{Q}^C$.
\end{thm}
\begin{proof}
According to Theorem~\ref{conditions_E}, we need to check conditions
(i)--(iii). Relation \eqref{rec_sch_E} implies that
$\sup_{0\le k \le n}{|Q_k^{(n,C)}|}\le\frac{b+CT}{n}+\si\sqrt
{\frac{TC}{n}}$. Hence, there exists a~constant $C_2>0$ such that $\sup_{0\le k\le n}{|Q_k^{(n,C)}|}\le\frac{C_2}{\sqrt{n}}$. This
means that condition~(i) is satisfied.
Furthermore, in order to establish (ii), we consider any fixed $a>0$
and $n\geq1$ such that $\frac{C_2}{\sqrt{n}}\le a$, that is,
$n\ge (\frac{C_2}{a} )^{2}$. For such $n$,
\begin{align}
\M \bigl(Q_k^{(n,C)}\I_{|Q_k^{(n,C)}|\le a} \big\rrvert \F_{k-1}^n \bigr)&=\M\bigl(Q_k^{(n,C)} \big\rrvert \F_{k-1}^n\bigr)\nonumber\\
&=\frac{(b-(X_{k-1}^{(n,C)}\wedge C))T}{n}+\si\M q_k^{(n)}\sqrt{X^{(n,C)}_{k-1}\wedge C}\nonumber\\
&=\frac{(b-(X_{k-1}^{(n,C)}\wedge C))T}{n}.\label{exp_Q}
\end{align}
For any $\epsilon>0$, we have\vspace*{-9pt}
\begin{align*}
&\lim\limits_{n}\pr^n \biggl(\sup\limits_{t\in\T}\biggl\llvert\sum\limits_{1\le k\le [\frac{nt}{T} ]}\M \bigl(Q_k^{(n,C)}\I_{|Q_k^{(n,C)}|\le a} \bigr\rrvert \F_{k-1}^n \bigr)-\int\limits_{0}^{t}\bigl(b-\bigl(X_s^{(n,C)}\wedge C\bigr)\bigr)ds\biggr\rrvert \ge\epsilon \biggr)\\
&\quad =\lim\limits_{n}\pr^n \biggl(\sup\limits_{t\in\T}\biggl\llvert\sum\limits_{1\le k\le [\frac{nt}{T} ]}\frac{(b-(X_{k-1}^{(n,C)}\wedge C))T}{n}-\sum\limits_{0\le k\le [\frac{nt}{T}]-1}\bigl(b-\bigl(X_{k}^{(n,C)}\wedge C\bigr)\bigr)\frac{T}{n}\\
&\qquad -\bigl(b-\bigl(X_{ [\frac{nt}{T} ]}^{(n,C)}\wedge C\bigr)\bigr) \biggl(t-\frac{ [\frac{nt}{T} ]T}{n} \biggr)\biggr\rrvert \ge\epsilon \biggr)\\
&\quad =\lim\limits_{n}\pr^n \biggl(\sup\limits_{t\in\T}\biggl\llvert\bigl(b-\bigl(X_{ [\frac{nt}{T} ]}^{(n,C)}\wedge C\bigr)\bigr) \biggl(t-\frac{ [\frac{nt}{T}]T}{n} \biggr)\biggr\rrvert \ge\epsilon \biggr)=0,
\end{align*}
and hence condition (ii) is satisfied. Now let us check condition
(iii). We have
\begin{align*}
&\M \bigl( \bigl(Q_k^{(n,C)} \bigr)^2\I_{|Q_k^{(n,C)}|\le a} \rrvert \F _{k-1}^n \bigr)=\M \bigl(\bigl(Q_k^{(n,C)} \bigr)^2 \rrvert \F_{k-1}^n \bigr)\\
&\quad = \biggl(\frac{(b-(X_{k-1}^{(n,C)}\wedge C))T}{n} \biggr)^2+2\frac{(b-(X_{k-1}^{(n,C)}\wedge C))T}{n}\si\M q_k^{(n)}\sqrt {X^{(n,C)}_{k-1}\wedge C}\\
&\qquad +\si^2 \M\bigl(q_k^{(n)}\bigr)^2\bigl(X^{(n,C)}_{k-1}\wedge C \bigr)\\
&\quad = \biggl(\frac{(b-(X_{k-1}^{(n,C)}\wedge C))T}{n}\biggr)^2{+}\,\si^2 \frac{T}{n} \bigl(X^{(n,C)}_{k-1}\wedge C \bigr).
\end{align*}
Therefore, for any $\epsilon>0$,
\begin{align*}
&\lim\limits_{n}\pr^n \biggl(\sup\limits_{t\in\T}\biggl\llvert\sum\limits_{1\le k\le [ \frac{nt}{T} ]} \Bigl(\M \bigl( \bigl(Q_k^{(n,C)}\bigr)^2\I_{|Q_k^{(n,C)}|\le a} \bigr\rrvert \F_{k-1}^n\bigr) \\
&\qquad - \bigl(\M \bigl(Q_k^{(n,C)}\I_{|Q_k^{(n,C)}|\le a} \big\rrvert \F_{k-1}^n \bigr) \bigr)^2 \Bigr)-\si^2\int\limits_{0}^{t}\bigl(X_s^{(n,C)}\wedge C\bigr) ds\biggr\rrvert \ge\epsilon \biggr)\\
&\quad =\lim\limits_{n}\pr^n \biggl(\sup\limits_{t\in\T}\biggl\llvert\sum\limits_{1\le k\le [ \frac{nt}{T} ]} \biggl( \biggl(\frac{(b-(X_{k-1}^{(n,C)}\wedge C) )T}{n}\biggr)^2+\si^2\frac{T}{n} \bigl(X_{k-1}^{(n,C)}\wedge C \bigr)\\
&\qquad - \biggl(\frac{ (b-(X_{k-1}^{(n,C)}\wedge C))T}{n} \biggr)^2 \biggr)-\sum\limits_{0\le k\le [ \frac{nt}{T}]-1} \biggl(\si^2 \frac{T}{n}\bigl(X_{k}^{(n,C)}\wedge C \bigr) \biggr)\\
&\qquad -\si^2 \bigl(X_{ [\frac{nt}{T} ]}^{(n,C)}\wedge C \bigr)\biggl(t-\frac{ [\frac{nt}{T} ]T}{n} \biggr)\biggr\rrvert \ge \epsilon \biggr)\\
&\quad =\lim\limits_{n}\pr^{n} \biggl(\sup\limits_{t\in\T} \biggl(\si^2 \bigl(X_{ [\frac{nt}{T} ]}^{(n,C)}\wedge C \bigr) \biggl(t-\frac{[\frac{nt}{T} ]T}{n} \biggr) \biggr)\ge\epsilon \biggr)=0.
\end{align*}
The theorem is proved.
\end{proof}
\begin{thm}\label{weak_con}
$\mathbb{Q}^n\stackrel{W}{\longrightarrow}\mathbb{Q}$, $n\rightarrow
\infty$.
\end{thm}
\begin{proof}
According to Theorem~\ref{bill} and Theorem~\ref{weak_conv_E}, it
suffices to prove that
\[
\lim\limits
_{C\rightarrow\infty}\overline{\lim\limits
_{n\rightarrow
\infty}}\pr \Bigl\{\sup\limits
_{0\le t\le T}
\bigl\llvert X_t^{(n)}- X_t^{(n,C)}\bigr
\rrvert \ge\epsilon \Bigr\}=0.
\]
However, due to Remark~\ref{zau3.1},
\begin{align*}
&\lim\limits_{C\rightarrow\infty}\overline{\lim\limits_{n\rightarrow\infty}}\pr \Bigl\{\sup\limits_{0\le t\le T}\bigl\llvert X_t^{(n)}- X_t^{(n,C)}\bigr\rrvert \ge\epsilon \Bigr\}\\
&\quad \le\lim\limits_{C\rightarrow\infty}\overline{\lim\limits_{n\rightarrow\infty}}\pr \bigl\{\exists t, t\in[0,T]: X_t^{(n)}\neq X_t^{(n,C)} \bigr\}=0.\qedhere
\end{align*}
\end{proof}
\section{Multiplicative scheme for Cox--Ingersoll--Ross process}\label{section4}
In this section, we construct a multiplicative discrete approximation
scheme for the process $e^{X_t}$, $t\in[0,T]$, where $X_t$ is the CIR
process given \xch{by}{by Eq.}~\eqref{explicit_CIR}. We construct the following
multiplicative process based on the discrete approximation scheme \mbox{\eqref
{rec_sch_CIR_E}--\eqref{rec_sch_E}}. We introduce limit and prelimit
processes as follows:
\begin{flalign*}
&S_t^{n,C}=\exp \{x_0 \}\prod\limits_{1\le k \le [\frac{tn}{T} ]} \bigl(1+Q_k^{(n,C)} \bigr), \quad t\in\mathbb{T},&\\
&S_t^C=\exp{ \biggl\{X_t^C-\frac{\si^2}{2}\int\limits_0^t\bigl(X_t^C\wedge C\bigr)dt \biggr\}}, \quad t\in\mathbb{T},&\\
&S_t^{n}=\exp \{x_0 \}\prod\limits_{1\le k \le [\frac{tn}{T} ]} \bigl(1+Q_k^{(n)} \bigr), \quad t\in \mathbb{T},&\\
&S_t=\exp{ \biggl\{X_t-\frac{\si^2}{2}\int\limits_0^t X_t dt \biggr\}}, \quad t\in\mathbb{T},&\\
&\widetilde{S}_t^{n}=\exp \{x_0 \}\prod\limits_{1\le k \le[\frac{tn}{T} ]} \biggl[ \bigl(1+Q_k^{(n,C)} \bigr)\exp\biggl\{\frac{\si^2}{2n}X_k^{(n)} \biggr\} \biggr], \quad t \in\mathbb{T},&\\
\mbox{and}\qquad \qquad &&\\
&\widetilde{S}_t=\exp \{X_t \}, \quad t\in\mathbb{T}.&
\end{flalign*}
Denote by $\G^{C}$, $\G^{n,C}$, $\G$, $\G^{n}$, $\widetilde{\G}$, and
$\widetilde{\G}^{n}$, $n\ge1$, the measures corresponding to the
processes $S_t^{C}$, $S_t^{n,C}$, $S_t$, $S_t^{n}$, $\widetilde{S}_t$,
and $\widetilde{S}_t^{n}$, $n\ge1$, respectively.
We apply Theorem~3.3 from \cite{Mishura3} to prove the weak convergence
of measures. This theorem can be formulated as follows.
\begin{thm}
\label{conditions}
Let the following conditions hold:
\begin{itemize}
\item[\rm(i)] $\sup_{1\le k \le n}|Q_k^{(n,C)}|\stackrel{\pr
}{\longrightarrow}0,\; n\rightarrow\infty$;
\item[\rm(ii)] For any $a\in (0,1] $,
\[
\lim\limits
_{D\rightarrow\infty}\overline{\lim\limits
_{n\rightarrow
\infty}}\pr^n \biggl(\sum
\limits
_{1\le k\le n}\M \bigl( \bigl(Q_k^{(n,C)}
\bigr)^2\I_{|Q_k^{(n,C)}|\le a} \big\rrvert \F _{k-1}^n
\bigr)\ge D \biggr)=0;
\]
\item[\rm(iii)] For any $a\in (0,1] $,
\[
\lim\limits
_{D\rightarrow\infty}\overline{\lim\limits
_{n\rightarrow
\infty}}\pr^n \biggl(\sum
\limits
_{1\le k\le n}\bigl\llvert \M \bigl(Q_k^{(n,C)}\I
_{|Q_k^{(n,C)}|\le a} \bigr\rrvert \F_{k-1}^n \bigr)\big\rrvert \ge D
\biggr)=0;
\]
\item[\rm(iv)] For any $\epsilon>0$ and $a\in (0,1] $,
\begin{align*}
&\lim\limits_{n}\pr^n \biggl(\sup\limits_{t\in\T}\biggl\llvert\sum\limits_{1\le k\le [\frac{nt}{T} ]}\M \bigl(Q_k^{(n,C)}\I_{|Q_k^{(n,C)}|\le a} \bigr\rrvert \F_{k-1}^n \bigr)\\
&\qquad -\int\limits_{0}^{t}\bigl(b-X_s^{(n,C)}\wedge C\bigr)ds\biggr\rrvert \ge \epsilon \biggr)=0;
\end{align*}
\item[\rm(v)] For any $\epsilon>0$ and $a\in (0,1] $,
\begin{align*}
&\lim\limits_{n}\pr^n \biggl(\sup\limits_{t\in\T}\biggl\llvert\sum\limits_{1\le k\le [ \frac{nt}{T} ]}\M \bigl( \bigl(Q_k^{(n,C)}\bigr)^2\I _{|Q_k^{(n,C)}|\le a} \bigr\rrvert \F_{k-1}^n\bigr)\\
&\qquad -\si^2\int\limits_{0}^{t}\bigl(X_s^{(n,C)}\wedge C\bigr) ds\biggr\rrvert \ge \epsilon \biggr)=0.
\end{align*}
\end{itemize}
Then
\[
\G^{n,C}\stackrel{W} {\longrightarrow}\G^C.
\]
\end{thm}
We prove the following result using Theorem~\ref{conditions}.
\begin{thm}
\label{weak_conv}
$\G^{n,C}\stackrel{W}{\longrightarrow}\G^C$.
\end{thm}
\begin{proof}
According to Theorem~\ref{conditions}, we need to check
conditions (i)--(v). It was established in the proof of Theorem~\ref
{weak_conv_E} that conditions (i) and (iv) are satisfied. Let us show
that condition (ii) holds. It was also established in the proof of
Theorem~\ref{weak_conv_E} that $\sup_{0\le k\le
n}{|Q_k^{(n,C)}|}\le\frac{C_2}{\sqrt{n}}$. So, for all $a\in
(0,1] $, starting from some number $n$, we have
\begin{equation*}
\begin{split}
&\sum\limits_{1\le k\le n}\M \bigl(\bigl(Q_k^{(n,C)} \bigr)^2\I _{|Q_k^{(n,C)}|\le a}\big\rrvert \F_{k-1}^n \bigr)\\
&\quad =\sum\limits_{1\le k\le n}\M \bigl( \bigl(Q_k^{(n,C)}\bigr)^2 \big\rrvert \F_{k-1}^n \bigr)\le\sum\limits_{1\le k\le n}\frac{C_2^2}{n} \le C_2^2,
\end{split}
\end{equation*}
whence condition (ii) holds. Now, \eqref{exp_Q} implies that, for all
$a\in (0,1] $, starting from some number $n$, we have
\[
\bigl\llvert \M \bigl(Q_k^{(n,C)}\I_{|Q_k^{(n,C)}|\le a} \bigr
\rrvert \F _{k-1}^n \bigr)\big\rrvert =\bigl\llvert \M
\bigl(Q_k^{(n,C)} \bigr\rrvert \F _{k-1}^n
\bigr)\big\rrvert \le\frac{C_3}{n},
\]
whence condition (iii) holds.
Let us check condition (v). For any $\epsilon>0$ and $a\in
.(0,1] $, we have
\begin{align*}
&\lim\limits_{n}\pr^n \biggl(\sup\limits_{t\in\T}\biggl\llvert\sum\limits_{1\le k\le [ \frac{nt}{T} ]}\M \bigl( \bigl(Q_k^{(n,C)}\bigr)^2\I _{|Q_k^{(n,C)}|\le a} \bigr\rrvert \F_{k-1}^n\bigr)-\si^2\int\limits_{0}^{t}\bigl(X_s^{(n,C)}\wedge C\bigr) ds\biggr\rrvert \ge\epsilon \biggr)\\
&\quad =\lim\limits_{n}\pr^n \biggl(\sup\limits_{t\in\T}\biggl\llvert\sum\limits_{1\le k\le [ \frac{nt}{T} ]} \biggl( \biggl(\frac{(b-(X_{k-1}^{(n,C)}\wedge C) )T}{n}\biggr)^2+\si^2\frac{T}{n} \bigl(X_{k-1}^{(n,C)}\wedge C \bigr) \biggr)\\
&\qquad - \sum\limits_{0\le k\le [ \frac{nt}{T} ]-1} \biggl(\si^2 \frac{T}{n}\bigl(X_{k}^{(n,C)}\wedge C \bigr) \biggr)-\si^2\bigl(X_{ [\frac{nt}{T} ]}^{(n,C)}\wedge C \bigr) \biggl(t-\frac{[\frac{nt}{T} ]T}{n}\biggr)\biggr\rrvert \ge\epsilon \biggr)\\
&\quad \le\lim\limits_{n}\pr^{n} \biggl(\sup\limits_{t\in\T} \biggl(\frac{(|b|+C)^2Tt}{n}+\si^2 \bigl(X_{ [\frac{nt}{T} ]}^{(n,C)}\wedge C\bigr) \biggl(t-\frac{ [\frac{nt}{T} ]T}{n} \biggr) \biggr)\ge \epsilon \biggr)=0.
\end{align*}
The theorem is proved.
\end{proof}
\begin{thm}
$\G^n\stackrel{W}{\longrightarrow}\G, \ n\rightarrow\infty$.
\end{thm}
\begin{proof}
The proof immediately follows from Theorem~\ref{bill}, Theorem~\ref{weak_conv}, and Remark~\ref{zau3.1}. Indeed,
\begin{align*}
&\lim\limits_{C\rightarrow\infty}\overline{\lim\limits_{n\rightarrow\infty}}\pr \Bigl\{\sup\limits_{0\le t\le T}\bigl\llvert X_t^{(n)}- X_t^{(n,C)}\bigr\rrvert \ge\epsilon \Bigr\}\\
&\quad \le\lim\limits_{C\rightarrow\infty}\overline{\lim\limits_{n\rightarrow\infty}}\pr \bigl\{\exists t, t\in[0,T]: X_t^{(n)}\neq X_t^{(n,C)} \bigr\}=0.\qedhere
\end{align*}
\end{proof}
\begin{zau}
The weak convergence
\[
\widetilde{\G}^n\stackrel{W} {\longrightarrow}\widetilde{\G}, \quad n \rightarrow\infty,
\]
can be proved in a similar way.
\end{zau}
|
1,941,325,220,505 | arxiv | \section{Introduction}
Recent observations \cite{castao_metastable_2003,kunz_field_2009,klaui_head-to-head_2008}
in thin ferromagnetic stripes show magnetization configurations in
which the magnetization makes a full $2\pi$ turn in a localized region
of the stripe, while the rest of the stripe is magnetized parallel
to the edges of the stripe. These structures are stable against small
applied external magnetic fields which makes them potentially useful
for information storage devices\cite{muratov_bit_2009}. The same phenomenon
has been observed in thin ferromagnetic annuli\cite{castao_metastable_2003,muratov_bit_2009}, which then results in
the existence of a hierarchy of equally spaced metastable states\cite{chaves-oflynn_micromagnetic_2009}.
Reliable control of domain wall structures is crucial in the successful
design of magnetic nanodevices\cite{allwood_magnetic_2005,parkin_magnetic_2008}.
Here we explore the stability of $2\pi$ domain walls in ferromagnetic
nanorings using the string method \cite{e_string,e_energy_2003,e_simplified_2007}.
We find the energy barrier $\Delta E$ separating two metastable configurations. This barrier determines the escape rate from a metastable state through the Arrhenius law,
to leading order $e^{-\frac{\Delta E}{k_B T}}$\cite{hnggi_reaction-rate_1990}.
The string method also gives the minimum energy path and transition
state.
We distinguish two types of domain walls by their winding number in
the global (local) coordinate system $\omega$ ($\Omega$). We compare
the energy barriers that separates each type of wall from the ground
state to the activation energy that separates the two vortex configurations
\cite{martens_magnetic_2006,chaves-oflynn_micromagnetic_2009}.
A current flowing along the axis of the ring produces a circumferential
field. The application of this field has two effects.
First, the degeneracy of the two vortex states is lifted. Second,
the field determines the width of the domain wall; in its absence
the $2\pi$ walls could dissociate into two independent $\pi$ transverse
walls.
\section{Method}
Following previous work \cite{martens_magnetic_2006,chaves-oflynn_micromagnetic_2009},
we study a permalloy ring with the following dimensions and material properties: outer
radius $R_{2}=220$ nm; inner radius $R_{1}=180$ nm, thickness $t=2$
nm, magnetization saturation $M_{s}=8\times10^{5}A/m$, and exchange
length given by $l_{\mathrm{ex}}=\sqrt{\frac{2A}{\mu_{0}M_{s}^{2}}}=5.6$
nm. A current flowing along the axis of the ring produces a field $\mathbf{H}(\mathbf{r})=(hH_{c}(R_{1}+R_{2})/2r)\hat{\theta}$
A/m. Here $h=H/H_c$ and the characteristic
field strength at midradius is $\mu_{0}H_{c}=73.9$ mT (for $H>H_{c}$, the clockwise vortex
state is no longer stable). The calculations were performed at $h$=0.1.
Precessional effects do not modify the location of the critical points
in the energy landscape: the exponential factor in the Arrhenius formula
is unaltered if we ignore them. We consider the overdamped case so that
the escape trajectory follows
the negative gradient of the energy. This is done by integrating only
the damping term of the Landau-Lifshitz-Gilbert equation
\begin{equation}
\frac{d\mathbf{M}}{dt}=-\frac{|\gamma|\alpha}{M_{s}}\mathbf{M}\times(\mathbf{M}\times\mathbf{H_{\mathrm{eff}}}).\end{equation}
Here $\alpha=1$ is the damping coefficient, $\gamma$ is the gyromagnetic
constant, and $\mathbf{H_{eff}}=-\nabla_{\mathbf{M}}E/\mu_{0}$ is
the effective magnetic field. The total micromagnetic energy $E$
is the sum of the exchange $E_{\mathrm{ex}}$, Zeeman $E_{\mathrm{Z}}$
and magnetostatic terms $E_{\mathrm{mag}}$.
The string method is necessary to calculate the minimum energy path between two
stable states ($\mathbf{M_{A}},\mathbf{M_{B}}$) when there is no a-priori knowledge of the transition state.
In practice, the path is discretized in N+1 images between $\mathbf{M_{A}}$ and
$\mathbf{M_{B}}$ denoted as $\mathbf{M}_{i}(t)\equiv\mathbf{M}_{i}(\mathbf{r},t)$ with
$i=0,...,N$. The images are updated using a two-step iteration procedure as follows: First, each image evolves using the publicly available
micromagnetic code OOMMF\cite{oommf} until the time reaches some
interval $\Delta t$ which we have selected to be 10 ps. This gives
a sequence of configurations:
\begin{equation}
\mathbf{M'}_{i}\equiv\mathbf{M'}_{i}(\mathbf{r})=\mathbf{M}_{i}(t)+\int_{t}^{t+\Delta t}\frac{d\mathbf{M}_{i}(t')}{dt}dt'\end{equation} Once all the $\mathbf{M'}_{i}(\mathbf{r})$ have been obtained, the
second step in the string method is a reparametrization step used to keep these images equidistant. First the complete
arc $s_{N}$ length of the trajectory is calculated by
\begin{equation}
s_{0}=0,s_{i}=s_{i-1}+|\mathbf{M'}_{i}-\mathbf{M'}_{i-1}|.\end{equation} The arc lengths are renormalized using $\alpha'_{i}=s_{i}/s_{N}$.
Finally we do a simple linear interpolation for all $i$ along the
trajectory so that \begin{equation}
\mathbf{M}_{i}(t+\Delta t)=\mathbf{M'}_{j(i)}+\frac{\mathbf{M'}_{j(i)+1}-\mathbf{M'}_{j(i)}}{\alpha'_{j(i)+1}-\alpha'_{j(i)}}(\frac{i}{N}-\alpha'_{j})\label{eq:interpolation}\end{equation} where $j(i)$ is the index of the string where $\alpha'_{j+1}\geq i/N\geq\alpha'_{j(i)}$.
During each step we observe the magnetic energy $E_{i}(t)$=$E_{\mathrm{ex}}(\mathbf{M}_{i}(\mathbf{r},t))+E_{\mathrm{Z}}(\mathbf{M}_{i}(\mathbf{r},t))+E_{\mathrm{mag}}(\mathbf{M}_{i}(\mathbf{r},t))$
as indicator of how far from convergence the current step is. The
iteration process is stopped when there is no visible change in the
function $E_{i}(t)$.
\section{Annihilation of $2\pi$ domain wall.}
We now present the results of the string method to find the minimum
energy path for destruction of a $2\pi$ wall for the two types of
$2\pi$ domain wall, $\Omega=\pm1$. Fig. \ref{fig:annihilationof2piwall}
\begin{figure}
\includegraphics[width=2.5in,angle=-90]{FIG1}
\caption{\label{fig:annihilationof2piwall}Energy barrier for annihilation
of $2\pi$ domain wall under an external field $h=0.1$. Fig. \ref{fig:annihilationof2piwalls} shows the configurations for images i=0,50,100.}
\end{figure}
and Fig. \ref{fig:annihilationof2piwalls} present the
string energies and configurations after relaxation of the string.
The $\Omega=+1$, $2\pi$ wall decays into the counterclockwise configuration
by the expulsion of a vortex from the inner hole. On the other hand,
the $\Omega=-1$, $2\pi$ wall decays into the counterclockwise configuration
by the expulsion of an antivortex. This observation shows a
correspondence between a topological defect crossing the stripe and
the signature of the $2\pi$ walls being annihilated.
\begin{figure}
\includegraphics[width=3in]{FIG2}\smallskip{}
\includegraphics[width=3in]{FIG3}
\caption{\label{fig:annihilationof2piwalls}Segment of the ring encompassing each $2\pi$ domain wall. Minimum energy path for the annihilation
of $2\pi$ domain walls in consideration. (Above) $2\pi$ wall with
topological index $\Omega=-1$, (below) $2\pi$ wall with topological
index $\Omega=+1$. The configurations shown correspond to the following images in the string (left) i=0, (center) i=50, (right) i=100.}
\end{figure}
For comparison purposes we provide magnitudes of the energy landscape
of ferromagnetic nanorings obtained with the string method. The lowest
energy barrier between the counterclockwise and the clockwise vortex
configurations passes through a configuration denoted as the instanton saddle \cite{martens_magnetic_2006}(with $\Omega=0$);
the activation energy of this event is $3.0\times10^{-19}$ J. This is
consistent with our previous work on nanorings \cite{chaves-oflynn_micromagnetic_2009}.
For a $2\pi$ wall with $\Omega=1$ the decay into the ground
state has an energy barrier equal to $1.9\times10^{-19}$ J. The annihilation of
a $2\pi$ wall with index $\Omega=-1$ has an energy barrier equal
to $2.8\times10^{-19}$ J. This shows the energy barrier to annihilate a $2\pi$
domain wall by the expulsion of a topological defect is comparable
to that of reversal between vortex states by a instanton fluctuation.
\section{Discussion}
For annuli with the dimensions being considered the magnetization
is constrained to lie in the plane of the ring; for the stable states
the magnetization can be considered to be independent of the radial
coordinate. The exchange energy $E_{\mathrm{ex}}$ of a stable state
is given by \cite{martens_magnetic_2006}:
\begin{equation}
E_{\mathrm{ex}}=\frac{\mu_{0}M_{s}^{2}tl_{\mathbf{\mathrm{ex}}}^{2}}{2}\ln\left(\frac{R_{2}}{R_{1}}\right)\left[2\pi(1+2\Omega)+\int_{0}^{2\pi}\left(\frac{\partial\phi}{\partial\theta}\right)^{2}d\theta\right]\label{eq:exchange}\end{equation}
where $\phi(\theta)$ is the angle that the magnetization makes with
the tangent of the ring at a given angle $\theta$; $\Omega$ is the
``winding number'' of the magnetization with respect to the local
coordinate system.
The difference in winding numbers ($\Delta\Omega$) of the $2\pi$
walls considered results in an exchange energy difference $\Delta E_{\mathrm{\Omega}}$
between these two states. Using Eq. (\ref{eq:exchange}) the difference
can been shown to be approximately:
\begin{equation}
\Delta E_{\mathrm{\Omega}}[\mathbf{M}(\mathbf{r})]\approx2\pi\mu_{0}M_{s}^{2}tl_{\mathbf{\mathrm{ex}}}^{2}\ln\left(\frac{R_{2}}{R_{1}}\right)\Delta\Omega=1.298\times10^{-19}\ \mathrm{J}.\label{eq:exchangedif}\end{equation}
The total exchange energy difference between these $2\pi$ walls
($\Omega=\pm1$) obtained from the micromagnetic simulation results
is $1.357\times10^{-19}\mathrm{J}$. Here we have ignored a contribution to this
difference of terms of the form $\int\left(\frac{\partial\phi}{\partial\theta}\right)^{2}d\theta$
since it is not a topological term. This shows a very good agreement
between an estimate obtained from the 1D model and the full numerical
simulation. The exchange energy term is the biggest contributor to
the difference between the total energies of the two domain walls:
the numerical values from the demagnetization and Zeeman energy are
10 times smaller. The main point is that most of the energy difference
between these two types of $2\pi$ domain wall is the result of their
respective topological windings. It is worth noting that this is a curvature
effect as can be seen from Eq. (\ref{eq:exchangedif}): the energy difference
tends to zero in the limit when the radii approaches infinity --i.e. in the limit of a straight ferromagnetic strip.
We now consider the question of how to experimentally produce these two types of
$2\pi$ walls. We use the information contained in their global topological
number $\omega$ and compare it to other known states of nanorings.
In particular, the well-known "onion" state has $\omega=0$. Since
the onion is the remanent magnetization after saturation by an inplane
uniform field, one can produce the $\Omega=-1$ wall by applying a
strong field in-plane followed by a circumferential field. The two walls will approach and form
a $2\pi,\Omega=-1$ wall. Changing the direction of either the in-plane
field or the circumferential field will only change the final position
of the $2\pi$ wall, not its topological index.
The $2\pi,\omega=2$ domain wall cannot be produced using only uniform
and circumferential fields. However, we propose the following technique
to produce that configuration in nanorings: apply a strong dipolar
field ($\omega=2$) in the interior of the ring. This could be produced
by a small current loop with its axis coplanar to the structure, or
by bringing a magnetic tip close to the device. If this dipolar field
is strong enough, two transverse walls would be produced at opposite
sides of the ring; the magnetization vector at the centers of the
wall will point in the opposite direction of the overall magnetization
of the ring. Activating the circumferential field as the dipole strength
is decreased will result in the desired configuration.
One final question is whether resistance measurements can distinguish
the difference between the two types of $2\pi$ structures reported
in this work. For instance, one could attempt to use anisotropic magnetoresistance
effect to read the overall winding number of the configuration.
An estimate of this effect can be obtained by integrating $\mathbf{J}\cdot\mathbf{M}$
along a certain segment of the ring that spans the whole $2\pi$ domain
wall. Since $\mathbf{J}$ runs along $\hat{\theta}$ the AMR would
be proportional to $\langle M_{\theta}/M_{s}\rangle$ which can be
directly calculated from the $2\pi$ domain structure. For two electrodes
located at the top and bottom of the segment shown in Fig. \ref{fig:annihilationof2piwalls}
these values are 0.042 and 0.016 for the $\Omega=+1$ and $\Omega=-1$
walls respectively. It therefore should be possible to apply a current to probe the winding
number of the $2\pi$ domain.
\section{Conclusion}
We have presented results on the thermal annihilation of $2\pi$ domain
walls. We differentiated between two types of $2\pi$ walls through
their winding number in curved nanowires. We have observed a simple
arithmetical relation between the topological index of the different
configurations and the processes by which each structure decays into
the ground state. The fact that the energy difference between the
two states is dominated by the exchange energy allows to identify
the states through their winding number. The transition path requires
the motion of a singularity through the bulk: an antivortex destroys
$\Omega=-1$ walls, and a vortex annihilates $\Omega=+1$ walls. Similar
behavior is expected to work in linear stripes. Topological
defects are known to play a role in certain types of phase transitions;
here we have determined the mechanism by which they destroy $2\pi$
wall structures.
The two types of $2\pi$ domain wall correspond to distinct metastable
states: the greatest contribution to the energy difference comes from
the exchange energy difference. The energy can be directly associated
with the topological signature of the magnetization configuration.
The stability of these two states is comparable to the stability of
the clockwise configuration. We suggest to use AMR noise
measurements as a way to verify the presence of the two types of wall
in magnetic nanowires and study their stability. Thermal fluctuations should
generate both types of wall which could be recognized as two separate
values of the AMR.
Further micromagnetic exploration of the $2\pi$ wall annihilation
problem presented here can be done moving away from the overdamped
regime by using a nonzero precessional term. The results presented
here will be interesting to compare to this case. While we have presented results on a 2nm
thick ring, we expect the observed transition states in thicker rings.
The key parameter is the ratio of the thickness to the mean radius
which should be less than approximately 0.1 \cite{martens_magnetic_2006}.
We plan to explore the energy barriers and transition states as a function of thickness and ring radii.
\section{acknowledgments}
This research was supported by NSF Grants No. NSF-DMR-0706522, NSF-PHY-0651077, NSF-DMS-0718172,
NSF-DMS-0708140, ONR grant ONR-N00014-04-1-6046 and the NYU Dean's Dissertation Fellowship.
|
1,941,325,220,506 | arxiv | \section{Introduction}
Applications using information on the user location are rapidly spreading, also to ensure that some services are obtained only in pre-determined areas.
In order to establish the user position we can rely on the device itself, requested to report the position provided by its GPS module. However, tempering with the GPS module or its software interface is relatively easy \cite{ceccato2018exploiting}. Thus, more reliable solutions must be explored. Location verification systems aim at verifying the position of devices \cite{Zeng-survey, 8376254}, possibly using distance measures obtained, for example, through the \ac{rss} at anchor nodes for signals transmitted by the terminal under verification. This problem is closely related to {\em user authentication} at the physical layer, where wireless channel features are exploited to verify the sender of a message \cite{7270404}.
We focus here on \ac{irlv}, the problem of deciding whether a message coming from a terminal over a wireless network has been originated from a specific physical region, e.g., a safe room, or not \cite{Zeng-survey}. \ac{irlv} can be seen as an hypothesis testing problem between two alternatives, namely being inside or outside the specific region. Among proposed solutions, we recall distance bounding techniques with rapid exchanges of packets between the verifier and the prover \cite{Brands}, also using radio-frequency and ultrasound signals \cite{Sastry}, and solutions based on anchor nodes and increasing transmit power by the sender \cite{Vora}. More recently, a delay-based verification technique has been proposed in \cite{7145434}, leveraging geometric properties of triangles, which prevent an adversary from manipulating measured delays.
In this paper, we consider the \ac{irlv} problem for a \ac{ue} connected to a set of network \acp{bs}. The decision on the user position is taken on the basis of observed features of the channel over which communication occurs. For exemplary purposes we focus here on the observation of the attenuation of the channels between the \ac{ue} and the \acp{bs}. We propose a \ac{ml} approach where i) channel measurements are collected by trusted nodes both inside and outside the \ac{roi}, ii) a machine is trained to take decisions between the two hypotheses, iii) the machine is exploited to take decisions on the unknown \acp{ue} in the exploitation phase. \ac{ml} techniques have already found application in user authentication (see \cite{xiao-2018} and references therein), however never in \ac{irlv}, to the best of authors' knowledge. The \ac{nn} training is based on the \ac{ce}, and, framing \ac{irlv} problem into an hypothesis testing problem, we establish the optimality of this criterion according to the \ac{np} lemma, in asymptotic conditions.
Then, we address the problem of optimum positioning of the \acp{bs} (network planning) for \ac{irlv}. Two metrics are considered for this optimization: a) the \ac{ce} of \ac{nn} training, and b) the \ac{auc} of the \ac{roc} of the hypothesis test. While \ac{ce} is directly related to the \ac{nn} training, the \ac{auc} is more connected to the final performance that we expect from \ac{irlv}. A limited number of neurons, as well as a limited size of the training set may provide different results for the two metrics. For the optimization of \acp{bs} position we propose a two-stage \ac{pso} algorithm, minimizing either the \ac{ce} or the \ac{roc} \ac{auc}. Simulation results over channels with shadow fading show the merits of the proposed solution, and its effectiveness in providing reliable \ac{irlv}.
\section{System Model}\label{sec:sys model}
We consider a cellular system with $N_{\rm AP}$ \acp{bs} covering a region $\mathcal{A}$ over a plane. We propose a \ac{irlv} system able to determine if a \ac{ue} is transmitting from inside an {\em authorized} sub-region $\mathcal{A}_0 \subset \mathcal{A}$. The dependency on location of the \ac{ue}-\acp{bs} channels is exploited to distinguish between transmissions from inside and outside $\mathcal{A}_0$. Transmissions are assumed to be narrowband and the channel feature used for \ac{irlv} is its attenuation.
The \ac{irlv} procedure comprises two phases. In the first phase, named identification or training, a trusted \ac{ue} transmits a training signal (known at the \acs{bs}) from various points inside region $\mathcal{A}_0$. The \acp{bs} estimate the channel attenuations and store them in association with $\mathcal{A}_0$. Some external authentication technique on the transmitted packet ensures that the received signal upon which the attenuation is estimated is actually transmitted by the trusted \ac{ue}. Similarly, attenuation values are collected when the trusted \ac{ue} transmits from the complementary region $\mathcal{A}_1$ and stored by the \acp{bs} in association to $\mathcal{A}_1 = \mathcal{A} \setminus \mathcal{A}_0$. In the second phase, named verification or exploitation, the \ac{ue} transmits a known training sequence from any point in $\mathcal{A}$ and the \ac{irlv} system must decide whether the \ac{ue} is in region $\mathcal{A}_0$ or $\mathcal{A}_1$.
\subsection{Channel Model}
Let $\bm{x}_{\rm ap}^{(n)} =(X_{\rm ap}^{(n)},Y_{\rm ap}^{(n)})$ be the position of the $n$-th \ac{bs}. For a \ac{ue} located at $\bm{x}_{\rm ue}=(X_u,Y_u)$, its distance from \ac{bs} $n$ is denoted as $L(\bm{x}_{\rm ue},\bm{x}_{\rm ap}^{(n)})$. We assume that the \ac{ue} transmits with constant power so that \ac{bs} $n$ can estimate the attenuation $a{(n)}$ incurred over the channel, including the effects of path-loss and shadowing. Let $\bm a = [a(1), a(2), \ldots, a(N_{\rm AP})]$ collect attenuation values from all \ac{bs}.
Denoting the path-loss coefficient as $a_{\rm PL}{(n)}$ the shadowing component is log-normally distributed, i.e., $\left(a_{\rm S}{(n)}\right)_{\rm dB} \sim \mathcal{N}(0,\sigma_s^2)$, and we have
\begin{equation}
\left(a{(n)}\right)_{\rm dB} = \left(a_{\rm PL}{(n)}\right)_{\rm dB} + \left(a_s{(n)}\right)_{\rm dB}.
\end{equation}
The channel model for path-loss and shadowing is derived from \cite{3gpp}.
For a \ac{los} link the path loss coefficient in dB is modelled as
\begin{equation}\label{eq:los}
a_{\rm PL-LOS}{(n)} = 20\log_{10}\left(\frac{f_0 4\pi L(\bm{x}_{\rm UE},\bm{x}_{\rm AP}^{(n)})}{c}\right),
\end{equation}
where $f_0$ is the carrier frequency, and $c$ is the speed of light. For a non-\ac{los} link the path loss coefficient in dB is defined as
\begin{equation}
\begin{split}
&a_{\rm PL-LOS}{(n)} = 40\left(1-4 \cdot 10^{-3} \left. h_{\rm AP} \right|_{\rm m} \right) \times \\
&\log_{10}\left (\left.\frac{L(\bm{x}_{\rm UE},\bm{x}_{\rm AP}^{(n)})}{10^3}\right|_{\rm m}\right ) -18\log_{10}\left.h_{\rm AP}\right|_{\rm m} +\\
&
+ 21\log_{10}\left(\frac{\left.f_0\right|_{\rm MHz}}{10^6}\right) + 80,
\end{split}
\end{equation}
where $\left.h_{\rm AP}\right|_{\rm m}$ is the \ac{bs} antenna elevation in meters, $\left. f_0 \right|_{\rm MHz}$ is the carrier frequency in MHz, and $\left. L(\bm{x}_{\rm UE},\bm{x}_{\rm AP}^{(n)})\right|$ is the \ac{ue}-\ac{bs} $n$ distance in meters.
We assume that correlation between shadowing coefficients $\left( a_{\rm S}(i)\right)_{\rm dB}$ and $\left( a_{\rm S}(j)\right)_{\rm dB}$ for two \acp{bs} located in $\bm{x}_i$ and $\bm{x}_j$ when the \ac{ue} is transmitting is
\begin{equation}\label{eq: coor mat}
\E{\bm x}{a_{\rm S}(i)a_{\rm S}(j)} = \sigma_s^2\exp \left({-\frac{L(\bm{x}_i,\bm{x}_j)}{d_c}}\right),
\end{equation}
where $d_c$ is the shadowing decorrelation distance and $\E{\bm x}{\cdot}$ is the expected value with respect to the distribution of $\bm x$.
\section{In-region Location Verification}\label{sec: ml}
Let us define the two hypotheses of the \ac{irlv} problem as
\begin{equation}
\mathcal H_0: \mbox{the \ac{ue} is in $\mathcal{A}_0$}, \quad \mathcal H_1: \mbox{the \ac{ue} is in $\mathcal{A}_1$}.
\end{equation}
In the training phase the \ac{ue} transmits from $S$ locations. For transmission $i=1,\ldots,S$, let $\bm{a}^{(i)}=[a^{(i)}(1),\ldots,a^{(i)}(N_{\rm AP})]$
be the vector of measured attenuations. We associate to $\bm a ^{(i)}$ the label $t_i=0$ if \ac{ue} is transmitting from inside $\mathcal{A}_0$ (hypothesis $\mathcal{H}_0$), and $t_i=1$ otherwise. Let also define $\mathcal{T} =
\{\bm{a}^{(1)}, \ldots , \bm{a}^{(S)} \}$.
By using these attenuation training vectors and labels, we aim at building a function
\begin{equation}
\hat{t} = g(\bm{a}) \in \{0,1\}\,
\end{equation}
that maps any attenuation vector $\bm{a}$ into a decision on the location of the \ac{ue}. We would like to have $\hat{t}=0$ if $\bm a$ was obtained when the \ac{ue} was inside $\mathcal{A}_0$ and $\hat{t}=1$ otherwise.
The performance of the \ac{irlv} system is assessed in terms of two error probabilities: the \ac{fa} probability, i.e., the probability that a \ac{ue} in $\mathcal A_0$ is declared outside this region, and the \ac{md} probability, i.e., the probability that a \ac{ue} outside $\mathcal A_0$ is declared inside the region. In formulas, denoting with $\pr{\cdot}$ the probability function,
\begin{equation}
P_{\rm FA} =\pr{\hat{t} =1 | \mathcal H_0}, \quad
P_{\rm MD}=\pr{\hat{t} = 0 | \mathcal H_1}.
\end{equation}
\subsection{Test for Known Attenuation Statistics}
The \ac{irlv} problem can be seen as an hypothesis testing problem between the two hypotheses $\mathcal H_0$ and $\mathcal H_1$. When the statistics of the attenuation vectors are known under the two hypotheses, the most powerful test for the \ac{irlv} problem is provided by the \ac{np} lemma. In particular, let us define the \ac{llr}
\begin{equation}\label{eq:lr}
\mathcal{L}{(\bm a)}=\log\left(\frac{p_{\bm a}(\bm a|\mathcal{H}_0)}{p_{\bm a}(\bm a|\mathcal{H}_1)}\right)\,,
\end{equation}
where $p_{\bm a|\mathcal{H}}(\bm a|\mathcal{H}_i)$ is the \ac{pdf} of the random vector $\bm a$ modelling all attenuation values $\bm a$, given that hypothesis $\mathcal H_i$ is verified, and $\log$ denotes the base-2 logarithm. The \ac{np} test function is
\begin{equation}
\label{eq:thrOpt}
\hat{t} = g(\bm a) = \begin{cases}
0 & \mathcal{L}{(\bm a)} \geq \theta\,, \\
1 & \mathcal{L}{(\bm a)} < \theta\,,
\end{cases}
\end{equation}
where $\theta$ is a threshold to be chosen in order to ensure the desired \ac{fa} probability. This test ensures that for the given \ac{fa} probability the \ac{md} probability is minimized.
\subsection{Example of \ac{np} Test}
\label{sec:los}
We now describe an example of application of the \ac{np} test, where we can easily obtain a close-form expression for $f(\bm a)$.
Let us define the overall network region as a circle $\mathcal{A}_c$ with radius $R_{\rm out}$ and consider a single \ac{bs} located at the center of $\mathcal{A}_c$. Consider $\mathcal{A}_{0}$ as a rectangle with nearest point to the center of $\mathcal{A}_c$ at a distance $R_{\rm min}$. The outside region is $\mathcal{A}_1 = \mathcal{A}_c \setminus \mathcal{A}_0$. In the \ac{los} scenario the scalar attenuation $a$ incurred by a \ac{ue} is given by path loss, which only depends on its relative distance to the \ac{bs}. Considering an attenuation value $a$, the \ac{ue}-\ac{bs} distance is given by
\begin{equation}
R = \frac{c a}{ 4 \pi f_0}.
\end{equation}
Therefore, instead of considering $p_{a|\mathcal{H}}(a|\mathcal H_i)$ we consider $p_{R|\mathcal{H}}(r|\mathcal H_i)$, where distance $R$ corresponds to attenuation $a$. We first derive the \ac{cdf} of $R$ in $\mathcal{A}_0$, i.e., the probability that the \ac{ue} is located in $\mathcal{A}_0$ at a distance $R\le r$ from the \ac{bs}. This is
\begin{equation}\label{eq:cdf}
\pr{R \le r|\mathcal{H}_0} = \frac{1}{|\mathcal{A}_0|}\int_{R_{\rm min}}^{r} \rho \alpha(\rho) d\rho,
\end{equation}
where $\alpha(R)$ denotes the angle of the circular sector measured from a distance $R$ and intersecting region $\mathcal{A}_0$ and $|\mathcal{A}_0|$ is the area of region $\mathcal{A}_0$. Then by taking the derivative of the \ac{cdf} (\ref{eq:cdf}) with respect to $r$ we obtain the \ac{pdf}
\begin{equation}\label{eq:num}
p_{R|\mathcal{H}}(r|\mathcal{H}_0) = \frac{1}{|\mathcal{A}_0|}r\alpha(r).
\end{equation}
Following the same reasoning and considering that the length of the circular sector with radius $r$ located in $\mathcal{A}_1$ is $2\pi - \alpha(r)$, we obtain the \ac{pdf} of transmission from a distance $r$ in $\mathcal{A}_1$ as
\begin{equation}\label{eq:den}
p_{R|\mathcal{H}}(r|\mathcal{H}_1) = \frac{1}{|\mathcal{A}_1|}r\left(2\pi-\alpha(r)\right).
\end{equation}
From (\ref{eq:num}) and (\ref{eq:den}) we obtain the \ac{llr} as a function of the \ac{ue}'s distance from the \ac{bs} as
\begin{equation}
\mathcal{L}{(a)}=\log\left[\frac{|\mathcal{A}_1|\alpha \left(\frac{c a}{f_0 4 \pi}\right)}{|\mathcal{A}_0|\left(2\pi-\alpha\left(\frac{c a}{f_0 4 \pi}\right)\right)}\right].
\end{equation}
\subsection{Neural Network Implementation}\label{sec:nn}
Under more complicated scenarios, it becomes hard to obtain close-form expressions for the \ac{llr}. Therefore, we reosrt to a \ac{ml} approach, using a \ac{nn} trained with attenuation vectors $\bm{a}^{(i)}$ and labels $t_i \in \{0,1\}$. In the verification phase the trained \ac{nn} is used on the test attenuation vectors $\bm a$ to provide the decision $\hat{t} \in \{0,1\}$. Now, $g(\cdot)$ is the function implemented by the \ac{nn}.
We now provide a short description of a \ac{nn}. A feed-forward \ac{nn} processes the input in stages, named layers, where the output of one layer is the input of the next layer. The input of the \ac{nn} is $\bm{y}^{(0)} = \bm{a}$, and layer $\ell-1$ has $N^{(\ell-1)}$ outputs obtained by processing the inputs with $N^{(\ell-1)}$ scalar functions named neurons. The output of the $n$-th neuron of the $\ell$-th layer is
\begin{equation}\label{eq:nonLin}
y_n^{(\ell)} = \sigma\left( \bm{w}_n^{(\ell -1)}\bm{y}^{(\ell-1)}+b_n^{(\ell)} \right),
\end{equation}
where $\bm{w}_n^{(\ell -1)}$ and $b_n^{(\ell)}$ are coefficients to be determined in the training phase, and $\sigma(\cdot)$ is the sigmoid activation function.
The last layer comprises only one neuron, $y^{(L)}$, and the final output of the \ac{nn} is the scalar
\begin{equation}
\tilde{t}(\bm a) \triangleq \sigma(y^{(L)}),
\end{equation}
where $L$ is the total number of layers. Finally, the test function is obtained by thresholding $\tilde{t}(\bm{a})$, i.e.,
\begin{equation}
\label{eq:decNN}
g(\bm{a}) = \begin{cases}
1 & \tilde{t}(\bm a) > \lambda \\
0 & \tilde{t}(\bm a) \leq \lambda.
\end{cases}
\end{equation}
By varying $\lambda$ we obtain different values of $P_{\rm FA}$ and $P_{\rm MD}$ for this \ac{irlv} test.
Various options have been proposed in the literature for \ac{nn} training. We consider here as objective function the empirical \ac{ce} between the \ac{nn} output and the labels $t_i$, defined as
\begin{equation}\label{eq:ce}
\begin{split}
\hatcross{p_{\mathcal{H}|\bm a}}{g} \triangleq& -\frac{1}{S} \sum_{i=1}^{S}\left[t_i\log \tilde{t}(\bm a^{(i)}) + \right.\\
&\left. +\left(1-t_i\right)\log\left(1-\tilde{t}(\bm a^{(i)})\right)\right].
\end{split}
\end{equation}
Training is performed with the gradient descent algorithm minimizing $\hatcross{p_{\mathcal{H}|\bm a}}{g}$.
In the following we show that a \ac{ce} based \ac{nn} is equivalent, in probability and for perfect training, to the \ac{np} solution. First, we prove that the output of the \ac{nn} can be interpreted as the class conditional probability.
\begin{theorem}
Let $\gy \in [0,1]$ be the output of a \ac{nn} obtained with perfect training, i.e., with infinite number of training points, layers and neurons. Let the training be performed with the \ac{ce} metric. Then
\begin{equation}
\gy = p_{\mathcal H|\bm a}(\mathcal{H}_0|\bm a)
\end{equation}
almost surely.
\end{theorem}
\begin{proof}
See Appendix A.
\end{proof}
Note that this approach does not require the knowledge of the statistics of $\bm{a}$ under the two hypotheses, while, instead, it requires a large enough set of training points to converge.
However, at convergence, the \ac{nn} achieves the same performance of the NP approach. This holds since \eqref{eq:decNN} is equivalent to \eqref{eq:thrOpt}, i.e., they provide the same \ac{roc}, with
\begin{align}
\label{eq:relation}
\tilde{t} &= p_{\mathcal{H}|\bm a}(\mathcal{H}_0|\bm a), \quad
\theta = \frac{1-\lambda}{\lambda} \frac{p_\mathcal{H}(\mathcal{H}_0)}{p_{\mathcal{H}}(\mathcal{H}_1)},
\end{align}
where \eqref{eq:relation} is a direct consequence of the Bayes rule applied as follows
\begin{equation}
p_{\mathcal{H}|\bm a}(\mathcal{H}_0| \bm a) = \left[1+ \frac{p_{\mathcal{H}}(\mathcal{H}_1)}{p_{\mathcal{H}}(\mathcal{H}_0)} 2^{\mathcal{L}(\bm a)}\right]^{-1}.
\end{equation}
\section{Network Planning}\label{sec:bsPos}
As the attenuation depends on the position of the \acp{bs} and on the surrounding environment, the performance of the authentication system depends on the number of \acp{bs} and on their location. In this Section, we derive an approach to optimally locate \acp{bs} (\emph{network planning}) so that the authentication system attains the best performance.
For \acp{bs} positioning we consider as performance metric a suitable trade-off between \ac{fa} and \ac{md} probabilities. In particular, the \ac{roc} curve associates the $P_{\rm MD}$ with the corresponding $P_{\rm FA}$, for all possible values of thresholds $\lambda$. However, as we aim at using a single performance measure without setting a priori $P_{\rm FA}$, we resort to the \ac{roc} \ac{auc} \cite{hanley-82}, defined as
\begin{equation}
C(\{\bm{x}_{\rm AP}^{(n)}\}) = \int_{0}^{1} P_{\rm MD}\left(P_{\rm FA}\right) d P_{\rm FA},
\label{defAUC}
\end{equation}
where $P_{\rm MD}\left(P_{\rm FA}\right)$ is the $P_{\rm MD}$ value as a function of $P_{\rm FA}$. In (\ref{defAUC}) we have highlighted the dependency of the AUC on the \ac{bs} positions. Note that $C(\{\bm{x}_{\rm AP}^{(n)}\}) $ is the integral of the \ac{roc} function. Therefore, the \ac{bs} position optimization aims at minimizing the \ac{auc}, i.e.
\begin{equation}
{\rm argmin}_{\{\bm{x}_{\rm AP}^{(n)}\}} C(\{\bm{x}_{\rm AP}^{(n)}\}).
\label{defoptpos}
\end{equation}
Note that minimizing the \ac{auc} is equivalent to minimizing the average $P_{\rm MD}$ under the assumption of a uniformly distributed $P_{\rm FA}$.
In practice, in order to compute the \ac{auc} we must run the \ac{nn} over the training set multiple times, with different thresholds and find the corresponding \ac{roc} curve, before performing its integral by numerical methods.
We hence propose to exploit the training process of the \ac{nn} and use the \ac{ce}, readily provided at the end of training, as a proxy of the system performance, avoiding an explicit estimation of \ac{roc} \ac{auc}. This is also motivated by Theorem 1, as the lower \ac{ce}, the more a \ac{nn} approaches \ac{np}, which is the optimal solution.
Recalling that the training minimization problem is non-convex, the same training set can lead to \acp{nn} with different classification performance and hence different \acp{auc}. However, we select the one minimizing the \ac{ce}, which is expected to have the minimum \ac{auc}.
\subsection{Particle Swarm Optimization}
In order to solve the network planning problem (\ref{defoptpos}) we resort to the \ac{pso} method \cite{Kennedy-11}, which is an iterative algorithm performing the simultaneous optimization of different points. This is similar to the multi-start solution for non-convex optimization, where local minima are avoided by selecting among different descent paths the one providing the minimum solution.
The \ac{pso} method is briefly recalled here. \ac{pso} is an iterative optimization algorithm based on social behavior of animals, e.g., birds flocking and fish schools. Consider $P$ particles, where particle $p=1, \ldots P$, is described by a vector of \acp{bs} positions $\bm{x}_p = [\bm{x}_{\rm AP}^{(1)}(p),\ldots, \bm{x}_{\rm AP}^{(N_{\rm ap})}(p)]$, and by its velocity $\bm{v}_p$. Each particle is a candidate solution of the optimization problem. Starting from particles at random positions and velocities, at each iteration both positions and velocities are updated. Two optimal values are defined in each iteration: the global optimum found so far in the entire particle population, and a local optimum for each particle, i.e., the optimal value found by the individual $p$ up to the current iteration. We define as $\bm{o}_{\rm G}$ the position of the the global optimal values and as $\bm{o}_p$ the position of the optimal value found by particle $p$ at the current iteration. The optimal values are those minimizing the selected objective function.
The position and velocity of the particles are updated at iteration $\ell$ as \cite{Kennedy-11}
\begin{equation}\label{eq: v up}
\begin{split}
\bm{v}_p(\ell) = \omega \bm{v}_p(\ell-1)+\phi_1(\ell)(\bm{o}_p(\ell-1)-\\
-\bm{x}_p(\ell-1))+\phi_2(\ell)(\bm{o}_{\rm G}(\ell-1)-\bm{x}_p(\ell-1));
\end{split}
\end{equation}
\begin{equation}\label{eq: p up}
\bm{x}_p(\ell) = \bm{x}_p(\ell-1) + \bm{v}_p(\ell),
\end{equation}
where $\omega$ is the inertia coefficient, and $\phi_1$ ($\phi_2$) is a random variable uniformly distributed in $[0,c_1]$ ($[0,c_2]$), where $c_1$ ($c_2$) is named the {\it acceleration constant}. The inertia coefficients and acceleration constants are parameters to be properly chosen.
\subsection{PSO-Based Network Planning}
As we have seen the \ac{roc} \ac{auc} well describes the overall behaviour of the \ac{roc} and is hence widely recognized as a valid synthetic metric for hypothesis testing. On the other hand, \ac{auc} computation is complicated by the need of performing extensive testing, while the \ac{ce} is immediately provided after the \ac{nn} training process.
In particular, the testing needed to compute \ac{auc} has an additional complexity (with respect to training that must be performed anyway), of
\begin{equation}
\mathcal{C}_{\rm test} = P \left( \mathcal{C}_{\rm out}+\mathcal{C}_{\rm ROC}+\mathcal{C}_{\rm AUC}\right),
\end{equation}
where $\mathcal{C}_{\rm out}$ denotes the complexity associated to running the \ac{nn} on the test points, $\mathcal{C}_{\rm ROC}$ denotes the complexity of building the \ac{roc} function, and $\mathcal{C}_{\rm AUC}$ denotes the complexity of integrating the \ac{roc}. The \ac{nn} running cost $\mathcal{C}_{\rm out}$ is given by the total number of multiplications and additions needed to compute the output value $y^{(L-1)}$ for all testing vectors, i.e.,
\begin{equation}
\mathcal{C}_{\rm out} = \left(2N_{\rm AP}N_{\rm h}+2N_{\rm h}^2N_{\rm L} + 2N_{\rm h}\right)\tau ,
\end{equation}
where $N_{\rm h}$ is the number of neurons in the hidden layer, $N_{\rm L}$ is the number of hidden layers, and $\tau$ is the size of the testing set.
The computation of the \ac{roc} curve requires the estimation of the $P_{\rm FA}$ and $P_{\rm MD}$ values for each threshold value $\lambda$, whereas the computation of the \ac{auc} requires the numerical integration of the \ac{roc} curve over $P_{\rm FA}$ values.
The proposed \ac{pso}-based network planning algorithm is reported in Algorithm 1. We denote as $\mathcal{B}$ the optimization metric and we initialize $P$ particles with random positions for each of the $N_{\rm AP}$ \acp{bs} in each particle. For each particle we train the \ac{nn} and compute $\mathcal{B}_p^{(0)}$. The global optimum value $\mathcal{B}_g$ is set to the minimum among all $\mathcal{B}_p^{(0)}$ values. Then, positions and velocities of the particles are updated via (\ref{eq: v up}) and (\ref{eq: p up}), and both the local and global optima are updated according to the obtained values at the current iteration. The algorithm stops when the global optimum converges.
\begin{algorithm}[b!]
\small
\KwData{ number of particles $P$, $N_{\rm AP}$}
\KwResult{optimal position }
Initialize particles\;
train the \ac{nn} algorithm for each particle\;
$\mathcal{B}_p^{(0)}$, $p=1,\ldots,N_p$\;
$\mathcal{B}_g=\underset{p=1,\ldots,N_p}{\min} \, \mathcal{B}_p^{(0)}$\;
$i = 0$\;
\Repeat{convergence of $\mathcal{B}_g$}{
$i = i + 1$\;
\For{$p=1,\ldots,P$}{
update velocity and position vector of particle via (\ref{eq: v up}) and (\ref{eq: p up})\;
train the \ac{nn} for each particle $\to$ $\mathcal{B}_p^{(i)}$\;
\If{$\mathcal{B}_p^{(i)} < \mathcal{B}_g$}{ $\mathcal{B}_g = \mathcal{B}_p^{(i)}$ \;}
}
}
\caption{Proposed \ac{ce}-based APs positioning algorithm.}
\end{algorithm}
Notice that, as the optimization problem is non-convex, \ac{pso} is similar to a multi-start optimization with $P$ different starting points, which is a standard method used to avoid local minima. As $P$ increases, the probability of finding only a local solution is reduced.
\section{Numerical Results}\label{sec: nr}
The considered scenario is depicted in Fig. \ref{fig:5bs}.
We consider a region represented by a square with side length $525$~m, where four buildings with side length $255$~m are located at the map corners, and separated by a road with width $15$~m. The \ac{roi} $\mathcal{A}_0$ is located inside the lower-left building, delimited by the dash-dotted line. Roads are considered as \ac{los} paths, whereas transmissions from \acp{ue} located in any other map position are in non-\ac{los} conditions. $N_{\rm AP}=5$ \acp{bs} are deployed, one for each street and one at the map center, to collect attenuation values. Each \ac{bs} sees the \ac{los} path of the street it is located in. For each \ac{bs} we generate a shadowing map with standard deviation $\sigma_s = 8$ dB and decorrelation distance $d_c = 75$~m. The \ac{ue} transmits with average unitary power at the frequency $f_0 = 2.12$~GHz.
\begin{figure}[h]
\centering
\includegraphics[width=0.7\columnwidth]{5bs}
\caption{\ac{irlv} deployment scenario. $N_{\rm AP}=5$ \acp{bs} are located in the streets separating 4 buildings. The \ac{roi} is located inside the lower-left building, delimited by the dash-dotted line.}
\label{fig:5bs}
\end{figure}
Results are averaged over different shadowing realizations. In particular, for each \ac{fa} probability value we compute the average \ac{md} probability over different shadowing maps.
\subsection{In-region Location Verification Results}
Fig. \ref{fig:n_neur} shows the average (over shadowing realizations) $P_{\rm MD}$ vs. $P_{\rm FA}$ of the proposed \ac{nn} \ac{irlv} system, with different numbers of neurons in the hidden layer $N_h$. Results have been obtained for a \ac{nn} with $N_L=3$ layers and with a training set of size $S= 10^5$. We notice that, as the number of neurons at the hidden layer increases, the average \ac{fa} probability decreases. When the number of neurons $N_h$ is higher than $8$ however we notice that results converge, meaning that increasing the network size does not lead to a performance improvement. Therefore, in the following we set $N_h=8$.
Fig. \ref{fig:n_train} shows the average (over shadowing realizations) $P_{\rm MD}$ vs. $P_{\rm FA}$ of the proposed \ac{nn} \ac{irlv} system trained with different numbers of training points $S$. Results have been obtained for a \ac{nn} with $L=3$ layers and $N_h=8$ neurons in the hidden layer. We see that the \ac{auc} decreases when increasing the number of training points and that, starting from $S=10^5$, the \ac{roc} does not significantly improve. This is due to the fact that, for the selected \ac{nn} architecture, training reaches convergence and hence adding further training points does not improve the \ac{nn} performance.
\begin{figure}[t]
\centering
\includegraphics[width=1\columnwidth]{res_avg_nNeur_1e5}
\caption{\ac{roc} of the \ac{nn} \ac{irlv} system with different numbers of neurons in the hidden layer $N_h$.}
\label{fig:n_neur}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=1\columnwidth]{res_avg_nTrain}
\caption{\ac{roc} of the \ac{nn} \ac{irlv} system trained with $S$ training points.}
\label{fig:n_train}
\end{figure}
Fig. \ref{fig:NP_comp} shows part of the \ac{roc} obtained with the \ac{np} test and with the \ac{nn}, using the model of Section \ref{sec:los}. In particular, we consider an overall circular region $\mathcal{A}$ with radius $R_{\rm out} = 40$~m, a square authentic region $\mathcal{A}_0$ of $L = H= 25$~m located inside $\mathcal{A}$, with upper left corner at a distance of $R_{\rm min} = 4$~m from the center of $\mathcal{A}$. We also report the results of \ac{np} theorem, that can be computed in close form for this simple scenario. We see that, even with a small number of neurons, in this simple problem, the \ac{nn} achieves the same \ac{roc} of the \ac{np} test, thus confirming Theorem 1.
\begin{figure}[h]
\centering
\includegraphics[width=0.9\columnwidth]{FA_MD_LOS}
\caption{\ac{roc} of the \ac{np} test and the the proposed \ac{ml} test, with different number of neurons in the hidden layer of the \ac{nn}.}
\label{fig:NP_comp}
\end{figure}
\subsection{Network Planning Performance}
We now consider the network planning problem, using the proposed Algorithm 1. We consider a \ac{pso} with $P=6$ particles, each composed by a set of $N_{\rm AP}=5$ \acp{bs} initialized with random positions. There exists a variety of implementations of the \ac{pso}, but the most general case for the parameter initialization is given by \cite{clerc2002}, where it is suggested to set $\omega=0.7298$, and $c_1=c_2=1.4961$. Results are averaged over different shadowing realizations. The used \acp{nn} are implemented with $L=3$ layers and $N_h=8$ neurons at the hidden layer. In order to validate the use of the \ac{ce} as a proxy for the \ac{auc}, we compare the performance of Algorithm 1 using either the CE or the AUC as objective function $\mathcal{B}$. For a performance assessment we show the \ac{auc} obtained with the various optimization approaches.
Fig. \ref{fig:CEvsAUC} shows the average \ac{auc} vs. the number of \ac{pso} iterations for the two implementations of Algorithm 1, and for different training set size $S$.
We recall that the result of Theorem 1 holds asymptotically and with perfect training (in terms of training set, number of layers and neurons), which are conditions that our experiments do not satisfy. Two observations are in place here.
First, the training set size should be sufficiently large so that \ac{ce} becomes a proxy of \ac{auc}: in fact we notice that for $S=10^3$, the mean \ac{auc} for \ac{pso} with \ac{ce} objective function increases with the number of iterations. This is due to the fact that convergence has not been reached because of the limited size of the training set, and hence the obtained \ac{ce} value is not representative of the \ac{irlv} system performance. This is solved by augmenting the training set size, as we see when $S=10^4$ or $10^5$.
Second, the \ac{auc}-based algorithm converges earlier than the \ac{ce}-based algorithm. As stated earlier, the \ac{nn} converges to the optimal \ac{np} solution only asymptotically. Therefore the obtained \ac{ce} is an approximation of the selected objective function, whose optimization does not necessarily entail the minimization of the \ac{auc}. Although requiring a larger number of iterations, the \ac{ce}-based solution is convenient as it does not require the explicit estimation of the \ac{roc} curve.
\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{res_PSO_comp}
\caption{Mean \ac{auc} vs. the number of \ac{pso} iterations for Algorithm 1, and two \ac{pso} algorithms using only the \ac{auc} and the \ac{ce} as objective functions. }
\label{fig:CEvsAUC}
\end{figure}
\section{Conclusions}
\label{sec:conc}
In this paper we formulated the \ac{irlv} problem as an hypothesis testing problem and proposed a \ac{ml} solution. We proved that the \ac{nn} implementation achieves the same performance of the optimal \ac{np} test with \ac{ce} as training objective function, and verified numerically this claim for a simple scenario. We also assessed the effects of the training set size over the \ac{roc} for a more realistic scenario. We then proposed a \ac{pso} algorithm for optimal \acp{bs} positioning, showing that there is a minimum training set size which allows to use the \ac{ce} as a proxy of the \ac{auc}.
\begin{appendices}
\section{Proof of Theorem 1}
Consider
\begin{equation}
\begin{split}
\hatcross{ p_{\mathcal{H}|\bm a}}{g} =& - \left[ \frac{n_0}{S} \frac{1}{n_0} \sum_{\bm a \in \mathcal{T}_0} \log(\gy) \right. \\
&\left. + \frac{n_1}{S} \frac{1}{n_1} \sum_{\bm a \in \mathcal{T}_1} \log(1-\gy) \right],
\end{split}
\end{equation}
where $\mathcal{T}_k = \{\bm a_i \in \mathcal{T} : t_i = k\}$, $|\mathcal{T}_k|=n_k$, $k \in \{0,1\}$.
\balance
Let $S \to \infty$. By the strong law of large number
\begin{equation}
\label{eq:as}
\begin{split}
\lim_{S \to \infty}& \hatcross{ p_{\mathcal{H}|\bm a}}{g} \simeq - \left[ p_{\mathcal H}(\mathcal{H}_0) \int_{\mathcal{Y}} p_{\bm a|\mathcal{H}}(\bm a|\mathcal{H}_0) \log (\gy) d\bm a \right. \\
& \left. + p_{\mathcal{H}}(\mathcal{H}_1) \int_{\mathcal{Y}} p_{\bm a|\mathcal{H}}(\bm a|\mathcal{H}_1) \log (1-\gy) d\bm a \right],
\end{split}
\end{equation}
where equality $\simeq$ holds in probability, i.e., \textit{almost surely}, as per the strong law of large numbers.
Rearranging terms with the Bayes rule we get
\begin{equation}
\label{eq:dim1}
\begin{split}
\lim_{S \to \infty} \hatcross{p_{\mathcal{H}|\bm a}}{g} \simeq - \left\{ \int_{\mathcal{Y}} \bigl[ p_{\mathcal{H}|\bm a}(\mathcal{H}_0|\bm a) \log \gy + \right. \bigr.\\
\bigl. \bigl.(1-p_{\mathcal{H}|\bm a}(\mathcal{H}_0|\bm a)) \log(1-\gy)\bigr] p(\bm a) d\bm a \biggr\}.
\end{split}
\end{equation}
By definition of expected value we can rewrite \eqref{eq:dim1} as
\begin{equation}
\label{eq:dim2}
\begin{split}
\lim_{S \to \infty} \hatcross{p_{\mathcal{H}|\bm a}}{g} \simeq - \mathbb{E}_{\bm a} \left[ p_{\mathcal{H}|\bm a}(\mathcal{H}_0|\bm a) \log \gy + \right.\\
\left. (1-p_{\mathcal{H}|\bm a}(\mathcal{H}_0|\bm a)) \log(1-\gy)\right].
\end{split}
\end{equation}
We introduce the Bernoulli random variable $\xi$ with alphabet $ \{0,1\}$ and \ac{pdf}
\begin{equation}
\label{eq:q}
p_\xi(0) = \gy, \quad
p_\xi(1) = 1- \gy.
\end{equation}
Note that $p_\xi$ is a valid \ac{pdf} since it sums to 1 and $\gy \in [0,1]$ by hypothesis.
Recall now that the cross entropy between two discrete \acp{pdf} $p_{W_1}(w)$ and $p_{W_2}(w)$ having the same alphabet $\mathcal{W}$ is defined as
\begin{equation}
\label{eq:defCross}
\cross{p_{W_1}}{W_2} = - \sum_{w \in \mathcal{W}} p_{W_1}(w) \log p_{W_2}(w),
\end{equation}
and an equivalent definition is
\begin{equation}
\label{eq:alternativeDef}
\cross{p_{W_1}}{W_2} = H (p_{W_1}) + D(p_{W_1}||p_{W_2}),
\end{equation}
where $D(\cdot||\cdot)$ is the Kullback-Leibler (K-L) divergence and $H(\cdot)$ is the entropy function.
Then, form \eqref{eq:dim2} and \eqref{eq:defCross}, we have
\begin{equation}
\lim_{S \to \infty} \hatcross{p_{\mathcal{H}|\bm a}}{g} \simeq \mathbb{E}_{\bm a} \left[ \cross{p_{\mathcal{H}|\bm a}}{\xi} \right],
\end{equation}
which from \eqref{eq:alternativeDef} yields
\begin{equation}
\label{eq:final}
\lim_{S \to \infty} \hatcross{p_{\mathcal{H}|\bm a}}{g} \simeq
\E{\bm a}{H ( p_{\mathcal{H}|\bm a}) + D( p_{\mathcal{H}|\bm a}|| p_\xi)}
\end{equation}
Recall that \ac{nn} training is performed by minimizing the left hand side of \eqref{eq:final} with respect to \ac{nn} parameters. In the right hand side of \eqref{eq:final} the only quantity depending on \ac{nn} parameters, through $\gy$ in \eqref{eq:q}, is $D( p_{\mathcal{H}|\bm a}||p_\xi)$. Then, with a infinite number of neurons (i.e., with the possibility of choosing any \ac{pdf} $p_\xi$, the minimum of the K-L divergence is attained for $D(p_{\mathcal{H}|\bm a}||p_\xi)=0$, that is when $p_\xi(i) = p_{\mathcal{H}|\bm a}(i|\bm{a})$.
\end{appendices}
\bibliographystyle{plain}
|
1,941,325,220,507 | arxiv | \section*{Appendix A}
\def\theequation{A.\arabic{equation}}
\setcounter{equation}0
In the appendix we describe the notation and definitions of geometric
quantities used in the paper. We use a smooth principal bundle which is an
ordered sequence
\beq{A.0}
\ul P=(P,F,G,E,\pi),
\end{equation}
where $P$ is a total bundle manifold, $F$ is typical fibre, $G$, a Lie group,
is a structural group, $E$~is a base manifold and $\pi$ is a projection. In our
case $G=\U(1)$, $E$~is a \spt, $\pi:P\to E$.
We have a map $\vf:P\times G\to P$ defining an
action of~$G$ on~$P$. Let $a,b\in G$ and $\ve$~be a unit element of the
group~$G$, then $\vf(a)\circ \vf(b)=\vf(ba)$, $\vf(\ve)=\id$, where $\vf(a)p
=\vf(p,a)$. Moreover, $\pi\circ\vf(a)=\pi$. For any open set $U\subset E$ we
have a local trivialization $U\times G\simeq \pi^{-1}(U)$. For any $x\in E$,
$\pi^{-1}(\{x\})=F_x \simeq G$, $F_x$ is a fibre over~$x$ and is equal to~$F$.
In our case we suppose $G=F$, i.e.\ a Lie group $G$ is a typical fibre.
$\o$~is a 1-form
of \cn\ on~$P$ with values in the algebra of~$G$, $\mathfrak G$. In the case
of $G=\U(1)$ we use a notation $\a$ (an \elm c \cn). Lie algebra of $U(1)$
is~$R$. Let $\vf'(a)$ be a
tangent map to $\vf(a)$ whereas $\vf^\ast(a)$ is the contragradient
to~$\vf'(a)$ at a point~$a$. The form $\o$ is a form of ad-type, i.e.
\beq{A.1}
\vf^\ast(a)\o=\ad{a^{-1}}'\o,
\end{equation}
where $\ad{a^{-1}}'$ is a tangent map to the internal automorphism of the
group~$G$
\beq{A.2}
\ad a(b)=aba^{-1}.
\end{equation}
In the case of $\U(1)$ (abelian) the condition \eqref{A.1} means
\beq{A.3}
\mathop{\cL}_{\z_5}\a=0,
\end{equation}
where $\z_5$ is a Killing vector corresponding to one generator of the group
$\U(1)$. Thus this is a vector tangent to the operation of the group $\U(1)$
on~$P$, i.e.\ to $\vf_{\exp(i\chi)}$, $\chi=\chi(x)$, $x\in E$,
$\mathop{\cL}\limits_{\z_5}$ is a Lie \dv\ along $\z_5$.
We may introduce the distribution (field) of linear elements $H_r$, $r\in P$,
where $H_r\subset T_r(P)$ is a subspace of the space tangent to~$P$ at a
point~$r$ and
\beq{A.4}
v\in H_r \iff \o_r(v)=0.
\end{equation}
So
\beq{A.5}
T_r(P)=V_r\oplus H_r,
\end{equation}
where $H_r$ is called a subspace of \ti{horizontal\/} vectors and $V_r$ of
\ti{vertical\/} vectors.
For vertical vectors $v\in V_r$ we have $\pi'(v)=0$. This means that $v$ is
tangent to the fibres.
Let
\beq{A.6}
v=\hor(v)+\ver(v),\quad \hor(v)\in H,\ \ver(v)\in V_r.
\end{equation}
It is proved that the distribution $H_r$ is equal to choosing a \cn~$\o$. We
use the operation $\hor$ for forms, i.e.
\beq{A.7}
(\hor\b)(X,Y)=\b(\hor X,\hor Y),
\end{equation}
where $X,Y\in T(P)$.
The 2-form of a curvature is defined as follows
\beq{A.8}
\O=\hor d\o=D\o,
\end{equation}
where $D$ means an exterior covariant \dv\ \wrt $\o$. This form is also of
ad-type.
For $\O$ the structural Cartan \e\ is valid
\beq{A.9}
\O=d\o+\tfrac12[\o,\o],
\end{equation}
where
\beq{A.10}
[\o,\o](X,Y)=[\o(X),\o(Y)].
\end{equation}
Bianchi's identity for $\o$ is as follows
\beq{A.11}
D\O=\hor d\O=0.
\end{equation}
The map $f:E\supset U\to P$ such that $f\circ \pi=\id$ is called a
\ti{section} ($U$ is an open set).
From physical point of view it means choosing a gauge. A~covariant \dv\
on~$P$ is defined as follows
\beq{A.12}
D\Ps=\hor d\Ps.
\end{equation}
This \dv\ is called a \ti{gauge \dv}. $\Ps$ can be a spinor field on~$P$.
In this paper we use also a linear \cn\ on manifolds $E$ and $P$, using the
formalism of differential forms. So the basic quantity is a one-form of the
\cn\ $\gd\o,A,B,$. The 2-form of curvature is as follows
\beq{A.13}
\gd\O,A,B,=d\gd\o,A,B,+\gd\o,A,C, \wedge \gd\o,C,B,
\end{equation}
and the two-form of torsion is
\beq{A.14}
\T^A=D\t^A,
\end{equation}
where $\t^A$ are basic forms and $D$ means exterior covariant \dv\ \wrt \cn\
$\gd\o,A,B,$. The following relations are established \cn s with generally met
symbols
\beq{A.15}
\bal
\gd\o,A,B,&=\gd\G,A,BC,\t^C\\
\T^A&=\tfrac12\gd Q,A,BC,\t^B\wedge \t^C\\
\gd Q,A,BC,&=\gd\G,A,BC,-\gd\G,A,CB,\\
\gd\O,A,B,&=\tfrac12 \gd R,A,BCD,\t^C \wedge \t^D,
\eal
\end{equation}
where $\gd\G,A,BC,$ are \cf s of \cn\ (they do not have to be \s\ in indices
$B$ and~$C$), $\gd R,A,BCD,$ is a tensor of a curvature, $\gd Q,A,BC,$ is a
tensor of a torsion in a holonomic frame. Covariant exterior derivation \wrt $\gd\o,A,B,$ is given
by the formula
\beq{A.16}
\bal
D\Xi^A&=d\Xi^A+\gd\o,A,C,\wedge \Xi^C\\
D\gd\Si,A,B,&=
d\gd\Si,A,B,+\gd\o,A,C,\wedge \gd\Si,C,B,-\gd\o,C,B,\wedge \gd\Si,A,C,.
\eal
\end{equation}
The forms of a curvature $\gd\O,A,B,$ and torsion $\T^A$ obey Bianchi's
identities
\beq{A.17}
\bal
{}&D\gd\O,A,B,=0\\
&D\T^A=\gd\O,A,B,\wedge \t^B.
\eal
\end{equation}
All quantities introduced here can be found in Ref.~\cite{KN}.
In this paper we use a formalism of a fibre bundle over a \spt~$E$ with an
\elm c \cn~$\a$ and traditional formalism of differential geometry for linear
\cn s on~$E$ and~$P$. In order to simplify the notation we do not use fibre
bundle formalism of frames over $E$ and~$P$. A~vocabulary connected geometrical
quantities and gauge fields (Yang--Mills fields) can be found in
Ref.~\cite{97}.
In Ref.~\cite{Wu} we have also a similar vocabulary (see Table~I, Translation
of terminology). Moreover, we consider a little different terminology. First
of all we distinguished between a gauge \pt\ and a \cn\ on a fibre bundle. In
our terminology a gauge \pt\ $A_\mu \ov\t{}^\mu$ is in a particular gauge $e$
(a~section of a bundle), i.e.
\beq{A.18}
A_\mu \ov\t{}^\mu=e^\ast\o
\end{equation}
where $A_\mu \ov\t{}^\mu$ is a 1-form defined on $E$ with values in a Lie
algebra $\mathfrak G$ of~$G$. In the case of a strength of a gauge field we have
similarly
\beq{A.19}
\tfrac12 F_\m \ov\t{}^\mu \wedge \ov\t{}^\nu=e^\ast\O
\end{equation}
where $F_\m \ov\t{}^\mu \wedge \ov\t{}^\nu$ is a 2-form defined on~$E$ with
values in a Lie algebra $\mathfrak G$ of~$G$.
Using generators of a Lie algebra $\mathfrak G$ of $G$ we get
\beq{A.20}
A=\gd A,a,\mu, \ov\t{}^\mu X_a=e^\ast \o \quad\hbox{and}\quad
F=\tfrac12\gd F,a,\m,\ov\t{}^\mu \wedge \ov\t{}^\nu X_a=e^\ast \O
\end{equation}
where
\beq{A.21}
[X_a,X_b]=\gd C,c,ab,X_c, \quad a,b,c=1,2,\dots,n, \ n=\dim G(=\dim \mathfrak G),
\end{equation}
are generators of $\mathfrak G$, $\gd C,c,ab,$ are structure \ct s of a Lie
algebra of~$G$, $\mathfrak G$, $[\cdot,\cdot]$ is a commutator of Lie algebra
elements.
In this paper we are using Latin lower case letters for 3-\di al space indices. Here
we are using Latin lower case letters as Lie algebra indices. It does not
result in any misunderstanding.
\beq{A.22}
\gd F,a,\m,=\pa_\mu\gd A,a,\nu,-\pa_\nu\gd A,a,\mu,+\gd C,a,bc,\gd A,b,\mu,
\gd A,c,\nu,.
\end{equation}
In the case of an \elm c \cn\ $\a$ the field strength~$F$ does not depend on
gauge (i.e.\ on a section of a~bundle).
Finally it is convenient to connect our approach using gauge \pt s $\gd
A,a,\mu,$ with usually met (see Ref.~\cite{Pok}) matrix valued gauge
quantities $A_\mu$ and $F_\m$. It is easy to see how to do it if we consider
Lie algebra generators $X_a$ as matrices. Usually one supposes that $X_a$ are
matrices of an adjoint representation of a Lie algebra~$\mathfrak G$, $T^a$
with a normalization condition
\beq{A.23}
{\rm Tr}(\{T^a,T^b\})=2\d^{ab},
\end{equation}
where $\{\cdot,\cdot\}$ means anticommutator in an adjoint representation.
In this way
\bea{A.24}
A_\mu&=&\gd A,a,\mu, T^a,\\
F_\m&=&\gd F,a,\m, T^a. \label{A.25}
\end{eqnarray}
One can easily see that if we take
\beq{A.26}
F_\m=\pa_\mu A_\nu - \pa_\nu A_\mu + [A_\mu,A_\nu]
\end{equation}
from Ref.~\cite{Pok} we get
\beq{A.26a}
F_\m=(\gd F,a,\m,)T^a,
\end{equation}
where $\gd F,a,\m,$ is given by \eqref{A.22}. From the other side if we take
a section $f$, $f:U\to P$, $U\subset E$, and corresponding to it
\bea{A.27}
\ov A=\gd\ov A,a,\mu, \ov\t{}^\mu X_a&=&f^\ast \o\\
\ov F=\tfrac12\gd\ov F,a,\m, \ov\t{}^\mu \wedge \ov\t{}^\nu X_a&=&f^\ast \O
\label{A.28}
\end{eqnarray}
and consider both sections $e$ and $f$ we get transformation from $\gd
A,a,\mu,$ to $\gd\ov A{},a,\mu,$ and from $\gd F,a,\m,$ to $\gd\ov F{},a,\m,$ in
the following way. For every $x\in U\subset E$ there is an element $g(x)\in
G$ such that
\beq{A.29}
f(x)=e(x)g(x)=\vf(e(x),g(x)).
\end{equation}
Due to \eqref{A.1} one gets
\bea{A.30}
\ov A(x)&=&\ad{g^{-1}(x)}'A(x)+{g^{-1}(x)}\,dg(x)\\
\ov F(x)&=&\ad{g^{-1}(x)}'F(x) \label{A.31}
\end{eqnarray}
where $\ov A(x),\ov F(x)$ are defined by \eqref{A.27}--\eqref{A.28} and
$A(x),F(x)$ by \eqref{A.20}. The formulae \eqref{A.30}--\eqref{A.31} give a
geometrical meaning of a gauge transformation (see Ref.~\cite{97}). In an
\elm c case $G=\U(1)$ we have similarly, if we change a local
section from $e$ to~$f$ we get
$$
f(x)=\vf(e(x), \exp(i\chi(x))) \quad (f:U\supset E\to P)
$$
and $\ov A=A+d\chi$.
Moreover,
in the traditional approach (see Ref.~\cite{Pok}) one gets
\bea{A.32}
\ov A_\mu(x)&=&U(x)^{-1}A_\mu(x)U(x)+U^{-1}(x)\pa_\mu U(x)\\
\ov F_\m(x)&=&U^{-1}(x)F_\m U(x), \label{A.33}
\end{eqnarray}
where $U(x)$ is the matrix of an adjoint representation of a Lie group $G$.
For an action of a group $G$ on $P$ is via \eqref{A.1}, $g(x)$ is exactly a
matrix of an adjoint representation of~$G$. In this way
\eqref{A.30}--\eqref{A.31} and \eqref{A.32}--\eqref{A.33} are equivalent.
Let us notice that usually a Lagrangian of a gauge field (Yang--Mills field)
is written as
\beq{A.34}
\cL_{\rm YM} \sim {\rm Tr}(F_\m F^\m)
\end{equation}
where $F_\m$ is given by \eqref{A.25}--\eqref{A.26}. It is easy to see that
one gets
\beq{A.35}
\cL_{\rm YM} \sim h_{ab}\gd F,a,\m, F^{b\m}
\end{equation}
where
\beq{A.36}
h_{ab}=\gd C,d,ac, \gd C,c,bd,
\end{equation}
is a Cartan--Killing tensor for a Lie algebra $\mathfrak G$, if we remember
that $X_a$ in adjoint representation are given by structure \ct s $\gd
C,c,ab,$.
Moreover, in Refs \cite{1,3} we use the notation
\beq{A.39}
\O=\tfrac12 \gd H,a,\m,\t^\mu \wedge \t^\nu X_a.
\end{equation}
In this language
\beq{A.40}
\cL_{\rm YM}=\tfrac1{8\pi} h_{ab}\gd H,a,\m, H^{b\m}.
\end{equation}
It is easy to see that
\beq{A.41}
e^\ast (\gd H,a,\m,\t^\mu \wedge \t^\nu X_a)=\gd F,a,\m,{\ov\t}^\mu
\wedge {\ov \t}^\nu X_a.
\end{equation}
Thus \eqref{A.40} is equivalent to \eqref{A.35} and to \eqref{A.34}.
\eqref{A.34} is invariant to a change of a gauge. \eqref{A.40} is invariant
\wrt the action of a group~$G$ on~$P$.
Let us notice that $h_{ab}\gd F,a,\m, F^{b\m}=h_{ab}\gd H,a,\m, \gd H,b,\m,$,
even $\gd H,a,\m,$ is defined on~$P$ and $\gd F,a,\m,$ on~$E$. In the
non-abelian case it is more natural to use $\gd H,a,\m,$ in place of $\gd
F,a,\m,$.
\section*{Appendix B}
\def\theequation{B.\arabic{equation}}
\setcounter{equation}0
\def\rp{representation}
\def\SL{{\rm SL}}
In this paper we consider two kinds of spinor fields $\Ps,\ov\Ps$ and
$\psi,\ov\psi$ defined respectively on~$P$ and~$E$. Spinor fields $\Ps$
and~$\ov\Ps$ transform according to $\Spin(1,4)$ and $\psi,\ov\psi$ according
to $\Spin(1,3)\simeq{\rm SL}(2,\C)$. We have
\beq{C.1}
U(g)\Psi(X)=D^F(g)\Psi(g^{-1}X), \q X\in M^{(1,4)}, \ g\in\SO(1,4).
\end{equation}
$\SO(1,4)$ acts linearly in $M^{(1,4)}$ (5-\di al Minkowski space). The
Lorentz group $\SO(1,3)\subset\SO(1,4)$. $D^F$ is a representation of
$\SO(1,4)$ (de~Sitter group) \st after a restriction to its subgroup
$\SO(1,3)$ we get
\beq{C.2}
D^F{}_{|\SO(1,3)}(\La)=L(\La),
\end{equation}
where
\beq{C.3}
L(\La)=D^{(1/2,0)}(\La)\oplus D^{(0,1/2)}(\La)
\end{equation}
is a Dirac representation of $\SO(1,3)$. More precisely, we deal with \rp s
of $\Spin(1,4)$ and $\Spin(1,3)\simeq \SL(2,\C)$ (see Ref.~\cite{52}). In
other words, we want spinor fields $\Ps$ and~$\ov\Ps$ to transform according
to such a \rp\ of $\Spin(1,4)$ which is induced by a Dirac \rp\
of~$\SL(2,\C)$. The complex \di s of both \rp s are the same:~4. The same are
also Clifford algebras
\beq{C.4}
C(1,4)\simeq C(1,3)
\end{equation}
(see Refs \cite{53}, \cite{54}).
One gets (up to a phase)
\beq{C.5}
\Ps_{|\SL(2,\C)}=\psi.
\end{equation}
Spinor fields $\psi$ and $\ov\psi$ transform according to Dirac \rp,
$\ov\psi=\psi^+B$. Our matrices $\g_\mu$ and $\g_A$ are \rp s of $C(1,3)$
($C(1,4)$). One can consider projective \rp s for $\Ps$ and~$\psi$, i.e.\
\rp s of $\Spin(1,3)\ot \U(1)$ and $\SL(2,\C)\ot \U(1)$. Moreover, we do not
develop this idea here.
In this paper we develop the following approach to spinor fields on~$E$ and
on~$P$. We introduce orthonormal frames on~$E$ ($dx^1$, $dx^2$, $dx^3$,
$dx^4$) and on~$P$ ($dX^1=\pi^\ast(dx^1)$, $dX^2=\pi^\ast(dx^2)$,
$dX^3=\pi^\ast(dx^3)$, $dX^4=\pi^\ast(dx^4)$, $dX^5$). Our spinors $\Ps$
on $(P,\g_\(AB))$ and $\psi$ on $(E,g_\(\a\b))$ are defined as complex bundles
$\C^4$ over~$P$ or~$E$ with homomorphisms $\rho:C(1,4)\to \cL(\C^4)$ (resp.\
$\rho:C(1,3)\to \cL(\C^4)$) of bundles of algebras over~$P$ (resp.~$E$) \st
for every $p\in P$ (resp.~$x\in E$), the restriction of~$\rho$ to the fiber
over~$p$ (resp.~$x$) is equivalent to spinor \rp\ of a Clifford algebra
$C(1,4)$ (resp.~$C(1,3)$), i.e.\ $D^F$ (resp.\ Dirac \rp, see Refs
\cite{55},~\cite{56}). (There is also a paper on a similar subject (see
Ref.~\cite{57}).) Spinor fields $\Ps$ and~$\psi$ are sections of these
bundles. There is also an approach to consider spinor bundles for~$\Ps$
and~$\psi$ as bundles associated to principal bundles of orthonormal frames
for $(P,\g_\(AB))$ or $(E,g_\(\a\b))$ (spin frames). Spinor fields $\Ps$ and
$\psi$ are sections of these bundles. In our case we consider spinor fields
$\Ps$ and~$\ov\PS$ transforming according to \eqref{4.13} and \eqref{4.14}.
In the case of $\psi$ and $\ov\psi$ we have
\beq{C.6}
\bga
\ov\t{}^{\a\prime}=\ov\t{}^\a+\d\ov\t{}^\a=\ov\t{}^\a-\gd\ve,\a,\b,
\ov\t{}^\b\\
\ov\ve_{\a\b}+\ov\ve_{\b\a}=0.
\ega
\end{equation}
If the spinor field $\psi$ corresponds to $\ov\t{}^\a$ and $\psi'$ to
$\ov\t{}^{\a\prime}$ we get
\beq{C.7}
\bal
\psi'&=\psi+\d\psi=\psi-\ov\ve{}^{\a\b}\si_{\a\b}\psi\\
\ov\psi{}'&=\ov\psi+\d\ov\psi=\ov\psi+\ov\psi\ov\ve{}^{\a\b}\si_{\a\b}.
\eal
\end{equation}
Spinor fields $\Ps$ and $\ov\Ps$ are $\psi$ and $\ov\psi$ in any section of a
bundle~$P$. Simultaneously we suppose conditions \eqref{4.2}.
Similarly as for $\Ps,\ov\PS$ one gets
\bg{C.8}
\bal
\wt{\ov D}\psi&=d\psi + \gd\wt{\ov w}{},\a,\b, \dg\si,\a,\b, \psi\\
\wt{\ov D}\,\ov\psi&=d\ov\psi - \gd\wt{\ov w}{},\a,\b, \ov\psi \dg\si,\a,\b,
\eal\\
\bal
\wt{\ov\cD}\psi=\hor\wt{\ov D}\psi=\gdv\psi
+ \gd\wt{\ov w}{},\a,\b,\dg\si,\a,\b,\psi\\
\wt{\ov\cD}\,\ov\psi=\hor\wt{\ov D}\,\ov\psi=\gdv\ov\psi
- \gd\wt{\ov w}{},\a,\b,\ov\psi\dg\si,\a,\b,.
\eal \label{C.9}
\end{gather}
\goodbreak
|
1,941,325,220,508 | arxiv | \section{Introduction}
In this paper, we will study the dynamics of a collection of particles which interact pairwise and which moves along the real line. We will also suppose that when particles collide, they undergo perfectly inelastic collisions. The equations of motion for this type of physical system are the {\it pressureless Euler equations}
\begin{equation}\label{PEE}
\begin{cases}
\hspace{.272in}\partial_t\rho +\partial_x(\rho v)=0\\
\partial_t(\rho v) +\partial_x(\rho v^2)=-\rho(W'*\rho),
\end{cases}
\end{equation}
which hold on $\mathbb{R}\times (0,\infty)$. The first equation expresses the conservation of mass, and the second expresses the conservation of momentum. Here $\rho$ and $v$ are the respective mass distribution and velocity field of particles and $W$ is the interaction energy.
\par The central goal of this work is to describe how to find a pair $\rho$ and $v$ which solves \eqref{PEE} for given initial conditions
\begin{equation}\label{Init}
\rho|_{t=0}=\rho_0\quad \text{and}\quad v|_{t=0}=v_0.
\end{equation}
We typically will assume $\rho_0$ belongs to the space ${\cal P}(\mathbb{R})$ of Borel probability measures on $\mathbb{R}$ and $v_0: \mathbb{R}\rightarrow \mathbb{R}$ is continuous.
To this end, we will first produce $X:[0,\infty)\rightarrow L^2(\rho_0)$ which satisfies the {\it pressureless Euler flow equation}
\begin{equation}\label{FlowMapEqn}
\dot X(t)=\E_{\rho_0}\left[v_0-\displaystyle\int^t_0(W'*\rho_s)(X(s))ds\bigg| X(t)\right], \quad a.e.\;t\ge 0
\end{equation}
and the initial condition
\begin{equation}\label{xInit}
X(0)=\text{id}_{\R}
\end{equation}
$\rho_0$ almost everywhere. Here
\begin{equation}\label{PushForwardMeasure}
\rho_t:=X(t)_{\#}\rho_0, \quad t\ge 0
\end{equation}
is the push forward of $\rho_0$ under $X(t)$, and $\E_{\rho_0}[ g |X(t)]$ is the conditional expectation of a Borel $g:\mathbb{R}\rightarrow \mathbb{R}$ given $X(t)$.
\par To emphasize that $X(t)$ is a function on $\mathbb{R}$, we will sometimes write
$$
X(t): \mathbb{R}\rightarrow \mathbb{R}; y\mapsto X(y,t).
$$
The quantity $X(y,t)$ represents the time $t$ position of a particle which was initially at position $y$. After showing a solution $X$ exists, we will argue that there is a Borel function $v: \mathbb{R}\times [0,\infty)\rightarrow \mathbb{R}$ such that
\begin{equation}\label{EulerLagrangeCoord}
\dot X(t)=v(X(t),t), \quad a.e.\;t\ge 0
\end{equation}
$\rho_0$ almost everywhere. In particular, we will see that
$$
\rho: [0,\infty)\rightarrow {\cal P}(\mathbb{R}); t\mapsto \rho_t
$$
and $v$ together comprise an appropriately defined weak solution pair for the pressureless Euler system.
\subsection{Main theorem}
Throughout this paper, we will assume the $\rho_0\in {\cal P}(\mathbb{R})$ has finite second moment
\begin{equation}
\int_{\mathbb{R}}x^2d\rho_0(x)<\infty
\end{equation}
and
\begin{equation}
v_0:\mathbb{R}\rightarrow\mathbb{R}\;\text{ is absolutely continuous}.
\end{equation}
We will also suppose $W:\mathbb{R}\rightarrow \mathbb{R}$ is continuously differentiable, $W$ is even
\begin{equation}\label{WEven}
W(x)=W(-x), \quad x\in \mathbb{R}
\end{equation}
and $W'$ grows at most linearly
\begin{equation}\label{WprimeGrowth}
\sup_{x\in \mathbb{R}}\frac{|W'(x)|}{1+|x|}<\infty.
\end{equation}
Moreover, we will suppose that $W$ is semiconvex. That is,
\begin{equation}\label{WSemiCon}
W(x)+\frac{c}{2}x^2\; \text{is convex}
\end{equation}
for some $c>0$. We recall that concave $W$ corresponds to repulsive interaction between particles. Assuming that $W$ is semiconvex forces $W''(x)\ge -c$ for Lebesgue almost every $x\in \mathbb{R}$, which in a sense limits repulsive interaction.
\begin{thm}\label{mainThm}
There is a locally Lipschitz continuous $X:[0,\infty)\rightarrow L^2(\rho_0)$ which satisfies the pressureless Euler flow equation \eqref{FlowMapEqn} and the initial condition \eqref{xInit}. Moreover, $X$ has the following properties.
\begin{enumerate}[(i)]
\item For Lebesgue almost every $t,s\in [0,\infty)$ with $s\le t$
$$
E(t)\le E(s),
$$
where
\begin{align*}
E(\tau):=\int_{\mathbb{R}}\frac{1}{2}\dot X(\tau)^2d\rho_0+\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{1}{2}W(X(y,\tau)-X(z,\tau))d\rho_0(y)d\rho_0(z).
\end{align*}
\item For $t\ge 0$ and $y,z\in\textup{supp}(\rho_0)$ with $y\ge z$,
\begin{equation}
0\le X(y,t)-X(z,t)\le \cosh(\sqrt{c}t)(y-z)+\frac{1}{\sqrt{c}}\sinh(\sqrt{c}t)\int^y_z|v'_0(x)|dx.
\end{equation}
\item For $0<s\le t$ and $y,z\in\textup{supp}(\rho_0)$,
\begin{equation}
\frac{|X(y,t)-X(z,t)|}{\sinh(\sqrt{c}t)}\le \frac{|X(y,s)-X(z,s)|}{\sinh(\sqrt{c}s)}.
\end{equation}
\end{enumerate}
\end{thm}
\par A few remarks about the statement of this theorem are in order. Locally Lipschitz means that $X:[0,T]\rightarrow L^2(\rho_0)$ is Lipschitz continuous for each $T\ge 0$, and consequently,
\begin{equation}
\dot X(t)=\lim_{\tau\rightarrow 0}\frac{X(t+\tau)-X(t)}{\tau}
\end{equation}
exists in $L^2(\rho_0)$ for almost almost every $t> 0$.
The function $E$ in condition $(i)$ represents the total energy of the physical system being modeled by the pressureless Euler flow equation. Condition $(ii)$ asserts that
$X(t)$ is nondecreasing and absolutely continuous on the support of $\rho_0$
$$
\text{supp}(\rho_0):=\{y\in \mathbb{R}: \rho_0((y-\delta,y+\delta))>0\;\text{for all $\delta>0$}\}.
$$
Property $(iii)$ asserts that $X$ is quantitatively ``sticky." That is, it quantifies the fact that if $X(y,s)=X(z,s)$, then $X(y,t)=X(z,t)$ for all $t\ge s$.
\par We will show that the existence of a weak solution of \eqref{PEE} for given initial conditions is a corollary of Theorem \ref{mainThm}. In particular, we will verify that $\rho$ defined in \eqref{PushForwardMeasure} and any Borel $v$ which satisfies \eqref{EulerLagrangeCoord} is a weak solution of the pressureless Euler system whose
energy
\begin{equation}
\int_{\mathbb{R}}\frac{1}{2}v(x,t)^2d\rho_t(x)+\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{1}{2}W(x-y)d\rho_t(x)d\rho_t(y)
\end{equation}
is essentially nonincreasing in $t$ and which satisfies the one sided Lipschitz condition
\begin{equation}
(v(x,t)-v(y,t))(x-y)\le \frac{\sqrt{c}}{\tanh(\sqrt{c}t)}(x-y)^2
\end{equation}
for $\rho_t$ almost every $x,y\in \mathbb{R}$.
\subsection{Prior work}
We have already established the existence of a weak solution pair to the pressureless Euler system with an even, continuously differentiable, semiconvex potential. In \cite{Hynd2}, we generated this solution via a Borel probability measure $\eta$ on the space of continuous paths $\Gamma:=C([0,\infty))$ endowed with the topology of local uniform convergence. Specifically, we constructed an $\eta$ which satisfies: $(i)$ for each bounded, continuous $h:\mathbb{R}\rightarrow \mathbb{R}$ and almost every $t\ge 0$
\begin{equation}
\int_{\Gamma}\dot\gamma(t) h(\gamma(t))d\eta(\gamma)=\int_{\Gamma}\left[v_0(\gamma(0))-\int^t_0(W'*\rho_s)(\gamma(s))ds \right]h(\gamma(t))d\eta(\gamma),
\end{equation}
where $\rho_s\in {\cal P}(\mathbb{R})$ is defined via
$$
\int_{\mathbb{R}}h(x)d\rho_s(x)=\int_{\Gamma}h(\gamma(s))d\eta(\gamma);
$$
$(ii)$ there is a Borel $v:\mathbb{R}\times[0,\infty)\rightarrow \mathbb{R}$, such that
$$
\dot\gamma(t)=v(\gamma(t),t),\quad\text{a.e.}\;t>0
$$
for $\eta$ almost every $\gamma\in \Gamma$. Then we checked that $\rho:[0,\infty)\rightarrow {\cal P}(\mathbb{R}); t\mapsto \rho_t$ and $v$ is indeed a weak solution pair.
\par Along the way, we derived some specific information on $\eta$ such as it is concentrated on absolutely continuous paths, it satisfies various energy estimates, and
\begin{equation}
\frac{|\gamma(t)-\xi(t)|}{\sinh(\sqrt{c}t)}\le \frac{|\gamma(s)-\xi(s)|}{\sinh(\sqrt{c}s)}.
\end{equation}
for $0<s\le t$ and $\eta$ almost every $\gamma,\xi\in\Gamma$. We consider Theorem \ref{mainThm} to be a
refinement of the main result in \cite{Hynd2} as it tells us that we can choose $\eta$ as the push forward of $\rho_0$ under the map $\mathbb{R}\mapsto \Gamma; y\mapsto X(y,\cdot)$. Here $X(y,\cdot)$ is the path $t\mapsto X(y,t)$, which is continuous for $\rho_0$ almost every $y\in\mathbb{R}$. That is, $\eta$ can be specified as
$$
\int_{\Gamma}F(\gamma)d\eta(\gamma)=\int_{\mathbb{R}}F(X(y,\cdot))d\rho_0(y)
$$
for each $F:\Gamma\rightarrow \mathbb{R}$ that is continuous and bounded.
\par There have been many other works on pressureless Euler type systems in one spatial dimension.
Especially since they are special cases of the multidimensional systems of equations which arise in the study of galaxy formation \cite{Gurbatov, Zeldovich}. One of the early mathematical works on this topic was by E, Rykov and Sinai \cite{ERykovSinai}, where they studied the case $W(x)=|x|$ which corresponds to gravitational interaction between a collection of interacting particles constrained to move along the real line. We acknowledge that the existence of solutions for this particular case does not follow from Theorem \ref{mainThm} as $W(x)=|x|$ isn't continuously differentiable. Nevertheless, we will revisit this particular case below.
\par Another very influential study on this topic was done by Brenier, Gangbo, Savar\'e and Westdickenberg \cite{BreGan}. In comparison to our work, they considered general interactions which could be attractive or repulsive. We also note that they recast the pressureless Euler equations in another coordinate system, and they were able to obtain precise information about solutions from the resulting differential inclusions. Other work with related approaches were done by Gangbo, Nguyen, and Tudorascu \cite{GNT} and Nguyen and Tudorascu \cite{NguTud} on the Euler-Poisson system and by Brenier and Grenier \cite{BreGre}, Natile and Savar\'e \cite{NatSav}, and
Cavalletti, Sedjro and Westdickenberg \cite{MR3296602} for the sticky particle system ($W\equiv 0$ in \eqref{PEE} or equation \eqref{SPS} below). We also recommend the additional references \cite{Bezard,Liu,Guo,Jabin,Jin,LeFloch,Makino,Shen} for results on stationary solutions, local existence, uniqueness, and hydrodynamic limits related to pressureless Euler type systems.
\par The particular approach we take in this paper is motivated by the work of Dermoune \cite{Dermoune} on the sticky particle system
\begin{equation}\label{SPS}
\begin{cases}
\hspace{.272in}\partial_t\rho +\partial_x(\rho v)=0\\
\partial_t(\rho v) +\partial_x(\rho v^2)=0.
\end{cases}
\end{equation}
In particular, Dermoune was the first to identify that
\begin{equation}\label{SPSFlowMapEqn}
\dot X(t)=\E_{\rho_0}\left[v_0| X(t)\right], \quad a.e.\;t\ge 0
\end{equation}
is the natural equation for the sticky particle system in Lagrangian variables. We performed a thorough analysis of \eqref{SPSFlowMapEqn} in \cite{MR4007615} and regard Theorem \ref{mainThm} as a significant generalization of the main results of \cite{MR4007615}.
\subsection{Euler-Poisson equations}
As mentioned above, we will also consider the {\it Euler-Poisson equations}
\begin{equation}\label{EP1D}
\begin{cases}
\hspace{.272in}\partial_t\rho +\partial_x(\rho v)=0\\
\partial_t(\rho v) +\partial_x(\rho v^2)=-\rho(\text{sgn}*\rho).
\end{cases}
\end{equation}
Here
\begin{equation}
\text{sgn}(x):=
\begin{cases}
1,\quad &x>0\\
0,\quad &x=0\\
-1,\quad &x<0,
\end{cases}
\end{equation}
and the associated interaction potential is $W(x)=|x|$. This system governs the dynamics of a collection of particles in which the force on each particle is proportional to the total mass to the right of the particle minus the total mass to the left of the particle; when particles collide, they undergo perfectly inelastic collisions \cite{BreGan, ERykovSinai}. This is a simple model for gravitationally interacting particles which are constrained to move on the real line.
\par As we did for the pressureless Euler system, we will design a trajectory mapping $X$ which satisfies the {\it Euler-Poisson flow equation}
\begin{equation}\label{FlowMapEqnEP}
\dot X(t)=\E_{\rho_0}\left[v_0-\displaystyle\int^t_0(\text{sgn}*\rho_s)(X(s))ds\bigg| X(t)\right], \quad a.e.\;t\ge 0
\end{equation}
and the initial condition \eqref{xInit}. However, since $\text{sgn}$ is not continuous, we will have to argue a bit differently than we did to prove Theorem \ref{mainThm} in order to obtain the following theorem.
\begin{thm}\label{secondThm}
There is a Lipschitz continuous $X:[0,\infty)\rightarrow L^2(\rho_0)$ which satisfies the Euler-Poisson flow equation \eqref{FlowMapEqnEP} and the initial condition \eqref{xInit}. Moreover, $X$ has the following properties.
\begin{enumerate}[(i)]
\item For Lebesgue almost every $t,s\in [0,\infty)$ with $s\le t$
$$
E(t)\le E(s),
$$
where
\begin{equation}
E(\tau):=\int_{\mathbb{R}}\frac{1}{2}\dot X(\tau)^2d\rho_0+\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{1}{2}|X(y,\tau)-X(z,\tau)|d\rho_0(y)d\rho_0(z).
\end{equation}
\item For $t\ge 0$ and $y,z\in\textup{supp}(\rho_0)$ with $y\ge z$,
\begin{equation}
0\le X(y,t)-X(z,t)\le y-z+t\int^y_z|v'_0(x)|dx.
\end{equation}
\item For $0<s\le t$ and $y,z\in\textup{supp}(\rho_0)$,
\begin{equation}
\frac{1}{t}|X(y,t)-X(z,t)|\le \frac{1}{s}|X(y,s)-X(z,s)|.
\end{equation}
\end{enumerate}
\end{thm}
\par As with the pressureless Euler system, we will be able to generate a weak solution pair of the Euler-Poisson system a the solution $X$ obtained in Theorem \ref{secondThm}. Namely, $\rho$ defined in \eqref{PushForwardMeasure} and any Borel $v$ which satisfies \eqref{EulerLagrangeCoord} is a weak solution of the Euler-Poisson system whose
total energy
\begin{equation}
\int_{\mathbb{R}}\frac{1}{2}v(x,t)^2d\rho_t(x)+\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{1}{2}|x-y|d\rho_t(x)d\rho_t(y)
\end{equation}
is nonincreasing in time and which fulfills the ``entropy" inequality
\begin{equation}
(v(x,t)-v(y,t))(x-y)\le \frac{1}{t}(x-y)^2
\end{equation}
for $\rho_t$ almost every $x,y\in \mathbb{R}$.
\\
\par The organization of this paper is as follows. First, we will review a few preliminaries needed for our study in section \ref{PrelimSect}. Then we will show by a near explicit construction how to solve the pressureless Euler flow equation when the support of $\rho_0$ is finite in section \ref{StickyParticleSect}. In section \ref{CompactSect}, we will analyze these special solutions and show they are compact in a certain sense. This compactness will allow us to solve the pressureless Euler flow equation for a general $\rho_0$ and consequently to solve the pressureless Euler equations for given initial conditions. Finally, in section \ref{EPsect}, we will show how to alter the arguments we used for the pressureless Euler flow equation to solve the flow equation associated with the Euler-Poisson equations.
\section{Preliminaries}\label{PrelimSect}
In this section, we will briefly recall the facts we will need regarding the convergence of probability measures and conditional expectation.
\subsection{Convergence of probability measures}
As in the introduction, we denote ${\cal P}(\mathbb{R}^d)$ as the space of Borel probability measures on $\mathbb{R}^d$. We will also write $C_b(\mathbb{R}^d)$ for the space of bounded continuous functions on $\mathbb{R}^d$. We will say that a sequence $(\mu^k)_{k\in\mathbb{N}}\subset {\cal P}(\mathbb{R}^d)$ converges to $\mu$ in ${\cal P}(\mathbb{R}^d)$ {\it narrowly} provided
\begin{equation}\label{gConvUndermukay}
\lim_{k\rightarrow\infty}\int_{\mathbb{R}^d}gd\mu^k=\int_{\mathbb{R}^d}gd\mu
\end{equation}
for every $g\in C_b(\mathbb{R}^d)$. It turns out that $(\mu^k)_{k\in\mathbb{N}}$ converges to $\mu$ narrowly if and only if $\lim_{k\rightarrow\infty}\mathcal{d}(\mu,\mu^k)=0$, where $\mathcal{d}$ is a metric of the form
\begin{equation}\label{NarrowMetric}
\mathcal{d}(\mu,\nu):=\sum^\infty_{j=1}\frac{1}{2^j}\left|\int_{\mathbb{R}^d}h_jd\mu-\int_{\mathbb{R}^d}h_jd\nu\right|,\quad \mu,\nu\in {\cal P}(\mathbb{R}^d).
\end{equation}
Here each $h_j:\mathbb{R}^d\rightarrow \mathbb{R}$ satisfies
\begin{equation}
|h_j(x)|\le 1\quad \text{and}\quad |h_j(x)-h_j(y)|\le |x-y|
\end{equation}
for $x,y\in \mathbb{R}^d$ (Remark 5.1.1 of \cite{AGS}). Furthermore, $({\cal P}(\mathbb{R}^d), \mathcal{d})$ is a complete metric space.
\par We will need to be able to identify when a sequence of measures in ${\cal P}(\mathbb{R}^d)$ has a narrowly convergent subsequence. Fortunately, Prokhorov's theorem provides a necessary and sufficient condition; it asserts that $(\mu^k)_{k\in\mathbb{N}}\subset {\cal P}(\mathbb{R}^d)$ has a narrowly convergent subsequence if and only if there is $\varphi: \mathbb{R}^d\rightarrow [0,\infty]$ with compact sublevel sets such that
\begin{equation}\label{prohorovCond}
\sup_{k\in \mathbb{N}}\int_{\mathbb{R}^d}\varphi d\mu^k<\infty
\end{equation}
(Theorem 5.1.3 of \cite{AGS}). In addition, we will need to know when \eqref{gConvUndermukay} holds for unbounded $g$. It turns out that if $g: \mathbb{R}^d\rightarrow \mathbb{R}$ is continuous and
$$
\lim_{R\rightarrow\infty}\int_{|g|\ge R}|g|d\mu^k=0
$$
uniformly in $k\in \mathbb{N}$, then \eqref{gConvUndermukay} holds (Lemma 5.1.7 of \cite{AGS}). In this case, we say that $|g|$ is {\it uniformly integrable} with respect to the sequence $(\mu^k)_{k\in\mathbb{N}}$.
\par The following lemma will also prove to be useful.
\begin{lem}[Lemma 2.1 of \cite{MR4007615}]\label{LemmaVariantNarrowCon}
Suppose $(g^k)_{k\in\mathbb{N}}$ is a sequence of continuous functions on $\mathbb{R}^d$ which converges locally uniformly to $g$ and $(\mu^k)_{k\in\mathbb{N}}\subset {\cal P}(\mathbb{R}^d)$ converges narrowly to $\mu$. Further assume there is $h:\mathbb{R}^d\rightarrow [0,\infty)$ with compact sublevel sets, which is uniformly integrable with respect to $(\mu^k)_{k\in\mathbb{N}}$ and satisfies
\begin{equation}\label{gkaydominatedbyH}
|g^k|\le h
\end{equation}
for each $k\in \mathbb{N}$. Then
\begin{equation}\label{UniformonvUndermukay}
\lim_{k\rightarrow\infty}\int_{\mathbb{R}^d}g^kd\mu^k=\int_{\mathbb{R}^d}gd\mu.
\end{equation}
\end{lem}
\subsection{The push-forward}
Suppose $f:\mathbb{R}^d\rightarrow \mathbb{R}^n$ is Borel measurable and $\mu\in {\cal P}(\mathbb{R}^d)$. We define the {\it push-forward} of $\mu$ through $f$ as the probability measure $f_{\#}\mu\in {\cal P}(\mathbb{R}^n)$ which satisfies
$$
\int_{\mathbb{R}^n}g(y)d(f_{\#}\mu)(y)=\int_{\mathbb{R}^d}g(f(x))d\mu(x)
$$
for every $g\in C_b(\mathbb{R}^n)$. We note
$$
f_{\#}\mu(A)=\mu(f^{-1}(A))
$$
for Borel $A\subset \mathbb{R}^n$. Moreover, if $f$ is continuous and $\mu^k\rightarrow \mu$ narrowly in ${\cal P}(\mathbb{R}^d)$, then
$$
f_{\#}\mu^k\rightarrow f_{\#}\mu
$$
in ${\cal P}(\mathbb{R}^n)$.
\subsection{Conditional expectation}
Suppose $\mu\in {\cal P}(\mathbb{R})$, $g\in L^2(\mu)$ and $Y: \mathbb{R}\rightarrow \mathbb{R}$ is a Borel measurable function.
A {\it conditional expectation} of $g$ with respect to $\mu$ given $Y$ is an $L^2(\mu)$ function $\E_{\mu}[g|Y]$ which satisfies two conditions: $(i)$
\begin{equation}\label{IntegralCondExpCond}
\displaystyle\int_{\mathbb{R}}\E_\mu[g|Y]\; h(Y)d\mu=\int_{\mathbb{R}}g\; h(Y)d\mu
\end{equation}
for each Borel $h:\mathbb{R}\rightarrow \mathbb{R}$ with $h(Y):=h\circ Y\in L^2(\mu)$; and $(ii)$
$$
\displaystyle\E_{\mu}[g|Y]=f(Y)
$$
$\mu$ almost everywhere for a Borel $f:\mathbb{R}\rightarrow \mathbb{R}$ with $f(Y)\in L^2(\mu)$. The existence and $\mu$ almost everywhere uniqueness of a conditional expectation can be proved using the Radon-Nikodym theorem.
\par We emphasize that $X:[0,\infty)\rightarrow L^2(\rho_0)$ satisfies the pressureless Euler flow equation \eqref{FlowMapEqn}, provided the following two conditions hold for almost every $t\ge 0$: $(i)$
\begin{equation}
\int_{\mathbb{R}}g(X(t))\dot X(t)d\rho_0=\int_{\mathbb{R}}g(X(t))\left[v_0-\displaystyle\int^t_0(W'*\rho_\tau)(X(\tau))d\tau\right]d\rho_0
\end{equation}
for each $g\in C_b(\mathbb{R})$; and $(ii)$ there exists a Borel $u: \mathbb{R}\rightarrow \mathbb{R}$ for which
$$
\dot X(t)=u(X(t))
$$
$\rho_0$ almost everywhere.
\section{Sticky particle trajectories}\label{StickyParticleSect}
In this section, we will assume that $\rho_0$ is a convex combination of Dirac measures
\begin{equation}\label{discreterhozero}
\rho_0:=\sum^N_{i=1}m_i\delta_{x_i}\in {\cal P}(\mathbb{R}).
\end{equation}
In particular, we suppose that $x_1,\dots, x_N\in \mathbb{R}$ are distinct and $m_1,\dots, m_N>0$ with $\sum^N_{i=1}m_i=1$. We also define
$$
v_i:=v_0(x_i)
$$
for $i=1,\dots, N$. It turns out that there is a natural ODE system related to the pressureless
Euler flow equation, which is
\begin{equation}\label{gammaODEt}
\ddot \gamma_i(t)=-\sum^N_{j=1}m_jW'(\gamma_i(t)-\gamma_j(t)).
\end{equation}
These are Newton's equations for $N$ interacting particles with masses $m_1,\dots, m_N$; the positions of these particles
are described by the trajectories $\gamma_1,\dots, \gamma_N$.
\par It turns out that a solution of the pressureless Euler flow equation can be built from these particle trajectories by first setting
$$
X(x_i,t)=\gamma_i(t), \quad t\ge 0.
$$
However, when trajectories intersect, we must
modify the paths. Remarkably, the natural thing to do is to require that the corresponding particles undergo perfectly
inelastic collisions when they collide. This amounts to requiring that the trajectories coincide and
that their slopes average from the moment they intersect. On any time interval when no collisions
occur, the resulting trajectories will satisfy \eqref{gammaODEt}. We will call these paths {\it sticky particle trajectories}
and we shall see that they are the building blocks for more general solutions.
\begin{figure}[h]
\centering
\includegraphics[width=.8\textwidth]{SPTrajFig}
\caption{A schematic of sticky particle trajectories for $N=6$. We have indicated the starting positions $x_1, \dots, x_6$ on the real line and we sketched the corresponding point masses larger than points to emphasize that their masses may be distinct. The path $\gamma_5$ that tracks point mass $m_5$ is shown in dashed along with its initial velocity $v_5$. Observe that there is a collision at time $s$ between the point masses $m_3, m_4$ and $m_5$. As a result, the slope $w$ satisfies $(m_3+m_4+m_5)w=m_3\dot\gamma_3(s-)+m_4\dot\gamma_4(s-)+m_5\dot\gamma_5(s-)$.
}\label{Fig1}
\end{figure}
\par The following proposition asserts that these trajectories exist and satisfy a few basic properties.
\begin{prop}[Proposition 2.1 \cite{Hynd2}]\label{StickyParticlesExist}
There are continuous, piecewise $C^2$ paths
$$
\gamma_1,\dots,\gamma_N : [0,\infty)\rightarrow \mathbb{R}
$$
with the following properties. \\
(i) For $i=1,\dots, N$ and all but finitely many $t\in (0,\infty)$, \eqref{gammaODEt} holds. \\
(ii) For $i=1,\dots, N$,
$$
\gamma_i(0)=x_i\quad \text{and}\quad \dot\gamma_i(0+)=v_i.
$$
(iii) For $i,j=1,\dots, N$, $0\le s\le t$ and $\gamma_i(s)=\gamma_j(s)$ imply
$$
\gamma_i(t)=\gamma_j(t).
$$
(iv) If $t>0$, $\{i_1,\dots, i_k\}\subset\{1,\dots, N\}$, and
$$
\gamma_{i_1}(t)=\dots=\gamma_{i_k}(t)\neq \gamma_i(t)
$$
for $i\not\in\{i_1,\dots, i_k\}$, then
$$
\dot\gamma_{i_j}(t+)=\frac{m_{i_1}\dot\gamma_{i_1}(t-)+\dots+m_{i_k}\dot\gamma_{i_k}(t-)}{m_{i_1}+\dots+m_{i_k}}
$$
for $j=1,\dots, k$.
\end{prop}
\begin{rem}\label{interchangeRemark}
Using property $(i)$, it is routine to check that $\dot\gamma_i(t\pm)$ both exist for each $t>0$ and $i=1,\dots, N$. Moreover,
$$
\dot\gamma_i(t\pm)=\lim_{h\rightarrow 0^\pm}\frac{\gamma_i(t+h)-\gamma_i(t)}{h}.
$$
\end{rem}
\par A corollary of property $(iv)$ above is the what we call the {\it averaging property}. It is a general assertion about the conservation
of momentum and is stated as follows.
\begin{cor}[Proposition 2.6 of \cite{Hynd2}]\label{AveragingProp}
Suppose $g:\mathbb{R}\rightarrow \mathbb{R}$ and $0\le s<t$. Then
\begin{align}\label{AveragingProp}
\sum^N_{i=1}m_ig(\gamma_i(t))\dot\gamma_i(t+)= \sum^N_{i=1}m_ig(\gamma_i(t))\left[\dot\gamma_i(s+)-\int^t_s\left(\sum^N_{j=1}m_jW'(\gamma_i(\tau)-\gamma_j(\tau))\right)d\tau \right]. \;\;\;
\end{align}
\end{cor}
\subsection{Quantitative stickiness }
Recall our standing assumption that there is a constant $c>0$ chosen so that $W(x)+(c/2)x^2$ is convex. In terms of this constant,
we can quantify $(iii)$ in Proposition \ref{StickyParticlesExist}. Namely, we can estimate the distance $|\gamma_i(t)-\gamma_j(t)|$ in terms of
the distance $|\gamma_i(s)-\gamma_j(s)|$ for $s\le t$. This is why we call the following assertion the {\it quantitative sticky particle property}.
\begin{prop}[Proposition 2.5 of \cite{Hynd2}]\label{PropQSPP}
For each $i,j=1,\dots,N$ and $0<s\le t$
\begin{equation}
\frac{|\gamma_i(t)-\gamma_j(t)|}{\sinh(\sqrt{c}t)}
\le \frac{|\gamma_i(s)-\gamma_j(s)|}{\sinh(\sqrt{c}s)}.
\end{equation}
\end{prop}
\par An immediate corollary is as follows.
\begin{prop}\label{measurabilityProp}
For each $0<s\le t$, there is a function $f_{t,s}:\mathbb{R}\rightarrow \mathbb{R}$ for which
$$
\gamma_i(t)=f_{t,s}(\gamma_i(s))
$$
for $i=1,\dots, N$ and
\begin{equation}\label{fteessLip}
|f_{t,s}(x)-f_{t,s}(y)|\le \frac{\sinh(\sqrt{c}t)}{\sinh(\sqrt{c}s)}|x-y|
\end{equation}
for $x,y\in \mathbb{R}.$
\end{prop}
\begin{proof}
By property $(iii)$ of Proposition \ref{StickyParticlesExist}, the cardinality of the set
$$
\{\gamma_1(t),\dots,\gamma_N(t)\}
$$
is nonincreasing in $t$. It follows that there is a surjective function
$$
g_{t,s}:\{\gamma_1(s),\dots,\gamma_N(s)\}\rightarrow \{\gamma_1(t),\dots,\gamma_N(t)\}; \gamma_i(s)\mapsto \gamma_i(t)
$$
for $0< s\le t$. By the quantitative sticky particle property, $g_{t,s}$ satisfies the Lipschitz condition \eqref{fteessLip}. We can then extend $g_{t,s}$ to all of $\mathbb{R}$ in order to obtain the desired Lipschitz function $f_{t,s}$.
\end{proof}
\subsection{Energy estimates}
Sticky particle trajectories have nonincreasing energy. That is,
\begin{equation}\label{ContNonincreaseEnergy}
\frac{1}{2}\sum^N_{i=1}m_i\dot\gamma_i(t+)^2+
\frac{1}{2}\sum^N_{i,j=1}m_im_jW(\gamma_i(t)-\gamma_j(t))\le \frac{1}{2}\sum^N_{i=1}m_i\dot\gamma_i(s+)^2+
\frac{1}{2}\sum^N_{i,j=1}m_im_jW(\gamma_i(s)-\gamma_j(s))
\end{equation}
for $0\le s<t$ (Proposition 2.8 of \cite{Hynd2}). Using the semiconvexity of $W$, we can derive the subsequent kinetic energy estimates. We will express this result in terms of the increasing function
$$
\vartheta(t):=e^{(c+1)t^2}\int^t_{0}e^{-(c+1)s^2}ds,\quad t\ge 0.
$$
\begin{lem}\label{EnergyEst}
For each $t\ge 0$,
\begin{equation}\label{DiscreteEnergyEst1}
\int^t_{0}\sum^N_{i=1}m_i\dot\gamma_i(s)^2ds\le \left(\sum^N_{i=1}m_iv_i^2+\frac{1}{2}\sum^N_{i,j=1}m_im_jW'(x_i-x_j)^2 \right)\vartheta(t).
\end{equation}
And for all but finitely many $t\ge 0$,
\begin{equation}\label{DiscreteEnergyEst2}
\sum^N_{i=1}m_i\dot\gamma_i(t)^2\le \left(\sum^N_{i=1}m_iv_i^2+\frac{1}{2}\sum^N_{i,j=1}m_im_jW'(x_i-x_j)^2 \right)\vartheta'(t).
\end{equation}
\end{lem}
\begin{proof}
Due to the convexity of $x\mapsto W(x)+(c/2)x^2$,\begin{align*}
W(\gamma_i(t)-\gamma_j(t))&\ge W(x_i-x_j)+ W'(x_i-x_j)(\gamma_i(t)-x_i-(\gamma_j(t)-x_j))\\
&\hspace{2in} -\frac{c}{2}(\gamma_i(t)-x_i-(\gamma_j(t)-x_j))^2 \\
&\ge W(x_i-x_j)- \frac{1}{2}W'(x_i-x_j)^2 -\frac{c+1}{2}(\gamma_i(t)-x_i-(\gamma_j(t)-x_j))^2 \\
&\ge W(x_i-x_j)- \frac{1}{2}W'(x_i-x_j)^2 -(c+1)((\gamma_i(t)-x_i)^2+(\gamma_j(t)-x_j)^2) \\
&\ge W(x_i-x_j)- \frac{1}{2}W'(x_i-x_j)^2 -(c+1)t\left( \int^t_{0}\dot\gamma_i(s)^2ds+\int^t_{0}\dot\gamma_j(s)^2ds \right).
\end{align*}
Combining these lower bounds with \eqref{ContNonincreaseEnergy} at $s=0$ gives
\begin{equation}\label{SteptoSecondGronwall}
\sum^N_{i=1}m_i\dot\gamma_i(t)^2\le \sum^N_{i=1}m_iv_i^2+\frac{1}{2}\sum^N_{i,j=1}m_im_jW'(x_i-x_j)^2+2(c+1)t\int^t_{0}\sum^N_{i=1}m_i\dot\gamma_i(s)^2ds
\end{equation}
for all but finitely many $t\ge 0$. As a result,
\begin{align*}
&\frac{d}{dt}e^{-(c+1)t^2}\int^t_{0}\sum^N_{i=1}m_i\dot\gamma_i(s)^2ds\\
&\quad =e^{-(c+1)t^2}\left(\sum^N_{i=1}m_i\dot\gamma_i(t)^2-2(c+1) t\int^t_{0}\sum^N_{i=1}m_i\dot\gamma_i(s)^2ds\right)\\
&\quad \le e^{-(c+1)t^2}\left(\sum^N_{i=1}m_iv_i^2 +\frac{1}{2}\sum^N_{i,j=1}m_im_jW'(x_i-x_j)^2\right)
\end{align*}
for all but finitely many $t\ge 0$. We can then integrate from $0$ to $t$ to derive \eqref{DiscreteEnergyEst1}. Inequality \eqref{DiscreteEnergyEst2} follows from \eqref{DiscreteEnergyEst1} and \eqref{SteptoSecondGronwall}.
\end{proof}
\subsection{Stability estimate}
We need one more estimate that depends on the following elementary lemma.
\begin{lem}\label{ODEineqLemma2}
Suppose $T>0$ and $y:[0,T)\rightarrow \mathbb{R}$ is continuous and piecewise $C^2$. Further assume
\begin{equation}\label{slopeDecreaseY}
\dot y(t+)\le \dot y(t-)
\end{equation}
for each $t\in (0,T)$ and that there is $c>0$ for which
\begin{equation}\label{YdoubleprimelessthanY}
\ddot y(t)\le c y(t)
\end{equation}
for all but finitely many $t\in (0,T)$. Then
\begin{equation}
y(t)\le \cosh(\sqrt{c}t)y(0)+\frac{1}{\sqrt{c}}\sinh(\sqrt{c}t)\dot y(0+).
\end{equation}
for $t\in [0,T)$.
\end{lem}
\begin{proof} By a routine scaling argument, it suffices to verify this assertion for $c=1$. To this end, we suppose in addition that there are times $0<t_1<\dots<t_n$ for which $y$ is $C^2$ on the intervals $(0,t_1),\dots, (t_n,T)$.
\par Define
$$
u(t):=\frac{y(t)}{\cosh(t)}, \quad t\in (0,T)
$$
so that $y(t)=u(t)\cosh(t).$ Observe
$$
\dot u(t)=\frac{\dot y(t)}{\cosh(t)}-\frac{y(t)}{\cosh(t)^2}\sinh(t)
$$
and
\begin{align*}
\ddot y(t) & = \ddot u(t)\cosh(t)+2\dot u(t)\sinh(t)+u(t)\cosh(t)\\
&=\ddot u(t)\cosh(t)+2\dot u(t)\sinh(t)+y(t)\\
&\le y(t)
\end{align*}
for $t\in (0,T)\setminus\{t_1,\dots, t_n\}$. Consequently,
\begin{equation}\label{UdotLessThan}
\frac{d}{dt}\left(\dot u(t)\cosh(t)^2\right)=\cosh(t)\left(\ddot u(t)\cosh(t)+2\dot u(t)\sinh(t)\right)\le 0
\end{equation}
for $t\in (0,T)\setminus\{t_1,\dots, t_n\}$.
\par In view of \eqref{UdotLessThan},
$$
\dot u(t)\cosh(t)^2\le \dot u(0+)= \dot y(0+)
$$
for $t\in (0,t_1).$ Multiplying through by $\text{sech}(t)^2$ and integrating from $0$ to $t$ gives
\begin{equation}\label{initialEstimateonU}
u(t)\le y(0)+\dot y(0+)\tanh(t)\quad t\in[0,t_1].
\end{equation}
That is,
\begin{equation}\label{initialEstimateonY}
y(t)=\cosh(t)u(t)\le
\cosh(t)y(0)+\sinh(t)\dot y(0+)
\end{equation}
for $t\in[0,t_1]$.
\par By \eqref{slopeDecreaseY} and \eqref{UdotLessThan}, we likewise have
$$
\dot u(t)\cosh(t)^2\le \dot u(t_1+)\cosh(t_1)^2\le \dot u(t_1-)\cosh(t_1)^2\le u(0+)= \dot y(0+)
$$
for $t\in (t_1,t_2)$. Again we multiply through by $\text{sech}(t)^2$ and integrate from $t_1$ to $t\in (t_1,t_2)$ to get
\begin{align*}
u(t)&\le u(t_1)+ \dot y(0+)(\tanh(t)-\tanh(t_1))\\
&\le y(0)+\dot y(0+)\tanh(t_1)+\dot y(0+)(\tanh(t)-\tanh(t_1))\\
&\le y(0)+\dot y(0+)\tanh(t).
\end{align*}
In particular, \eqref{initialEstimateonY} holds for $t\in [t_1,t_2]$. We can argue similarly to show that
\eqref{initialEstimateonY} also holds on the intervals $[t_2,t_3], \dots, [t_n,T)$ .
\end{proof}
This leads to a stability estimate.
\begin{prop}\label{StabilityProp}
Suppose $i,j\in\{1,\dots, N\}$, $x_i\ge x_j$ and $t\ge 0$. Then
\begin{equation}\label{timeZeroEst}
\gamma_i(t)-\gamma_j(t)\le \cosh(\sqrt{c}t)(x_i-x_j)+\frac{1}{\sqrt{c}}\sinh(\sqrt{c}t)\int^{x_i}_{x_j}|v_0'(x)|dx.
\end{equation}
\end{prop}
\begin{proof}
Without loss of generality, we may assume $x_1\le\dots\le x_N$ so that the sticky particle trajectories are ordered $\gamma_1\le \dots\le\gamma_N$. Under this assumption, it suffices to verify
\begin{equation}\label{timeZeroEstiplusone}
\gamma_{i+1}(t)-\gamma_i(t)\le \cosh(\sqrt{c}t)(x_{i+1}-x_i)+\frac{1}{\sqrt{c}}\sinh(\sqrt{c}t)|v_{i+1}-v_i|
\end{equation}
for $t\ge 0$. For if $j,k\in \{1,\dots, N\}$ with $k>j$,
\begin{align}
\gamma_k(t)-\gamma_j(t)&= \sum^{k-1}_{i=j}(\gamma_{i+1}(t)-\gamma_i(t))\\
&\le \sum^{k-1}_{i=j}\left(\cosh(\sqrt{c}t)(x_{i+1}-x_i)+\frac{1}{\sqrt{c}}\sinh(\sqrt{c}t)
|v_{i+1}-v_i|\right)\\
&= \cosh(\sqrt{c}t)(x_k-x_j)+\frac{1}{\sqrt{c}}\sinh(\sqrt{c}t)\sum^{k-1}_{i=j}|v_{i+1}-v_i|\\
&\le \cosh(\sqrt{c}t)(x_k-x_j)+\frac{1}{\sqrt{c}}\sinh(\sqrt{c}t)\sum^{k-1}_{i=j}\int^{x_{i+1}}_{x_i}|v_0'(x)|dx\\
&= \cosh(\sqrt{c}t)(x_k-x_j)+\frac{1}{\sqrt{c}}\sinh(\sqrt{c}t)\int^{x_{k}}_{x_j}|v_0'(x)|dx.
\end{align}
\par To this end, we fix $i\in\{1,\dots, N\}$ and set
$$
T:=\inf\{t\ge 0: \gamma_{i+1}(t)-\gamma_i(t)=0\}.
$$
In order to verify \eqref{timeZeroEstiplusone}, it is enough to show
\begin{equation}\label{LasttimeZeroEst}
\gamma_{i+1}(t)-\gamma_i(t)\le \cosh(\sqrt{c}t)(x_{i+1}-x_i)+\frac{1}{\sqrt{c}}\sinh(\sqrt{c}t)|v_{i+1}-v_i|,\quad t\in [0,T].
\end{equation}
We will do so by applying the previous lemma to the restriction of the function
$$
y(t):=\gamma_{i+1}(t)-\gamma_i(t), \quad t\ge 0
$$
to $[0,T)$. In particular, we note that $y(t)=0$ for $t\ge T$ whenever $T$ is finite.
\par We first claim
\begin{equation}\label{DotGmamaiplusone}
\dot\gamma_{i+1}(s+)\le \dot\gamma_{i+1}(s-).
\end{equation}
Note that if $\gamma_{i+1}$ does not have a first intersection time at $s\in (0,T)$, then $\gamma_{i+1}$ is $C^1$ near $s$ and so
$$
\dot\gamma_{i+1}(s)=\dot\gamma_{i+1}(s+)=\dot\gamma_{i+1}(s-).
$$
Alternatively let us suppose $\gamma_{i+1}$ has a first intersection time at $s$. As a result, there are trajectories $\gamma_{i+2}, \dots, \gamma_{i+r}$ $(r\ge 2)$ such that
$$
\gamma_{i+1}(s)=\gamma_{i+2}(s)=\dots= \gamma_{i+r}(s)
$$
and
\begin{equation}\label{AveragingForQSPS}
\dot\gamma_{i+j}(s+)=\frac{m_{i+1}\dot\gamma_{i+1}(s-)+\dots + m_{i+r}\dot\gamma_{i+r}(s-)}{m_{i+1}+\dots+m_{i+r}}
\end{equation}
$j=1,\dots, r$.
\par Also note that as $\gamma_{i+1}\le \gamma_{i+j}$ for $j=2,\dots, r$,
$$
\frac{\gamma_{i+1}(s+h)-\gamma_{i+1}(s)}{h}\ge \frac{\gamma_{i+j}(s+h)-\gamma_{i+j}(s)}{h}
$$
for all $h<0$ small. By Remark \ref{interchangeRemark}, we can send $h\rightarrow 0^-$ and conclude
$$
\dot\gamma_{i+1}(s-)\ge \dot\gamma_{i+j}(s-).
$$
It then follows from \eqref{AveragingForQSPS} (with $j=1$) that
$$
\dot \gamma_{i+1}(s+)\le\frac{m_{i+1}\dot\gamma_{i+1}(s-)+\dots + m_{i+r}\dot\gamma_{i+1}(s-)}{m_{i+1}+\dots+m_{i+r}}=\dot\gamma_{i+1}(s-),
$$
which is \eqref{DotGmamaiplusone}. A similar argument gives
\begin{equation}\label{DotGmamajusti}
\dot\gamma_{i}(s+)\ge \dot\gamma_{i}(s-)
\end{equation}
for each $s\in (0,T)$. Combining \eqref{DotGmamaiplusone} and \eqref{DotGmamajusti}
\begin{equation}\label{dotYnotgoingup}
\dot y(s+)=\dot\gamma_{i+1}(s+)-\dot\gamma_{i}(s+)\le \dot\gamma_{i+1}(s-)-\dot\gamma_{i}(s-)= \dot y(s-)
\end{equation}
for all $s\in (0,T)$.
\par As $x\mapsto W'(x)+cx$ is nondecreasing,
\begin{align*}
\ddot y(t)&=\ddot\gamma_i(t)-\ddot\gamma_j(t)\\
&=-\sum_{k=1}^Nm_k\left(W'(\gamma_i(t)-\gamma_k(t))-W'(\gamma_j(t)-\gamma_k(t))\right) \\
&\le \sum_{k=1}^Nm_k c\left(\gamma_i(t)-\gamma_j(t)\right)\\
&= c(\gamma_i(t)-\gamma_j(t))\\
&=y(t)
\end{align*}
for all but finitely many $t>0$. Therefore, \eqref{LasttimeZeroEst} follows from Lemma \ref{ODEineqLemma2}.
\end{proof}
\subsection{Associated trajectory map}
We are now ready to show how to design a solution of \eqref{FlowMapEqn} with $\rho_0$ given by \eqref{discreterhozero}. For $t\ge0$, we define
$$
X(t): \{x_1,\dots, x_N\}\rightarrow \mathbb{R}; x_i\mapsto \gamma_i(t).
$$
We will also write
$$
X(x_i,t)=\gamma_i(t)
$$
for $ i=1,\dots, N$ and $t\ge 0$. The following proposition details all the important features of $X$.
\begin{prop}\label{DiscreteSolnProp}
The mapping $X$ has the following properties.
\begin{enumerate}[(i)]
\item $X(0)=\textup{id}_\mathbb{R}$ and
\begin{equation}
\dot X(t)=\E_{\rho_0}\left[v_0-\displaystyle\int^t_0(W'*\rho_s)(X(s))ds\bigg| X(t)\right]
\end{equation}
for all but finitely many $t\ge 0$. Both equalities hold on the support of $\rho_0$.
\item $E(t)\le E(s)$, for $s\le t$. Here
\begin{align*}
E(\tau):=\int_{\mathbb{R}}\frac{1}{2}\dot X(\tau+)^2d\rho_0+\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{1}{2}W(X(y,\tau)-X(z,\tau))d\rho_0(y)d\rho_0(z).
\end{align*}
\item $X: [0,\infty)\rightarrow L^2(\rho_0); t\mapsto X(t)$ is locally Lipschitz continuous.
\item For $t\ge 0$ and $y,z\in\textup{supp}(\rho_0)$ with $y\le z$,
$$
0\le X(z,t)-X(y,t)\le \cosh(\sqrt{c}t)(z-y)+\frac{1}{\sqrt{c}}\sinh(\sqrt{c}t)\int^z_y|v_0'(x)|dx.
$$
\item For each $0<s\le t$ and $y,z\in\textup{supp}(\rho_0)$
$$
\frac{|X(y,t)-X(z,t)|}{\sinh(\sqrt{c}t)}\le \frac{|X(y,s)-X(z,s)|}{\sinh(\sqrt{c}s)}.
$$
\item For each $0<s\le t$, there is a function $f_{t,s}:\mathbb{R}\rightarrow \mathbb{R}$ which satisfies the Lipschitz condition \eqref{fteessLip}
and
$$
X(y,t)=f_{t,s}(X(y,s))
$$
for $y\in\textup{supp}(\rho_0)$.
\end{enumerate}
\end{prop}
\begin{proof}
\underline{Part $(i)$}: As $X(x_i,0)=x_i$,
$$
X(0)=\text{id}_{\R}
$$
on $\text{supp}(\rho_0)$. Also note that if $g:\mathbb{R}\rightarrow \mathbb{R}$ and $t\ge 0$, then Corollary \ref{AveragingProp} gives
\begin{align*}
\int_{\mathbb{R}}g(X(t))\dot X(t+)d\rho_0&=\sum^N_{i=1}m_ig(\gamma_i(t))\dot\gamma_i(t+)\\
&= \sum^N_{i=1}m_ig(\gamma_i(t))\left[\dot\gamma_i(0+)-\int^t_0\left(\sum^N_{j=1}m_jW'(\gamma_i(\tau)-\gamma_j(\tau))\right)d\tau \right]\\
&=\int_{\mathbb{R}}g(X(t))\left[v_0-\displaystyle\int^t_0(W'*\rho_\tau)(X(\tau))d\tau\right]d\rho_0.
\end{align*}
In particular,
$$
\int_{\mathbb{R}}g(X(t))\dot X(t)d\rho_0=\int_{\mathbb{R}}g(X(t))\left[v_0-\displaystyle\int^t_0(W'*\rho_\tau)(X(\tau))d\tau\right]d\rho_0
$$
for all but finitely many $t\ge 0$.
\par Define
\begin{equation}
\begin{cases}
v(x,t)=\dot\gamma_i(t+), \quad &x=\gamma_i(t)\\
\hspace{.48in}=0, \quad & \text{otherwise}.
\end{cases}
\end{equation}
By parts $(iii)$ and $(iv)$ of Proposition \ref{StickyParticlesExist}, $v$ is well defined. Moreover, it is routine to check that $v:\mathbb{R}\times[0,\infty)\rightarrow \mathbb{R}$ is Borel measurable. Furthermore,
$$
\dot X(t)=v(X(t),t)
$$
on the support of $\rho_0$ for all but finitely many $t\ge 0$. It follows that $X$ satisfies the pressureless Euler flow equation \eqref{FlowMapEqn} for all but finitely many $t\ge 0$.
\par \underline{Part $(ii)$}: In view of \eqref{ContNonincreaseEnergy},
\begin{align*}
E(t)&=\int_{\mathbb{R}}\frac{1}{2}\dot X(t+)^2d\rho_0+\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{1}{2}W(X(y,t)-X(z,t))d\rho_0(y)d\rho_0(z)\\
&=\frac{1}{2}\sum^N_{i=1}m_i\dot\gamma_i(t+)^2+\frac{1}{2}\sum^N_{i,j=1}m_im_jW(\gamma_i(t)-\gamma_j(t))\\
&\le \frac{1}{2}\sum^N_{i=1}m_i\dot\gamma_i(s+)^2+\frac{1}{2}\sum^N_{i,j=1}m_im_jW(\gamma_i(s)-\gamma_j(s))\\
&=\int_{\mathbb{R}}\frac{1}{2}\dot X(t+)^2d\rho_0+\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{1}{2}W(X(y,t)-X(z,t))d\rho_0(y)d\rho_0(z)\\
&=E(s).
\end{align*}
\par \underline{Part $(iii)$}: By the energy estimate \eqref{DiscreteEnergyEst2},
\begin{align}\label{LipEst}
\int_{\mathbb{R}}(X(t)-X(s))^2d\rho_0&\le (t-s)\int^t_s\int_{\mathbb{R}}\dot X(\tau)^2d\rho_0 d\tau\\
&= (t-s)\int^t_s\sum^N_{i=1}m_i\dot\gamma_i(\tau)^2d\tau\\
&\le (t-s)(\vartheta(t)-\vartheta(s))\left(\sum^N_{i=1}m_iv_i^2+\frac{1}{2}\sum^N_{i,j=1}m_im_jW'(x_i-x_j)^2 \right)\\
&\le (t-s)(\vartheta(t)-\vartheta(s))\left(\int_{\mathbb{R}}v_0^2d\rho_0+\frac{1}{2}\int_{\mathbb{R}}\int_{\mathbb{R}}W'(x-y)^2d\rho_0(x)d\rho_0(y)\right).
\end{align}
Since $\vartheta$ is smooth, $X:[0,\infty)\rightarrow L^2(\rho_0)$ is locally Lipschitz continuous.
\par \underline{Part $(iv)$, Part $(v)$ and $(vi)$}: Part $(iv)$ follows from Proposition \ref{StabilityProp}, Part $(v)$ is a corollary of Proposition \ref{PropQSPP}, and part $(vi)$ is due to Corollary \ref{measurabilityProp}.
\end{proof}
\par Observe that as $v_0:\mathbb{R}\rightarrow \mathbb{R}$ is absolutely continuous,
\begin{equation}\label{modulusOmega}
\omega(r):=\sup\left\{\int^b_a|v_0'(x)|dx \;:\; 0\le b-a\le r\right\}
\end{equation}
tends to $0$ as $r\rightarrow 0^+$ and is sublinear. By part $(iv)$ of the above proposition,
\begin{equation}\label{XdiscreteUnifCont}
|X(y,t)-X(z,t)|\le \cosh(\sqrt{c}t)|y-z|+\frac{1}{\sqrt{c}}\sinh(\sqrt{c}t)\omega(|y-z|)
\end{equation}
for $y,z$ belonging to the support of $\rho_0$. As a result, $X(t)$ is uniformly continuous on the support of $\rho_0$. In particular, we may extend $X(t)$ to a uniformly continuous function on $\mathbb{R}$ which satisfies \eqref{XdiscreteUnifCont} and agrees with $X(t)$ on the support of $\rho_0$. Without any loss of generality, we will identify $X(t)$ with this extension and assume $X(t)$ is uniformly continuous on $\mathbb{R}$.
\section{Existence of solutions}\label{CompactSect}
Now let $\rho_0\in {\cal P}(\mathbb{R})$ with
\begin{equation}
\int_{\mathbb{R}}x^2d\rho_0(x)<\infty.
\end{equation}
We can select a sequence $(\rho_0^k)_{k\in \mathbb{N}}\subset {\cal P}(\mathbb{R})$ such that each $\rho^k_0$ is a convex combination of Dirac measures, $\rho_0^k\rightarrow \rho_0$ narrowly, and
\begin{equation}\label{rhoQuadraticConv}
\lim_{k\rightarrow\infty}\int_{\mathbb{R}}x^2d\rho^k_0(x)=\int_{\mathbb{R}}x^2d\rho_0(x).
\end{equation}
We recommend, for instance, the reference \cite{Bolley} for a discussion on how to design such a sequence.
\par In view of Proposition \ref{DiscreteSolnProp}, there is a locally Lipschitz continuous mapping $X^k:[0,\infty)\rightarrow L^2(\rho^k_0)$ which satisfies
\begin{equation}
\begin{cases}
\dot X^k(t)=\E_{\rho_0^k}\left[v_0-\displaystyle\int^t_0(W'*\rho^k_s)(X^k(s))ds\bigg| X^k(t)\right], \quad a.e.\;t\ge 0\\
X^k(0)=\text{id}_{\R}
\end{cases}
\end{equation}
for each $k\in \mathbb{N}$. Here
$$
\rho^k_t:=X^k(t)_{\#}\rho^k_0, \quad t\ge 0.
$$
In this section, we will show that there is a subsequence of $(X^k)_{k\in \mathbb{N}}$ which converges in an appropriate sense to a solution of the pressureless Euler flow equation \eqref{FlowMapEqn} and satisfies \eqref{xInit}. Then we will show how to use this solution to generate a corresponding weak solution of the pressureless Euler equations.
\subsection{Compactness}
We will prove that $(X^{k})_{k\in \mathbb{N}}$ has a subsequence which converges in a strong sense. The limit mapping will be our candidate for a solution of the pressureless Euler flow equation.
\begin{prop}\label{StrongCompactXkayjay}
There is a subsequence $(X^{k_j})_{j\in \mathbb{N}}$ and a locally Lipschitz $X: [0,\infty)\rightarrow L^2(\rho_0)$ such that
\begin{equation}\label{StrongConvLimXkayJay}
\lim_{j\rightarrow\infty}\int_{\mathbb{R}}h(\textup{id}_\mathbb{R},X^{k_j}(t))d\rho^{k_j}_0=\int_{\mathbb{R}}h(\textup{id}_\mathbb{R},X(t))d\rho_0
\end{equation}
for each $t\ge 0$ and continuous $h:\mathbb{R}^2\rightarrow \mathbb{R}$ with
\begin{equation}
\sup_{(x,y)\in \mathbb{R}^2}\frac{|h(x,y)|}{1+x^2+y^2}<\infty.
\end{equation}
Furthermore, $X$ has the following properties.
\begin{enumerate}[(i)]
\item For $t\ge 0$ and $y,z\in\textup{supp}(\rho_0)$ with $y\le z$,
$$
0\le X(z,t)-X(y,t)\le \cosh(\sqrt{c}t)(z-y)+\frac{1}{\sqrt{c}}\sinh(\sqrt{c}t)\int^z_y|v_0'(x)|dx.
$$
\item For each $0<s\le t$ and $y,z\in\textup{supp}(\rho_0)$
$$
\frac{|X(y,t)-X(z,t)|}{\sinh(\sqrt{c}t)}\le \frac{|X(y,s)-X(z,s)|}{\sinh(\sqrt{c}s)}.
$$
\item For each $0<s\le t$, there is a function $f_{t,s}:\mathbb{R}\rightarrow \mathbb{R}$ which satisfies the Lipschitz condition \eqref{fteessLip}
and
$$
X(y,t)=f_{t,s}(X(y,s))
$$
for $y\in\textup{supp}(\rho_0)$.
\end{enumerate}
\end{prop}
\begin{proof}
\underline{1. Weak convergence}: Define
\begin{equation}
\sigma^k_t:=(\text{id}_{\R},X^{k}(t))_{\#}\rho^k_0
\end{equation}
for $t\ge 0$ and $k\in \mathbb{N}$. By \eqref{LipEst},
\begin{align*}
\left(\int_{\mathbb{R}}(X^k(t))^2d\rho^k_0\right)^{1/2}&\le\left(\int_{\mathbb{R}}(X^k(t)-X^k(0))^2d\rho_0^k\right)^{1/2}+\left(\int_{\mathbb{R}}(X^k(0))^2d\rho^k_0\right)^{1/2}\\
&\le \sqrt{t\vartheta(t)}\left(\int_{\mathbb{R}}v_0^2d\rho^k_0+\frac{1}{2}\int_{\mathbb{R}}\int_{\mathbb{R}}W'(x-y)^2d\rho^k_0(x)d\rho^k_0(y)\right)^{1/2}\\
&\hspace{1in}+\left(\int_{\mathbb{R}}x^2d\rho^k_0(x)\right)^{1/2}.
\end{align*}
And as $v_0$ and $W'$ grow at most linearly, there are constants $A, B\ge 0$
$$
\left(\int_{\mathbb{R}}(X^k(t))^2d\rho^k_0\right)^{1/2}\le A\sqrt{t\vartheta(t)} +B
$$
for each $t\ge 0$ and $k\in \mathbb{N}$.
\par It follows that
\begin{equation}\label{SecondMomentEst}
\sup_{k\in\mathbb{N}}\iint_{\mathbb{R}^2}(x^2+y^2)d\sigma^k_t(x,y)<\infty
\end{equation}
for each $t\ge 0.$ As a result, $(\sigma^k_t)_{k\in \mathbb{N}}$ is narrowly precompact. Moreover, for Lipschitz continuous $h:\mathbb{R}^2\rightarrow \mathbb{R}$,
\begin{align*}
\iint_{\mathbb{R}^2}h(x,y)d\sigma^k_t(x,y)-\iint_{\mathbb{R}^2}h(x,y)d\sigma^k_s(x,y)&=\int_{\mathbb{R}}h(\text{id}_{\R},X^k(t))d\rho^k_0-\int_{\mathbb{R}}h(\text{id}_{\R},X^k(s))d\rho^k_0\\
&\le \text{Lip}(h) \int_{\mathbb{R}}|X^k(t)-X^k(s)| d\rho^k_0\\
&\le \text{Lip}(h) \int^t_s\left(\int_{\mathbb{R}}|\dot X^k(\tau)| d\rho^k_0\right)d\tau\\
&\le \text{Lip}(h) \int^t_s (A\sqrt{\tau\vartheta(\tau)} +B)d\tau.
\end{align*}
In terms of
the metric \eqref{NarrowMetric}, we then have
\begin{equation}
\mathcal{d}(\sigma^k_t,\sigma^k_s)\le \int^t_s (A\sqrt{\tau\vartheta(\tau)} +B)d\tau
\end{equation}
for $k\in \mathbb{N}$ and $0\le s\le t$.
\par By the Arzel\`a-Ascoli, there is a subsequence $(\sigma^{k_j})_{j\in \mathbb{N}}$ and a narrowly continuous path $\sigma:[0,\infty)\rightarrow {\cal P}(\mathbb{R});t\mapsto \sigma_t$ such that $\sigma^{k_j}_t\rightarrow \sigma_t$ uniformly for $t$ belonging to compact subsets of $[0,\infty)$. We can also use this narrow convergence and \eqref{SecondMomentEst} to show
\begin{equation}
\lim_{j\rightarrow\infty}\iint_{\mathbb{R}^2}|y|d\sigma^{k_j}_t(x,y)=\iint_{\mathbb{R}^2}|y|d\sigma_t(x,y)
\end{equation}
for each $t\ge 0$. Further,
\begin{equation}
\int_{\mathbb{R}}\phi d\rho_0 =\lim_{j\rightarrow\infty}\int_{\mathbb{R}}\phi d\rho^{k_j}_0 =\lim_{j\rightarrow\infty}\iint_{\mathbb{R}^2}\phi(x)d\sigma^{k_j}_t(x,y)=\iint_{\mathbb{R}^2}\phi(x)d\sigma_t(x,y)
\end{equation}
for $\phi\in C_b(\mathbb{R})$.
\par By the disintegration theorem (Theorem 5.3.1 of \cite{AGS}), there is a family of probability measures $(\zeta^x_t)_{x\in \mathbb{R}}$ for which
\begin{equation}
\iint_{\mathbb{R}^2}h(x,y)d\sigma_t(x,y)=\int_{\mathbb{R}}\left(\int_{\mathbb{R}}h(x,y)d\zeta^x_t(y)\right)d\rho_0(x).
\end{equation}
Set
$$
X(x,t):=\int_{\mathbb{R}}yd\zeta^x_t(y), \quad (x,t)\in \mathbb{R}\times[0,\infty),
$$
and as before we will write $X(t):\mathbb{R}\rightarrow \mathbb{R}; x\mapsto X(x,t)$. Observe
\begin{align}\label{WeakConvXkayJay}
\lim_{j\rightarrow\infty}\int_{\mathbb{R}}\phi X^{k_j}(t)d\rho^{k_j}_0&=\lim_{j\rightarrow\infty}\int_{\mathbb{R}^2 }\phi(x) yd\sigma^{k_j}_t(x,y)\nonumber \\
&=\int_{\mathbb{R}^2 }\phi(x) yd\sigma_t(x,y)\nonumber\\
&=\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\phi(x) yd\zeta^x_t(y)\right)d\rho_0(x)\nonumber\\
&=\int_{\mathbb{R}}\phi(x)\left(\int_{\mathbb{R}}yd\zeta^x_t(y)\right)d\rho_0(x)\nonumber\\
&=\int_{\mathbb{R}}\phi X(t)d\rho_0
\end{align}
for each $t\ge 0$.
\par \underline{2. Strong convergence}: By \eqref{XdiscreteUnifCont},
\begin{equation}
|X^k(y,t)-X^k(z,t)|\le \cosh(\sqrt{c}t)|y-z|+\frac{1}{\sqrt{c}}\sinh(\sqrt{c}t)\omega(|y-z|)
\end{equation}
for $y,z\in \mathbb{R}$. As $\lim_{r\rightarrow 0^+}\omega(r)=0$, $(X^k(t))_{k\in \mathbb{N}}$ is uniformly equicontinuous. Moreover,
\begin{equation}
|X^k(y,t)|\le |X^k(y,t)-X^k(z,t)|+|X^k(z,t)|
\end{equation}
and integrating over $z\in \mathbb{R}$ gives
\begin{align*}
|X^k(y,t)|&\le \int_{\mathbb{R}}|X^k(y,t)-X^k(z,t)|d\rho^k_0(z)+\int_{\mathbb{R}}|X^k(z,t)|d\rho^k_0(z) \\
&\le \cosh(\sqrt{c}t)\int_{\mathbb{R}}|y-z|d\rho^k_0(z)+\frac{1}{\sqrt{c}}\sinh(\sqrt{c}t) \int_{\mathbb{R}}\omega(|y-z|)d\rho^k_0(z)\\
&\quad +\int_{\mathbb{R}}|X^k(z,t)|d\rho^k_0(z) \\
&\le \cosh(\sqrt{c}t)\int_{\mathbb{R}}|y-z|d\rho^k_0(z)+\frac{1}{\sqrt{c}}\sinh(\sqrt{c}t) \int_{\mathbb{R}}\omega(|y-z|)d\rho^k_0(z)\\
&\quad +A\sqrt{t\vartheta(t)} +B.
\end{align*}
By \eqref{rhoQuadraticConv} and the at most sublinear growth of $\omega$, there are constants $a,b\ge 1$ such that
\begin{equation}\label{XkayUpperBound}
|X^k(y,t)|\le a e^{b t^2}(|y|+1)
\end{equation}
for $k\in \mathbb{N}$, $y\in \mathbb{R}$ and $t\ge0$.
\par It follows that a subsequence of $(X^{k_j}(t))_{j\in \mathbb{N}}$ (which we will not relabel) converges locally uniformly on $\mathbb{R}$ to a continuous function $Y:\mathbb{R}\rightarrow \mathbb{R}$. It is easy to check
\begin{equation}
\lim_{j\rightarrow\infty}\int_{\mathbb{R}}\phi X^{k_j}(t)d\rho^{k_j}_0=\int_{\mathbb{R}}\phi Yd\rho_0
\end{equation}
for each $\phi\in C_b(\mathbb{R})$. So if $(X^{k_j}(t))_{j\in \mathbb{N}}$ has another subsequence which converges locally uniformly on $\mathbb{R}$ to $Z$, then
\begin{equation}
\int_{\mathbb{R}}\phi Yd\rho_0=\int_{\mathbb{R}}\phi Zd\rho_0.
\end{equation}
In particular, $Y=Z$ $\rho_0$ almost everywhere. Since $Y$ and $Z$ are continuous, it is routine to check that $Y=Z$ on the entire support of $\rho_0$.
\par By \eqref{WeakConvXkayJay}, we also have that $X(t)=Y$ $\rho_0$ almost everywhere. Without loss of generality, we can redefine $X(t)=Y$ to ensure that $X^{k_j}(t)\rightarrow X(t)$ locally uniformly on $\mathbb{R}$. By estimate \eqref{XkayUpperBound} and Proposition \ref{LemmaVariantNarrowCon}, we also have
\begin{equation}
\lim_{j\rightarrow\infty}\int_{\mathbb{R}}X^{k_j}(t)^2d\rho^{k_j}_0=\int_{\mathbb{R}}X(t)^2d\rho_0.
\end{equation}
In particular,
\begin{equation}
\lim_{j\rightarrow\infty}\iint_{\mathbb{R}^2}(x^2+y^2)d\sigma^{k_j}_t(x,y)=\iint_{\mathbb{R}^2}(x^2+y^2)d\sigma_t(x,y),
\end{equation}
which when combined the narrow converge $\sigma^{k_j}_t\rightarrow \sigma_t$ in ${\cal P}(\mathbb{R}^2)$ gives \eqref{StrongConvLimXkayJay}.
\par In view of \eqref{LipEst}, we also have
\begin{align*}
\int_{\mathbb{R}}(X(t)-X(s))^2d\rho_0&=\lim_{j\rightarrow\infty}\int_{\mathbb{R}}(X^{k_j}(t)-X^{k_j}(s))^2d\rho^{k_j}_0\\
&\le (t-s)(\vartheta(t)-\vartheta(s))\lim_{j\rightarrow\infty}\left(\int_{\mathbb{R}}v_0^2d\rho^{k_j}_0+\frac{1}{2}\int_{\mathbb{R}}\int_{\mathbb{R}}W'(x-y)^2d\rho^{k_j}_0(x)d\rho^{k_j}_0(y)\right)\\
&\le (t-s)(\vartheta(t)-\vartheta(s))\left(\int_{\mathbb{R}}v_0^2d\rho_0+\frac{1}{2}\int_{\mathbb{R}}\int_{\mathbb{R}}W'(x-y)^2d\rho_0(x)d\rho_0(y)\right).
\end{align*}
Here we used that $v_0$ and $W'$ are continuous and grow at most linearly. Therefore, the mapping $X:[0,\infty)\rightarrow L^2(\rho_0)$ is locally Lipschitz continuous.
\par \underline{3. Properties of the limit}: Suppose $y,z\in\text{supp}(\rho_0)$ with $y<z$. As $\rho^{k_j}\rightarrow \rho_0$ narrowly in ${\cal P}(\mathbb{R})$,
there are sequences $y^j,z^j\in\text{supp}(\rho^{k_j}_0)$ such that $y^j\rightarrow y$ and $z^j\rightarrow z$ (Proposition 5.1.8 in \cite{AGS}). Without any loss of generality, we may
suppose that $y^j<z^j$ for all $j\in \mathbb{N}$ as this occurs for all $j$ large enough. By part $(iv)$ of Proposition \ref{DiscreteSolnProp},
\begin{equation}
0\le X^{k_j}(z^j,t)-X^{k_j}(y^j,t)\le \cosh(\sqrt{c}t)(z^j-y^j)+\frac{1}{\sqrt{c}}\sinh(\sqrt{c}t)\int^{z^j}_{y^j}|v_0'(x)|dx.
\end{equation}
Since $X^{k^j}(t)\rightarrow X(t)$ locally uniformly, we can send $j\rightarrow\infty$ in order to conclude property $(i)$ of this theorem. Property $(ii)$ can be proved similarly.
\par Now let $0<s\le t$ and $y\in\text{supp}(\rho_0)$. As above, we may select $y^j\in\text{supp}(\rho^{k_j}_0)$ such that $y^j\rightarrow y$. Appealing to part $(vi)$ of Proposition \ref{DiscreteSolnProp}, there is a sequence of functions $(f^{k_j}_{t,s})_{j\in \mathbb{N}}$ in which each element satisfies the Lipschitz condition \eqref{fteessLip} and
\begin{equation}\label{XkayJayFinalest}
X^{k_j}(y^j,t)=f^{k_j}_{t,s}(X^{k_j}(y^j,s))
\end{equation}
for each $j\in \mathbb{N}$. In particular,
\begin{align*}
|f^{k_j}_{t,s}(x)|&\le |f^{k_j}_{t,s}(x)-f^{k_j}_{t,s}(X^{k_j}(y^j,s))|+|f^{k_j}_{t,s}(X^{k_j}(y^j,s))|\\
&\le \frac{\sinh(\sqrt{c}t)}{\sinh(\sqrt{c}s)}|x-X^{k_j}(y^j,s)|+|X^{k_j}(y^j,t)|.
\end{align*}
As $X^{k^j}(t)\rightarrow X(t)$ and $X^{k^j}(s)\rightarrow X(s)$ locally uniformly, $|f^{k_j}_{t,s}(x)|$ is bounded above for $x$ belonging to compact subsets of $\mathbb{R}$ independently of $j\in \mathbb{N}$. As a result, $(f^{k_j}_{t,s})_{j\in \mathbb{N}}$ has a locally uniformly convergent subsequence. Therefore, we can send $j\rightarrow\infty$ along an appropriate subsequence in \eqref{XkayJayFinalest} to obtain part $(iii)$.
\end{proof}
For the remainder of this section, let $X$ denote the mapping obtained in Proposition \ref{StrongCompactXkayjay}.
\begin{cor}\label{dotXfunctionofX}
For Lebesgue almost every $t>0$, there is a Borel function $u:\mathbb{R}\rightarrow \mathbb{R}$ for which
\begin{equation}
\dot X(t)=u(X(t))
\end{equation}
$\rho_0$ almost everywhere.
\end{cor}
\begin{proof}
Let $t$ be a time such that
\begin{equation}\label{dotXlimit}
\dot X(t)=\lim_{n\rightarrow \infty}n\left(X(t+1/n)-X(t)\right)
\end{equation}
in $L^2(\rho_0)$. We recall that since $X:[0,\infty)\rightarrow L^2(\rho_0)$ is locally Lipschitz, the set of all such $t$ has full measure in $[0,\infty)$. Moreover, we may suppose that the limit \eqref{dotXlimit} holds $\rho_0$ almost everywhere in $\mathbb{R}$ as it does for a subsequence. By Theorem 1.19 in \cite{Folland}, we may further assume that the limit \eqref{dotXlimit} holds everywhere on some Borel $S\subset \mathbb{R}$ with $\rho_0(S)=1$.
\par In view of part $(iii)$ of Proposition \ref{StrongCompactXkayjay},
$$
\dot X(t)=\lim_{n\rightarrow \infty}u_n\left(X(t)\right)
$$
on $S$. Here
$$
u_n:=n(f_{t+1/n,t}-\text{id}_{\R})
$$
is a Borel measurable function for each $n\in \mathbb{N}$. Consequently, $u_n\left(X(t)\right)|_S$ is measurable with respect to the Borel sigma-sub-algebra generated by $X(t)|_S$
$$
{\cal F}:=\{\{y\in S: X(y,t)\in B\}: B\subset \mathbb{R}\;\text{Borel}\}.
$$
As $\dot X(t)|_S$ is a pointwise limit of ${\cal F}$ measurable functions, it is ${\cal F}$ measurable itself (Proposition 2.7 \cite{Folland}). It follows that there is a Borel function $u:\mathbb{R} \rightarrow \mathbb{R}$ such that
$$
\dot X(t)|_S=u(X(t))|_S.
$$
That is, $\dot X(t)=u(X(t))$ $\rho_0$ almost everywhere.
\end{proof}
\begin{proof}[Proof of Theorem \ref{mainThm}] 1. \underline{Initial condition}: The limit \eqref{StrongConvLimXkayJay} taken when $t=0$ implies $X(0)=\text{id}_{\R}$ $\rho_0$ almost everywhere.
\par 2. \underline{Flow equation}: We next claim
\begin{equation}\label{SolnGoal}
\int^t_s\int_{\mathbb{R}}\dot X(\tau) h(X(\tau))d\rho_0d\tau=\int^t_s\int_{\mathbb{R}}\left[v_0-\displaystyle\int^\tau_0(W'*\rho_\xi)(X(\xi))d\xi\right]h(X(t))d\rho_0d\tau
\end{equation}
for each continuous $h:\mathbb{R}\rightarrow\mathbb{R}$ which satisfies
\begin{equation}
\sup_{x\in \mathbb{R}}\frac{|h(x)|}{1+|x|}<\infty
\end{equation}
and each $0\le s\le t$. In view of Corollary \ref{dotXfunctionofX}, this would imply that $X$ is a solution of the pressureless Euler flow equation.
To this end, we set $F(y):=\int^y_0 h(x)dx$ for $y\in \mathbb{R}$ and note that $F$ grows at most quadratically. By Proposition \ref{StrongCompactXkayjay},
\begin{align*}
\lim_{j\rightarrow\infty}\int^t_s\int_{\mathbb{R}}\dot X^{k_j}(\tau) h(X^{k_j}(\tau))d\rho^{k_j}_0d\tau&=\lim_{j\rightarrow\infty}\int_{\mathbb{R}}\left(\int^t_s\dot X^{k_j}(\tau) h(X^{k_j}(\tau))d\tau\right)d\rho^{k_j}_0\\
&=\lim_{j\rightarrow\infty}\int_{\mathbb{R}}\left(\int^t_s\frac{d}{d\tau}F(X^{k_j}(\tau))d\tau\right)d\rho^{k_j}_0\\
&=\lim_{j\rightarrow\infty}\int_{\mathbb{R}}\left(F(X^{k_j}(t))-F(X^{k_j}(s))\right)d\rho^{k_j}_0\\
&=\int_{\mathbb{R}}\left(F(X(t))-F(X(s))\right)d\rho_0.
\end{align*}
Consequently,
\begin{equation}\label{WeakConvDotXkayjay}
\lim_{j\rightarrow\infty}\int^t_s\int_{\mathbb{R}}\dot X^{k_j}(\tau) h(X^{k_j}(\tau))d\rho^{k_j}_0d\tau=\int^t_s\int_{\mathbb{R}}\dot X(\tau) h(X(\tau))d\rho_0d\tau.
\end{equation}
\par Let us fix $\tau\ge 0$ for the moment and consider the integral
\begin{align*}
&\int_{\mathbb{R}}\left[v_0-\displaystyle\int^\tau_0(W'*\rho^{k_j}_\xi)(X^{k_j}(\xi))d\xi\right] h(X^{k_j}(\tau))d\rho^{k_j}_0\\
&\quad = \int_{\mathbb{R}}v_0 h(X^{k_j}(\tau))d\rho^{k_j}_0-\int^\tau_0\left[\int_{\mathbb{R}}\int_{\mathbb{R}}W'(X^{k_j}(y,\xi)-X^{k_j}(z,\xi)) h(X^{k_j}(y,\tau)) d\rho^{k_j}_0(z)d\rho^{k_j}_0(y)\right]d\xi.
\end{align*}
In view of Proposition \ref{StrongCompactXkayjay} and the at most linear growth of $v_0$,
\begin{equation}
\lim_{j\rightarrow\infty}\int_{\mathbb{R}}v_0 h(X^{k_j}(\tau))d\rho^{k_j}_0=\int_{\mathbb{R}}v_0 h(X(\tau))d\rho_0
\end{equation}
as $j\rightarrow\infty$. Also observe
\begin{equation}
\lim_{j\rightarrow\infty}W'(X^{k_j}(y,\xi)-X^{k_j}(z,\xi)) h(X^{k_j}(y,t))=W'(X(y,\xi)-X(z,\xi)) h(X(y,\tau))
\end{equation}
locally uniformly for $(y,z)\in \mathbb{R}^2$ and each fixed $\xi\in [0,\tau]$.
\par Choosing $C$ so large that
$$
|h(x)|+|W'(x)|\le C(1+|x|)\quad (x\in \mathbb{R})
$$
gives
\begin{align}\label{SimpleConvBoundWprime}
&|W'(X^{k_j}(y,\xi)-X^{k_j}(z,\xi)) h(X^{k_j}(y,\tau))| \nonumber \\
&\quad\quad \le C(1+|X^{k_j}(y,\xi)|+ |X^{k_j}(z,\xi)|)\cdot C(1+|X^{k_j}(y,\tau)|)\nonumber \\
&\quad\quad \le C(1+a e^{b \xi^2}(|y|+1)+ a e^{b \xi^2}(|z|+1))\cdot C(1+a e^{b \tau^2}(|y|+1))\nonumber \\
&\quad\quad \le C^2(1+a e^{b \tau^2}(|y|+1)+ a e^{b \tau^2}(|z|+1))^2 \nonumber\\
&\quad\quad \le (Ca e^{b \tau^2})^2(3+|y|+|z|)^2 \nonumber \\
&\quad\quad \le (Ca e^{b \tau^2})^22(9+y^2+z^2)\nonumber \\
&\quad\quad \le 18(Ca e^{b \tau^2})^2(1+y^2+z^2)
\end{align}
for $j\in \mathbb{N}$ and $\xi\in [0,\tau]$. Here $a, b$ are the constants from inequality \eqref{XkayUpperBound}.
\par We can then appeal to Proposition \ref{LemmaVariantNarrowCon} to conclude
\begin{align}\label{FirstThingWeNeedforKineticEnergyLimit}
&\lim_{j\rightarrow\infty}\int_{\mathbb{R}}\int_{\mathbb{R}}W'(X^{k_j}(y,\xi)-X^{k_j}(z,\xi)) h(X^{k_j}(y,\tau)) d\rho^{k_j}_0(z)d\rho^{k_j}_0(y)\\
&\quad =\int_{\mathbb{R}}\int_{\mathbb{R}}W'(X(y,\xi)-X(z,\xi)) h(X(y,\tau)) d\rho_0(z)d\rho_0(y)
\end{align}
for each $\xi\in [0,\tau]$. What's more, \eqref{SimpleConvBoundWprime} implies
\begin{equation}
\left|\int_{\mathbb{R}}\int_{\mathbb{R}}W'(X^{k_j}(y,\xi)-X^{k_j}(z,\xi)) h(X^{k_j}(y,\tau)) d\rho^{k_j}_0(z)d\rho^{k_j}_0(y)\right|
\le 18(Ca e^{b \tau^2})^2\left(1+2\int_{\mathbb{R}}y^2d\rho^{k_j}_0(y)\right)
\end{equation}
for $\xi\in [0,\tau]$. The limit \eqref{rhoQuadraticConv} and a standard variant of dominated convergence (Theorem 1.20 in \cite{EvaGar}) together give
\begin{align}
&\lim_{j\rightarrow\infty}\int^\tau_0\int_{\mathbb{R}}\int_{\mathbb{R}}W'(X^{k_j}(y,\xi)-X^{k_j}(z,\xi)) h(X^{k_j}(y,\tau)) d\rho^{k_j}_0(z)d\rho^{k_j}_0(y)d\xi\\
&\quad =\int^\tau_0\int_{\mathbb{R}}\int_{\mathbb{R}}W'(X(y,\xi)-X(z,\xi)) h(X(y,\tau)) d\rho_0(z)d\rho_0(y)d\xi.
\end{align}
\par As a result,
\begin{align}\label{HlimitvzeroMinus}
&\lim_{j\rightarrow\infty}\int_{\mathbb{R}}\left[v_0-\displaystyle\int^\tau_0(W'*\rho^{k_j}_\xi)(X^{k_j}(\xi))d\xi\right] h(X^{k_j}(\tau))d\rho^{k_j}_0\\
&\hspace{1in} =\int_{\mathbb{R}}\left[v_0-\displaystyle\int^\tau_0(W'*\rho_\xi)(X(\xi))d\xi\right] h(X(\tau))d\rho_0.
\end{align}
Since
\begin{align*}
&\left|\int_{\mathbb{R}}\left[v_0-\displaystyle\int^\tau_0(W'*\rho^{k_j}_\xi)(X^{k_j}(\xi))d\xi\right] h(X^{k_j}(\tau))d\rho^{k_j}_0\right|\\
&\quad \le \int_{\mathbb{R}}|h(X^{k_j}(\tau))||v_0|d\rho^{k_j}_0+
\tau\cdot 18(Ca e^{b \tau^2})^2\left(1+2\int_{\mathbb{R}}y^2d\rho^{k_j}_0(y)\right),
\end{align*}
and the integrals $\int_{\mathbb{R}}|h(X^{k_j}(\tau))||v_0|d\rho^{k_j}_0,\int_{\mathbb{R}}y^2d\rho^{k_j}_0(y)$ are bounded and converge for each $\tau\in [s,t]$ as $j\rightarrow\infty$, we likewise have
\begin{align*}
&\lim_{j\rightarrow\infty}\int^t_s\int_{\mathbb{R}}\left[v_0-\displaystyle\int^\tau_0(W'*\rho^{k_j}_\xi)(X^{k_j}(\xi))d\xi\right] h(X^{k_j}(\tau))d\rho^{k_j}_0d\tau\\
&\quad =\int^t_s\int_{\mathbb{R}}\left[v_0-\displaystyle\int^\tau_0(W'*\rho_\xi)(X(\tau))d\xi\right] h(X(\tau))d\rho_0d\tau.
\end{align*}
The claim \eqref{SolnGoal} then follows by sending $j\rightarrow\infty$ in
\begin{equation}\label{solnconditionXkayjay}
\int^t_s\int_{\mathbb{R}}\dot X^{k_j}(\tau) h(X^{k_j}(\tau))d\rho^{k_j}_0d\tau=\int^t_s\int_{\mathbb{R}}\left[v_0-\displaystyle\int^\tau_0(W'*\rho^{k_j}_\xi)(X^{k_j}(\xi))d\xi\right]h(X^{k_j}(\tau))d\rho^{k_j}_0d\tau.
\end{equation}
\par 3. \underline{Properties of the solution}: Assertions $(ii)$ and $(iii)$ of this claim follow from parts $(i)$ and $(ii)$ of Proposition \ref{StrongCompactXkayjay}, respectively. So
we will now focus on verifying assertion $(i)$ which states that
\begin{equation}
E(\tau):=\int_{\mathbb{R}}\frac{1}{2}\dot X(\tau)^2d\rho_0+\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{1}{2}W(X(y,\tau)-X(z,\tau))d\rho_0(y)d\rho_0(z)
\end{equation}
is essentially nonincreasing on $[0,\infty)$. Recall that
\begin{equation}
E^j(\tau):=\int_{\mathbb{R}}\frac{1}{2}\dot X^{k_j}(\tau+)^2d\rho^{k_j}_0+\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{1}{2}W(X^{k_j}(y,\tau)-X^{k_j}(z,\tau))d\rho^{k_j}_0(y)d\rho^{k_j}_0(z)
\end{equation}
is nonincreasing by part $(iii)$ of Proposition \ref{StrongCompactXkayjay}.
\par As $X^{k_j}$ solves the pressureless Euler flow equation, we can integrate by parts to find
\begin{align*}
\int^t_s\int_{\mathbb{R}}\dot X^{k_j}(\tau)^2d\rho^{k_j}_0d\tau&= \int^t_s\int_{\mathbb{R}}\dot X^{k_j}(\tau)\left[v_0-\int^\tau_0(W'*\rho^{k_j}_s)(X^{k_j}(s))ds\right]d\rho^{k_j}_0d\tau\\
& = \int_{\mathbb{R}}\left(\int^t_s\dot X^{k_j}(\tau)\left[v_0-\int^\tau_0(W'*\rho^{k_j}_s)(X^{k_j}(s))ds\right]d\tau \right)d\rho^{k_j}_0\\
& = \left.\int_{\mathbb{R}}X^{k_j}(\tau)\left[v_0-\int^\tau_0(W'*\rho^{k_j}_s)(X^{k_j}(s))ds\right] d\rho^{k_j}_0 \right|^{\tau=t}_{\tau=s}\\
& \hspace{1in} +\int^t_s\int_{\mathbb{R}} X^{k_j}(\tau)(W'*\rho^{k_j}_\tau)(X^{k_j}(\tau)) d\rho^{k_j}_0d\tau.
\end{align*}
The limit \eqref{FirstThingWeNeedforKineticEnergyLimit} with $\tau=t,s$ and $h=\text{id}_{\R}$ and the limit \eqref{HlimitvzeroMinus} with $h=\text{id}_{\R}$ give
\begin{align}\label{KineticEnergyLimit}
\lim_{j\rightarrow\infty}\int^t_s\int_{\mathbb{R}}\dot X^{k_j}(\tau)^2d\rho^{k_j}_0d\tau&=\left.\int_{\mathbb{R}}X(\tau)\left[v_0-\int^\tau_0(W'*\rho_s)(X(s))ds\right] d\rho_0 \right|^{\tau=t}_{\tau=s}\\
& \hspace{1in} +\int^t_s\int_{\mathbb{R}} X(\tau)(W'*\rho_\tau)(X(\tau)) d\rho_0d\tau\\
&=\int^t_s\int_{\mathbb{R}}\dot X(\tau)^2d\rho_0d\tau
\end{align}
for each $0\le s\le t$.
\par Next, we claim that
\begin{equation}\label{SecondPartEnergyConv}
\lim_{j\rightarrow\infty}\int_{\mathbb{R}}\int_{\mathbb{R}}W(X^{k_j}(y,\tau)-X^{k_j}(z,\tau))d\rho^{k_j}_0(y)d\rho^{k_j}_0(z)=\int_{\mathbb{R}}\int_{\mathbb{R}}W(X(y,\tau)-X(z,\tau))d\rho_0(y)d\rho_0(z)
\end{equation}
for each $\tau\ge 0$. Note that $W$ grows at most quadratically. Indeed,
\begin{equation}
W(x)\ge W(0)+W'(0)x-\frac{c}{2}x^2
\end{equation}
for $x\in \mathbb{R}$ as $W$ is semiconvex. Likewise
\begin{equation}
W(0)\ge W(x)+W'(x)(0-x)-c\frac{(0-x)^2}{2},
\end{equation}
and so
\begin{equation}
W(x)\le W(0)+xW'(x)+\frac{c}{2}x^2.
\end{equation}
Since $W'(x)$ grows at most linearly, it must be that
\begin{equation}\label{Wquadratic}
|W(x)|\le D(1+x^2)
\end{equation}
for some constant $D>0$.
\par Combining \eqref{Wquadratic} and \eqref{XkayUpperBound}, it is routine to check
\begin{equation}\label{WXkayjayUpper}
|W(X^{k_j}(y,\tau)-X^{k_j}(z,\tau))|\le 12D a^2 e^{2b \tau^2}(1+y^2+z^2)
\end{equation}
for each $j\in \mathbb{N}$, $y,z\in \mathbb{R}$, $\tau\ge 0$. Note that
\begin{equation}
\lim_{j\rightarrow\infty}W(X^{k_j}(y,\tau)-X^{k_j}(z,\tau))=W(X(y,\tau)-X(z,\tau))
\end{equation}
locally uniformly for $y,z\in \mathbb{R}$ for each $\tau\ge 0$. Also observe that $\rho^{k_j}_0\times \rho^{k_j}_0\rightarrow \rho_0\times \rho_0$ narrowly in ${\cal P}(\mathbb{R}^2)$, and
\begin{equation}
\lim_{j\rightarrow\infty}\iint_{\mathbb{R}^2}(y^2+z^2)d\rho^{k_j}_0(y)d\rho^{k_j}_0(z)=\iint_{\mathbb{R}^2}(y^2+z^2)d\rho_0(y)d\rho_0(z),
\end{equation}
which implies that $(y,z)\mapsto y^2+z^2$ is uniformly integrable with respect to $(\rho^{k_j}_0\times \rho^{k_j}_0)_{j\in \mathbb{N}}$. It follows from Lemma \ref{LemmaVariantNarrowCon} that
\eqref{SecondPartEnergyConv} holds for each $\tau\ge 0$.
\par In view of estimate \eqref{WXkayjayUpper}, we can also apply a standard variant of dominated convergence to find
\begin{align}\label{SecondPartEnergyConvPart2}
&\lim_{j\rightarrow\infty}\int^t_s\int_{\mathbb{R}}\int_{\mathbb{R}}W(X^{k_j}(y,\tau)-X^{k_j}(z,\tau))d\rho^{k_j}_0(y)d\rho^{k_j}_0(z)d\tau\\
&\hspace{1in} =\int^t_s\int_{\mathbb{R}}\int_{\mathbb{R}}W(X(y,\tau)-X(z,\tau))d\rho_0(y)d\rho_0(z)d\tau
\end{align}
for each $0\le s\le t$. Combining with \eqref{KineticEnergyLimit} gives
\begin{equation}\label{EjayLimit}
\lim_{j\rightarrow\infty}\int^t_sE^j(\tau)d\tau=\int^t_sE(\tau)d\tau
\end{equation}
for each $0\le s\le t$. As $E^j:[0,\infty)\rightarrow [0,\infty)$ is uniformly bounded and nonincreasing, $E^j(t)$ also converges for each $t\ge 0$ by Helly's selection theorem (Lemma 3.3.3 in \cite{AGS}). By \eqref{EjayLimit}, $\lim_{j\rightarrow\infty}E^j(t)=E(t)$ for almost every $t\in [0,\infty)$. We then conclude that for almost every $t,s\in[0,\infty)$ with $t\ge s$
$$
E(t)=\lim_{j\rightarrow\infty}E^j(t)\le \lim_{j\rightarrow\infty}E^j(s)=E(s).
$$
\end{proof}
\subsection{Solution of the pressureless Euler equations}
We are now in position to establish the existence of a weak solution of the pressureless Euler equations \eqref{PEE} which satisfy the
initial conditions \eqref{Init}. These types of solutions are defined as follows.
\begin{defn}\label{WeakSolnDefn}
A narrowly continuous
$\rho: [0,\infty)\rightarrow {\cal P}(\mathbb{R}); t\mapsto \rho_t$ and a Borel measurable
$v:\mathbb{R}\times[0,\infty)\rightarrow\mathbb{R}$ is a {\it weak solution pair of the pressureless Euler equations \eqref{PEE}}
which satisfies the initial conditions \eqref{Init} if the following hold. \\
$(i)$ For each $T>0$,
$$
\int^T_0\left\{\int_{\mathbb{R}}v^2d\rho_t+\iint_{\mathbb{R}^2}|W'(x-y)|d\rho_t(x)d\rho_t(y)\right\}dt<\infty.
$$
$(ii)$ For each $\phi\in C^\infty_c(\mathbb{R}\times[0,\infty))$,
$$
\int^\infty_0\int_{\mathbb{R}}(\partial_t\phi+v\partial_x\phi)d\rho_tdt+\int_{\mathbb{R}}\phi(\cdot,0)d\rho_0=0.
$$
$(iii)$ For each $\phi\in C^\infty_c(\mathbb{R}\times[0,\infty))$,
$$
\int^\infty_0\int_{\mathbb{R}}(v\partial_t\phi +v^2\partial_x\phi)d\rho_tdt+\int_{\mathbb{R}}\phi(\cdot,0)v_0d\rho_0=\int^\infty_0\int_{\mathbb{R}}\phi(W'*\rho_t)d\rho_tdt.
$$
\end{defn}
\begin{cor}\label{PEEexistCor}
There exists a weak solution pair $\rho$ and $v$ of the pressureless Euler equations \eqref{PEE} which satisfies the initial conditions \eqref{Init}. Moreover, this solution pair has the following two properties.
\begin{enumerate}[(i)]
\item For almost every $0\le s\le t$,
\begin{align*}
&\int_{\mathbb{R}}\frac{1}{2}v(x,t)^2d\rho_t(x)+\iint_{\mathbb{R}^2}\frac{1}{2}W(x-y)d\rho_t(x)d\rho_t(y)\\
&\hspace{1in} \le \int_{\mathbb{R}}\frac{1}{2}v(x,s)^2d\rho_s(x)+\iint_{\mathbb{R}^2}\frac{1}{2}W(x-y)d\rho_s(x)d\rho_s(y).
\end{align*}
\item For almost every $t>0$ and $\rho_t$ almost every $x,y\in \mathbb{R}$,
\begin{equation}
(v(x,t)-v(y,t))(x-y)\le \frac{\sqrt{c}}{\tanh(\sqrt{c}t)}(x-y)^2.
\end{equation}
\end{enumerate}
\end{cor}
\begin{proof}
\underline{1. Specifying $v$}: Let $X$ denote the solution of the pressureless Euler flow equation \eqref{FlowMapEqn} which satisfies \eqref{xInit} as described in Theorem \ref{mainThm} and define
$$
\nu(S):=\int^\infty_0\int_{\mathbb{R}}\left[v_0-\displaystyle\int^t_0(W'*\rho_s)(X(s))ds\right]\chi_{S}(X(t),t) d\rho_0dt
$$
for Borel $S\subset \mathbb{R}\times [0,\infty)$. This clearly defines a signed Borel measure on $\mathbb{R}\times [0,\infty)$, and it is not hard to check that $\nu$ is sigma finite. Let us also set
$$
\mu(S):=\int^\infty_0\int_{\mathbb{R}}\chi_{S} d\rho_tdt=\int^\infty_0\int_{\mathbb{R}}\chi_{S}(X(t),t) d\rho_0dt
$$
for Borel $S\subset \mathbb{R}\times [0,\infty)$. It is clear that $\mu$ it is also sigma finite and that $\nu$ is absolutely continuous with respect to $\mu$.
\par By the Radon-Nikodym theorem, there is a Borel measurable $v:\mathbb{R}\times[0,\infty)\rightarrow \mathbb{R}$
such that
$$
\nu(S)=\int_S vd\mu.
$$
Note in particular that
\begin{align*}
\int^\infty_0\int_{\mathbb{R}}\phi d\nu&=\int^\infty_0\int_{\mathbb{R}}\phi(X(t),t) \left[v_0-\displaystyle\int^t_0(W'*\rho_s)(X(s))ds\right] d\rho_0dt\\
&=\int^\infty_0\int_{\mathbb{R}}\phi(X(t),t)\dot X(t) d\rho_0dt\\
&=\int^\infty_0\int_{\mathbb{R}}\phi vd\mu\\
&=\int^\infty_0\int_{\mathbb{R}}\phi(X(t),t) v(X(t),t)d\rho_0dt\\
\end{align*}
for each continuous $\phi:\mathbb{R}\times [0,\infty)\rightarrow \mathbb{R}$ with compact support. It follows that
\begin{equation}\label{FindingV}
\dot X(t)=v(X(t),t)=\E_{\rho_0}\left[v_0-\displaystyle\int^t_0(W'*\rho_s)(X(s))ds\bigg| X(t)\right]\quad \text{a.e.}\; t>0
\end{equation}
$\rho_0$ almost everywhere.
\par \underline{2. Integrability}: Since $X: [0,\infty)\rightarrow L^2(\rho_0)$ is locally Lipschitz,
\begin{equation}
\int^T_0\int_{\mathbb{R}}v^2d\rho_tdt=\int^T_0\int_{\mathbb{R}}\dot X(t)^2d\rho_0dt<\infty
\end{equation}
for each $T>0$. Lipschitz continuity also implies that $\int_{\mathbb{R}}|X(t)|d\rho_0$ is bounded on $[0,T]$, so
\begin{align*}
\int^T_0\iint_{\mathbb{R}^2}|W'(x-y)|d\rho_t(x)d\rho_t(y)dt&=\int^T_0\iint_{\mathbb{R}^2}|W'(X(w,t)-X(z,t))|d\rho_0(w)d\rho_0(z)dt\\
&\le C\int^T_0\iint_{\mathbb{R}^2}(|X(w,t)-X(z,t)|+1)d\rho_0(w)d\rho_0(z)dt\\
&\le C\int^T_0\iint_{\mathbb{R}^2}(|X(w,t)|+|X(z,t)|+1)d\rho_0(w)d\rho_0(z)dt\\
&\le 2C\left(\int^T_0\int_{\mathbb{R}}|X(t)|d\rho_0dt+1\right)<\infty.
\end{align*}
Thus, $\rho$ and $v$ satisfy part $(i)$ of Definition \ref{WeakSolnDefn}.
\par \underline{3. Weak solution property}: Suppose $\phi\in C^\infty_c(\mathbb{R}\times [0,\infty))$
\begin{align*}
&\int^\infty_0\int_\mathbb{R}\left(\partial_t\phi+v\partial_x\phi\right)d\rho_tdt\\
&\quad=\int^\infty_0\int_\mathbb{R}\left(\partial_t\phi(X(t),t)+v(X(t),t)\partial_x\phi(X(t),t)\right)d\rho_0dt\\
&\quad=\int^\infty_0\int_\mathbb{R}\left(\partial_t\phi(X(t),t)+\dot X(t)\partial_x\phi(X(t),t)\right)d\rho_0dt\\
&\quad=\int^\infty_0\int_\mathbb{R}\frac{d}{dt}\phi(X(t),t)d\rho_0dt\\
&\quad=\int_\mathbb{R}\int^\infty_0\frac{d}{dt}\phi(X(t),t)dt d\rho_0\\
&\quad=-\int_\mathbb{R}\phi(X(0),0) d\rho_0\\
&\quad=-\int_\mathbb{R}\phi(\cdot,0)d\rho_0.
\end{align*}
This proves part $(ii)$ of Definition \ref{WeakSolnDefn}. As for part $(iii)$ that definition,
\begin{align*}
&\int^\infty_0\int_\mathbb{R}\left(\partial_t\phi+v\partial_x\phi\right)vd\rho_tdt\\
&\quad=\int^\infty_0\int_\mathbb{R}\left(\partial_t\phi(X(t),t)+v(X(t),t)\partial_x\phi(X(t),t)\right)v(X(t),t)d\rho_0dt\\
&\quad=\int^\infty_0\int_\mathbb{R}\left(\partial_t\phi(X(t),t)+v(X(t),t)\partial_x\phi(X(t),t)\right)\dot X(t)d\rho_0dt\\
&\quad=\int^\infty_0\int_\mathbb{R}\frac{d}{dt}\phi(X(t),t) \dot X(t)d\rho_0dt\\
&\quad=\int^\infty_0\int_\mathbb{R}\frac{d}{dt}\phi(X(t),t) \left[v_0-\displaystyle\int^t_0(W'*\rho_s)(X(s))ds\right]d\rho_0dt\\
&\quad=\int_\mathbb{R}\int^\infty_0\frac{d}{dt}\phi(X(t),t) \left[v_0-\displaystyle\int^t_0(W'*\rho_s)(X(s))ds\right]dt d\rho_0\\
&\quad=\int^\infty_0\int_\mathbb{R}\phi(X(t),t)(W'*\rho_t)(X(t)) d\rho_0 dt -\int_\mathbb{R}\phi(X(0),0)v_0 d\rho_0\\
&\quad=\int^\infty_0\int_\mathbb{R}\phi(W'*\rho_t) d\rho_t dt -\int_\mathbb{R}\phi(\cdot,0)v_0 d\rho_0.
\end{align*}
\underline{4. Nonincreasing energy and entropy inequality}: Since
\begin{align*}
{\cal E}(\tau)&:=\int_{\mathbb{R}}\frac{1}{2}v(x,\tau)^2d\rho_\tau(x)+\iint_{\mathbb{R}^2}\frac{1}{2}W(x-y)d\rho_\tau(x)d\rho_\tau(y)\\
&=\int_{\mathbb{R}}\frac{1}{2}v(X(\tau),\tau)^2d\rho_0+\iint_{\mathbb{R}^2}\frac{1}{2}W(X(y,\tau)-X(z,\tau))d\rho_0(y)d\rho_0(z)\\
&=\int_{\mathbb{R}}\frac{1}{2}\dot X(\tau)^2d\rho_0+\iint_{\mathbb{R}^2}\frac{1}{2}W(X(y,\tau)-X(z,\tau))d\rho_0(y)d\rho_0(z)
\end{align*}
for almost every $\tau\ge 0$, ${\cal E}(t)\le {\cal E}(s)$ for almost every $t,s\in[0,\infty)$ with $s\le t$. Here, of course, we are
employing conclusion $(i)$ of Theorem \ref{mainThm}.
\par Recall that $[0,\infty)\ni t\mapsto X(y,t)$ is absolutely continuous on any compact interval within $[0,\infty)$ for $\rho_0$ almost every $y\in \mathbb{R}$. Let us denote this set of $y$ as $Q\subset \mathbb{R}$, and we emphasize that $Q$ is $\rho_0$ measurable and $\rho_0(Q)=1$. By conclusion $(iii)$ of Theorem \ref{mainThm},
\begin{align*}
0&\ge \frac{d}{dt}\frac{(X(y,t)-X(z,t))^2}{\sinh(\sqrt{c}t)^2}\\
&= \frac{2(X(y,t)-X(z,t))(\partial_tX(y,t)-\partial_tX(z,t))}{\sinh(\sqrt{c}t)^2}-2\frac{\sqrt{c}\cosh(\sqrt{c}t)}{\sinh(\sqrt{c}t)^3}(X(y,t)-X(z,t))^2\\
&= \frac{2}{\sinh(\sqrt{c}t)^2}\left[(X(y,t)-X(z,t))(v(X(y,t),t)-v(X(z,t),t))-\frac{\sqrt{c}}{\tanh(\sqrt{c}t)}(X(y,t)-X(z,t))^2 \right]
\end{align*}
for Lebesgue almost every $t>0$ and $y,z\in Q$.
\par As a result, we have proved part $(ii)$ of this assertion for $x,y$ belonging to the image of $Q$ under $X(t)$
\begin{equation}
X(t)(Q)=\{X(y,t)\in \mathbb{R}: y\in Q\}
\end{equation}
for almost every $t>0$. Without loss of generality, we may suppose $Q$ is a countable union of closed sets (Theorem 1.19 of \cite{Folland}). By part $(ii)$ of Theorem \ref{mainThm}, we may as well assume $X(t)$ is continuous on $\mathbb{R}$. It follows that $X(t)(Q)$ is Borel measurable (Proposition A.1 in \cite{MR4007615}). As $Q\subset X(t)^{-1}[X(t)(Q)]$,
\begin{equation}
\rho_t(X(t)(Q))=\rho_0(X(t)^{-1}[X(t)(Q)])\ge \rho_0(Q)=1.
\end{equation}
We conclude that part $(ii)$ of this assertion holds for Lebesgue almost every $t>0$ and $x,y$ belonging to a Borel set of full measure for $\rho_t$.
\end{proof}
\section{Euler-Poisson equations in 1D}\label{EPsect}
For the remainder of this paper, we will assume
\begin{equation}
W(x)=|x|, \quad x\in \mathbb{R}.
\end{equation}
As $W$ is not continuously differentiable,
we will also consider the closely related interaction potential
\begin{equation}
W_\epsilon(x)=(x^2+\epsilon^2)^{1/2}, \quad x\in \mathbb{R}
\end{equation}
for $\epsilon>0$ and small. Clearly,
\begin{equation}\label{WepsUniform}
W(x)\le W_\epsilon(x)\le W(x)+\epsilon,\quad x\in \mathbb{R}.
\end{equation}
Also observe that $W_\epsilon$ is convex, even, and continuously differentiable with
\begin{equation}
|W_\epsilon'(x)|\le 1.
\end{equation}
\par Therefore, there is a locally Lipschitz continuous $X^\epsilon:[0,\infty)\rightarrow L^2(\rho_0)$ which satisfies
\begin{equation}\label{XepsilonEquation}
\begin{cases}
\dot X^\epsilon(t)=\E_{\rho_0}\left[v_0-\displaystyle\int^t_0(W_\epsilon'*\rho^\epsilon_s)(X^\epsilon(s))ds\bigg| X^\epsilon(t)\right], \quad a.e.\;t\ge 0\\
X^\epsilon(0)=\text{id}_{\R}
\end{cases}
\end{equation}
for each $\epsilon>0$. Here
$$
\rho^\epsilon_t:=X^\epsilon(t)_{\#}\rho_0, \quad t\ge 0.
$$
We will argue below that there is a sequence of positive numbers $\epsilon_k\rightarrow 0$ and $(X^{\epsilon_k})_{k\in\mathbb{N}}$ which converges in a strong sense to
a solution $X$ of the Euler-Poisson flow equation \eqref{FlowMapEqnEP} which satisfies the initial condition \eqref{xInit}. We will then make a final remark on the existence of weak solution pairs to the Euler-Poisson system \eqref{EP1D}.
\subsection{A strongly convergent subsequence}
Let us begin by recalling a few facts we have already established for $X^\epsilon$.
\\
\par {\bf Lipschitz continuity in time}. In view of Theorem \ref{mainThm}, $X^\epsilon$ satisfies
\begin{align}
\int_{\mathbb{R}}\frac{1}{2}(\dot X^\epsilon(t))^2d\rho_0&\le \int_{\mathbb{R}}\frac{1}{2}(\dot X^\epsilon(t))^2d\rho_0+\iint_{\mathbb{R}^2}\frac{1}{2}W_\epsilon(X^\epsilon(y,t)-X^\epsilon(z,t))d\rho_0(y)d\rho_0(z)\\
&\le \int_{\mathbb{R}}\frac{1}{2}v_0^2d\rho_0+\iint_{\mathbb{R}^2}\frac{1}{2}W_\epsilon(y-z)d\rho_0(y)d\rho_0(z)\\
&\le \int_{\mathbb{R}}\frac{1}{2}v_0^2d\rho_0+\iint_{\mathbb{R}^2}\frac{1}{2}(|y-z|+\epsilon)d\rho_0(y)d\rho_0(z)\\
&= \int_{\mathbb{R}}\frac{1}{2}v_0^2d\rho_0+\iint_{\mathbb{R}^2}\frac{1}{2}|y-z|d\rho_0(y)d\rho_0(z)+\frac{1}{2}\epsilon
\end{align}
for almost every $t\ge 0$. Therefore, $X^\epsilon:[0,\infty)\rightarrow L^2(\rho_0)$ is uniformly Lipschitz continuous.
\\
\par {\bf Uniform spatial continuity}. Theorem \ref{mainThm} also gives
\begin{equation}
0\le X^\epsilon(y,t)-X^\epsilon(z,t)\le y-z +t\int^y_z|v_0'(x)|dx
\end{equation}
for each $y,z\in\text{supp}(\rho_0)$ with $y\ge z$ and for each $t\ge 0$. As a result, we may was well suppose $X^\epsilon(t):\mathbb{R}\rightarrow \mathbb{R}$ is uniformly continuous and satisfies
\begin{equation}
|X^\epsilon(y,t)-X^\epsilon(z,t)|\le |y-z| +t\omega(|y-z|)
\end{equation}
for every $y,z\in \mathbb{R}$ and $t\ge 0$. Here $\omega$ is the modulus of continuity defined in \eqref{modulusOmega}.
\\
\par {\bf Quantitative stickiness}. For $0<s\le t$, there is a function $f^\epsilon_{t,s}:\mathbb{R}\rightarrow \mathbb{R}$ which satisfies
\begin{equation}\label{SimpleLipCond}
|f^\epsilon_{t,s}(x)-f^\epsilon_{t,s}(y)|\le \frac{t}{s}|x-y|,\quad x,y\in \mathbb{R}
\end{equation}
such that
\begin{equation}\label{SemiGroupPropXeps}
X^\epsilon(y,t)=f^\epsilon_{t,s}(X^\epsilon(y,s))
\end{equation}
for $y\in\text{supp}(\rho_0)$. In particular,
\begin{equation}
\frac{1}{t}|X^\epsilon(y,t)-X^\epsilon(z,t)|\le \frac{1}{s}|X^\epsilon(y,s)-X^\epsilon(z,s)|
\end{equation}
for $y,z\in\text{supp}(\rho_0)$ and $0<s\le t$.
\\
\par We can use this Lipschitz continuity in time, uniform spatial continuity and quantitative stickiness of $(X^\epsilon)_{\epsilon>0}$, along with the arguments we used to prove Proposition \ref{StrongCompactXkayjay} and Corollary \ref{dotXfunctionofX}, to establish the following assertion.
\begin{prop}\label{XepsKayCompactness}
There is a sequence of positive numbers $\epsilon_k\rightarrow 0$ and a Lipschitz $X: [0,\infty)\rightarrow L^2(\rho_0)$ such that for each $t\ge 0$,
\begin{equation}
X^{\epsilon_k}(t)\rightarrow X(t)
\end{equation}
locally uniformly on $\mathbb{R}$ and
\begin{equation}
\lim_{j\rightarrow\infty}\int_{\mathbb{R}}h(\textup{id}_\mathbb{R},X^{\epsilon_k}(t))d\rho_0=\int_{\mathbb{R}}h(\textup{id}_\mathbb{R},X(t))d\rho_0
\end{equation}
for continuous $h:\mathbb{R}^2\rightarrow \mathbb{R}$ with
\begin{equation}
\sup_{(x,y)\in \mathbb{R}^2}\frac{|h(x,y)|}{1+x^2+y^2}<\infty.
\end{equation}
In addition, $X$ has the following properties.
\begin{enumerate}[(i)]
\item For $t\ge 0$ and $y,z\in\textup{supp}(\rho_0)$ with $y\le z$,
$$
0\le X(z,t)-X(y,t)\le z-y+t\int^z_y|v_0'(x)|dx
$$
\item For each $0<s\le t$ and $y,z\in\textup{supp}(\rho_0)$
\begin{equation}
\frac{1}{t}|X(y,t)-X(z,t)|\le \frac{1}{s}|X(y,s)-X(z,s)|
\end{equation}
\item For each $0<s\le t$, $(f^{\epsilon_k}_{t,s})_{k\in \mathbb{N}}$ has a subsequence which converges locally uniformly to a function $f_{t,s}:\mathbb{R}\rightarrow \mathbb{R}$ which satisfies the Lipschitz condition \eqref{SimpleLipCond}
and
$$
X(y,t)=f_{t,s}(X(y,s))
$$
for $y\in\textup{supp}(\rho_0)$.
\item For almost every $t>0$, there is a Borel function $u:\mathbb{R}\rightarrow \mathbb{R}$ for which
$$
\dot X(t)=u(X(t))
$$
for $\rho_0$ almost everywhere.
\end{enumerate}
\end{prop}
We also have the following immediate corollary which can be proved the same way that we justified \eqref{WeakConvDotXkayjay}.
\begin{cor}\label{WeakconvDerXepsKay}
Suppose $g:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and
\begin{equation}
\sup_{x\in \mathbb{R}}\frac{|g(x)|}{1+|x|}<\infty.
\end{equation}
Then
\begin{equation}
\lim_{k\rightarrow\infty}\int^t_s\int_{\mathbb{R}}\dot X^{\epsilon_k}(\tau) g(X^{\epsilon_k}(\tau))d\rho_0d\tau=\int^t_s\int_{\mathbb{R}}\dot X(\tau) g(X(\tau))d\rho_0d\tau
\end{equation}
for $0\le s\le t$.
\end{cor}
This is as far as we can go with the convergence arguments we used to establish Theorem \ref{mainThm}. We will need to identify another mechanism
which will allow us to pass to the limit in the term with $W_\epsilon'$ in equation \eqref{XepsilonEquation}. This will be the topic of the following subsection.
\subsection{A convergence lemma}
Let us recall the definition of $\text{sgn}$
\begin{equation}
\text{sgn}(x)=
\begin{cases}
1,\quad &x>0\\
0,\quad &x=0\\
-1,\quad &x<0.
\end{cases}
\end{equation}
We will also fix a sequence $(\mu^k)_{k\in \mathbb{N}}\subset{\cal P}(\mathbb{R})$ which converges narrowly to $\mu\in {\cal P}(\mathbb{R})$ and additionally satisfies
\begin{equation}\label{myookayConvStrong}
\lim_{k\rightarrow\infty}\int_{\mathbb{R}}x^2d\mu^k(x)=\int_{\mathbb{R}}x^2d\mu(x).
\end{equation}
The central assertion of this subsection is as follows.
\begin{lem}\label{SubdifferentialConProp}
Suppose $g:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and
\begin{equation}\label{geeGrowAtMostLinearly}
\sup_{x\in \mathbb{R}}\frac{|g(x)|}{1+|x|}<\infty.
\end{equation}
Then
\begin{equation}
\lim_{k\rightarrow\infty}\iint_{\mathbb{R}^2}W'_{\epsilon_k}(x-y)g(x)d\mu^k(x)d\mu^k(y)
=\iint_{\mathbb{R}^2}\textup{\text{sgn}}(x-y)g(x)d\mu(x)d\mu(y).
\end{equation}
\end{lem}
\par We will first verify an elementary observation, which is ultimately due to the convexity of the absolute value function. In particular, we will employ
\begin{equation}\label{sgnSubdiff}
|y|\ge |x|+\text{sgn}(x)(y-x)
\end{equation}
for each $x,y\in \mathbb{R}$.
\begin{lem}\label{SubdiffLemma}
The following are equivalent for $\xi\in L^2(\mu)$. \\
(i) For $\mu$ almost every $x\in \mathbb{R}$,
$$
\xi(x)=\textup{\text{sgn}}*\mu(x).
$$
(ii) For each continuous $g:\mathbb{R}\rightarrow \mathbb{R}$ which satisfies \eqref{geeGrowAtMostLinearly},
$$
\iint_{\mathbb{R}^2}\frac{1}{2}|x-y+(g(x)-g(y))|d\mu(x)d\mu(y)\ge \iint_{\mathbb{R}^2}\frac{1}{2}|x-y|d\mu(x)d\mu(y)+\int_{\mathbb{R}}\xi gd\mu.
$$
\end{lem}
\begin{proof}
Suppose $g:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function which grows at most linearly as $|x|\rightarrow\infty$.
\par $(i)\Longrightarrow (ii)$ Employing \eqref{sgnSubdiff} and noting that $\text{sgn}$ is odd gives
\begin{align*}
&\iint_{\mathbb{R}^2}\frac{1}{2}|x-y+(g(x)-g(y))|d\mu(x)d\mu(y)\\
&\quad\quad \ge \iint_{\mathbb{R}^2}\frac{1}{2}|x-y|d\mu(x)d\mu(y)+\frac{1}{2}\iint_{\mathbb{R}^2}\text{sgn}(x-y)(g(x)-g(y))d\mu(x)d\mu(y)\\
&\quad\quad=\iint_{\mathbb{R}^2}\frac{1}{2}|x-y|d\mu(x)d\mu(y)+\frac{1}{2}\iint_{\mathbb{R}^2}\text{sgn}(x-y)g(x)d\mu(x)d\mu(y)\\
&\hspace{1in} -\frac{1}{2}\iint_{\mathbb{R}^2}\text{sgn}(x-y)g(y)d\mu(x)d\mu(y)\\
&\quad\quad= \iint_{\mathbb{R}^2}\frac{1}{2}|x-y|d\mu(x)d\mu(y)+\frac{1}{2}\iint_{\mathbb{R}^2}\text{sgn}(x-y)g(x)d\mu(x)d\mu(y)\\
&\hspace{1in} +\frac{1}{2}\iint_{\mathbb{R}^2}\text{sgn}(y-x)g(y)d\mu(x)d\mu(y)\\
&\quad\quad= \iint_{\mathbb{R}^2}\frac{1}{2}|x-y|d\mu(x)d\mu(y)+\int_{\mathbb{R}}\left(\int_{\mathbb{R}}\text{sgn}(x-y)d\mu(y)\right)g(y)d\mu(x)\\
&\quad\quad= \iint_{\mathbb{R}^2}\frac{1}{2}|x-y|d\mu(x)d\mu(y)+\int_{\mathbb{R}}(\text{sgn}*\mu) gd\mu.
\end{align*}
\par $(ii)\Longrightarrow (i)$ By assumption,
\begin{equation}\label{PhiConvexIneq}
\frac{1}{2}\int_{\mathbb{R}}\frac{|x-y+t(g(x)-g(y))|-|x-y|}{t}d\mu(x)d\mu(y)\ge \int_{\mathbb{R}}g \xi d\mu.
\end{equation}
Also notice
$$
\left|\frac{|x-y+t(g(x)-g(y))|-|x-y|}{t}\right|\le |g(x)-g(y)|
$$
and
$$
\lim_{t\rightarrow 0^+}\frac{|x-y+t(g(x)-g(y))|-|x-y|}{t}=\text{sgn}(x-y)(g(x)-g(y)).
$$
for $x,y\in \mathbb{R}$. By dominated convergence, we can send $t\rightarrow 0^+$ in \eqref{PhiConvexIneq} to find
$$
\int_{\mathbb{R}}g \xi d\mu\le \frac{1}{2}\int_{\mathbb{R}}\int_{\mathbb{R}}\text{sgn}(x-y)(g(x)-g(y))d\mu(x)d\mu(y)=\int_{\mathbb{R}}(\text{sgn}*\mu) gd\mu.
$$
Replacing $g$ with $-g$ gives
$$
\int_{\mathbb{R}}g (\xi -\text{sgn}*\mu)d\mu=0.
$$
As $g$ is arbitrary, $\xi - \text{sgn}*\mu$ vanishes $\mu$ almost everywhere.
\end{proof}
\begin{proof}[Proof of Lemma \ref{SubdifferentialConProp}]
Using the same method to prove $(i)\Longrightarrow (ii)$ in Lemma \ref{SubdiffLemma}, we find
\begin{align*}
&\iint_{\mathbb{R}^2}\frac{1}{2}W_{\epsilon_k}(x-y+(g(x)-g(y)))d\mu^k(x)d\mu^k(y)\\
&\hspace{1in}\ge \iint_{\mathbb{R}^2}\frac{1}{2}W_{\epsilon_k}(x-y)d\mu^k(x)d\mu^k(y)+\int_{\mathbb{R}}(W_{\epsilon_k}'*\mu^k) gd\mu^k
\end{align*}
for each continuous and at most linearly growing $g:\mathbb{R}\rightarrow \mathbb{R}$ . And by \eqref{WepsUniform},
\begin{align}\label{approximateSubdiffLim}
&\frac{1}{2}\epsilon_k+\iint_{\mathbb{R}^2}\frac{1}{2}|x-y+(g(x)-g(y))|d\mu^k(x)d\mu^k(y)\\
&\hspace{1in}\ge \iint_{\mathbb{R}^2}\frac{1}{2}|x-y|d\mu^k(x)d\mu^k(y)+\int_{\mathbb{R}}(W_{\epsilon_k}'*\mu^k) gd\mu^k.
\end{align}
\par Since $|W'_{\epsilon_k}|\le 1$,
$$
|(W_{\epsilon_k}'*\mu^k)(z)|\le 1, \quad z\in \mathbb{R}.
$$
Combining this fact with \eqref{myookayConvStrong} provides a subsequence $(W_{\epsilon_{k_j}}'*\mu^{k_j})_{j\in \mathbb{N}}$ and $\xi\in L^2(\mu)$ such that
$$
\lim_{j\rightarrow\infty}\int_{\mathbb{R}}(W_{\epsilon_{k_j}}'*\mu^{k_j}) gd\mu^{k_j}=\int_{\mathbb{R}}\xi gd\mu
$$
for each continuous and at most linearly growing $g:\mathbb{R}\rightarrow \mathbb{R}$ (Theorem 5.4.4 of \cite{AGS}). Sending $k=k_j\rightarrow\infty$ in \eqref{approximateSubdiffLim} gives
$$
\iint_{\mathbb{R}^2}\frac{1}{2}|x-y+(g(x)-g(y))|d\mu(x)d\mu(y)\ge \iint_{\mathbb{R}^2}\frac{1}{2}|x-y|d\mu(x)d\mu(y)+\int_{\mathbb{R}}\xi gd\mu.
$$
Lemma \ref{SubdiffLemma} implies
$$
\xi=\text{sgn}*\mu.
$$
Since this limit is independent of the subsequence,
\begin{align*}
\lim_{k\rightarrow\infty}\iint_{\mathbb{R}^2}W'_{\epsilon_k}(x-y)g(x)d\mu^k(x)d\mu^k(y)&=\lim_{k\rightarrow\infty}\int_{\mathbb{R}}(W_{\epsilon_k}'*\mu^k) gd\mu^{k}\\
&=\int_{\mathbb{R}}(\text{sgn}*\mu) gd\mu\\
&=\iint_{\mathbb{R}^2}\text{sgn}(x-y)g(x)d\mu(x)d\mu(y)
\end{align*}
for each continuous $g:\mathbb{R}\rightarrow \mathbb{R}$ which satisfies \eqref{SubdifferentialConProp}.
\end{proof}
We will actually need a minor refinement of Lemma \ref{SubdiffLemma} in our proof of Theorem \ref{secondThm}.
\begin{cor}\label{SubdifferentialConProp}
Suppose $(g^k)_{k\in \mathbb{N}}$ is a sequence of continuous functions on $\mathbb{R}$ which satisfies
\begin{equation}\label{uniformLinearGrowthGeeKay}
\sup_{x\in \mathbb{R}}\frac{|g^k(x)|}{1+|x|}\le C
\end{equation}
for some $C$ and which converges locally uniformly to $g: \mathbb{R}\rightarrow \mathbb{R}$. Then
\begin{equation}
\lim_{k\rightarrow\infty}\iint_{\mathbb{R}^2}W'_{\epsilon_k}(x-y)g^k(x)d\mu^k(x)d\mu^k(y)=\iint_{\mathbb{R}^2}\textup{\text{sgn}}(x-y)g(x)d\mu(x)d\mu(y).
\end{equation}
\end{cor}
\begin{proof}
As $g^k(x)\rightarrow g(x)$ for each $x\in \mathbb{R}$, we also have
$$
\sup_{x\in \mathbb{R}}\frac{|g(x)|}{1+|x|}\le C.
$$
Fix $\delta>0$ and choose a compact interval $K_\delta\subset \mathbb{R}$ such that
\begin{equation}\label{ChoiceofConstantMyooKay}
\int_{\mathbb{R}\setminus K_\delta}(1+|x|)d\mu^k(x)\le \frac{\delta}{2C}
\end{equation}
for $k\in \mathbb{N}$; such an interval exists as $1+|x|$ is uniformly integrable with respect to $(\mu^k)_{k\in \mathbb{N}}$ by assumption \eqref{myookayConvStrong}. In view of Lemma \ref{SubdiffLemma},
\begin{align*}
\iint_{\mathbb{R}^2}W'_{\epsilon_k}(x-y)g^k(x)d\mu^k(x)d\mu^k(y)&=\int_{\mathbb{R}}(W'_{\epsilon_k}*\mu^k)g^kd\mu^k \\
&=\int_{\mathbb{R}}(W'_{\epsilon_k}*\mu^k)gd\mu^k +\int_{\mathbb{R}}(W'_{\epsilon_k}*\mu^k)(g^k-g)d\mu^k\\
&=\int_{\mathbb{R}}(\textup{\text{sgn}}*\mu) gd\mu + o(1)+\int_{\mathbb{R}}(W'_{\epsilon_k}*\mu^k)(g^k-g)d\mu^k\\
&=\int_{\mathbb{R}}(\textup{\text{sgn}}*\mu) gd\mu + o(1)+\int_{K_\delta}(W'_{\epsilon_k}*\mu^k)(g^k-g)d\mu^k\\
&\hspace{1in}+\int_{\mathbb{R}\setminus K_\delta}(W'_{\epsilon_k}*\mu^k)(g^k-g)d\mu^k\\
&=\iint_{\mathbb{R}^2}\textup{\text{sgn}}(x-y)g(x)d\mu(x)d\mu(y) + o(1)\\
&\quad +\int_{K_\delta}(W'_{\epsilon_k}*\mu^k)(g^k-g)d\mu^k+\int_{\mathbb{R}\setminus K_\delta}(W'_{\epsilon_k}*\mu^k)(g^k-g)d\mu^k
\end{align*}
as $k\rightarrow\infty$.
\par Observe
$$
\left|\int_{K_\delta}(W'_{\epsilon_k}*\mu^k)(g^k-g)d\mu^k\right|\le \max_{K_\delta}|g^k-g|.
$$
And by \eqref{ChoiceofConstantMyooKay},
$$
\left|\int_{\mathbb{R}\setminus K_\delta}(W'_{\epsilon_k}*\mu^k)(g^k-g)d\mu^k\right|\le 2C\int_{\mathbb{R}\setminus K_\delta}(1+|x|)d\mu^k(x)\le \delta.
$$
As a result,
$$
\limsup_{k\rightarrow\infty}\left|\iint_{\mathbb{R}^2}W'_{\epsilon_k}(x-y)g^k(x)d\mu^k(x)d\mu^k(y)-\iint_{\mathbb{R}^2}\textup{\text{sgn}}(x-y)g(x)d\mu(x)d\mu(y)\right|\le \delta.
$$
The claim follows as $\delta>0$ was arbitrarily chosen.
\end{proof}
\subsection{Solution of the flow equation}
This subsection is dedicated to the proof of Theorem \ref{secondThm}. Here, we will show that the mapping $X$ obtained in Proposition \ref{XepsKayCompactness} is
a solution flow equation \eqref{FlowMapEqnEP} which has all of the required properties. First note that since $X^{\epsilon_k}(0)=\text{id}_{\R}$ and $X^{\epsilon_k}(0)\rightarrow X(0)$
in $L^2(\rho_0)$ as $k\rightarrow\infty$, then $X(0)=\text{id}_{\R}$. Next we claim that $X$ satisfies flow equation \eqref{FlowMapEqnEP}. It suffices to let $0\le s\le t$, fix a continuous $h:\mathbb{R}\rightarrow \mathbb{R}$ which satisfies
$$
\sup_{x\in \mathbb{R}}\frac{|h(x)|}{1+|x|}\le C,
$$
and show
\begin{equation}\label{SolnGoalEP}
\int^t_s\int_{\mathbb{R}}\dot X(\tau) h(X(\tau))d\rho_0d\tau=\int^t_s\int_{\mathbb{R}}\left[v_0-\displaystyle\int^\tau_0(\text{sgn}*\rho_\xi)(X(\xi))d\xi\right]h(X(t))d\rho_0d\tau.
\end{equation}
Once we establish this identity, parts $(i), (ii)$, and $(iii)$ of Theorem \ref{secondThm} would follow by minor variations of the arguments we gave in our proof of
Theorem \ref{mainThm}.
\par To this end, we recall that for each $k\in N$,
\begin{equation}\label{solnconditionXepskay}
\int^t_s\int_{\mathbb{R}}\dot X^{\epsilon_k}(\tau) h(X^{\epsilon_k}(\tau))d\rho_0d\tau=\int^t_s\int_{\mathbb{R}}\left[v_0-\displaystyle\int^\tau_0(W_{\epsilon_k}'*\rho^{\epsilon_k}_\xi)(X^{\epsilon_k}(\xi))d\xi\right]h(X^{\epsilon_k}(\tau))d\rho_0d\tau.
\end{equation}
Moreover, Proposition \ref{XepsKayCompactness} implies
\begin{equation}\label{FinalRhoNarrowConv}
\rho_t=X(t)_{\#}\rho_0=\lim_{k\rightarrow\infty}X^{\epsilon_k}(t)_{\#}\rho_0=\lim_{k\rightarrow\infty}\rho^{\epsilon_k}_t
\end{equation}
narrowly and
\begin{equation}\label{FinalRhoStrongConv}
\lim_{k\rightarrow\infty}\int_{\mathbb{R}}x^2d\rho^{\epsilon_k}_t(x)=\lim_{k\rightarrow\infty}\int_{\mathbb{R}}(X^{\epsilon_k}(t))^2d\rho_0=\int_{\mathbb{R}}(X(t))^2d\rho_0=\int_{\mathbb{R}}x^2d\rho_t(x)
\end{equation}
for each $t\ge 0$.
\par Proposition \ref{XepsKayCompactness} also can be used to show
\begin{equation}
\lim_{k\rightarrow\infty}\int^t_s\int_{\mathbb{R}} v_0 h(X^{\epsilon_k}(\tau))d\rho_0d\tau=\int^t_s\int_{\mathbb{R}}v_0 h(X(t))d\rho_0d\tau.
\end{equation}
Furthermore,
\begin{equation}
\lim_{k\rightarrow\infty}\int^t_s\int_{\mathbb{R}}\dot X^{\epsilon_k}(\tau) h(X^{\epsilon_k}(\tau))d\rho_0d\tau=\int^t_s\int_{\mathbb{R}}\dot X(\tau) h(X(\tau))d\rho_0d\tau
\end{equation}
as noted in Corollary \ref{WeakconvDerXepsKay}. As a result, we are left to justify the limit
\begin{align}\label{FinalSolutionLimitXepsKay}
&\lim_{k\rightarrow\infty}\int^t_s\int_{\mathbb{R}}\left(\int^\tau_0(W_{\epsilon_k}'*\rho^{\epsilon_k}_\xi)(X^{\epsilon_k}(\xi))d\xi \right)h(X^{\epsilon_k}(\tau))d\rho_0d\tau\\
&\hspace{1in}=
\int^t_s\int_{\mathbb{R}}\left(\int^\tau_0(\text{sgn}*\rho_\xi)(X(\xi))d\xi \right)h(X(\tau))d\rho_0d\tau.
\end{align}
Then we would be able to send $k\rightarrow \infty$ in \eqref{solnconditionXepskay} to conclude \eqref{SolnGoalEP}.
\par So we will now focus on establishing \eqref{FinalSolutionLimitXepsKay}. Observe
\begin{align}
&\int^t_s\int_{\mathbb{R}}\left(\int^\tau_0(W_{\epsilon_k}'*\rho^{\epsilon_k}_\xi)(X^{\epsilon_k}(\xi))d\xi \right)h(X^{\epsilon_k}(\tau))d\rho_0d\tau\\
&\hspace{1in}=\int^t_s\int^\tau_0\left[\int_{\mathbb{R}}(W_{\epsilon_k}'*\rho^{\epsilon_k}_\xi)(X^{\epsilon_k}(\xi)) h(X^{\epsilon_k}(\tau))d\rho_0\right]d\xi d\tau.
\end{align}
Since $W'_{\epsilon_k}$ is uniformly bounded and $h$ grows at most linearly, we just need to show
\begin{equation}\label{FinalSolutionLimitXepsKayToo}
\lim_{k\rightarrow\infty}\int_{\mathbb{R}}(W_{\epsilon_k}'*\rho^{\epsilon_k}_\xi)(X^{\epsilon_k}(\xi)) h(X^{\epsilon_k}(\tau))d\rho_0=
\int_{\mathbb{R}}(\text{sgn}*\rho_\xi)(X(\xi)) h(X(\tau))d\rho_0
\end{equation}
for each $\xi,\tau> 0$ with $\xi\le \tau$. For if \eqref{FinalSolutionLimitXepsKayToo} holds, \eqref{FinalSolutionLimitXepsKay} would follow from a simple application of the dominated convergence theorem.
\par In view of \eqref{SemiGroupPropXeps},
\begin{align}
\int_{\mathbb{R}}(W_{\epsilon_k}'*\rho^{\epsilon_k}_\xi)(X^{\epsilon_k}(\xi)) h(X^{\epsilon_k}(\tau))d\rho_0&=\int_{\mathbb{R}}(W_{\epsilon_k}'*\rho^{\epsilon_k}_\xi)(X^{\epsilon_k}(\xi))\; h\circ f_{\tau,\xi}^{\epsilon_k}(X^{\epsilon_k}(\xi))d\rho_0\\
&=\int_{\mathbb{R}}(W_{\epsilon_k}'*\rho^{\epsilon_k}_\xi)\; h\circ f_{\tau,\xi}^{\epsilon_k} d\rho^{\epsilon_k}_\xi\\
&=\iint_{\mathbb{R}^2}W_{\epsilon_k}'(x-y)\; h\circ f_{\tau,\xi}^{\epsilon_k}(y) d\rho^{\epsilon_k}_\xi(x)d\rho^{\epsilon_k}_\xi(y).
\end{align}
By part $(iii)$ of Proposition \ref{XepsKayCompactness}, $f_{\tau,\xi}^{\epsilon_k}\rightarrow f_{\tau,\xi}$ locally uniformly on $\mathbb{R}$ (up to a subsequence that we will not relabel) and
$$
X(\tau)=f_{\tau,\xi}(X(\xi))
$$
$\rho_0$ almost everywhere. It follows that $h\circ f_{\tau,\xi}^{\epsilon_k}$ converges locally uniformly to $h\circ f_{\tau,\xi}$. We also have the limits
\eqref{FinalRhoNarrowConv} and \eqref{FinalRhoStrongConv} for each $t=\xi$. We can then apply Corollary \ref{SubdifferentialConProp} once
we know $h\circ f_{\tau,\xi}^{\epsilon_k}(y)$ grows at most linearly in $|y|$ in a uniform way.
\par Fix $z_0\in\text{supp}(\rho_0)$ and observe
\begin{align}
|h\circ f_{\tau,\xi}^{\epsilon_k}(y)|&\le C(1+|f_{\tau,\xi}^{\epsilon_k}(y)|)\\
&\le C\left(1+|f_{\tau,\xi}^{\epsilon_k}(y)-f_{\tau,\xi}^{\epsilon_k}(X^{\epsilon_k}(z_0,\xi))|+|f_{\tau,\xi}^{\epsilon_k}(X^{\epsilon_k}(z_0,\xi))|\right)\\
&\le C\left(1+\frac{\tau}{\xi}|y-X^{\epsilon_k}(z_0,\xi)|+|X^{\epsilon_k}(z_0,\tau)|\right)
\end{align}
for all $y\in\mathbb{R}$. Since $(X^{\epsilon_k}(z_0,\xi), X^{\epsilon_k}(z_0,\tau))\rightarrow (X(z_0,\xi), X(z_0,\tau))$ as $k\rightarrow\infty$, it must be that
\begin{equation}
\sup_{k\in \mathbb{N}}\left\{\sup_{y\in \mathbb{R}}\frac{|h\circ f_{\tau,\xi}^{\epsilon_k}(y)|}{1+|y|}\right\}<\infty.
\end{equation}
Corollary \ref{SubdifferentialConProp} then gives
\begin{align}
\lim_{k\rightarrow\infty}\int_{\mathbb{R}}(W_{\epsilon_k}'*\rho^{\epsilon_k}_\xi)(X^{\epsilon_k}(\xi)) h(X^{\epsilon_k}(\tau))d\rho_0&=\lim_{k\rightarrow\infty}\iint_{\mathbb{R}^2}W_{\epsilon_k}'(x-y)\; h\circ f_{\tau,\xi}^{\epsilon_k}(y) d\rho^{\epsilon_k}_\xi(x)d\rho^{\epsilon_k}_\xi(y)\\
&=\iint_{\mathbb{R}^2}\text{sgn}(x-y)\; h\circ f_{\tau,\xi}(y) d\rho_\xi(x)d\rho_\xi(y)\\
&=\int_{\mathbb{R}}(\text{sgn}*\rho_\xi)(X(\xi))\; h\circ f_{\tau,\xi}(X(\xi))d\rho_0\\
&=\int_{\mathbb{R}}(\text{sgn}*\rho_\xi)(X(\xi))\; h(X(\tau))d\rho_0.
\end{align}
We conclude \eqref{FinalSolutionLimitXepsKayToo} and in turn that $X$ is a solution of the flow equation \eqref{FlowMapEqnEP}.
\subsection{Solving the Euler-Poisson equations }
Weak solution pairs of the Euler-Poisson system \eqref{EP1D} which satisfy given initial conditions \eqref{Init} are defined as follows.
\begin{defn}\label{EPWeakSolnDefn}
A narrowly continuous $\rho: [0,\infty)\rightarrow {\cal P}(\mathbb{R}); t\mapsto \rho_t$ and a Borel measurable
$v:\mathbb{R}\times[0,\infty)\rightarrow\mathbb{R}$ is a {\it weak solution pair of the Euler-Poisson equations \eqref{EP1D}}
which satisfies the initial conditions \eqref{Init} if the following hold. \\
$(i)$ For each $T>0$,
$$
\int^T_0\int_{\mathbb{R}}v^2d\rho_t dt<\infty.
$$
$(ii)$ For each $\phi\in C^\infty_c(\mathbb{R}\times[0,\infty))$,
$$
\int^\infty_0\int_{\mathbb{R}}(\partial_t\phi+v\partial_x\phi)d\rho_tdt+\int_{\mathbb{R}}\phi(\cdot,0)d\rho_0=0.
$$
$(iii)$ For each $\phi\in C^\infty_c(\mathbb{R}\times[0,\infty))$,
$$
\int^\infty_0\int_{\mathbb{R}}(v\partial_t\phi +v^2\partial_x\phi)d\rho_tdt+\int_{\mathbb{R}}\phi(\cdot,0)v_0d\rho_0=\int^\infty_0\int_{\mathbb{R}}\phi(\text{sgn}*\rho_t)d\rho_tdt.
$$
\end{defn}
Employing the same method used to prove Corollary \ref{PEEexistCor} from Theorem \ref{mainThm}, we have the subsequent corollary to Theorem \ref{secondThm}.
\begin{cor}
There exists a weak solution pair $\rho$ and $v$ of the Euler-Poisson equations \eqref{EP1D} which satisfies the initial conditions \eqref{Init}. Moreover, this solution pair additionally has the following features.
\begin{enumerate}[(i)]
\item For almost every $t,s\ge 0$ with $0\le s\le t$,
\begin{align*}
&\int_{\mathbb{R}}\frac{1}{2}v(x,t)^2d\rho_t(x)+\iint_{\mathbb{R}^2}\frac{1}{2}|x-y|d\rho_t(x)d\rho_t(y)\\
&\hspace{1in} \le \int_{\mathbb{R}}\frac{1}{2}v(x,s)^2d\rho_s(x)+\iint_{\mathbb{R}^2}\frac{1}{2}|x-y|d\rho_s(x)d\rho_s(y).
\end{align*}
\item For almost every $t>0$ and $\rho_t$ almost every $x,y\in \mathbb{R}$,
\begin{equation}
(v(x,t)-v(y,t))(x-y)\le \frac{1}{t}(x-y)^2.
\end{equation}
\end{enumerate}
\end{cor}
|
1,941,325,220,509 | arxiv | \section{Introduction}
In many problems tackled by genetic programming (GP), each individual is evaluated on a number of different \textit{test cases} to determine how well it solves the problem. In order to determine which individuals perform better and should be selected as parents, we calculate a \textit{fitness error} for each individual on each test case by comparing the distance between each output of the program and the correct output.
Most parent selection methods, including tournament selection, aggregate all of the errors of an individual into a single \textit{fitness value}, which is used to determine which individuals to select as parents.
Lexicase selection is a parent selection algorithm that does not combine errors into a single fitness value, but instead treats them all separately~\cite{LexicaseUncomprimising, LexicaseOriginal}. Lexicase selection often selects \textit{specialists}, or individuals that perform extremely well on one or several test cases, but may not perform well on others, and thus may have a poor aggregate fitness value~\cite{Helmuth:2019:GECCO:Specialists}. Specialists are usually good at solving test cases with similar behavior, but may not have figured out how to solve test cases with a different pattern. Lexicase selection, by favoring specialists, allows the GP system to often explore many areas of the search space simultaneously instead of converging on one promising-looking area of behavior. Lexicase selection has been shown to increase the diversity of the population, and improves performance on several types of problems, including software synthesis problems~\cite{LexicaseDiversity, lexicaseVtourn}.
With a similar motivation of avoiding premature convergence of evolutionary computation to areas of the search space with no solutions, Lehman and Stanley proposed novelty search~\cite{Lehmanoriginal, Lehman_Efficiently}.
Novelty search aims to explore the search space by selecting individuals with unique or uncommon behavior to be parents, without considering the fitness errors of the potential parents. In this way, novelty search does not drive selection towards an objective, but instead towards individuals that are different from those seen previously.
Novelty search has proven itself as an excellent selection method for solving problems with deceptive fitness landscapes~\cite{Lehmanoriginal, Lehman_Efficiently, Lehman11, Revising, Empirical_Study}, but has not yet been tested on general program synthesis problems.
In this paper, we synthesize novelty search and lexicase selection into \textit{novelty-lexicase} selection, with the aim of exploring and exploiting the search space by encouraging novel and specialized behavior simultaneously. We combine these objectives by adding ``novelty scores'' to the normal error values used by lexicase selection. Novelty-lexicase selection at times emphasizes selection based on novelty; other times it emphasizes good performance on the test cases; often it selects a single individual based on some aspects of both.
We conducted experiments comparing novelty-lexicase selection with novelty search, lexicase selection, and tournament selection on a suite of general program synthesis benchmark problems, a new problem domain for novelty search. While experiments capped at 300 generations failed to differentiate novelty-lexicase and lexicase selections, other metrics suggested that longer runs may benefit novelty-lexicase. Indeed, on runs capped at 1000 generations, novelty-lexicase selection significantly outperformed lexicase selection. This indicates that although novelty-lexicase selection is slower to find solutions on average, it continues to effectively search for solutions longer than standard lexicase selection.
The following section gives background information, and we give the details of novelty-lexicase selection in section~\ref{sec:novlex}. We next explain our experiments with novelty-lexicase selection in section~\ref{sec:experiments}, give the results of those experiments in section~\ref{sec:results}, and discuss the results in section~\ref{sec:discussion}.
\section{Related Work}
We will discuss behavioral diversity, a method for measuring population diversity, and the two selection methods motivating novelty-lexicase selection.
\subsection{Behaviors and Diversity}
We define the \textit{behavior} of an individual to be the vector of its outputs over the inputs.
Behavioral diversity is defined as the proportion of the population with distinct behaviors~\cite{JacksonPaper}. In order to calculate this, we tally the number of distinct behaviors within the population, and then divide by the number of individuals. Behavioral diversity is a phenotypic measure of diversity---meaning it is a diversity measure based on the functionality of programs, rather than their code or structure. An increased behavioral diversity allows the GP system to search programs in various parts of the search space simultaneously, and explore multiple methods of solving the problem. Previous studies have shown behavioral diversity to correlate well with problem-solving performance~\cite{JacksonPaper}.
\subsection{Lexicase Selection} \label{sec:lexicase}
\begin{algorithm}[t]
\caption{Lexicase Selection \textit{(to select one parent)}}
\label{alg:lexicase}
\begin{algorithmic}
\STATE Inputs: $candidates$, the entire population;
\STATE $\phantom{Inputs: } cases$, a list of test cases
\STATE Shuffle $cases$ into a random order
\LOOP
\STATE Set $first$ be the first case in $cases$
\STATE Set $best$ be the best performance of any individual in $candidates$ on the $first$ test case
\STATE Set $candidates$ to be the subset of
$candidates$ that have exactly $best$ performance on $first$
\IF {$|candidates| = 1$}
\STATE Return the only individual in $candidates$
\ENDIF
\IF {$|cases| = 1$}
\STATE Return a randomly selected ind. from $candidates$
\ENDIF
\STATE Remove the first case from $cases$
\ENDLOOP
\end{algorithmic}
\end{algorithm}
Lexicase selection can be used in any setting where individuals are evaluated on multiple test cases; in our study, test cases are given as input/output pairs, as in supervised learning. Lexicase selection chooses a parent by pruning the population over a number of steps. During each step, it compares the remaining candidate pool on a single randomly-selected test case and removes any individuals that do not have the very best error on this test case. This process continues until one individual remains or all test cases have been used, at which point it returns a randomly selected candidate from those remaining. Algorithm~\ref{alg:lexicase} formalizes lexicase selection.
While lexicase selection has typically been used over the errors of the individuals on each test case, it could be used on any type of scores assigned to the members of the population.
Previous studies have shown lexicase selection outperforming tournament selection and other parent selection methods in areas such as automatic program synthesis~\cite{ProblemInfo, Forstenlechner:2017:EuroGP}, boolean logic and finite algebras~\cite{LexicaseUncomprimising, Helmuth:2013:GECCOcomp, Liskowski:2015:GECCOcomp}, evolutionary robotics~\cite{moore:2017:ecal}, and boolean constraint satisfaction using genetic algorithms~\cite{Metevier2019}. Additionally, $\epsilon$-lexicase, a relaxed version of lexicase selection that at each step keeps any individuals within some threshold of the best individual, has performed well on symbolic regression problems~\cite{LaCava:2018:ecj, LaCava:2016:GECCO}. Studies showing that lexicase selection produces higher population diversity than other methods help explain its improved performance; it explores more widely in the search space, not converging to local optima, while still applying pressure for improvement on the test cases~\cite{LexicaseDiversity, lexicaseVtourn}.
\subsection{Novelty Search}
In 2008, Lehman and Stanley first presented novelty search as a way to avoid convergence to local optima when searching for a solution to a problem with a deceptive fitness landscape~\cite{Lehmanoriginal}. Fitness-seeking algorithms tend towards convergence, narrowing their search towards the most promising part of the search space. In many problems though, such as maze solving, converging towards the seemingly best solution can often lead to local optima where there is in fact no solution to be found. Novelty search originated from a need to prevent this convergence to local optima and increase genetic diversity within the population~\cite{Lehmanoriginal, Lehman_Efficiently, Lehman11}. Instead of selecting individuals with good performance, novelty search selects the most novel individuals to be parents for the next generation.
Novelty is calculated as the distance between an individual's behavior vector and that of its $k$ closest neighbors in the population~\cite{Lehman_Efficiently, Empirical_Study}. Some studies maintain a novelty archive consisting of individuals from earlier populations; these archives are also considered when finding the $k$ closest neighbors~\cite{Lehmanoriginal, Empirical_Study}. Once the novelty scores have been calculated, novelty search selects parents with tournament selection, selecting the most novel individual in each tournament regardless of their fitness. Novelty search has typically been applied to problems where behaviors have a small number of dimensions, allowing for inexpensive distance calculations compared to problems with behaviors over many dimensions.
Novelty search has proven useful in a variety of fields, such as Grammatical Evolution \cite{GEinSantaFeTrail}, Soft Robotics \cite{LaCava:2018:ALIFE} and Swarm Robotics \cite{Gomes2013}, though in soft robotics and swarm robotics it did not significantly improve results. Additionally, novelty search has been proven effective for solving problems in GP, such as difficult symbolic regression problems \cite{Martinez:2013:CEC}.
In previous studies, novelty search has proven more effective than fitness search for problems with deceptive fitness landscapes, such as the maze and artificial ant problems\cite{Lehman_Efficiently, GEinSantaFeTrail}.
Additionally, a family of related search techniques, termed ``quality diversity'' algorithms, aim to combine the ideals of novelty search with performance optimization~\cite{QualityDiversity}. Quality diversity algorithms are motivated by the same issues with novelty search that motivate us here to combine novelty search's ability to explore the search space while also providing pressure toward solving the problem. These quality diversity algorithms include MAP-elites~\cite{DBLP:journals/corr/MouretC15}, using a weighted sum of an individual's novelty and fitness scores, multiobjectivisation of novelty and fitness scores, and progressive minimal criteria novelty search (PMCNS) where individual's novelty scores are penalized for being below the 50th percentile in fitness score~\cite{Empirical_Study}. MAP-Elites aims to optimize performance within bins that emphasize maintaining diversity through novelty in the population~\cite{DBLP:journals/corr/MouretC15, Dolson}. Multiobjectivisation of novelty and fitness scores and the weighted score methods worked well on problems with deceptive fitness landscapes, but the PMCNS method did not~\cite{Empirical_Study}.
\subsection{Combining Novelty and Lexicase}
Perhaps the most similar work to that discussed here is another effort to combine novelty search and objective performance, called knobelty~\cite{kelly2019}. This work takes a different approach, with each parent selection event choosing to either select based on novelty (using tournament selection) or based on performance (using lexicase selection). Thus the novelty-based selection does not use the lexicase algorithm. They showed that knobelty, when paired with grammatical evolution and a variety of novelty metrics, performed well compared to vanilla lexicase selection on three program synthesis benchmark problems.
\section{Novelty-Lexicase Selection} \label{sec:novlex}
Novelty-lexicase selection uses the standard lexicase selection algorithm presented in section~\ref{sec:lexicase}, except that we add novelty scores to the list of error values normally used by lexicase selection. Similarly to how lexicase selection typically uses one error value per test case, we create one novelty score per test case. Thus, the list of cases used by novelty-lexicase selection is a randomly shuffled combination of errors and novelty scores, and each step of the algorithm compares the population either on one error value or one novelty score. Since lexicase selection puts the most emphasis on performance of the first handful of cases in the shuffled case list, some uses of novelty-lexicase will randomly emphasize excellence on novelty scores, some will emphasize excellence on fitness errors, and some will emphasize a mix of both. In this way, novelty-lexicase selection drives the population simultaneously towards novel behavior and more fit behavior.
Each novelty score should measure how similar a program's behavior is to other behaviors in the population.
In previous novelty search studies, the novelty of an individual is calculated as the average distance to the nearest $k$ neighbors, where the distance is measured between the individuals' behavior vectors~\cite{Empirical_Study,Enhancements,Lehman_Efficiently,Revising}. In order to use lexicase, we would instead like to calculate novelty scores for each test case. As computing the average distance to the $k$ closest neighbors over hundreds of test cases for each individual each generation incurs considerable expense when comparing outputs such as strings, we instead use a simpler novelty metric. We define the \textit{novelty score} of one behavior (output) of an individual to be the number of individuals in the population and archive that produce the same behavior on that test case. A lower novelty score indicates increased novelty, since fewer individuals produce the same output---thus novelty-lexicase selection will prefer lower novelty scores, as it prefers lower errors.
As recommended with novelty search, novelty-lexicase selection maintains an archive of individuals, adding one randomly selected individual each generation. We calculate novelty scores with respect to the combined population and archive, encouraging individuals to be novel not only with respect to their population, but also to areas of the search space explored in prior generations.
Novelty-lexicase selection never combines novelty scores into a single aggregate value, distinguishing it from novelty search and other quality diversity algorithms. As with lexicase selection not combining errors into a single fitness value, this allows novelty-lexicase to differentiate between the performance/novelty of different outputs. Thus an individual that exhibits extremely novel behavior on some test cases but very common behavior on other test cases may receive the attention of novelty-lexicase selection, where it would have received little or no attention by novelty search. This allows novelty-lexicase to select individuals that specialize in not only excellent performance on some test cases, but also excellent novelty on some test cases.
There is no asymptotic difference in time complexity between novelty-lexicase and lexicase selections, as they simply follow the same algorithm, except that novelty-lexicase has twice as many test cases in its list of test cases. The time for calculating novelty scores is negligible compared to the run time of the lexicase algorithm.
\section{Experimental Design} \label{sec:experiments}
Below we discuss the benchmark problems, GP system, and GP parameters used in our experiments.
\subsection{Benchmark Problems}
\begin{table
\centering
\caption{Benchmark problem input and output types. CSL is the Compare String Lengths problem, and RSWN is the Replace Space With Newline problem.}
\label{table:Problems}
\taburowcolors[2] 2{white .. light-gray}
\begin{tabu}{l ll}
\toprule
\textbf{Problem} & \textbf{Input Type} & \textbf{Output Type}\tabularnewline
\midrule
CSL & 3 Strings & Boolean \tabularnewline
Double Letters & String & Printed String \tabularnewline
Last Index of Zero & Vector of Integers & Integer \tabularnewline
Mirror Image & 2 Vectors & Boolean \tabularnewline
Negative to Zero & Vector of Integers & Vector of Integers \tabularnewline
RSWN & String & Printed String, Int \tabularnewline
Scrabble Score & String & Integer \tabularnewline
Syllables & String & Printed String \tabularnewline
Vector Average & Vector of Floats & Float \tabularnewline
X-Word Lines & String, Integer & Printed String \tabularnewline
\bottomrule
\end{tabu}
\end{table}
We compare selection strategies on ten representative program synthesis benchmark problems~\cite{ProblemInfo}. These problems challenge the GP system to produce programs that mimic introductory computer science student code, requiring various control flow structures and data types. We chose this subset of problems from the benchmark suite as they represent a wide range of difficulties, with some rarely solved in previous studies while others have been solved more than 70\% of the time using standard lexicase selection. Additionally, we chose problems with diverse data type requirements; we list the input and output types for each problem in Table~\ref{table:Problems}. We decided to use program synthesis benchmark problems instead of problems with deceptive fitness landscapes (such as the maze problems that have frequently been used with novelty search~\cite{Lehman_Efficiently, MinCriterion, QualityDiversity, illuminationalgos, Empirical_Study}) to highlight the differences between novelty search and novelty-lexicase selection on problems with highly-dimensional behaviors---the problems here each have 100 or more test cases used for evaluation.
For novelty search, we need to define the distance metrics used for the different possible output types listed in Table~\ref{table:Problems}.
The problems Compare String Lengths (CSL) and Mirror Image produce boolean outputs, meaning distances between behavior vectors are comparing vectors of booleans, for which it is natural to use Hamming distance. For the string and integer outputs of Replace Space With Newline (RSWN), we used Hamming distance simply comparing whether each output was equal or not to the other individual's output. The Last Index of Zero and Vector Average problems output integers/floats respectively, allowing us to use the Manhattan distance between vectors of numbers. Manhattan distance gives a more accurate estimate of the distance between two output vectors. However, we found Manhattan distance to be prohibitively expensive to calculate on problems with string output types, since it requires a Levenshtein distance calculation on every pair of string outputs for the same test case in the population. For this reason, we use a simple Hamming distance comparison for string outputs. For the Negative to Zero problem, which outputs a vector of integers for each test case, it was unclear to us how one would best calculate distances between vectors of vectors of integers.
\subsection{GP System and Parameters}
Our experiments with novelty-lexicase selection use PushGP, a GP system that evolves programs in the Push programming language~\cite{1068292, spector:2002:GPEM}.
Push maintains a stack for each data type, taking arguments from the stacks and pushing return values to the stacks. Additionally, it allows for a variety of standard and exotic control flow structures, aided by the fact that the running Push program itself is stored on a stack.
Push was specifically designed for use in GP, contributing to its flexibility with data types and its simple syntax, in which every nested list of instructions and literals is a valid Push program.
We use the Clojure\footnote{\url{https://github.com/lspector/Clojush}} implementation of PushGP, which has previously been used in a number of program synthesis experiments~\cite{ProblemInfo, LexicaseDiversity, lexicaseVtourn}. The source code used in our experiments is freely available.\footnote{\url{https://github.com/thelmuth/Clojush/releases/tag/Novelty-Lexicase}}
To measure the performance of each selection method, we follow the recommendation of the benchmark suite to measure the number of successful runs out of 100 GP runs~\cite{ProblemInfo}. We stop a GP run if it finds a program that perfectly passes all of the test cases used as training data. Additionally, for a program to count as a \textit{generalizing} success, it must perfectly pass a large number of withheld test cases not used during evolution. Before testing for generalization, we automatically simplify each solution program, which has been shown to significantly raise generalization rates~\cite{Helmuth:2017:GECCO}. To test for significant differences in success rates, we use a pairwise chi-square test with Holm correction and a 0.05 significance level.
\begin{table
\centering
\caption{GP system parameters and the usage rates of genetic operators.}
\label{table:PushGP-params}
\begin{tabular}{l r}
\toprule
\textbf{Parameter} & \textbf{Value} \tabularnewline
\midrule
population size & 1000 \tabularnewline
max number of generations & 300, 1000 \tabularnewline
tournament size for tournament selection & 7 \tabularnewline
tournament size for novelty search & 2 \tabularnewline
individuals added to archive per generation & 1 \tabularnewline
$k$ nearest neighbors for novelty search & 25 \tabularnewline
\midrule
\textbf{Genetic Operator Rates} & \textbf{Prob} \tabularnewline
\midrule
alternation & 0.2 \tabularnewline
uniform mutation & 0.2 \tabularnewline
uniform close mutation & 0.1 \tabularnewline
alternation followed by uniform mutation & 0.5 \tabularnewline
\bottomrule
\end{tabular}
\end{table}
We conducted 100 runs in each configuration on each benchmark problem using the GP parameters in Table~\ref{table:PushGP-params}. Except for where stated otherwise, we use the same parameters and recommendations given in the description of the program synthesis benchmark suite~\cite{ProblemInfo}. While we initially ran each run for a maximum of 300 generations, the results suggested that longer evolution times might prove interesting (as we will discuss below), so we also conducted experiments using a maximum of 1000 generations. For novelty search and novelty-lexicase selection, we add one randomly selected individual to the archive each generation, as recommended by Gomes et al.~\cite{Empirical_Study}. For novelty search, we assign novelty as the average distance between an individual and its $k = 25$ nearest neighbors, use a tournament size of 2, and a population of 1000, as recommended by Lehman and Stanley~\cite{Lehman_Efficiently}.
\section{Results} \label{sec:results}
\begin{table}
\centering
\caption{Number of successes out of 100 runs for novelty-lexicase selection, lexicase selection, tournament selection, and novelty search with a maximum of 300 generations. We have not run novelty search on Negative to Zero. Bold values are significantly better than all others in that row.}
\label{tab:completionrates}
\taburowcolors[2] 2{white .. light-gray}
\begin{tabu} {l |rrrr}
\toprule
\textbf{Problem} & \textbf{Novelty-lex} & \textbf{Lex} & \textbf{Tourn} & \textbf{Novelty}\\
\midrule
CSL & 3 & 5 & 3 & 0 \\
Double Letters & 4 & 1 & 0 & 0 \\
Last Index of Zero & 35 & 29 & 4 & 1 \\
Mirror Image & \textbf{100} & 87 & 45 & 72 \\
Negative to Zero & 61 & 62 & 6 & - \\
RSWN & 71 & 58 & 15 & 0 \\
Scrabble Score & 1 & 4 & 0 & 0 \\
Syllables & 11 & 24 & 2 & 0 \\
Vector Average & 35 & 43 & 14 & 0 \\
X-Word Lines & 18 & 18 & 0 & 0 \\
\bottomrule
\end{tabu}
\end{table}
Table~\ref{tab:completionrates} gives the number of successes out of 100 runs on each problem for each selection method with a maximum of 300 generations. Both lexicase selection and novelty-lexicase selection outperformed tournament selection and novelty search consistently. The lexicase selection and novelty-lexicase selection results are very similar, with the only significant difference between them on the Mirror Image problem. Novelty search performed particularly poorly here, only finding solutions to two of the nine problems on which we ran it, and only outperforming tournament selection on one problem.
\begin{figure}
\centering
\includegraphics[width=0.99\linewidth]{rswn-behavioral-diversity.pdf}
\caption{Mean population behavioral diversity across generations of each selection method on the Replace Space with Newline problem.}
\label{fig:bdrswn}
\end{figure}
\begin{figure}
\includegraphics[width=0.99\linewidth]{syllables-behavioral-diversity.pdf}
\caption{Mean population behavioral diversity across generations of each selection method on the Syllables problem.}
\label{fig:bdsyllables}
\end{figure}
Figures \ref{fig:bdrswn} and \ref{fig:bdsyllables} show the behavioral diversity of each selection method over the maximum 300 generations on two representative problems, Replace Space with Newline and Syllables.\footnote{The initial dip in behavioral diversity in the first few generations is a known phenomenon, caused by a few randomly generated individuals dominating the selection, and should be ignored in favor of the later trends~\cite{Helmuth:2016:GECCO}.} The levels of diversity in these plots mirror those for other problems, which we do not present. Although novelty-lexicase selection did not achieve a significantly better success rate than lexicase selection, the behavioral diversity of the novelty-lexicase populations was close to 1 for both problems, much higher than lexicase selection, which in turn is much higher than tournament selection. Both novelty search and novelty-lexicase selection rapidly evolve diverse populations, while lexicase selection cultivates diversity more slowly on Replace Space with Newline. When the diversity of novelty search peaks and levels off around generation 30, the diversity of novelty-lexicase continues to rise throughout evolution. As has been noted previously~\cite{lexicaseVtourn}, tournament selection does an extremely poor job of maintaining population diversity compared to the other methods.
Even when solving the Syllables problem, where novelty-lexicase selection produced fewer solutions than lexicase selection, novelty-lexicase selection created more behaviorally diverse populations than lexicase or tournament selection. In this case, lexicase and novelty-lexicase selection's behavioral diversity grew significantly in the first twenty generations, but then lexicase plateaued just under 0.80 behavioral diversity, while novelty-lexicase hovered over 0.90. Novelty search grows diversity quicker than any other method, but is again overtaken slightly by novelty-lexicase selection my the end of the runs. Tournament selection's behavioral diversity was, at it's highest point near the end of the run, approximately 0.40.
\begin{table}
\centering
\caption{Average generation of solution discovery for novelty-lexicase selection, lexicase selection, tournament selection, and novelty search with a maximum of 300 generations.}
\label{tab:gensolution}
\taburowcolors[2] 2{white .. light-gray}
\begin{tabu} {l | cccc}
\toprule
\textbf{Problem} & \textbf{Novelty-lex} & \textbf{Lex} & \textbf{Tourn} & \textbf{Novelty} \\
\midrule
CSL & 199 & 13 & 28 & - \\
Double Letters & 270 & 203 & - & - \\
Last Index of Zero & 83 & 81 & 15 & 250 \\
Mirror Image & 44 & 18 & 43 & 100 \\
Negative to Zero & 85 & 80 & 141 & - \\
Scrabble Score & 239 & 128 & - & - \\
Syllables & 237 & 159 & 84 & - \\
RSWN & 96 & 93 & 122 & - \\
Vector Average & 150 & 138 & 66 & - \\
X-Word Lines & 239 & 199 & - & - \\
\bottomrule
\end{tabu}
\end{table}
The behavioral diversity of the novelty-lexicase populations were consistently higher than those produced by tournament selection or lexicase selection; higher than lexicase selection by more than 5\% consistently. This higher diversity raises the question of which algorithm is finding solutions faster; whether the diversity of novelty-lexicase selection allows it to locate important areas of the search space first, or if it is putting more effort into exploration and less into exploitation of good programs, leading to slower traversal of gradients toward solutions. We calculated the average generation of solution discovery in Table~\ref{tab:gensolution}. Novelty-lexicase selection was generally slower than lexicase and tournament selections, only twice finding solutions faster than tournament selection. Lexicase selection was consistently faster at finding solutions than novelty-lexicase selection, although on the Last Index of Zero, Negative to Zero, and Replace Space With Newline problems the difference was small.
Since we see that novelty-lexicase selection produces increased diversity and takes more generations to find solutions, we hypothesize that with a larger number of generations novelty-lexicase will continue to find solutions, while lexicase selection will slow down as remaining runs sometimes converge to local optima. Thus we conducted a second experiment, where novelty-lexicase and lexicase systems run for 1000 generations.
\subsection{Maximum of 1000 Generations}
Table~\ref{tab:completionrates1000} presents the results of extended, 1000-generation runs. While both novelty-lexicase and lexicase selection increased their solution rates on most problems compared to 300 generations, many of lexicase's increases were insignificant while novelty-lexicase produced large increases on most problems. Novelty-lexicase selection performed significantly better than lexicase selection on five problems, while lexicase only performed significantly better on one.
\begin{figure}
\includegraphics[width=0.99\linewidth]{successes-lexicase-new}
\caption{The accumulated number of solutions found by lexicase selection over 1000 generations for every problem.}
\label{fig:lexicase-gen}
\end{figure}
\begin{table}
\centering
\caption{Number of successes out of 100 runs for novelty-lexicase selection and lexicase selection with a maximum of 1000 generations. Results that are significantly better at the 0.05 level are in bold.}
\label{tab:completionrates1000}
\taburowcolors[2] 2{white .. light-gray}
\begin{tabu} {l |cc}
\toprule
\textbf{Problem} & \textbf{Novelty-lex} & \textbf{Lex}\\
\midrule
Compare String Lengths & \textbf{39} & 9 \\
Double Letters & 17 & 14 \\
Last Index of Zero & \textbf{51} & 30 \\
Mirror Image & \textbf{100} & 88 \\
Negative to Zero & \textbf{83} & 60 \\
RSWN & 77 & 77 \\
Scrabble Score & 18 & 17 \\
Syllables & 50 & 44 \\
Vector Average & 62 & \textbf{78} \\
X-Word Lines & \textbf{65} & 47 \\
\bottomrule
\end{tabu}
\end{table}
\begin{figure}
\includegraphics[width=0.99\linewidth]{successes-novelty-lexicase-new}
\caption{The accumulated number of solutions found by novelty-lexicase selection over 1000 generations for every problem.}
\label{fig:nov-lex-gen}
\end{figure}
\begin{figure}
\includegraphics[width=0.99\linewidth]{generalization.pdf}
\caption{Generalization rates of novelty-lexicase and lexicase selections (after simplification).}
\label{fig:generalization}
\end{figure}
To get a better understanding of when during search GP finds solutions, we plot the accumulated number of solutions found over all 1000 generations in Figures~\ref{fig:lexicase-gen} and \ref{fig:nov-lex-gen}. Figure~\ref{fig:lexicase-gen} shows that lexicase selection finds many of its solutions in the first 500 generations. On the other hand, novelty-lexicase continues finding solutions to most problems right up to generation 1000, and may even continue finding solutions past generation 1000 if it were given the chance. Take, for example, the Compare String Lengths (CSL) problem. As can be seen in Figure \ref{fig:lexicase-gen}, the number of successes for lexicase selection plateaus around generation 200, while the number of successes found by novelty-lexicase selection in Figure \ref{fig:nov-lex-gen} steadily increases over all 1000 generations.
In general, lexicase selection plateaus and stops finding new solutions around generation 500, while novelty-lexicase selection does not have a clear plateau, except for the Mirror Image and Negative To Zero problems. The plateau in solutions found for the Mirror Image problem is due to all runs having found solutions by this point. The Negative To Zero problem plateaus for a similar reason: 98 out of 100 runs found a solution that passed the training data, and hence stopped evolution. However, only about 80\% of these solutions generalized and counted as a success. So although this set of runs plateaus at 83 successes, only 2 runs continued beyond generation 600 and could have contributed to more successes.
Although novelty-lexicase selection had some difficulty on that problem with generalization, overall novelty-lexicase had better generalization of training solutions to unseen test data than lexicase selection, as shown in Figure~\ref{fig:generalization}. The difference in generalization rate is significant in the favor of novelty-lexicase selection on the Compare String Lengths, Last Index of Zero, Mirror Image, Negative to Zero, and X-Word Lines problems, while lexicase selection has significantly better generalization on only the Vector Average problem. For whatever reason, lexicase selection finds many more solutions to the training data that do not generalize to unseen data. While we have not explored exactly what causes this difference, we offer a hypothesis in the next section. Regardless of why novelty-lexicase generalizes better, its ability to generalize contributes significantly to its better success rate on many of these problems.
\section{Discussion} \label{sec:discussion}
When comparing the diversity rates produced by each selection method, we found that novelty search and novelty-lexicase have many similarities, with both displaying higher levels of diversity than lexicase selection. By shuffling the errors normally used by lexicase selection with novelty scores on each test case, we have added extra objectives that focus on individuals that not only perform well, but behave differently from other individuals in the population, leading to increased diversity.
Is GP exploring additional interesting and useful parts of the search space by creating more novel individuals, or simply increasing novelty for novelty's sake while using up program evaluations that could instead be used for refining promising individuals? The low success rates produced by novelty search indicate that in the highly-dimensional and unbounded output spaces of the program synthesis problems considered here, novelty search often produces a huge variety of individuals without finding a solution. Novelty search does not apply selection pressure toward better individuals by design; while this enables it to produce interesting and different individuals, as well as solve problems in constrained, low-dimensional output spaces, there are simply too many ways to be novel without being good when a behavior contains 100 string or integer outputs. We note that novelty search may perform better on these problems with a more carefully crafted method for measuring behavior than simply listing the outputs of the program on the test cases, but with no obvious alternative, this nontrivial task is beyond the scope of this paper.
Turning to novelty-lexicase selection, when running GP for a maximum of 300 generations, we found that the increased diversity did not immediately translate into more successes than lexicase selection, though they behaved comparably. However, given more time to evolve, novelty-lexicase selection converged to local optima much less often than lexicase selection, finding solutions throughout all 1000 generations while lexicase selection often failed to find many solutions after 300-500 generations. The lexicase results mirror some extended 600 generation runs on this same benchmark suite using grammar-guided GP, where lexicase selection often plateaued after approximately 300 generations~\cite{Forstenlechner:2018:CEC}. The ability of novelty-lexicase selection to avoid premature convergence and continue effectively exploring the search space shows great promise whenever evolution requires many generations to construct complex solution programs.
In addition to avoiding local optima, novelty-lexicase selection tends to evolve solutions that more likely generalize than those produced by lexicase selection, as shown in Figure~\ref{fig:generalization}. We hypothesize that the strong selection pressure that lexicase selection applies toward finding solutions encourages solutions that sometimes ``memorize'' edge cases in a way that does not generalize. Additionally, because novelty-lexicase selection gives the lexicase algorithm a combination of novelty scores and fitness errors, any one fitness error will appear in the first $n$ shuffled cases half as often as with standard lexicase selection. Thus novelty-lexicase selection will reward memorizing the answer to a single edge case less often than lexicase selection.
One interesting and confounding observation from Figure~\ref{fig:generalization} is how generalization changes for the two problems with boolean program outputs, Compare String Lengths and Mirror Image. Both of these problems, especially Compare String Lengths, exhibited significantly better generalization with novelty-lexicase selection compared to lexicase selection. Let's consider how the novelty scores work for a boolean output problem. Since such a program has only two possible outputs, all of the novelty scores for each test case will have one of two values; whether a novelty score is good or not will depend entirely on whether the output is the same as the majority of the population or the minority of the population.
Now consider how the error case and novelty score case allow an individual to continue or not in the candidates pool during the lexicase process. If the output is correct and is in the minority of outputs in the population, the individual will continue in candidates when either case appears in lexicase. If the output is correct but in the majority, the individual will continue in candidates when the error value appears, but not the novelty score case. On the other hand, if the output is wrong but in the minority, the individual will continue in candidates when the novelty score appears, but not the error case. Thus, being wrong and in the minority is just as good as being right and in the majority, if not better, since the case that allows continuation will have fewer other candidates also continuing! This is quite confounding, as it seems like it rewards being wrong as long as most other individuals are right. It is not clear if this observation contributes to better generalization, and if so, how.
Novelty-lexicase selection not only continually emphasizes unseen parts of the search space, but also targets regions that show promise for solutions, combining the core strengths of both novelty search and lexicase selection. It additionally improves generalization of the solutions it does find. By taking the best parts of novelty search and lexicase selection, novelty-lexicase selection considers more of the search space than lexicase selection but concentrates on objectives more than novelty search, leading to better problem-solving performance than either.
\section{Conclusions and Future Work}
In this paper, we synthesized novelty search and lexicase selection in order to create a new parent selection method, novelty-lexicase selection. In doing so, we hoped to combine the diverging and exploratory qualities of novelty search with fitness seeking of lexicase selection in order to target multiple promising parts of the search space. For novelty-lexicase selection, we calculate the novelty score of every output of every program by simply counting the number of other programs with the same output, instead of calculating the difference between error vectors in the style of novelty search. This makes novelty-lexicase selection less computationally expensive for output types that require significant computation for finding distances, such as strings, making it feasible to run on problems with any output type. To select a parent, novelty-lexicase combines the lists of novelty scores and fitness errors into one shuffled list before applying the lexicase algorithm. Since the lexicase algorithm applies more pressure toward the cases appearing earlier in the shuffled list, novelty-lexicase distributes selection pressure to both individuals with excellent errors and those with novel behaviors, most often selecting individuals that perform well on a combination of the two. This pressure drives evolution to seek novel behavior in productive parts of the search space.
We tested the performance of novelty-lexicase selection by comparing its success rate on ten program synthesis benchmark problems against lexicase selection, tournament selection, and novelty search. We compared the performance of these selection methods using PushGP for a maximum of 300 generations, and found that lexicase selection and novelty-lexicase selection were competitive in solution discovery, with novelty search and tournament selection performing poorly in comparison; additionally, novelty-lexicase selection improved population diversity compared to lexicase selection. We were unable to run novelty search on problems with vector outputs, and using the most informative string distance calculations via Levenshtein distance proved prohibitively slow. Novelty search with Hamming distances between string outputs proved ineffectual. We continued our experimentation by running lexicase selection and novelty-lexicase selection for 1000 generations, where novelty-lexicase selection found significantly more solutions on several of the problems and showed much better generalization rates.
To continue this research, we could imagine changing the novelty-lexicase implementation to more closely follow the traditional novelty calculation. In modifying novelty search to be compatible with lexicase selection, we deviated from the $k$-closest neighbor novelty scoring method. In our novelty score calculation, we simply compare an individual's output on a test case to each other individual's output on the same test case, and check for equality. To give a finer-grained measure of distance to other individuals, we could instead calculate the novelty error as the \textit{distance} from the individual's output on a test case to the $k$ closest outputs on the same test case. We could imagine this more detailed version of novelty distance could provide a more accurate representation of the novelty of each output, leading to better exploration of novel programs. However, these real-valued distances provide a new challenge for the lexicase algorithm, which performs best if some or many individuals have tied best values on a test case. With these novelty errors, it would likely therefore be important to use $\epsilon$-lexicase selection, which has been shown to handle real-valued errors much better than standard lexicase selection~\cite{LaCava:2018:ecj, LaCava:2016:GECCO}.
In order to better understand the mechanics of novelty-lexicase selection, we would find it fascinating to track the number of individuals that make it past each step of the lexicase algorithm. In particular, we could compare the number of individuals knocked out of contention by novelty scores compared to fitness errors. It is possible that many times, one or a small number of individuals achieve the best novelty score on each test case; on the other hand, it is just as possible that many individuals have equally good novelty on each score, leading to few individuals being knocked out by novelty scores. A better understanding of the influence of each type of test case on selection could lead to an improved weighted shuffling~\cite{Troise:2017:GPTP} of the novelty scores and fitness errors, instead of simply using uniform shuffling.
In this experiment we used a combination of novelty scores and fitness errors in the list of test cases for the lexicase algorithm, with the goal of simultaneously driving the population toward fit and novel solutions. This combination of novelty and errors improved the novelty-lexicase selection success rate on the problems in our experiments, but could potentially hinder solution discovery for problems with more deceptive (and constrained) fitness landscapes, such as maze problems. Using only novelty scores without fitness errors in lexicase selection could potentially improve upon the performance of novelty search on these types of problems.
\begin{acks}
We would like to thank Alexander Dennis and Lee Spector for conversations that helped shape this work, and to Steven Young for systems support.
\end{acks}
|
1,941,325,220,510 | arxiv | \section{Introduction}
On 4 July 2005 UT, the Deep Impact spacecraft impacted comet 9P/Tempel 1.
The goal of the impact experiment was to expose deep layers of the
comet nucleus to study the chemical composition
of those layers and to ascertain the structural properties of
the nucleus.
Comets are the remnant building blocks from the outer solar nebula.
The majority of their lifetimes are spent at large heliocentric
distances so they undergo little chemical evolution. However, even
far from the Sun, the outer layers are altered via radiation
darkening (Strazzula and Johnson 1991). Once comets enter the inner
Solar System, sublimation
removes outer layers, while mantling covers the exterior with dust.
\nocite {stjo91}
Comet 9P/Tempel 1 is a Jupiter family comet which has made many
passages into the inner Solar System and has not been extremely
far from the Sun in many orbits. Thus, the expectation was that
there was an altered outer layer to the comet, but that the interior
was pristine. The outer layer is thought to be too deep
for a small drill carried by a spacecraft to penetrate to pristine
material. In order to reach fresh, unaltered material,
the Deep Impact mission proposed to create a large crater by placing a
small (370kg) spacecraft in the path of the comet, setting up an impact.
The impact occurred on 4 July 2005 at 05:44:36 UT (Earth-received time
of 05:52:02 UT). The impact speed was 10.3 km/sec resulting in an impact
kinetic energy of 19 GJ (A'Hearn {\it et al.} 2005).
\nocite{ahdeepimpact}
The impactor was carried by a larger flyby spacecraft which was able
to monitor the impact with high spatial resolution. However, while the
flyby spacecraft carried an IR spectrometer,
it was not able to perform optical spectroscopic observations of
the impact,
nor was it able to monitor the comet for many days after impact.
Thus, a ground-based campaign was established in order to monitor
changes in the comet (Meech {\it et al.} 2005). \nocite{medeepimpact}
We observed the impact and its aftermath from the Keck 1 telescope using the
HIRES spectrograph as part of the ground-based campaign.
High resolution spectra could be used to monitor aspects of the impact which
could not be observed at lower spectral resolution. These include
monitoring isotope ratios to determine whether the ices in the interior
of the comet are isotopically similar to the ices we normally
see subliming from the surface. Jehin {\it et al.} (2006) reported the
isotopic ratios for $^{12}$C/$^{13}$C and $^{14}$N/$^{15}$N, both
during the first few hours after impact and at times other than
the impact. They showed that the excavated material is isotopically
similar to the normal outgassing. High resolution spectra
allow for detailed examination of the rotational states of the molecules.
This allowed us to determine whether any prompt emissions were
seen or whether the impact produced any rotationally hot lines.
High resolution also allowed for detection of any new, unexpected
species. High resolution data are more sensitive to very weak lines
than low resolution spectra because higher spectral resolving power disperses
the underlying continuum to a greater extent, resulting in higher
line-to-continuum contrast (our spectral resolution of $\sim$6 km\,sec$^{-1}$
was not sufficient to resolve typical cometary lines with widths of
$\sim$1 km\,sec$^{-1}$).
\nocite{jedeepimpact}
\section{Observations and Reductions}
The data were obtained with the Keck 1 telescope and HIRES
spectrograph (Vogt 1992) on 30 May 2005 and 4--6 July 2005 UT
(see table \ref{log} \nocite{vogthires}
for details of the observations). The Keck HIRES operates with three
$2048 \times 4096$
Lincoln Lab CCD detectors with 15~$\mu$m pixels. Typically, the detectors are
binned in the spatial direction, resulting in 1024 spatial pixels of 0.24
arcsec projected on the sky. The spectrograph has two cross dispersing
gratings to preferentially image the blue or the red part of the spectrum.
We used the UV cross disperser (despite the wavelength
covered, the convention is to call the chips blue, green and red and we will
use these names throughout this paper; blue covers 3047--3951\AA, green
covers 3971--4926\AA, and red covers 4977--5894\AA).
The slit was the ``C1" slit, which images $7 \times 0.86$ arcsec
of the sky (or $4570\times562$\,km at the comet's distance).
This results in a nominal resolving power,
$\lambda/\Delta\lambda = 47,000$ with the slit projecting to 4.8 pixels on
the detector.
The slit was set at the parallactic angle, which means that it rotated
on the sky during
the night. The slit orientation is listed in table~\ref{log}. By
setting the slit at the parallactic angle, we ensured that
atmospheric dispersion changed the position of the comet along the
slit, minimizing loss of light on the slit. However, it did mean that
we sampled slightly different coma regions throughout the night.
Thus, we could be sampling overdense regions differently at different
times.
On all nights, bias frames, flat field and ThAr lamps were observed for
calibration (darks were not obtained because these chips typically have
almost no dark current; however, as we will discuss below, this turned out
not to be the case in July). During the July observations we also observed
a solar analogue
star (HD159222 on 4 July (Hardorp 1978) and 18 Sco (Porto de Mello 1997)
on 5 and 6 July) to be able
to remove the dust continuum and a rapidly rotating hot star (HR 7279
on 4 July and HR 7415 on 5 and 6 July) to be able to remove telluric
features. On 30 May and 4 July a flux standard was observed.
\nocite{podmesi97,hardorp78}
On 30 May 2005, three 20-minute spectra were obtained as part
of the Keck Director's time. These observations were intended
to give a pre-impact baseline of how the spectrum of the
coma looked in its normal state.
The observations of 4 July 2005 started during nautical
twilight, at 05:36:15, so that we could obtain a pre-impact spectrum.
Eight degree twilight was at 05:29UT; 12 degree twilight (nautical) was at
05:58UT; 18 degree twilight (astronomical) was at 06:28.
Since it was not fully dark, the signal
from the sky was substantial in this first spectrum. The second
spectrum was started at approximately the impact time (start at 05:55:18~UT,
about 3 minutes after impact). We were
guiding directly on the comet so we could watch the coma brightening.
The exposure time chosen was a compromise between obtaining sufficient
signal/noise and keeping as rapid a cadence as possible.
The observations continued until 09:41:10 UT. The exposure
times were lengthened as the comet dimmed again after the impact and
the airmass of the comet increased.
We observed the comet to brighten rapidly within a few minutes of
impact and to gradually fade as we continued to observe. This
is in accord with other observers (Meech {\it et al.} 2005;
Schleicher {\it et al.} 2006). \nocite{medeepimpact,scbaba06}
We will discuss the trends of what we observed more in the next
section.
The observations of 4 July 2005 were obtained as Keck Director's
time. The observations of 30 May and 4 July are available to the
public from the Keck archive. Because of the data reduction issues
discussed below, we suggest that people interested in obtaining data
from the archive download raw and not processed data.
We also obtained observations on 5 and 6 July 2005 UT to follow up on
the behavior of the comet after the impact. For some of
the spectra on these two nights, we offset the telescope from the
optocenter to be able to probe the distribution of the gas in
the coma. The offset positions are noted in the log.
The night of 6 July had some clouds, whereas 30 May and 4 and 5 July
were photometric. Because of the clouds on 6 July, these observations were
used only for the isotopic abundances but not for monitoring the cometary
activity.
Keck Observatory has a data reduction pipeline, MAKEE, which many
observers use. However, since it assumes that the target is a
point source and the ends of the slit are sky, it is not appropriate
for the reduction of cometary data. Note that the reduced versions
of the data in the archive have been reduced with the MAKEE pipeline
so should be used with caution.
We reduced all four nights' data with the {\it echelle} package in {\it IRAF}.
Each chip was split out to a separate file and the chips were processed
in parallel. Zero and bias correction and trimming were first performed.
The first 7 pixels and pixels greater than 4000 in the spectral direction
are no good for any chip. All the images were then flat fielded.
While building flat field images for the July data, we discovered an
enhanced background in the spectral images. This manifests itself as a signal
that is measurable in the inter-order gaps. At first, we assumed
that this was scattered light in the spectrograph. However,
inspection of the comet frames showed that this background scaled
with exposure time, as if there was a high dark current. This effect
was absent in the May data. We cannot confirm whether the darks were indeed
elevated for our observations since the dewar had been opened for
unrelated reasons before we realized the extent of the background problem.
Regardless of the reason for the background, we needed to
remove this from all the spectra or it would have added an unwanted
offset to all spectra. For the green and red chip
this was not difficult. It could be removed with the
{\it apscatter} routine for the flat fields.
For the cometary spectra, we fit a surface to the
inter-order region, working in small sections around each order.
Handling the blue chip background was more complicated because the blue-most
orders overlap one another slightly. We could define the trace
of the spectrum on the chip with a slit which was less tall than
the 7 arcsec slit we were using for targets. However, with no
inter-order gap between some orders in the actual comet spectra,
there was no background to fit a
surface. Therefore, we built a ``background" image from a
flat obtained with the shorter slit, scaled that image to
the background of the individual cometary spectra and subtracted
it from the comet spectra.
The fact that the background fit to a flat field spectrum was the same
shape as that underneath the cometary spectra argues again for a
dark current-like source, rather than scattered light since the comet and
the flat field are different colors.
Once the data were flat fielded, the spectra for each order were
extracted. We used variance weighting to collapse
the spectrum spatially.
We defined the extent of the slit on the chip with the
flat field images. We fit the dispersion curve to the
ThAr lamp spectra. The typical rms for the wavelength
solution for each chip was 0.004\AA. The cometary spectra
were Doppler shifted to the spectrograph rest frame using
the orbital information.
In order to determine the behavior of the dust and coma gas as
a function of time, we first corrected the spectra for
extinction using mean extinction coefficients for Mauna Kea Observatory
(as given on the Keck Telescope web site; Hodapp (personal communication)
reported that the
extinction was normal for Mauna Kea on 4 July and inspection of our solar
analogue spectra confirm this).
This results in spectra as if seen outside the atmosphere. Then,
since we obtained the spectra with a variety of exposure times,
each spectrum was normalized to the equivalent count rate for a
15 minute exposure (our most common exposure time on 4 July).
Cometary spectra consist of three component parts: the emission spectrum
of the gaseous coma, mostly as the result of resonance fluorescence;
the continuous spectrum which underlies the emission that results
from the solar light being reflected from the dust; the telluric
spectrum being superposed on the comet spectrum. The telluric
spectrum only shows up in some orders.
In order to study the gaseous component, we used the spectra of
the solar analogue stars as a proxy for the solar spectrum and
shifted these spectra to align the absorption features with the
cometary absorption features. The stellar spectra were scaled
to match the continuum of the comet spectra and then subtracted.
The process was done interactively, with the aim of having
no features left in the continuum regions.
This shift and scale process was performed on each order
independently.
There were several problems which we encountered with this process.
For some of the orders, especially the bluest orders and observations
at high airmass, the continuum in the comet spectra was very weak, so
it was difficult to align the stellar and cometary spectra. In those
cases we relied on shifts derived at lower airmass or redder
wavelengths, assuming that all spectra in a given spectral image and night
had the same shift. This
assumption is accurate to the nearest pixel but not to sub-pixel
values of the shifts. However, since in these cases the solar
continuum contributed little to the cometary spectra, incorrect
shifts at this level introduce very little additional noise.
For the first spectrum on 4 July, which was obtained with a bright
sky, we actually had two continuum spectra imposed on each gas
spectrum. The first was the cometary. The second was the Earth's blue
sky. The sky spectrum dominated the signal and was not at the
same Doppler shift as the cometary spectrum. Thus, when removing
the continuum, we preferentially removed the sky spectrum but
would end up with a noisy comet spectrum from the slight wavelength
offset between the comet and sky spectra.
The most important problem which we encountered for interpretation of
what happened during the aftermath of the impact was that the
continuum brightened much more and faster than did the gas. For most
orders, this was just taken into account by the scaling process.
However, we had problems removing the continuum for the spectral
order which contains the CN $\Delta v = 0$ band. In that order,
in the first few spectra where the continuum was strongest, we
found that if we used the criterion that there should be no
absorptions left between the P and R branches, it would appear
that we had over-subtracted continuum to the blue and red of the
band and had imposed ``emission" features. If we ended up with
a smooth continuum to the blue and (especially) red, then we ended
up with absorptions left between the R and P branches. We
compromised and ended up with some absorption between the branches.
This effect is illustrated in Fig.~\ref{cnprob}
and most affected spectra 67--70.
In general, the continuum was strongest in spectra 67--70 and these
spectra had relatively short exposure times. Thus, these are
the spectra for which the spectral regions between the lines are
noisiest and therefore, these are the spectra with the lowest
signal/noise.
Once the continuum has been removed, the resultant spectra represent
the pure gaseous emission from the comet. We can subtract these
gas spectra from the original spectra, yielding pure continuum spectra.
These continuum spectra give a measure of the dust.
\section{Results}
\subsection{Dust}
Once we have separate dust and gas spectra we can investigate
the trend of each with time after the impact. Since there are
no distinct dust features, we maximized the signal/noise by
averaging the dust spectra by chip. In practice, to avoid
edge effects, we averaged the middle 2000 pixels in wavelength
and all except the first and last order on each chip.
For each spectral image, this results in three average values
centered at $\sim3460$, $\sim4475$, and $\sim5425$\AA\ (the
``bandpasses" are approximately 750\AA\ wide).
The upper panel of Fig.~\ref{dustplot} shows the trend of the dust in the
three chips as a function of time. Note that the average counts
peaked approximately 2300\,sec after the impact and then declined.
The peak is less distinct in the blue chip data than the green
and red chips. This may or may not be significant since the signal
is so much lower at these blue wavelengths.
The error bars on these data are based on Poisson statistics of the
average counts. However, since each average is computed from 18,000 --
38,000 pixels, these error bars are extremely conservative.
The initial rise in the average counts indicates that the impact
released a large amount of material, increasing the scattering
cross-section. This brightening can come from both an increase
in the number of particles, as well as from a change in particle
size distribution to a larger fraction of smaller particles.
A change in particle size distribution
could potentially result in a change in dust color, depending
on the particle sizes relative to the wavelength of observation.
We can look at the color evolution of the dust by ratioing
the average counts of the different chips.
These colors are shown in the lower panel of Fig.~\ref{dustplot}.
We see no change in the color of the dust during our observations
of 4 July (note that the 4 July observations span a large range of airmass
so the lack of a color change corroborates the reasonableness of the
adopted extinction coefficients).
We also show the average values for 30 May and 5 July for both the individual
chip counts and for the colors (averaging only those observations
when the slit was centered on the optocenter).
The colors are similar on all three nights, though the
4 July data may be slightly redder.
However, within the error bars, the dust is the same color on 4 July
as on 30 May or 5 July.
This would be in contrast
with the Spitzer observations, where they report a ``bluening"
of the scattered light (Lisse {\it et al.} 2006),
and with Schleicher {\it et al.} (2006), who noted that the material
was redder in color than the general inner coma in the first
15 minutes. \nocite{lilpsc06}
The average counts for each chip show that the continuum count
rate was lower on 5 July than when we stopped
obtaining spectra on 4 July.
The 5 July count rate was similar to the 30 May count rate so probably
represents the ambient level.
If the particles are around 0.5--1$\mu$m in size,
as observed by Sugita {\it et al.} (2005), or even in the 0.5--2.5$\mu$m
size range as observed by Schleicher {\it et al.} (2005), we would have seen
a change in color if the particles changed size substantially.
Since we do not see a change in color, there was not substantial
fragmentation or sublimation of the dust in these first two days.
\nocite{sudeepimpact, scbaba06}
\subsection{Gas}
The observations spanned a large wavelength range so we were able
to observe many molecular emission features. These included
OH, NH, CN, C$_{3}$, CH, C$_{2}$ and NH$_{2}$. All of these
species are radicals and are believed to be either daughter or
granddaughter species. The emission spectra are formed
via resonance fluorescence.
When the impact occurred, some volatile and non-volatile
material was thrown out of the crater.
Keller {\it et al.} (2005) argue that the impact
energy was insufficient, by a factor of several hundred (K\"{u}ppers
{\it et al}. 2005) to sublimate the observed amount of water
molecules, so there was unlikely to be vapor released because
of the impact. \nocite{kuetal2005}
Then, the ices needed to sublime to form
the parent gases and finally they dissociated to form the daughter
and granddaughter species. Each dissociation step has a
characteristic lifetime, which is typically longer than the time
for the material to flow out of our, rather small ($0.86\times7$ arcsec
or 562$\times$4570\,km),
aperture. However, some of any parent will be
dissociated instantly, while other molecules of that parent will take
more time, so that the lifetime is an e-folding time for the dissociation
process to take place. The signal which we observe
is that which is imposed upon the ambient spectrum of the cometary coma had
there been no impact.
We can achieve the highest signal/noise for extracted spectra by summing
the whole length of the 7\,arcsec slit. However, we then must try to deconvolve
the effects of outflow from the effects of the impact.
In order to understand better the impact effects, we extracted the
spectra only over the spatial resolution of the telescope -- the
inner 0.7\,arcsec (457 km) or 3 pixels in the spatial direction (the
slit width of 0.86 arcsec or 562 km still defines the spectral
resolution of the observations). These spectra from the inner
pixels still incorporate some of the coma at large distances
along the line of sight but the relative contribution of the near
nucleus region to the coma is increased over spectra extracted
along the full length of the slit. As would be expected, these spectra are
considerably fainter than whole slit spectra (a 10 meter telescope does have its limitations!)
and have lower signal/noise. In addition, we are prone to the
effects of noise spikes (radiation events) in the extracted spectra
since the cleaning algorithm for such spikes is not effective when
only sampling 3 pixels. We had to edit out noise spikes by hand and
some of the smaller spikes remain (unfortunately not all the noise
spikes are single pixels since radiation events can strike the chip
obliquely).
Fig.~\ref{spectra-3pix} shows pieces
of the spectra for each of the species listed above. In this
figure, we show the first spectrum after the impact (file 66),
another spectrum towards the end of 4 July (file 75) and a
spectrum at the start of 5 July (file 129).
From inspection of Fig.~\ref{spectra-3pix} some trends emerge.
First, the comet was not very gaseous to begin with.
If we difference the last pre-impact spectrum (number 65) from
the first post-impact spectrum (number 66), we find there are no
additional emissions seen over the ambient cometary spectrum. Thus,
in the first 10 minutes, no increased emission was detected and the left-hand
panel of Fig.~\ref{spectra-3pix} for each molecule shows, in essence, the
ambient cometary spectrum.
All of the species show an increase in line strength in response
to the impact. However, the relaxation timescale back to the
ambient spectrum was different for different species.
The CN increases dramatically in strength from the impact
to the spectrum 2 hours later, but by 5 July it is back to its pre-impact
level. Note, however, that the rotational structure of the emissions
does not change due to the impact. This is true for all species and
argues that there is no prompt emission component due to the impact
and that all species continue to be excited only by resonance fluorescence.
The OH spectra shown here are extremely noisy. This is especially true
of the spectrum at 08:16 on 4 July because of the large airmass. While we
corrected for the extinction, these emissions would be the ones {\it most}
affected by any errors in the extinction coefficients applied. Also, if the
extinction is severe enough that most photons are blocked before they reach
the detector, no correction will repair the signal. We will discuss OH
more when we discuss full slit extractions below.
C$_{3}$ shows an increase in brightness on 5 July from the
first spectrum on 4 July but this perceived increase may be
due to the noise in spectrum 66 (recall that this spectrum was obtained
in the first 10 minutes after the impact).
Inspection of Fig.~\ref{spectra-3pix} shows
that the C$_{2}$ was essentially unchanged (except for the noise level) from the end
of 4 July to the beginning of 5 July (note that the strong line
at around 5155\AA\ is an NH$_{2}$ line, not C$_{2}$). More can be said
about the C$_2$ if we can increase the signal/noise by summing along
the whole slit. This is discussed below.
NH is the only molecule
which is essentially as bright on 5 July as it was near the end
of 4 July. The NH$_2$, however, shows a decay back to the ambient
level by 5 July. If, as we believe, NH$_3$ is the parent of NH$_2$
and the grand-parent of NH, the earlier decay of the NH$_2$ would
be expected.
CH does not appear to be present in either the first or last spectra.
It is, however, present in the spectrum at 08:16 on 4 July. This molecule's
spectrum
is helped tremendously by extraction along the complete length of the slit.
In order to boost the signal/noise, we next re-extracted the spectra,
collapsing
the spectra from the full length of the slit. These spectra are shown
in Fig.~\ref{spectra}. Spectra are shown at the same three times
as in Fig.~\ref{spectra-3pix}.
In these full-slit extracted spectra, we see the same trend for CN
as we did in the inner three pixels. Recall that spectrum 66, the
first shown here, is essentially an ambient cometary spectrum.
The OH appears to continue to increase in line strength for the 5 July
spectrum. This must be viewed with caution because OH is the
molecule which is most affected by the increasing airmass of the
observations on 4 July. The observation at 08:08~UT was while
the comet was at $\sim1.8$ airmasses and the extinction is
around 1.4 magnitudes/airmass at this wavelength, even at the high
elevation of Mauna Kea. Small errors in the extinction coefficient
used would have a large effect. However, it is apparent that the
OH did not decrease much, if at all, by 5 July.
Inspection of Fig.~\ref{spectra} shows that C$_2$ was essentially unchanged
from the end of 4 July to the beginning of 5 July, but is stronger than
the ambient spectrum. Recall that
C$_{2}$ has no dipole moment and thus has no easy relaxation pathway.
This is also why we normally see such high J-levels for C$_{2}$.
All of the apparent ``noise" to the blue of the bandhead is C$_{2}$ lines
which are so weak in the first spectrum that they are not easily
discerned.
The behavior of C$_{3}$ is intermediate to that of the CN and C$_{2}$.
It is clearly still stronger on 5 July than the first spectrum, but it
is only half as strong as it was at 08:08~UT.
The NH$_{2}$ is just as strong on 5 July as it
was at the end of 4 July when integrated along the whole slit,
while the NH has decreased in strength.
With the very blue wavelength of the NH emissions, it would
be affected more by extinction than would the NH$_{2}$, which
would argue that it might be even slightly brighter for the
late spectrum on 4 July. Recall that when we examined the spectra
of these two species in just the inner three pixels spatially,
the NH$_{2}$ actually dimmed sooner than NH, as would be
expected if NH$_{3}$ is the dominant parent of NH. Thus, the reversed
trend we see here must be a function of the dissociation of the species.
CH behaved like the CN, having almost returned to its pre-impact
value by the next night. It seems to be only slightly stronger
at the start of 5 July than the first observation on 4 July.
In order to follow the trend of the various species with time in more detail,
we integrated select emission lines for each molecule. In general,
we fit a continuum under the emission lines and added up the
flux above the continuum in each line individually and summed
a series of lines. For C$_{2}$, it is impossible to
use individual lines so we summed the bandhead. For C$_{3}$,
we used the Q- and part of the R-branch. The integrated intensities
are shown in Fig.~\ref{gasplot}. It is again obvious that not
all species behaved alike. As with Fig.~\ref{spectra}, the values
shown here are for the spectra integrated along the full length of
the slit. All of the species are normalized to the value for that
species in spectrum 66, the first post-impact spectrum. However, remember
that we saw no emission increase in this spectrum so the value of 1 in this
plot represents the ambient cometary emission.
We measured 9 R-branch lines and 10 P-branch lines of CN [the
``lines" are actually unresolved blends of two closely spaced lines e.g.
R$_1$(11) + R$_2$(10)]. After the impact, the CN lines
grew stronger for the first 4500 seconds and then the intensity
of the lines began to decrease. At their maximum, the integrated
line intensity was 3 times greater than at the time of impact.
Recall that the slit was $0.86 \times 7$ arcsec, which corresponds
to $562 \times 4570$ km at the comet's distance. If all of the
CN is produced instantaneously and no more is produced after the
impact, then 4500 sec represents the time
for the material to start flowing out of the aperture. Since the
impact was a distinct impulse which caused a release of
material from the impact site, what we measured was the increasing
cross section as the material expanded away from the
nucleus. Eventually, the area which was fluorescing grew larger
than the footprint of our slit on the cometary coma and we
were no longer able to measure all of the material, though our
aperture does still include the material coming directly towards
us. That point came at approximately 4500 sec after the impact, causing
the diminution of light we observe.
If the CN was only produced instantaneously, then the diminution
at 4500 sec indicates that the material was flowing outwards
at 0.51~km~sec$^{-1}$. This is twice as large as the outflow velocity of
0.2~km~sec$^{-1}$ and of 0.23~km~sec$^{-1}$ found for dust by Meech {\it et al}
and by Schleicher {\it et al.} (2006), respectively.
Keller {\it et al.} (2005) found a dust velocity of $>0.16$~km~sec$^{-1}$, with
the fastest particles moving at $>0.4$~km~sec$^{-1}$. There are no reported
gas outflow velocities. Keller {\it et al}.
assumed a CN outflow velocity of 0.7~km~sec$^{-1}$ at 1au heliocentric
distance or 0.6~km~sec$^{-1}$ at the comet's distance.
\nocite{kedeepimpact05}
At the time of impact, the first materials released are the parent ices
which must first sublime (unless vaporized by the impact) to the vapor
state before they dissociate to the daughter species.
Initially, the CN represents the
dissociation product of only a very small amount of the parent, implying
that a large quantity of the parent was excavated.
As more CN is dissociated, the intensity of the CN increases.
But this material is still flowing out
of the aperture. The combination of the initial component and
the delayed component means that the velocity we detect is a lower
limit to the outflow velocity.
The OH, NH and CH all appear to increase until around 4500~sec and
then decrease. CH and NH were back to their pre-impact levels
by 5 July. In Fig.~\ref{gasplot}, the OH appears to reach
its pre-impact levels by the end of our observations of 4 July.
However, this is likely not real. As the airmass increased, the
extinction at OH increased dramatically. In the UV, extinction
increases by Rayleigh scattering and follows a $\lambda^{-4}$ law.
While we applied an extinction correction to all the data, at the
larger airmasses essentially no cometary light can penetrate
the atmosphere so the correction fails. In addition, atmospheric
dispersion begins to cause the image of the comet at the wavelength
of OH to be pushed out of the slit so that we are no longer centered
on the optocenter at OH. The observations of 5 July show
that the OH emission was still about 1.5 times its pre-impact level.
The behavior of the C$_{3}$, C$_{2}$ and NH$_{2}$ emissions is different
than the other four molecules already discussed. These molecules
show a monotonic increase in their emissions for the whole time
of our observations on 4 July and were only about 20\% below
their maximum levels of 4 July when we resumed observations on
5 July. This indicates that they are being produced faster
than they can flow out of the aperture and are not being dissociated
further in the timescale of observations. As discussed above,
the C$_{2}$ result is expected since relaxation is a slow process for C$_{2}$.
This is similarly true for C$_{3}$. In addition, the very high line
density of the C$_{3}$ bands makes C$_{3}$ easier to see in noisy spectra
because more lines are included per resolution element.
C$_{3}$ and C$_{2}$ are also grand-daughter species, so should take longer
to form in the first place than daughter species.
However, the monotonic increase is unexpected for NH$_{2}$, generally
thought to be a daughter species.
Unfortunately, integrating
the lines in the spectra which are extracted over only the inner three
pixels produced curves such as those seen in Fig.~\ref{gasplot}
which were too noisy to use to study meaningfully any trends.
Within the noise, it looks like all of the molecules followed
more or less the same trends with time after the initial few minutes.
We have carefully inspected the spectra at all times from these
extractions of the inner three pixels. Table~\ref{3pixtrends}
summarizes what we see in those spectra, including the times
of peak brightness for each molecule.
For CN, we note that there
may have been a change in the relative brightness of a few of the lines
relative to the others. This may be a signature of the Greenstein
effect. Further comment on this will have to wait until we can
compute a fluorescence model of the molecule.
\section{Summary}
The spatial extent of the Keck HIRES slit allowed us to monitor the
material as it flowed outwards superposed on the ambient cometary spectrum.
All species will eventually dissociate as they flow outwards. However, the
Keck slit is very short and narrow compared with the scale lengths of the
observed species. Thus, the spatial profiles are essentially flat along the
slit. The off-optocenter spectra of 5 and 6 July may allow us to probe
parent-daughter relationships. This is beyond the scope of this paper.
Prior to the Deep Impact Mission, our model of comets such as Tempel~1
was that they had a mostly pristine inner nucleus with a mantle
of altered material on the outside. The depth of this outer mantle
was believed to be a few to tens of meters thick. By crashing
a spacecraft into the nucleus, the intention was to probe the inner
regions of the cometary nucleus to release more primitive materials.
Our observations of the impact using the HIRES spectrograph on the
Keck 1 telescope were intended to monitor the spectral changes.
The impact certainly caused an increase in brightness from the
release of material from the nucleus. As the ices sublimed and
dissociated, we would have been sensitive to any changes in the
relative strengths of individual lines which would have resulted
if there had been any prompt emission or energy driven disequilibrium.
We saw none of these (with the possible exception of the CN lines
mentioned above). All of the molecules brightened and then
dimmed with approximately the same trend. Thus, we did not even
see a change in relative chemistry as a result of the impact.
Feldman {\it et al.} (2006) observed the impact using the Advanced
Camera for Surveys on HST and reported that CO/H$_{2}$O was
unchanged by the impact from the ambient, consistent with our
observations of the molecules in our bandpass.
These results are in contrast with the spacecraft, which saw a change
in the release of organics, HCN and CO$_2$ relative
to H$_{2}$O (A'Hearn {\it et al.} 2005) and Keck 2, which
saw an enrichment in ethane from the impact (Mumma {\it et al.} 2005).
We would not have been sensitive to these species.
\nocite{mudeepimpact2005,fedeepimpact2006}
We have summed all of the spectra in the first few hours after
the impact in order to increase the signal/noise to look for any
new species which were not apparent in the pre-impact spectra
of 30 May. None are readily apparent. We plan to investigate
this further by extracting spectra over different cometocentric
distances within our aperture in a future publication.
The impactor succeeded in knocking a large crater in the nucleus,
ejecting $1.5\times10^{32}$ water molecules or $4.4\times10^6$ kg of
H$_{2}$O (Keller {\it et al.} 2005) and
10$^6$~kg of dust (Sugita {\it et al.} 2005). This should have
exposed fresh material and yet, we observed no chemical changes.
This can only lead to one of two conclusions: 1) The crater was not
deep enough to penetrate the mantle to primitive material, i.e.
the mantle is thicker than we had supposed; or 2) The cometary
material on the outside of the nucleus is not altered significantly
from the interior materials. Groussin {\it et al.} (2006)
showed that the nucleus has very low thermal inertia.
Thus, neither the diurnal heat wave or the heat wave from
the extended passage into the inner solar system would
penetrate deeply into the nucleus. This would leave pristine
material near the nucleus. Thus, as we saw with our Keck
data, the interior of the comet did not look substantially
different than the exterior layers and the outer layers
must be very thin.
\nocite{grlpsc06}
\vspace*{0.5in}
\begin{center}Acknowledgements\end{center}
We thank Dr. Fred Chaffee for making Director's Discretionary time available
pre-impact. Our thanks to Dr. Hien Tran for obtaining the observations
of 30 May and 4 July and for help on 5 and 6 July.
This work was funded by NASA Grant NNG04G162G (ALC), NASA Grant
NNG06A67G (WMJ), NSF Grant CHE-0503765 (WMJ) and by support
from the University of Maryland through a subcontract 2667702 (KJM), which
was awarded under prime contract NASW-00004 from NASA .
\newpage
|
1,941,325,220,511 | arxiv | \section{Introduction}
Recent observational evidence of intergalactic magnetic fields from $\gamma$-ray observations of blazars and constraints imposed by CMB power spectrum suggest the existence of magnetic fields created in the early Universe \cite{1,2,3,4,5}. Interesting theoretical models have been proposed to explain generation processes which gave rise to this likely primordial field. Some of them are originated by causal
requirements via cosmological phase transitions or by non-linear evolution of primordial density perturbations\cite{4a,4b,4c}; and the other ones could be generated during inflation, with them being the most appealing models due to the production
of large-scale magnetic fields beyond the horizon scale \cite{6,7,8,9,9a,10,11,12,13,14,15,16,17,17a,17b,17c,17d,18,18l,18l1,18l2}. Moreover, several additional attempts from string theory or extra dimensions have been done to generate the seed magnetic fields needed to be coherent on cosmological scales \cite{19n1,19n2,19n3,19n4}.
One way to find out the process which gave rise to this field and determine its main features is by making theoretical predictions about the signatures in the CMB
from primordial magnetic fields (PMFs) \cite{19,20,20a,21,21a}. Indeed, some authors have shown that vector modes dominate all the temperature and polarization anisotropies for higher multipolar numbers while the scalar mode contribution is larger for lower $l$ \cite{21bn,21cn,23}.
Other effect of PMFs on the CMB comes from the non-Gaussian (NG) signals because its contribution to energy-momentum tensor is quadratic in the fields. The relevant NG signal from PMFs with an amplitude similar to the curvature turns out to be a feature that is important for constraining mean-field amplitude depending on what signal is induced by passive or compensated modes.
Studies of NG signals via bispectrum on CMB have found upper limits of PMFs around $2-22$ nG derived from scalar magnetic modes, and $3.2-10$ nG derived from vector-tensor magnetic modes smoothed on a scale of $1$ Mpc \cite{24,25,26,27,28,29,30,31,32}. The Planck Collaboration also reported limits on the amplitude of $B_{1Mpc}< 3$nG for $n_B=-2.9$; $B_{1Mpc}< 0.07$nG for $n_B=-2$; and $B_{1Mpc}<0.04$nG for $n_B=2$ from compensated modes; and $B_{1Mpc}<4.5$nG for $n_B=-2.9$ from the passive-scalar mode \cite{33}. PMFs have also been constrained by the POLARBEAR experiment, where they reported that the PMF amplitude from the two-point correlation functions is less than 3.9 nG at the 95\% confidence level \cite{33n}.
On the other hand, distinct signatures on the parity-odd CMB cross correlations would carry valuable clues about a primordial magnetic helicity.
In fact, helical contribution in the field (in the perfect conductivity limit) has been widely studied because it produces efficient transference of power from smaller to larger scales, and thus be able to explain the actual observed magnetic fields\cite{4a,33n1}. Further, observational evidence of helical primordial magnetic fields would offer
a window for probing physics beyond the standard models of particle physics, particularly processes of parity violation in the early Universe\cite{33n2,33n3,33n4}.\\
The study of helical fields via NG will give a deeper understanding of the magnetic field generation model and help us to strengthen the constraints of PMF amplitude.
Thus, the main goal of this paper is to investigate the effects of helical PMFs on the CMB bispectrum. Following previous formulation for calculating the bispectrum, we have found signals that arise by considering a minimal cut-off and we observed that local-type shape contains the biggest contribution to the bispectrum. This paper is organized as follows. Sec. \ref{sec:intro} describes the statistical properties of PMFs. In Sec. \ref{bissection} we define the magnetic bispectrum and in Sec.\ref{redbisa4} through the numerical computation, we solve exactly the bispectrum for a collinear configuration and observe some important signals. Sec. \ref{redbisa5} is defined the reduced bispectrum and in Sec. \ref{redbisa} we present numerical results of the primary bispectrum sourced by helical PMFs. Sec. \ref{discu} is devoted to further discussion and conclusions.
\section{ Statistical Aspects of Primordial Magnetic Fields.}
\label{sec:intro}
We consider a stochastic primordial magnetic field (PMF) generated in the very early Universe which could have been produced during inflation (non-causal field) or after inflation (causal field).
This field acts like a source of fluctuations on the CMB anisotropies under a FLRW background Universe described by the metric
\begin{equation}
ds^2=a^2(t)(-dt^2+\delta_{ij}dx^idx^j)
\end{equation}
with $t$ being the conformal time. The PMF power spectrum which is defined as the Fourier transform of the two point correlation can be written as
\begin{equation}\label{PMFespectro1}
\langle B_i(\mathbf{k})B_j^*(\mathbf{k}^{\prime})\rangle =(2\pi)^3\delta^3(\mathbf{k}-\mathbf{k}^{\prime})\bigg( P_{ij}(k)P_B(k)+i \epsilon_{ijl}\hat{k}^l P_H(k) \bigg),
\end{equation}
where $P_{ij}(k)=\delta_{ij}-\hat{k}_i\hat{k}_j$ is a projector onto the transverse plane\footnote{This projector has the property $P_{ij}\hat{k}^i=0$ with $\hat{\mathbf{k}}^i=\frac{\mathbf{k}^i}{k}$ and we
use the Fourier transform notation $ B_i(\mathbf{k})=\int d^3 x\exp^{i\mathbf{k}\cdot x} B_i(\mathbf{x})$. }, $\epsilon_{ijk}$ is the 3D Levi-Civita tensor and, $P_B(k)$, $P_H(k)$ are the
symmetric/anti-symmetric parts of the power spectrum which represent the magnetic field energy density and absolute value of the kinetic helicity respectively \cite{23}
\begin{eqnarray}\label{PMFespectro}
\langle B_i(\mathbf{k})B_i^*(\mathbf{k}^{\prime})\rangle& =&2(2\pi)^3\delta^3(\mathbf{k}-\mathbf{k}^{\prime})P_B(k),\\
-i\langle \epsilon_{ijl}\hat{k}^lB_i(\mathbf{k})B_j^*(\mathbf{k}^{\prime})\rangle& =&2(2\pi)^3\delta^3(\mathbf{k}-\mathbf{k}^{\prime})P_H(k).
\end{eqnarray}
We assume that power spectrum scales as a simple power law \cite{21}
\begin{equation}\label{powerPMF1}
P_B(k)=A_Bk^{n_B}, \quad \mbox{ with} \quad A_B=\frac{B^2_{\lambda}2\pi^2\lambda^{n_B+3}}{\Gamma(\frac{n_B+3}{2})},
\end{equation}
\begin{equation}\label{powerPMF2}
P_H(k)=A_H k^{n_H}, \quad \mbox{ with} \quad A_H=\frac{H^2_{\lambda}2\pi^2\lambda^{n_H+3}}{\Gamma(\frac{n_H+4}{2})},
\end{equation}
with $\Gamma$ being the Gamma function. In order to avoid infrared divergences (when we do not consider an infrared cutoff), $n_B > -3$, $n_H > -4$. Also, $B_\lambda$, $H_\lambda$ are the comoving PMF strength and magnetic helicity smoothing over a Gaussian sphere of comoving radius $\lambda$ \cite{34,35}.
The more general case of the power spectrum for magnetic fields can be studied if we assume that it is defined for $k_{m} \leq k \leq k_{D}$, being $k_{D}$ an ultraviolet cut-off corresponding to damping scale where the field is suppressed on small scales\cite{3} as $k_D \sim\mathcal{O}(10)\text{Mpc}^{-1}$ and we also consider a possible dependence on an infrared cut-off, $k_m$.
Given the Schwarz inequality\cite{34},
\begin{equation}
\lim\limits_{k^\prime \rightarrow k}\langle \mathbf{B}(k)\cdot \mathbf{B}^*(k^\prime) \rangle \ge |\lim\limits_{k^\prime \rightarrow k}\langle (\hat{\mathbf{k}} \times \mathbf{B}(k))\cdot \mathbf{B}^*(k^\prime) \rangle |,
\end{equation}
an additional constraint is found for these fields
\begin{equation}\label{cond1a}
|A_H| \le A_Bk^{n_B-n_H}.
\end{equation}
In the case where $A_H=A_B$ and $n_B=n_H$ we define the maximal helicity condition. We will also use the procedure in \cite{21} to parametrize the infrared cut-off by a single constant parameter $\alpha$,
\begin{equation}\label{mcut-off}
k_m=\alpha k_D,\quad 0\leq\alpha<1
\end{equation}
which in the case of inflationary scenarios would correspond to the wave mode that exits the horizon at inflation epoch and for causal modes would be important when this scale is larger than the wavenumber of interest (as claimed by Kim et.al. \cite{21d}). Thus, this infrared cut-off would be important in order to constrain PMF parameters and magnetogenesis models\cite{21,21b,21bb,21c,21d}.
Equation (\ref{mcut-off}) gives only an useful mathematical representation to constrain these cut-off values via cosmological datasets (for this case, the parameter space would be given by $(\alpha, k_D, B_\lambda, H_\lambda, n_H, n_B)$), and therefore we want to point out that latter expression does not state any physical relation between both wave numbers. In \cite{21b,21bb}, they showed constraints on the maximum wave number $k_D$ as a function of $n_B$ via big bang nucleosynthesis (BBN), and they considered the maximum and minimum wave numbers as independent parameters. In fact, we have found out that the integration scheme used for calculating the spectrum and bispectrum of PMFs is exactly the same if we parametrize $k_m$ as seen in (\ref{mcut-off}), or if we consider $(k_m, k_D, B_\lambda, H_\lambda, n_H, n_B)$ as independent parameters. Thus the inclusion of $k_m$ is done only for studying at a phenomenological level its effects on the CMB bispectrum.
On the other hand, the equations for the adimensional energy density of magnetic field and spatial part of the electromagnetic energy momentum tensor respectively written in Fourier space are given as
\begin{equation}
\rho_B(\mathbf{k1})=\frac{1}{8\pi \rho_{\gamma,0}}\int \frac{d^3p}{(2\pi)^3}B_l(\mathbf{p})B^l(\mathbf{k1}-\mathbf{p}),\nonumber
\end{equation}
\begin{equation}
\Pi_{ij}(\mathbf{k1})=\frac{1}{4\pi \rho_{\gamma,0}} \int \frac{d^3p}{(2\pi)^3}\left[\frac{\delta_{ij}}{2}B_l(\mathbf{p})B^l(\mathbf{k1}-\mathbf{p})-B_i(\mathbf{p})B_j(\mathbf{k1}-\mathbf{p}) \right],\label{convdensity}
\end{equation}
where in the last expressions we are considering high conductivity so, the electric field is suppressed; the magnetic field evolves as $B^2\sim a^{-4}(t)$, and therefore we can express each component
of the energy momentum tensor in terms of photon energy density $\rho_{\gamma}=\rho_{\gamma,0}a^{-4}$, with $\rho_{\gamma,0}$ being its present value.\footnote{The adimensional energy density of magnetic field showed here is written with different notation in \cite{31}: $\Omega_B\equiv\frac{B^2}{8 \pi a^4\rho_\gamma}$ and in \cite{30,25}: $\Delta_B\equiv\frac{B^2}{8 \pi a^4\rho_\gamma}$.}
Given that spatial electromagnetic energy momentum tensor is symmetric, we can decompose this tensor into the two scalar ($\rho_B$, $\Pi^{(S)}$), one vector ($\Pi^{(V)}_i$) and one tensor ($\Pi^{(T)}_{ij}$) components as
\begin{equation}
\Pi_{ij}=\frac{1}{3}\delta_{ij}\rho_B+(\hat{k}_i\hat{k}_j-\frac{1}{3}\delta_{ij})\Pi^{(S)}+(\hat{k}_i\Pi_j^{(V)}+\hat{k}_j\Pi_i^{(V)})+\Pi_{ij}^{(T)}\label{tensor1}
\end{equation}
which obey to $\hat{k}^i\Pi_i^{(V)}=\hat{k}^i\Pi_{ij}^{(T)}=\Pi_{ii}^{(T)}=0$ \cite{21,36,37}. The components of this tensor are recovered by applying projector operators defined as
\begin{eqnarray}\label{projecto}
\rho_B &=&\delta^{ij}\Pi_{ij}\nonumber \\
\Pi^{(S)}&=&(\delta^{ij}-\frac{3}{2}P^{ij})\Pi_{ij}= \mathcal{P}^{ij} \Pi_{ij} \nonumber \\
\Pi^{(V)}_i&=& \hat{k}^{(j} P^{l)}_i \Pi_{l\,j}=\mathcal{Q}_{i}^{j\,l} \Pi_{l\,j} \nonumber \\
\Pi^{(T)}_{ij}&=& (P^{(a}_iP^{b)}_j-\frac{1}{2}P^{ab}P_{ij})\Pi_{ab}=\mathcal{P}^{ab}_{ij}\Pi_{ab},
\end{eqnarray}
where $(..)$ in the indices denotes symmetrization\cite{24}.
\section{The Magnetic Bispectrum}\label{bissection}
Since the magnetic field is assumed as a Gaussianly-distributed stochastic helical field and the electromagnetic energy momentum tensor is quadratic in the fields, the statistics must be non-Gaussian and the bispectrum is non-zero as was claimed by \cite{24}. Using eq. (\ref{convdensity}) we have that three-point correlation function is expressed as
\begin{eqnarray}
\langle \Pi_{ij}(\mathbf{k1}) \Pi_{tl}(\mathbf{k2}) \Pi_{mn}(\mathbf{k3})\rangle&=&\frac{-1}{(4\pi \rho_{\gamma,0})^3 }\int \frac{d^3p}{(2\pi)^3}\int \frac{d^3q}{(2\pi)^3} \int \frac{d^3s}{(2\pi)^3} \times \nonumber \\
&&\langle B_i(\mathbf{p})
B_j(\mathbf{k1}-\mathbf{p})B_l(\mathbf{k2}-\mathbf{q})B_n(\mathbf{k3}-\mathbf{s})B_t(\mathbf{q})B_m(\mathbf{s})\rangle \nonumber \\
&-&\frac{\delta_{ij}}{2}\langle ... \rangle-\frac{\delta_{tl}}{2}\langle ... \rangle-\frac{\delta_{mn}}{2}\langle ... \rangle+\frac{\delta_{ij}\delta_{mn}}{4}\langle ... \rangle+\frac{\delta_{ij}\delta_{tl}}{4}\langle ... \rangle \nonumber\\
&+&\frac{\delta_{tl}\delta_{mn}}{4}\langle ... \rangle-\frac{\delta_{ij}\delta_{mn}\delta_{tl}}{8}\langle ... \rangle. \label{TEMbis}
\end{eqnarray}
Where $\langle ... \rangle$ describes an ensemble average over six stochastic fields. We can use Wicks theorem to decompose the six point correlation function into products of the magnetic field power spectrum expressed in eq. (\ref{PMFespectro}).
Eight of fifteen terms contribute to the bispectrum and they are proportional to $\delta(\mathbf{k1}+\mathbf{k2}+\mathbf{k3})$ due to the homogeneity condition.
In \cite{37} they point out that expression (\ref{TEMbis}) can be reduce to just one contribution if the projection tensor used for extracting each one of the contribution is symmetric in ($ij$), ($tl$) and ($mn$). Therefore one can write the bispectrum as follows
\begin{eqnarray}
\langle \Pi_{ij}(\mathbf{k1}) \Pi_{tl}(\mathbf{k2}) \Pi_{mn}(\mathbf{k3})\rangle&=& \delta(\mathbf{k1}+\mathbf{k2}+\mathbf{k3}) \times \nonumber \\
&&\frac{8}{(4\pi \rho_{\gamma,0})^3 }\int \frac{d^3p}{(2\pi)^3} F_{it}(\mathbf{p})F_{jm}(\mathbf{k1}-\mathbf{p})F_{ln}(\mathbf{k2}+\mathbf{p}), \label{TEMbis1}
\end{eqnarray}
being $F_{ij}(\mathbf{k})=P_{ij}(\mathbf{k})P_{B}(\mathbf{k})+i\epsilon_{ijm}\hat{\mathbf{k}}^m P_H(\mathbf{k})$. Wavevectors that appear in eq.(\ref{TEMbis1}) generate a tetrahedron configuration (see figure \ref{ref}) such that fifteen angles must be included for calculating the bispectrum. So, in order to make comparison with previos works, we use non only the geometry configuration for bispectrum but as well the notation of these angles defined in \cite{37} given as
\begin{eqnarray}\label{angles}
&&\beta=\hat{\mathbf{p}}\cdot\widehat{\mathbf{k1}-\mathbf{p}},\quad \gamma=\hat{\mathbf{p}}\cdot\widehat{\mathbf{k2}+\mathbf{p}}, \quad \mu=\widehat{\mathbf{k1}-\mathbf{p}}\cdot\widehat{\mathbf{k2}+\mathbf{p}}, \quad \theta_{kp}=\hat{\mathbf{k1}}\cdot\hat{\mathbf{k2}} \nonumber \\
&&\theta_{kq}=\hat{\mathbf{k1}}\cdot\hat{\mathbf{k3}},\quad \theta_{pq}=\hat{\mathbf{k2}}\cdot\hat{\mathbf{k3}}, \quad \alpha_{k}=\hat{\mathbf{k1}}\cdot \hat{\mathbf{p}},\quad \alpha_{p}=\hat{\mathbf{k2}}\cdot \hat{\mathbf{p}},\quad \alpha_{q}=\hat{\mathbf{k3}}\cdot \hat{\mathbf{p}}\nonumber \\
&& \beta_{k}=\hat{\mathbf{k1}}\cdot \widehat{\mathbf{k1}-\mathbf{p}}, \beta_{p}=\hat{\mathbf{k2}}\cdot \widehat{\mathbf{k1}-\mathbf{p}}, \quad,\beta_{q}=\hat{\mathbf{k3}}\cdot \widehat{\mathbf{k1}-\mathbf{p}}, \quad \gamma_{k}=\hat{\mathbf{k1}}\cdot \widehat{\mathbf{k2}+\mathbf{p}}\nonumber\\
&&\gamma_{p}=\hat{\mathbf{k2}}\cdot \widehat{\mathbf{k2}+\mathbf{p}}, \quad \gamma_{q}=\hat{\mathbf{k3}}\cdot \widehat{\mathbf{k2}+\mathbf{p}}.
\end{eqnarray}
\begin{figure}
\centering
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{plotsn/Ref1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{plotsn/Ref2}
\end{subfigure}
\caption{ Geometrical configuration for bispectrum. The wavevectors $\mathbf{k1}$, $\mathbf{k2}$ and $\mathbf{k3}$ are free, while $\mathbf{p}$ is the integration mode.}
\label{ref}
\end{figure}
As we will see, the bispectrum has two main contributions, the first contribution contains terms proportional to $A_B^3$ or $A_BA^2_H$ and this is called the even contribution denoted here with $B^{(S)}$. The second contribution is proportional to terms like $A_H^3$ or $A_B^2A_H$ and it is called the odd contribution denoted as $\mathcal{B}^{(A)}$. Hence, we can define the three-point correlation for the scalar modes as
\begin{equation}
\langle Z_1(\mathbf{k1}) Z_2(\mathbf{k2}) Z_3(\mathbf{k3})\rangle \equiv \delta\left(\sum_{i=1}^3\mathbf{ki}\right)\left(B_{Z_1Z_2Z_3}^{(S)}-i\epsilon_{ljk}\mathcal{B}_{Z_1Z_2Z_3}^{(A\,)\,ljk}\right),\label{rrr}
\end{equation}
where $Z_{\{1,2,3\}}=\{\rho_B,\Pi^{(S)}_B\}$. We begin calculating the bispectrum of the magnetic energy density. To do so, we shall apply the projector defined in eq.(\ref{projecto}) three times $\delta_{ij}\delta_{tl}\delta_{mn}$ on eq.( \ref{TEMbis}) to obtain the following
\begin{equation}
\langle \Pi_{ij}(\mathbf{k1}) \Pi_{tl}(\mathbf{k2}) \Pi_{mn}(\mathbf{k3})\rangle\delta_{ij}\delta_{tl}\delta_{mn}=\langle \rho_B(\mathbf{k1}) \rho_B(\mathbf{k2}) \rho_B(\mathbf{k3})\rangle,
\end{equation}
using eq.(\ref{rrr}), a straightforward calculation gives the following expression
\begin{eqnarray}\label{passiveee1}
B_{\rho_B\rho_B\rho_B}^{(S)}&=&\frac{8}{(2\pi)^3(4\pi \rho_{\gamma,0})^3 }\int d^3p \left(P_B(p)P_B(\left| \mathbf{k1}-\mathbf{p}\right|)P_B(\left|\mathbf{p}+\mathbf{k2}\right|)F_{\rho \rho \rho}^{1} \nonumber \right.\\
&+&P_H(\left| \mathbf{p}+\mathbf{k2}\right|)(-P_B(p)P_H(\left|\mathbf{k1}-\mathbf{p}\right|)F_{\rho \rho \rho}^{2}+P_B(\left|\mathbf{k1}-\mathbf{p}\right|)P_H(p)F_{\rho \rho \rho}^{3} )\nonumber \\
&-& \left. P_B(\left|\mathbf{p}+\mathbf{k2}\right|) P_H(p) P_H(\left| \mathbf{k1}-\mathbf{p}\right|)F_{\rho \rho \rho}^{4} \right),
\end{eqnarray}
for the even contribution. The values of $F_{\rho\rho\rho}$ are shown along with the odd contribution in appendix \ref{A1}. In orden to find the three-point cross-correlation between scalar anisotropic stress and magnetic energy density, we will apply the three projections $\delta_{ij}\delta_{tl}\mathcal{P}_{mn}$ defined in eq.(\ref{projecto}) on eq.(\ref{TEMbis}) which gives us
\begin{equation}
\langle \Pi_{ij}(\mathbf{k1}) \Pi_{tl}(\mathbf{k2}) \Pi_{mn}(\mathbf{k3})\rangle\delta_{ij}\delta_{tl}\mathcal{P}_{mn}=\langle \rho_B(\mathbf{k1}) \rho_B(\mathbf{k2}) \Pi_B^{(S)}(\mathbf{k3})\rangle,
\end{equation}
and using the three-point correlation eq.(\ref{rrr}),
the even contribution yields
\begin{eqnarray}
B_{\rho_B\rho_B\Pi_B^{(S)}}^{(S)}&=&\frac{8}{2(2\pi)^3(4\pi \rho_{\gamma,0})^3 }\int d^3p \left(-P_B(p)P_B(\left| \mathbf{k1}-\mathbf{p}\right|)P_B(\left|\mathbf{p}+\mathbf{k2}\right|)F_{\rho \rho \Pi^{(S)}}^{1} \right. \nonumber \\
&-&P_B(p)P_H(\left|\mathbf{k1}-\mathbf{p}\right|)P_H(\left|\mathbf{p}+\mathbf{k2}\right|)F_{\rho \rho \Pi^{(S)}}^{2}\nonumber \\
&+&P_H(p)\left(-P_H(\left| \mathbf{p}+\mathbf{k2}\right|)P_B(\left|\mathbf{k1}-\mathbf{p}\right|)F_{\rho \rho \Pi^{(S)}}^{3} \right. \nonumber \\
&+&\left.\left. P_B(\left| \mathbf{p}+\mathbf{k2}\right|)P_H(\left|\mathbf{k1}-\mathbf{p}\right|)F_{\rho \rho \Pi^{(S)}}^{4} \right) \right).
\end{eqnarray}
Other cross-bispectra is obtained by applying $\delta_{ij}\mathcal{P}_{tl}\mathcal{P}_{mn}$ on eq.(\ref{TEMbis}) and this gives
\begin{equation}
\langle \Pi_{ij}(\mathbf{k1}) \Pi_{tl}(\mathbf{k2}) \Pi_{mn}(\mathbf{k3})\rangle\delta_{ij}\mathcal{P}_{tl}\mathcal{P}_{mn}=\langle \rho_B(\mathbf{k1}) \Pi_B^{(S)}(\mathbf{k2}) \Pi_B^{(S)}(\mathbf{k3})\rangle,
\end{equation}
as result we found the following expression
\begin{eqnarray}
B_{\rho_B\Pi_B^{(S)}\Pi_B^{(S)}}^{(S)}&=&\frac{8}{4(2\pi)^3(4\pi \rho_{\gamma,0})^3}\int d^3p \left(P_B(p)P_B(\left| \mathbf{k1}-\mathbf{p}\right|)P_B(\left|\mathbf{p}+\mathbf{k2}\right|)F_{\rho \Pi^{(S)}\Pi^{(S)}}^{1} \right. \nonumber \\
&+&P_B(p)P_H(\left| \mathbf{p}+\mathbf{k2}\right|)P_H(\left|\mathbf{k1}-\mathbf{p}\right|)F_{\rho \Pi^{(S)}\Pi^{(S)}}^{2} \nonumber \\
&-&P_H(p)P_H(\left| \mathbf{p}+\mathbf{k2}\right|)P_B(\left|\mathbf{k1}-\mathbf{p}\right|)F_{\rho \Pi^{(S)}\Pi^{(S)}}^{3}\nonumber \\
&-&\left. P_H(p)P_H(\left|\mathbf{k1}-\mathbf{p}\right|)P_B(\left|\mathbf{p}+\mathbf{k2}\right|)F_{\rho \Pi^{(S)}\Pi^{(S)}}^{4} \right).
\end{eqnarray}
Finally the three-point correlation of scalar anisotropic stress is obtained by applying $\mathcal{P}_{ij}\mathcal{P}_{tl}\mathcal{P}_{mn}$ on eq.(\ref{TEMbis}) finding that
\begin{equation}
\langle \Pi_{ij}(\mathbf{k1}) \Pi_{tl}(\mathbf{k2}) \Pi_{mn}(\mathbf{k3})\rangle\mathcal{P}_{ij}\mathcal{P}_{tl}\mathcal{P}_{mn}=\langle \Pi_B^{(S)}(\mathbf{k1}) \Pi_B^{(S)}(\mathbf{k2}) \Pi_B^{(S)}(\mathbf{k3})\rangle,
\end{equation}
thus, the expression for the bispectrum for that contribution is
\begin{eqnarray}\label{passiveee}
B_{\Pi_B^{(S)}\Pi_B^{(S)}\Pi_B^{(S)}}^{(S)}&=&\frac{1}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}\int d^3p \left(-P_B(p)P_B(\left| \mathbf{k1}-\mathbf{p}\right|)P_B(\left|\mathbf{p}+\mathbf{k2}\right|)F_{\Pi^{(S)} \Pi^{(S)}\Pi^{(S)}}^{1} \right. \nonumber \\
&-& P_B(p)P_H(\left| \mathbf{p}+\mathbf{k2}\right|)P_H(\left|\mathbf{k1}-\mathbf{p}\right|)F_{\Pi^{(S)} \Pi^{(S)}\Pi^{(S)}}^{2} \nonumber\\
&+& P_H(p)P_B(\left| \mathbf{p}+\mathbf{k2}\right|)P_H(\left|\mathbf{k1}-\mathbf{p}\right|) F_{\Pi^{(S)} \Pi^{(S)}\Pi^{(S)}}^{3} \nonumber \\
&+& \left. P_H(p)P_B(\left|\mathbf{k1}-\mathbf{p}\right|)P_H(\left|\mathbf{p}+\mathbf{k2}\right|) F_{\Pi^{(S)} \Pi^{(S)}\Pi^{(S)}}^{4} \right).
\end{eqnarray}
Again, the $F_{\Pi\Pi\Pi}$'s values can be checked along with the odd contribution in appendix \ref{A1}.
In the case where the helicity of the field is not considered $(A_H=0)$, the only contribution that remains is sourced from $A^3_B$. The results of this contribution were reported in \cite{37} and we have found the same expressions. Therefore, we have generalized those previous results to even and odd contributions of the PMFs bispectrum and thus these findings are the first results of the paper.
\section{Full Evaluation}\label{redbisa4}
With the derivation of the angular part of the three-point correlation for each component of the magnetic tensor, we proceed to make the evaluation of the above integrals. Due to the complexity of the calculation,
we are going to follow the exact methodology proposed in \cite{25,26,27}. They consider two cases for finding solution of the correlator. In the first case, the terms dependent on the integration vector $\mathbf{p}$
are not considered in the evaluation. For the second case, the squeezed collinear configuration is used to make predictions. We will use five representative shapes of the bispectrum which are shown in the Figure \ref{fig2}.
The odd signal arising from $\mathcal{B}^{(A\,)}$ will not be considered here but will be deferred for later work.
\begin{figure}
\centering
\includegraphics[width=.8\linewidth]{plotsn/prueba1}
\caption{Geometrical representations for the bispectrum. The figure shows a visual representation of the triangles and the collinear configuration of the bispectrum shape. The table in the bottom-right panel describes the values
of the $\mathbf{p}$-independent terms for each configuration.}
\label{fig2}
\end{figure}
\subsection{$\mathbf{p}$-independent}\label{pindep}
For this case, the only terms which appear in the evaluation are those angles given in eq.(\ref{angles}) independent of $\mathbf{p}$; they are ($\theta_{kp}$, $\theta_{kq}$, $\theta_{pq}$). The values of these angles for each configuration are shown in figure \ref{fig2}. The $F$'s functions defined above take the following values under this approximation
\begin{equation}
F_{\rho \rho\rho}^{1}=\mu^2, \quad F_{\rho \rho\rho}^{2}=\mu, \quad F_{\rho \rho\rho}^{3}=F_{\rho \rho\rho}^{4}=0,
\end{equation}
\begin{equation}
F_{\rho \rho\Pi}^{1}=\mu^2-3, \quad F_{\rho \rho\Pi}^{2}=-\mu, \quad F_{\rho \rho\Pi}^{3}=F_{\rho \rho\Pi}^{4}=0,
\end{equation}
\begin{equation}
F_{\rho \Pi\Pi}^{1}=\mu^2-6+9\theta_{pq}^2, \quad F_{\rho \Pi\Pi}^{2}=(-7+9\theta_{pq}^2)\mu, \quad F_{\rho \Pi\Pi}^{3}=F_{\rho \Pi\Pi}^{4}=0,
\end{equation}
\begin{eqnarray}
F_{\Pi \Pi\Pi}^{1}&=&-9+9\theta_{kp}^2-27\theta_{kp}\theta_{kq}\theta_{pq}+9\theta_{pq}^2+\mu^2+9\theta_{kq}^2, \quad F_{\Pi \Pi\Pi}^{3}=0, \nonumber \\
F_{\Pi \Pi\Pi}^{2}&=&(-13+18\theta_{kp}^2+9\theta_{kq}^2-27\theta_{kp}\theta_{kq}\theta_{pq}+9\theta_{pq}^2)\mu, \quad F_{\Pi \Pi\Pi}^{4}=0,
\end{eqnarray}
where the result given for $F_{\Pi \Pi\Pi}^{1}$ is in agreement with the reported in \cite{25}\footnote{There is a difference with a minus sign because we are taking a different signature in the metric.}.
The values of $F$ for each geometrical representation of the bispectrum are shown in table \ref{table1} and $\mu=0$ for all configuration except to squeezed configuration where it takes $\mu \sim -1$.
\begin{table}[!hbt]
\begin{center}
\begin{tabular}[b]{| l | c|c|c|c|c|c|c| r |}
\hline
\textbf{Configuration} & $F_{\rho \rho\rho}^{1}$ & $F_{\rho \rho\rho}^{2}$ & $F_{\rho\rho\Pi}^{1}$ & $F_{\rho \rho\Pi}^{2}$ & $F_{\rho \Pi\Pi}^{1}$ & $F_{\rho \Pi\Pi}^{2}$ & $F_{\Pi \Pi\Pi}^{1}$ & $F_{\Pi \Pi\Pi}^{2}$ \\
\hline
Squeezed triangle & 0 & 0 & -3 & 0 & -6 & 0 & 0 & 0 \\
\hline
Equilateral triangle & 0 & 0& -3 & 0 & -15/4 & 0 & -45/8 & 0 \\
\hline
Folded triangle & 0 & 0 & -3 & 0 & 3 & 0 & -9 & 0 \\
\hline
Squeezed collinear & 1 & -1 & -2 & 1 & 4 & -2 & -8 & 4 \\
\hline
Flattened & 0 & 0 & -3 & 0 & 3 & 0 & -9 & 0 \\
\hline
\end{tabular}
\caption{Values of $F$ for different geometrical configurations in the $\mathbf{p}$-independent case.}\label{table1}
\end{center}
\end{table}
\subsection{Squeezed Collinear Configuration}
n this approximation, the magnitude of one wave vector ($\hat{\mathbf{k3}}$) is small while the others have equal magnitudes but have opposing directions ($\hat{\mathbf{k1}}=-\hat{\mathbf{k2}}$) as shown in figure \ref{fig2}.
With this assumption the angles can be reduced to
\begin{eqnarray}
\beta&=&\hat{\mathbf{p}}\cdot\widehat{\mathbf{k1}-\mathbf{p}}\sim -\hat{\mathbf{p}}\cdot\widehat{\mathbf{k1}-\mathbf{p}}\sim-\gamma, \quad \mu\sim-\widehat{\mathbf{k1}-\mathbf{p}}\cdot\widehat{\mathbf{k1}-\mathbf{p}}\sim -1, \nonumber\\
\alpha_k&\sim& -\alpha_p\sim \alpha_q, \quad \beta_k \sim -\beta_p\sim \beta_q\sim -\gamma_k\sim \gamma_p\sim-\gamma_q.
\end{eqnarray}
By using this approximation the values of the $F$'s are simplified to
\begin{equation}
F_{\rho \rho\rho}^{1}=1+\beta^2, \quad F_{\rho \rho\rho}^{2}=-(1+\beta^2), \quad F_{\rho \rho\rho}^{3}=-2\beta,\quad F_{\rho \rho\rho}^{4}=2\beta,
\end{equation}
\begin{eqnarray}
F_{\rho \rho\Pi}^{1}&=&-2+3\alpha_k^2+\beta^2-6\alpha_k\beta\beta_k+3\beta_k^2+3\beta^2\beta_k^2, \quad F_{\rho \rho\Pi}^{2}=1-3\alpha_k^2-2\beta^2+6\alpha_k\beta\beta_k, \nonumber \\
F_{\rho \rho\Pi}^{3}&=&-\beta(-1+3\beta_k^2), \quad F_{\rho \rho\Pi}^{4}=\beta(-1+3\beta_k^2),
\end{eqnarray}
\begin{eqnarray}
F_{\rho \Pi\Pi}^{1}&=&-9\alpha_k\beta\beta_k^3+(2-3\beta_k^2)^2+\beta^2(1+3\beta_k^2)+\alpha_k^2(-3+9\beta_k^2), \quad F_{\rho \Pi\Pi}^{3}=3\alpha_k\beta_k-\beta(4-3\beta_k^2), \nonumber\\
F_{\rho \Pi\Pi}^{2}&=&-2-2\beta^2+3\alpha_k\beta\beta_k+3\beta_k^2,\quad F_{\rho \Pi\Pi}^{4}=-6\alpha_k\beta_k+9\alpha_k\beta_k^3-\beta(-5+6\beta_k^2),
\end{eqnarray}
\begin{eqnarray}
F_{\Pi \Pi\Pi}^{1}&=&-8+\beta^2+18\beta_k^2+3\beta^2\beta_k^2-9\beta_k^4+6\alpha_k\beta\beta_k(1-3\beta_k^2)+9\alpha_k^2(1-3\beta_k^2+3\beta_k^4), \nonumber \\
F_{\Pi \Pi\Pi}^{2}&=& 4-2\beta^2-3\beta_k^2+\alpha_k^2(-6+9\beta_k^2), \quad F_{\Pi \Pi\Pi}^{3}=F_{\Pi \Pi\Pi}^{4}=(2\beta-3\alpha_k\beta_k)(1-3\beta_k^2).
\end{eqnarray}
Same results have been obtained in \cite{25} for $F_{\Pi \Pi\Pi}^{1}$ (case where there is not helicity). The angular part of the integrals must be written in spherical coordinates $d^3p=p^2 dp d\alpha_kd\theta$, where $\theta$ is the azimuthal angle. Since we consider an upper cut-off $k_D$,
we must introduce the ($k1,k2$)-dependence on the angular integration domain; this implies that we should split the integral domain
in different regions such as $0 <k1,k2< 2k_D$.
The integration domains we use for calculating the integrals are shown in appendix \ref{apenb}. By using the power spectrum expression for the magnetic fields
eqs.(\ref{powerPMF1})-(\ref{powerPMF2}) and the F's values for the $\mathbf{p}-$independent case given above, we get the causal magnetic bispectrum ($n_B=n_H=2$) which is shown in the Figure \ref{figparte1}. We see that the most contribution for the bispectrum occurs when $k1 \sim k2$ and $\langle \Pi_B \Pi_B \Pi_B \rangle$ generates the largest value for scalar mode. Hence, we conclude that the shape of the non-Gaussian associated with the PMF can be classified into the local-type configuration as was previously reported in \cite{28} for a scale invariant shape. We also observe that effects from $A_BA_H^2$ contribution are smaller with respect to $A_B^3$.
The figures \ref{fig1aa}, \ref{fig2aa} show the cross-correlation of the bispectrum obtaining the same behavior and a large contribution (with respect to the energy density bispectrum).
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{plots1/rrrNHAp.pdf}
\caption{{\footnotesize Three-point correlation of $\langle \rho_B \rho_B \rho_B \rangle$ in units of $\frac{8}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}$ only for $A_B^3$.}}
\label{fig:gull}
\end{subfigure}
~
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{plots1/rrrOHAp.pdf}
\caption{{\footnotesize Three-point correlation of $\langle \rho_B \rho_B \rho_B \rangle$ in units of $\frac{8}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}$ only with $A_BA_H^2$.}}
\label{fig:tiger}
\end{subfigure}
~
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{plots1/rrrTAp.pdf}
\caption{{\footnotesize Even contribution of three-point correlation of $\langle \rho_B \rho_B \rho_B \rangle$ in units of $\frac{8}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}$.}}
\label{fig:mouse}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{plots1/rrpTAp.pdf}
\caption{{\footnotesize Even contribution of three-point correlation of $\langle \rho_B \rho_B \Pi_B \rangle$ in units of $\frac{4}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}$.}}
\label{fig1aa}
\end{subfigure}
~
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{plots1/rppTAp.pdf}
\caption{{\footnotesize Even contribution of three-point correlation of $\langle \rho_B \Pi_B \Pi_B \rangle$ in units of $\frac{2}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}$.}}
\label{fig2aa}
\end{subfigure}
~
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{plots1/pppTAp.pdf}
\caption{{\footnotesize Even contribution of three-point correlation of $\langle \Pi_B \Pi_B \Pi_B \rangle$ in units of $\frac{1}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}$.}}
\label{fig:mouse}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{plots1/rrpEquTAp.pdf}
\caption{{\footnotesize Even contribution of three-point correlation of $\langle \rho_B \rho_B \Pi_B \rangle$ in units of $\frac{4}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}$ in the equilateral configuration.}}
\label{fig1aa}
\end{subfigure}
~
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{plots1/rppEquTAp.pdf}
\caption{{\footnotesize Even contribution of three-point correlation of $\langle \rho_B \Pi_B \Pi_B \rangle$ in units of $\frac{2}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}$ in the equilateral configuration.}}
\label{fig2aa}
\end{subfigure}
~
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{plots1/pppEquTAp.pdf}
\caption{{\footnotesize Even contribution of three-point correlation of $\langle \Pi_B \Pi_B \Pi_B \rangle$ in units of $\frac{1}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}$ in the equilateral configuration.}}
\label{fig:mouse}
\end{subfigure}
\caption{{\footnotesize
Total contribution of three-point correlation of all scalar modes described in the text using the $\textbf{p}$-independent approximation. The figures (a), (b) (c) show the three-point correlation of the energy density of the magnetic field without, with $A_BA_H^2$ and full contribution respectively. The figures (d), (e) and (f) show the cross three-point correlation of the field. Finally, the figures (g), (h) (i) show the cross three-point correlation field in the equilateral configuration, where the $\langle \rho_B \Pi_B \Pi_B \rangle$ has a total negative contribution. We can see that largest contribution to the bispectrum is obtained when $k1\sim k2$. We can also see that biggest contributions to the scalar modes is given by $\langle \Pi_B \Pi_B \Pi_B \rangle$ when we consider a squeezed configuration. }}\label{figparte1}
\end{figure}
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{plots1/rrrNHG.pdf}
\caption{{\footnotesize Three-point correlation of $\langle \rho_B \rho_B \rho_B \rangle$ in units of $\frac{8}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}$ without $A_BA_H^2$.}}
\label{fig:gull}
\end{subfigure}
~
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{plots1/rrrOHG.pdf}
\caption{{\footnotesize Three-point correlation of $\langle \rho_B \rho_B \rho_B \rangle$ in units of $\frac{8}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}$ only with $A_BA_H^2$.}}
\label{fig:tiger}
\end{subfigure}
~
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{plots1/rrrTG.pdf}
\caption{{\footnotesize Even contribution of three-point correlation of $\langle \rho_B \rho_B \rho_B \rangle$ in units of $\frac{8}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}$.}}
\label{fig:mouse}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{plots1/pppNHG.pdf}
\caption{{\footnotesize Three-point correlation of $\langle \Pi_B \Pi_B \Pi_B \rangle$ in units of $\frac{1}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}$ without $A_BA_H^2$.}}
\label{fig:gull}
\end{subfigure}
~
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{plots1/pppOHG.pdf}
\caption{{\footnotesize Three-point correlation of $\langle \Pi_B \Pi_B \Pi_B \rangle$ in units of $\frac{1}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}$ only with $A_BA_H^2$.}}
\label{fig:tiger}
\end{subfigure}
~
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{plots1/pppTG.pdf}
\caption{{\footnotesize Even contribution of three-point correlation of $\langle \Pi_B \Pi_B \Pi_B \rangle$ in units of $\frac{1}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}$.}}
\label{fig:mouse}
\end{subfigure}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{plots1/rrpTG.pdf}
\caption{{\footnotesize Three-point correlation of $\langle \rho_B \rho_B \Pi_B \rangle$ in units of $\frac{4}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}$.}}
\label{fig:gull}
\end{subfigure}
~
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{plots1/rppTG.pdf}
\caption{{\footnotesize Three-point correlation of $\langle \rho_B \Pi_B \Pi_B \rangle$ in units of $\frac{2}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}$ only with $A_BA_H^2$.}}
\label{fig:tiger}
\end{subfigure}
~
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=\textwidth]{plots1/rrpTTG.pdf}
\caption{{\footnotesize Even contribution of three-point correlation of $\langle \rho_B \Pi_B \Pi_B \rangle$ in units of $\frac{2}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}$.}}
\label{fig:mouse}
\end{subfigure}
\caption{{\footnotesize Total contribution of three-point correlation of non-crossing scalar modes described in the text using the squeezed collinear configuration. The figures (a), (b) (c) show the three-point correlation of the energy density of the magnetic field without, with $A_BA_H^2$ and full contribution respectively; while figures (d), (e) and (f) show the three-point correlation of the anisotropic stress of the magnetic field without, with $A_BA_H^2$ and full contribution respectively. On the other hand, the figures (g), (h) and (i) show the cross three-point correlation of the field in this configuration. We can see that $k1\sim k2$ has the biggest contribution to the bispectrum. One important feature of the anisotropic stress mode is the negative contribution to the total bispectrum for wavevenumbers larger than $K_D/2$.
}}\label{figparte2}
\end{figure}
On the other hand, in figure \ref{figparte2} we present the results under the squeezed collinear configuration driven also by causal fields. Here, we see that the bispectrum for this configuration is less than the $\mathbf{p}-$independent case. We also see that magnetic anisotropic stress change the sign under this configuration for wavenumbers larger than $k_D/2$, while the $A_BA_H^2$ contribution is practically negative. Note that all the palettes shown here are sequential colour palettes, so they are very well suited to show the amplitude of the bispectrum.
Finally, for negative spectral indices, an approximate solution for the bispectrum can be found using the formula (5.13) in \cite{31} (see also equation (\ref{apenb1}) in appendix \ref{apenb}). Assuming ($n_B=n_H=n$) along with $k2<k1<k_D$ , the expression for bispectrum becomes
\begin{eqnarray}
B_{\rho_B^{(S)}\rho_B^{(S)}\rho_B^{(S)}}^{(S)}&\sim&\frac{16}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}\left(\frac{nk1^nk2^{2n+3}}{(n+3)(2n+3)}+\frac{nk1^{3n+3}}{(2n+3)(3n+3)}+\frac{k_D^{3n+3}}{(3n+3)}\right) \nonumber\\
&\times&\left(A_s^3 F_{\rho \rho\rho}^{1}+A_H^2A_s\left(F_{\rho\rho\rho}^{3}-F_{\rho\rho\rho}^{4}-F_{\rho\rho\rho}^{2}\right)\right),
\end{eqnarray}
\begin{eqnarray}
B_{\Pi_B^{(S)}\Pi_B^{(S)}\Pi_B^{(S)}}^{(S)}&\sim&\frac{2}{(2\pi)^3(4\pi \rho_{\gamma,0})^3}\left(\frac{nk1^nk2^{2n+3}}{(n+3)(2n+3)}+\frac{nk1^{3n+3}}{(2n+3)(3n+3)}+\frac{k_D^{3n+3}}{(3n+3)}\right) \nonumber\\
&\times&\left(A_s^3 F_{\Pi \Pi\Pi}^{1}+A_H^2A_s\left(F_{\Pi \Pi\Pi}^{3}+F_{\Pi \Pi\Pi}^{4}-F_{\Pi \Pi\Pi}^{2}\right)\right).
\end{eqnarray}
\subsection{Infrared Cut-off}
We now analyze the effect of an infrared cut-off parametrized by $\alpha$ on the magnetic bispectrum (see equation (\ref{mcut-off})). We saw that the non-Gaussianity peaks at $k1 \sim k2$ under a squeezed configuration, so we compute the magnetic bispectrum using the strategy adopted in \cite{21} and in appendix \ref{apenc}. In figure \ref{figparte22}, the effect of this IR cut-off for causal fields is illustrated.
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/2017fig6a.png}
\caption{{\footnotesize Change of $k1^3\langle \rho_B \rho_B \rho_B \rangle$ respect to the infrared cut-off without $A_BA_H^2$.}}
\label{fig6a}
\end{subfigure}
~
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/2017fig6b.png}
\caption{{\footnotesize \footnotesize Change of $k1^3\langle \rho_B \rho_B \rho_B \rangle$ respect to the infrared cut-off only with $A_BA_H^2$.}}
\label{fig6b}
\end{subfigure}
~
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/newfi6c.png}
\caption{{\footnotesize Change of $k1^3\langle \Pi_B \Pi_B \Pi_B \rangle$ respect to the infrared cut-off without $A_BA_H^2$.}}
\label{fig6c}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/newfi6d.png}
\caption{{\footnotesize Change of $k1^3\langle \Pi_B \Pi_B \Pi_B \rangle$ respect to the infrared cut-off only with $A_BA_H^2$.}}
\label{fig6d}
\end{subfigure}
\caption{{\footnotesize Effects of a lower cut-off at the three-point correlation of non-crossing scalar modes described in the text using the squeezed collinear configuration. The figures (a), (b) show the three-point correlation of the energy density of the magnetic field without and only with $A_BA_H^2$ respectively, while figures (c) and (d) show the three-point correlation of the anisotropic stress of the magnetic field without and only with $A_BA_H^2$ respectively. The black, red, blue, violet and green lines refer to lower cut-off for $\alpha=0.01$, $\alpha=0.4$, $\alpha=0.6$, $\alpha=0.7$, $\alpha=0.9$ respectively. Here the units are normalized respect to the values in figure (a) and we use $n_B=n_H=2$.
}}\label{figparte22}
\end{figure}
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/fig6aneg.png}
\caption{{\footnotesize Change of $k1^3\langle \rho_B \rho_B \rho_B \rangle$ respect to the infrared cut-off without $A_BA_H^2$.}}
\label{fig6aa}
\end{subfigure}
~
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/fig6bneg.png}
\caption{{\footnotesize \footnotesize Change of $k1^3\langle \rho_B \rho_B \rho_B \rangle$ respect to the infrared cut-off only with $A_BA_H^2$.}}
\label{fig6bb}
\end{subfigure}
~
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/fig6dnew.png}
\caption{{\footnotesize Change of $k1^3\langle \rho_B \rho_B \rho_B \rangle$ respect to the infrared cut-off without $A_BA_H^2$.}}
\label{fig6cc}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/fig6cnew.png}
\caption{{\footnotesize Change of $k1^3\langle \rho_B \rho_B \rho_B \rangle$ respect to the infrared cut-off only with $A_BA_H^2$.}}
\label{fig6dd}
\end{subfigure}
\caption{{\footnotesize Effects of a lower cut-off at the three-point correlation of non-crossing scalar modes described in the text using the squeezed collinear configuration. The figures (a), (b) show the three-point correlation of the energy density of the magnetic field for $n_B=n_H=-5/2$ without and only with $A_BA_H^2$ respectively, while figures (c) and (d) show the three-point correlation of the energy density of the magnetic field for $n_B=n_H=-1.9$ without and only with $A_BA_H^2$ respectively. The black, red, blue, violet and green lines refer to lower cut-off for $\alpha=0.01$, $\alpha=0.4$, $\alpha=0.6$, $\alpha=0.7$, $\alpha=0.9$ respectively. Here the units are normalized respect to the values in figure (\ref{fig6a}).
}}\label{figparte22a}
\end{figure}
The top figures (\ref{fig6a}) and (\ref{fig6b}) show the effect of $\alpha$ on $k1^3\langle \rho_B \rho_B \rho_B \rangle$ where the lines refer to different values of the cut-off, while the bottom figures (\ref{fig6c}) and (\ref{fig6d}) show the effect of $\alpha$ on $k1^3\langle \Pi_B \Pi_B \Pi_B \rangle$. What we read off these figures is how the peak of the bispectrum moves to high wavenumbers when we increase the value of $k_m$ in the same way that magnetic power spectrum (and its amplitude decreases also due to reduction of the wavenumber space) and how the effects of the $A_BA_H^2$ contributions PMF are tiny compared with the non-helical case. On the other hand, the figure (\ref{figparte22a}) shows the effect of the infrared cut-off when we are considering non-causal fields. The top panel shows the three-point correlation of the energy density of the magnetic field for $n_H=n_B=-5/2$ while the bottom panel shows the three-point correlation for $n_H=n_B=-1.9$. Here we can see that the contribution driven by $A_BA_H^2$ is bigger than $A_B^3$, meaning that for negative spectral indices the effect of helicity becomes relevant for our studies.
\section{Reduced Bispectrum from PMF} \label{redbisa5}
In this section, we estimate the reduced bispectrum and give a careful review the results of \cite{25},\cite{38},\cite{39}.
The CMB temperature perturbation at a direction of photon momentum $\hat{\mathbf{n}}$ can be expanded into spherical harmonics
\begin{equation}
\frac{\Delta T^{(Z)}}{T}(\hat{\mathbf{n}})=\sum_{lm}a^{(Z)}_{lm}Y_{lm}(\hat{\mathbf{n}}),
\end{equation}
where $Z=S,V,T$ refers to the contribution given by scalar , vector or tensor perturbations. The coefficient $a^{(Z)}_{lm}$ is written as \cite{30}
\begin{eqnarray}\label{alm}
a^{Z}_{lm}&=&4\pi(-i)^{l}\int\frac{d^3\mathbf{k}}{(2\pi)^3}\Delta^{Z}_l(k)\sum_\lambda[sgn(\lambda)]^\lambda\xi_{lm}^{(\lambda)}(k), \nonumber\\
\xi_{lm}^{(\lambda)}(k)&=&\int d^2\hat{\mathbf{k}}\xi^\lambda(k) _{-\lambda}\!Y^*_{lm}(\hat{\mathbf{k}}),
\end{eqnarray}
where $\lambda=0,\pm1,\pm2$ describes the helicity of the scalar, vector, tensor mode; $_{-\lambda}\!Y^*_{lm}$ is the spin-weight spherical harmonics; $\xi^\lambda(k)$ is the primordial perturbation and $\Delta^{Z}_l(k)$ is the
transfer function. Let us define the CMB angular bispectrum as
\begin{equation}\label{bis1}
B_{l_1\,l_2\,l_3}^{m_1\,m_2\,m_3}=\bigg\langle \prod_{n=1}^3 a_{l_n\,m_n}^{(Z)} \bigg\rangle,
\end{equation}
where only scalar perturbations (Z=S) will be considered in the paper. Now, by substituting the equation (\ref{alm}) into equation (\ref{bis1}) we can find
\begin{equation}\label{bisalm}
\bigg\langle \prod_{n=1}^3 a_{l_n\,m_n} \bigg\rangle=\left[\prod_{n=1}^3 4\pi(-i)^{l_n}\int \frac{d^3\mathbf{kn}}{(2\pi)^3}\Delta_{l_n}(kn)Y^*_{l_n\,m_n}(\hat{\mathbf{n}})\right]\times\bigg\langle \prod_{n=1}^3 \xi(kn) \bigg\rangle.
\end{equation}
We also consider a rough approximation for the transfer function that works quite well at large angular scales and for primordial adiabatic perturbations given by $\Delta_l(k)=\frac{1}{3}j_l(k(\eta_0-\eta_*))$, where $j_l(x)$ the spherical Bessel function, $\eta_0=14.38$ Gpc the conformal time at present and $\eta_*=284.85$ Mpc the conformal time at the recombination epoch \cite{39}. This is the large scale Sachs Wolfe effect.
Since we want to evaluate the contribution of the bispectrum by PMFs, the three-point correlator for the primordial perturbation must satisfy the relation
\begin{equation}\label{pmf1}
\bigg\langle \prod_{n=1}^3 \xi(kn) \bigg\rangle=A_P \delta(\mathbf{k1}+\mathbf{k2}+\mathbf{k3})B_{\Pi_B\Pi_B\Pi_B}^{(S)}.
\end{equation}
Here, $A_P$ is a constant that depends on the type of perturbation (passive or compensated magnetic mode) and $B_{\Pi_B\Pi_B\Pi_B}^{(S)}$ is the magnetic bispectrum computed above. Using the fo\-llowing relations \cite{38}
\begin{equation}
\delta(\mathbf{k1}+\mathbf{k2}+\mathbf{k3})=\frac{1}{(2\pi)^3}\int_{-\infty}^{\infty} \exp^{i(\mathbf{k1}+\mathbf{k2}+\mathbf{k3})\cdot \mathbf{x}}d^3x,
\end{equation}
\begin{equation}
\exp^{i\mathbf{k}\cdot \mathbf{x}}=4\pi\sum_li^lj_l(kx)\sum_m Y_{lm}(\hat{\mathbf{k}})Y_{lm}^*(\hat{\mathbf{x}}),
\end{equation}
with the Gaunt integral $\mathcal{G}_{l_1l_2l_3}^{m_1m_2m_3}$ defined by
\begin{eqnarray}
\mathcal{G}_{l_1l_2l_3}^{m_1m_2m_3}&\equiv& \int d^2 \hat{\mathbf{n}}Y_{l_1m_1}(\hat{\mathbf{n}})Y_{l_2m_2}(\hat{\mathbf{n}})Y_{l_3m_3}(\hat{\mathbf{n}})\nonumber\\
&=&\sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}}\bigg(\begin{array}{ccc}
l_1 & l_2 & l_3 \\
0 & 0 & 0 \end{array} \bigg)\bigg(\begin{array}{ccc}
l_1 & l_2 & l_3 \\
m_1 & m_2 & m_3 \end{array} \bigg),
\end{eqnarray}
and along with the equation (\ref{pmf1}), the equation (\ref{bisalm}) takes the form
\begin{eqnarray}
\bigg\langle \prod_{n=1}^3 a_{l_n\,m_n} \bigg\rangle&=&A_P\sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}}\bigg(\begin{array}{ccc}
l_1 & l_2 & l_3 \\
0 & 0 & 0 \end{array} \bigg)\bigg(\begin{array}{ccc}
l_1 & l_2 & l_3 \\
m_1 & m_2 & m_3 \end{array} \bigg)\times \nonumber \\
&\times&\left[\prod_{n=1}^3 \frac{1}{3\pi^2}\int kn^2\int j_{l_n}(kn\,x)j_{l_n}(kn(\eta_0-\eta_*))dkn\right]B_{\Pi_B\Pi_B\Pi_B}^{(S)}x^2dx.
\end{eqnarray}
Here the matrix is the Wigner-3j symbol and it vanishes unless the selection rules are satisfied\footnote{The Wigner-3j symbols satisfy that: $|m_1|\leq l_1$, $|m_2|\leq l_2$, $|m_3|\leq l_3$, $m_1+m_2+m_3=0$, $l_1+l_2+l_3=\mathbb{Z}$ and $|l_1-l_2|\leq l_3 \leq l_1+l_2$ \cite{30}. }.
Given the rotational invariance of the Universe, Komatsu-Spergel \cite{38} defined a real symmetric function of $l_i$ called the reduced bispectrum $b_{l_1l_2l_3}$
\begin{equation}
\bigg\langle \prod_{n=1}^3 a_{l_n\,m_n} \bigg\rangle \equiv \mathcal{G}_{l_1l_2l_3}^{m_1m_2m_3} b_{l_1l_2l_3}.
\end{equation}
Checking the last two equations, the properties of the bispectrum generated by PMFs can be expressed via the reduced bispectrum as
\begin{eqnarray}\label{redbis}
b_{l_1l_2l_3}=A_P\left[\prod_{n=1}^3 \frac{1}{3\pi^2}\int kn^2\int j_{l_n}(kn\,x)j_{l_n}(kn(\eta_0-\eta_*))dkn\right]B_{\Pi_B\Pi_B\Pi_B}^{(S)}x^2dx.
\end{eqnarray}
In order to calculate $A_P$, we must clarify the sources of primordial perturbations. Prior to neutrino decoupling ($\tau_\nu=1$MeV$^{-1}$), the Universe is dominated by radiation and it is tightly coupled to baryons such that they cannot have any anisotropic
stress contribution. Since we are also considering magnetic fields, they would be the only ones that develop anisotropic stress and therefore at superhorizon scales the curvature perturbation depends on the primordial magnetic source\cite{30}. But after neutrino decoupling, neutrinos generated anisotropic stress which compensates the one coming
from PMF finishing the growth of the perturbations. Shaw-Lewis \cite{40} showed that curvature perturbation is given by
\begin{equation}
\xi(k)\sim -\frac{1}{3}R_\gamma \ln\left(\frac{\tau_\nu}{\tau_B}\right)\Pi_B^{(S)}(k),
\end{equation}
commonly known as passive mode, where $R_\gamma=\frac{\rho_\gamma}{\rho}\sim 0.6$ and $\tau_B$ is the epoch of magnetic field generation. Another contribution comes from the density-sourced
mode with unperturbed anisotropic stresses, the magnetic compensated scalar mode, this is proportional to the amplitude of the perturbed magnetic density just as the magnetic Sachs Wolfe effect \cite{40}.
So, if the primordial perturbation is associated with the initial gravitational potential, in the limit on large-angular scales the compensated modes is expressed as \cite{25,42}
\begin{equation}
\xi(k)\sim \frac{1}{4}R_\gamma \rho_B(k).
\end{equation}
Therefore if we use the passive mode contribution, $B_{\Pi_B\Pi_B\Pi_B}^{(S)}$ is given by $B_{\Pi_B^{(S)}\Pi_B^{(S)}\Pi_B^{(S)}}^{(S)}$ (see equation (\ref{passiveee})) with $A_P=\left(-\frac{1}{3}R_\gamma \ln\left(\frac{\tau_\nu}{\tau_B}\right)\right)^3$, whilst compensated mode the primordial three-point correlation is described by $B_{\rho_B\rho_B\rho_B}^{(S)}$ (see equation (\ref{passiveee1})) with $A_P=\left(\frac{1}{4}R_\gamma\right)^3$.
Since the magnetic bispectrum only depends on $(k1;k2)$, the $k3$ integral in the equation (\ref{redbis}) gives $\frac{\pi}{2x^2} \delta(x-(\eta_0-\eta_*))$ due to the closure relation \cite{25}, and integrating out the delta function one finally obtains
\begin{eqnarray}\label{redbis1}
b_{l_1l_2l_3}=A_P\frac{\pi}{2}\left[\prod_{n=1}^2 \frac{1}{3\pi^2}\int kn^2 j_{l_n}(kn(\eta_0-\eta_*))^2 dkn\right]B_{\Pi_B\Pi_B\Pi_B}^{(S)}.
\end{eqnarray}
This is the master formula that we shall use in the following section in order to calculate the CMB reduced bispectrum.
\section{Analysis}\label{redbisa}
In this section we show the numerical results of the CMB reduced bispectrum produced by helical PMFs. In order to numerically solve eq.(\ref{redbis1}), we use the adaptive strategy implemented in Mathematica called Levin-type rule which estimates the integral of an oscillatory function with a good accuracy\cite{mathem}.
\subsection{Causal Fields}
Figure \ref{fig7s} presents the CMB reduced bispectrum generated by compensated PMFs modes under collinear configuration.
In this figure we observe the signal produced by only the $A_B^3$ contribution (\ref{fig7a}), as welll as the signal by the whole(\ref{fig7b}). We found that $A_BA_H^2$ contribution (helical) is smaller than non-helical part $A_B^3$. Here we plot the change of the reduced bispectrum with respect to $l_1$, finding a large contribution for larges values of $l_1$. We also see that helical contribution reaches a maximum around $l_2 \sim 400$ whilst non-helical contribution tends to increase at least until $l_2 \sim 500$.
In figures (\ref{fig7c}) and (\ref{fig7d}), the effect of an IR cut-off on the reduced bispectrum are shown. Each of these plots show the signal for different values of $l_1$. We see that signal is biggest for small $\alpha$ values (being the biggest contribution for spectrum without IR cut-off) similar to the found with the power spectrum case\cite{21}.
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/newfig7a.png}
\caption{{\footnotesize Reduced bispectrum given by $A_B^3$ contribution of compensated PMFs.}}
\label{fig7a}
\end{subfigure}
~
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/fig8c.png}
\caption{{\footnotesize \footnotesize Reduced bispectrum given by total contribution of compensated PMFs.}}
\label{fig7b}
\end{subfigure}
~
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/fig8d.png}
\caption{{\footnotesize Effects of infrared cut-off on the reduced bispectrum with $l_1=61$.}}
\label{fig7c}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/fig8f.png}
\caption{{\footnotesize Effects of infrared cut-off on the reduced bispectrum with $l_1=210$.}}
\label{fig7d}
\end{subfigure}
\caption{{\footnotesize Reduced bispectrum seeded by compensated PMFs with $n=2$ using the squeezed collinear configuration. The figures (a) shows the reduced bispectrum of the magnetic field with only $A_B^3$, while figure (b) shows the total contribution of the compensated mode; here the lines refers to different values of $l_1$, violet($l_1=11$), black($l_1=61$), red($l_1=110$), green($l_1=161$) and blue line($l_1=210$). The figures (c), (d) show the effects of an infrared cut-off on the reduced bispectrum for different values of multipolar numbers $l_1$.
Black; red; green; blue; and violet lines refer to lower cut-off of $\alpha=0.001$, $\alpha=0.4$, $\alpha=0.5$, $\alpha=0.7$, $\alpha=0.8$ respectively.
The reduced bispectrum is in units of $4\pi10^{-8}A_P/(8\pi^2 \rho_{\gamma,0})^3$.
}}\label{fig7s}
\end{figure}
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/fig8a1.png}
\caption{{\footnotesize Reduced bispectrum given by only $A_B^3$ contribution of passive PMFs.}}
\label{fig8a}
\end{subfigure}
~
~
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/fig8ac.png}
\caption{{\footnotesize Reduced bispectrum given by total contribution of passive PMFs. }}
\label{fig8b}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/fig8ad.png}
\caption{{\footnotesize Effects of infrared cut-off on the reduced bispectrum with $l_1=61$.}}
\label{fig8c}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/fig8af.png}
\caption{{\footnotesize Effects of infrared cut-off on the reduced bispectrum with $l_1=210$.}}
\label{fig8d}
\end{subfigure}
\caption{{\footnotesize Reduced bispectrum seeded by passive PMFs with $n_B=n_H=n=2$ using the squeezed collinear configuration. The figure (a) shows reduced bispectrum of the magnetic field with only $A_B^3$, while figure (b) shows the total contribution of the passive mode, here the lines refer to different values of $l_1$, violet($l_1=11$), black($l_1=61$), red($l_1=110$), green($l_1=161$) and blue line($l_1=210$). The figures (c) and (d) show the effects of an infrared cut-off on the reduced bispectrum for difference values of multipolar numbers $l_1$.
Black; red; green; blue; and violet lines refer to lower cut-off of $\alpha=0.001$, $\alpha=0.4$, $\alpha=0.5$, $\alpha=0.7$, $\alpha=0.8$ respectively. The reduced bispectrum is in units of $4\pi10^{-8}A_P/2(8\pi^2 \rho_{\gamma,0})^3$.
}}\label{fig8}
\end{figure}
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/fig9a.png}
\caption{{\footnotesize Effects of infrared cut-off on the reduced bispectrum with $l_1=161$ seeded by total contribution of compensated helical PMFs.}}
\label{fig9a}
\end{subfigure}
~
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/fig9b.png}
\caption{{\footnotesize \footnotesize Effects of infrared cut-off on the reduced bispectrum with $l_1=161$ seeded by total contribution of passive helical PMFs.}}
\label{fig9b}
\end{subfigure}
~
\caption{{\footnotesize Reduced bispectrum seeded by compensated (a) and passive (b) helical PMFs with only $A_BA_H^2$ contribution using the squeezed collinear configuration.
Black; red; green; blue; and violet lines refer to lower cut-off of $\alpha=0.001$, $\alpha=0.4$, $\alpha=0.5$, $\alpha=0.7$, $\alpha=0.8$ respectively. The reduced bispectrum is in units of $4\pi10^{-8}A_P/2(8\pi^2 \rho_{\gamma,0})^3$.
}}\label{fighel}
\end{figure}
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/newfig10a.png}
\caption{{\footnotesize Reduced bispectrum given by $A_B^3$ contribution of compensated PMFs.}}
\label{fig10a}
\end{subfigure}
~
~
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/newfig10c.png}
\caption{{\footnotesize Reduced bispectrum given by total contribution of compensated PMFs.}}
\label{fig10b}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/newfig10d.png}
\caption{{\footnotesize Effects of infrared cut-off on the reduced bispectrum with $l_1=111$.}}
\label{fig10c}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/newfig10f.png}
\caption{{\footnotesize Effects of infrared cut-off on the reduced bispectrum with $l_1=210$.}}
\label{fig10d}
\end{subfigure}
\caption{{\footnotesize Absolute value of reduced bispectrum seeded by compensated PMFs with $n=-5/2$ using the squeezed collinear configuration. The figure (a) shows reduced bispectrum of the magnetic field with only $A_B^3$ contribution, while figure (b) shows the total contribution of the compensated mode; here the lines refers to different values of $l_1$, violet($l_1=11$), black($l_1=61$), red($l_1=110$), green($l_1=161$) and blue line($l_1=210$). The figures (c) and (d) show the effects of an infrared cut-off on the reduced bispectrum for difference values of multipolar numbers $l_1$.
Black; red; green; and violet lines refer to lower cut-off of $\alpha=0.001$, $\alpha=0.4$, $\alpha=0.6$, $\alpha=0.8$ respectively. The reduced bispectrum is in units of $4\pi10^{16}A_P/(8\pi^2 \rho_{\gamma,0})^3$.
}}\label{fig7s1}
\end{figure}
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/newfig11a.png}
\caption{{\footnotesize Reduced bispectrum of compensated PMFs for $n=-3/2$.}}
\label{fig11a}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/newfig11b.png}
\caption{{\footnotesize Reduced bispectrum of compensated PMFs for $n=-1.9$.}}
\label{fig11b}
\end{subfigure}
~
~
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/newfig11e.png}
\caption{{\footnotesize Effects of infrared cut-off on the reduced bispectrum with $l_1=161$ for $n=-3/2$.}}
\label{fig11c}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/newfig11f.png}
\caption{{\footnotesize Effects of infrared cut-off on the reduced bispectrum with $l_1=161$ for $n=-1.9$.}}
\label{fig11d}
\end{subfigure}
\caption{{\footnotesize Absolute values of reduced bispectrum seeded by compensated PMFs using the squeezed collinear configuration. The figures (a), (b) show the total contribution of the compensated mode of the magnetic field for $n=-3/2$ and $n=-1.9$ respectively; here the lines refers to different values of $l_1$, violet($l_1=11$), black($l_1=61$), red($l_1=110$), green($l_1=161$) and blue line($l_1=210$). The last figures (c),(d) show the effects of an infrared cut-off on the reduced bispectrum for difference values of multipolar numbers $l_1$. Blue; green; red; and black lines refer to lower cut-off of $\alpha=0.001$, $\alpha=0.4$, $\alpha=0.6$, $\alpha=0.8$, respectively.
The bispectrum is in units of $4\pi10^{16}A_P/(8\pi^2\rho_{\gamma,0})^3$.
}}\label{fig7s2}
\end{figure}
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/newfig12a.png}
\caption{{\footnotesize Reduced bispectrum of compensated PMFs for $n=-5/2$.}}
\label{fig12a}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/newfig12b.png}
\caption{{\footnotesize Reduced bispectrum of compensated PMFs for $n=-3/2$.}}
\label{fig12b}
\end{subfigure}
~
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/newfig12c.png}
\caption{{\footnotesize \footnotesize Reduced bispectrum of passive PMFs for $n=-5/2$.}}
\label{fig12c}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{plotsn/newfig12d.png}
\caption{{\footnotesize Reduced bispectrum of passive PMFs for $n=-3/2$.}}
\label{fig12d}
\end{subfigure}
~
\caption{{\footnotesize Absolute values of reduced bispectrum seeded by passive and compensated PMFs under p-independent approximation. The figures (a), (b) show the total contribution of the compensated mode of the magnetic field for $n=-5/2$ and $n=-3/2$ respectively. The figures (c), (d) show the total contribution of the passive mode of the magnetic field for $n=-5/2$ and $n=-3/2$ respectively. Here the lines refer to different values of $l_1$, violet($l_1=11$), black($l_1=61$), red($l_1=110$), green($l_1=161$) and blue line($l_1=210$).
}}\label{fig7s3}
\end{figure}
In Figure (\ref{fig8}) we show the CMB reduced bispectrum generated by PMFs passive modes under collinear configuration. Meanwhile, figures (\ref{fig8a}) and (\ref{fig8b}) describe the signal for the non-helical and total contribution respectively. An interesting feature is that reduced bispectrum generated by helical contribution is totally negative, then, by using this unusual behavior we would have direct evidence of a helical component in the field.
Other important result of this paper is reported in figures (\ref{fig8c}) and (\ref{fig8d}). Here we show again the effect of an IR cut-off on the reduced bispectrum and we have found out that the biggest contribution of the bispectrum comes from an IR cut-off near $\alpha \sim 0.5$ instead of $\alpha = 0$. This peak might correspond to a type of dynamics in large scales and help us to determine the nature of PMFs (Since we are trying with causal fields, this infrared cut-off would correspond to the maximum scale in which magnetic fields may be generated at later times). Therefore the evidence of this cut-off in the bispectrum would reveal an interesting
signal from passive magnetic scalar mode.
In addition, the change of the reduced bispectrum for helical PMFs in presence of an IR cut-off is showed in figure (\ref{fighel}). Here we observe how the signal decreases when the IR cut-off increases for the compensated mode and how change the behavior for the passive case.
We want to remark on some approximations used so far. Since we assume that effects of PMFs are important for small multipolar numbers, we write the transfer functions in terms of spherical Bessel function. Previous papers have worked without this approximation. For instance, \cite{32} computes the full radiation transfer function taking into account the effect of PMFs. The full numerical integration of the bispectrum can be done via second-order Einstein-Boltzmann codes like SONG\cite{tram} improving the estimation of the amplitude of PMFs.
Moreover, we must note that \cite{43} found a WMAP bound on non-helical passive mode for tensor temperature bispectrum of $B_{1Mpc}<3.1$nG and the Planck paper \cite{33} reported $B_{1Mpc} < 2.8$nG all of them for scale-invariant fields. Actually, the tensor mode is dominant in the passive mode, and can give quite tighter constraint of the PMF amplitude than the scalar mode.
Thus, tensor mode contribution and a full transfer function determined by the presence of these fields will improve our results and therefore they will be interesting subjects of our future work.
\subsection{Non-causal Fields}
Now let us consider a red magnetic spectrum. Figure (\ref{fig7s1}) shows the reduced bispectrum for compensated mode with $n_H=n_B=n=-5/2$. Again, we plot the non-helical (\ref{fig10a}) and total (\ref{fig10b}) contribution of the bispectrum while plots (\ref{fig10c}) and (\ref{fig10d}) correspond to the change of the signal due to an IR cut-off. Since the PMFs bispectrum is almost determined by the poles in each $k$, the value of it peaked for $l_1=l_2=l_3$ as we can observe in figures (\ref{fig7s1}),(\ref{fig7s2}), (\ref{fig7s3}). Additionally, the figure (\ref{fig7s2}) shows signals for $n_H=n_B=n=-3/2$ and $n_H=n_B=n=-1.9$. As a matter of fact, some mechanisms of inflationary magnetogenesis with parity violating terms which lead to helical magnetic field ($n=-1.9$), stand for the lower bound for the which the field can satisfy the intensity of magnetic fields in the intergalactic medium; thereby the signal described in figure (\ref{fig7s2}) constraints models for providing the seed for galactic magnetic fields \cite{18l2}. On the other hand, figure (\ref{fig7s3}) shows the reduced bispectrum taking into account the p-independent approximation implemented in section \ref{pindep}. Due to complexity of the angular structure on the PMF bispectrum for passive mode, the numerical computation for reduced bispectrum requires a great deal of time. To avoid this problem we can use the p-independent approximation by reducing the PMF bispectrum to an independent angular form as we
studied above. From figures (\ref{fig12a}) and (\ref{fig12b}) we observe an increase in the signal as we expected due to the lack of angular terms in the bispectrum. Finally, figures (\ref{fig12c}) and (\ref{fig12d}) show the total contribution of passive
modes. Note that the amplitude is larger than the compensated mode. This behavior have also been reported in \cite{25}. This result is interesting because an estimation of $B_\lambda$ through a local-type primordial NG in curvature perturbation generates constraints stronger than the compensated ones. Finally, if we compare the results reported in this section with the ones shown in above section for causal fields, we can observe that effect of $k_m$ is more significant in negative spectral indices, specially for nearly scale invariant scale fields. We then conclude that $k_m$ plays an important role in the study of these non-causal fields and this generates the possibility of determining some important clues in the mechanisms of magnetogenesis.
\subsection{Estimation of the Magnetic Field Amplitude}
In fact, it is possible to obtain a rough estimate of $B_\lambda$ using the formula for the primary reduced bispectrum found in \cite{38}
\begin{equation}
b_{lll}\sim l^{-4}\times2\times10^{-17}f_{NL},
\end{equation}
where the non-Gaussianity (local) is fully specified by a single constant parameter $f_{NL}$. As we mentioned above, the k-dependence on the magnetic bispectrum is similar to the CMB bispectrum arising from the local type NG of curvature perturbations, therefore, by comparing the last equation with eq.(\ref{redbis1}), allow us to express a simple relation between $F_{NL}$ and $B_\lambda$ given by
\begin{equation}\label{eqng}
f_{NL}\propto \left( \frac{B_\lambda}{10^{-9}\text{G}}\right)^6.
\end{equation}
In Table \ref{tabla2} we present the constant of proportionality of the last expression. Here we use $\frac{\tau_\nu}{\tau_B}=10^{17}$ which corresponds to the PMF generated at the grand unification energy scale(GUT) scale, $\left(\frac{B_\lambda^2}{8\pi\rho_{\gamma,0}}\right)\sim10^{-7}\left(\frac{B_\lambda}{10^{-9}\text{G}}\right)^2$, and $R_\gamma\sim0.6$. In order to constrain the smoothed amplitude of the magnetic field on a scale of $1$Mpc ($B_1$), we will use the $f_{NL}$ value reported by Planck Collaboration \cite{pl2} of $f_{NL}<5.8$ at $68\%$ CL. The results of $B_1$ are shown in table \ref{tabla3}. Our results for compensated modes lead to upper bounds on the PMF smoothed amplitude which are consistent with the 2015 Planck analysis \cite{33}, but for passive modes our results are slightly tigher because these were based on a rough estimation and may involve some uncertainties (except for the causal case where the bound coincides with Planck analysis), however notice that for passive modes the limits are almost 10 times more stringent than the compensated ones as was reported in \cite{25} for no-helical and scale invariant case, hence CMB-observation are sensitive to the magnetic induced modes. Since our results were obtained under the Sachs-Wolfe approximation, we expected a lower value of $B_1$ for causal fields with respect to the non causal fields, and therefore the blue spectra generated by these fields is strongly disfavoured by the CMB bispectrum.
On the other hand, we constrain $B_1$ through the helical contribution and we observed an enhance of its amplitude. We see this same effect in the two point correlation as was reported by Planck analysis $B_1<5.6$nG at 95\% CL \cite{33} for that contribution. In the tables also show the bounds when a high value of the IR cut-off ($\alpha\sim 0.8$) is used. Since cut-off reduces the amplitude magnetic bispectrum signal, the upper limit of $B_\lambda$ becomes somewhat relaxed and therefore, we are able to illustrate the impact $k_m$ can have on constraints on the PMF amplitude.
Although this effect becomes very small for a tiny value of $k_m$, the presence of this scale in the analysis of NG are complementary to the ones found by the two point correlation case and will provide new insight into the nature of primordial magnetic fields.
\begin{table}[htb]
\centering
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
& \multicolumn{2}{c|}{n$=-\frac{5}{2}$} & \multicolumn{1}{c|}{n$=-1.9$}& \multicolumn{2}{c|}{n$=-\frac{3}{2}$}& \multicolumn{2}{c|}{n$=2$} \\
\cline{2-8}
& Comp. & Passive & Comp & Comp. & Passive & Comp & Passive \\
\hline \hline
Helical & 0.86(1.92) & 0.12 & 0.47(0.80) & 0.35(0.49) & 0.008 & 0.021 & 0.0069\\ \cline{1-8}
N-Helical & 1.04(2.43) & 0.13 & 0.38 & 0.33 & 0.067 & 0.018 & 0.0050\\ \cline{1-8}
Total & 0.92(1.85) & 0.14 & 0.40(0.73) & 0.38(0.44) & 0.064 & 0.017(0.021) & 0.0051(0.006) \\ \cline{1-8}
\end{tabular}
\caption{Constant of proportionality of the eq.(\ref{eqng}) for different spectral indices and modes(compensated(comp) or passive) of the PMF without considering an IR cut-off. Parentheses are used to represent this value for $\frac{k_m}{k_D}\sim0.8$.}
\label{tabla2}
\end{table}
\begin{table}[htb]
\centering
\begin{tabular}{|l|l|l|l|l|l|l|l|}
\hline
& \multicolumn{2}{c|}{n$=-\frac{5}{2}$} & \multicolumn{1}{c|}{n$=-1.9$}& \multicolumn{2}{c|}{n$=-\frac{3}{2}$}& \multicolumn{2}{c|}{n$=2$} \\
\cline{2-8}
& Comp. & Passive & Comp & Comp. & Passive & Comp & Passive \\
\hline \hline
Helical & 1.15(2.58) & 0.16 & 0.64(1.07) & 0.47(0.66) & 0.098 & 0.029 & 0.0092\\ \cline{1-8}
N-Helical & 1.39(3.25) & 0.17 & 0.52 & 0.44 & 0.089 & 0.024 & 0.0068\\ \cline{1-8}
Total & 1.24(2.48) & 0.19 & 0.54(0.98) & 0.52(0.59) & 0.085 & 0.023(0.029) & 0.0068(0.0083) \\ \cline{1-8}
\end{tabular}
\caption{Bound on smoothed amplitude of the magnetic field on a scale of $1$Mpc ($B_1$ in units of nG) for different spectral indices and modes(compensated(comp) or passive) of the PMF without considering an IR cut-off. Parentheses are used to represent $B_1$ for $\frac{k_m}{k_D}\sim0.8$.
Here we use $f_{NL}<5.8$ reported by Planck Collaboration, 2016 \cite{33}.}
\label{tabla3}
\end{table}
\section{Discussion and Conclusions} \label{discu}
In this paper we investigate the effects of helical PMFs in the CMB reduced bispectrum.
One of the main motivations to introduce the helicity comes from the fact that the these fields are good observables to probe parity-violation in the early stages of the Universe.
Furthermore, since magnetic fields depend quadratically on the field, it must induce non-Gaussian signals on CMB anisotropies at lower order instead of the standard inflationary mechanisms where this signal appears only at high orders\cite{27}.
We started our work deriving the full even and odd parts of the bispectrum which comes from the helical magnetic fields, thus extending the previous results reported in \cite{37}.
We obtained the full expression for the PMF bispectrum but, we did not consider modes that arise from odd intensity-intensity-intensity bispectrum. Although these signals are smaller than the even ones, the evidence of the odd signals would be a decisive observable to probe parity-violating processes in the early Universe\cite{28}. We will provide more details of the parity-odd signals in a future paper.
Then, through the methodology used in \cite{25}, \cite{26}, we found that
PMFs bispectrum peaks at $k1 \sim k2$ under a squeezed configuration implying that statistical properties of the PMFs are similar to those of the local-type NG of curvature perturbations.
By calculating the bispectrum given by PMFs anisotropic stress, we observed that its amplitude is larger that the density one and also has a negative contribution for values less than $k_D$.
Through numerical calculations of the intensity-intensity-intensity reduced bispectra of the scalar modes, we also studied the total contributions of the helical PMF bispectrum and the presence of an IR cut-off in the convolution integrals. Here we observe the same behavior seen in the power spectral case and the presence of negative contribution due to helical terms $A_BA_H^2$.
Nevertheless, in the computation in the passive modes for causal fields we observed an unusual behavior in the bispectrum. Indeed, in Figure (\ref{fig8}) we have found out that biggest contribution of the bispectrum comes from an IR cut-off near to $\alpha \sim 0.5$ instead of $\alpha = 0$ . Since $k_m$ is dependent on PMF generation model, this behavior might set strong limits on PMF amplitude. Finally, we investigated the effects of $k_m$ on the reduced bispectrum for $n<0$. Due to the fact that the magnetic field intensity can be enhanced when we use passive modes, it is expected that those modes determine a very strong constraints on the amplitude of the magnetic field on a given characteristic scale $\lambda$. We verify this statement by using the primary reduced bispectrum found in \cite{38} and calculating $B_\lambda$.
Our results showed in tables \ref{tabla2}, \ref{tabla3} reflect the fact that the corresponding bound on the mean amplitude of the field is dependent on strong values of the minimal cut-off and the helical contribution relaxing the constraints of $B_\lambda$. We also found that for passive modes the limits are almost 10 times more stringent than the compensated ones for both helical and hon helical contribution,
this result was also reported in \cite{27} for non helical fields.
However, we can observe that effect of $k_m$ is more significant in the magnetic bispectrum driven by negative spectral indices. Hence, the presence of $k_m$ plays an important role in the analysis of the signatures that these non-causal fields may leave in cosmological observations.\\
In conclusion, we have studied the effects of helicity and a minimal cut-off on the constraints of the PMFs amplitude by computing the CMB reduced bispectrum induced on large angular scales by those fields.
Even though $k_m$ for causal modes would be important when this scale is larger
than the wavenumber of interest, for non-causal modes it is related with the horizon scale of the beginning of inflation \cite{tk,21}, and therefore the study of this cut-off on the bispectrum give us information about the PMF generation mechanisms.
|
1,941,325,220,512 | arxiv | \section*{Introduction}
Deformation of structure theories is useful to formalize quantum physics. The algebraic structure considered by classical mechanics is the associative commutative algebra of smooth functions over a symplectic, or more generally, a Poisson manifold. Deformation quantization consists to construct an associative non-commutative multiplication (more precisely a $\star$-product) on the formal series in $\hbar$ (which represents Planck's constant) with coefficients in this algebra, which encodes the Poisson bracket in the first order. The Poisson structure is then called the classical limit and the deformation is the $\star$-product. This point of view, initiated in 1978 in \cite{BFFLS78}, tries to consider quantum mechanics as a deformation from classical mechanics. This pioneering work raises the fundamental questions about the existence and the uniqueness of a deformation quantization for a given Poisson manifold.
The first results treated the case of symplectics manifolds. The general case of Poisson manifolds was solved by Kontsevich in 1997 in \cite{K03}. He deduced the result by proving a much more general statement, which he called \og{}formality conjecture\fg{}. Endowed with the Gerstenhaber bracket, the Hochschild complex of the algebra of smooth functions over a Poisson manifold admits a graded Lie algebraic structure by shift, which controls the deformations of the Poisson bracket. Kontsevich shows that this complex is linked with its cohomology --- which therefore controls the same deformations --- by a $L_\infty$-quasi-isomorphism, called a formality map. Using this formality map, he shows of to deform the commutative product of functions in a $\star$-product.
Konstevich shows in particular that the formality criterion is true for symmetric algebras over a finite-dimensional vector space are formal. Bordemann and Makhlouf have examined in \cite{BM08} a slight generalization to the case of universal enveloping algebras. They showed (implicit in Kontsevich's work) that the formality for a Lie algebra is equivalent to those of its universal enveloping algebra. They also proved that there is formality for the universal enveloping algebra of an affine Lie algebra. These methods were used in \cite{BMP05} and give informations about the rigidity of universal enveloping algebras. \\
In \cite{El12}, we detailed the definitions of formality for associative and Lie algebras, and we studied the formality equations for the associative free algebra, and for the Lie algebra $\mathfrak{so}(3)$. It turns out that in these cases, there is no formality in Kontsevich's sense, but one has to add an extra term to the graded Lie bracket on the cohomology to obtain a perturbed formality map.
This work is part of this line of research. We study here the formality for the three-dimensional Heisenberg Lie algebra. Although the Lie bracket is more simple than the one of $\mathfrak{so}(3)$, the cohomology is more important, and therefore the formality equations contains more terms. Actually, to obtain a perturbed formality map in this case, we have to add infinitely many terms to the Schouten bracket on the cohomology. \\
The paper is organised as follows. In \autoref{Sec:formality-Lie}, we set the notations and recall the graded Lie structure (Schouten bracket) on the Chevalley-Eilenberg complex of a Lie algebra and its cohomology. We also give the definition of formality for this complex and write down the formality equations. In \autoref{Sec:Heisenberg-cohomology}, we compute the cohomology of the Heisenberg algebra. \autoref{Schouten-cochain} and \autoref{Schouten-cohomology} are devoted to the computation of the Schouten brackets on cochain level and cohomology, in order to solve the formality equations in \autoref{Sec:formality-eq}. \\
I wish to thank gratefully Mr Makhlouf who showed me his first computations of the cohomology for the Heisenberg algebra and for his advice to pursue the study of the formality.
\section{Formality of Lie algebras} \label{Sec:formality-Lie}
Let $(\mathfrak{g},[~,~])$ be a Lie $\numberset{K}$-algebra. We consider the Chevalley-Eilenberg differential complex $(\ChEcochains(\mathfrak{g},\Sym\mathfrak{g}),\delta_{CE})$ with values in $\Sym\mathfrak{g}$
\begin{gather}
C^0 \coloneqq \ChEcochains[0](\mathfrak{g},\Sym\mathfrak{g}) \coloneqq \Sym\mathfrak{g} \qquad C^k \coloneqq \ChEcochains[k](\mathfrak{g},\Sym\mathfrak{g}) \coloneqq \Hom\big({\bigwedge}^k \mathfrak{g},\Sym\mathfrak{g}\big) \quad \text{for $k \in \numberset{N}^\star$} \\
\delta_{CE}^k : C^k \to C^{k+1} \notag
\end{gather}
and its cohomology
\( \displaystyle
\ChE(\mathfrak{g},\Sym\mathfrak{g}) \coloneqq \bigoplus_{k \in \numberset{N}} \ChE[k](\mathfrak{g},\Sym\mathfrak{g})
\)
where for $k \in \numberset{N}$, the $k^\mathrm{th}$ Chevalley-Eilenberg cohomology group with coefficients in $\Sym\mathfrak{g}$ is
\begin{equation}
H^k \coloneqq \ChE[k](\mathfrak{g},\Sym\mathfrak{g}) \coloneqq \frac{\Kr(\delta_{CE}^k)}{\Img(\delta_{CE}^{k-1})} \coloneqq Z^k/B^k
\end{equation}
The coboundary operator $\delta_{CE}$ equals $[\pi,~]_s$, where $\pi = [~,~] \in \Hom\big({\bigwedge}^2 \mathfrak{g},\mathfrak{g}\big)$ is the Lie bracket from $\mathfrak{g}$ and $[~,~]_s$ is the Schouten bracket on the complex $\ChEcochains(\mathfrak{g},\Sym\mathfrak{g})$.
As the Gerstenhaber bracket, the Schouten bracket defines a graded Lie bracket on the shifted space $\ChEcochains(\mathfrak{g},\Sym\mathfrak{g})[1]$. Therefore the graded antisymmetry and graded Jacobi identities: for $\xi,\eta,\zeta \in \ChEcochains(\mathfrak{g},\Sym\mathfrak{g})[1]$ we have
\begin{gather}
[\eta,\xi]_s = -(-1)^{|\eta| |\xi|}[\xi,\eta]_s, \label{gr-antisymmetry} \\
(-1)^{|\xi| |\zeta|} [[\xi,\eta]_s,\zeta]_s + (-1)^{|\eta| |\xi|} [[\eta,\zeta]_s,\xi]_s + (-1)^{|\zeta| |\eta|} [[\zeta,\xi]_s,\eta]_s = 0 \label{gr-Jacobi},
\end{gather}
degrees being taken in $\ChEcochains(\mathfrak{g},\Sym\mathfrak{g})[1]$.
On the other hand there is the pointwise exterior multiplication $\wedge$ which makes $\ChEcochains(\mathfrak{g},\Sym\mathfrak{g})$ a graded commutative algebra. The Schouten bracket and the exterior multiplication are compatible by the graded Leibniz rule
\begin{equation}
[\xi,\eta \wedge \zeta]_s = [\xi,\eta]_s \wedge \zeta + (-1)^{|\xi| (|\eta|+1)} \eta \wedge [\xi,\zeta]_s, \label{gr-Leibniz}
\end{equation}
(if $\eta$ is of degree $|\eta|$ in $\ChEcochains(\mathfrak{g},\Sym\mathfrak{g})[1]$ then it is of degree $|\eta| + 1$ in $\ChEcochains(\mathfrak{g},\Sym\mathfrak{g})$). \\
Throughout the computations, we will use the fact that $\ChEcochains(\mathfrak{g},\Sym\mathfrak{g})$ is an $\Sym \mathfrak{g}$-module \textit{i.e.} \/ for $\xi \in \ChEcochains(\mathfrak{g},\Sym\mathfrak{g}),\ f \in \Sym\mathfrak{g}$, $f \wedge \xi = f \xi$. We also give some examples of Schouten brackets which will be used later.
\begin{example}
The space $C^1 \coloneqq \ChEcochains[1](\mathfrak{g},\Sym\mathfrak{g})$ is the space of vectors fields, \textit{i.e.} \/ the derivations of $\Sym\mathfrak{g} = C^0$. For $f,g \in \Sym\mathfrak{g},\ X,Y \in C^1$
\begin{gather*}
[f,g]_s = 0, \\
[X,f]_s = X(f) = \sum_{i=1}^n X_i \partial_i f, \\
[X,Y]_s = [X,Y] = \sum_{i=1}^n X_i \partial_i Y - \sum_{i=1}^n Y_i \partial_i X,
\shortintertext{with in particular}
[f \partial_i,g \partial_j] = f (\partial_i g) \partial_j - g (\partial_j f) \partial_i.
\end{gather*}
\end{example}
The Schouten bracket $[~,~]_s$ on the shifted complex $C[1] \coloneqq \ChEcochains(\mathfrak{g},\Sym\mathfrak{g})[1]$ induces a bracket $[~,~]_s'$ on the shifted cohomology complex $H[1] \coloneqq \ChE(\mathfrak{g},\Sym\mathfrak{g})[1]$, which is also a graded Lie algebra.
We consider $\overline{d}$ the unique coderivation of $\Sym(H[2])$ induced by $[~,~]'_s[1]$ and $\overline{\delta_{CE} + D}$, the unique coderivation of $\Sym(C[2])$ induced by $\delta_{CE} + [~,~]_s[1]$. They are also differentials, equipping respectively $(H[1],\overline{d})$ and $(C[1],\overline{\delta_{CE}+D})$ of structures of $L_\infty$-algebras.
\begin{defi}
The Chevalley-Eilenberg complex $C[1]$ associated to the Lie algebra $(\mathfrak{g},[~,~])$ is said to be \emph{formal} if there is a $L_\infty$-quasi-isomorphism $\Phi : \Sym(H[2]) \to \Sym(C[2])$, \textit{i.e.} \/
\begin{equation}
(\Phi \otimes \Phi) \circ \Delta_{\Sym{} H[2]} = \Delta_{\Sym{} C[2]} \circ \Phi \qquad \textrm{et} \qquad \overline{\delta_{CE}+D} \circ \Phi = \Phi \circ \overline{d},
\end{equation}
such that the restriction $\Phi_1$ of $\Phi$ to $C[2]$ is a section. The map $\Phi$ is called a \emph{formality map}.
\end{defi}
Even if there is no formality map between $\Sym(H[2])$ and $\Sym(C[2])$, we can always modify the $L_\infty$ structure induced by $d = d_2 = [~,~]'_s[1]$ in $d = \sum_{n \geqslant 2} d_n$ and by induction build the components of the differential $\overline{d}$ and of the $L_\infty$-morphism $\Phi = \overline{\varphi}$. This construction was done by Bordemann et al. in \cite[A.4, Proposition A.3]{BGHHW05}. This result can also be obtained by a more general point of view using the \emph{Perturbation Lemma} for contraction of differential complexes. This was done by induction by Huebschmann for dg-Lie and $L_\infty$-algebras in \cite{Hueb10,Hueb11}, and Bordemann gives in \cite{B08} a closed formula. \\
The obtained $L_\infty$ morphism $\Phi = \overline{\varphi}$ and new differential $\overline{d}$ satisfy the equation
\begin{equation}
\Phi \circ \overline{d} = \overline{\delta_{CE}+D} \circ \Phi,
\end{equation}
which writes, after projection on $C[2]$ and using that $D = D_2 = [~,~]_s[1]$,
\begin{equation}
\varphi \circ (d \tilde{\star} id_{\Sym(H[2])}) = \delta_{CE} \circ \varphi + \frac{1}{2} D_2 \circ \varphi \tilde{\star} \varphi.
\end{equation}
Since $d = \sum_{n \geqslant 2} d_n$, evaluating on $y_1 \bullet \dotsb \bullet y_{k+1} \in \Sym(H[2])$ gives
\begin{equation} \label{eqs_formality_shifted_twice}
\begin{split}
\sum_{a=2}^{k+1} & \sum_{1 \leqslant i_1 < \dotsb < i_a \leqslant k+1} \nu_a(y_1,\dotsc,y_{k+1}) \\
& \cdotp \varphi_{k+2-a}(d_a(y_{i_1} \bullet \dotsc \bullet y_{i_a}) \bullet y_1 \bullet \dotsc \bullet \widehat{y_{i_1}} \bullet \dotsc \bullet \widehat{y_{i_a}} \bullet \dotsc \bullet y_{k+1}) \\
={} & \delta_{CE} \varphi_{k+1}(y_1 \bullet \dotsc \bullet y_{k+1}) \\
+ \frac{1}{2} & \sum_{a=1}^k \sum_{1 \leqslant i_1 < \dotsb < i_a \leqslant k+1} \nu_a(y_1,\dotsc,y_{k+1}) \\
& \cdotp D_2(\varphi_a(y_{i_1} \bullet \dotsc \bullet y_{i_a}) \bullet \varphi_{k+1-a}(y_1 \bullet \dotsc \bullet \widehat{y_{i_1}} \bullet \dotsc \bullet \widehat{y_{i_a}} \bullet \dotsc \bullet y_{k+1}).
\end{split}
\end{equation}
The maps $\varphi_k$ are of degree $0$, and the signs
\begin{equation*}
\nu_a(y_1,\dotsc,y_{k+1}) = \prod_{r=1}^a (-1)^{|y_{i_r}|(|y_1|+ \dotsb + \widehat{|y_{i_1}|} + \dotsb + \widehat{|y_{i_{r-1}}|} + \dotsb + |y_{i_r-1}|)}
\end{equation*}
for $1 \leqslant a \leqslant k+1$ only come from the permutation which place the elements of indices $i_1,\dotsc,i_a$ at the beginning. \\
However, care of signs must be taken when working with the shifted maps. Denoting by $s : C[1] \to C$ the suspension map which adds one to the degree of elements, we have for example
\begin{equation*}
D_2(x,y) = [x,y]_s[1] = s^{-1} \circ [~,~]_s \circ (s \otimes s)(x \otimes y) = (-1)^{|x|} s^{-1} [s(x),s(y)]_s = (-1)^{|x|}[x,y]_s.
\end{equation*}
\section{Cohomology of the Heisenberg algebra} \label{Sec:Heisenberg-cohomology}
From now on, we take $\mathfrak{g} = \mathfrak{h}_3$ the three-dimensional Heisenberg Lie algebra
\[
\mathfrak{h}_3 = <x,y,z \ \ |\ \ [x,y]=z>
\]
or, writing the Lie bracket as a bivector $[~,~] = \pi = z \partial_x \wedge \partial_y \in \Hom\big({\bigwedge}^2 \mathfrak{g},\mathfrak{g}\big)$.
Since $\dim \mathfrak{g} = 3$, the degree zero part $C^0 = \Sym\mathfrak{g}$ of the Chevalley-Eilenberg complex $\ChEcochains(\mathfrak{g},\Sym\mathfrak{g})$ identifies with $\numberset{K}[x,y,z]$. Moreover, the cohomology $\ChE(\mathfrak{g},\Sym\mathfrak{g})$ is concentrated only in degrees $0,1,2,3$. Indeed, $C^k = \Hom\big({\bigwedge}^k \mathfrak{g},\Sym\mathfrak{g}\big) = \{0\}$ for $k \geqslant 4$, because $\partial_x \wedge \partial_y \wedge \partial_z \wedge \partial_t = 0$ for $t \in \{x,y,z\}$. \\
In the following computations, we mostly use the three relations satisfied by the Schouten bracket and the exterior multiplication $\wedge$, namely the graded antisymmetry \eqref{gr-antisymmetry}, the graded Jacobi identity \eqref{gr-Jacobi} and the graded Leibniz rule \eqref{gr-Leibniz}.
\addtocounter{subsection}{-1}
\subsection{\texorpdfstring{Degree $0$}{Degree 0}}
In degree $0$, we have $\ChE[0](\mathfrak{g},\Sym\mathfrak{g}) = \Sym\mathfrak{g}^\mathfrak{g} = \{f \in \Sym\mathfrak{g},\ \delta_{CE}^0(f) = 0\}$. Let $f \in \Sym\mathfrak{g} = \numberset{K}[x,y,z]$,
\begin{equation*}
\delta_{CE}^0(f) = [\pi,f]_s = [z \partial_x \wedge \partial_y,f]_s = z \partial_y f \partial_x - z \partial_x f \partial_y,
\end{equation*}
so $\delta_{CE}^0(f) = 0 \Leftrightarrow \left\{
\begin{aligned}
z \partial_y f &= 0 \\
-z \partial_x f &= 0
\end{aligned}
\right. \
\Leftrightarrow f(x,y,z) = \varphi(z)$ is a function of $z$ only. \\[1ex]
Thus $\ChE[0](\mathfrak{g},\Sym\mathfrak{g}) = \numberset{K}[z]$.
\subsection{\texorpdfstring{Degree $1$}{Degree 1}}
The $1$-cocycles are $Z^1 = \{X : \mathfrak{g} \to \Sym\mathfrak{g},\ \delta_{CE}^1(X) = 0\}$. We write $X \in C^1 = \Hom(\mathfrak{g},\Sym\mathfrak{g})$ as $X = X_1 \partial_x + X_2 \partial_y + X_3 \partial_z$, with $X_i \in \numberset{K}[x,y,z]$.
\begin{align*}
\delta_{CE}^1(X) = [\pi,X]_s &= [z \partial_x \wedge \partial_y,X_1 \partial_x + X_2 \partial_y + X_3 \partial_z]_s \\
&= \left(z \partial_x X_1 + z \partial_y X_2 - X_3\right) \partial_x \wedge \partial_y + z \partial_y X_3\ \partial_x \wedge \partial_z - z \partial_x X_3\ \partial_y \wedge \partial_z,
\end{align*}
\begin{equation*}
\text{so}\quad \delta_{CE}^1(X) = 0 \Leftrightarrow \left\{
\begin{aligned}
z \partial_x X_1 + z \partial_y X_2 - X_3 &= 0 \\
z \partial_y X_3 &= 0 \\
- z \partial_x X_3 &= 0
\end{aligned}
\right..
\end{equation*}
The last two equations imply that $X_3(x,y,z) = \varphi(z)$ is a function of $z$ only. The first equation shows then that $X_3 = \varphi(z) = z \psi(z)$ is a multiple of $z$; and can be rewritten as
\begin{equation}
\partial_x X_1 + \partial_y X_2 = \psi. \label{eqZ1}
\end{equation}
Setting
\(
\left\{
\begin{gathered}
X_1 = - \partial_y g + Y_1 \psi \\
X_2 = \partial_x g + Y_2 \psi
\end{gathered}
\right.,
\)
with $g \in \Sym\mathfrak{g}$, solving \eqref{eqZ1} in $X_1,X_2$ is equivalent to solve in $Y_1,Y_2$ the equation
\begin{equation}
\partial_x Y_1 + \partial_y Y_2 = 1. \label{eqZ11}
\end{equation}
The solutions of \eqref{eqZ11} are of the form \(
\left\{
\begin{aligned}
Y_1 &= a x + K_1(y,z) \\
Y_2 &= (1-a) y + K_2(x,z)
\end{aligned}
\right.,
\)
with $a \in \numberset{K}$, and a good choice of the function $g$, gives
\(
\left\{
\begin{aligned}
X_1 &= - \partial_y g + a x \psi \\
X_2 &= \partial_x g + (1-a) y \psi
\end{aligned}
\right..
\)
\\[1ex]
We obtained the following expression for $1$-cocycles
\begin{equation}
X = \left(a x \psi(z) - \partial_y g\right) \partial_x + \big((1-a) y \psi(z) + \partial_x g\big) \partial_y + z \psi(z) \partial_z. \label{1cocycle-expr}
\end{equation}
The $1$-coboundaries are $B^1 = \left\{X : \mathfrak{g} \to \Sym\mathfrak{g},\ \exists\ F \in \Sym\mathfrak{g},\ X = \delta_{CE}^0(F)\right\}$. Since we have $\delta_{CE}^0(F) = z \partial_y F \partial_x - z \partial_x F \partial_y$,\quad $W = \partial_y g \partial_x - \partial_x g \partial_y$ is an element of $B^1$ if and only if $g = z F + \text{cst}$.
Writing $g(x,y,z) = g_0(x,y) + z g_1(x,y,z)$, we have
\[
W = \partial_y g \partial_x - \partial_x g \partial_y = \underbrace{\partial_y g_0(x,y) \partial_x - \partial_x g_0(x,y) \partial_y}_{\in\ Z^1 \setminus B^1} + \delta_{CE}^0(g_1).
\]
Since the $z$ multiple part is a coboundary, we will often use subscript $0$ for the constant part in $z$ and subscript $1$ for the $z$-part. We also use the following notations: $X_g \coloneqq - \partial_y g \partial_x + \partial_x g \partial_y$ for the Lie derivative associated to the function $g$ and $D_a~\coloneqq~a x \partial_x + (1-a) y \partial_y + z \partial_z$. The expression \eqref{1cocycle-expr} rewrites
\begin{equation*}
X = \psi D_a + X_{g_0} - \delta_{CE}^0(g_1),
\end{equation*}
hence the description of the degree one part of the cohomology:
\begin{equation}
\ChE[1](\mathfrak{g},\Sym\mathfrak{g}) = \left\{[X] : \mathfrak{g} \to \Sym\mathfrak{g},\ X = \psi(z) D_a + X_{g_0}\right\},
\end{equation}
with $\psi \in \numberset{K}[z]$ and $g_0 \in \numberset{K}[x,y]$. (The element $[X]$ is the class of cohomology of $X \in Z^1$.) \\
Let $X = \varphi D_a + X_{g_0}$ and $Y = \psi D_b + X_{h_0}$ two elements of $Z^1$, with $\varphi,\psi \in \numberset{K}[z]$, $g_0,h_0 \in \numberset{K}[x,y]$ and $a,b \in \numberset{K}$. Like said before, we note
\(
\left\{
\begin{aligned}
\varphi(z) &= \varphi_0 + z \varphi_1(z) \\
\psi(z) &= \psi_0 + z \psi_1(z)
\end{aligned}
\right..
\)
We have $D_{1/2} = \dfrac{1}{2} x \partial_x + \dfrac{1}{2} y \partial_y + z \partial_z$, and setting
\begin{flalign*}
& & k(x,y,z) &= \left(\dfrac{1}{2}-a\right)x y \varphi_1(z) &
\tilde{k}(x,y,z) &= \left(\dfrac{1}{2}-b\right)x y \psi_1(z) & &
\end{flalign*}
we get
\begin{align*}
X &= \left(a x \varphi_0 - \partial_y g_0 + \dfrac{1}{2} x z \varphi_1(z)\right) \partial_x + \left((1-a) y \varphi_0 + \partial_x g_0 + \dfrac{1}{2} y z \varphi_1(z)\right) \partial_y + z \varphi(z) \partial_z + \delta_{CE}^0(k) \\
Y &= \left(b x \psi_0 - \partial_y h_0 + \dfrac{1}{2} x z \psi_1(z)\right) \partial_x + \left((1-b) y \psi_0 + \partial_x h_0 + \dfrac{1}{2} y z \psi_1(z)\right) \partial_y + z \psi(z) \partial_z + \delta_{CE}^0(\tilde{k}).
\end{align*}
Thus $X$ and $Y$ are in the same cohomology class if, and only if
\begin{align*}
[X] = [Y] &\Leftrightarrow \left\{
\begin{aligned}
a x \varphi_0 - \partial_y g_0 + \dfrac{1}{2} x z \varphi_1(z) &= b x \psi_0 - \partial_y h_0 + \dfrac{1}{2} x z \psi_1(z) \\
(1-a) y \varphi_0 + \partial_x g_0 + \dfrac{1}{2} y z \varphi_1(z) &= (1-b) y \psi_0 + \partial_x h_0 + \dfrac{1}{2} y z \psi_1(z) \\
z \varphi(z) &= z \psi(z)
\end{aligned}
\right. \\
&\Leftrightarrow \varphi = \psi \qquad \text{and} \qquad g_0 - h_0 = (a-b)x y \varphi_0,
\end{align*}
and $Y \in [X] \Leftrightarrow \psi = \varphi$ and $h_0 = g_0 - (a-b)x y \varphi_0$. \\
So up to changing the function $g_0 \in \numberset{K}[x,y]$, when considering an element $[X] \in \ChE[1](\mathfrak{g},\Sym\mathfrak{g})$, we can always assume that $X = X_{g_0} + \varphi D_a$, with the same $a \in \numberset{K}$ fixed once for all.
\subsection{\texorpdfstring{Degree $2$}{Degree 2}}
The $2$-cocycles are $Z^2 = \{\xi : \mathfrak{g} \wedge \mathfrak{g} \to \Sym\mathfrak{g},\ \delta_{CE}^2(\xi) = 0\}$. We write $\xi \in C^2 = \Hom(\mathfrak{g} \wedge \mathfrak{g},\Sym\mathfrak{g})$ as $\xi_{12}\ \partial_x \wedge \partial_y + \xi_{13}\ \partial_x \wedge \partial_z + \xi_{23}\ \partial_y \wedge \partial_z$, with $\xi_{ij} \in \numberset{K}[x,y,z]$.
\begin{align*}
\delta_{CE}^2(\xi) = [\pi,\xi]_s &= [z \partial_x \wedge \partial_y,\xi_{12}\ \partial_x \wedge \partial_y + \xi_{13}\ \partial_x \wedge \partial_z + \xi_{23}\ \partial_y \wedge \partial_z]_s \\
&= z \left(\partial_x \xi_{13} + \partial_y \xi_{23}\right) \partial_x \wedge \partial_y \wedge \partial_z.
\end{align*}
\begin{equation*}
\text{so}\quad \delta_{CE}^2(\xi) = 0 \Leftrightarrow z \left(\partial_x \xi_{13} + \partial_y \xi_{23}\right) = 0 \Leftrightarrow \xi_{13} = - \partial_y g\quad \text{and}\quad \xi_{23} = \partial_x g
\end{equation*}
We can express the $2$-cocycles as
\begin{equation}
\xi = h\ \partial_x \wedge \partial_y - \partial_y g\ \partial_x \wedge \partial_z + \partial_x g\ \partial_y \wedge \partial_z,
\end{equation}
with $g,h$ arbitrary functions in $\numberset{K}[x,y,z]$. \\
The $2$-coboundaries are $B^2 = \left\{\xi : \mathfrak{g} \wedge \mathfrak{g} \to \Sym\mathfrak{g},\ \exists\ X \in \ChEcochains[1](\mathfrak{g},\Sym\mathfrak{g}),\ \xi = \delta_{CE}^1(X)\right\}$.
Since $\delta_{CE}^1(X) = \left(z \partial_x X_1 + z \partial_y X_2 - X_3\right) \partial_x \wedge \partial_y + z \partial_y X_3\ \partial_x \wedge \partial_z - z \partial_x X_3\ \partial_y \wedge \partial_z,$
\[
\xi = \delta_{CE}^1(X) \Leftrightarrow \left\{
\begin{aligned}
h &= z \partial_x X_1 + z \partial_y X_2 - X_3 \\
- \partial_y g &= z \partial_y X_3 \\
\partial_x g &= z \partial_x X_3
\end{aligned}
\right.
\]
We note
\begin{equation*}
\begin{aligned}
g(x,y,z) &= g_0(x,y) + z g_1(x,y,z) \\
&= g_0(x,y) + z \left(g_{10}(x,y) + z g_{11}(x,y,z)\right)
\end{aligned} \qquad \text{and} \qquad
\begin{aligned}
h(x,y,z) &= \widetilde{h_0}(x,y) + z h_1(x,y,z) \\
h_0 &= \widetilde{h_0} - g_{10},
\end{aligned}
\end{equation*}
and we take $X = X_1 \partial_x + X_2 \partial_y + X_3 \partial_z$ with $X_3 = g_1$ and $X_1,X_2$ such that $\partial_x X_1 + \partial_y X_2 = g_{11} - h_1$.
We then have
\[
\xi + \delta_{CE}^1(X) = h_0\ \partial_x \wedge \partial_y - \partial_y g_0\ \partial_x \wedge \partial_z + \partial_x g_0\ \partial_y \wedge \partial_z,
\]
with $g_0,h_0$ function of $x$ and $y$ only. Further setting $Y = h_0 \partial_z$, we get
\begin{align*}
\xi + \delta_{CE}^1(X) + \delta_{CE}^1(Y) &= - \partial_y (g_0 - z h_0)\ \partial_x \wedge \partial_z + \partial_x (g_0 - z h_0)\ \partial_y \wedge \partial_z \\
&= X_{g_0 - z h_0} \wedge \partial_z,
\end{align*}
recalling that $X_f = - \partial_y f \partial_x + \partial_x f \partial_y$.
Hence the description of the degree two part of the cohomology:
\begin{equation}
\ChE[2](\mathfrak{g},\Sym\mathfrak{g}) = \left\{[\xi] : \mathfrak{g} \wedge \mathfrak{g} \to \Sym\mathfrak{g},\ \xi = X_g \wedge \partial_z\right\},
\end{equation}
with $g \in \numberset{K}[x,y,z]$.
\subsection{\texorpdfstring{Degree $3$}{Degree 3}}
Since the Schouten bracket $[~,~]_s$ is of degree $0$, for $\xi \in C^k$, $\delta_{CE}(\xi) = [\pi,\xi]_s$ is of degree $1+k-1=k$. (Recall that in $C[1]$, elements of $C^k = \ChEcochains[k](\mathfrak{g},\Sym\mathfrak{g})$ are of degree $k-1$.)
Therefore, all elements of $C^3=\Hom\big({\bigwedge}^3 \mathfrak{g},\Sym\mathfrak{g}\big)$ are cocycles, since they are of degree $3$ and that $C^4 = \{0\}$, thus $Z^3 = \ChEcochains[3](\mathfrak{g},\Sym\mathfrak{g})$. \\
The $3$-coboundaries are $B^3 = \left\{\xi : \mathfrak{g} \wedge \mathfrak{g} \wedge \mathfrak{g} \to \Sym\mathfrak{g},\ \exists\ \phi \in \ChEcochains[2](\mathfrak{g},\Sym\mathfrak{g}),\ \xi = \delta_{CE}^2(\phi)\right\}$.
\begin{gather*}
\xi = \delta_{CE}^2(\phi) \Leftrightarrow \xi_{123}\ \partial_x \wedge \partial_y \wedge \partial_z = z \left(\partial_x \phi_{13} + \partial_y \phi_{23}\right) \partial_x \wedge \partial_y \wedge \partial_z
\end{gather*}
Setting $\xi_{123} = \xi_0(x,y) + z \xi_1(x,y,z)$, we take $\phi = \phi_{12}\ \partial_x \wedge \partial_y + \phi_{13}\ \partial_x \wedge \partial_z + \phi_{23}\ \partial_y \wedge \partial_z$ such that $\partial_x \phi_{13} + \partial_y \phi_{23} = - \xi_1$, so that
\begin{equation*}
\xi + \delta_{CE}^2(\phi) = \xi_0\ \partial_x \wedge \partial_y \wedge \partial_z.
\end{equation*}
Hence the description of the degree three part of the cohomology:
\begin{equation}
\ChE[3](\mathfrak{g},\Sym\mathfrak{g}) = \left\{[\xi] : \mathfrak{g} \wedge \mathfrak{g} \wedge \mathfrak{g} \to \Sym\mathfrak{g},\ \xi = \xi_0\ \partial_x \wedge \partial_y \wedge \partial_z\right\},
\end{equation}
with $\xi_0 \in \numberset{K}[x,y]$.
\section{Schouten brackets on cochain level} \label{Schouten-cochain}
We work with the graded Lie algebra $C[1] = \ChEcochains(\mathfrak{g},\Sym\mathfrak{g})[1]$ where $\mathfrak{g} = \mathfrak{h}_3$. We compute in this section the various Schouten brackets $[C^i,C^j]_s$, for $0 \leqslant i \leqslant j \leqslant 3$. In the computations, we do not use the Lie structure of the Heisenberg algebra, all that is used is that $\dim \mathfrak{g} = 3$. \\
Reasoning on degrees, for $f,g \in C^0 = \Sym\mathfrak{g}$, we have
\begin{equation} \label{cochain00}
[f,g]_s = 0.
\end{equation}
\medskip
Let $f \in C^0$ and $X = X_1 \partial_x + X_2 \partial_y + X_3 \partial_z \in C^1$, with $X_i \in \numberset{K}[x,y,z]$.
\begin{equation} \label{cochain01}
[X,f]_s = X(f) = X_1 \partial_x f + X_2 \partial_y f + X_3 \partial_z f
\end{equation}
\medskip
Let $f \in C^0$ and $\xi = \xi_{12}\ \partial_x \wedge \partial_y + \xi_{13}\ \partial_x \wedge \partial_z + \xi_{23}\ \partial_y \wedge \partial_z \in C^2$, with $\xi_{ij} \in \numberset{K}[x,y,z]$.
\begin{equation} \label{cochain02}
[f,\xi]_s = \xi_{12} \left(\partial_y f \partial_x - \partial_x f \partial_y\right) + \xi_{13} \left(\partial_z f \partial_x - \partial_x f \partial_z\right) + \xi_{23} \left(\partial_z f \partial_y - \partial_y f \partial_z\right)
\end{equation}
\medskip
Let $f \in C^0$ and $\xi = \xi_{123}\ \partial_x \wedge \partial_y \wedge \partial_z \in C^3$, with $\xi_{123} \in \numberset{K}[x,y,z]$.
\begin{equation} \label{cochain03}
[f,\xi]_s = \xi_{123} \left(- \partial_z f\ \partial_x \wedge \partial_y + \partial_y f\ \partial_x \wedge \partial_z - \partial_x f\ \partial_y \wedge \partial_z\right)
\end{equation}
\medskip
Let $X,Y \in C^1$, with $X = X_1 \partial_x + X_2 \partial_y + X_3 \partial_z$ and $Y = Y_1 \partial_x + Y_2 \partial_y + Y_3 \partial_z$
\begin{align} \label{cochain11}
[X,Y]_s ={}& \quad\!\! \left(X_1 \partial_x Y_1 - Y_1 \partial_x X_1 + X_2 \partial_y Y_1 - Y_2 \partial_y X_1 + X_3 \partial_z Y_1 - Y_3 \partial_z X_1\right) \partial_x \notag \\
&+ \left(X_1 \partial_x Y_2 - Y_1 \partial_x X_2 + X_2 \partial_y Y_2 - Y_2 \partial_y X_2 + X_3 \partial_z Y_2 - Y_3 \partial_z X_2\right) \partial_y \\
&+ \left(X_1 \partial_x Y_3 - Y_1 \partial_x X_3 + X_2 \partial_y Y_3 - Y_2 \partial_y X_3 + X_3 \partial_z Y_3 - Y_3 \partial_z X_3\right) \partial_y \notag
\end{align}
\medskip
Let $X = X_1 \partial_x + X_2 \partial_y + X_3 \partial_z \in C^1$ and $\xi = \xi_{12}\ \partial_x \wedge \partial_y + \xi_{13}\ \partial_x \wedge \partial_z + \xi_{23}\ \partial_y \wedge \partial_z \in C^2$.
\begin{adjustwidth}{-1cm}{-1cm} \vspace*{-1em}
\begin{align} \label{cochain12}
[X,\xi]_s ={}& \quad\!\! \left(X_1 \partial_x \xi_{12} - \xi_{12} \partial_x X_1 + X_2 \partial_y \xi_{12} - \xi_{12} \partial_y X_2 + X_3 \partial_z \xi_{12} - \xi_{13} \partial_z X_2 + \xi_{23} \partial_z X_1\right) \partial_x \wedge \partial_y \notag \\
&+ \left(X_1 \partial_x \xi_{13} - \xi_{13} \partial_x X_1 + X_2 \partial_y \xi_{13} - \xi_{12} \partial_y X_3 + X_3 \partial_z \xi_{13} - \xi_{13} \partial_z X_3 - \xi_{23} \partial_y X_1\right) \partial_x \wedge \partial_z \\
&+ \left(X_1 \partial_x \xi_{23} - \xi_{13} \partial_x X_2 + X_2 \partial_y \xi_{23} - \xi_{23} \partial_y X_2 + X_3 \partial_z \xi_{23} - \xi_{23} \partial_z X_3 + \xi_{12} \partial_x X_3\right) \partial_y \wedge \partial_z \notag
\end{align}
\end{adjustwidth}
\medskip
Let $X = X_1 \partial_x + X_2 \partial_y + X_3 \partial_z \in C^1$ and $\xi = \xi_{123}\ \partial_x \wedge \partial_y \wedge \partial_z \in C^3$.
\begin{equation} \label{cochain13}
[X,\xi]_s = \big(X_1 \partial_x \xi_{123} + X_2 \partial_y \xi_{123} + X_3 \partial_z \xi_{123} - \xi_{123} \left(\partial_x X_1 + \partial_y X_2 + \partial_z X_3\right) \big) \partial_x \wedge \partial_y \wedge \partial_z
\end{equation}
\medskip
Let $\xi,\eta \in C^2$, with $\xi = \xi_{12}\ \partial_x \wedge \partial_y + \xi_{13}\ \partial_x \wedge \partial_z + \xi_{23}\ \partial_y \wedge \partial_z$ and $\eta = \eta_{12}\ \partial_x \wedge \partial_y + \eta_{13}\ \partial_x \wedge \partial_z + \eta_{23}\ \partial_y \wedge \partial_z$.
\begin{equation} \label{cochain22}
\begin{split}
[\xi,\eta]_s ={}& \bigg( \xi_{12} \partial_x \eta_{13} - \eta_{13} \partial_x \xi_{12} + \xi_{12} \partial_y \eta_{23} - \eta_{23} \partial_y \xi_{12} + \eta_{12} \partial_x \xi_{13} - \xi_{13} \partial_x \eta_{12} \\
\quad & + \xi_{13} \partial_z \eta_{23} - \eta_{23} \partial_z \xi_{13} + \eta_{12} \partial_y \xi_{23} - \xi_{23} \partial_y \eta_{12} + \eta_{13} \partial_z \xi_{23} - \xi_{23} \partial_z \eta_{13} \bigg) \partial_x \wedge \partial_y \wedge \partial_z
\end{split}
\end{equation}
Reasoning on degrees, we already had that $[\xi,\eta]_s = 0$ for $\xi \in C^2$ and $\eta \in C^3$ and it is also the case for $\xi \in C^3$ and $\eta \in C^3$.
\section{Schouten brackets on cohomology} \label{Schouten-cohomology}
We will now compute the various Schouten brackets $[C^i,C^j]'_s$ on cohomology, for $0 \leqslant i \leqslant j \leqslant 3$. Since we use cocycle representatives, we will actually do the computations with the Schouten bracket $[~,~]_s$ on cochain level by taking care of coboundaries. When writing elements of the cohomology, we will abusively omit the class notation to lighten the expressions. The equations on cochain level with the coboundaries will be useful for expressing formality equations in \autoref{Sec:formality-eq}. \\
Let $\varphi,\psi \in H^0 = \ChE[0](\mathfrak{g},\Sym\mathfrak{g}) = \numberset{K}[z]$. Like for cochains,
\begin{equation} \label{cohomology00}
[\varphi,\psi]'_s = 0.
\end{equation}
\medskip
Let $\varphi \in H^0$ and $X_{g_0} + \psi D_a \in H^1$, with $\psi \in \numberset{K}[z]$ and $g_0 \in \numberset{K}[x,y]$.
\begin{equation} \label{cohomology01}
[\varphi,X_{g_0} + \psi D_a]_s = [\varphi,z \psi \partial_z]_s = - z \psi \partial_z \varphi
\end{equation}
\medskip
Let $\varphi \in H^0$ and $X_g \wedge \partial_z \in H^2$, with $g \in \numberset{K}[x,y,z]$.
\begin{equation} \label{cohomology02}
[\varphi,X_g \wedge \partial_z]_s = X_{g \varphi'_z} = X_{(g \varphi'_z)_0} + \delta_{CE}^0\left((g \varphi'_z)_1\right)
\end{equation}
Here, $\varphi'_z$ is the derivative of $\varphi$ with respect to $z$, and $g \varphi'_z = (g \varphi'_z)_0 + z (g \varphi'_z)_1$ are the constant term (in $z$) and remaining $z$-part. Hence on the cohomology, we have
\begin{equation}
[\varphi,X_g \wedge \partial_z]'_s = X_{(g \varphi'_z)_0} = X_{g_0 \varphi'_z(0)}.
\end{equation}
\medskip
Let $\varphi \in H^0$ and $P\ \partial_x \wedge \partial_y \wedge \partial_z \in H^3$, with $P \in \numberset{K}[x,y]$.
\begin{equation} \label{cohomology03}
\begin{split}
[\varphi,P\ \partial_x \wedge \partial_y \wedge \partial_z]_s ={} & - P \varphi'_z \partial_x \wedge \partial_y \\
={} & \delta_{CE}^1(P \varphi'_z \partial_z) + X_{P z \varphi'_z} \wedge \partial_z
\end{split}
\end{equation}
Hence on the cohomology,
\begin{equation}
[\varphi,P\ \partial_x \wedge \partial_y \wedge \partial_z]'_s = X_{P z \varphi'_z} \wedge \partial_z.
\end{equation}
\medskip
Let $X_{g_0} + \varphi D_a,\ X_{h_0} + \psi D_a \in H^1$, with $g_0,h_0 \in \numberset{K}[x,y]$ and $\varphi,\psi \in \numberset{K}[z]$.
\begin{equation} \label{cohomology11}
\begin{split}
[X_{g_0} &+ \varphi D_a, X_{h_0} + \psi D_a]_s \\ ={}& X_{\{g_0,h_0\} + \varphi_0 (D_a(h_0) - h_0) - \psi_0 (D_a(g_0) - g_0)} + z(\varphi \psi'_z - \varphi'_z \psi) D_a \\
&+ \delta_{CE}^0\big(\psi'_z (D_a(g_0) - g_0) - \varphi'_z (D_a(h_0) - h_0)\big)
\end{split}
\end{equation}
The Poisson bracket $\{~,~\}$ is the one for functions of two variables $x$ and $y$: we note $\{g,h\} = \partial_x g \partial_y h - \partial_x h \partial_y g$; and we have $[X_g,X_h] = X_{\{g,h\}}$.
On cohomology we get
\begin{equation}
\begin{split}
[X_{g_0} &+ \varphi D_a, X_{h_0} + \psi D_a]'_s \\ ={}& X_{\{g_0,h_0\} + \varphi_0 (D_a(h_0) - h_0) - \psi_0 (D_a(g_0) - g_0)} + z(\varphi \psi'_z - \varphi'_z \psi) D_a \\
\eqqcolon & X_{g_0,\varphi,h_0,\psi} + z(\varphi \psi'_z - \varphi'_z \psi) D_a,
\end{split}
\end{equation}
with the notation $(g_0,\varphi,h_0,\psi)$ standing for the long function in subscript.
\medskip
Let $X_{g_0} + \varphi D_a \in H^1$ and $X_p \wedge \partial_z \in H^2$, with $g_0 \in \numberset{K}[x,y]$, $\varphi \in \numberset{K}[z]$ and $p \in \numberset{K}[x,y,z]$.
\begin{equation} \label{cohomology12}
\begin{split}
[X_{g_0} + \varphi D_a,X_p \wedge \partial_z]_s ={}& X_{\{g_0,p\} + \varphi (D_a(p) - p) - p (\varphi'_z + \varphi) - z \varphi'_z (D_a(p) - z \partial_z p)} \wedge \partial_z \\
& + \delta_{CE}^1\big(- \varphi'_z (D_a(p) - z \partial_z p) \partial_z\big)
\end{split}
\end{equation}
So we get
\begin{equation}
\begin{split}
[X_{g_0} + \varphi D_a,X_p \wedge \partial_z]_s ={}& X_{\{g_0,p\} + \varphi (D_a(p) - p) - p (\varphi'_z + \varphi) - z \varphi'_z (D_a(p) - z \partial_z p)} \wedge \partial_z \\
\eqqcolon & X_{g_0,\varphi,p} \wedge \partial_z
\end{split}
\end{equation}
with the notation $(g_0,\varphi,p)$ standing for the long function in subscript. (It cannot be confused with the preceding notation since it has only three arguments.)
\medskip
Let $X_{g_0} + \varphi D_a \in H^1$ and $P\ \partial_x \wedge \partial_y \wedge \partial_z \in H^3$, with $g_0,P \in \numberset{K}[x,y]$ and $\varphi \in \numberset{K}[z]$.
\begin{equation} \label{cohomology13}
\begin{split}
[X_{g_0} + \varphi D_a,P\ \partial_x \wedge \partial_y \wedge \partial_z]_s ={}& \left(X_{g_0}(P) + \varphi (D_a(P) - 2P) - P z \varphi'_z\right) \partial_x \wedge \partial_y \wedge \partial_z \\
={}& \left(X_{g_0}(P) + \varphi_0 (D_a(P) - 2P)\right) \partial_x \wedge \partial_y \wedge \partial_z \\
& + \delta_{CE}^2(\phi),
\end{split}
\end{equation}
with $\phi = \phi_{13}\ \partial_x \wedge \partial_z + \phi_{23} \partial_y \wedge \partial_z$ such that $\partial_x \phi_{13} + \partial_y \phi_{23} = \varphi_1 (D_a(P) - 2P) - P \varphi'_z$.
So we have on the cohomology
\begin{equation}[X_{g_0} + \varphi D_a,P\ \partial_x \wedge \partial_y \wedge \partial_z]'_s = \left(X_{g_0}(P) + \varphi_0 (D_a(P) - 2P)\right) \partial_x \wedge \partial_y \wedge \partial_z.
\end{equation}
\medskip
Let $X_f \wedge \partial_z,\ X_g \wedge \partial_z \in H^2$, with $f,g \in \numberset{K}[x,y,z]$. Writing
\(
\left\{
\begin{aligned}
f(x,y,z) &= f_0(x,y) + z f_1(x,y,z) \\
g(x,y,z) &= g_0(x,y) + z g_1(x,y,z)
\end{aligned}
\right.
\) we get
\begin{equation} \label{cohomology22}
[X_f \wedge \partial_z,X_g \wedge \partial_z]_s = [X_f \wedge \partial_z,X_g \wedge \partial_z]'_s = \big(\{f_0,g_1\} + \{g_0,f_1\}\big)\partial_x \wedge \partial_y \wedge \partial_z.
\end{equation}
For completeness, we write the last two cases were the brackets are trivial. Let $X_f \wedge \partial_z \in H^2$ and $P\ \partial_x \wedge \partial_y \wedge \partial_z,\ Q\ \partial_x \wedge \partial_y \wedge \partial_z \in H^3$, with $f \in \numberset{K}[x,y,z]$ and $P,Q \in \numberset{K}[x,y]$.
\begin{align}\label{cohomology233}
[X_f \wedge \partial_z,P\ \partial_x \wedge \partial_y \wedge \partial_z]'_s ={}& 0 \\
[P\ \partial_x \wedge \partial_y \wedge \partial_z,Q\ \partial_x \wedge \partial_y \wedge \partial_z]'_s ={}& 0
\end{align}
\section{Formality equations} \label{Sec:formality-eq}
In this section, we will explicit the formality equations \eqref{eqs_formality_shifted_twice} for the first orders. In particular, we will obtain non-trivial terms $d_k$ of arity $k=3$ and higher showing that the Chevalley-Eilenberg complex of the Heisenberg algebra $\mathfrak{h}_3$ is not formal in Kontsevich sense, \textit{i.e.} \/ the $L_\infty$ structure on the cohomology is not induced by the Schouten bracket $d_2 = [~,~]'_s[1]$ only.
Note that in equations \eqref{eqs_formality_shifted_twice}, we work with elements $y_i \in H[2]$ so the degrees are shifted again. \\
The maps $\varphi_k : \Sym^k(H[2]) \to C[2]$ are of degree zero, so for $i_1,\dotsc,i_n \in \text{\textlbrackdbl}-2,1\text{\textrbrackdbl}$
\begin{equation*}
\varphi_k(H[2]^{i_1} \bullet \dotsb \bullet H[2]^{i_n}) \subset C[2]^{i_1+\dotsb+i_n}.
\end{equation*}
Sometimes, we will consider the maps $\varphi_k : H[2]^{\otimes k} \to C[2]$ having source in the tensorial power instead of the symmetric algebra of the space, and write $\varphi_k(H[2]^{i_1},\dotsc,H[2]^{i_n})$ with periods to separate arguments.
Since the cohomology is non-trivial in degrees $0,1,2,3$, we can always choose $k$ elements in $H[2]$ of various degrees such that the image by the map $\varphi_k$ is in the non-trivial part of $C[2]$, it suffice to choose $k$ numbers in $\text{\textlbrackdbl}-2,1\text{\textrbrackdbl}$ such that their sum is again in $\text{\textlbrackdbl}-2,1\text{\textrbrackdbl}$. The situation here is more complicated that for the Lie algebra $\mathfrak{so}(3)$ (see \cite{El12}), since in this case we have terms $\varphi_k$, and hence terms $d_k$ of all arities greater than two.
\addtocounter{subsection}{-1}
\subsection{\texorpdfstring{Order $k=0$}{Order k=0}}
For $k=0$ and $y_1 \in H[2]$, we have
\begin{equation}
0 = \delta_{CE}(\varphi_1(y_1))
\end{equation}
which express that $\varphi_1$ is a section from the cohomology into the cocycles. We choose $\varphi_1 = \operatorname{inc}$, the inclusion given by the choice of representatives of classes given in \autoref{Sec:Heisenberg-cohomology}.
\subsection{\texorpdfstring{Order $k=1$}{Order k=1}}
For $k=1$ and $y_1 \bullet y_2 \in \Sym^2(H[2])$, we have
\begin{equation} \label{formality-eq-k1}
\varphi_1(d_2(y_1 \bullet y_2)) = \delta_{CE} \varphi_2(y_1 \bullet y_2) + D(\varphi_1(y_1) \bullet \varphi_1(y_2)).
\end{equation}
This equation shows that the inclusion $\varphi_1$ is not a morphism of the graded Lie algebras $(H[2],d_2) \to (C[2],D)$, since there is the coboundary $\delta_{CE} \varphi_2(y_1 \bullet y_2)$. We will explicit the term $\varphi_2$ and higher terms to obtain an $L_\infty$-morphism between these two $L_\infty$-algebras.
To simplify the expressions, we will omit the inclusion $\varphi_1$ in the remaining computations. The preceding equation rewrites
\begin{equation}
d_2(y_1 \bullet y_2) = \delta_{CE} \varphi_2(y_1 \bullet y_2) + D(y_1 \bullet y_2).
\end{equation} \\
Let $y_1=\varphi,y_2=\psi \in H^0 = \numberset{K}[z]$. We have
\(
d_2(\varphi \bullet \psi) = (-1)^{-2} [\varphi,\psi]'_s = 0
\)
and also
\(
D(\varphi \bullet \psi) = (-1)^{-2} [\varphi,\psi]'_s = 0
\)
so $\delta_{CE} \varphi_2(\varphi,\psi) = 0$ and we can set
\begin{equation}
\varphi_2(\varphi,\psi) = 0.
\end{equation}
\medskip
Let $y_1 = \varphi \in H^0$ and $y_2 = X_{g_0} + \psi D_a \in H^1$ with $\varphi,\psi \in \numberset{K}[z]$ and $g_0 \in \numberset{K}[x,y]$. As before
\(
[\varphi,X_{g_0} + \psi D_a]_s = [\varphi,X_{g_0} + \psi D_a]'_s = - z \psi \varphi'_z,
\)
and we set again
\begin{equation}
\varphi_2(\varphi,X_{g_0} + \psi D_a) = 0.
\end{equation}
\medskip
Let $y_1 = \varphi \in H^0$ and $y_2 = X_g \wedge \partial_z \in H^2$, with $\varphi \in \numberset{K}[z]$ and $g \in \numberset{K}[x,y,z]$. Equation \eqref{cohomology02} rewrites
\(
D(\varphi \bullet X_g \wedge \partial_z) = d_2(\varphi \bullet X_g \wedge \partial_z) + \delta_{CE}^0((g\varphi'_z)_1),
\)
so to obtain the formality equation \eqref{formality-eq-k1} in this case, we can set
\begin{equation}
\varphi_2(\varphi,X_g \wedge \partial_z) = - (g\varphi'_z)_1.
\end{equation}
\medskip
Let $y_1 = \varphi \in H^0$ and $y_3 = P \partial_x \wedge \partial_y \wedge \partial_z \in H^3$, with $\varphi \in \numberset{K}[z]$ and $P \in \numberset{K}[x,y]$. Equation \eqref{cohomology03} rewrites
\(
D(\varphi \bullet P \partial_x \wedge \partial_y \wedge \partial_z) = d_2(\varphi \bullet P \partial_x \wedge \partial_y \wedge \partial_z) + \delta_{CE}^1(P \varphi'_z \partial_z),
\)
so we set
\begin{equation}
\varphi_2(\varphi,P \partial_x \wedge \partial_y \wedge \partial_z) = - P \varphi'_z \partial_z.
\end{equation}
\medskip
Let $y_1 = X_{g_0} + \varphi D_a,\ y_2 = X_{h_0} + \psi D_a \in H^1$, with $g_0,h_0 \in \numberset{K}[x,y]$ and $\varphi,\psi \in \numberset{K}[z]$. As in the previous cases, equation \eqref{cohomology11} gives
\begin{equation}
\varphi_2(X_{g_0} + \varphi D_a,X_{h_0} + \psi D_a) = \psi'_z (D_a(g_0) - g_0) - \varphi'_z (D_a(h_0) - h_0).
\end{equation}
\medskip
For $y_1 = X_{g_0} + \varphi D_a \in H^1$ and $y_2 = X_p \wedge \partial_z \in H^2$, with $\varphi \in \numberset{K}[z]$ and $g_0,p \in \numberset{K}[x,y]$, equation \eqref{cohomology12} gives
\begin{equation}
\varphi_2(X_{g_0} + \varphi D_a,X_p \wedge \partial_z) = - \varphi'_z (D_a(p) - z \partial_z p) \partial_z
\end{equation}
\medskip
For $y_1 = X_{g_0} + \varphi D_a \in H^1$ and $y_2 = P\ \partial_x \wedge \partial_y \wedge \partial_z \in H^3$, with $\varphi \in \numberset{K}[z]$ and $g_0,P \in \numberset{K}[x,y]$, equation \eqref{cohomology13} gives
\begin{equation}
\varphi_2(X_{g_0} + \varphi D_a,P\ \partial_x \wedge \partial_y \wedge \partial_z) = \phi_{13}\ \partial_x \wedge \partial_z + \phi_{23} \partial_y \wedge \partial_z
\end{equation}
such that $\partial_x \phi_{13} + \partial_y \phi_{23} = \varphi_1 (D_a(P) - 2P) - P \varphi'_z$.
\medskip
Let $X_f \wedge \partial_z,\ X_g \wedge \partial_z \in H^2$, with $f,g \in \numberset{K}[x,y,z]$, and $P\ \partial_x \wedge \partial_y \wedge \partial_z,\ Q\ \partial_x \wedge \partial_y \wedge \partial_z \in H^3$, with $P,Q \in \numberset{K}[x,y]$. Since in equation \eqref{cohomology22} the brackets are the same on cochain level and cohomology, we have
\begin{equation}
\varphi_2(X_f \wedge \partial_z,X_g \wedge \partial_z) = 0,
\end{equation}
and the last two brackets are trivial in both levels, so
\begin{align}
\varphi_2(X_f \wedge \partial_z,P\ \partial_x \wedge \partial_y \wedge \partial_z) = 0 \\
\varphi_2(P\ \partial_x \wedge \partial_y \wedge \partial_z,Q\ \partial_x \wedge \partial_y \wedge \partial_z) = 0
\end{align}
\subsection{\texorpdfstring{Order $k=2$}{Order k=2}}
For $k=2$ and $y_1 \bullet y_2 \bullet y_3 \in \Sym^3(H[2])$, we have
\begin{equation} \label{formality-eq-k2}
\begin{split}
d_3(& y_1 \bullet y_2 \bullet y_3) \\
& + \varphi_2(d_2(y_1 \bullet y_2) \bullet y_3) + (-1)^{|y_2| |y_3|} \varphi_2(d_2(y_1 \bullet y_3) \bullet y_2) + (-1)^{|y_1|(|y_2|+|y_3|)} \varphi_2(d_2(y_2 \bullet y_3) \bullet y_1) \\
={} & \delta_{CE} \varphi_3(y_1 \bullet y_2 \bullet y_3) \\
& + D(y_1 \bullet \varphi_2(y_2 \bullet y_3)) + (-1)^{|y_1| |y_2|} D(y_2 \bullet \varphi_2(y_1 \bullet y_3)) + (-1)^{|y_3|(|y_1|+|y_2|)} D(y_3 \bullet \varphi_2(y_1 \bullet y_2)).
\end{split}
\end{equation}
Recall that the maps $\delta_{CE}$, $D$ and $d_k$ are of degree~$1$ and the maps $\varphi_k$ are of degree~$0$. Thus, the left-hand side and the right-hand side of the equation are of degree one more than the sum of the degrees of the three elements $y_1,y_2,y_3$.
We will keep the same notation as before for elements of various degrees in $H[2]$, without specifying further the types of the functions used (of $x$ and $y$, of $z$ only, of $x,y,z$).
Although we work with double-shifted spaces, we keep the original degree for the Chevalley-Eilenberg coboundary operator, so
\[
\delta_{CE}^k : C^k = \ChEcochains[k](\mathfrak{g},\Sym\mathfrak{g}) \to C^{k+1} = \ChEcochains[k+1](\mathfrak{g},\Sym\mathfrak{g}).
\]
By graded symmetry, it will be enough to study the cases $-2 \leqslant i \leqslant j \leqslant k \leqslant 1$, where $i,j,k$ are the degrees of $y_1,y_2,y_3 \in H[2]$. We note $\omega \coloneqq \partial_x \wedge \partial_y \wedge \partial_z$. \\
Let $y_1 = \varphi, y_2 = \psi \in H^0$ and $P \omega \in H^3$. The equation \eqref{formality-eq-k2} rewrites
\begin{equation*}
d_3(\varphi \bullet \psi \bullet P \omega) - 2(Pz\varphi'_z\psi'_z)_1 = \delta_{CE} \varphi_3(\varphi \bullet \psi \bullet P \omega) + 2P\varphi'_z\psi'_z
\end{equation*}
The elements of this equation are of degree $|\varphi|+|\psi|+|P \omega| + 1 = -2-2+1+1 = -2$ and since $\delta_{CE}$ is of degree $+1$, the element $\varphi_3(\varphi \bullet \psi \bullet P \omega) \in C[2]^{-3} = C^{-1}$ has image in $C^0$, so here $\delta_{CE} = \delta_{CE}^{-1} : C^{-1} \to C^0$. Of course, $C^{-1} = \{0\}$, so we get
\begin{equation*}
\varphi_3(\varphi \bullet \psi \bullet P \omega) = 0 \quad \text{and} \quad d_3(\varphi \bullet \psi \bullet P \omega) = 4P\varphi'_z\psi'_z.
\end{equation*}
Even if we had chosen other expressions for $\varphi_2(\varphi,P \omega)$ and $\varphi_2(\varphi,X_g \wedge \partial_z)$ adding cocycles, there is no possibility to cancel the constant part in $z$ coming from the function $P$. Therefore, this case alone shows that the Chevalley-Eilenberg complex of the Heisenberg algebra is not formal. \\
Let $y_1 = \varphi \in H^0$, $y_2 = X_{g_0} + \psi D_a \in H^1$ and $y_3 = X_h \wedge \partial_z \in H^2$. The equation \eqref{formality-eq-k2} rewrites
\begin{gather*}
\begin{aligned}
d_3(& \varphi \bullet X_{g_0} + \psi D_a \bullet X_h \wedge \partial_z) + \psi'_z \big(D_a(h_0 \varphi'_z(0))-h_0 \varphi'_z(0)\big) + \big((g_0,\psi,h)\varphi'_z\big)_1 \\
={}& \delta_{CE}^{-1}(\varphi \bullet X_{g_0} + \psi D_a \bullet X_h \wedge \partial_z) - \psi'_z \varphi'_z(z\partial_z h-D_a(h))-z \psi \partial_z (h \psi'_z)_1
\end{aligned} \\[1ex]
\begin{aligned}
\!\!\Leftrightarrow d_3(\varphi \bullet X_{g_0} + \psi D_a \bullet X_h \wedge \partial_z) ={}& \psi'_z \big(h_0 \varphi'_z(0) - D_a(h_0 \varphi'_z(0)) - \varphi'_z(z\partial_z h-D_a(h))\big) \\
& - \big((g_0,\psi,h)\varphi'_z\big)_1 - z \psi \partial_z (h \psi'_z)_1.
\end{aligned}
\end{gather*}
Let $y_1 = \varphi \in H^0$, $y_2 = X_{g_0} + \psi D_a \in H^1$ and $y_3 = P \omega \in H^3$. The equation \eqref{formality-eq-k2} rewrites
\begin{equation*}
\begin{split}
d_3(&\varphi \bullet X_{g_0} + \psi D_a \bullet P \omega) \\ ={}& \delta_{CE}^0 \varphi_3(\varphi \bullet X_{g_0} + \psi D_a \bullet P \omega) - \varphi'_z (P \psi'_z a x + \phi_{13}) \partial_x - \varphi'_z (P \psi'_z (1-a) y + \phi_{23}) \partial_y \\
& + \big( X_{g_0}(P)(1-\varphi'_z) + \left(\psi - \psi_0 \varphi'_z - \psi'_z \varphi'_z z\right) D_a(P) - 2 P \varphi'_z(\psi + \psi_0) - \psi'_z \varphi'_z z P \big) \partial_z,
\end{split}
\end{equation*}
where $\phi_{13},\phi_{23}$ come from
\begin{equation*}
\varphi_2(X_{g_0} + \psi D_a,P\ \partial_x \wedge \partial_y \wedge \partial_z) = \phi_{13}\ \partial_x \wedge \partial_z + \phi_{23} \partial_y \wedge \partial_z
\end{equation*}
with $\partial_x \phi_{13} + \partial_y \phi_{23} = \psi_1 (D_a(P) - 2P) - P \psi'_z$.
We can choose the value of $\varphi_3(\varphi \bullet X_{g_0} + \psi D_a \bullet P \omega)$ such that its image by $\delta_{CE}^0$ cancels the coboundary $- \varphi'_z (P \psi'_z a x + \phi_{13}) \partial_x - \varphi'_z (P \psi'_z (1-a) y + \phi_{23}) \partial_y$ and the remaining term in $\partial_z$ is then the value of $d_3(\varphi \bullet X_{g_0} + \psi D_a \bullet P \omega)$. \\
Let $y_1 = \varphi \in H^0$, $y_2 = X_f \wedge \partial_z,\ y_3 = X_g \wedge \partial_z \in H^2$. The equation \eqref{formality-eq-k2} rewrites
\begin{equation*}
d_3(\varphi \bullet X_f \wedge \partial_z \bullet X_g \wedge \partial_z) = \delta_{CE}^0 \varphi_3(\varphi \bullet X_f \wedge \partial_z \bullet X_g \wedge \partial_z) + \varphi'_z \big(\{f_0,g_1\}+\{g_0,f_1\}\big) \partial_z.
\end{equation*}
The term in $\partial_z$ is not a coboundary, so we take it as the value of the $d_3$ term and set the value of the $\varphi_3$ term to zero. \\
Let $y_1 = \varphi \in H^0$, $y_2 = X_g \wedge \partial_z \in H^2$ and $y_3 = P \omega \in H^3$. The equation \eqref{formality-eq-k2} rewrites
\begin{equation*}
\begin{split}
d_3(& \varphi \bullet X_g \wedge \partial_z \bullet P\omega) \\
={}& \delta_{CE}^1 \varphi_3(\varphi \bullet X_g \wedge \partial_z \bullet P\omega) + (P \varphi'_z X_{\partial_z g} - P \varphi''_z X_g - \phi_{13} \partial_x - \phi_{23} \partial_y) \wedge \partial_z \\
& - P \partial_z (g\varphi'_z)_1 \partial_x \wedge \partial_y + P \partial_y (g\varphi'_z)_1 \partial_x \wedge \partial_z - P \partial_x (g\varphi'_z)_1 \partial_y \wedge \partial_z,
\end{split}
\end{equation*}
with $\phi_{13},\phi_{23}$ coming from $\varphi_2(X_{g_0 \varphi'_z(0)},P\omega)$. Here the term in $\partial_x \wedge \partial_y$ is a coboundary and can be cancelled, the remaining expression is the value of $d_3$. \\
Let $y_1 = \varphi \in H^0$, $y_2 = P \omega,\ y_3 = Q \omega \in H^3$. The equation \eqref{formality-eq-k2} gives
\begin{equation*}
d_3(\varphi \bullet P \omega \bullet Q \omega) = \delta_{CE}^2 \varphi_3(\varphi \bullet P \omega \bullet Q \omega) - 2 PQ\varphi''_z \omega
\end{equation*}
We can cancel the coboundary $(- 2 PQ\varphi''_z)_1 \omega$ and it remains $d_3(\varphi \bullet P \omega \bullet Q \omega) = - 2 PQ\varphi''_z(0) \omega$. \\
Let $y_1 = X_{f_0} + \varphi D_a,\ y_2 = X_{g_0} + \psi D_a,\ y_3 = X_{h_0} + \xi D_a \in H^1$. The equation \eqref{formality-eq-k2} gives
\begin{equation*}
\begin{split}
d_3(& X_{f_0} + \varphi D_a,\bullet X_{g_0} + \psi D_a, \bullet X_{h_0} + \xi D_a) = - \xi'_z \big(D_a((f_0,\varphi,g_0,\psi)) - (f_0,\varphi,g_0,\psi)\big) \\
& + \psi'_z \big(D_a((f_0,\varphi,h_0,\xi)) - (f_0,\varphi,h_0,\xi)\big) - \varphi'_z \big(D_a((g_0,\psi,h_0,\xi)) - (g_0,\psi,h_0,\xi)\big) \\
& + (\varphi \psi'_z - \varphi'_z \psi + 2z (\varphi \psi''_z - \varphi''_z \psi))(D_a(h_0)-h_0) - (\varphi \xi'_z - \varphi'_z \xi - 2z (\varphi \xi''_z - \varphi''_z \xi))(D_a(g_0)-g_0) \\
& + (\psi \xi'_z - \psi'_z \xi + 2z (\psi \xi''_z - \psi''_z \xi))(D_a(f_0)-f_0). \\
\end{split}
\end{equation*}
The other cases also give very long expressions when plugged in equation \eqref{formality-eq-k2}.
\begin{gather*}
y_1 = X_{f_0} + \varphi D_a,\quad y_2 = X_{g_0} + \psi D_a \in H^1 \qquad y_3 = X_h \wedge \partial_z \in H^2 \\
y_1 = X_{f_0} + \varphi D_a,\quad y_2 = X_{g_0} + \psi D_a \in H^1 \qquad y_3 = P \omega \in H^3 \\
y_1 = X_{f_0} + \varphi D_a \in H^1,\qquad y_2 = X_g \wedge \partial_z,\quad y_3 = X_h \wedge \partial_z \in H^2 \\
y_1 = X_{f_0} + \varphi D_a \in H^1,\qquad y_2 = X_g \wedge \partial_z \in H^2,\qquad y_3 = P \omega \in H^3 \\
y_1 = X_f \wedge \partial_z,\quad y_2 = X_g \wedge \partial_z,\quad y_3 = X_h \wedge \partial_z \in H^2
\end{gather*}
Each time, the map $\varphi_3$ can be used to cancel as many term that is possible to obtain with some coboundary, but the remaining term for $d_3$ are still big expressions.
In the same way, we could obtain explicit values for higher maps $\varphi_k$ and $d_k$. Since there are several possibilities to choose the degrees of the elements to land in the non-trivial part of the complex $C[2]$, the $L_\infty$ structure obtained by solving inductively the formality equations has non-zero terms in each order.
\bibliographystyle{amsalpha}
\nocite{*}
|
1,941,325,220,513 | arxiv | \section{Introduction}
\subsection{Preliminaries: Conformal symmetry}
The techniques of conformal field theory (CFT) \cite{DiFrancesco:1997nk, Blumenhagen:2009zz} have been central to our understanding of various diverse subjects including the theory of phase transitions in statistical physics, the understanding of string theory from worldsheet symmetries, the study of quantum gravity in AdS (and dS) spacetimes through the celebrated gauge-gravity correspondence, as well as explorations in cosmology. These techniques hinge on the extra symmetries CFTs possess vis-a-vis usual relativistic quantum field theories (QFTs). Of particular importance is the procedure of conformal bootstrap \cite{Ferrara:1973yt, Polyakov:1974gs, Poland:2016chs, Simmons-Duffin:2016gjk}, which aims to solve CFTs by imposing an infinite set of consistency conditions on the theory that stem from crossing symmetry of the four-point function. This programme has seen a resurgence over the last decade, following the seminal work \cite{Rattazzi:2008pe}.
\medskip
\noindent
In general $d$ dimensions, the symmetry algebra that underlies a CFT is $so(d,2)$ as opposed to $iso(d-1,1)$ of a Poincare-invariant QFT in the same dimension. In two dimensional CFTs, the underlying symmetry enhances to two copies of the infinite dimensional Virasoro algebra. Following the lead of Belavin, Polyakov and Zamolodchikov \cite{Belavin:1984vu}, over the years, 2d CFTs have been the most fertile of all theories for performing analytic calculations. The power of infinite symmetry coupled with the might of the machinery of complex analysis has helped with an enormous amount of analytic control on calculations in 2d CFTs in general, without having to refer to any particular Lagrangian description. Before the current resurgence mentioned above, conformal bootstrap techniques were used in 2d to exactly solve a class of CFTs, called the 2d Minimal models.
\medskip
\noindent
In addition to consistency from crossing symmetry of 4-point function, in 2d CFTs there are further constraints that arise from modular invariance on the torus. These help in constraining the possible parameter space in 2d CFTs even further. Modular invariance has the additional very useful feature of relating the low energy spectrum of the theory to the high energy spectrum. This leads to the famous Cardy formula for entropy counting in 2d CFTs \cite{Cardy:1986ie}, which spectacularly matches up with the Bekenstein-Hawking entropy of Banados-Teitelboim-Zanelli black holes in AdS$_3$, and serves as one of the very early successes of the holographic principle \cite{Strominger:1997eq,Carlip:1998qw}.
\subsection{Flat spacetimes, BMS and Holography}
\noindent
Holography for AdS$_3$ uses the power of the infinite dimensional Virasoro symmetries, which are also the asymptotic symmetries of the bulk spacetime \cite{Brown:1986nw}. However, infinite asymptotic symmetries are not exclusive to AdS$_3$. Bondi, van der Burg, Mezner and, independently, Sachs in the 1960's found rather surprisingly that the symmetries of 4d asymptotically flat spacetimes at null infinity ($\mathscr{I}^\pm$) enhances from the expected Poincare group to to an infinite dimensional group, named BMS after the discoverers \cite{Bondi:1962px, Sachs}. There has been a recent explosion in activity in research connecting the infrared physics of gauge theories and gravity to the BMS group \cite{Strominger:2017zoo}. The existence of an infinite dimensional asymptotic symmetry algebra is also true in 3d asymptotically flat spacetimes and the symmetry algebra in question is the BMS$_3$ \cite{Barnich:2006av}:
\begin{subequations}{}\label{bms3}
\begin{eqnarray}{}
&& [L_n, L_m] = (n-m) L_{n+m} + c_L (n^3-n) \delta_{n+m, 0} \\
&& [L_n, M_m] = (n-m) M_{n+m} + c_M (n^3-n) \delta_{n+m, 0} \\
&& [M_n, M_m] = 0.
\end{eqnarray}
\end{subequations}
Here $L_n$'s are the so-called super-rotations, which are the diffeomorphisms of the circle at infinity. $M_n$'s are called super-translations and these are angle dependent translations along the null directions of $\mathscr{I}^+$. $c_L, c_M$ are two allowed central extensions of the theory. For Einstein gravity, $c_L = 0, c_M= 1/4G$, where $G$ is the Newton's constant. One can generate other values of the central term by considering more general theories of gravity e.g. by adding gravitational Chern-Simons term to the Einstein-Hilbert action \cite{Bagchi:2012yk}.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.5]{penrose_flat_space}
\end{center}
\caption{Penrose diagram for three-dimensional Minkowski spacetime. }
\end{figure}
\medskip
\noindent It is natural, following lessons of AdS$_3$, to attempt constructions of holography in 3d asymptotically flat spacetimes, using the algebra \refb{bms3}. Specifically, the putative dual 2d field theory should be one which lives on the null boundary, say $\mathscr{I}^+$, and be invariant under the symmetry algebra \refb{bms3} \cite{Bagchi:2010zz, Bagchi:2012cy}. As we will go on to describe in the next section, the BMS$_3$ algebra can be obtained by an In{\"o}n{\"u}-Wigner contraction of two copies of the Virasoro algebra. This contraction which is the infinite radius limit of AdS, manifests itself on the boundary theory as an ultra-relativistic or a Carrollian limit on the parent 2d CFT, where the speed of light goes to zero. This rather strange limit results in the degeneration of the background (flat) metric that the field theory lives on and the Riemannian structures are replaced by Carrollian structures. These Carrollian manifolds are generically fibre-bundles. In the case of $\mathscr{I}^+$, the structure is of course a direct product ${\rm I\!R}_u \times S^1$, where $u$ is the null direction. This is a flat Carrollian manifold. We would be interested in 2d conformal field theories defined on these manifolds, the so called conformal Carrollian theories. Conformal Carrollian symmetries have been shown to be isomorphic to BMS symmetries \cite{Duval:2014uva}.
\bigskip
\noindent In this paper, continuing with our earlier endeavours, we take forward our construction of the modular properties of BMS-invariant field theories (henceforth abbreviated as BMSFTs) \cite{Bagchi:2012xr, Bagchi:2013qva, Bagchi:2019unf}. As mentioned earlier, the BMS algebra can be arrived at from a limit of (two copies of) the Virasoro algebra. Hence it is conceivable that in this limit, some of the modular properties of 2d CFTs are also inherited by 2d BMSFTs. Following this line of argument, a BMS-Cardy formula \cite{Bagchi:2012xr} was proposed using these contracted modular properties{\footnote{See \cite{Barnich:2012xq} for an alternative derivation.}}. BTZ black holes are orbifolds of AdS$_3$. Similar orbifolds exist for 3d Minkowski spacetimes. The so-called shifted boost orbifold turns out to be the analogue of the non-extremal BTZ and is a cosmological solution with a cosmological horizon \cite{Cornalba:2003kd}. The thermodynamics of these cosmologies, also called Flat Space Cosmologies (FSCs) have been studied \cite{Bagchi:2012xr, Barnich:2012xq, Bagchi:2013lma, Detournay:2014fva} and in particular, the Bekenstein-Hawking entropy of these FSCs have been reproduced by the BMS-Cardy formula adding the significant feather to the cap of the programme of construction of 3d flat holography \cite{Bagchi:2012xr}. We will review modular properties and the BMS-Cardy formula in Sec.~2, and the FSC story briefly in Sec.~\refb{bulk}.
Some other interesting advances in the study of holography for three dimensional asymptotic flat spacetimes include the construction of the flat limit of Liouville theory as an explicit putative boundary theory \cite{Barnich:2012rz}, matching of correlation functions between bulk and boundary for stress-energy tensors \cite{Bagchi:2015wna} and generic fields \cite{Hijano:2017eii}, aspects of entanglement entropy \cite{Bagchi:2014iea, Basu:2015evh, Hijano:2017eii, Jiang:2017ecm, Grumiller:2019xna, Fareghbal:2019czx, Godet:2019wje, Apolo:2020bld, Apolo:2020qjm}, understanding holographic reconstruction \cite{Hartong:2015usd}, and the flat version of the fluid-gravity correspondence \cite{Ciambelli:2018wre}.
\subsection{Goal and summary of results of current work}
The conformal bootstrap programme has been instrumental in constraining the landscape of known CFTs, and thereby relativistic QFT as renormalisation group flows away from CFTs, and ultimately aims to chart out all know and consistent relativistic theories. The main idea of the bootstrap is simple: any CFT is completely fixed by the dimension of primary operators and the structure constants characterising the three-point functions of primary operators. In 2d CFTs, an arbitrary correlation function or the partition function on an arbitrary Riemann surface can be completely fixed by this data. In two dimensions, imposing crossing symmetry of four-point functions on the sphere and modular covariance of one-point function on the torus implies the imposition of crossing and modular invariance to higher points and higher genus \cite{Moore:1988uz,Sonoda:1988fq}. Hence investigation of properties of the torus one-point function is an important programme, which was initiated recently by Kraus and Maloney in \cite{Kraus:2016nwo}. This has interesting holographic ramifications as well. Like the Cardy formula, the asymptotic behaviour of the structure constants or three-point coefficients of the theory is again fixed by the low lying spectrum. From the dual bulk, the field theory structure constants match with the corresponding expectation values in the dual BTZ geometry \cite{Kraus:2016nwo}. See \cite{Alkalaev:2016ptm} for a different slightly different bulk interpretation.
\medskip
\noindent
The BMS bootstrap programme has been initiated in \cite{Bagchi:2016geg} and elaborated in \cite{Bagchi:2017cpu}, together with other relevant developments \cite{Hijano:2017eii, Hijano:2018nhq, Lodato:2018gyp, Merbis:2019wgk}, in an effort to constrain possible 2d BMSFTs. We wish to take this programme of constraining BMSFTs further by focusing on modular invariance. Following \cite{Kraus:2016nwo}, the analysis of the torus one-point function has been generalised in many contexts, viz. to charged 2d CFTs \cite{Das:2017vej}, warped CFTs \cite{Song:2019txa}. In this paper, we work out the generalisation of the Kraus-Maloney analysis to the case of 2d BMS invariant field theories. In the process, we develop a lot of new machinery for BMSFTs, in particular derive expressions for the torus blocks of BMSFTs. Finally, we also show that the our expressions can be reproduced by looking at a one-point function in the putative dual FSC solution.
\subsection{Outline of the paper and a quick look at the main results}
Our paper is organised as follows. In Sec.~\refb{bmsft}, we review properties of 2d BMSFTs including in particular modular invariance of BMSFTs. In Sec~\refb{1pt}, we provide a definition of the BMS torus one-point function and derive its modular properties.
\medskip
\noindent
In Sec~\refb{sconst}, we discuss the derivation of the asymptotic form of the BMS structure constants by exploiting the modular properties of the torus one-point function. Here we employ two methods, one a saddle-point analysis and the second based on inverting integral transforms that improves on the saddle point analysis. These methods don't assume much about the representations the states fall into in the sums used in the definition of the torus one-point function. Our principal result in this section is
\begin{eqnarray}\label{cipi}
C_{i p i} \approx D(\Delta_{\chi},\xi_{\chi}) C_{\chi p \chi} \exp \left(-\frac{\Delta_i \xi _p }{2\xi_i }\right) \exp\left(-2\pi \frac{\Delta_i \xi_{\chi}}{\sqrt{2c_M \xi_i}}\right). \\
\nonumber \end{eqnarray}
This above result is the outcome of the saddle point approximation and here presented for the case $c_L=0$. Here $C_{ipi}\equiv\<\Delta_i,\xi_i|\phi_p|\Delta_i,\xi_i\>$ is the three point-coefficient, where $\phi_p$ is a BMS primary with weights $\Delta_p$ and $\xi_p$. $\phi_\chi$ is the lightest BMS primary in the spectrum above the vacuum and has weights $\Delta_\chi$ and $\xi_\chi$, and $D(\Delta_\chi,\xi_\chi)$ is its degeneracy.
\smallskip
\noindent
There are further improvements on this result in Sec~\refb{sconst} to include a non-zero $c_L$, to increase the range of validity of the result and to include generalisations when the vacuum has a degeneracy as well as where the vacuum is not the lightest primary. We will give details of this in the main body of the paper.
\medskip
\noindent
In the latter half of the paper, we refine our analysis to look at the torus one-point function of BMS primaries in the highest weight representation. This entails the development of a lot of new results in the context of BMS highest weight representations in Sec.~\refb{hw}. Our primary achievement in this part of the paper outlined in Sec~\refb{tblock} which details the construction of the BMS torus blocks. These are defined by
\begin{equation}
\mathcal{F}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(\sigma,\rho) = \frac{q^{-\Delta_A+\frac{c_L}{2}}y^{-\xi_A+\frac{c_M}{2}}}{C_{ApA}} {\rm Tr}_{\Delta_A,\xi_A}\left(\phi_p(0,1)q^{L_0-\frac{c_L}{2}}y^{M_0-\frac{c_M}{2}}\right).
\end{equation}
Here the trace is over $\{A\}$, the collection of all the the primary states with dimension $(\Delta_A,\xi_A)$. Also $q=e^{2\pi i \sigma}, \ y=e^{2\pi i \rho}$ are the exponentiation of the BMS modular parameters $\sigma, \rho$. We present expressions of the BMS torus blocks in a certain limit of large BMS weights (large $\xi_A$) for these states. In this approximation, as detailed in the main body of the paper, the result for the BMS torus block reads:
\begin{eqnarray}
&& \mathcal{F}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(\sigma,\rho)\equiv \sum_N q^N \mathcal{F}_{N}(\Delta_p,\xi_p;\Delta_A,\xi_A|c_L,c_M|\rho), \quad \text{where}\\
&& \mathcal{F}_{N}=\left(1+\frac{\xi_p(\Delta_p-1)}{2\xi_A}N\right)\widetilde{\text{dim}}_N +\pi i\rho\frac{\xi_p^2}{\xi_A} \sum_{k=0}^Np(N-k)p(k)(N-k)(N-2k) + \mathcal{O}(\xi_A^{-2}). \nonumber
\end{eqnarray}
In the above, $p(N)$ is the partition of the integer $N$ and $\widetilde{\text{dim}}_N$ is the partition of the same integer $N$ using two colors. As an aside, using a differential equation arising from the quadratic Casimirs of the global BMS algebra (or the Poincare algebra), we also derive a closed form expression for the BMS global torus block.
\medskip
\noindent
Using all of these tools, in Sec~\refb{onepointsec}, we arrive at the asymptotic structure constants for the BMS highest weight primaries. The form of this primary structure constant is almost identical to the one obtained in the analysis for general states in the earlier section with the identification
\begin{equation}
c_L \to c_L - \frac{1}{6}
\end{equation}
This indicates a quantum shift in the central term $c_L$ and no such correction in $c_M$, in keeping with recent literature \cite{Merbis:2019wgk}. We have some more remarks at the end of Sec~\refb{onepointsec}.
\medskip
\noindent
Sec.~\refb{bulk} contains a bulk computation that matches our field theory results, specifically \refb{cipi}. Here we first review the Flat Space Cosmology (FSC) solutions and then use a geodesic approximation to compute a probe one-point function in this FSC background. We end in Sec.~\refb{conc} with some discussions and future directions. Complementing the analysis in the main text, there are twelve (no, really twelve!) appendices which outline various detailed computations omitted in the main body of the paper.
\bigskip \bigskip
\newpage
\section{BMS field theories and modular invariance}\label{bmsft}
In this section, we review aspects of BMS invariant field theories in 2d, or in short 2d BMSFTs. We start with algebraic preliminaries and then look at aspects of representation theory. We then discuss modular invariance in 2d BMSFTs. Since this is basically all review material, readers familiar with earlier work on BMSFT can skip over to the next section.
\subsection{Algebra and contraction}
As we have stated in the introduction, we will be interested in 2d quantum field theories invariant under the BMS$_3$ algebra, rewritten here for convenience:
\begin{subequations}{}\label{bms}
\begin{eqnarray}{}
&& [L_n, L_m] = (n-m) L_{n+m} + c_L (n^3-n) \delta_{n+m, 0} \\
&& [L_n, M_m] = (n-m) M_{n+m} + c_M (n^3-n) \delta_{n+m, 0} \\
&& [M_n, M_m] = 0.
\end{eqnarray}
\end{subequations}
This algebra \refb{bms} can be obtained from two copies of the Virasoro algebra
\begin{subequations}{}\label{vir}
\begin{eqnarray}{}
&& [\L_n, \L_m] = (n-m) \L_{n+m} + \frac{c}{12}(n^3-n) \delta_{n+m, 0} \\
&& [\bar{\mathcal{L}}_n, \bar{\mathcal{L}}_m] = (n-m) \bar{\mathcal{L}}_{n+m} + \frac{\bar{c}}{12}(n^3-n) \delta_{n+m, 0} \\
&& [\L_n, \bar{\mathcal{L}}_m] = 0.
\end{eqnarray}
\end{subequations}
by an In{\"o}n{\"u}-Wigner contraction defined by
\begin{equation} \label{v2b}
L_n = \L_n - \bar{\mathcal{L}}_{-n}, \quad M_n = \epsilon\left( \L_n + \bar{\mathcal{L}}_{-n} \right), \quad \epsilon \to 0.
\end{equation}
We shall refer to this limit as the ultra-relativistic (UR) or Carrollian limit. In terms of coordinates, this has the interpretation of sending the speed of light to infinity in the dual field theory. We shall see this shortly. From the point of the bulk, the limit \refb{v2b} can be identified as taking the radius of AdS ($\ell$) to infinity. This can be easily seen by considering the identification $\epsilon = 1/\ell$. Remembering that the Brown-Henneaux central terms of Einstein gravity in AdS$_3$ is $c= \bar{c}= 3 \ell/2G$, it is interesting to note here that the limit \refb{v2b} along with the identification of the contraction parameter to the inverse of the AdS radius, leads to
\begin{equation}
c_L=\frac{1}{12}(c - \bar{c})=0, \quad c_M= \lim_{\ell \to \infty} \frac{1}{12 \ell}(c + \bar{c}) = \frac{1}{4G}
\end{equation}
for Einstein gravity, as mentioned in the introduction. This can also be obtained from an independent asymptotic symmetry analysis in 3d asymptotically flat spacetimes \cite{Barnich:2006av}.
In order to see the ultrarelativistic nature of the limit \refb{v2b}, one can look at the Virasoro generators on the cylinder (which also corresponds to the conformal boundary of global AdS$_3$ spacetime).
\begin{equation}
\L_n = e^{in \omega} \partial_\omega, \quad \bar{\mathcal{L}}_n = e^{in \bar{\omega}} \partial_{\bar{\omega}}, \quad \mbox{where} \quad \omega, \bar{\omega} = u \pm \phi \nonumber
\end{equation}
The linear combinations required for taking the limit \refb{v2b} means that on the coordinates, the contraction manifests itself as \cite{Bagchi:2012cy}
\begin{equation}{}
\phi \to \phi, u \to \epsilon u \quad \epsilon \to 0.
\end{equation}
In terms of velocities, this means $v/c \sim \sigma/\t \to \infty$, where $c$ is the speed of light, which is the same as a Carrollian or UR limit of $c\to 0$. In terms of the coordinates on the null cylinder $(u, \phi)$, the generators of the BMS algebra take the following form:
\begin{equation}\label{genc}
L_n = ie^{in \phi} (\partial_\phi + in u \partial_u), \quad M_n = ie^{in \phi} \partial_u.
\end{equation}
It is of importance to note here that the coordinate $\phi$ representing the circle at infinity is compact, whereas the null direction $u$ which serves as the time direction in the dual field theory is non-compact. We will be interested in another representation of the BMS algebra where both directions are non-compact. This we will call the ``plane" representation. The generator on the plane take the form
\begin{equation}\label{genp}
L_n =- x^{n+1}\partial_x - (n+1) x^n t \partial_t, \quad M_n = x^{n+1} \partial_t.
\end{equation}
It can be checked that these again close to form the (centreless) BMS algebra \refb{bms}. We can map the cylinder to the plane by the coordinate transformation
\begin{equation}\label{c2p}
t= -iu e^{i\phi}, \quad x = e^{i \phi}
\end{equation}
Again it is straight-forward to check that the generators \refb{genc} and \refb{genp} go into each other under this transformation \refb{c2p}.
\subsection{A brief look at representation theory}
In 2d CFTs, states are labelled by the eigenvalues of the $\L_0$ and $\bar{\mathcal{L}}_0$ operators:
\begin{equation}
\L_0|h, {\bar h}\> = h |h, {\bar h}\>, \quad \bar{\mathcal{L}}_0|h, {\bar h}\> = {\bar h} |h, {\bar h}\>.
\end{equation}
Demanding the spectrum to be bounded from below, and noting that $\L_n, \bar{\mathcal{L}}_n$ for $n>0$ lowers the $(h, {\bar h})$ values, we are led to the definition of a primary state
\begin{equation}
\L_n|h, {\bar h}\>_p = \bar{\mathcal{L}}_n|h, {\bar h}\>_p = 0, \quad \forall n>0.
\end{equation}
A Virasoro highest weight module is built out of these primary states by acting with raising operators. An arbitrary descendant state is constructed by hitting a primary state with a bunch of raising operators $\L_{-n}, \bar{\mathcal{L}}_{-n}$ for $n>0$. The primary states form the basis of these highest weight representations and as we mentioned in the introduction, a particular CFT is completely fixed by the spectrum of its primary states and the coefficients associated with the three point functions of these primary states.
Looking at the structure of the BMS algebra \refb{bms}, we could attempt a similar construction of highest weight representations \cite{Bagchi:2009pe}. Now the states would be labelled under $L_0$ and $M_0$.
\begin{equation}
L_0|\Delta, \xi\>= \Delta |\Delta, \xi\>, \quad M_0|\Delta, \xi\> = \xi |\Delta, \xi\>.
\end{equation}
Demanding that the spectrum of $\Delta$ be bounded from below, we can define a BMS primary $|\Delta, \xi\>_p$ as
\begin{equation}
L_n |\Delta, \xi\>_p = M_n |\Delta, \xi\>_p = 0 \quad \forall n>0.
\end{equation}
In a manner analogous to the 2d CFT case, we can build BMS highest weight modules from these primary states by acting with raising operators $L_{-n}, M_{-n}$ for $n>0$.
\medskip
\noindent It is important to notice that the Virasoro highest weight representation does not contract to the BMS highest weight representation in the UR limit \refb{v2b}. To see this directly, we will look at the primary state conditions. We first notice that there is no problem with the labelling of the states in the limit.
\begin{subequations}
\begin{eqnarray}
&& (\L_0 - \bar{\mathcal{L}}_0) |h, {\bar h}\> = (h - {\bar h}) |h, {\bar h}\> \Rightarrow L_0 |\Delta, \xi\>= \Delta |\Delta, \xi\>, \, \mbox{where} \ \Delta= h - {\bar h}, \\
&& (\L_0 + \bar{\mathcal{L}}_0) |h, {\bar h}\> = (h + {\bar h}) |h, {\bar h}\> \Rightarrow M_0 |\Delta, \xi\>= \xi |\Delta, \xi\>, \, \mbox{where} \ \xi= \epsilon(h + {\bar h}).
\end{eqnarray}
\end{subequations}
In the above, the state $|h, {\bar h}\>$ goes over to the state $|\Delta, \xi\>$ in the limit. Now let us look at the Virasoro primary state conditions:
\begin{subequations}
\begin{eqnarray}
\L_n |h, {\bar h}\> = (L_n + \frac{1}{\epsilon} M_n) |h, {\bar h}\> &=& 0, \, \, \forall \, n>0 \Rightarrow M_n |\Delta, \xi\>= 0, \, \, \forall \, n>0 \\
\bar{\mathcal{L}}_n |h, {\bar h}\> = (- L_{-n} + \frac{1}{\epsilon} M_{-n}) |h, {\bar h}\> &=& 0, \, \, \forall \, n>0 \Rightarrow M_{-n} |\Delta, \xi\>= 0, \, \, \forall \, n>0.
\end{eqnarray}
\end{subequations}
So, Virasoro highest weight representations under the UR contraction become BMS representations with the following characteristics
\begin{equation}
L_0 |\Delta, \xi\>= \Delta |\Delta, \xi\>, \, M_0 |\Delta, \xi\>= \xi |\Delta, \xi\>, \, M_n |\Delta, \xi\>= 0 \ \forall n \neq 0.
\end{equation}
These representations are the so-called induced representations, studied in detail in \cite{Barnich:2014kra, Barnich:2015uva, Campoleoni:2016vsh}. It should be pretty obvious that these representations are very different from the above described BMS highest weight representations.
\medskip
\noindent The BMS highest weights are reproduced from another limit of the Virasoro which of course again yield the same algebra. This is a non-relativistic (NR) contraction, as opposed to the UR one described earlier. This different contraction is given by \cite{Bagchi:2009pe}
\begin{equation}
L_n = \L_n + \bar{\mathcal{L}}_n, \quad M_n = -\epsilon (\L_n - \bar{\mathcal{L}}_n)
\end{equation}
The isomorphism between the algebras obtained by these two contractions is a surprising fact in two dimensions \cite{Bagchi:2010zz}. Following this limit on the cylinder Virasoro generators, we can see that the speed of light in this contraction goes to infinity thus justifying the non-relativistic nomenclature. Showing that the highest weights go to highest weights is also a simple exercise.
\subsection{BMS modular invariance}\label{bmsmod}
In a 2d CFT, the partition function of the theory is defined as
\begin{equation}
Z_{\mbox{\tiny{CFT}}}(\t, \bar{\t}) = \mbox{Tr} \left(q^{\L_0 - c/24} \ \bar{q}^{\bar{\mathcal{L}}_0 - \bar{c}/24}\right).
\end{equation}
Here $\t, \bar{\t}$ are the modular parameters and $q= e^{2\pi i \t}, \, \bar{q} = e^{-2\pi i \bar{\t}}$. In a similar vein, we will define the partition function of a 2d BMSFT as
\begin{equation}
Z_{\mbox{\tiny{BMS}}}(\sigma, \rho) = \mbox{Tr} \left(e^{2\pi i \sigma(L_0 - c_L/24)} \ e^{2\pi i \rho(M_0 - c_M/24)}\right).
\end{equation}
Here the BMS modular parameters are $\sigma, \rho$. We demand that the BMS partition function arises as a smooth limit from the CFT partition function. This necessitates the following identification:
\begin{equation}\label{v2bmp}
\t = \sigma + \epsilon \rho, \, \bar{\tau} = \sigma - \epsilon \rho.
\end{equation}
The modular transformation on a 2d CFT reads
\begin{equation}
\t \to \frac{a \t + b}{c \t + d}, \quad \bar{\tau} \to \frac{a \bar{\tau} + b}{c \bar{\tau} + d}
\end{equation}
Following the identification \refb{v2bmp}, we find the BMS version of modular transformations \cite{Bagchi:2013qva}:
\begin{equation}
\sigma \to \frac{a \sigma + b}{c \sigma + d}, \quad \rho \to \frac{\rho}{(c \sigma + d)^2}.
\end{equation}
Demanding that the CFT partition function be invariant under modular transformations, particularly the S-transformation
$$ \t \to - \frac{1}{\t}, $$
one relates the very high energy spectrum of the theory to the low energy sector and in particular derives an expression for the entropy of states of the theory, given by Cardy's famous formula
\begin{equation}\label{car}
S_{\mbox{\tiny{Cardy}}} = 2 \pi \left( \sqrt{\frac{c h}{6}} + \sqrt{\frac{\bar{c} {\bar h}}{6}} \right),
\end{equation}
where $h, {\bar h}$ are conformal weights of the states and $c, \bar{c}$ are the central charges of the 2d CFT. Using similar arguments and the BMS S-transformation
$$ \sigma \to - \frac{1}{\sigma}, \, \rho \to \frac{\rho}{\sigma^2}, $$
we arrive at the BMS-Cardy formula \cite{Bagchi:2013qva}
\begin{equation}\label{bcar}
S_{\mbox{\tiny{BMS-Cardy}}} = 2 \pi \left( c_L \sqrt{\frac{2 \xi}{c_M}} + \Delta \sqrt{\frac{2 c_M}{\xi}} \right).
\end{equation}
One can also obtain the above formula \refb{bcar} directly from \refb{car} by looking carefully at the limit \cite{Riegler:2014bia,Fareghbal:2014qga}\footnote{This crucially involves a minus sign between the two factors in parenthesis in \refb{car}, a fact which can be attributed to the inner horizon of the original BTZ solution from where the bulk dual FSC is obtained.} or by an exchange of spatial and temporal circles on the BMS torus following methods outlined in \cite{Detournay:2012pc} (see e.g. \cite{Jiang:2017ecm}). The validity of \refb{bcar} can be verified by the fact that this formula correctly reproduces the Bekenstein-Hawking entropy of the putative bulk dual Flatspace Cosmological solutions. We will have more to say about this when we do a bulk calculation in Sec~\refb{bulk}.
\bigskip
\section{BMS one-point function on the torus} \label{1pt}
Having set the stage, we now move on to our object of interest in the current paper, the torus one-point function of a 2d BMSFT. We expect that in keeping with properties of 2d CFTs, the modular covariance of this quantity, together with crossing symmetry on the sphere, would crucial to ensuring higher point crossing symmetry and higher genus modular invariance in a 2d BMSFT.
\subsection{Torus 1-pt function: Definition}
For a local primary field $\phi_{\rm{cyl}}(u, \phi)$ (where cyl in subscript denotes `cylinder') on the BMS torus with modular parameters $\sigma,\rho$, we define a {\em{BMS torus one-point function}} as:
\begin{equation}
\langle \phi_{\rm cyl}(u, \phi)\rangle_{(\sigma, \rho)} = {\rm Tr}\left[ \phi_{\rm{cyl}}(u, \phi) e^{-\beta H} e^{i \theta P }\right],
\end{equation}
where $H$ is the Hamiltonian which generates translation along the longitude of the torus and $P$ generates translation along the meridian (circumference) of the torus. We think of the torus as a cylinder with length $\beta$ in which the two ends of the cylinder are glued together. Before gluing, we twist one of the ends by an angle $\theta$.
\medskip
\noindent
We will start out by showing that $\langle \phi_{\rm cyl}(u, \phi)\rangle_{(\sigma, \rho)} $ is translation invariant. To see this, let us take the basis states to be orthonormal states $|E_i,P_i\rangle$ which are simultaneous eigenstates of $H$ and $P$
\begin{equation}
H|E_i,P_i\rangle = E_i |E_i,P_i\rangle ,\,\,\,\, P|E_i,P_i\rangle = P_i |E_i,P_i\rangle.
\end{equation}
Then
\begin{equation}
\langle \phi_{\rm cyl}(u, \phi)\rangle_{(\sigma, \rho)} = \sum_i \langle E_i,P_i| \phi_{\rm{cyl}}(u, \phi) e^{-\beta H} e^{i \theta P }|E_i,P_i\rangle.
\end{equation}
Using the fact that $P$ and $H$ are operators for spatial and temporal translations we have
\begin{eqnarray}
\langle \phi_{\rm cyl}(u+u^{\prime}, \phi+\phi^{\prime})\rangle_{(\sigma, \rho)} &=& \sum_i \langle E_i,P_i| e^{iu^{\prime} H} e^{i \phi^{\prime} P }\phi_{\rm{cyl}}(u, \phi) e^{-iu^{\prime} H} e^{-i \phi^{\prime} P } e^{-\beta H} e^{i \theta P }|E_i,P_i\rangle \cr
&=& \sum_i e^{iu^{\prime} E_i} e^{i \phi^{\prime} P_i } e^{-iu^{\prime} E_i} e^{-i \phi^{\prime} P_i } \langle E_i,P_i| \phi_{\rm{cyl}}(u, \phi) e^{-\beta H} e^{i \theta P }|E_i,P_i\rangle \cr
&=& \langle \phi_{\rm cyl}(u, \phi)\rangle_{(\sigma, \rho)}. \nonumber
\end{eqnarray}
Thus, $\langle \phi_{\rm cyl}(u, \phi)\rangle_{(\sigma, \rho)}$ is invariant under translation. Using this, we may write
\begin{equation}
\langle \phi_{\rm cyl}(u, \phi)\rangle_{(\sigma, \rho)} = \langle \phi_{\rm cyl}(0,0)\rangle_{(\sigma, \rho)}.
\end{equation}
As we have seen earlier, we can map the cylinder to a plane with coordinates $(t,x) $ by
\begin{equation}\label{ptc}
t=-iue^{i\phi},\quad\quad x=e^{i\phi}.
\end{equation}
Since we have built the torus by identifying the ends of a cylinder, this map stays the same.
The operator $H$ and $P$ can be written in terms of operators $L_0$ and $M_0$ on the plane. Then, the one point functions can be written as
\begin{align}
\langle \phi_{\rm cyl}(u, \phi)\rangle_{(\sigma, \rho)}={\rm{Tr}}\left[\phi_{\rm{cyl}}(u, \phi)e^{2\pi i\sigma(L_0-\frac{c_L}{2})}e^{2\pi i \rho(M_0-\frac{c_M}{2})}\right].
\end{align}
The primary field in the cylinder $\phi_{\rm cyl}(u,\phi)$ are related to the primary field on the plane $\phi(x,t)$ as
\begin{equation}\label{ptceq}
\phi_{\rm cyl}(u,\phi) = e^{i\phi\Delta}e^{-i u\xi}\phi(t,x).
\end{equation}
The detailed derivation along with the BMS$_3$ transformation for primary fields are done in appendix \ref{Finite ${BMS}_3$ transformations}.
Thus the one-point function is given by
\begin{equation}
\boxed{\langle \phi_{\rm cyl}(u, \phi)\rangle_{(\sigma, \rho)} = {\rm{Tr}}\left[\phi(0,1)\ e^{2\pi i\sigma(L_0-\frac{c_L}{2})}e^{2\pi i \rho(M_0-\frac{c_M}{2})}\right].}
\label{one_point_function}
\end{equation}
This will be our working definition of the BMS torus one-point function.
\subsection{Torus 1-pt function: Modular transformation property}
We will now derive the modular transformation properties of a BMS torus one-point function. We will take inspiration from 2d CFTs, like reviewed in Sec.~{\refb{bmsmod}}. Modular transformation in relativistic CFT is given by
\begin{equation}
\tau \rightarrow \gamma. \tau \equiv \frac{a\tau+b}{c\tau+d},\quad \bar{\tau} \rightarrow \gamma. \bar{\tau} \equiv \frac{a\bar{\tau}+b}{c\bar{\tau}+d},
\end{equation}
where $a, b, c, d$ are integers satisfying $ad-bc=1$. Under this transformation, the elliptic co-ordinates $(w,\bar{w})$ of the torus transforms as
\begin{equation}
w\rightarrow \gamma. w\equiv \frac{w}{c\tau+d},\quad \bar{w}\rightarrow \gamma. \bar{w}\equiv \frac{\bar{w}}{c\bar{\tau}+d}.
\label{elliptic}
\end{equation}
Thus primary fields transforms as
\begin{eqnarray}
\phi_{h,\bar{h}}(\gamma.w,\bar{\gamma.w})|_{\gamma.\tau} &=& \left(\frac{\partial (\gamma.w)}{\partial w}\right)^{-h}\left(\frac{\partial (\gamma.\bar{w})}{\partial\bar{w}}\right)^{-\bar{h}}\phi_{h,\bar{h}}(w,\bar{w})
\end{eqnarray}
Using this and the modular invariance of torus partition function, we can deduce the following transformation property for one point torus function
\begin{eqnarray}
\<\phi_{h,\bar{h}}(\gamma.w,\gamma.\bar{w})\>_{(\gamma.\tau,\gamma.\bar{\tau})} = (c\tau+d)^{-h}(c\bar{\tau}+d)^{-\bar{h}}\<\phi_{h,\bar{h}}(w,\bar{w})\>_{(\tau,\bar{\tau})}
\label{cft_1pt_tran}
\end{eqnarray}
\medskip
\noindent To find the transformation property of BMS torus one-point function under BMS modular transformation, we adopt a similar prescription. As stated before, the BMS version of modular transformation reads
\begin{equation}
\sigma \rightarrow \gamma.\sigma = \frac{a\sigma +b }{c\sigma + d},\quad\quad \rho \rightarrow \gamma.\rho=\frac{\rho}{(c\sigma+d)^2}.
\end{equation}
For the coordinate $u$ and $\phi$, we can find the analog of \eqref{elliptic} for by taking its UR limit.
For this, we have to use the contraction
\begin{equation}
(w,\bar{w})\rightarrow \epsilon u \pm \phi, \quad\quad(\tau,\bar{\tau})\rightarrow \sigma \pm \epsilon\rho.
\end{equation}
We know that
\begin{equation}
u =\frac{w+\bar{w}}{2\epsilon},\quad\quad \phi = \frac{w-\bar{w}}{2}.
\end{equation}
Thus
\begin{equation}
\gamma.u = \frac{\gamma.w+\gamma.\bar{w}}{2\epsilon},\quad\quad \tilde{\phi}\equiv\gamma.\phi = \frac{\gamma.w - \gamma.\bar{w}}{2}.
\end{equation}
We expand $ \gamma.w$ and $\gamma.\bar{w}$ in powers of $\epsilon$ with the above definitions to obtain:
\begin{eqnarray}
&& \gamma.w = \frac{w}{c\tau + d} = \frac{\epsilon u + \phi}{c(\sigma + \epsilon \rho) + d}
=\epsilon \left(\frac{u}{c\sigma + d}-\frac{\phi\rho}{(c\sigma +d)^2}\right) + \frac{\phi}{c\sigma + d} +\mathcal{O}(\epsilon^2). \\
&& \gamma.\bar{w} = \epsilon \left(\frac{u}{c\sigma + d}-\frac{\phi\rho}{(c\sigma +d)^2}\right) - \frac{\phi}{c\sigma + d} +\mathcal{O}(\epsilon^2).
\end{eqnarray}
Thus we have
\begin{equation}
\gamma.u = \frac{u}{c\sigma + d}-\frac{\phi\rho}{(c\sigma +d)^2},\quad \quad \gamma.\phi= \frac{\phi}{c\sigma + d}.
\label{bms_ellip_trans}
\end{equation}
Note that this is a finite BMS transformation of the form: $u \rightarrow u \partial_{\phi} f(\phi) + g(\phi),\,\phi \rightarrow f(\phi)$.
\bigskip \bigskip
\noindent The BMS primary field with weight $\Delta$ and $\xi$ transform under
$u \rightarrow \gamma.u,\,\phi \rightarrow \gamma.\phi$
as
\begin{equation}
\phi_{\Delta,\xi}(\gamma.u,\gamma.\phi)|_{\gamma.\sigma,\gamma.\rho}= (\partial_{\phi}(\gamma.\phi))^{-\Delta}e^{\xi \frac{\partial_{\phi}(\gamma.u)}{\partial_{\phi}(\gamma.\phi)}}\phi_{\Delta,\xi}(u,\phi)_{(\sigma,\rho)}.
\end{equation}
Thus the one point function for BMS primary field would transform as
\begin{eqnarray}
\<\phi_{\Delta,\xi}(\gamma.u,\gamma.\phi)\>_{(\gamma.\sigma,\gamma.\rho)} &=& (\partial_{\phi}(\gamma.\phi))^{-\Delta}e^{\xi \frac{\partial_{\phi}(\gamma.u)}{\partial_{\phi}(\gamma.\phi)}}\<\phi_{\Delta,\xi}(u,\phi)\>_{(\sigma,\rho)}.
\label{bms_1pt_tran}
\end{eqnarray}
Substituting \eqref{bms_ellip_trans} in the above equation we get
\begin{eqnarray}
\<\phi_{\Delta,\xi}(\gamma.u,\gamma.\phi)\>_{(\gamma.\sigma,\gamma.\rho)}
&=& (c\sigma + d)^{\Delta}e^{- \frac{\xi c\rho}{(c\sigma+d)}} \<\phi_{\Delta,\xi}(u,\phi)\>_{(\sigma,\rho)}.
\end{eqnarray}
In particular for modular S-transformation $(a=0,b=-1,c=1,d=0)$, we have
\begin{equation}
\left<\phi_{\Delta,\xi}\right>_{(-1/\sigma, \ \rho/\sigma^2)} = \sigma^{\Delta} e^{-\frac{\xi\rho}{\sigma}} \<\phi_{\Delta,\xi}\>_{(\sigma,\rho)}.
\label{s_mod_trans}
\end{equation}
We will use the above to derive an asymptotic form of the BMS structure constants in the next section{\footnote{We note that analogues of the above formulae could have of course been derived completely in the NR limit, with a flip in the time and space directions.}}.
\bigskip
\newpage
\section{Asymptotic form of BMS structure constants} \label{sconst}
In this section, we present two different ways of obtaining the asymptotic form of the BMS structure constants. The first is a saddle-point method, which generalises the BMS-Cardy analysis of e.g. \cite{Bagchi:2012xr} from zero-point functions, i.e. partition functions, to one-point functions on the torus. The second relies on inverting a combination of a Laplace and Fourier transformations and is an improvement on the saddle point analysis.
\medskip
\noindent It is important to state here that the analyses in both methods are independent of basis. We will just assume that the states of the BMSFT are eigenstates of $L_0$ and $M_0$, and the forms of the BMS modular transformations discussed in the previous section. So, our analysis should be independent of which of the two representations discussed earlier, the highest weight or the induced, the states of the BMSFT fall into. This is a statement we don't prove in our paper, as we will not work with induced representations. We will later see that an explicit computation with BMS primaries and highest weight representations, based on the construction of BMS torus blocks yields identical results when the appropriate limits are considered.
\medskip
\noindent
The statement about the equivalence of the two representations may seem like an over-reach at this point, but we point the reader to our earlier work on BMS characters in \cite{Bagchi:2019unf}, which remarkably showed that the expressions for characters in these two very different looking representations are actually identical. We expect a similar story to emerge here for the torus one-point function as well. In later sections, we will use the BMS torus blocks to rewrite our torus 1-point function. The leading piece of the expression of the BMS torus blocks in the highest weight representation is again the character. It is not unreasonable to expect that a similar (but admittedly more difficult) calculation of the induced BMS torus block in the same limit may also yield the character as the leading piece. These are one and the same as shown in \cite{Bagchi:2019unf}. We will not have anything to say explicitly about calculations in the induced representation in this work. But the enigmatic equivalence between the two very different representations is something we wish to return to in the near future.
\medskip
\noindent
Now, we concentrate on finding the asymptotic form of the BMS structure constants from the modular properties of the one-point function that we derived in the previous section. We first use a saddle-point analysis.
\subsection{The saddle-point method}
\label{sec:simultaneous_L_M}
Let us emphasise again that the trace on the left hand side of \eqref{one_point_function} does not depend on the representation which the states of the 2d BMSFT fall into. If we are using a bases $|\Delta_i,\xi_i\>$ which are both eigenstates of $L_0$ and $M_0$
\begin{equation}
L_0|\Delta_i,\xi_i\>=\Delta_i|\Delta_i,\xi_i\>,\quad\quad M_0|\Delta_i,\xi_i\>=\xi_i|\Delta_i,\xi_i\>,
\end{equation}
which is true for both the highest weight and the induced representations, we see that the one-point function for a primary field $\phi_p$ is given by{\footnote{The notion of a primary state would of course depend on the representation and definition we are working with stems from the highest weight representation. So this may be a point of dispute. We can choose to define the 1-point function of a torus by \refb{t1pt}, which will make this a representation independent statement.}}
\begin{equation}\label{t1pt}
\langle \phi_p\rangle_{(\sigma, \rho)} = \sum_{i}\<\Delta_i,\xi_i|\phi_p(0,1)|\Delta_i,\xi_i\> D(\Delta_i,\xi_i)e^{2\pi i\sigma(\Delta_i-\frac{c_L}{2})}e^{2\pi i \rho(\xi_i-\frac{c_M}{2})},
\end{equation}
where $D(\Delta_i,\xi_i)$ is the density of states for $|\Delta_i,\xi_i\>$. We called $\<\Delta_i,\xi_i|\phi_p(0,1)|\Delta_i,\xi_i\>$ a one-point coefficient and we will use $C_{ipi}$ to denote it. Thus
\begin{equation}
\langle \phi_p\rangle_{(\sigma, \rho)} = e^{-2\pi i \left(\sigma\frac{c_L}{2}+\rho\frac{c_M}{2}\right)}\sum_{i} D(\Delta_i,\xi_i)C_{ipi}e^{2\pi i(\sigma\Delta_i+\rho \xi_i)}.
\label{one_point_function_rep1}
\end{equation}
In this section we will deduce the form of three-point coefficient
\begin{equation}
C_{ipi}\equiv\<\Delta_i,\xi_i|\phi_p(0,1)|\Delta_i,\xi_i\>,
\end{equation}
for large $\Delta_i$ and $\xi_i$. Sometime we will drop the index $i$ and use the notation $C_{\Delta p \Delta }\equiv\<\Delta_,\xi|\phi_p(0,1)|\Delta,\xi\>$.
We define
\begin{equation}\label{tilp}
\widetilde{\langle \phi_p\rangle}_{(\sigma, \rho)}=e^{2\pi i \left(\sigma\frac{c_L}{2}+\rho\frac{c_M}{2}\right)}\,\langle \phi_p\rangle_{(\sigma, \rho)}.
\end{equation}
Thus from \eqref{one_point_function_rep1} and \refb{tilp} we have
\begin{equation}
\widetilde{\langle \phi_p\rangle}_{(\sigma, \rho)}= \sum_{i} D(\Delta_i,\xi_i)C_{ipi}e^{2\pi i(\sigma\Delta_i+\rho \xi_i)},
\label{phitilde}
\end{equation}
The S-modular transformation property of $\widetilde{\langle \phi_p\rangle}_{(\sigma, \rho)}$ can be worked out from that of $\langle \phi_p\rangle_{(\sigma, \rho)}$ given in \eqref{s_mod_trans}
\begin{eqnarray}
\widetilde{\langle \phi_p\rangle}_{{(-\frac{1}{\sigma},\frac{\rho}{\sigma^2})}}&=& e^{2\pi i \left(-\frac{1}{\sigma}\frac{c_L}{2}+\frac{\rho}{\sigma^2}\frac{c_M}{2}\right)}\left<\phi_p\right>_{(-\frac{1}{\sigma},\frac{\rho}{\sigma^2})} \cr
&=& e^{2\pi i \left(-\frac{1}{\sigma}\frac{c_L}{2}+\frac{\rho}{\sigma^2}\frac{c_M}{2}\right)}\sigma^{\Delta_p} e^{-\xi_p\frac{\rho}{\sigma}} \<\phi_p\>_{(\sigma,\rho)}\cr
&=& e^{2\pi i \left(-\frac{1}{\sigma}\frac{c_L}{2}+\frac{\rho}{\sigma^2}\frac{c_M}{2}\right)}\sigma^{\Delta_p} e^{-\xi_p\frac{\rho}{\sigma}} e^{-2\pi i \left(\sigma\frac{c_L}{2}+\rho\frac{c_M}{2}\right)}\,\widetilde{\langle \phi_p\rangle}_{(\sigma, \rho)}\cr
&=& e^{-2\pi i \left(\frac{1}{\sigma}\frac{c_L}{2}-\frac{\rho}{\sigma^2}\frac{c_M}{2}+\sigma\frac{c_L}{2}+\rho\frac{c_M}{2}\right)}e^{-\xi_p\frac{\rho}{\sigma}}\sigma^{\Delta_p}\widetilde{\langle \phi_p\rangle}_{(\sigma, \rho)}.
\end{eqnarray}
Thus the S-modular transformation of $\widetilde{\langle \phi_p\rangle}_{(\sigma, \rho)}$ is given by
\begin{equation}
\widetilde{\langle \phi_p\rangle}_{(\sigma, \rho)} = e^{2\pi i \left(\frac{1}{\sigma}\frac{c_L}{2}-\frac{\rho}{\sigma^2}\frac{c_M}{2}+\sigma\frac{c_L}{2}+\rho\frac{c_M}{2}+\frac{\xi_p}{2\pi i}\frac{\rho}{\sigma}-\frac{1}{2\pi i}\Delta_p\log \sigma\right)}\widetilde{\langle \phi_p\rangle}_{{(-\frac{1}{\sigma},\frac{\rho}{\sigma^2})}}.
\end{equation}
\newpage \noindent
We invert \eqref{phitilde} to obtain
\begin{eqnarray}
D(\Delta,\xi)C_{\Delta p \Delta} &=& \int d\sigma d\rho \,e^{-2\pi i(\sigma\Delta+\rho \xi)} \widetilde{\langle \phi_p\rangle}_{(\sigma, \rho)}\cr
&=& \int d\sigma d\rho \, e^{-2\pi i(\sigma\Delta+\rho \xi)} \, e^{2\pi i \left(\frac{1}{\sigma}\frac{c_L}{2}-\frac{\rho}{\sigma^2}\frac{c_M}{2}+\sigma\frac{c_L}{2}+\rho\frac{c_M}{2}+\frac{\xi_p}{2\pi i}\frac{\rho}{\sigma}-\frac{1}{2\pi i}\Delta_p\log \sigma\right)}\widetilde{\langle \phi_p\rangle}_{{(-\frac{1}{\sigma},\frac{\rho}{\sigma^2})}}\cr
&=& \int d\sigma d\rho \,\widetilde{\langle \phi_p\rangle}_{{(-\frac{1}{\sigma},\frac{\rho}{\sigma^2})}} \,\,e^{f(\sigma,\rho)},
\label{M_integral}
\end{eqnarray}
where
\begin{equation}
f(\sigma,\rho)=2\pi i \left(\frac{1}{\sigma}\frac{c_L}{2}-\frac{\rho}{\sigma^2}\frac{c_M}{2}+\sigma\frac{c_L}{2}+\rho\frac{c_M}{2}+\frac{\xi_p}{2\pi i}\frac{\rho}{\sigma}-\frac{1}{2\pi i}\Delta_p\log \sigma-\sigma\Delta-\rho \xi\right).
\end{equation}
For large $(\Delta, \xi)$, we use the saddle point approximation to evaluate the integral \eqref{M_integral}
\begin{equation}
D(\Delta,\xi)C_{\Delta p \Delta} \approx \widetilde{\langle \phi_p\rangle}_{{(-\frac{1}{\sigma_c},\frac{\rho_c}{\sigma^2_c})}} \,e^{f(\sigma_c,\rho_c)},
\label{saddle_approx}
\end{equation}
where $\sigma_c$ and $\rho_c$ are the critical points of the function $f(\sigma,\rho)$: $$\partial_{\sigma}f(\sigma_c,\rho_c)=0,\,\partial_{\rho}f(\sigma_c,\rho_c)=0.$$ These are given by:
\begin{subequations}\begin{eqnarray}
&& -\frac{c_L}{2 \sigma ^2}+\frac{c_L}{2}+\frac{\rho c_M}{\sigma ^3}-\Delta +\frac{i \Delta _p}{2 \pi \sigma }+\frac{i \rho \xi _p}{2 \pi \sigma ^2}=0, \\
&& -\frac{c_M}{2 \sigma ^2}+\frac{c_M}{2}-\xi -\frac{i \xi _p}{2 \pi \sigma }=0.
\end{eqnarray}
\end{subequations}
We have two solutions for the above equations. For $(\Delta, \xi)\gg(c_L,c_M,\Delta_p,\xi_p)$, these are given by
\begin{equation}
(\sigma_{c}, \rho_c) \approx \left(i\sqrt{\frac{c_M}{2\xi}}, i\frac{ \left( \xi c_L - \Delta c_M \right)}{ (2 \xi )^{3/2} \sqrt{c_M}}\right);
\quad
(\tilde{\sigma}_c, \tilde{\rho}_c) \approx \left(-i\sqrt{\frac{c_M}{2\xi}}, -i\frac{ \left(\xi c_L - \Delta c_M \right)}{(2 \xi )^{3/2} \sqrt{c_M}}\right).
\end{equation}
Among this, $(\sigma_{c},\rho_c)$ is the solution we are interested in. It can be checked that the saddle-point approximation breaks down for the other solution and we will disregard it. For more details of the validity of the saddle-point analysis, we point the interested reader to \cite{Bagchi:2019unf}. Substituting $(\sigma_{_c},\rho_c)$ in \eqref{saddle_approx}, we have
\begin{equation}
D(\Delta,\xi)C_{\Delta p \Delta} \approx \widetilde{\langle \phi_p\rangle}_{{(-\frac{1}{\sigma_c},\frac{\rho_c}{\sigma^2_c})}} i^{-\Delta_p}\left(\frac{2\xi}{c_M}\right)^{\frac{\Delta_p}{2}} e^{2\pi\left(\sqrt{\frac{\xi}{2c_M}}c_L+\Delta\sqrt{\frac{c_M}{2\xi}}\right)}e^{\frac{\xi _p}{2} \left(-\frac{\Delta }{\xi }+\frac{c_L}{c_M}\right)}.
\label{MC}
\end{equation}
Now
\begin{equation}
-\frac{1}{\sigma_c} = i\sqrt{\frac{2\xi}{c_M}},\quad\quad \frac{\rho_c}{\sigma_c^2} = i\sqrt{\frac{\xi}{2c_M}} \left(\frac{\Delta}{\xi}-\frac{c_L}{c_M}\right)
\end{equation}
So, we have
\begin{eqnarray}
\widetilde{\langle \phi_p\rangle}_{{(-\frac{1}{\sigma_c},\frac{\rho_c}{\sigma^2_c})}} &=& \sum_{i} D(\Delta_i,\xi_i)C_{i p i}e^{2\pi i(-\frac{1}{\sigma_c}\Delta_i+\frac{\rho_c}{\sigma_c^2} \xi_i)}\cr
&=& \sum_{i} D(\Delta_i,\xi_i)C_{i p i}\,e^{-2\pi\sqrt{\frac{2\xi}{c_M}}\Delta_i-2\pi\sqrt{\frac{\xi}{2c_M}} \left(\frac{\Delta}{\xi} - \frac{c_L}{c_M}\right) \xi_i}. \label{tp}
\end{eqnarray}
In the above equation, $C_{i p i}=\<\Delta_i,\xi_i|\phi_p(0,1)|\Delta_i,\xi_i\>$ is zero when $|\Delta_i,\xi_i\>$ is the vacuum $|0,0\>$. Hence the vacuum would not contribute to the sum on the right hand side of \refb{tp}. Since the exponent is decreasing very fast due to large value of $\Delta$ and $\xi$, the leading contribution comes from lightest state in the spectrum, excluding vacuum i.e, state with smallest values of the conformal weights. We denote this state by $|\chi\>$ and its weights by $(\Delta_{\chi},\xi_{\chi})$. (Equivalently, $|\chi\> = |\Delta_{\chi},\xi_{\chi}\>$.) Thus
\begin{equation}
\widetilde{\langle \phi_p\rangle}_{{(-\frac{1}{\sigma_c},\frac{\rho_c}{\sigma^2_c})}} \approx D(\Delta_{\chi},\xi_{\chi})C_{\chi p \chi}\,e^{-2\pi\left(\sqrt{\frac{2\xi}{c_M}}\Delta_{\chi}+\sqrt{\frac{\xi}{2c_M}} \left(-\frac{c_L}{c_M} + \frac{\Delta}{\xi}\right) \xi_{\chi}\right)}.
\end{equation}
Substituting the above equation in \eqref{MC}, we have
\begin{eqnarray}
D(\Delta,\xi)C_{\Delta p \Delta} &\approx & D(\Delta_{\chi},\xi_{\chi})C_{\chi p \chi} i^{-\Delta_p}\left(\frac{2\xi}{c_M}\right)^{\frac{\Delta_p}{2}} e^{2\pi\left(\sqrt{\frac{\xi}{2c_M}}c_L+\Delta\sqrt{\frac{c_M}{2\xi}}\right)}e^{\frac{\xi _p}{2} \left(-\frac{\Delta }{\xi }+\frac{c_L}{c_M}\right)}\cr
&&\times \,e^{-2\pi\left(\sqrt{\frac{2\xi}{c_M}}\Delta_{\chi}+\sqrt{\frac{\xi}{2c_M}} \left(\frac{\Delta}{\xi}-\frac{c_L}{c_M}\right) \xi_{\chi}\right)}.
\end{eqnarray}
Then the three-point coefficient is given by
\begin{eqnarray}
C_{\Delta p \Delta} &\approx & \frac {D(\Delta_{\chi},\xi_{\chi})}{ D(\Delta,\xi)}C_{\chi p \chi} i^{-\Delta_p}\left(\frac{2\xi}{c_M}\right)^{\frac{\Delta_p}{2}} e^{2\pi\left(c_L \sqrt{\frac{\xi}{2c_M}}+\Delta\sqrt{\frac{c_M}{2\xi}}\right)}e^{\frac{\xi _p}{2} \left(-\frac{\Delta }{\xi }+\frac{c_L}{c_M}\right)}\nonumber\\
&&\times \,e^{-2\pi\left(\sqrt{\frac{2\xi}{c_M}}\Delta_{\chi}+\sqrt{\frac{\xi}{2c_M}} \left(\frac{\Delta}{\xi} -\frac{c_L}{c_M} \right) \xi_{\chi}\right)}.
\end{eqnarray}
The density of states $D(\Delta,\xi)$ for large $\Delta$ and $\xi$ is known (this is basically just the BMS-Cardy formula \cite{Bagchi:2012xr, Bagchi:2013qva,Bagchi:2019unf}) and given by
\begin{equation}
D(\Delta,\xi) \approx D(0,0)e^{2\pi \left(c_L\sqrt{\frac{\xi}{2c_M}}+\Delta\sqrt{\frac{c_M}{2\xi}}\right)}.
\end{equation}
Substituting this in the previous equation we get
\bigskip
\noindent {\fbox{
\addtolength{\linewidth}{+2\fboxsep}%
\addtolength{\linewidth}{+2\fboxrule}%
\begin{minipage}{\linewidth}
\begin{eqnarray}\label{asymptotic_carlip}
C_{i p i} \approx \frac {D(\Delta_{\chi},\xi_{\chi})}{ D(0,0)}C_{\chi p \chi} i^{-\Delta_p}\left(\frac{2\xi_i}{c_M}\right)^{\frac{\Delta_p}{2}} e^{\frac{\xi _p}{2} \left(-\frac{\Delta_i }{\xi_i }+\frac{c_L}{c_M}\right)} e^{-2\pi\left(\sqrt{\frac{2\xi_i}{c_M}}\Delta_{\chi}+\sqrt{\frac{\xi_i}{2c_M}} \left(\frac{\Delta_i}{\xi_i}-\frac{c_L}{c_M}\right) \xi_{\chi}\right)}. \cr
\end{eqnarray}
\end{minipage}
}}
\bigskip
\noindent This formula is the first of our main results of this paper. In the next subsection, we will improve on this result with a more refined analysis.
\medskip
\noindent
For the case when $|\phi_p\>$ and the lightest state $|\chi\>$ are both scalars, we have $\Delta_p=0$ and $\Delta_{\chi}=0$. Additionally, for Einstein gravity in 3d asymptotically flat spacetimes, we also have $c_L=0$ in the BMS algebra. For this special case we get
\begin{equation}
C_{i p i} \equiv \<\Delta_i,\xi_i|\phi_p(0,1)|\Delta_i,\xi_i\>\approx \frac {D(\Delta_{\chi},\xi_{\chi})}{ D(0,0)}C_{\chi p \chi} e^{-\frac{\xi _p}{2}\frac{\Delta_i }{\xi_i }}e^{-2\pi \frac{\Delta_i}{\sqrt{2c_M \xi_i}}\xi_{\chi}}.
\label{three_point_ceoff_limit}
\end{equation}
We would be comparing the dual bulk analysis in Sec.~\refb{bulk} to the above formula.
\subsection{An improved method}\label{onepointsec1}
In the previous subsection, we derived an asymptotic form of the one-point coefficient $C_{ipi}\equiv \<\Delta_i,\xi_i|\phi_p(0,1)|\Delta_i,\xi_i\>$ where we used bases that are simultaneous eigenstates of $L_0$ and $M_0$. Here we will obtain the asymptotic formula using a different method that is based on inverting integral transformations. As before, the BMS one-point function is given by
\begin{equation}{}
\langle \phi_p\rangle_{(\sigma, \rho)}=\text{Tr}\left(\phi_p(0,1)e^{2\pi i\sigma(L_0-\frac{c_L}{2})}e^{2\pi i\rho(M_0-\frac{c_M}{2})}\right)=\sum_{i} D(\Delta, \xi) q^{\Delta_i-\frac{c_L}{2}}y^{\xi_i-\frac{c_M}{2}}C_{ipi}, \nonumber
\end{equation}
where we have $q= e^{2\pi i \sigma}$ and $y= e^{2\pi i \rho}$.
From \eqref{s_mod_trans}, $S$-transformation of $\langle \phi_p\rangle$ is given as:
\begin{subequations}
\begin{eqnarray}
\langle \phi_p\rangle_{(\sigma, \rho)} &=& \sum_{i} D(\Delta_i,\xi_i)C_{ipi}e^{2\pi i\sigma(\Delta_i-\frac{c_L}{2})}e^{2\pi i\rho(\xi_i-\frac{c_M}{2})} \label{a1} \\
&=&\sigma^{-\Delta_p}e^{\xi_p\frac{\rho}{\sigma}}\sum_{i} D(\Delta_i,\xi_i)C_{ipi} e^{-\frac{2\pi i}{\sigma}(\Delta_i-\frac{c_L}{2})}e^{2\pi i\frac{\rho}{\sigma^2}(\xi_i-\frac{c_M}{2})}. \label{a2}
\end{eqnarray}
\end{subequations}
We choose $\sigma=i\beta$ and consider the limit $\beta\rightarrow0^+$ and $|\rho|\rightarrow0^+$. In this limit, the summation in \refb{a1} over the discrete\footnote{Let's denote the set of all $\Delta_A$ as $\Delta$ and the set of all $\xi_A$ as $\xi$. We assume the spectra to be such that any mapping from (domain) $\Delta$ to (codomain) $\xi$ is a surjection; otherwise a little reformulation is needed.} spectra can be approximated as a double-integral:
\begin{equation}
\langle \phi_p\rangle_{(\sigma, \rho)} = \int_{\xi_\chi}^\infty d\xi_i\int_{\Delta_\chi}^\infty d\Delta_i \ D(\Delta_i,\xi_i)C_{ipi} e^{-2\pi \beta(\Delta_i-\frac{c_L}{2})}e^{2\pi i\rho(\xi_i-\frac{c_M}{2})} \label{q1}
\end{equation}
while \refb{a2} can be approximated by keeping only the most-dominating summand, which is the contribution due to the lightest-primary (having the lowest $\Delta$) $\chi$ of the theory such that $C_{\chi p\chi}\neq0$:
\begin{equation}
\langle \phi_p\rangle_{(\sigma, \rho)} = (i\beta)^{-\Delta_p} e^{-i\xi_p\frac{\rho}{\beta}} D(\Delta_\chi,\xi_\chi)C_{\chi p\chi} e^{-\frac{2\pi }{\beta}(\Delta_\chi-\frac{c_L}{2})}e^{-2\pi i\frac{\rho}{\beta^2}(\xi_\chi-\frac{c_M}{2})} \label{q2}
\end{equation}
We now denote
$$T_p(\Delta_i,\xi_i)\equiv D(\Delta_i,\xi_i)C_{ipi}.$$
The above equations \refb{q1}, \refb{q2} then lead to the following equality
\begin{eqnarray}
\int_{0}^\infty d{\xi_i^\prime} \ \int_{0}^\infty d{\Delta_i^\prime} && T_p^\prime({\Delta_i^\prime},{\xi_i^\prime}) e^{-2\pi \beta{\Delta_i^\prime}}e^{2\pi i\rho{\xi_i^\prime}} \cr
&& \hspace{-1cm} = (i\beta)^{-\Delta_p}T_p(\Delta_\chi,\xi_\chi) e^{2\pi (\beta-\frac{1}{\beta})(\Delta_\chi-\frac{c_L}{2})}e^{-2\pi i\rho\{(1+\frac{1}{\beta^2})(\xi_\chi-\frac{c_M}{2})+\frac{\xi_p}{2\pi \beta} \}},
\label{double_integral_fin}
\end{eqnarray}
The equation \refb{double_integral_fin} can be thought of as a Laplace transformation of ${T}_p^\prime({\Delta_i^\prime},{\xi_i^\prime})$ with respect to ${\Delta_i^\prime}$ and one-sided Fourier transformation of the same of ${T}_p^\prime({\Delta_i^\prime},{\xi_i^\prime})$ with respect to ${\xi_i^\prime} \ (\xi_i^\prime \geqslant0)$. We will invert the integral transformations \footnote{In hindsight, we know that $T_p^\prime({\Delta_i^\prime},{\xi_i^\prime})$ is of exponential order, so the inversion as done is justified.} one-by-one. We will first concentrate on the Fourier transform. This leads us to
\begin{eqnarray}
\int_{0}^\infty d\Delta_i^\prime {T}_p^\prime({\Delta_i^\prime},{\xi_i^\prime}) e^{-2\pi \beta\Delta_i^\prime}&=& \frac{T_p(\Delta_\chi,\xi_\chi)}{(i\beta)^{\Delta_p}} e^{2\pi (\beta-\frac{1}{\beta})(\Delta_\chi-\frac{c_L}{2})} \int_{-\infty}^{\infty}d(\text{Re}\,\rho)e^{-2\pi i\rho[(1+\frac{1}{\beta^2})(\xi_\chi-\frac{c_M}{2})+\frac{\xi_p}{2\pi \beta}]} \nonumber\\
&=& \frac{T_p(\Delta_\chi,\xi_\chi)}{(i\beta)^{\Delta_p}} e^{2\pi (\beta-\frac{1}{\beta})(\Delta_\chi-\frac{c_L}{2})}e^{2\pi\,(\text{Im}\,\rho) (\xi_i^\prime+(1+\frac{1}{\beta^2})(\xi_\chi-\frac{c_M}{2})+\frac{\xi_p}{2\pi\beta})} \nonumber\\
&& \quad \times \delta\left[\xi_i^\prime+(1+\frac{1}{\beta^2})(\xi_\chi-\frac{c_M}{2})+\frac{\xi_p}{2\pi\beta}\right].
\end{eqnarray}
Let us denote the two roots of the argument of the $\delta$-function as $\beta_{\pm}$:
\begin{eqnarray}
\beta_\pm = \frac{-\frac{\xi_p}{2\pi}\pm \sqrt{\left(\frac{\xi_p}{2\pi}\right)^2+4(\frac{c_M}{2}-\xi_\chi)(\xi_i-\frac{c_M}{2})}}{2(\xi_i-\frac{c_M}{2})}.
\end{eqnarray}
where $\xi_i = \xi_i'+\xi_\chi$. Then, using the property of the $\delta$-function, we are led to
\begin{eqnarray}
\hspace{-0.5 cm} \int_{0}^\infty d\Delta_i^\prime {T}_p^\prime({\Delta_i^\prime},{\xi_i^\prime}) e^{-2\pi \beta\Delta_i^\prime}&=&\sum_{\star=+,-}\frac{T_p(\Delta_\chi,\xi_\chi)}{(i\beta_\star)^{\Delta_p}} e^{2\pi (\beta_\star-\frac{1}{\beta_\star})(\Delta_\chi-\frac{c_L}{2})}\frac{\delta(\beta-\beta_\star)}{|\frac{c_M-2\xi_\chi}{\beta_\star^3}-\frac{\xi_p}{2\pi\beta_\star^2}|}.
\label{to_invert_1}
\end{eqnarray}
We started out with the assumption that $\beta$ is real. Hence, for consistency, we require $\beta_\pm$ to be real.
\bigskip
\noindent Now we concentrate on the Laplace transformation. To invert the Laplace-transform in \eqref{to_invert_1}, we need to know how to analytically continue the $\delta$-function which is defined for real arguments. Let's consider the following equivalence of the Cauchy integral theorem and the $\delta$-function (with $x_0$ being real and $F$ being an entire ($\equiv{C}^\infty$) function such that $\lim_{r\rightarrow\infty}F(re^{i\theta})=0$):
\begin{align}
\frac{1}{2\pi i}\oint_C dz\frac{F(z)}{z-x_0}=F(x_0)=\int_{-\infty}^b dxF(x)\delta(x-x_0).
\end{align}
where $C$ is the contour in Figure \ref{fig:contour} (relevant for inverse-Laplace transformation) enclosing $x_0$ where $-\infty<x_0<b$.
\begin{figure}[t]
\begin{center}
\begin{tikzpicture}[decoration={markings,
mark=at position 1.8cm with {\arrow[line width=1pt]{>}},
mark=at position 7.7cm with {\arrow[line width=1pt]{>}}}]
\draw[help lines,->] (-3,0) -- (3,0) coordinate (xaxis);
\draw[help lines,->] (0,-3.0) -- (0,3.0) coordinate (yaxis);
\path[draw,line width=0.8pt,postaction=decorate] (0.8,-2.4) -- (0.8,2.4) arc (90:270:2.4);
\node[below] at (xaxis) {$x$};
\node[left] at (yaxis) {$y$};
\draw (0.4,0) node{$\bullet$};
\node at (0.4,0.3) {$x_0$};
\node at (1.3,-0.3) {$C_{x=b}$};
\node at (1.9,2.4) {$z=b+i\infty$};
\node at (1.9,-2.4) {$z=b-i\infty$};
\node at (-1.8,1.7) {$C_{r\rightarrow\infty}$};
\end{tikzpicture}
\end{center}
\caption{Contour used for inverting the Laplace transformation}
\label{fig:contour}
\end{figure}
The semicircle part $C_{r\rightarrow\infty}$ of the closed contour has the equation $|z-b|=\lim_{r\rightarrow\infty}r$ and by the demanded property of $F(z)$, the contribution from $C_{r\rightarrow\infty}$ vanishes. Thus we are left with:
\begin{align}
\int_{-\infty}^b dxF(x)\delta(x-x_0)=\frac{1}{2\pi i }\int_{b-i\infty}^{b+i\infty} dz\frac{F(z)}{z-x_0}.
\end{align}
As $F(z)$ is any function belonging to the class of functions having the above specified properties, we conclude that for the problem (and contour) in hand, it is justified to write:
\begin{align}
\delta(x-x_0)\xrightarrow [\text{continuation}]{\text{Analytic}}\frac{1}{2\pi i }\frac{1}{z-x_0}.
\label{delta_ac}
\end{align}
Inverting the Laplace Laplace-transformation in \eqref{to_invert_1} using \eqref{delta_ac}, we have
\begin{eqnarray}
\hspace{-0.5cm}\frac{{T}_p^\prime({\Delta_i^\prime},{\xi_i^\prime})}{T_p(\Delta_\chi,\xi_\chi)} &=&\sum_{\star=+,-}\frac{e^{2\pi (\beta_\star-\frac{1}{\beta_\star})(\Delta_\chi-\frac{c_L}{2})}}{(i\beta_\star)^{\Delta_p} } \frac{1}{|\frac{c_M-2\xi_\chi}{\beta_\star^3}-\frac{\xi_p}{2\pi\beta_\star^2}|}\int_{b-i\infty}^{b+i\infty} \frac{d\beta}{i}\frac{e^{2\pi \beta\Delta_i^\prime}}{2\pi i (\beta-\beta^{*})},
\end{eqnarray}
where, $b>\beta_+$, so that all of the singularities of the integrand lie at the left of the vertical integration-path. Also, when $|\beta-b|=\lim_{r\rightarrow\infty}r$,
\begin{align}
\lim_{r\rightarrow\infty}r\frac{e^{2\pi \Delta_A^\prime(b+r\cos{\theta}+ir\sin\theta)}}{b+re^{i\theta}-\beta_\star}=0
\end{align}
as along the semi-circle $C_{r\rightarrow\infty}$, $\cos\theta<0$ being on the left of the vertical path $C_{x=b}$ and $\Delta_A^\prime\geqslant0$. Thus, we can close the contour from the left to get the depicted contour $C$ whose semi-circle part doesn't contribute to the integral. Hence,
\begin{eqnarray}
\frac{{T}_p^\prime({\Delta_i^\prime},{\xi_i^\prime})}{T_p(\Delta_\chi,\xi_\chi)}&=& \sum_{\star=+,-}(i\beta_\star)^{-\Delta_p} e^{2\pi (\beta_\star-\frac{1}{\beta_\star})(\Delta_\chi-\frac{c_L}{2})}\frac{1}{|\frac{c_M-2\xi_\chi}{\beta_\star^3}-\frac{\xi_p}{2\pi\beta_\star^2}|}\oint_{C} \frac{d\beta}{i}\frac{e^{2\pi \beta\Delta_i^\prime}}{2\pi i (\beta-\beta^{*})} \cr
&=&\sum_{\star=+,-}(i\beta_\star)^{-\Delta_p}e^{2\pi (\beta_\star-\frac{1}{\beta_\star})(\Delta_\chi-\frac{c_L}{2})}\frac{e^{2\pi \beta_\star\Delta_i^\prime}}{i|\frac{c_M-2\xi_\chi}{\beta_\star^3}-\frac{\xi_p}{2\pi\beta_\star^2}|}.
\label{laplace_fin_0}
\end{eqnarray}
Thus we note that upon inverting the double-integral transformation in \eqref{double_integral_fin}, the resultant inverse integrand has very sharp ($\delta$-function like) peaks at two different real values $\beta_\pm$ that are the sole contributors to the inverse transformation \eqref{laplace_fin_0}.
To make the initial consideration of $\beta\rightarrow0^+$ consistent with the final result, $|\beta_\pm|\rightarrow0^+$ must be satisfied. This is same as the demand that $|\beta_\pm|\ll1$. We also require $\beta$ to be real. Thus, we require $\xi_i\gg c_M-\xi_\chi\pm \frac{\xi_p}{2\pi}$ with the assumption of the existence of a $\xi_\chi<\frac{c_M}{2}$. This implies that $$\xi_i\gg \frac{c_M}{2}\pm \frac{\xi_p}{2\pi}.$$ In this limit, $$\beta_\pm\approx-\frac{\xi_p}{4\pi\xi_i}\pm\sqrt{\frac{{c_M}-2\xi_\chi}{2\xi_i}}.$$
As $|\beta_\pm|\rightarrow0^+$, in each summand in \eqref{laplace_fin_0} the most dominating contribution comes from the factor $e^{2\pi (-\frac{1}{\beta\star})(\Delta_\chi-\frac{c_L}{2})}$. Also, as $\sigma$ lies on the upper-half plane, to start with, we had $\beta>0$; hence, it is desirable to have $\beta_-$'s contribution to \eqref{laplace_fin_0} negligible in comparison to that of $\beta_+$. So, we also need to assume the existence of a $\Delta_\chi<\frac{c_L}{2}$. Hence, we finally reach
\begin{eqnarray}
&&\frac{{T}_p({\Delta_i},{\xi_i})}{T_p(\Delta_\chi,\xi_\chi)}\approx\frac{(-i)^{\Delta_p+1}}{\beta_+^{\Delta_p-2}}e^{ \frac{2\pi}{\beta_+}(\frac{c_L}{2}-\Delta_\chi)}\frac{e^{2\pi \beta_+(\Delta_i-\frac{c_L}{2})}}{\frac{c_M-2\xi_\chi}{\beta_+}-\frac{\xi_p}{2\pi}}\nonumber\\
&\approx& \frac{(-i)^{\Delta_p+1}}{2\xi_i}{\left(\frac{\xi_i}{\frac{c_M}{2}-\xi_\chi}\right)}^{\frac{\Delta_p-1}{2}}
e^{\frac{\xi_p}{2}\left(-\frac{\Delta_i-\frac{c_L}{2}}{\xi_i}+\frac{\frac{c_L}{2}-\Delta_\chi}{\frac{c_M}{2}-\xi_\chi}\right)+2\pi[\sqrt{\frac{\frac{c_M}{2}-\xi_\chi}{\xi_i}}(\Delta_i-\frac{c_L}{2})+\sqrt{\frac{\xi_i}{\frac{c_M}{2}-\xi_\chi}}(\frac{c_L}{2}-\Delta_\chi)]}.\nonumber\\
\label{eqn:tpbytp}
\end{eqnarray}
\textit{Density of states}: Let us call the lightest primary field in the theory $\O$. This may be the vacuum, but we are further generalising our results by considering cases where it may not be so and $\Delta_\chi\geqslant\Delta_\O\geqslant0$. Taking a cue out of Liouville theory, a BMS version of Liouville theory could be amenable to such a generalisation. We assume the existence of $\Delta_\O<\frac{c_L}{2}$ and $\xi_\O<\frac{c_M}{2}$. Clearly, putting $\Delta_p=0=\xi_p$ directly in \eqref{eqn:tpbytp} and replacing the primary field $\chi$ by $\O$, we get the asymptotic formula for density of states in the large $\xi_i$ limit:
\begin{align}
\frac{D(\Delta_i,\xi_i)}{D(\Delta_\O,\xi_\O)}=\frac{-i}{2\xi_i}\sqrt{\frac{\frac{c_M}{2}-\xi_\O}{\xi_i}} \exp 2\pi \left({\sqrt{\frac{\frac{c_M}{2}-\xi_\O}{\xi_i}}\left(\Delta_i-\frac{c_L}{2}\right)+\sqrt{\frac{\xi_i}{\frac{c_M}{2}-\xi_\O}}\left(\frac{c_L}{2}-\Delta_\O\right)}\right).
\label{den_states_large_xi}
\end{align}
Substituting \eqref{den_states_large_xi} in \eqref{eqn:tpbytp}, we find an asymptotic formula for the 3-point coefficient $C_{ipi}$ in the large $\xi_i$ limit:
\bigskip
\noindent {\fbox{
\addtolength{\linewidth}{+2\fboxsep}%
\addtolength{\linewidth}{+2\fboxrule}%
\begin{minipage}{\linewidth}
\begin{eqnarray}\label{impr}
\hspace{-0.5cm} C_{ipi}&\approx &(-i)^{\Delta_p}C_{\chi p\chi}\frac{D(\Delta_\chi,\xi_\chi)}{D(\Delta_\O,\xi_\O)}\left(\frac{2\xi_i}{c_M-2\xi_\chi}\right)^{\frac{\Delta_p-1}{2}}\sqrt{\frac{2\xi_i}{c_M-2\xi_\O}}\,e^{\frac{\xi_p}{2}\left(-\frac{2\Delta_i-c_L}{2\xi_i}+\frac{c_L-2\Delta_\chi}{c_M-2\xi_\chi}\right)}\nonumber\\
&&\times e^{2\pi\left(\sqrt{\frac{2\xi_i}{c_M-2\xi_\chi}}\left(\frac{c_L}{2}-\Delta_\chi\right)-\sqrt{\frac{2\xi_i}{c_M-2\xi_\O}}\left(\frac{c_L}{2}-\Delta_\O\right)+\left(\sqrt{\frac{c_M- 2\xi_\chi}{2\xi_i}}-\sqrt{\frac{ c_M- 2\xi_\O}{2\xi_i}}\,\right)\left(\Delta_i-\frac{c_L}{2}\right)\right)}. \\
\nonumber
\end{eqnarray}
\end{minipage}
}}
\bigskip
\noindent Note that the above asymptotic formula is valid irrespective of the order of magnitude of $\frac{\Delta_i}{\xi_i}$. Furthermore, if we take the field $(\Delta_\O,\xi_\O)$ to be the vacuum and consider the limit in which $c_M \gg \xi_{\chi}$, then we get back the asymptotic formula \eqref{asymptotic_carlip} obtained in the previous section using a saddle-point analysis. So, in this sense, the range of validity of the above formula is broader than that of \eqref{asymptotic_carlip}.
\newpage
\section{Machinery for BMS highest weights}\label{hw}
We have, in the preceding section, derived the asymptotic form of the BMS torus one-point function using general analyses, that did not rely on the explicit representations the states of the BMSFT fall into, except for the fact that the states were simultaneous eigenstates of $L_0$ and $M_0$. We now want to concentrate on BMS highest weight representations and use this to find the expression for the torus one-point function of primary fields. In this section and the next, we will build machinery for our ultimate calculation, which will be described in Sec~\refb{onepointsec}.
\subsection{BMS highest weight modules}
As a basis for calculating the trace in the partition function or the one-point function on the torus \refb{one_point_function}, we can use the BMS primary states and their descendants. As we have briefly reviewed in Sec.~2, the primary states are eigenstates of $L_0$ and $M_0$
\begin{equation}
L_0 |\Delta_A,\xi_A\>= \Delta_A |\Delta_A,\xi_A\>,\quad\quad M_0 |\Delta_A,\xi_A\>= \xi_A |\Delta_A,\xi_A\>,
\end{equation}
which are annihilated by $L_n$ and $M_n$ for $n>0$
\begin{equation}
L_n |\Delta_A,\xi_A\> =0,\quad\quad M_n |\Delta_A,\xi_A\> =0,\quad n>0.
\end{equation}
By acting the operator $L_{-n}$ and $M_{-n}$ on the primary states we could raise the eigenvalue of $L_0$. Such states which are obtained by acting a series of $L_{-n}$ and $M_{-n}$ operators on the primary states are called descendant states. A BMS module of a primary state $|\Delta_A,\xi_A\>$ is the set which contain the primary states and all its descendant. States in the module have the form
\begin{equation}
L_{-1}^{k_1}L_{-2}^{k_2}....L_{-l}^{k_l} M_{-1}^{q_1}M_{-2}^{q_2}....M_{-r}^{q_r}|\Delta_A,\xi_A \> \equiv L_{\overrightarrow{k}}M_{\overrightarrow{q}}|\Delta_A,\xi_A \> ,
\end{equation}
where $\overrightarrow{k}=(k_1,k_2,....,k_l)$ and $\overrightarrow{q}=(q_1,q_2,....,q_r)$ and its $L_0$ eigenvalue is given by
\begin{equation}
L_0 L_{\overrightarrow{k}}M_{\overrightarrow{q}}|\Delta,\xi \> = (N + \Delta_A) L_{\overrightarrow{k}}M_{\overrightarrow{q}}|\Delta_A,\xi_A \>, \quad \mbox{where} \ N=\sum_{i}ik_i + \sum_j jq_j.
\end{equation}
$N$ is called the level of the state. To simplify or shorten equations, we will sometimes use the shorthand notation $|\Delta_A,\xi_A\> \equiv |A\>$.
We can group the states according to their level because inner product of sates having different level is zero. In general we can write the states at level $N$ as
\begin{equation}
|\Psi\>= \left(\prod_{j=1}^{N} (L_{-j})^{N_j-k_j} \right)\left(\prod_{j=1}^{N} (M_{-j})^{k_j} \right) |A\>,
\end{equation}
with $\sum_j j N_j=N$. In the table below we shown the basis states for up to level 3.
\begin{table}[t]
\begin{tabular}{|l|l|}
\hline
Level & States\tabularnewline
\hline
\hline
N=0 & $|\Psi_{1}\rangle=|\Delta_A,\xi_A\rangle$\tabularnewline
\hline
N=1 & $|\Psi_{1}\rangle=L_{-1}|\Delta_A,\xi_A\rangle,\,\,\,|\Psi_{2}\rangle=M_{-1}|\Delta_A,\xi_A\rangle$\tabularnewline
\hline
N=2 & $\begin{array}{l}
|\Psi_{1}\rangle=L_{-1}^{2}|\Delta_A,\xi_A\rangle,\,\,\,|\Psi_{2}\rangle=L_{-2}|\Delta_A,\xi_A\rangle,\,\,\,|\Psi_{3}\rangle=L_{-1}M_{-1}|\Delta_A,\xi_A\rangle,\,\,\, \\
|\Psi_{4}\rangle = M_{-2}|\Delta_A,\xi_A\rangle,\,\,\,
|\Psi_{5}\rangle=M_{-1}^{2}|\Delta_A,\xi_A\rangle
\end{array}$\tabularnewline
\hline
N=3 & $\begin{array}{l}
|\Psi_{1}\rangle=L_{-1}^3|\Delta_A,\xi_A\rangle,\,\,\,|\Psi_{2}\rangle=L_{-1}L_{-2}|\Delta_A,\xi_A\rangle,\,\,\,|\Psi_{3}\rangle=L_{-1}^2M_{-1}|\Delta_A,\xi_A\rangle,\\
|\Psi_{4}\rangle=L_{-3}|\Delta_A,\xi_A\rangle,\,\,\,|\Psi_{5}\rangle=L_{-2}M_{-1}|\Delta_A,\xi_A\rangle,\,\,\,|\Psi_{6}\rangle=L_{-1}M_{-2}|\Delta_A,\xi_A\rangle,\\
|\Psi_{7}\rangle=M_{-3}|\Delta_A,\xi_A\rangle,\,\,\,|\Psi_{8}\rangle=L_{-1}M_{-1}^2|\Delta_A,\xi_A\rangle,\,\,\,|\Psi_{9}\rangle=M_{-1}M_{-2}|\Delta_A,\xi_A\rangle,\\|\Psi_{10}\rangle=M_{-1}^3|\Delta_A,\xi_A\rangle
\end{array}$\tabularnewline
\hline
\end{tabular}
\caption{States in the BMS module upto level 3}
\label{BMS_states}
\end{table}
We will use $|\Delta_A,\xi_A;N,i\>$ to denote the $i$-th basis state at level $N$. Sometimes, we will simply this further as $|A;N,i\>$.
\subsection*{Gram matrix and their inverse}
For a given module we can form a matrix by taking the inner product of the basis states. This is known as the Gram matrix. Since states in different levels are orthogonal to each other, the Gram matrix will be block diagonal with each block for each level. We use the notation $K^{(N)}$ for the Gram matrix at level $N$. Since we have two basis states at level 2, the The Gram matrix for level 1 will be a $2\times 2$ matrix
\begin{align}
K^{(1)} = \left[ \begin{array}{cc}
\langle\Delta_A,\xi_A|L_{1}L_{-1}|\Delta_A,\xi_A\rangle & \langle\Delta_A,\xi_A|L_{1}M_{-1}|\Delta_A,\xi_A\rangle\\
\langle\Delta_A,\xi_A|M_{1}L_{-1}|\Delta_A,\xi_A\rangle & \langle\Delta_A,\xi_A|M_{1}M_{-1}|\Delta_A,\xi_A\rangle
\end{array} \right]
&= \left [ \begin{array}{cc}
2\Delta_A & 2\xi_A\\
2\xi_A & 0
\end{array} \right].
\label{GM_level1}
\end{align}
The Gram matrix $K^{(2)}$ for level 2 is a $5\times 5$ matrix given by
\begin{eqnarray}
K^{(2)} = \left[ \begin{array}{ccccc}
4\Delta_A(2\Delta_A+1) & 6\Delta_A & 4\xi_A(2\Delta_A+1) & 6\xi_A & 8\xi^{2}_A\\
6\Delta_A & 4\Delta_A+6c_L & 6\xi_A & 4\xi_A+6c_M & 0\\
4\xi_A(2\Delta_A+1) & 6\xi_A & 4\xi^{2}_A & 0 & 0\\
6\xi_A & 4\xi_A+6c_M & 0 & 0 & 0\\
8\xi^{2}_A & 0 & 0 & 0 & 0
\end{array} \right].
\label{GM_level2}
\end{eqnarray}
Note that the Gram matrix has a peculiar triangular structure in which all the matrix elements below the anti-diagonal elements are zero. This is due to way we order our basis \cite{Bagchi:2019unf}. In fact for any level we can order our basis states such that the Gram matrix will have the following form:
\begin{equation}
K^{(N=\text{odd})} = \left(\begin{array}{ccccc}
K_{1,1} & & \cdots & & A_{\widetilde{\rm dim}_N}\\
& & & \iddots & 0\\
\vdots & & \iddots & & \vdots\\
& A_{2}\\
A_{1} & 0 & \cdots & & 0
\end{array}\right) ,
\label{oddmatrix}
\end{equation}
\begin{equation}
K^{(N=\text{even})}=\left(\begin{array}{cccccccccc}
K_{1,1} & \cdots & \cdots & & \cdots & & \cdots & & \cdots & A_{\widetilde{{\rm dim}}_{N}}\\
\vdots & \ddots & & & & & & & \iddots & 0\\
& & \ddots & & & & & A_{k+l} & & \vdots\\
& & & D_{1} & 0 & \cdots & 0\\
\vdots & & & 0 & D_{2} & \cdots & 0 & & & \vdots\\
\vdots & & & \vdots & \vdots & \ddots & \vdots & & & \vdots\\
& & & 0 & 0 & \cdots & D_{l}\\
& & A_{k} & & & & & 0\\
\vdots & \iddots & & & & & & & \ddots & \vdots\\
A_{1} & 0 & \cdots & & \cdots & & \cdots & & \cdots & 0
\end{array}\right).
\label{evenmatrix}
\end{equation}
In the above, the non-zero Gram matrix in the anti-diagonal line, labelled by $A$, and the diagonal elements labelled $D$ which appear for even $N$ are of the form
\begin{equation}
\langle A|(\prod_{j=N}^1 M_j^{N_j-k_j})(\prod_{j=N}^1 L_j^{k_j})(\prod_{j=1}^N L_{-j}^{N_j-k_j})(\prod_{j=1}^N M_{-j}^{k_j})|A\rangle,
\end{equation}
where $\sum_{j=1}^{N}jN_j = N$. These are the matrix elements for which the number of $L$-operators and number of $M$-operators in the inner product are equal. In particular, the elements of $D$ are the norm of self conjugate states $(\prod_{j=1}^N L_{-j}^{N_j})(\prod_{j=1}^N M_{-j}^{N_j})|A\rangle$ (states which looks the same when $L$ and $M$ are exchanged, for example, state like $L_{-1}M_{-1}|A\>$). For more details about the above structure, we refer the interested reader to \cite{Bagchi:2019unf}.
\medskip
In a later section, we would like to take the large $\xi_A$ limit of the BMS torus blocks (which are of course to be defined later). So, let us point out some important properties of the matrix elements of the form given above. In explicitly calculating the matrix elements, we make use of the BMS algebra. The highest power in $\xi_A$ comes from the term is which each $L_{j}$ contracted with $M_{-j}$ and each $M_{j}$ contracted with $L_{-j}$ to form $M_0$ which will then act on the vacuum to give $\xi_A$. Thus the highest power in $\xi_A$ is equal to the number of $L$-operators or $M$-operators inside the inner product. This is given by $\sum_{i=1}^N N_j$.
We also note down the following factorization property of the inner product
\begin{eqnarray}
\langle A| \prod_{j=N}^1 M_j^{N_j-k_j} \prod_{j=N}^1 L_j^{k_j} \prod_{j=1}^N L_{-j}^{N_j-k_j} \prod_{j=1}^N M_{-j}^{k_j}|A\rangle =\prod_{j=1}^N \langle A| M_j^{N_j-k_j}L_{-j}^{N_j-k_j}|A\>\<A| L_j^{k_j}M_{-j}^{k_j}|A\rangle. \cr
\end{eqnarray}
The terms in the above equation are given by
\begin{eqnarray}
\<A| M_j^{N_j-k_j}L_{-j}^{N_j-k_j}|A\> &=& (N_j-k_j)!\<A|[M_j,L_{-j}]|A\> = (N_j-k_j)! (\xi_A -c_L (j^3-j)),\nonumber \\
\<A| L_j^{k_j}M_{-j}^{k_j}|A\> &=& k_j!(\xi_A -c_L (j^3-j)),
\end{eqnarray}
where the factorial account for the number of ways to contracts the $L$ and $M$ operators.
From these properties we can deduce another factorisation formula which we will use later. Before we present a formula for generic matrix elements, we demonstrate it with the following example
\begin{eqnarray}
\<A| M_2 M_{1}^3 L_{2}^2L_{-1}^3L_{-2}M_{-2}^2|A\> &=& 3 \<A|M_{1}L_{-1}|A\>\<A| M_2 M_{1}^2L_{2}^2L_{-1}^2L_{-2}M_{-2}^2|A\> \cr
\<A| M_2 M_{1}^3 L_{2}^2L_{-1}^3L_{-2}M_{-2}^2|A\> &=& \<A|M_{2}L_{-2}|A\>\<A|M_{1}^3 L_{2}^2L_{-1}^3M_{-2}^2|A\> \cr
\<A| M_2 M_{1}^3 L_{2}^2L_{-1}^3L_{-2}M_{-2}^2|A\> &=& 2\<A|L_{2}M_{-2}|A\>\<A|M_2 M_{1}^3 L_{2}L_{-1}^3L_{-2}M_{-2}|A\>.
\end{eqnarray}
In general we have
\begin{eqnarray}
\hspace{-1cm}\frac{\langle A| \prod_{j=N}^1 M_j^{N_j-k_j-\delta_{j,r}} \prod_{j=N}^1 L_j^{k_j} \prod_{j=1}^N L_{-j}^{N_j-k_j-\delta_{j,r}} \prod_{j=1}^N M_{-j}^{k_j} |A\rangle}{\langle A| \prod_{j=N}^1 M_j^{N_j-k_j} \prod_{j=N}^1 L_j^{k_j} \prod_{j=1}^N L_{-j}^{N_j-k_j} \prod_{j=1}^N M_{-j}^{k_j}|A\rangle} = \frac{(N_r-k_r)^{-1}}{\<A|M_r L_{-r}|A\>}.
\label{gram_matrix_property3}
\end{eqnarray}
\begin{eqnarray}
\hspace{-1cm} \frac{\langle A| \prod_{j=N}^1 M_j^{N_j-k_j} \prod_{j=N}^1 L_j^{k_j-\delta_{j,r}} \prod_{j=1}^N L_{-j}^{N_j-k_j} \prod_{j=1}^N M_{-j}^{k_j-\delta_{j,r}}|A\rangle}{\langle A| \prod_{j=N}^1 M_j^{N_j-k_j} \prod_{j=N}^1 L_j^{k_j} \prod_{j=1}^N L_{-j}^{N_j-k_j} \prod_{j=1}^N M_{-j}^{k_j}|A\rangle}
= \frac{1}{k_r\<A|M_r L_{-r}|A\>}.
\label{gram_matrix_property4}
\end{eqnarray}
\bigskip
\noindent We use $K_{(N)}$ to denote the inverse of the Gram matrix $K^{(N)}$ and specify the matrix elements by $K^{ab}_{(N)}$. Due to the structure of the matrix given above, the inverse Gram matrix $K_{(N)}$ will have the following structure:
\begin{equation}
K_{(N=\text{odd})} = \left(\begin{array}{ccccc}
0 & \cdots & 0 & \cdots & \frac{1}{A_{\widetilde{\rm dim}_N}}\\
0 & & & \iddots & \vdots\\
\vdots & & \iddots & & \vdots\\
& \frac{1}{A_{2}}\\
\frac{1}{A_{1}} & \iddots & \cdots & & K^{\widetilde{\rm dim}_N,\widetilde{\rm dim}_N}
\end{array}\right) ,
\label{oddmatrix_in}
\end{equation}
\begin{equation}
K_{(N=\text{even})}=\left(\begin{array}{cccccccccc}
0 & \cdots & \cdots & & \cdots & & 0 & & \cdots & \frac{1}{A_{\widetilde{{\rm dim}}_{N}}}\\
\vdots & \ddots & & & & & & & \iddots & \\
& & \ddots & & & & & \frac{1}{A_{k+l}} & & \vdots\\
& & & \frac{1}{D_{1}} & 0 & \cdots & 0\\
\vdots & & & 0 & \frac{1}{D_{2}} & \cdots & 0 & & & \vdots\\
0 & & & \vdots & \vdots & \ddots & \vdots & & & \vdots\\
& & & 0 & 0 & \cdots & \frac{1}{D_{l}}\\
& & \frac{1}{A_{k}} & & & & & \ddots \\
\vdots & \iddots & & & & & & & \ddots & \vdots\\
\frac{1}{A_{1}} & & \cdots & & \cdots & & \cdots & & \cdots & K^{\widetilde{\rm dim}_N,\widetilde{\rm dim}_N}
\end{array}\right).
\label{evenmatrix_in}
\end{equation}
\medskip
\noindent Note that for matrix elements in the anti-diagonal line as well as the diagonal elements of the matrix at the centre for even level, $K_{(N)}^{ab}$ are simply given by $1/{K^{(N)}_{ab}}$. This fact will be important when we study large $\xi_A$ limit of the BMS torus blocks. Even for non-zero, non-anti-diagonal elements $K_{(N)}^{ab}$, the leading term in $\xi_A$ is given by $1/K^{(N)}_{ab}$.
\subsection{BMS mode expansion}
Inspired from 2d CFT, we propose that a local field $\phi_p(t,x)$ with dimensions $(\Delta,\xi)$ in a 2d BMSFT on the 2d Euclidean plane (endowed with a degenerate metric) has the following mode-expansion
\begin{align}
\phi_p(t,x)=\sum_{m\in \mathbb{Z}}\sum_{n\in \mathbb{Z}-\mathbb{N}} {x}^{-\Delta-m}e^{\frac{t}{x}\xi}(-1)^n{\left(\frac{t}{x}\right)}^{-n}\phi_{m,n},
\label{mode_exp}
\end{align}
where the quasi-primary field satisfies the following Hermitian conjugation relation
\begin{align}
\phi_p^{\dagger}(t,x)=x^{-2\Delta}e^{2\xi\frac{t}{x}}\phi_p\left(\frac{1}{x},\frac{t}{x^2}\right).
\end{align}
It is interesting to note here the $n$ index in \eqref{mode_exp} runs over negative integers only. This condition is imposed on $n$ to prevent blowing up of $\phi_p$ at $(t=0,x=1)$ which is mapped to the point $(0,0)$ on the cylinder.
Putting the mode-expansion ansatz into the above conjugation equation, we get:
\begin{align}
\sum_{m\in \mathbb{Z}}\sum_{n\in \mathbb{Z}-\mathbb{N}} x^{-\Delta-m}e^{\xi\frac{t}{x}}(-1)^n{\left(\frac{t}{x}\right)}^{-n}\phi_{m,n}^{\dagger}=\sum_{m\in \mathbb{Z}}\sum_{n\in \mathbb{Z}-\mathbb{N}} {x}^{-\Delta+m}e^{\xi\frac{t}{x}}(-1)^n{\left(\frac{t}{x}\right)}^{-n}\phi_{m,n}.
\end{align}
Comparing the coefficients on the both sides, we conclude that the BMS quasi-primary modes satisfy the following Hermitian-conjugation condition:
\begin{align}
\phi_{m,n}^{\dagger}=\phi_{-m,n}
\end{align}
The asymptotic `in'-state $|\Delta,\xi\rangle$ is created from the vacuum $|0\rangle$ of the theory as below:
\begin{align}
|\Delta,\xi\rangle\equiv \lim_{x\to 0}\lim_{\frac{t}{x}\to 1}\phi_p(t,x)|0\rangle.
\end{align}
Clearly, for the `in' states to be well-defined, the vacuum must satisfy
\begin{align}
\phi_{m,n}|0\rangle=0 \text{ , for $m>-\Delta$}\Longleftrightarrow\langle0|\phi_{m,n}=0 \text{ , for $m<\Delta$}.
\end{align}
From the plane-representation of $BMS_3$ algebra, we know that for a local primary field $\phi_p$ on 2d Euclidean plane (for $n\geqslant -1$)
\begin{eqnarray}
[L_n,\phi_{\Delta,\xi}(t,x)]&=&[x^{n+1}\partial_x + (n+1)x^n t\partial_t +(n+1)(\Delta x^n -n \xi x^{n-1} t )]\phi_{\Delta,\xi}(t,x) \cr
[M_n,\phi_{\Delta,\xi}(t,x)] &=& [-x^{n+1}\partial_t + (n+1)\xi x^n]\phi_{\Delta,\xi}(t,x).
\end{eqnarray}
Substituting the mode-expansion ansatz into the above representation and comparing the coefficients on both sides, we obtain the following mode-commutation relations (for $n\geqslant -1$) only for primary fields:
\begin{align}
&[L_n,\phi_{j,k}]=[n(\Delta-1-k)-j]\phi_{j+n,k}+n^2\xi\phi_{j+n,k+1}, \quad \forall k\leqslant -1,\nonumber\\
&[L_n,\phi_{j,0}]=[n(\Delta-1)-j]\phi_{j+n,0},\nonumber\\
&[M_n,\phi_{j,k}]=n\xi\phi_{j+n,k}+(k-1)\phi_{j+n,k-1}, \quad \forall k\leqslant 0.
\label{commutation_mode}
\end{align}
Although we shall not be highlighting this in the main text, most of the intrinsic analysis of 2d BMSFT done in this paper, and elaborated in the appendices, would be done with the help of the mode expansions defined in this section.
\newpage
\section{BMS Torus blocks}\label{tblock}
In this section, using the tools we have developed for the BMS highest weight states, we construct the torus blocks for 2d BMSFTs. We start by defining these blocks. We then go onto calculating the contribution for low orders and then work in a particular limit, which we will call the large $\xi_A$ limit, which will be crucial for our analysis of the one-point functions in the following section.
\subsection{Highest weight torus blocks}
For our basis we can use the primary states and their descendants. Then taking the trace with respect to this basis in \eqref{one_point_function}, the one-point function for primary field $\phi_p$ is given by
\begin{eqnarray}
\<\phi_p\>_{(\sigma,\rho)} &=& {\rm{Tr}}\left(\phi_p(0,1) e^{2\pi i\sigma (L_{0}-c_{L}/2)}e^{2\pi i\rho (M_{0}-c_{M}/2)} \right) \cr
&=& e^{-2\pi i (\sigma c_L/2+\rho c_M/2)}\sum_{a,b} K^{ab} \<\Psi_a| \phi_p(0,1) \, e^{2\pi i\sigma L_{0}}e^{2\pi i\rho M_{0}}|\Psi_b\>,
\end{eqnarray}
where $|\Psi_a\>$ are the primary states and their descendant. $K^{ab}$ is the element of the inverse Gram Matrix formed by taking inner products of the basis states. Since inner products between states in different modules vanish, we can separate out the contribution of each module $|\Delta,\xi\>$ to the one-point function. So, with a bit of hindsight, we may write the one point function as
\begin{eqnarray}
\<\phi_p\>_{A} &=& e^{-\pi i (\sigma c_L+\rho c_M)}\sum_{A} D(\Delta_A,\xi_A) e^{2\pi i \sigma \Delta_A} e^{2\pi i \rho \xi_A} \<A|\phi_p(0,1)|A\> \mathcal{F}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(\sigma,\rho),\cr
& \equiv & e^{-\pi i (\sigma c_L+\rho c_M)}\sum_{A} D(\Delta_A,\xi_A) e^{2\pi i \sigma \Delta_A} e^{2\pi i \rho \xi_A} C_{ApA}\,\mathcal{F}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(\sigma,\rho), \label{onepointwblocks}
\end{eqnarray}
where $\{A\}$ is the collection of all the the primary states and $D(\Delta_A,\xi_A)$ is the number of primary states (density of states, or, multiplicity) with dimension $(\Delta_A,\xi_A)$. As before, $C_{ApA} \equiv \<A|\phi_p(0,1)|A\>$ is the three point coefficient. We called $\mathcal F$ the BMS torus block. This is completely determined by BMS symmetry alone.
To simplify our notation, let us use $q=e^{2\pi i \sigma}$ and $y=e^{2\pi i \rho}$. Then, the BMS torus block $\mathcal F$ is formally given by
\begin{eqnarray}
{\boxed{\mathcal{F}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(\sigma,\rho) = \frac{{\rm Tr}_{\Delta_A,\xi_A}\left(\phi_p(0,1)q^{L_0-\frac{c_L}{2}}y^{M_0-\frac{c_M}{2}}\right)}{q^{\Delta_A-\frac{c_L}{2}}y^{\xi_A-\frac{c_M}{2}}C_{ApA}}.}}
\label{bms_torus_block}
\end{eqnarray}
The trace, ${\rm Tr}_{\Delta_A,\xi_A}$, is over the states in the module of $|\Delta_A,\xi_A\>$. Thus we have
\begin{eqnarray}
{\rm Tr}_{\Delta_A,\xi_A}\left(\phi_p(0,1)q^{L_0-\frac{c_L}{2}}y^{M_0-\frac{c_M}{2}}\right) = \sum_{N; i,j} K^{ij}_{(N)} \< A,N,i|\phi_p \ q^{L_0-\frac{c_L}{2}}y^{M_0-\frac{c_M}{2}}|A,N,j\>.
\label{trace_module}
\end{eqnarray}
For a state at level $N$
\begin{equation}
L_0|A;N,i\> = (N+\Delta_A)|A;N,i\> \Rightarrow q^{L_0} |A;N,i\> = q^{N+\Delta_A} |A;N,i\>.
\end{equation}
Using this in \eqref{trace_module}, we get
\begin{eqnarray}
{\rm Tr} \left( \phi_p(0,1)q^{L_0-\frac{c_L}{2}}y^{M_0-\frac{c_M}{2}}\right) = \sum_{N}q^{N+\Delta_A -\frac{c_L}{2}}y^{-\frac{c_M}{2}}\sum_{i,j} K^{ij}_{(N)} \<A;N,i|\phi_p y^{M_0}|A;N,j\>.
\end{eqnarray}
Further substituting this in \eqref{bms_torus_block}, we have
\begin{eqnarray}
\mathcal{F}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(\sigma,\rho) &=& \sum_N q^N\sum_{i,j} K^{ij}_{(N)} \frac{\<A;N,i|\phi_p(0,1)y^{M_0}|A;N,j\>}{y^{\xi_A}C_{ApA}}\cr
&\equiv& \sum_N q^N \mathcal{F}_{N}(\Delta_p,\xi_p;\Delta_A,\xi_A|c_L,c_M|\rho),
\label{bms_block_fin}
\end{eqnarray}
where
\begin{eqnarray}
\mathcal{F}_{N}(\Delta_p,\xi_p;\Delta_A,\xi_A|c_L,c_M|\rho)&=&\frac{1}{y^{\xi_A}C_{ApA}}\sum_{i,j} K^{ij}_{(N)} \<A;N,i|\phi_p(0,1)e^{2\pi i M_0}|A;N,j\>.
\label{F_N}
\end{eqnarray}
From \eqref{bms_block_fin} can see that $q^N \mathcal{F}_N$ is the contribution of level $N$ to the torus BMS block. In what follows, we give an explicit expression for the BMS torus block for the first couple of levels.
\subsection*{$\mathcal{F}_N$ for level 1}
As an example let us calculate $\mathcal{F}_N$ for level 1.
For this level, the states are
\begin{equation}
|\Delta_A,\xi_A;1,1\> \equiv |\Psi_1\>= L_{-1}|\Delta_A,\xi_A\>,\quad\quad\quad|\Delta_A,\xi_A;1,2\> \equiv|\Psi_2\> = M_{-1}|\Delta_A,\xi_A\>,
\end{equation}
and the Gram matrix is given in \eqref{GM_level1}.
The inverse of the Gram matrix is
\begin{align}
K_{(1)} = \left[
\begin{array}{cc}
0 & \frac{1}{2 \xi_A } \\
\frac{1}{2 \xi_A } & -\frac{\Delta_A }{2 \xi ^2_A} \\
\end{array}
\right].
\end{align}
Now,
\begin{eqnarray}
&& \mathcal{F}_{1}(\Delta_p,\xi_p;\Delta_A,\xi_A|c_L,c_M|\rho)\cr
=&&\frac{1}{y^{\xi_A}C_{ApA}}\left( K^{11}_{(1)} \<\Psi_1|\phi_p(0,1)e^{2\pi i M_0}|\Psi_1\> + K^{12}_{(1)} \<\Psi_1|\phi_p(0,1)e^{2\pi i M_0}|\Psi_2\>\right.\cr
&&\quad\quad\quad\quad \quad\quad \left.
+ K^{21}_{(1)} \<\Psi_2|\phi_p(0,1)e^{2\pi i M_0}|\Psi_1\> + K^{22}_{(1)} \<\Psi_2|\phi_p(0,1)e^{2\pi i M_0}|\Psi_2\>
\right).
\label{block_level1_exp}
\end{eqnarray}
Since $K^{11}_{(1)}$ is zero, we don't have to calculate $\<\Psi_1|\phi_p(0,1)e^{2\pi i M_0}|\Psi_1\>$. The other three matrix elements are listed below
\begin{eqnarray}
\<\Psi_1|\phi_p(0,1)e^{2\pi i M_0}|\Psi_2\> &=& C_{ApA}\,y^{\xi_A} (\xi_p\Delta_p - \xi_p + 2\xi_A),\cr
\<\Psi_2|\phi_p(0,1)e^{2\pi i M_0}|\Psi_1\>&=& y^{\xi_A} C_{ApA}(\xi_p\Delta_p-\xi_p + 2\xi_A + 2\pi i \rho \xi_p^2),\cr
\<\Psi_2|\phi_p(0,1)e^{2\pi i M_0}|\Psi_2\> &=& y^{\xi_A} C_{ApA} \, \xi_p^2. \label{eq5}
\end{eqnarray}
These can be obtained using the method given in Appendix \ref{sec:useful_identities} or as done in Appendix \ref{sec:matrxi_ele_level1}.
Substituting the above matrix elements and $K_{(2)}^{ij}$ in \eqref{block_level1_exp}, we obtain
\begin{equation}
\mathcal{F}_{1}(\Delta_p,\xi_p;\Delta_A,\xi_A|c_L,c_M|\rho) = 2+\frac{\xi_p(\Delta_p-1)}{\xi_A}+ \pi i \rho \frac{\xi_p^2}{\xi_A}-\frac{\Delta_A}{2\xi^2_A}\xi_p^2 .
\label{level1_bms}
\end{equation}
Note that this is a polynomial in $\xi_A$, $\Delta_A$, $\xi_p$ and $\Delta_p$. In later section we would be interested in the large $\xi_A$ limit. In this limit we could see that the leading term is 2, which is equal to the number of states and the subleading term is $\frac{\xi_p(\Delta_p-1)}{\xi_A}+ \pi i \rho \frac{\xi_p^2}{\xi_A}$. Similarly, we have calculated some of the details of the torus block for level 2. They are described in appendix \ref{appendix:identities}.
\subsection{Large $\xi_A$ limit of BMS torus blocks}
\label{sec:large_xi}
While deriving an asymptotic form for the BMS 3-point coefficients, we will need to consider the torus block of a heavy exchanged operator with dimension $$\xi_A\gg c_M,\xi_p,\Delta_p.$$
We shall call this the {\em{large $\xi_A$ limit}}. It is not required to know the relative values (ratios) of $\xi_A$, $\Delta_A$ and $c_L$ for the same, so, the asymptotic formula thus obtained will be valid for all values of $\Delta_A$, irrespective of any comparison to $\xi_A$ and $c_L$. We will find the leading and the sub-leading terms in the asymptotic expansion in $\frac{1}{\xi_A}$ of the torus block $\mathcal{F}_N$ in the large $\xi_A$ limit along with $\xi_A\gg \Delta_A$.
\subsubsection{Summary of results}
Since this particular part of the paper may become difficult to read, for the sanity of the reader who is mainly interested in the results, we will present the main results of the section.
\bigskip
\noindent{\em{Leading term in BMS torus block}}
\smallskip
\noindent The leading term of the BMS torus block in the large $\xi_A$ limit is given by
\begin{eqnarray}
\mathcal{F}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(\sigma,\rho) = \sum_N q^N \widetilde{\text{dim}}_N + \mathcal{O}\left(\frac{1}{\xi_A}\right) = \frac{q^{\frac{1}{12}}}{\eta(\sigma)^2} + \mathcal{O}\left(\frac{1}{\xi_A}\right),
\label{f_leading}
\end{eqnarray}
where $$\eta(\sigma)=e^{\frac{2\pi i\sigma}{24}}\prod_{n=1}^{\infty}(1-e^{2\pi i \sigma n})=q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^n).$$
\bigskip
\noindent{\em{Subleading term in BMS torus block}}
\smallskip
\noindent
We calculate the sub-leading terms in the BMS torus one-point function by intrinsic methods in a 2d BMSFT and the combined result of the leading and sub-leading orders is given by:
\begin{equation}
\mathcal{F}_{N}=\left(1+\frac{\xi_p(\Delta_p-1)}{2\xi_A}N\right)\widetilde{\text{dim}}_N +\pi i\rho\frac{\xi_p^2}{\xi_A} \sum_{k=0}^Np(N-k)p(k)(N-k)(N-2k) + \mathcal{O}(\xi_A^{-2}).
\end{equation}
Here $\mathcal{F}_{N} = \mathcal{F}_{N}(\Delta_p,\xi_p;\Delta_A,\xi_A|c_L,c_M|\rho)$ which is defined by \refb{F_N}. $p(N)$ counts the number of partitions of a given number $N$. The formula above is the result of a long and tedious calculation and a number theoretic conjecture \refb{conjecture} which we provide evidence for in Appendix \ref{secconjecture}. We reproduce the same answer in Appendix \ref{gensub} as a limit of analogous answers in 2d CFT (relevant details of which are reviewed in Appendix \ref{ApA}).
\subsubsection{Back to details}
We now dive head-long into the details of the computation of the BMS torus blocks. A number of appendices would give further details of the computations carried out in the following analysis. We make heavy use of the BMS mode expansions developed earlier to prove several of these relations. Some of the formulae useful for what we will calculate and their proofs are presented in Appendix C.
\medskip
\noindent For computing the BMS torus block to an arbitrary order, one of the principal tasks is to compute the inverse Gram-matrix to that order. Following our discussions of the generic form of the Gram matrix in Sec. 3, we will investigate this in detail now.
\medskip
\noindent
We are at first interested in isolating the elements of the inverse Gram matrix that would contribute to our computation of the torus block. This matrix has all its elements vanishing above the anti-diagonal barring those that are situated at the positions corresponding to the Gram-matrix elements that are norm-squared of the self-conjugate states. So, we don't have to know the value of $\<A;N,a|\phi_p(0,1)e^{2\pi i M_0}|A;N,b\>$ for which $K^{ab}_{(N)}=0$ since they will not contribute to $\mathcal{F}_N$ in \eqref{bms_block_fin}. In other words we only have to know the value of $\<A;N,a|\phi_p(0,1)e^{2\pi i \rho M_0}|A;N,b\>$ for which the corresponding Gram matrix elements
\begin{equation}
K_{ab}^{(N)} =\<A;N,a|A;N,b\> = \langle A|(\prod_{j=1}^N M_j^{k_j^\prime})(\prod_{j=1}^N L_j^{N_j^\prime-k_j^\prime})(\prod_{j=1}^N L_{-j}^{N_j-k_j})(\prod_{j=1}^N M_{-j}^{k_j})|A\rangle,
\end{equation}
lie on the anti-diagonal Gram-matrix-elements or vanishing Gram-matrix-elements lying below the anti-diagonal or are norm-squared of the self-conjugate states. Such a Gram-matrix element must have: $$\text{len }M\geqslant\text{min}\{\sum_{j=1}^NN_j,\sum_{j=1}^NN_j^\prime\}\geqslant\text{len }L.$$
Here $\text{len }M$ and $\text{len }L$ are the number of $M$-operators and $L$-operators in the inner product $\<A;N,a|A;N,b\>$ i.e.,
\begin{equation}
\text{len }M=\sum_{j=1}^N (k_j+k_j^\prime),\quad \text{len }L=\sum_{j=1}^N (N_j + N_{j}^{\prime}-k_j-k_j^\prime).
\end{equation}
We observed that in the large $\xi_A$ limit with $\xi_A\gg \Delta_A$, non-zero $K^{ab}_{(N)}$ will be of the order of $\mathcal{O}(\frac{1}{{\xi_A}^{\text{len }M}})$.
Next, we note that in the one point matrix
\begin{align}
\langle A|(\prod_{j=N}^1 M_j^{k_j^\prime})(\prod_{j=N}^1 L_j^{N_j^\prime-k_j^\prime})\phi_p(0,1)(\prod_{j=1}^N L_{-j}^{N_j-k_j})\prod_{j=1}^N (M_{-j}^{k_j})|A\rangle
\end{align}
the `uncontracted' $M$-operators give rise to factors of $\xi_p$ through the commutator with $\phi_p(0,1)$.
\newpage
\noindent As an example let us look at $\<A|L_1L_2\phi_p(0,1)M_{-1}M_{-1}M_{-1}|A\>$. After a bit of calculation, we have
\begin{eqnarray}
\label{uncontracted_M}
\<A|L_2L_1\phi_p(0,1) M_{-1}M_{-1}M_{-1}|A\> = && 3\xi_A [M_1,[\phi_p(0,1),M_{-1}]] \cr
&+& 9\<A|[M_1,[L_1[[\phi_p(0,1),M_{-1}],M_{-1}]]]|A\> \cr
&+& 3 \xi_A \<A|[L_2,[[\phi_p(0,1),M_{-1}],M_{-1}]]|A\> \cr
&+& 3\<A|[M_2,[[\phi_p(0,1),M_{-1}],M_{-1}]]|A\>\\
&+& \<A|[L_2,[L_1,[[[\phi_p(0,1),M_{-1}],M_{-1}],M_{-1}]]]|A\>.\nonumber
\end{eqnarray}
The factor of $\xi_A$ is due to $M_0$ which we get due to contraction of $L$'s with $M$'s. The commutator of the operators with $\phi_p$ are due to all the uncontracted operators. Each $[\phi_p,M]$ and $[\phi_p,L]$ will give a factor of $\xi_p$ and $\Delta_p$ respectively.
Hence, the generic element $\<A;N,a|\phi_p(0,1)|A;N,b\>$
is a polynomial in both $\xi_A$ and $\xi_p$ (and also in $\Delta_A^\prime$ and $\Delta_p$), with each monomial's $\xi$-dependence being of the form $~{\xi_A^\prime}^r{\xi_p}^s$, such that $r+s=\text{len }M$. Clearly, $r$ is the number of `contracted' $M$-operators while $s$ denotes the number of `uncontracted' $M$-operators for each monomial, with no two monomials having the same $r$, $s$. This is because these monomials have varying number of available $L$-operators for contracting $M$-operators as some $L$-operators are used in commutation with the $\phi_p(0,1)$-field but the total number of $M$-operators in each of them is same($=\text{len }M$).
\medskip
\noindent
Note that $$\<A|L_1L_2M_{-1}M_{-1}M_{-1}|A\>=0$$
and we can see from \eqref{uncontracted_M} that the number of uncontracted M-operators is equal to 2 or more. We gave a proof in Appendix \ref{sec:minimum_uncontracted} that this is true for any $\<A;N,a|\phi_p(0,1)|A;N,b\>$ for which the Gram matrix vanish, i.e.
$$ \<A;N,a|\phi_p(0,1)|A;N,b\> = 0 \quad \text{if} \, \<A;N,a|A;N,b\>=0 \, \text{and $\#$ of uncontracted} \, M_n>2.$$ So, the one point matrix elements corresponding to the vanishing Gram matrix will be a polynomial in which the exponents of $\xi_p$ for each term is greater than or equal to 2.
\medskip
\noindent
To this end, we note that the generic matrix element at level $N$ can be rewritten using \eqref{bch} as below
\begin{align}
&\langle A|(\prod_{j=N}^1 M_j^{k_j^\prime})(\prod_{j=N}^1 L_j^{N_j^\prime-k_j^\prime})\phi_p(0,1)y^{M_0}(\prod_{j=1}^N L_{-j}^{N_j-k_j})(\prod_{j=1}^N M_{-j}^{k_j})|A\rangle y^{-\xi_A}\nonumber\\
=&\langle A|(\prod_{j=N}^1 M_j^{k_j^\prime})(\prod_{j=N}^1 L_j^{N_j^\prime-k_j^\prime})\phi_p(0,1)(\prod_{j=1}^N (L_{-j}+2\pi i\rho jM_{-j})^{N_j-k_j})\prod_{j=1}^N (M_{-j}^{k_j})|A\rangle\nonumber\\
=&\langle A|(\prod_{j=N}^1 M_j^{k_j^\prime})(\prod_{j=N}^1 L_j^{N_j^\prime-k_j^\prime})\phi_p(0,1)(\prod_{j=1}^N L_{-j}^{N_j-k_j})\prod_{j=1}^N (M_{-j}^{k_j})|A\rangle\nonumber\\
&+(\text{terms with less right-$L$-string length and more right-$M$-string length}).
\end{align}
Clearly, for the matrix elements contributing to $\mathcal{F}_{N}(\Delta_p,\xi_p;\Delta_A,\xi_A|c_L,c_M|\rho)$, the first term is of the highest order($\leqslant\text{len }L$) in $\xi_A$ among all the terms (due to maximum number of $L$-operators and minimum number of $M$-operators, under the restriction $\text{len }M\geqslant\text{len }L$), though other terms may be of this same order. Thus, the contribution of a generic matrix element that corresponds to a vanishing Gram-matrix element lying below the anti-diagonal is $\mathcal{O}(\frac{1}{{\xi_A}^s})$ in the said limit, where $s$ is the number of uncontracted $M$-operators in the first term (clearly, other terms have $s^\prime\geqslant s$). We already mentioned that $s\geqslant2$ for vanishing Gram-matrix elements. Thus, in this limit, $K^{ab}_{(N)}\<A;N,a|\phi_p(0,1)y^{M_0}|A;N,b\>$ corresponding to any vanishing Gram-matrix element, do not contribute to the leading and sub-leading pieces of interest in $\mathcal{F}_N$.
Thus, only the matrix elements corresponding to the non-vanishing anti-diagonal Gram-elements or squared-norms of the self-conjugate states contribute to the said pieces in this limit.
The generic matrix element corresponding to a non-vanishing anti-diagonal Gram-element is given by:
\begin{align}
&\langle A|(\prod_{j=N}^1 M_j^{N_j-k_j})(\prod_{j=N}^1 L_j^{k_j})\phi_p(0,1)y^{M_0}(\prod_{j=1}^N L_{-j}^{N_j-k_j})(\prod_{j=1}^N M_{-j}^{k_j})|A\rangle y^{-\xi_A}\nonumber\\
=&\langle A|(\prod_{j=N}^1 M_j^{N_j-k_j})(\prod_{j=N}^1 L_j^{k_j})\phi_p(0,1)(\prod_{j=1}^N (L_{-j}+2\pi i\rho jM_{-j})^{N_j-k_j})(\prod_{j=1}^N M_{-j}^{k_j})|A\rangle\nonumber\\
=&\langle A|(\prod_{j=N}^1 M_j^{N_j-k_j})(\prod_{j=N}^1 L_j^{k_j})\phi_p(0,1)(\prod_{j=1}^N L_{-j}^{N_j-k_j})(\prod_{j=1}^N M_{-j}^{k_j})|A\rangle+\sum_{r=1}^N r(N_r-k_r)\nonumber\\
&\times (2\pi i\rho)\langle A|(\prod_{j=N}^1 M_j^{N_j-k_j})(\prod_{j=N}^1 L_j^{k_j})\phi_p(0,1)(\prod_{j=1}^{N} L_{-j}^{N_j-k_j-\delta_{j,r}}(\prod_{j=1}^N M_{-j}^{k_j+\delta_{j,r}})|A\rangle\nonumber\\
&+(\text{terms with less right-$L$-string length and more right-$M$-string length})
\label{one_point_matrix_limit}
\end{align}
The corresponding element in the inverse Gram-matrix is $\mathcal{O}({\xi_A^{-\sum_{j=1}^NN_j}})$. It is simply the inverse of the corresponding anti-diagonal Gram-element. As the terms not explicitly written above are $\mathcal{O}(\xi_A^{\text{len }L})$, with $\text{len }L=\sum_{j=1}^NN_j-r$, with $r\geqslant2$, their contribution to $\mathcal{F}_{N}(\Delta_p,\xi_p;\Delta_A,\xi_A|c_L,c_M|\rho)$ is $\mathcal{O}(\frac{1}{\xi_A^r})$; thus they don't contribute to the leading or sub-leading pieces in this limit. On the other hand, the first term has $r=0$ and the second one has $r=1$; hence they contribute.
\subsubsection{Leading term}
Now we will construct the leading term in the BMS torus block in the large $\xi_A$ expansion. For this, let us look at the first term in \eqref{one_point_matrix_limit}
\begin{equation}
\langle A|(\prod_{j=N}^1 M_j^{N_j-k_j})(\prod_{j=N}^1 L_j^{k_j})\phi_p(0,1)(\prod_{j=1}^N L_{-j}^{N_j-k_j})(\prod_{j=1}^N M_{-j}^{k_j})|A\rangle.
\label{leading_first_term}
\end{equation}
The leading term in $\xi_A$ will be the one in which all the $L$-operators and $M$-operators are contracted with each other
\begin{eqnarray}
&& \langle A|(\prod_{j=N}^1 M_j^{N_j-k_j})(\prod_{j=N}^1 L_j^{k_j})(\prod_{j=1}^N L_{-j}^{N_j-k_j})(\prod_{j=1}^N M_{-j}^{k_j})\phi_p(1,0)|A\rangle\cr
=&& \langle A|(\prod_{j=N}^1 M_j^{N_j-k_j})(\prod_{j=N}^1 L_j^{k_j})(\prod_{j=1}^N L_{-j}^{N_j-k_j})(\prod_{j=1}^N M_{-j}^{k_j})|A\rangle\langle A|\phi_p(1,0)|A\rangle\cr
= && C_{ApA} \langle A|(\prod_{j=N}^1 M_j^{N_j-k_j})(\prod_{j=N}^1 L_j^{k_j})(\prod_{j=1}^N L_{-j}^{N_j-k_j})(\prod_{j=1}^N M_{-j}^{k_j})|A\rangle.
\label{matrix_leading}
\end{eqnarray}
The inverse Gram matrix elements $K_{(N)}^{ab}$ for
\begin{equation}
K_{ab}^{(N)}=\langle A|(\prod_{j=N}^1 M_j^{N_j-k_j})(\prod_{j=N}^1 L_j^{k_j})(\prod_{j=1}^N L_{-j}^{N_j-k_j})(\prod_{j=1}^N M_{-j}^{k_j})|A\rangle
\end{equation}
are simply given by
\begin{equation}
K_{(N)}^{ab} = \left[{\langle A|(\prod_{j=N}^1 M_j^{N_j-k_j})(\prod_{j=N}^1 L_j^{k_j})(\prod_{j=1}^N L_{-j}^{N_j-k_j})(\prod_{j=1}^N M_{-j}^{k_j})|A\rangle}\right]^{-1}.
\end{equation}
Thus we have
\begin{eqnarray}
&& \text{Leading order in }K^{ab}_{(N)}\<A;N,a|\phi_p(0,1)y^{M_0}|A;N,b\> \nonumber\\ \cr
&& = y^{\xi_A}\frac{\langle A|(\prod_{j=N}^1 M_j^{N_j-k_j})(\prod_{j=N}^1 L_j^{k_j})(\prod_{j=1}^N L_{-j}^{N_j-k_j})(\prod_{j=1}^N M_{-j}^{k_j})|A\rangle\langle A|\phi_p(1,0)|A\rangle}{\langle A|(\prod_{j=N}^1 M_j^{N_j-k_j})(\prod_{j=N}^1 L_j^{k_j})(\prod_{j=1}^N L_{-j}^{N_j-k_j})(\prod_{j=1}^N M_{-j}^{k_j})|A\rangle} \nonumber\\ \cr
&&= y^{\xi_A}\langle A|\phi_p(0,1)|A\rangle = y^{\xi_A}C_{ApA}.
\end{eqnarray}
Each non-zero anti-diagonal terms in the Gram matrix and diagonal elements corresponding to self-conjugate states contributes the same factor as above.
Since we have $\widetilde{\text{dim}}_N$ number of such elements, the leading term in $\mathcal{F}_{N}$ is given by
\begin{equation}
\text{Leading term} = \frac{1}{y^{\xi_A}C_{ApA}} y^{\xi_A}C_{ApA} \widetilde{\text{dim}}_N = \widetilde{\text{dim}}_N.
\end{equation}
So, we have
\begin{equation}
\mathcal{F}_N = \widetilde{\text{dim}}_N + \mathcal{O}\left(\frac{1}{\xi_A}\right).
\end{equation}
Using this, we can see from \eqref{bms_block_fin} that
\begin{eqnarray}
\mathcal{F}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(\sigma,\rho) = \sum_N q^N \widetilde{\text{dim}}_N + \mathcal{O}\left(\frac{1}{\xi_A}\right) = \frac{q^{\frac{1}{12}}}{\eta(\sigma)^2} + \mathcal{O}\left(\frac{1}{\xi_A}\right),
\label{f_leading}
\end{eqnarray}
where $$\eta(\sigma)=e^{\frac{2\pi i\sigma}{24}}\prod_{n=1}^{\infty}(1-e^{2\pi i \sigma n})=q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^n).$$
It is of interest to note here that if we modify the definition of the BMS torus block \refb{bms_torus_block} to keep the exponential factors $q^{\Delta_A-c_L/2}$ and $y^{\xi_A-c_M/2}$, the expression of the leading term BMS torus block is exactly the expression for the BMS character found in \cite{Bagchi:2019unf, Oblak:2015sea, Barnich:2015mui, Garbarz:2015lua}.
\subsubsection{Subleading term}
Now, we focus our attention on the sub-leading pieces in the BMS torus block in our large $\xi_A$ expansion. We shall see that there are actually two sources for the sub-leading correction. We being our descriptions with an example.
\medskip
\noindent The sub-leading contribution to \eqref{leading_first_term} is due to the terms in which all the $L$-operators and $M$-operators contract to produce $M_0$ except for one pair of $L$ and $M$. This will have one less $\xi_A$ as compared to the leading term. The un-contracted pair of $L$ and $M$ appear in the commutator with $\phi_p$ in the form $[L_k,[\phi_p(0,1),M_{-k}]]$ or $[M_k,[\phi_p(0,1),L_{-k}]]$. Let us consider an example:
\begin{eqnarray}
&&\text{Sub-leading term in } \<A|M_1^3L_2^2\phi_p(0,1)L_{-1}^3M_{-2}^3|A\> \nonumber\\
&=& 9\<A|M_1^2L_2^2L_{-1}^2M_{-2}^3[M_1,[\phi_p(0,1),L_{-1}]|A\> + 6\<A|M_1^3L_2^1L_{-1}^3M_{-2}^2[L_2,[\phi_p(0,1),M_{-2}]|A\> \cr
&=& 9\<A|M_1^2L_2^2L_{-1}^2M_{-2}^3|A\>\<A|[M_1,[\phi_p(0,1),L_{-1}]|A\> + 6\<A|M_1^3L_2^1L_{-1}^3M_{-2}^2|A\>\<A|[L_2,[\phi_p(0,1),M_{-2}]|A\>. \nonumber
\end{eqnarray}
The factor of 9 is the number of choosing a pair of $M_1$ and $L_{-1}$ that appear in the commutator $\<A|[M_1,[\phi_p(0,1),L_{-1}]|A\>$. Since we have 3 $M_1^3$ and 3 $L_{-1}^3$, the total of pairs is given by $3\times 3 =9$. Similarly the factor of 6 is the number of choosing a pair of $L_2$ and $M_{-2}$ that appear in the commutator $\<A|[L_2,[\phi_p(0,1),M_{-2}]|A\>$.
\medskip
\noindent As mentioned, we will have two types of terms contributing at the next-to-leading order to the BMS torus block. Following the example above, we can deduce the first type of term, which arises from the sub-leading term for a generic anti-diagonal element \eqref{leading_first_term}. This first type consist of terms with
$$\<A|[M_j,[\phi_p(0,1),L_{-j}]|A\> \quad \text{or} \quad \<A|[L_j,[\phi_p(0,1),M_{-j}]|A\>.$$
We can see that there are $(N_j-k_j)^2$ ways of choosing a pair of $M_j$ and $L_{-j}$ and $k_j^2$ number of ways of choosing a pair of $L_j$ and $M_{-j}$. Thus the subleading term is given by
\begin{eqnarray}
&&\text{Type 1 Sub-leading term (SLT$_1$)} = \nonumber\\
&&\quad \sum_{r=1}^{N}(N_r-k_r)^2 \langle A|\prod_{j=N}^1 M_j^{N_j-k_j-\delta_{j,r}}\prod_{j=N}^1 L_j^{k_j}\prod_{j=1}^N L_{-j}^{N_j-k_j-\delta_{j,r}}\prod_{j=1}^N M_{-j}^{k_j}|A\rangle \<A|[M_r,[\phi_p(0,1),L_{-r}]]|A\> \cr
&& \quad +\sum_{r=1}^{N}k_r^2 \langle A|(\prod_{j=N}^1 M_j^{N_j-k_j})(\prod_{j=N}^1 L_j^{k_j-\delta_{j,r}})(\prod_{j=1}^N L_{-j}^{N_j-k_j})(\prod_{j=1}^N M_{-j}^{k_j-\delta_{j,r}})|A\rangle \<A|[L_r,[\phi_p(0,1),M_{-r}]]|A\>. \nonumber\\
\end{eqnarray}
Now, we have
\begin{equation}
\<A|[M_r,[\phi_p(0,1),L_{-r}]]|A\> =
\<A|[L_r,[\phi_p(0,1),M_{-r}]]|A\> = C_{ApA}\,r^2 \xi_p(\Delta_p-1) .
\end{equation}
Substituting this in the previous equation we get
\begin{eqnarray}
\text{SLT$_1$} &=&C_{ApA}\sum_{r=1}^{N}(N_r-k_r)^2 r^2 \xi_p(\Delta_p-1) \langle A|\prod_{j=N}^1 M_j^{N_j-k_j-\delta_{j,r}} \prod_{j=N}^1 L_j^{k_j}\prod_{j=1}^N L_{-j}^{N_j-k_j-\delta_{j,r}}\prod_{j=1}^N M_{-j}^{k_j}|A\rangle \cr
&& +C_{ApA}\sum_{r=1}^{N}k_r^2 r^2 \xi_p(\Delta_p-1) \langle A|\prod_{j=N}^1 M_j^{N_j-k_j}\prod_{j=N}^1 L_j^{k_j-\delta_{j,r}}\prod_{j=1}^N L_{-j}^{N_j-k_j}\prod_{j=1}^N M_{-j}^{k_j-\delta_{j,r}}|A\rangle.
\end{eqnarray}
We have to multiply this by the corresponding inverse Gram matrix $K^{ab}_{(N)}$. As we have already pointed out, for the one point matrices of interest, these are simply given by $K^{ab}_{(N)}=\frac{1}{K_{ab}^{(N)}}$.
So, we get
\begin{eqnarray}
\text{SLT$_1$} &=& C_{ApA}\sum_{r=1}^{N}\frac{(N_r-k_r)^2 r^2 \xi_p(\Delta_p-1) \prod_{j=1}^N \<A|M_j^{N_j-k_j-\delta_{j,r}}L_{-j}^{N_j-k_j-\delta_{j,r}}|A\>\<A|L_j^{k_j}M_{-j}^{k_j}|A\>}{\prod_{j=1}^N \<A|M_j^{N_j-k_j}L_{-j}^{N_j-k_j}|A\>\<A|L_j^{k_j}M_{-j}^{k_j}|A\>}\nonumber\\
&& +C_{ApA}\sum_{r=1}^{N}\frac{k_r^2 r^2 \xi_p(\Delta_p-1) \prod_{j=1}^N \<A|M_j^{N_j-k_j}L_{-j}^{N_j-k_j}|A\>\<A|L_j^{k_j-\delta_{j,r}}M_{-j}^{k_j-\delta_{j,r}}|A\>}{{\prod_{j=1}^N \<A|M_j^{N_j-k_j}L_{-j}^{N_j-k_j}|A\>\<A|L_j^{k_j}M_{-j}^{k_j}|A\>}},
\end{eqnarray}
Using \eqref{gram_matrix_property3} and \eqref{gram_matrix_property4} the above expression is given by
\begin{eqnarray}
\text{SLT$_1$}&=& C_{ApA}\sum_{r=1}^{N}\frac{(N_r-k_r) r^2 \xi_p(\Delta_p-1) }{\<A|M_rL{-r}|A\>} + C_{ApA}\sum_{r=1}^{N}\frac{k_r r^2 \xi_p(\Delta_p-1) }{\<A|L_rM{-r}|A\>}.
\end{eqnarray}
In the large $\xi_A$ limit we have
\begin{equation}
\<A|L_rM{-r}|A\> = 2r\xi_A -c_M(r^3-r) \approx 2r\xi_A.
\end{equation}
Substituting this in the previous equation, we get
\begin{eqnarray}
\text{SLT$_1$}&=& C_{ApA}\sum_{r=1}^{N}\frac{(N_r-k_r) r \xi_p(\Delta_p-1) }{2\xi_A} + C_{ApA}\sum_{r=1}^{N}\frac{k_r r \xi_p(\Delta_p-1) }{2\xi_A} \cr
&& \quad = C_{ApA} \frac{\xi_p(\Delta_p-1)}{2\xi_A} \sum_{r=1}^{N}rN_r = C_{ApA} \frac{\xi_p(\Delta_p-1) }{2\xi_A} N
\end{eqnarray}
This is only for one term. Thus the total subleading contribution of these Type 1 subleading terms in $\mathcal{F}_{N}$ due to \eqref{leading_first_term}, after taking care of the factors of $C_{ApA}$ and $y^{\xi_A}$ is
\begin{equation}
\text{Total contribution from all SLT$_1$} = N\frac{\xi_p(\Delta_p-1) }{2\xi_A} \widetilde{\text{dim}}_N.
\end{equation}
The other sub-leading piece in the BMS torus block comes from the second term in \eqref{one_point_matrix_limit}
\begin{eqnarray}
&&\text{Type 2 Sub-leading term (SLT$_2$)} = \nonumber\\
&& \quad 2\pi i\rho \sum_{r=1}^N r(N_r-k_r)\langle A| \prod_{j=N}^1 M_j^{N_j-k_j} \prod_{j=N}^1 L_j^{k_j} \ \phi_p(0,1) \ \prod_{j=1}^{N} L_{-j}^{N_j-k_j-\delta_{j,r}} \prod_{j=1}^N M_{-j}^{k_j+\delta_{j,r}}|A\rangle \cr
\label{sublead_2}
&&
\end{eqnarray}
The leading term in this is will be subleading in $\mathcal{F}_N$. These are the term in which one $M_{j}$ from $ M_j^{N_j-k_j}$ and an $M_{-j}$ from $M_{-j}^{k_j+\delta_{j,r}}$ are used in the commutator with $\phi_p$ i,e., $[M_j,[\phi_p,M_{-j}]]$. With this, the remaining $M$-operators is equal to the number of $L$-operators and thus can be contracted with no uncontracted $M$-operators. More explicitly, this is given by
\begin{eqnarray}
\text{SLT$_2$} = && 2\pi i\rho \sum_{r=1}^N r(N_r-k_r) \langle A|\prod_{j=N}^1 M_j^{N_j-k_j-\delta_{j,r}} \prod_{j=N}^1 L_j^{k_j} \prod_{j=1}^{N} L_{-j}^{N_j-k_j-\delta_{j,r}}\prod_{j=1}^N M_{-j}^{k_j}|A\rangle \nonumber\\
&& \quad \times \<A|\phi(0,1)|A\>(N_r-k_r)(k_r+1) \<A|[M_r,[\phi_p(0,1),M_{-r}]]|A\>,
\end{eqnarray}
where $(N_r-k_r)(k_r+1)$ is the number of ways of choosing a pair of $M_r$ and $M_{-r}$.
We have to multiply the above by the inverse Gram matrix elements. From our previous calculation we have
\begin{eqnarray}
&& \frac{\langle A|(\prod_{j=N}^1 M_j^{N_j-k_j-\delta_{j,r}})(\prod_{j=N}^1 L_j^{k_j})(\prod_{j=1}^{N} L_{-j}^{N_j-k_j-\delta_{j,r}}(\prod_{j=1}^N M_{-j}^{k_j})|A\rangle}{\langle A|(\prod_{j=N}^1 M_j^{N_j-k_j})(\prod_{j=N}^1 L_j^{k_j})(\prod_{j=1}^{N} L_{-j}^{N_j-k_j}(\prod_{j=1}^N M_{-j}^{k_j})|A\rangle}\nonumber\\
&&\qquad = \frac{1}{(N_r-k_r)\<A|[M_r,L_{-r}]|A\>} \approx \frac{1}{(N_r-k_r)2r\xi_A}.
\end{eqnarray}
Using $\<A|[M_r,[\phi_p(0,1),M_{-r}]]|A\>=r^2\xi_p^2$, the type 2 sub-leading term due to \eqref{sublead_2} is given by
\begin{eqnarray}
\text{SLT$_2$}=2\pi i\rho \sum_{r=1}^N \frac{\,r^3(N_r-k_r)^2(k_r+1)}{(N_r-k_r)2r\xi_A}= (2\pi i\rho)\frac{\xi_p^2}{2\xi_A}\sum_{r=1}^N r^2(N_r-k_r)(k_r+1).
\end{eqnarray}
Let us change our notation a bit, we will replace $N_r-k_r$ by $l_r$ and $k_r$ by $m_r$. With this new notation, we have
\begin{eqnarray}
\text{SLT$_2$}= (2\pi i\rho)\frac{\xi_p^2}{2\xi_A}\sum_{r=1}^N r^2l_r(m_r+1).
\end{eqnarray}
This is the contribution due to only one matrix elements. We have to sum up for all the sets $\{\{l_r\},\{m_r\}\}$ for which
\begin{equation}
\sum_{r=1}^N r(l_r+m_r)=N,
\end{equation}
Let us denote this set by $\mathcal{A}$. Thus the subleading term due to \eqref{sublead_2} is
\begin{equation}
\text{Total contribution from all SLT$_2$}= (2\pi i\rho)\frac{\xi_p^2}{2\xi_A}\sum_{\{\{l_r\},\{m_r\}\}\in \mathcal{A}}\sum_{r=1}^N r^2l_r(m_r+1).
\end{equation}
Combining all our results, we get
\begin{align}
\mathcal{F}_{N}(\Delta_p,\xi_p;\Delta_A,\xi_A|c_L,c_M|\rho)=&\left(1+\frac{\xi_p(\Delta_p-1)}{2\xi_A}N\right)\widetilde{\text{dim}}_N\nonumber\\
&+(2\pi i\rho)\frac{\xi_p^2}{2\xi_A}\sum_{\{\{l_r\},\{m_r\}\}\in A}\sum_{r=1}^N r^2l_r(m_r+1)+\mathcal{O}(\frac{1}{\xi_A^2}).
\label{blocks_large_limit}
\end{align}
\subsection*{A conjecture and a rewriting}
We will now conjecture an identity involving the partition of integers using two colours which will enable us rewrite our earlier equation \refb{blocks_large_limit}. Consider the partition of an integer $N$ using two colours in the form: $$\sum_{j=1}^Nj(l_j+m_j)=N, \quad \forall \, l_j,m_j\geqslant0.$$
Let us denote the set of all such different partitions as $A$ and denote an element in $A$ as $\{\{l_j\},\{m_j\}\}$. Now, we conjecture the following relation:
\begin{align}
\sum_{\{\{l_j\},\{m_j\}\}\in A}\,\,\sum_{j=1}^N j^2l_j(m_j+1)&=\frac{1}{2}\sum_{k=0}^Np(N-k)p(k)(N-2k)^2\cr
&= \sum_{k=0}^Np(N-k)p(k)(N-k)(N-2k).
\label{conjecture}
\end{align}
Although we don't have a proof of this conjecture, we have checked for the validity of \eqref{conjecture} upto level 5. Some of the details are described in appendix \ref{secconjecture}. Substituting \eqref{conjecture} in \eqref{blocks_large_limit}, we have
\bigskip
\noindent {\fbox{
\addtolength{\linewidth}{+2\fboxsep}%
\addtolength{\linewidth}{+2\fboxrule}%
\begin{minipage}{\linewidth}
\begin{eqnarray}
\mathcal{F}_{N}(\Delta_p,\xi_p;\Delta_A,\xi_A|c_L,c_M|\rho)=&&\left(1+\frac{\xi_p(\Delta_p-1)}{2\xi_A}N\right)\widetilde{\text{dim}}_N \\
&&+\pi i\rho\frac{\xi_p^2}{\xi_A} \sum_{k=0}^Np(N-k)p(k)(N-k)(N-2k) + \mathcal{O}\left(\frac{1}{\xi_A^2}\right). \nonumber
\label{onepoint5}
\end{eqnarray}
\end{minipage}
}}
\bigskip
\subsubsection{The special case of $\xi_p=0$}\label{tbxip0}
As noted previously, the uncontracted $M$-operators in the relevant matrix elements of \eqref{F_N} give rise to factors of $\xi_p$. So, when $\xi_p=0$, only the matrix elements corresponding to the non-vanishing anti-diagonal Gram-elements or squared-norms of the self-conjugate states give non-vanishing contribution to $\mathcal{F}_{N}(\Delta_p,\xi_p;\Delta_A,\xi_A|c_L,c_M|\rho)$. From \eqref{one_point_matrix_limit} and \eqref{matrix_leading}, it is evident that for $\xi_p=0$:
\begin{align}
&\langle A|(\prod_{j=N}^1 M_j^{N_j-k_j})(\prod_{j=N}^1 L_j^{k_j})\phi_p(0,1)y^{M_0}(\prod_{j=1}^N L_{-j}^{N_j-k_j})(\prod_{j=1}^N M_{-j}^{k_j})|A\rangle y^{-\xi_A}\nonumber\\
&=\langle A|(\prod_{j=N}^1 M_j^{N_j-k_j})(\prod_{j=N}^1 L_j^{k_j})\phi_p(0,1)(\prod_{j=1}^N L_{-j}^{N_j-k_j})(\prod_{j=1}^N M_{-j}^{k_j})|A\rangle\nonumber\\
&\quad +(\text{terms with atleast 1 uncontracted $M$-operators, hence giving 0})\nonumber\\
&=\langle A|(\prod_{j=N}^1 M_j^{N_j-k_j})(\prod_{j=N}^1 L_j^{k_j})(\prod_{j=1}^N L_{-j}^{N_j-k_j})(\prod_{j=1}^N M_{-j}^{k_j})|A\rangle y^{-\xi_A}\langle A|\phi_p(0,1)|A\rangle+0.
\end{align}
Thus the leading term in $\xi_A$ is the exact result in a closed-form for $\xi_p=0$. Consequently, we have
\begin{align}
\mathcal{F}_{N}(\Delta_p,\xi_p=0;\Delta_A,\xi_A|c_L,c_M|\rho)_{A\neq Id.}=\widetilde{\text{dim}}_N =\sum_{n=0}^Np(n)p(N-n).
\end{align}
Thus the `torus block' for a primary with $\xi_p=0$ is given by
\begin{align}
\mathcal{F}^{\Delta_p,\xi_p=0}_{\Delta_A,\xi_A:A\neq Id.,c_L,c_M}(\sigma,\rho)=\sum_{N=0}^{\infty}q^N\widetilde{\text{dim}}_N=\frac{q^\frac{1}{12}}{\eta(\sigma)^2}.
\end{align}
So, for $\xi_p=0$, the torus block doesn't depend on $\Delta_p$. Hence, the one-point function for a primary-field\footnote{We restrict the BMSFT to only contain primary-fields with $\Delta_p>0$, with the exception of the Identity being the only primary-field with $\Delta_p=0$.} with $\Delta_p>0$ and $\xi_p=0$ is as follows (the vacuum block doesn't contribute for $\Delta_p>0$, as $\langle 0| \phi_p(1,0)|0\rangle=\delta_{\Delta_p,0}$)
\begin{align}
\langle \phi_{\Delta_p>0,\xi_p=0}\rangle_{(\sigma, \rho)}=\frac{q^{\frac{1}{12}-\frac{c_L}{2}}y^{-\frac{c_M}{2}}}{\eta(\sigma)^2}\sum_{A\neq Id.} D(\Delta_A,\xi_A)\langle A| \phi_p(0,1)|A\rangle q^{\Delta_A}y^{\xi_A}.
\label{one_pt_zero_xi}
\end{align}
\bigskip
\subsection{Minkowski torus block from Casimir equation}
We will now digress a bit from the main theme of the paper to discuss an important point, which we will not use later in the paper. In this section we will find the global BMS torus block or the Minkowski torus block, using differential equation from the Casimir operators. Of course, the name Minkowski torus block is because the BMS$_3$ algebra, restricted to the ``global" part $n=0, \pm1$ is just $iso(2,1)$, the Poincare algebra in three bulk dimensions. We are interested, as always, in a 2d field theory with $iso(2,1)$ as its symmetries.
\medskip
\noindent We will adopt the ideas and methods of \cite{Kraus:2017ezw} with suitable modifications to obtain the BMS versions of the analogous 2d CFT results. These methods are also similar to the ones in \cite{Bagchi:2016geg} used to obtain the BMS four-point blocks.
\medskip
\noindent We begin by rewriting the block a bit as shown below
\begin{eqnarray}
\mathcal{F}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(q,y) &=& \frac{{\rm Tr}_{\Delta_A,\xi_A}\left(\phi_p(u,\phi)q^{L_0-\frac{c_L}{2}}y^{M_0-\frac{c_M}{2}}\right)}{q^{\Delta_A-\frac{c_L}{2}}y^{\xi_A-\frac{c_M}{2}}C_{ApA}}\cr
&=&\frac{q^{-\Delta_A}y^{-\xi_A}}{C_{ApA}}{\rm Tr}_{\Delta_A,\xi_A}\left(q^{L_0}y^{M_0}\phi_p(u,\phi)\right)\cr
&\equiv& \frac{q^{-\Delta_A}y^{-\xi_A}}{C_{ApA}}\tilde{\mathcal{F}}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(q,y).
\end{eqnarray}
Note that $\phi_p(u,\phi)$ is the primary field on the cylinder and instead of expressing the blocks as a function of $\sigma$ and $\rho$, we are using the variables $q=e^{2\pi i \sigma}$ and $y=e^{2\pi i \rho}$.
We also use the cyclicity property of trace to move $q^{L_0}y^{M_0}$ to the front (also remember that $L_0$ and $M_0$ commute).
Let $P_A$ be the projector operator for the BMS module of the highest weight state $|\Delta_A,\xi_A\>$ i.e., it acts as an identity operator for any state in the module and annihilate any other states which does not belong to the module. It is given by
\begin{equation}
P_A = \sum_{N=0}^{\infty} K_{(N)}^{ij}|A;N,i\>\<A;N,j|.
\label{projector}
\end{equation}
We can use $P_A $ to project out the contribution of the BMS module of $|\Delta_A,\xi_A\>$ to the one-point function. So, the blocks $\tilde{\mathcal{F}}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(q,y)$ is given by
\begin{eqnarray}
\tilde{\mathcal{F}}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(q,y)= {\rm{Tr}}_{\Delta_A,\xi_A}\left(q^{L_0} y^{M_0} \phi_p(u,\phi) \right)={\rm{Tr}}\left(P_A\,q^{L_0} y^{M_0} \phi_p(u,\phi) \right),
\label{torus_block_proj}
\end{eqnarray}
where $\rm{Tr}$, without the subscript $\Delta_A$ and $\xi_A$, is trace over the basis of the whole Hilbert space of the theory.
\medskip
\noindent Just like what we have in the case of four-point conformal block, in the large central charge limit, the leading term of the torus block in \eqref{torus_block_proj} is given by tracing over states of the form $L_{-1}^k M_{-1}^q|\Delta_A,\xi_A\>$. So, at level $N$, we have to consider only states of the form $\sum_{k=0}^N (L_{-1})^{N-k}(M_{-1})^k|\Delta_A,\xi_A\>$. Likewise, the states $|A;N,i\>$ appearing in the projector in \eqref{projector} will only be of this form in the large central charge limit. Thus, the global part of $\tilde{\mathcal{F}}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(q,y)$ is schematically given by
\begin{eqnarray}
\tilde{f}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(q,y) = \text{Tr}_g\left(P_A^g\,q^{L_0} y^{M_0}\phi_p(u,\phi) \right)=\text{Tr}_{g,A}\left(q^{L_0} y^{M_0} \phi_p(u,\phi) \right),
\label{global_block_def}
\end{eqnarray}
where the index $g$ in $\text{Tr}_g$ and $P_A^g$ means that we are tracing only over states of the form $L_{-1}^k M_{-1}^q|\Delta,\xi\>$ and the projector also consists only of states of this form.
Likewise, $\text{Tr}_{g,A}$ is over states of the form $L_{-1}^k M_{-1}^q|\Delta_A,\xi_A\>$.
\medskip
\noindent
The global subgroup of BMS is generated by the set of generators $\{L_{0,\pm 1} 1,M_{0,\pm 1}\}$ which from a closed sub-algebra. The quadratic Casimir's of these generators are given by
\begin{eqnarray}
\mathcal{C}_1 &=& M_0^2-M_{-1}M_1 \\
\mathcal{C}_2 &=& 2L_0M_0-\frac{1}{2}(L_{-1}M_1+L_1M_{-1}+M_1L_{-1}+M_{-1}L_1)\cr
&=& 2L_0M_0 - L_{-1}M_1 - M_{-1}L_1 -2M_0.
\end{eqnarray}
Since $L_{-1}$ and $M_{-1}$ commutes with $\mathcal{C}_1$ and $\mathcal{C}_2$, the states $L_{-1}^k M_{-1}^q|\Delta_A,\xi_A\>$ are eigenstates of these operators
\begin{equation}
\mathcal{C}_{1,2} L_{-1}^k M_{-1}^q|\Delta_A,\xi_A\> = \lambda_{1,2} L_{-1}^k M_{-1}^q|\Delta_A,\xi_A\>,
\end{equation}
with eigenvalues given by
\begin{equation}
\lambda_1 = \xi_A^2, \quad \lambda_2 = 2\xi_A(\Delta_A-1).
\end{equation}
Thus we have
\begin{equation}
\mathcal{C}_{1,2} P_A^g = \lambda_{1,2} P_A^g,\quad P_A^g \mathcal{C}_{1,2}= \lambda_{1,2} P_A^g.
\end{equation}
Using the above equation in \eqref{global_block_def}, we obtain
\begin{eqnarray}
\text{Tr}_g\left( P_A^g \,\mathcal{C}_{1,2}\,q^{L_0} y^{M_0}\phi_p(u,\phi) \right) &=& \lambda_{1,2} \tilde{f}(q,y)\cr
\implies \text{Tr}_{g,A}\left(\mathcal{C}_{1,2}\,q^{L_0} y^{M_0}\phi_p(u,\phi) \right) &=& \lambda_{1,2} \tilde{f}(q,y),
\end{eqnarray}
where we denote $\tilde{f}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(q,y)$ by $\tilde{f}(q,y)$ to reduce cluttering of indices.
In the LHS, we can move $\mathcal{C}_{1,2}$ toward the right by commuting with $q^{L_0}$ and $y^{M_0}$. For this, we will make use of the identities given below
\begin{eqnarray}
&& L_n q^{L_0}=q^{n+L_0}L_n,\quad L_n y^{M_0} = y^{M_0}L_n + n\log y\, y^{M_0} M_n,\quad M_n q^{L_0}=q^{L_0+n}M_n.\cr
&&
\label{identities_qy}
\end{eqnarray}
Using the fact that
\begin{equation}
L_0 q^{L_0} = q \partial_q q^{L_0}, \quad M_0 y^{M_0} = y \partial_y y^{M_0},
\end{equation}
the insertion of $L_0$ and $M_0$ is same as acting with differential operator $q \partial_q$ and $y \partial_y$
\begin{eqnarray}
\text{Tr}_{g,A}\left(L_0\,q^{L_0} y^{M_0}\phi_p(u,\phi) \right) &=& q \partial_q \text{Tr}_{g,A}\left(q^{L_0} y^{M_0}\phi_p(u,\phi)\right),\cr
\text{Tr}_{g,A}\left(M_0\,q^{L_0} y^{M_0}\phi_p(u,\phi) \right) &=& y \partial_y \text{Tr}_{g,A}\left(q^{L_0} y^{M_0}\phi_p(u,\phi)\right).
\end{eqnarray}
For the insertion of $M_n$, we use \eqref{identities_qy} as well as the cyclicity property of trace to obtain
\begin{eqnarray}
\text{Tr}_{g,A}&& \hspace{-.5cm}\left(M_n\,q^{L_0} y^{M_0}\phi_p(u,\phi) \right) = q^n \text{Tr}_{g,A}\left(\,q^{L_0} y^{M_0}M_n\phi_p(u,\phi) \right) \nonumber\\
&=& q^n \text{Tr}_{g,A}\left(\,q^{L_0} y^{M_0}\phi_p(u,\phi)M_n \right) + q^n \text{Tr}_{g,A}\left(\,q^{L_0} y^{M_0}[M_n,\phi_p(u,\phi)] \right) \nonumber\\
&=& q^n \text{Tr}_{g,A}\left(M_n\,q^{L_0} y^{M_0}\phi_p(u,\phi)M_n \right) + q^n \text{Tr}_{g,A}\left(\,q^{L_0} y^{M_0}\mathcal{M}_n \phi_p(u,\phi) \right)\nonumber\\
&=& q^n \text{Tr}_{g,A}\left(M_n\,q^{L_0} y^{M_0}\phi_p(u,\phi)M_n \right) + q^n \mathcal{M}_n \text{Tr}_{g,A}\left(\,q^{L_0} y^{M_0} \phi_p(u,\phi) \right),
\end{eqnarray}
where the differential operator $\mathcal{M}_n$ is defined as
\begin{equation}
[M_n,\phi_p(u,\phi)] \equiv \mathcal{M}_n \phi_p(u,\phi).
\end{equation}
Similarly, we define the differential operator $\mathscr{L}_n$ by
\begin{equation}
[L_n,\phi_p(u,\phi)] \equiv \mathscr{L}_n \phi_p(u,\phi).
\end{equation}
Rearranging the above equation we get
\begin{equation}
\text{Tr}_{g,A}\left(M_n\,q^{L_0} y^{M_0}\phi_p(u,\phi) \right) = \frac{q^n}{1-q^n} \mathcal{M}_n \tilde{f}(q,y).
\end{equation}
We also have
\begin{eqnarray}
\text{Tr}_{g,A}\left(L_n\,q^{L_0} y^{M_0}\phi_p(u,\phi) \right) =\frac{q^n}{1-q^n} \mathscr{L}_n \tilde{f}(q,y) + \frac{q^n n \log y}{(1-q^n)^2}\mathcal{M}_n \tilde{f}(q,y).
\end{eqnarray}
We could insert an additional $M$ or $L$ in the above two equations to obtain the following identities (note that $n=1,-1$)
\begin{equation}
\text{Tr}_{g,A}\left(M_{-n}M_n\,q^{L_0} y^{M_0}\phi_p(u,\phi) \right) = \frac{1}{(1-q^n)(1-q^{-n})}\mathcal{M}_n \mathcal{M}_{-n}\tilde{f}(q,y),
\end{equation}
\newpage
\begin{eqnarray}
\text{Tr}_{g,A}\left(M_{-n}L_n\,q^{L_0} y^{M_0}\phi_p(u,\phi) \right) =&& \frac{1}{(1-q^n)(1-q^{-n})}\mathscr{L}_n\mathcal{M}_{-n}\tilde{f}(q,y) \\
&& + \frac{2n q^n y}{1-q^n}\partial_y \tilde{f}(q,y)+ \frac{n \log y}{(1-q^n)^2(1-q^{-n})}\mathcal{M}_n\mathcal{M}_{-n}\tilde{f}(q,y), \nonumber \\
\text{Tr}_{g,A}\left(L_{-n}M_n\,q^{L_0} y^{M_0}\phi_p(u,\phi) \right)
=&& \frac{1}{(1-q^n)(1-q^{-n})}\mathcal{M}_n\mathscr{L}_{-n}\tilde{f}(q,y) \\
&& + \frac{2n q^n y}{1-q^n}\partial_y \tilde{f}(q,y)- \frac{n \log y}{(1-q^n)(1-q^{-n})^2}\mathcal{M}_n\mathcal{M}_{-n}\tilde{f}(q,y).\nonumber
\end{eqnarray}
Now that we have the required identities, let us find the differential equation for $\tilde{f}(q,y)$ using the quadratic Casimir $\mathcal{C}_1$. For this we have
\begin{eqnarray}
\text{Tr}_{g,A}\left(\mathcal{C}_1\,q^{L_0} y^{M_0}\phi_p(u,\phi) \right) &=& \text{Tr}_{g,A}\left(M_0^2\,q^{L_0} y^{M_0}\phi_p(u,\phi) \right)-\text{Tr}_{g,A}\left(M_{-1}M_1\,q^{L_0} y^{M_0}\phi_p(u,\phi) \right),\cr
&&
\end{eqnarray}
which gives us
\begin{eqnarray}
\xi_A^2 f(q,y)&=& y\partial y \left(y\partial y f(q,y)\right) - \frac{1}{(1-q)(1-q^{-1})}\mathcal{M}_{1}\mathcal{M}_{-1}f(q,y).
\end{eqnarray}
We know that $\tilde{f}(q,y)$ is invariant under translation. Thus
\begin{equation}
\mathcal{M}_0 \tilde{f}(q,t)=\partial_u \tilde{f}(q,t) = 0.
\end{equation}
So, we may add $\mathcal{M}_0^2 \tilde{f}(q,y)=0$ in the above equation
\begin{eqnarray}
\xi_A^2 \tilde{f}(q,y)&=& y\partial y \left(y\partial y \tilde{f}(q,y)\right) + \frac{1}{(1-q)(1-q^{-1})}(\mathcal{M}_0^2-\mathcal{M}_{1}\mathcal{M}_{-1})\tilde{f}(q,y).
\end{eqnarray}
Then using
\begin{equation}
[\mathcal{C}_1,\phi_p(u,\phi)]=(\mathcal{M}_0^2-\mathcal{M}_{1}\mathcal{M}_{-1})\phi_p(u,\phi)=\xi_p^2 \phi_p(u,\phi),
\end{equation}
in the above differential equation, and substituting $$Y=\log y, \quad \tilde{f}(q,Y)=\frac{q}{(1-q)^2}f(q,Y)$$ we obtain{\footnote{Interestingly, the differential equation stays the same in $\tilde{f}$ as well. }}:
\begin{equation}
\partial_Y^2 f(q,Y) - \left(\xi_p^2\frac{q}{(1-q)^2}+\xi_A^2\right)f(q,Y)=0.
\end{equation}
The general solution of this equation is given by
\begin{equation}
f(q,y) = A(q)y^{\xi_A\sqrt{1+\frac{\xi_p^2}{\xi_A^2}\frac{q}{(1-q)^2}}}+B(q)y^{-\xi_A\sqrt{1+\frac{\xi_p^2}{\xi_A^2}\frac{q}{(1-q)^2}}}
\end{equation}
In the above, we have resubistuted $Y$ in terms of $y$. We know that $ f(q,y)$ varies like $y^{\xi_A}$. Thus, we have to drop the second term in the above equation which is not of this form. Thus, we have
\begin{equation}
f(q,Y) = A(q)e^{Y\sqrt{\xi_A^2+\xi_p^2\frac{q}{(1-q)^2}}}=A(q)y^{\sqrt{\xi_A^2+\xi_p^2\frac{q}{(1-q)^2}}}.
\end{equation}
\bigskip
\noindent Next, let us look at the differential equation obtained from the Casimir $\mathcal{C}_2$
\begin{eqnarray}
2\xi_A(\Delta_A-1) \tilde{f}(q,y) &=& \text{Tr}_{g,A}\left((2L_0M_0 - L_{-1}M_1 - M_{-1}L_1 -2M_0)\,q^{L_0} y^{M_0}\phi_p(u,\phi) \right)\cr
&=& 2q\partial_q \partial_Y \tilde{f}(q,Y) - 2\frac{1+q}{1-q}\partial_Y\tilde{f}(q,Y)+\frac{q}{(1-q)^2}(\mathcal{M}_1\mathscr{L}_{-1}+\mathscr{L}_1\mathcal{M}_{-1})\tilde{f}(q,Y)\cr
&& +Y\frac{q(1+q)}{(1-q)^3}\mathcal{M}_1\mathcal{M}_{-1}\tilde{f}(q,Y).
\end{eqnarray}
Translational invariance of $\tilde{f}(q,Y)$ gives us $\mathcal{M}_0\tilde{f}(q,Y)=\mathcal{M}_0^2\tilde{f}(q,Y)=\mathcal{M}_0\mathscr{L}_0\tilde{f}(q,Y)=0$. Thus, the above equation is same as
\begin{eqnarray}
&& 2q\partial_q \partial_Y \tilde{f}(q,Y) - 2\frac{1+q}{1-q}\partial_Y\tilde{f}(q,Y)+\frac{q}{(1-q)^2}(-2\mathcal{M}_0\mathscr{L}_0 + \mathcal{M}_1\mathscr{L}_{-1}+\mathscr{L}_1\mathcal{M}_{-1} + 2 \mathcal{M}_0)\tilde{f}(q,Y)\cr
&& +Y\frac{q(1+q)}{(1-q)^3}(-\mathcal{M}_0^2+\mathcal{M}_1\mathcal{M}_{-1})f(q,Y)=2\xi_A(\Delta_A-1) f(q,y).
\end{eqnarray}
Then we substitute
\begin{eqnarray}
&&(\mathcal{M}_0^2-\mathcal{M}_1\mathcal{M}_{-1})\tilde{f}(q,Y)=\xi_p^2 \tilde{f}(q,Y)\cr
&&(2\mathcal{M}_0\mathscr{L}_0 - \mathcal{M}_1\mathscr{L}_{-1}-\mathscr{L}_1\mathcal{M}_{-1} -2 \mathcal{M}_0)\tilde{f}(q,Y) = 2\xi_p(\Delta_p-1) \tilde{f}(q,Y),
\end{eqnarray}
in the above equation to obtain
\begin{equation}
q\partial_q \partial_Y \tilde{f}(q,Y) - \frac{1+q}{1-q}\partial_Y\tilde{f}(q,Y)-\left(\frac{q}{(1-q)^2} \xi_p(\Delta_p-1) +Y\frac{q(1+q)}{(1-q)^3}\frac{\xi_p^2}{2}\right)\tilde{f}(q,y)= \xi_A(\Delta_A-1) \tilde{f}(q,Y).
\end{equation}
Substituting $\tilde{f}(q,Y)=\frac{q}{(1-q)^2}f(q,Y)$, the differential equation simplifies to
\begin{equation}
q\partial_q \partial_Y f(q,Y) -\left(\frac{q}{(1-q)^2} \xi_p(\Delta_p-1) +Y\frac{q(1+q)}{(1-q)^3}\frac{\xi_p^2}{2}\right)f(q,y)= \xi_A(\Delta_A-1) f(q,Y).
\label{diff_eqn_sim_1}
\end{equation}
Now
\begin{equation}
\partial_Y f(q,Y)= \sqrt{\xi_A^2+\xi_p^2\frac{q}{(1-q)^2}} f(q,Y) \equiv \tilde{A}(q) e^{Y\sqrt{\xi_A^2+\xi_p^2\frac{q}{(1-q)^2}}},
\end{equation}
where
\begin{equation}
\tilde{A}(q) = \sqrt{\xi_A^2+\xi_p^2\frac{q}{(1-q)^2}} A(q).
\end{equation}
Then we have
\begin{equation}
q\partial_q \partial_Y f(q,Y)= \frac{Y q \left(1+q\right) \xi _p^2 e^{Y \sqrt{\xi _A^2+\frac{q \xi _p^2}{\left(1-q\right)^2}}}}{\left(1-q\right)^3 \left(2 \sqrt{\xi _A^2+\frac{q \xi _p^2}{\left(1-q\right)^2}}\right)} \tilde{A}(q) + e^{Y\sqrt{\xi_A^2+\xi_p^2\frac{q}{(1-q)^2}}}\, q \partial_q \tilde{A}(q)
\end{equation}
Using this, the differential equation \eqref{diff_eqn_sim_1} further simplifies to
\begin{equation}
q\partial_q \tilde{A}(q)-\frac{1}{\sqrt{\xi_A^2+\xi_p^2\frac{q}{(1-q)^2}}}\left( \frac{q}{(1-q)^2}\xi_p(\Delta_p-1) +\xi_A(\Delta_A-1)\right) \tilde{A}(q)=0.
\label{diff_eqn_sim_2}
\end{equation}
Let $\tilde{A}(q)=G(q)\tilde{B}(q)$ such that
\begin{equation}
q\partial_q G(q) = \frac{q}{(1-q)^2\sqrt{\xi_p^2+\xi_p^2\frac{q}{(1-q)^2}}}\xi_p(\Delta_p-1) G(q).
\end{equation}
We can solve this to find $G(q)$ which is given by
\begin{eqnarray}
G(q)
&=& K_1\left(2 \sqrt{\xi _A^2+\frac{q \xi _p^2}{(1-q)^2}}+\xi_p\frac{1+q}{1-q}\right)^{\Delta_p-1},
\end{eqnarray}
where $K_1$ is a constant.
Substituting $\tilde{A}(q)=G(q)\tilde{B}(q) \tilde{B}(q)$ in \eqref{diff_eqn_sim_2}, we get the differential equation for $\tilde{B}(q)$
\begin{equation}
q \partial_q \tilde{B}(q) - \frac{1}{\sqrt{\xi_A^2+\xi_p^2\frac{q}{(1-q)^2}}}\xi_A(\Delta_A-1)\tilde{B}(q)=0,
\end{equation}
whose solution is
\begin{equation}
\tilde{B}(q)=K_2\xi _p^{2(-\Delta_A+1)}\left(\sqrt{(1-q)^2 \xi _A^2+q \xi _p^2}+(q+1) \xi _A\right)^{2(-\Delta_A+1)}q^{\Delta_A-1}
\end{equation}
Combining all our results we have
\begin{eqnarray}
\tilde{f}(q,y) &=& \frac{K q}{(1-q)^2\sqrt{\xi_A^2+\xi_p^2\frac{q}{(1-q)^2}}}y^{\sqrt{\xi_A^2+\xi_p^2\frac{q}{(1-q)^2}}} \left(2 \sqrt{\xi _A^2+\frac{q \xi _p^2}{(1-q)^2}}+\xi_p\frac{1+q}{1-q}\right)^{\Delta_p-1}\cr
&&\times \,\left(\sqrt{(1-q)^2 \xi _A^2+q \xi _p^2}+(q+1) \xi _A\right)^{-2(\Delta_A-1)}q^{\Delta_A-1}.\cr
&&
\label{fin_sol_tgb}
\end{eqnarray}
\bigskip
\noindent Finally we would like to find the integration constant $K$. We will do this by expanding the above solution around $q=0$ and compare the first few terms with the contribution from level $0$ and $1$ to $\tilde{f}(q,y)$. The expansion of \eqref{fin_sol_tgb} up to order in $q^1$ (mod $q^{\Delta_A}$) is given by
\begin{equation}
K\,2^{2-2 \Delta _A} \xi _A^{1-2 \Delta _A} \left(2 \xi _A+\xi _p\right){}^{\Delta _p-1} q^{\Delta_A}y^{\xi_A}\left(1+q\left(2+\frac{\xi_p(\Delta_p-1)}{\xi_A}+\pi i \rho \frac{\xi_p^2}{\xi_A}-\frac{\Delta_A}{2\xi_A^2}\xi_p^2\right)+\mathcal{O}(q^2)\right).
\end{equation}
Each level comes with a factor of $q^{N+\Delta_A}$ and the contribution of level 0 and level 1 to the global block $\tilde{f}(q,y)$ is
\begin{equation}
C_{ApA}\,q^0 q^{\Delta_A} y^{\xi_A} + C_{ApA}\,q^{\Delta_A +1 }y^{\xi_A}\left(2+\frac{\xi_p(\Delta_p-1)}{\xi_A}+\pi i \rho \frac{\xi_p^2}{\xi_A}-\frac{\Delta_A}{2\xi_A^2}\xi_p^2\right).
\end{equation}
Comparing this with the above equation we can see that
\begin{equation}
K = C_{ApA}\,4^{\Delta _A-1} \xi _A^{2 \Delta _A-1}\left(2 \xi _A+\xi _p\right){}^{1-\Delta _p}.
\end{equation}
Substituting this back in \eqref{fin_sol_tgb}, the form of the global torus block is given by
\begin{eqnarray}
\tilde{f}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(q,y) &=& C_{ApA}\,4^{\Delta _A-1} \xi _A^{2 \Delta _A-1}\left(2 \xi _A+\xi _p\right){}^{1-\Delta _p} \left(\sqrt{(1-q)^2 \xi _A^2+q \xi _p^2}+(q+1) \xi _A\right)^{-2(\Delta_A-1)} \cr
&& \times \frac{q^{\Delta_A}}{(1-q)^2\sqrt{\xi_A^2+\xi_p^2\frac{q}{(1-q)^2}}}y^{\sqrt{\xi_A^2+\xi_p^2\frac{q}{(1-q)^2}}} \left(2 \sqrt{\xi _A^2+\frac{q \xi _p^2}{(1-q)^2}}+\xi_p\frac{1+q}{1-q}\right)^{\Delta_p-1}\cr
&&
\end{eqnarray}
This is our final expression for the global BMS torus block, or the Minkowski torus block.
\medskip
Before ending this subsection, we would like to point out that there is some potential subtlety between the BMS global blocks and the large $c_M$ limit of the BMS blocks. The expectation from 2d CFT on general grounds is that in the large central charge limit, since the norm of the states at level 2 and higher depend inversely on the central charge, the Virasoro blocks would reduce to the global blocks. However, as shown in \cite{Alkalaev:2016fok}, this is not true for the torus CFT in contrast to the CFT on the sphere. One may introduce the so-called ``light blocks" which are defined as the leading approximation in $1/c$ expansion. On the other hand, by truncating Virasoro algebra to its global $sl(2)$ sub-algebra one is led to global block. On the sphere, these two functions are equal, while on torus and higher genus surfaces these are different. It is likely that BMS torus blocks share the same property, i.e. one generally distinguishes between BMS global and BMS light blocks for field theories defined on higher genus surfaces. We thank Konstantin Alkalaev for pointing this out to us. We hope to come back to this issue in the future and resolve it.
\bigskip \bigskip
\newpage
\section{Structure constants for BMS highest weight primaries}\label{onepointsec}
We derived expressions for the BMS torus blocks in the previous section. Armed with this knowledge, we now move on to calculating the asymptotic behaviour of three-point structure constants $C_{ApA}$ for primary fields in the highest weight representations. We will evaluate this in the large $\xi_A$ limit, which was also the limit we used to derive the expressions of the BMS torus blocks.
\medskip
\noindent We note that $C_{ApA}$ is the coordinate independent constant in the three point function $\<\phi_{\Delta_A,\xi_A}(t_1,x_1)\phi_p(t_2,x_2)\phi_{\Delta_A,\xi_A}(t_3,x_3)\>$. The analysis in this section will closely follow to the one used in Section \ref{onepointsec1}. We will find the asymptotic formula for a generic primary field $\phi_p$. First of all let us rewrite the S-modular transformation \eqref{s_mod_trans} as
\begin{eqnarray} \label{701}
\<\phi_{\Delta,\xi}\>_{(\sigma,\rho)}&=&\sigma^{-\Delta} e^{\frac{\xi\rho}{\sigma}}\left<\phi_{\Delta,\xi}\right>_{(-\frac{1}{\sigma},\frac{\rho}{\sigma^2})}.
\end{eqnarray}
We expand the left hand side (LHS) of \refb{701} in terms of the torus blocks of the primaries, where as the right hand side (RHS) of \refb{701} is expanded using the eigenbasis of $L_0$ and $M_0$ as in Section \ref{onepointsec1}. This leads to
\begin{eqnarray}
&&e^{-2\pi i (\sigma c_L/2+\rho c_M/2)}\sum_{A} D(\Delta_A,\xi_A) e^{2\pi i \sigma \Delta_A} e^{2\pi i \rho \xi_A} C_{ApA}\,\mathcal{F}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(\sigma,\rho)\nonumber\\
&& \quad =\sigma^{-\Delta_p}e^{\xi_p\frac{\rho}{\sigma}}\sum_{i} D(\Delta_i,\xi_i)C_{ipi}
e^{-\frac{2\pi i}{\sigma}(\Delta_i-\frac{c_L}{2})}e^{2\pi i\frac{\rho}{\sigma^2}(\xi_i-\frac{c_M}{2})}.
\end{eqnarray}
We will choose $\sigma=i\beta$ and consider the limit $\beta\rightarrow0^+$ and $|\rho|\rightarrow0^+$. In this limit, as before, we replace the LHS by a double integral while the RHS is dominated by contribution from the lightest field $\chi$ such that $C_{\chi p \chi}\neq 0$:
\begin{eqnarray}
&& e^{2\pi \beta \frac{c_L}{2}}e^{-2\pi i \rho c_M/2}\int d\Delta_A \int d\xi_A T_p(\Delta_A,\xi_A) e^{-2\pi \beta \Delta_A} e^{2\pi i \rho \xi_A} \,\mathcal{F}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(i\beta,\rho)\nonumber\\
&& \quad \approx (i\beta)^{-\Delta_p} e^{-i\xi_p\frac{\rho}{\beta}} T_p(\Delta_\chi,\xi_\chi)e^{-\frac{2\pi }{\beta}(\Delta_\chi-\frac{c_L}{2})}e^{-2\pi i\frac{\rho}{\beta^2}(\xi_\chi-\frac{c_M}{2})}.
\label{non_zero_xip_ti}
\end{eqnarray}
Here $T_i(\Delta_A,\xi_A) \equiv D(\Delta_i,\xi_i)C_{ipi}$. For small $\beta$, the LHS is dominated by torus blocks with large $\xi_A$. So, to a good approximation, we may replace the integral by integral over only large $\xi_A$. From \eqref{f_leading}, for large $\xi_A$ we have
\begin{equation}
\mathcal{F}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(\sigma,\rho)
=\frac{q^\frac{1}{12}}{\eta(\sigma)^2} + \mathcal{O}\left(\frac{1}{\xi_A}\right)+ \ldots
\Rightarrow \mathcal{F}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(i\beta,\rho)= \frac{e^{-\frac{2\pi\beta}{12}}}{\eta(i\beta)^2}+ \mathcal{O}\left(\frac{1}{\xi_A}\right)+ \ldots
\end{equation}
For the special case of $\xi_p=0$, as described in Sec~\ref{tbxip0}, there are no subleading corrections to the block. For a generic $\xi_p \neq 0$, we will work also in the leading $\xi_A$ approximation. So we will keep only the first term in the above equation and neglect the $\mathcal{O}(\frac{1}{\xi_A})$ term and other higher order terms in $\xi_A$. For this to be possible i.e., for the subleading and other higher order terms to be suppressed in comparison to the leading term, we need the additional requirement that $\beta^2 \xi_A \gg1$. This is shown in Appendix \ref{justif}.
\medskip
\noindent Next, we use the small $\beta$ asymptotic of the Dedekind eta-function to obtain
\begin{equation}
\mathcal{F}_{\Delta_A,\xi_A,c_L,c_M}^{\Delta_p,\xi_p}(i\beta,\rho) \approx 2\pi \beta \exp \left\{-\frac{\pi}{6}\left(\beta-\frac{1}{\beta}\right)\right\}+...
\end{equation}
Substituting the above expression in \eqref{non_zero_xip_ti}, we have
\begin{eqnarray}
&& e^{2\pi \beta \frac{c_L}{2}}e^{-2\pi i \rho \frac{c_M}{2}}2\pi \beta e^{\frac{\pi}{6}\left(-\beta+\frac{1}{\beta}\right)}\int d\Delta_A \int d\xi_A T_p(\Delta_A,\xi_A) e^{-2\pi \beta \Delta_A} e^{2\pi i \rho \xi_A}\nonumber\\
&& \quad \approx (i\beta)^{-\Delta_p} e^{-i\xi_p\frac{\rho}{\beta}} \ T_p(\Delta_\chi,\xi_\chi)e^{-\frac{2\pi }{\beta}(\Delta_\chi-\frac{c_L}{2})}e^{-2\pi i\frac{\rho}{\beta^2}(\xi_\chi-\frac{c_M}{2})}\nonumber
\end{eqnarray}
This leads to
\begin{eqnarray}
\int d\Delta_A^{\prime} \int d\xi_A^{\prime} \ T_p(\Delta_A^{\prime},\xi_A^{\prime}) e^{-2\pi \beta \Delta_A^{\prime}} e^{2\pi i \rho \xi_A^{\prime}}
\approx && \frac{\beta^{-\Delta_p-1}}{2\pi i^{\Delta_p}} T_p(\Delta_\chi,\xi_\chi) e^{2\pi \beta (-\frac{c_L}{2}+\frac{1}{12})} \\
&& \times e^{-2\pi i \rho (\frac{1}{\beta^2}(\xi_\chi-\frac{c_M}{2})-\frac{c_M}{2}+\frac{\xi_p}{2\pi \beta})}e^{-\frac{2\pi }{\beta}(\Delta_\chi-\frac{c_L}{2}+\frac{1}{12})} .\nonumber
\end{eqnarray}
We multiply both sides with $e^{-2\pi i \xi_A (\text{Re}\,\rho)}$ and integrate over real part of $\rho$. On the LHS we have a $\delta(\xi_A^{\prime}-\xi_A)$ which when integrated over $\xi_A^{\prime}$ will give us
$${\text{LHS}} = \int d\Delta_A^{\prime} T_p(\Delta_A^{\prime},\xi_A) e^{-2\pi \beta \Delta_A^{\prime}} e^{-2\pi (\text{Im}\,\rho) \xi_A}.$$
On the RHS, the integral over $(\text{Re}\,\rho)$ will give us a delta-function. Specifically, we have
\begin{eqnarray}
\int_0^{\infty} d\Delta_A^{\prime} T_p(\Delta_A^{\prime},\xi_A) e^{-2\pi \beta \Delta_A^{\prime}} &\approx& \frac{\beta^{-\Delta_p-1}}{2\pi i^{\Delta_p}} e^{2\pi \beta (-\frac{c_L}{2}+\frac{1}{12})}e^{-\frac{2\pi }{\beta}(\Delta_\chi-\frac{c_L}{2}+\frac{1}{12})} \ T_p(\Delta_\chi,\xi_\chi)\\
&& {\hspace{-2cm}} \times e^{2\pi (\text{Im}\,\rho)(\frac{1}{\beta^2}(\xi_\chi-\frac{c_M}{2})-\frac{c_M}{2}+\frac{\xi_p}{2\pi \beta}+\xi_A)}\delta\left\{\frac{1}{\beta^2}(\xi_\chi-\frac{c_M}{2})-\frac{c_M}{2}+\frac{\xi_p}{2\pi \beta}+\xi_A\right\}. \nonumber
\end{eqnarray}
The two roots of the argument of the $\delta$-function are
\begin{equation}
\beta_{\pm}=\frac{-\frac{\xi _p}{2 \pi }\pm\sqrt{\left(\frac{\xi _p}{2 \pi }\right)^2+4 \left(\frac{c_M}{2}-\xi _{\chi }\right)\left(\xi _A-\frac{c_M}{2}\right)}}{2 \left(\xi _A-\frac{c_M}{2}\right)}.
\label{roots_beta_3}
\end{equation}
Using the property of the $\delta$-function we have
\begin{eqnarray}
\int_0^{\infty} d\Delta_A^{\prime} T_p(\Delta_A^{\prime},\xi_A) e^{-2\pi \beta \Delta_A^{\prime}} = && \sum_{*=+,-}\frac{\beta^{-\Delta_p-1}}{2\pi i^{\Delta_p}}e^{2\pi \beta_* (-\frac{c_L}{2}+\frac{1}{12}) -\frac{2\pi }{\beta_*}(\Delta_\chi-\frac{c_L}{2}+\frac{1}{12})} T_p(\Delta_\chi,\xi_\chi) \nonumber\\
&& \qquad \times e^{2\pi (\text{Im}\,\rho)}\frac{\delta(\beta-\beta_\star)}{|\frac{c_M-2\xi_\chi}{\beta_\star^3}-\frac{\xi_p}{2\pi\beta_\star^2}|}.
\end{eqnarray}
We then invert the above Laplace-transformation using the analytic continuation of the $\delta$-function \eqref{delta_ac} to obtain
\begin{eqnarray}
\frac{T_p(\Delta_A,\xi_A)}{T_p(\Delta_\chi,\xi_\chi)} &=&\sum_{*=+,-}\frac{\beta^{-\Delta_p-1}}{2\pi i^{\Delta_p}}{\frac{e^{2\pi \beta_* (-\frac{c_L}{2}+\frac{1}{12})}e^{-\frac{2\pi }{\beta_*}(\Delta_\chi-\frac{c_L}{2}+\frac{1}{12})}}{|\frac{c_M-2\xi_\chi}{\beta_\star^3}-\frac{\xi_p}{2\pi\beta_\star^2}|}} \int_{b-i\infty}^{b+i\infty} \frac{d\beta}{i}\frac{e^{2\pi \beta\Delta_A^\prime}}{2\pi i (\beta-\beta^{*})}\cr
&=& \sum_{*=+,-}\frac{1}{2\pi}(i\beta_*)^{-\Delta_p-1} e^{2\pi \beta_* (-\frac{c_L}{2}+\frac{1}{12})}e^{-\frac{2\pi }{\beta_*}(\Delta_\chi-\frac{c_L}{2}+\frac{1}{12})} \frac{e^{2\pi \beta_*\Delta_A}}{|\frac{c_M-2\xi_\chi}{\beta_\star^3}-\frac{\xi_p}{2\pi\beta_\star^2}|}.\cr
&&
\end{eqnarray}
Note that we start out with the condition that $\beta$ is small and $\xi_A$ is large. Indeed, from \eqref{roots_beta_3} we see that for $\xi_A \gg \frac{\xi_p}{2\pi} \pm \frac{c_M}{2}$, $\beta_{\pm}$ is small. We also assume $\beta$ to be real for which we require $\xi_{\chi} < \frac{c_M}{2}$. In these limit $$\beta_\pm\approx-\frac{\xi_p}{4\pi\xi_A}\pm\sqrt{\frac{{c_M}-2\xi_\chi}{2\xi_A}}.$$
We also require the contribution from $\beta_+$ to dominate and the contribution from $\beta_-$ to be negligible. This is achieved by the condition that $$\Delta_{\chi}<\frac{c_L}{2}-\frac{1}{12}.$$ So, we have
\begin{eqnarray}
\frac{T_p(\Delta_A,\xi_A)}{T_p(\Delta_\chi,\xi_\chi)}
&\approx & \frac{1}{2\pi}\frac{(-i)^{\Delta_p+1}}{\beta_+^{\Delta_P-1}} e^{2\pi \beta_+ (\Delta_A-\frac{c_L}{2}+\frac{1}{12})}e^{\frac{2\pi }{\beta_+}(\frac{c_L}{2}-\Delta_\chi-\frac{1}{12})} \frac{1}{\frac{c_M-2\xi_\chi}{\beta_+}-\frac{\xi_p}{2\pi}}. \cr
&\approx& \frac{1}{2\pi}\frac{(-i)^{\Delta_p+1}}{2\xi_A}\left(\frac{\xi_A}{\frac{c_M}{2}-\xi_{\chi}}\right)^{\frac{\Delta_p}{2}}e^{\frac{\xi_p}{2}\left(-\frac{\Delta_A-\frac{c_L}{2}+\frac{1}{12}}{\xi_A}-\frac{\frac{c_L}{2}-\Delta_\chi-\frac{1}{12}}{\frac{c_M}{2}-\xi_{\chi}}\right)}\cr
&&\times e^{2\pi\left(\sqrt{\frac{\frac{c_M}{2}-\xi_{\chi}}{\xi_A}}(\Delta_A-\frac{c_L}{2}+\frac{1}{12})+\sqrt{\frac{\xi_A}{\frac{c_M}{2}-\xi_{\chi}}})(\frac{c_L}{2}-\Delta_{\chi}-\frac{1}{12})\right)}.
\label{onepoint_non_zero_xi}
\end{eqnarray}
To find the asymptotic formula for density of states in the large $\xi_A$ limit, we put $\Delta_p=0=\xi_p$ directly in \eqref{onepoint_non_zero_xi} and also replace the primary field $\chi$ by the lightest primary field $\O$ present in the theory
\begin{eqnarray}
\frac{D(\Delta_A,\xi_A)}{D(\Delta_\O,\xi_\O)}
&\approx& - \frac{i}{4\pi\xi_A}
e^{2\pi\left[(\Delta_A-\frac{c_L}{2}+\frac{1}{12}) \sqrt{\frac{\frac{c_M}{2}-\xi_\O}{\xi_A}}+(\frac{c_L}{2}-\Delta_\O-\frac{1}{12})\sqrt{\frac{\xi_A}{\frac{c_M}{2}-\xi_\O}}\right]}. \label{dos}
\end{eqnarray}
It is satisfying to note that in the limit $\Delta_A\gg c_L$ and with $\O$ as the vacuum (so $\Delta_\O= \xi_\O = 0$), taking the logarithm of the above equation \refb{dos}, we reproduce the Cardy formula for BMS primaries, which was worked out in the saddle point method in \cite{Bagchi:2019unf}.
Substituting \refb{dos} back in \eqref{onepoint_non_zero_xi} we obtain the asymptotic formula of three-point coefficient for primary fields for large $\xi_A$
\bigskip
\noindent {\fbox{
\addtolength{\linewidth}{+2\fboxsep}%
\addtolength{\linewidth}{+2\fboxrule}%
\begin{minipage}{\linewidth}
\begin{eqnarray} \label{q}
C_{ApA}&\approx&(-i)^{\Delta_p}C_{\chi A \chi}\frac{D(\Delta_{\chi},\xi_{\chi})}{D(\Delta_\O,\xi_\O)}\left(\frac{\xi_A}{\frac{c_M}{2}-\xi_{\chi}}\right)^{\frac{\Delta_p}{2}}e^{\frac{\xi_p}{2}\left(-\frac{\Delta_A-\frac{c_L}{2}+\frac{1}{12}}{\xi_A}-\frac{\frac{c_L}{2}-\Delta_\chi-\frac{1}{12}}{\frac{c_M}{2}-\xi_{\chi}}\right)} \times \\
&&e^{2\pi\left(\sqrt{\frac{\xi_A}{\frac{c_M}{2}-\xi_\chi}}(\frac{c_L}{2}-\frac{1}{12}-\Delta_\chi)-\sqrt{\frac{\xi_A}{\frac{c_M}{2}-\xi_\O}}(\frac{c_L}{2}-\frac{1}{12}-\Delta_\O)+(\sqrt{\frac{\frac{c_M}{2}-\xi_\chi}{\xi_A}}-\sqrt{\frac{\frac{c_M}{2}-\xi_\O}{\xi_A}})(\Delta_A+\frac{1}{12}-\frac{c_L}{2})\right)}. \nonumber\\\nonumber
\end{eqnarray}
\end{minipage}
}}
\bigskip
\noindent If we consider the limit where $c_M\gg \xi_\chi, \xi_\O$, then we see that the asymptotic structure constants for the BMS primaries matches the general analysis we have considered in earlier sections (e.g. comparing \refb{q} and \refb{impr}), with the identification
\begin{equation} \label{shift}
c_L \to c_L -\frac{1}{6}.
\end{equation}
The shift in central charge due to descendants is similar to that which has been observed in 2d CFTs. This shift can be viewed as a one-loop renormalisation of the bulk central charge due to the presence of BMS descendants. It is interesting to see that the central term $c_M$ does not get shifted. This is reminiscent of the recent analysis of \cite{Merbis:2019wgk}, where the authors also found a one-loop shift of only $c_L$ and not $c_M$. In fact the shift of the central charge $c_L$ found in \cite{Merbis:2019wgk} exactly matches with our result above \refb{shift} (after accounting for the difference in normalisations between our work and \cite{Merbis:2019wgk}).{\footnote{The authors in \cite{Merbis:2019wgk} speak about two different shifts in the central extension $c_L$ corresponding to two different classes of co-adjoint orbits of the BMS group, viz. the vacuum orbit and a generic orbit. The case we are interested in is the generic orbit since on the boundary side we are interested in thermal states and in the bulk dual we deal with the FSC solution, and not the vacuum Minkowski solution.}}
\bigskip
\newpage
\section{A Quick Bulk analysis}\label{bulk}
\subsection{Flatspace Cosmologies}
In a holographic duality, the thermal states on the field theory are equivalent to geometries with horizons in the bulk dual. For a 2d CFT, thermal states are dual to BTZ black holes in AdS$_3$. For non-extremal BTZ solutions, the metric is given by
\begin{equation} \label{btz}
ds^2_{\text{\tiny BTZ}} = - \frac{(r^2 - r_+^2)(r^2 - r_-^2)}{r^2 \ell^2}dt^2 + \frac{r^2 \ell^2}{(r^2 - r_+^2)(r^2 - r_-^2)}dr^2 + r^2\left(d\phi - \frac{r_+ r_-}{\ell r^2} dt\right)^2.
\end{equation}
Here $r_\pm$ are the outer and inner horizons, which are given in terms of the mass ($M$) and angular momentum ($J$) of the BTZ black hole by
\begin{equation}
r_\pm = \sqrt{2G\ell(\ell M +J)} \pm \sqrt{2G\ell(\ell M -J)}.
\end{equation}
In the above, $\ell$ is the radius of AdS$_3$. BTZ black hole solutions are locally AdS$_3$ and can be viewed as orbifolds of AdS$_3$.
The holographic duals of thermal states in 2d BMSFTs are the so-called Flat Space Cosmologies (FSC). The metric for these cosmological solutions are given by
\begin{equation}\label{fsc1}
ds^2_{\text{\tiny FSC$_1$}} = M du^2 - 2du dr + J du d\psi +r^2 d\psi^2
\end{equation}
Here $u$ is the retarded time and $M$ and $J$ label the mass and angular momentum of these solutions. These are the zero modes of the most generic metric
\begin{equation}
ds^2_{\text{\tiny FSC$_1$}} = \Theta(\phi) du^2 - 2du dr + \left[\Xi(\phi) + \partial_\phi \Theta(\phi)\right] du d\psi +r^2 d\psi^2
\end{equation}
for asymptotically flat boundary conditions for which one gets the BMS$_3$ algebra as the asymptotic symmetry algebra on the null boundary.
$\Theta(\phi)$ and $\Xi(\phi)$ are functions called the mass and angular momentum aspects which reduce to $M$ and $J/2$ for the FSC solution. FSC solutions are locally flat and can also be obtained by quotienting 3d Minkowski spacetime by a boost and a translation. For this reason, they are also known as shifted-boost orbifolds \cite{Cornalba:2003kd}. The Penrose diagram of FSC spacetimes is given in Figure \ref{fig2}.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=.022]{penrose_diagram}
\end{center}
\caption{Penrose diagram for Flat Space Cosmologies.}
\label{fig2}
\end{figure}
\medskip
\noindent Minkowski spacetimes can be viewed as an infinite radius of AdS. One can similarly obtain the FSC solutions by starting with \refb{btz} and taking $\ell \to \infty$. The metric here, written in $(t,r, \phi)$ coordinates, takes the form
\begin{equation}\label{fsc2}
ds^2_{\text{\tiny FSC$_2$}}=\hat{r}^2_{+}dt^2-\frac{r^2}{\hat{r}^2_{+}(r^2-r_0^2)}dr^2 + 2\hat{r}_+ r_0 d\phi dt + r^2 d\phi^2,
\end{equation}
where
\begin{equation}
\hat{r}_+ = \sqrt{8GM},\,\,\,r_0 = \sqrt{\frac{2G}{M}}J.
\end{equation}
Going from \refb{fsc1} to \refb{fsc2} is done by the following change of variables:
\begin{equation}
d\psi = d\phi + \frac{r_0 dr}{\hat{r}_+(r^2-r_0^2)}, \quad du = dt + \frac{r^2 dr}{\hat{r}_+^2(r^2-r_0^2)}
\end{equation}
The cosmological nature of the metric \refb{fsc2} manifests itself in the limit. When $\ell \to \infty$, the outer horizon of the original BTZ black hole goes out to infinity leaving behind just the interior of the black hole as the whole solution. Here thus the roles of the radial and temporal directions flip. The solution \refb{fsc2} is dependent on $r$, which in this case is the temporal direction. $r=r_0$ is the remnant of the inner BTZ horizon that turns into a cosmological horizon in this singular limit.
To understand the cosmological nature of the FSC intrinsically without the aid of a limiting procedure, it is best to look at another coordinate system where the metric can be rewritten as
\begin{equation}\label{fsc3}
ds^2_{\text{\tiny FSC$_3$}}=-d\t^2 + \frac{(E \t)^2}{1+ (E \t)^2} dx^2 + \left(1+ (E \t)^2\right)\left( dy + \frac{(E \t)^2}{1+ (E \t)^2} dx\right)^2
\end{equation}
In the above, $y$ is identified as $y \sim y + 2\pi r_0$. It can be checked that the above is a solution to vacuum Einstein's equation in three dimensions with a vanishing cosmological constant. The obvious temporal dependence makes this a cosmological solution. For positive $\t$, \refb{fsc3} describes an expanding universe from a cosmological horizon at $\t=0$. The map between \refb{fsc3} and \refb{fsc2} is the following
\begin{equation}
\hat{r}_+ t = x, \quad r_0 \phi = y+x, \quad \left(\frac{r}{r_0}\right)^2 = 1+ (E \t)^2, \, \text{with} \, E= \frac{\hat{r}_+}{r_0}.
\end{equation}
For our calculations in the next subsection, we will focus on using \refb{fsc2} as our metric of choice.
\subsection{One point functions in FSC background}
We now wish to calculate $\<E|O|E\>$, for high energy $E$, from the bulk side. The dual to the finite temperature BMSFT thermal state is a FSC solution. The BMS weights of the FSC are
\begin{equation}
\xi_{\text{\tiny FSC}}=M+\frac{c_M}{2},\,\,\,\Delta_{\text{\tiny FSC}}=J+\frac{c_L}{2}.
\end{equation}
We also know that for 3d Einstein gravity $c_L=0,\,\,c_M=\frac{1}{4G}$. For large $M$ and $J$, we have
\begin{equation}
\xi_{\text{\tiny FSC}} \approx M,\,\,\, r_0 \approx \frac{\Delta_{\text{\tiny FSC}}}{\sqrt{2c_M \xi_{\text{\tiny FSC}}}},\,\,\, \hat{r}_+ \approx \sqrt{\frac{2\xi_{\text{\tiny FSC}}}{c_M}}.
\end{equation}
It is of interest to note here that with these identifications of the BMS weights of the FSC, the Bekenstein-Hawking entropy associated with the cosmological horizon matches precisely with the BMS-Cardy formula \refb{bcar} presented in Sec.~\refb{bmsft} \cite{Bagchi:2012xr}:
\begin{equation}
S_{\text{\tiny{Bekenstein-Hawking}}} = \frac{\text{Area of horizon}}{4G} = \frac{2 \pi r_0}{4G} = S_{\mbox{\tiny{BMS-Cardy}}}.
\end{equation}
We now wish to take this correspondence one step further and match the asymptotic formula for the structure constants that we have obtained to a bulk analysis.
\bigskip
\noindent Our object of interest, $\<E|O|E\>$, in the bulk side denotes the calculation of a one-point function of a light operator $O$, or a probe, in the background of a heavy state $|E\>$, which is given by a FSC solution. Of course, we don't expect a single microstate to have a geometric description, so in a sense the FSC geometry arises on course graining over a family of such microstates. We are working in the probe limit and hence we will discount the backreaction of $O$ on the FSC geometry.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=1]{fsc_oneloop}
\end{center}
\caption{One-loop contribution to $\<E|O|E\>$}
\label{oneloop}
\end{figure}
\medskip
\noindent We are interested in the contribution to $\<E|O|E\>$ that comes from the diagram shown in Figure \ref{oneloop}. The field $\phi_O$ comes from boundary at infinity and splits into a pair of $\phi_{\chi}$ which wraps around the cosmological horizon $r_0$. We are working in the probe limit where $\xi_O,\xi_{\chi} \gg 1$ but these fields are light in the sense that $\xi_O,\xi_{\chi} \ll c_M$. We also have $\xi_{\text{\tiny FSC}} \gg c_M$. Even in the case of asymptotically flat spacetimes, for bulk scalar field with mass $m$, the two point function is given by $e^{-mL}$, where $L$ is the length of the geodesic connecting the two points \cite{Hijano:2017eii}. So, for $\phi_{\chi}$, the contribution to the one loop diagram is given by
\begin{equation}
\exp({-\xi_{\chi}2\pi r_0}) \approx \exp\left({-\frac{2\pi\xi_{\chi}\Delta_{\text{\tiny FSC}}}{\sqrt{2\xi_{\text{\tiny FSC}} c_M}}}\right).
\end{equation}
In the above equation we make use of the fact that for a primary field $(\Delta_b,\xi_b)$ on the boundary, the corresponding field in the bulk has a mass $\xi_b$. This follows from a matching of Casimirs from the point of the bulk and the boundary.
\medskip
\noindent Now let us calculate the contribution from the geodesic along the $r$ direction $r = r_0$ to $r = \infty$. In general, the length of this geodesic from $r = 0$ to $r=\Lambda$ is given by
\begin{eqnarray}
L = \int_{r_0}^{\Lambda} \frac{r dr}{\hat{r}_+\sqrt{r^2-r_0^2}} = \frac{\sqrt{\Lambda^2-r_0^2}}{\hat{r}_+^2} \, \Rightarrow \log{L} = \frac{1}{2}\log \left(\frac{\Lambda^2}{r_0^2}-1\right) - \log\left(\frac{\hat{r}_+}{r_0}\right).
\end{eqnarray}
In the above expression, we can see that $\log L$ diverges when $\Lambda$ and hence $\log\left(\frac{\Lambda}{r_0}-1\right)$ goes to infinity. We regulate this it by removing the first term. The renormalized geodesic length is thus given by
\begin{equation}{}
\log L = -\log\left(\frac{\hat{r}_+}{r_0}\right) \implies L = \frac{r_0}{\hat{r}_+} \approx \frac{\Delta_{\text{\tiny FSC}}}{2\xi_{\text{\tiny FSC}}}.
\end{equation}
So, the contribution of this geodesic to to the one-point function is given by
\begin{equation}
\exp(-\xi_O L) \approx \exp\left(-{\xi_{O}\frac{r_0}{\hat{r}_+}}\right) \approx \exp\left(-\frac{\xi_{O} \Delta_{\text{\tiny FSC}}}{2\xi_{\text{\tiny FSC}}}\right).
\end{equation}
Combining the above calculations, we have
\begin{equation}
\boxed{\< E|O|E \> \approx \langle \chi|O|\chi \rangle \exp \left(-{\frac{\xi_{O} \Delta_{\text{\tiny FSC}}}{2\xi_{\text{\tiny FSC}}}} - {\frac{2\pi\xi_{\chi}\Delta_{\text{\tiny FSC}}}{\sqrt{2\xi_{\text{\tiny FSC}} c_M}}}\right).}
\label{1p_bulk}
\end{equation}
This matches with the field theory calculation for asymptotic formula for three-point coefficient given in \eqref{three_point_ceoff_limit}.
\medskip
\noindent From the other field theory calculation in \eqref{impr}, we have
\begin{eqnarray}
{\langle E|O|E} \rangle &\approx & \frac{\rho(\chi)}{\rho(O)}C_{\chi O \chi}\left(\frac{\xi}{\frac{c_M}{2}-\xi_{\chi}}\right)^{\frac{\Delta_O-1}{2}}\sqrt{\frac{\xi}{\frac{c_M}{2}-\xi_l}}\,\,e^{\frac{\xi_O}{2}\left(-\frac{\Delta-\frac{c_L}{2}}{\xi}+\frac{\frac{c_L}{2}-\Delta_\chi}{\frac{c_M}{2}-\xi_\chi}\right)} \\
&& \times e^{2\pi\left(\sqrt{\frac{\xi}{\frac{c_M}{2}-\xi_\chi}}\left(\frac{c_L}{2}-\Delta_\chi\right)-\sqrt{\frac{\xi}{\frac{c_M}{2}-\xi_l}}\left(\frac{c_L}{2}-\Delta_l\right)+\left(\sqrt{\frac{\frac{c_M}{2}-\xi_\chi}{\xi}}-\sqrt{\frac{\frac{c_M}{2}-\xi_l}{\xi}}\right)\left(\Delta-\frac{c_L}{2}\right)\right)}.\nonumber
\label{1p_field_theory}
\end{eqnarray}
Here $l$ is the lightest field in the theory with weights $(\xi_l,\Delta_l)$. We will take this field to be the vacuum, so we have $\xi_l=\Delta_l=0$. Since we consider the case where the corresponding bulk fields $\phi_O$ and $\phi_{\chi}$ are scalars, we have $\Delta_O=\Delta_{\chi}=0$. For 3d Einstein gravity, we also have $c_L=0$. So, for these cases, \eqref{1p_field_theory} reduces to
\begin{eqnarray}
\langle E|O|E \rangle &\approx & \frac{\rho(\chi)}{\rho(O)}C_{\chi O \chi}\,\exp\left(-{\frac{\xi_O\Delta_{\text{\tiny FSC}}}{2\xi_{\text{\tiny FSC}}}} {-\frac{2\pi\xi_{\chi}\Delta_{\text{\tiny FSC}}}{\sqrt{2\xi_{\text{\tiny FSC}} c_M}}}\right),
\end{eqnarray}
matching with the bulk analysis answer \eqref{1p_bulk}.
\newpage
\section{Conclusions}\label{conc}
\subsection{Summary}
Let us summarise for the reader what we have achieved in the current paper. We have been looking 2d field theories invariant under the BMS$_3$ algebra. These are theories which are putative duals to 3d asymptotically flat spacetimes.
\medskip
\noindent We concentrated on modular properties of these 2d BMSFTs and specifically those of the torus one-point function. Using the BMS modular transformations of the torus one-point function, we computed an asymptotic formula for the three-point structure constants. To begin with, we used two distinct methods, one a saddle-point analysis and the other which hinged on inverting integral transformations, to obtain expressions for the structure constants. We found that the second method worked for a wider range of parameters than the saddle point method. Both methods were however blind to the exact states that the trace was performed on.
\medskip
\noindent
We then computed the asymptotic form of structure constants when the states were in the BMS highest weight representation. In order to do this, we needed to develop quite a lot of new machinery for these BMSFTs. A particularly important development was the derivation of the BMS torus blocks. We found expressions for the leading and subleading terms in the BMS torus blocks in a particular limit of large weights. These developments were put in use when we recomputed the torus one-point function for the BMS primaries.
\medskip
\noindent
Finally, we used a bulk analysis that depended on a geodesic approximation in terms of a probe scalar in the background of a flat space cosmological solution to reproduce the field theory results. Additionally, we have a large number of appendices with a lot of more technical details.
\subsection{Discussions and Future Directions}
We now turn to discuss various aspects and future directions of our findings in this paper.
\medskip
\noindent {\em{More flat space holography}}
\smallskip
\noindent One of our principal objectives in this work has been to further the advancement of holography for 3d asymptotically flat spacetimes. We have added the asymptotic three-point structure constants to the list of quantities that now match between the bulk and the putative boundary theory. Our bulk analysis can be improved by looking at a Witten diagram like procedure instead of the geodesic approximation \cite{Kraus:2016nwo, Kraus:2017ezw}. One can also look at the Chern-Simons formulation to understand this in a different way, following e.g. \cite{Alkalaev:2020yvq}.
\smallskip
\noindent There are obvious generalisations to our work in this paper to theories with higher spin \cite{Afshar:2013vka, Gonzalez:2013oaa}, and supersymmetric BMS theories (see e.g. \cite{Barnich:2014cwa, Lodato:2016alv, Banerjee:2018hbl}), and other extensions of BMS e.g. with extra currents \cite{Basu:2017aqn}.
\smallskip
\noindent An important direction is the generalisation to higher point functions on the torus. For the torus two point function, there are important applications. These would be connected to quasinormal modes in the dual bulk spacetime and to the eigenstate thermalisation hypothesis \cite{Brehm:2018ipf, Romero-Bermudez:2018dim, Hikida:2018khg} in the context of BMSFTs {\footnote{For further discussions of higher point torus functions for 2d CFTs and their holographic interpretation, the reader is pointed to \cite{Alkalaev:2017bzx}.}}.
\smallskip
\noindent The modular bootstrap programme in 2d CFTs initiated in \cite{Hellerman:2009bu} helps constrain theories further. The S-modular transformation is particularly useful in this regard. A review of current developments is summarised in \cite{Brehm:2019pcx}. A similar programme is currently being attempted currently for 2d BMSFTs. This is something we hope to report on in the near future.
\smallskip
\noindent It is of interest to see how the one-point torus amplitude calculation generalises in the case of Flatspace Chiral gravity (F$\chi$G) \cite{Bagchi:2012yk} (and its supersymmetric cousin \cite{Bagchi:2018ryy}). F$\chi$G relates a theory of Chern-Simons gravity with asymptotically flat boundary conditions to a chiral 2d conformal field theory. This theory admits FSC solutions. So it is natural to ponder what sort of bulk construction would reproduce what should be a chiral half of the Kraus-Maloney CFT answer.
\bigskip
\noindent {\em{Black holes in general dimensions}}
\smallskip
\noindent In \cite{Carlip:2017xne, Carlip:2019dbu}, Carlip has argued that the algebra of horizon preserving diffeomorphisms are enhanced to form BMS$_3$ for generic black holes and has used the BMS-Cardy formula \cite{Bagchi:2012xr} to derive the Bekenstein-Hawking entropy for these black holes. Along the same lines, it could thus be argued that the considerations of this paper would also be applicable to black holes in general dimensions. In particular, higher dimensional black holes will also get their ``spots" \cite{Kraus:2016nwo} through the mechanism elucidated in this paper.
\bigskip
\noindent {\em{Tensionless string theory and BMS modular invariance}}
\smallskip
\noindent BMS$_3$ algebra arises as residual gauge symmetries on the worldsheet of the tensionless bosonic closed string, in the equivalent of the conformal gauge \cite{Isberg:1993av, Bagchi:2013bga}. There has been recent efforts in trying to formulate the theory of tensionless strings from the point of BMS symmetries, mirroring the development of usual string theory by methods of conformal symmetry. Our modular explorations are hence of importance when one attempts to understand tensionless string theory on higher genus surfaces. One of the principal aims of the programme initiated in \cite{Bagchi:2013bga, Bagchi:2015nca} is the understanding of string amplitudes in this very high energy limit by purely worldsheet methods and comparing it to the seminal analysis of Gross and Mende \cite{Gross:1987kza, Gross:1987ar, Gross:1988ue}. Modular properties of 2d BMSFTs would be central to understanding one loop amplitudes in this tensionless regime.
\smallskip
\noindent Relatedly, ambitwistor strings have been shown to be tensionless strings in disguise \cite{Casali:2016atr}. Efforts in attempting to construct partition functions for these ambitwistor strings have been made \cite{Casali:2017zkz} using the modular transformations we have worked extensively with in this work. It is conceivable that more can be understood for ambitwistor string theory using techniques developed in this paper.
\bigskip
\newpage
\noindent {\em{Non-relativistic conformal symmetry}}
\smallskip
\noindent As remarked earlier, BMS$_3$ is isomorphic to the 2d Galilean conformal algebra (GCA) \cite{Bagchi:2010zz}, which can be obtained by a non-relativistic limit on the parent relativistic conformal theory. This remarkable isomorphism means that all our computations in this paper are equally valid for non-relativistic conformal systems in 2d.
\smallskip
\noindent
There has been recent interesting progress in the understanding of non-relativistic strong gravity \cite{Hansen:2020pqs}, where non-relativistic black hole solutions have been discussed. Building on the initial proposal of a non-relativistic limit of AdS/CFT, one could put our results in context. Starting with AdS$_3$/CFT$_2$, following \cite{Bagchi:2009my}, one would end up with a non-relativistic Newton-Cartan like AdS$_2 \times {\rm I\!R}$ which would be dual to a 2d Galilean CFT. There would presumably be solutions in the bulk corresponding to the BTZ black hole, which like the Schwarzschild solutions in \cite{Hansen:2020pqs} would have horizons. The modular properties described in this paper would help reproduce the entropy and probe reactions to these solutions. It would be interesting to investigate this in further detail.
\smallskip
\noindent
The 2d GCA also appears on non-relativistic string worldsheets corresponding to strings moving on target spaces with Newton-Cartan geometries \cite{Harmark:2018cdl}. Again, properties of 2d field theories with these symmetries, particularly modular properties discussed in this paper, would be important there. Construction of partition functions and modular invariance in the context of these string theories would rely heavily on the things we have developed here.
\bigskip
\noindent {\em{Higher dimensional explorations}}
\smallskip
\noindent
It would be very useful to try and generalise these considerations to higher dimensions. The general idea of holography stays the same, viz. we propose that the $d$-dimensional field theory would inherit the asymptotic symmetries of the $(d+1)$ dimensional asymptotically flat bulk, which is the BMS$_{d+1}$ group \cite{Bagchi:2016bcd, Bagchi:2019xfx} {\footnote{For other recent proposals, see e.g. \cite{Ball:2019atb, Laddha:2020kvp}. See \cite{Banerjee:2020kaa, Banerjee:2018gce} for interesting work in trying to understand scattering in four dimensional asymptotically flat spacetimes with a potential three dimensional dual building on the work relating scattering with a celestial 2d CFT in \cite{Pasterski:2016qvg, Pasterski:2017kqt}.}}. In particular, if some of these methods can be applied to 3d field theories with BMS$_4$ symmetries, we would be able to connect with quantities of very significant physical interest, S-matrix elements.
BMS$_4$ symmetries however are more involved and the structure of $\mathscr{I}^\pm$ which is given by ${\rm I\!R}_u \times S^2$, now admits two copies of the (non-centrally extended) Virasoro algebra along with supertranslations {\footnote{This is one of the proposed infinite dimensional extensions of BMS$_4$ \cite{Barnich:2010eb}. For another proposal, involving the group Diff$(S^2)$, see \cite{Campiglia:2014yka}.}}. It is not entirely obvious how the methods described in this paper would generalise.
\smallskip
\noindent As we mentioned, the isomorphism between the BMS$_3$ and the GCA$_2$ means that the results we derived here are equally applicable to non-relativistic conformal systems. In higher dimensions, the structure of GCA$_d$ remains very similar to the $d=2$ case, with only additional vectorial labels on the $M$ generators and the existence of rotations. These symmetries have been shown to exist in non-relativistic versions of electrodynamics \cite{Bagchi:2014ysa, Festuccia:2016caf} and Yang-Mills \cite{Bagchi:2015qcw}, with arbitrary massless matter couplings as well \cite{Bagchi:2017yvj}. Our methods developed in this paper may be more amenable to generalisation to higher dimensions in this context given the similar structures of the $d=2$ case with the general case.
\bigskip
\bigskip \bigskip
\subsection*{Acknowledgements}
It is a pleasure to thank Rudranil Basu, Daniel Grumiller, Max Reigler, Joan Simon for interesting conversations and comments on a draft of the paper. We also thank Konstantin Alkalaev for helpful correspondence and an anonymous referee for pointing out some inconsistencies in an earlier version of the paper.
\medskip
\noindent AB's research is supported by a Swarnajayanti fellowship of the Department of Science and Technology and the Science and Engineering Research Board (SERB), India. AB is further supported by the following grants from SERB: EMR/2016/008037, ERC/2017/000873, MTR/2017/000740. PN is supported by a Fulbright-Nehru Doctoral Research fellowship (Grantee ID E0609781). PN would also like to thank Mukund Rangamani and the hospitality of UC Davis during the course of this work. She would further like to thank Sankha Bhattacharya and Nivedita Bhattacharya for their unconditional support and hospitality at San Jose during a difficult time of emergency in middle of the worldwide pandemic.
\newpage
\section*{APPENDICES}
\bigskip
|
1,941,325,220,514 | arxiv | \section{Introduction}
Let $\Sigma$ be a compact orientable hyperbolic surface whose boundary, if any, is geodesic, and let $G$ denote its fundamental group.
By a standard generating set $S$ for $G$ we mean the following:
when $G$ is free (i.e. when $\partial \Sigma \neq \emptyset$) $S$ is a free basis for $G$. Otherwise, $\Sigma$ is a closed orientable surface of genus $g \ge 2$ and $S$ is the generating set used in the usual presentation $G = \langle a_1,\ldots,a_g,b_1, \ldots b_g : \prod_i [a_i,b_i] = 1 \rangle$.
Now fix a standard generating set $S$ of $G$, and let $|g |$ be the word length of $g$ with respect to $S$.
For each $g \in G$, let $[g]$ denote its conjugacy class, and for any conjugacy class $\gamma = [g]$ define its conjugacy length
$\Vert \gamma \Vert = \Vert g \Vert := \min_{[g] = \gamma} | g |$ to
be the minimum word length over all elements representing $\gamma$.
Any conjugacy class $\gamma$ is represented by a closed geodesic in $\Sigma$, and let $\tau(\gamma)$ denote the length of this geodesic in the hyperbolic metric.
Let $\mu_n$ denote the uniform distribution on the set $\mathcal{F}_n$ of conjugacy classes of length $n$.
The goal of this note is to prove the following central limit theorem:
\begin{theorem} \label{T:main}
There exist constants $L > 0$, $\sigma > 0$ such that for any $a, b \in \mathbb{R}$ with $a < b$ we have
$$\mu_n\left(\gamma \ : \ \frac{\tau(\gamma) - n L}{\sigma \sqrt{n}} \in [a, b] \right) \to \frac{1}{\sqrt{2 \pi}} \int_a^b e^{-\frac{x^2}{2}} \ dx$$
as $n \to \infty$.
\end{theorem}
Motivated by experimental evidence, Chas--Li--Maskit \cite{CLM} conjectured that the conclusion of Theorem \ref{T:main} holds for a hyperbolic pair of pants.
This followed an earlier central limit theorem by Chas--Lalley \cite{CL} for the distribution of self-intersection numbers of random geodesics.
The proof of Theorem \ref{T:main} uses the central limit theorem for H\"older continuous observables on a mixing Markov chain following Ruelle \cite{Ruelle} and Bowen \cite{Bowen}
(see also Pollicott--Sharp \cite{PS} and Calegari \cite{Ca}), combined with estimates on Gromov products by the authors \cite{GTT}. The Markov chain which encodes closed geodesics on a surface is provided by Series \cite{Se} (see also Wroten \cite{Wr}).
\smallskip
We note that P. Park has recently written up a related result where the uniform distribution on conjugacy classes is replaced by the $n^{th}$ step distribution of a
simple random walk on $G$ \cite{Park}. Let us note that counting for the simple random walk and counting with respect to balls in the Cayley graph are in general different, and many authors addressed the question of how they are related on various groups.
\section{Preliminaries}
\subsection*{The geometric setup}
Throughout we consider $G$ as a discrete group of isometries of the hyperbolic plane $\mathbb{H}^2$. When $\Sigma$ is closed, $G$ is a cocompact Fuchsian group. Otherwise, $\partial \Sigma \neq \emptyset$ and $G$ is the free group $F_N$ for some $N \ge 2$.
In this case, we have that $\Sigma$ is the convex core of $\mathbb{H}^2/G$, where $G$ acts on $\mathbb{H}^2$ as a Schottky group.
Fix a base point $z \in \mathbb{H}^2$.
Then for $\gamma = [g]$, $\tau(\gamma) = \tau(g)$ equals the (stable) translation length of $g$ on $\mathbb{H}^2$. Hence, one has the formula (see e.g. \cite[Proposition 5.8]{MT})
\begin{equation} \label{E:tau}
\tau(g) = d(gz, z) - 2 (gz, g^{-1} z)_z + O(\delta)
\end{equation}
where $(x, y)_z$ denotes the Gromov product. Here $A = B + O(\delta)$ means that there is a constant $C$, which depends only on the hyperbolicity constant $\delta$ of $\mathbb{H}^2$, such that $|A - B| \leq C$.
\subsection*{Some basic probability}
We begin by recording a few basic lemmas that will be needed for our arguments.
\begin{lemma} \label{L:basic}
Let $(A_n)$ be any sequence of measurable sets in a probability space, $(\mathbb{P}_n)$ a sequence of probability measures, and let $(B_n)$ a sequence such that
$$\lim_{n \to \infty} \mathbb{P}_n(B_n) = 1.$$
Then
$$\limsup_n | \mathbb{P}_n(A_n) - \mathbb{P}_n(A_n \cap B_n) | = 0.$$
\end{lemma}
\begin{proof}
By elementary set theory,
$$\mathbb{P}_n(A_n) = \mathbb{P}_n(A_n \cap B_n) + \mathbb{P}_n(A_n \setminus B_n) \leq \mathbb{P}_n(A_n \cap B_n) + \mathbb{P}_n(B_n^c)$$
which yields the claim.
\end{proof}
\begin{lemma} \label{L:sum}
Let $(\mathbb{P}_n)$ be a sequence of probability measures on a Borel space $X$, and let $F, G : X \to \mathbb{R}$ be two measurable functions.
Suppose that $\mathbb{P}_n(|G(x)| \geq \epsilon) \to 0$ for any $\epsilon > 0$, and let $(c_n)$ be a sequence of positive real numbers with $\lim_{n \to \infty} c_n = 1$.
Suppose that there exists a continuous function $\rho : \mathbb{R} \to \mathbb{R}^+$ such that
$$\lim_{n \to \infty} \mathbb{P}_n(F(x) \in [a, b]) = \int_a^b \rho(x) \ dx.$$
Then
$$\lim_{n \to \infty} \mathbb{P}_n \left( \frac{F(x) + G(x)}{c_n} \in [a, b] \right) = \int_a^b \rho(x) \ dx.$$
\end{lemma}
\begin{proof}
Fix $\epsilon > 0$, and denote $\Phi_{a, b} := \int_a^b \rho(x) \ dx$. If $n$ is sufficiently large, then by setting $Y_n := \frac{F + G}{c_n}$ we have
$$\{ F(x) \in [a+\epsilon, b - \epsilon] \textup{ and } |G(x)| \leq \epsilon/2 \} \subseteq \{Y_n \in [a, b] \}$$
hence using Lemma \ref{L:basic}
$$\liminf \mathbb{P}_n( Y_n \in [a, b] ) \geq \liminf \mathbb{P}_n(
F \in [a+\epsilon, b - \epsilon] \textup{ and } |G| \leq \epsilon/2) = $$
$$= \liminf \mathbb{P}_n(F \in [a+\epsilon, b - \epsilon]) = \Phi_{a+\epsilon, b - \epsilon}.$$
On the other hand
$$\{ Y_n \in [a, b] \textup{ and } |G| \leq \epsilon/2 \} \subseteq \{F \in [a - \epsilon, b + \epsilon] \}$$
hence
$$\limsup \mathbb{P}_n( Y_n \in [a, b] )= \limsup \mathbb{P}_n(
Y_n \in [a, b] \textup{ and } |G| \leq \epsilon/2) \leq $$
$$\leq \limsup \mathbb{P}_n(F \in [a- \epsilon, b + \epsilon]) = \Phi_{a-\epsilon, b + \epsilon}$$
and taking $\epsilon \to 0$ completes the proof.
\end{proof}
\begin{lemma} \label{L:TV}
If $\lambda, \nu$ are purely atomic probability measures on a set $X$ and $\lambda$ is absolutely continuous with respect to $\nu$, then
$$\Vert \lambda - \nu \Vert_{TV} \leq \left \Vert \frac{d \lambda}{d\nu} - 1 \right\Vert_\infty,$$
where $\Vert \cdot \Vert_{TV}$ denotes the total variation of a measure.
\end{lemma}
\begin{proof}
Since the measures are atomic,
$$\frac{d\lambda}{d\nu}(x) = \frac{\lambda(x)}{\nu(x)} \qquad \textup{ for any }x \in X.$$
Then
$$\Vert \lambda - \nu \Vert_{TV} = \sup_{A \subseteq X} |\lambda(A)-\nu(A)| \leq \sup_A \sum_{x \in A} |\lambda(x) - \nu(x)| \leq $$
$$\leq \sup_A \sum_{x \in A} \left| \frac{\lambda(x)}{\nu(x)} - 1 \right| \nu(x) \leq \sup_A \left( \left\Vert \frac{d\lambda}{d\nu} - 1 \right\Vert_{\infty} \nu(A) \right)\leq
\left\Vert \frac{d\lambda}{d\nu} - 1 \right\Vert_{\infty},$$
completing the proof.
\end{proof}
\section{The central limit theorem for displacement}
The proof of Theorem \ref{T:main} uses equation \eqref{E:tau}: basically, one proves a CLT for the displacement function $d(z, gz)$, and then shows that the contribution of the term $2 (gz, g^{-1}z)_z$ tends to zero. These two facts will suffice by Lemma \ref{L:sum}.
We will start by establishing the CLT for displacement.
\subsection*{Coding for closed geodesics}
First of all, we use that conjugacy classes in free and surface groups can be encoded by a finite graph.
\begin{lemma} \label{L:code}
Let $G = \pi_1(\Sigma)$ where $\Sigma$ is an orientable hyperbolic surface of finite type.
Let $S$ be a standard generating set for $G$. Then there exists an oriented graph $\Gamma$ whose edges are labeled by elements of $S \cup S^{-1}$ and
such that:
\begin{enumerate}
\item cycles in $\Gamma$ of length $n$ are in bijection with conjugacy classes $C(G)$ of length $n$ in the group, except for finitely many exceptions;
\item a conjugacy class is primitive if and only if the corresponding cycle in the graph is primitive, i.e. not the power of a shorter cycle;
\item the adjacency matrix $M$ for this graph is aperiodic.
\end{enumerate}
\end{lemma}
For closed surface groups, Lemma \ref{L:code} is precisely (\cite{PS3}, Lemma 1.1), which is a reformulation of results of Series \cite{Se}.
Another coding for closed surfaces is given by Wroten \cite{Wr} and also yields this result. The much easier case of free groups is briefly explained below.
Recall that a matrix $M$ is \emph{aperiodic} if there exists an integer $k \geq 1$ such that all entries of $M^k$ are positive.
By the Perron-Frobenius theorem, the matrix $M$ has a unique, simple eigenvalue $\lambda > 1$ of maximum modulus.
Also, if $\mathrm{ev}$ denotes the map which reads the labels off of oriented edges of $\Gamma$, then $\mathrm{ev}$ extends to the \emph{evaluation map} from directed paths in $\Gamma$ to $G$. In details, if a directed path $p$ is a concatenation of orientated edges $e_1,\ldots, e_n$, then $\mathrm{ev}(p) = \mathrm{ev}(e_1) \cdots \mathrm{ev}(e_n)$ in $G$.
It is this map that induces the bijection in item $(1)$ of Lemma \ref{L:code}. We further remark that Lemma \ref{L:code} implies that directed paths in $\Gamma$ map to geodesics in $G$; that is, if $p$ is a directed path of length $n$ in $\Gamma$, then $|\mathrm{ev}(p)| = n$ with respect to the generating set $S$.
We note that for free groups, the construction of the graph $\Gamma$ is immediate.
Let $G = F_N$ and fix a basis $\{a_1, \dots, a_N\}$ of $F_N$. Then the graph $\Gamma$ has $2N$ vertices, labelled $a_i^\epsilon$ with $i = 1, \dots, N$, $\epsilon = \pm$. For each vertex $v = a_i^\epsilon$, there exists an edge labelled $a_j^\eta$ to the vertex $a_j^\eta$ unless $i = j$ and $\epsilon = - \eta$. In this case, nontrivial cyclically reduced words are in bijection with oriented based closed paths since such a word can only be read as a \emph{closed} path by starting at the vertex corresponding to the label on its last letter. As two cyclically reduced words represent conjugate elements if and only if they differ by cyclic permutation, which corresponds to changing the basepoint of the loop, this establishes items $(1)$ and $(2)$. (In this case, the only exception in item $(1)$ is for the trivial conjugacy class, which can be read as the trivial cycle based at each vertex.)
Item $(3)$ is clear from the construction.
\medskip
Let $\lambda_n$ be the uniform distribution on the set $\mathcal{C}_n$ of (based) closed paths of length $n$ in $\Gamma$.
A short counting argument shows the following, which is a variation of (\cite{CL}, Lemma 4.1):
\begin{lemma} \label{L:push}
Let $p_n \colon \mathcal{C}_n \to \mathcal{F}_n$ be the map from closed paths to conjugacy classes induced by the evaluation map. Then
$$\Vert (p_n)_\star \lambda_n - \mu_n \Vert_{TV} \to 0\qquad \textup{as }n \to \infty,$$
where $\Vert \cdot \Vert_{TV}$ denotes the total variation of a measure.
\end{lemma}
\begin{proof}
Let $\Gamma_n$ denote
the set of primitive cycles of length $n$ (without remembering the basepoint). Note that $\textup{Tr }M^n = \lambda^n + \sum_{i =1}^k \lambda_i^n$, where $|\lambda_i| < \lambda$.
Hence $\lambda^n \leq \textup{Tr }M^n \leq c \lambda^n$ where $c$ only depends on $M$, and $\lim_{n \to \infty}\frac{\textup{Tr }M^n}{\lambda^n} = 1$.
Moreover, by definition $\sum_{d | n} d (\# \Gamma_d) =\textup{Tr }M^n$, so $n (\# \Gamma_n) \leq \textup{Tr }M^n \leq c \lambda^n$.
Hence, the set of non-primitive closed paths of length $n$ has cardinality
$$\textup{Tr }M^n - n (\# \Gamma_n) = \sum_{d|n, d \neq n} d (\#\Gamma_d) \leq \frac{c n}{2} \lambda^{n/2}$$
and so
$$\lim_{n \to \infty}\frac{n (\# \Gamma_n)}{\lambda^n} = 1.$$
Then for any set $A \subseteq \mathcal{F}_n$ and any $n \geq 1$ sufficiently large
(using the bijection provided by Lemma \ref{L:code} with $n$ large enough to avoid the exceptions)
$$\mu_n(A) = \frac{\sum_{d | n} \#(A \cap \Gamma_d)}{\sum_{d | n} (\# \Gamma_d)} = \frac{\#(A \cap \Gamma_n) + O(n \lambda^{n/2})}{(\# \Gamma_n) + O(n \lambda^{n/2})}$$
and, since the map $p_n$ is exactly $d$-to-$1$ on the preimage of $\Gamma_d$,
$$\lambda_n(p_n^{-1}(A))= \frac{\sum_{d | n} d \#(A \cap \Gamma_d)}{\sum_{d | n} d (\# \Gamma_d)} = \frac{n \#(A \cap \Gamma_n) + O(n \lambda^{n/2})}{n (\# \Gamma_n) + O(n \lambda^{n/2})}.$$
Hence
$$\left| \mu_n(A) - \lambda_n(p_n^{-1}(A)) \right| = O\left(\frac{n \lambda^{n/2}}{(\# \Gamma_n)}\right) \to 0, $$
as $n\to \infty$. This completes the proof.
\end{proof}
\subsection*{Markov chains}
Let $\Gamma$ be a directed graph with vertex set $V = V(\Gamma)$, edge set $E = E(\Gamma)$, and aperiodic transition matrix $M$.
For $n \ge0$, we let $\Omega^n$ denote the set of paths of length $n$ in $\Gamma$ starting at any vertex, and $\Omega^\star = \bigcup_{n \geq 1} \Omega^n$
the set of all paths of finite length. Also denote by $\Omega$ the set of all \emph{infinite} directed paths. If we wish to focus on the subset of paths that start at the vertex $v \in V$, then we use $v$ as a subscript as in $\Omega_v$ or $\Omega^n_v$.
We will associate to $\Gamma$ a probability on each edge and a probability on each vertex so that the corresponding measure on the space of infinite paths in $\Gamma$ is shift-invariant.
(The reader is directed to \cite[Section 8]{Walters} for additional details of this standard construction.)
Recall that if $x = (x_i)_{i\ge 0} \in \Omega$ is a directed path in $\Gamma$, then the image of $x$ under the shift $T \colon \Omega \to \Omega$ is $Tx = (x_i)_{i\ge 1}$. The assumption that $M$ is aperiodic translates to the fact that the shift is topologically mixing.
For $x\in \Omega^n$, we continue to use the notation $T^ix$ to denote the image of $x$ under the shift for $i\le n$.
Recall that $\lambda >1$ is the unique (simple) eigenvalue of $M$ of maximum modulus.
Fix a right eigenvector $\textbf{v}$ for $M$ of eigenvalue $\lambda$ and a left eigenvector $\textbf{u}$ of the same eigenvalue, normalized so that
$\textbf{u}^T \textbf{v} = 1$.
Let us now define the measure $(\pi_i)_{i \in V}$ on the set of vertices of $\Gamma$ where $\pi_i = u_i v_i$,
and for each edge in $\Gamma$ from $i$ to $j$ let us define the probability $q_{ij} = \frac{m_{ij} v_j}{\lambda v_i}$.
This defines a Markov chain on $\Gamma$, where $\pi_i$ is the probability of starting at vertex $v_i$, and $q_{ij}$
is the probability of going from $v_i$ to $v_j$.
This construction defines a shift invariant measure $\nu$ (the so-called Parry measure) on the set of infinite paths $\Omega$. Namely, let
$C(i_0, i_1, \dots, i_k)$ denote the set of infinite paths which start with the path $v_{i_0} \to v_{i_1} \to \dots \to v_{i_k}$.
This is a cylinder set
of $\Omega$ and we set its measure to be
$$\nu(C(i_0, i_1, \dots, i_k)) = \pi_{i_0} q_{i_0 i_1} q_{i_1 i_2} \dots q_{i_{k-1} i_k},$$
and this determines a shift invariant measure $\nu$ on $\Omega$.
(In fact, this defines the measure of maximal entropy for the shift $T \colon \Omega \to \Omega$ \cite[Theorem 8.10]{Walters}.)
Now for each $n$, let $\nu_n$ be the pushforward of $\nu$ with respect to the map $\Omega \to \Omega^n$ which takes an infinite path to its
prefix of length $n$.
The measure $\nu_n$ is the distribution of the $n^{th}$ step of the Markov chain whose initial distribution is $(\pi_i)_{i \in V}$, and $\nu_n$ is supported on the set of paths of length $n$.
As before $\mathrm{ev} : E(\Gamma) \to G$ is the evaluation map which associates to each edge its label in $S \cup S^{-1} \subset G$. By concatenation, the map extends to a map $\mathrm{ev} : \Omega^\star \to G$
from the set of all finite paths to $G$. Hence, if $x \in \Omega^n$ is a path of length $n$ given as a sequence of vertices $(x_i)_{i=0}^{n}$ associated to the edge path $e_1, \ldots, e_n$, then $\mathrm{ev}(x) = \mathrm{ev}(e_1) \cdots \mathrm{ev}(e_n)$ in $G$. In particular, if $x$ is a single vertex (i.e. a path of length $0$) then $\mathrm{ev}(x) = 1$.
\subsection{The Central Limit Theorem for H\"older observables}
Here we briefly recall the classical CLT from Thermodynamic Formalism as we will need it. We closely follow Bowen \cite[Chapter 1]{Bowen}.
Let $T \colon \Omega \to \Omega$ be a topologically mixing shift of finite type and $\nu$ a shift-invariant Gibbs measure on $\Omega$. (For us, $\nu$ will always be the measure of maximal entropy on the Markov chain $\Omega$, defined as above.) We have a metric $d_\Omega$ on the space $\Omega^\star \cup \Omega$
of all directed paths defined by $d_\Omega(x,y) = 2^{-K}$ where $K$ is the largest nonnegative integer such that $x_i = y_i$ for $i < K$.
We note that with this metric, $\Omega$ is the set of limit points of the discrete set $\Omega^\star$. Topologically, $\Omega$ is a Cantor set.
A function $f \colon \Omega \to \mathbb{R}$ is \emph{H\"older continuous}
if for all $x,y \in \Omega$, $|f(x)-f(y)| \le Cd_\Omega(x,y)^\epsilon$ for some $C, \epsilon >0$.
The following result is a combination of Theorem 1.27 of \cite{Bowen} and the remark that follows it. We will use the notation $N_{a, b} :=\frac{1}{\sqrt{2 \pi}} \int_a^b e^{-\frac{x^2}{2}} \ dx$.
\begin{theorem} \label{T:CLT}
Suppose that $f \colon \Omega \to \mathbb{R}$ is H\"older and that there does not exist a function $u \colon \Omega \to \mathbb{R}$ such that
$f = u - u \circ T + \int f d\nu$.
Then there is a constant $\sigma>0$ such that for any $a<b$,
$$\nu \left( x \in \Omega \ : \ \frac{\sum_{i=0}^{n-1}f(T^ix) - n \int f d\nu }{\sigma \sqrt{n}} \in [a, b] \right) \to N_{a, b}.$$
\end{theorem}
\subsection{Displacement and the CLT}
The main result of this section is the following central limit theorem for displacement along the Markov chain.
First, fix a basepoint $z\in \mathbb{H}^2$.
\begin{theorem} \label{T:disp}
There exist constants $L > 0$ and $\sigma > 0$ such that for any $a, b \in \mathbb{R}$ with $a < b$ we have
$$\nu_n \left( x \in \Omega^n \ : \ \frac{d(\mathrm{ev}(x)z, z) - n L }{\sigma \sqrt{n}} \in [a, b] \right) \to N_{a, b}.$$
\end{theorem}
The proof of Theorem \ref{T:disp} requires the following setup, which approximately follows the discussion in Calegari \cite[Section 3.7]{Ca}.
For $g \in G$ and generator $s\in S$, define $D_s F(g) = d(z,gz) - d(z,sgz)$.
For a \emph{finite} path $x \in \Omega^\star$, let $DF(x) = D_{s^{-1}}F(\mathrm{ev}(x))$, where $s$ labels the first edge of $x$. This defines a function $DF \colon \Omega^\star \to \mathbb{R}$ on the set of all finite paths such that
\begin{align*}
DF(x) = d(z,\mathrm{ev}(x)z) - d(z,\mathrm{ev}(T x)z).
\end{align*}
In particular, if $x \in \Omega^n$, then we note that
\begin{align} \label{sum:1}
\sum_{i=0}^{n-1}DF(T^ix) = d(z,\mathrm{ev}(x)z).
\end{align}
To apply the central limit theorem (Theorem \ref{T:CLT}), one needs to verify the H\"older continuity property of the observable. For this, we need the following:
\begin{lemma}[\cite{PS}, Proposition 1; \cite{PS2}, Lemma 1 and Proposition 3] \label{L:Holder}
Let $X$ be a $\mathrm{CAT}(-k)$ metric space, with $k > 0$. Then there exist constants $C > 0$ and $\alpha > 1$ such that for any $h, g \in G$, any $s \in S$
\begin{align}\label{e:holder}
|DF_s(g) - DF_s(h)| \le C \alpha^{- (g, h)}.
\end{align}
\end{lemma}
Since the evaluation map $\mathrm{ev} \colon \Omega^\star \to G$ is geodesic, then for any $x, y \in \Omega^\star$
$$\alpha^{-(\mathrm{ev}(x), \mathrm{ev}(y))} \leq d_\Omega(x, y)^\eta$$
where $\eta = \log_2(\alpha)$. Hence, the function $DF$ on $\Omega^\star$ is H\"older continuous and therefore has a unique continuous extension to the H\"older function $DF \colon \Omega \to \mathbb{R}$ on the space of all \emph{infinite} paths $\Omega$.
For $x \in \Omega$, define
\begin{align*}
F_n(x) = DF(x) + DF(Tx) + \ldots +DF(T^{n-1}x).
\end{align*}
If we let $x^n \in \Omega^n$ denote the prefix of $x$ of length $n$, we have that
\begin{align*}
F_n(x^n) = d(z,\mathrm{ev}(x^n)z),
\end{align*}
by (\ref{sum:1}). Whereas for an infinite path $x \in \Omega$, we have
\begin{align*}
F_n(x) = \lim_{k \to \infty} \big (d(z,\mathrm{ev}(x^k)z) - d(z,\mathrm{ev}(T^n x^k)z) \big).
\end{align*}
However, the following lemma bounds how far $F_n(x)$ can be from the displacement $F_n(x^n) = d(z,\mathrm{ev}(x^n)z)$.
\begin{lemma}\label{lem:almost_displacement}
The difference $| F_n(x) - F_n(x^n)|$ is uniformly bounded, independent of $x \in \Omega$ and $n \ge 1$.
\end{lemma}
\begin{proof}
Write $x = y_1y_2 \ldots$, where the $y_i$ are edges of $\Gamma$, i.e. we represent $x$ as an edge path in $\Gamma$. Then $x^k = y_1y_2\ldots y_k$. Also, let $\mathrm{ev}(y_i) = g_i$. Then
\begin{align*}
|F_n(x^n) - F_n(x)| &= \lim_{k\to \infty} \big( d(z,g_1\ldots g_n z) + d(z, g_{n+1}\ldots g_k z) - d(z, g_1\ldots g_k z) \big) \\
& = \lim_{k\to \infty} \big( d(z,g_1\ldots g_n z) + d(g_1\ldots g_n z, g_{1}\ldots g_k z) - d(z, g_1\ldots g_k z) \big) \\
&= 2 \lim_{k \to \infty} \big(z,\mathrm{ev}(x^k)z \big)_{\mathrm{ev}(x^n)z}.
\end{align*}
Since $G \curvearrowright \mathbb{H}^2$ is convex cocompact and the path $i \to \mathrm{ev}(x^i) = g_1\ldots g_i$ is geodesic in $G$, the path $i \to \mathrm{ev}(x^i) z$ is a uniform quasigeodesic.
That is, $i \to \mathrm{ev}(x^i) z$ is a $(K,C)$--quasigeodesic in $X$ for $K\ge 1$ and $C\ge 0$ not depending on $x$. This (together with the stability of quasigeodesics in the hyperbolic space $\mathbb{H}^2$) immediately implies that the quantity $\big(z,\mathrm{ev}(x^k)z \big)_{\mathrm{ev}(x^n)z}$ is uniformly bounded for every $k \ge n$. This completes the proof.
\end{proof}
Finally, the following lemma is needed to establish positivity of $\sigma$ in Theorem \ref{T:disp}.
\begin{lemma} \label{lem:sig_pos}
There does not exist a function $u : \Omega \to \mathbb{R}$ and $L \in \mathbb{R}$ such that
$$DF = u - u \circ T + L.$$
\end{lemma}
\begin{proof}
Suppose not. Then
$$\sum_{i = 0}^{n-1} DF(T^i(x)) = u(x) - u(T^n(x)) + nL$$
Thus, if $x \in \Omega$ is a periodic point of period $n$ for $T$, then
$$F_n(x) = \sum_{i = 0}^{n-1} DF(T^i(x)) = n L .$$
Next, a direct computation
shows that if $g = \mathrm{ev}(x^n)$, then
\[
\tau(g) = \tau(\mathrm{ev}(x^n)) = F_n(x) = nL.
\]
Indeed, as in \cite[Lemma 4.2]{HS}, since $T^nx = x$, $F_{nk}(x) = k F_n(x)$ and we see that $F_{nk}(x) = F_{nk}(x^{nk}) + O(1)$ (Lemma \ref{lem:almost_displacement}) implies
\[
F_n(x) = \lim_{k \to \infty} \frac{1}{k}F_{nk}(x^{nk}) = \lim_{k\to \infty} \frac{1}{k}d(z,g^kz),
\]
which, by definition, equals the translation length $\tau(g)$.
Hence, we conclude using Lemma \ref{L:code} that $\tau([g]) = L \Vert g \Vert$ for all but finitely many conjugacy classes in $G$. This, however, contradicts the fact that there exists two closed geodesics on $\Sigma$ with incommensurable lengths (see, for example, \cite{Dal'bo}).
\end{proof}
\begin{proof}[Proof of Theorem \ref{T:disp}]
Since $DF$ is a H\"older continuous function on a mixing shift of finite type,
Theorem \ref{T:CLT} along with Lemma \ref{lem:sig_pos}
give
\[
\nu \left( x \ : \ \frac{F_n(x) - n L }{\sigma \sqrt{n}} \in [a, b] \right) \to N_{a, b},
\]
where as before $F_n(x) = DF(x) + DF(Tx) + \ldots +DF(T^{n-1}x)$.
But by Lemma \ref{lem:almost_displacement}, the probability
\[
\nu \left(x \ : \frac{|F_n(x^n) - F_n(x)|}{\sqrt{n}} \ge \epsilon \right) \to 0,
\]
as $n \to \infty$, for any $\epsilon > 0$.
Hence, applying Lemma \ref{L:sum} gives the CLT for the displacement $F_n(x^n) = d(z,\mathrm{ev}(x^n)z)$ for $x \in \Omega$ with respect to the measure $\nu$. Since the distribution of $x^n \in \Omega^n$ is $\nu_n$, this completes the proof of the theorem.
\end{proof}
\section{Convergence to the counting measure for closed paths}
Next we use the Theorem \ref{T:disp} (which is about the Markov chain) to study the distribution of closed paths.
Given a path $x$ of length $n$, let $\widehat{x}$ denote the prefix of $x$ of length $n - \log n$.
Recall that $\lambda_n$ is the uniform distribution on the set $\mathcal{C}_n$ of based closed paths of length $n$ in $\Gamma$. Since $\mathcal{C}_n \subset \Omega^n$, $\lambda_n$ defines a measure on $\Omega^n$ supported on $\mathcal{C}_n$.
Let $\lambda_{n, m}$ denote the distribution of the prefix of length $n - m$ of a uniformly chosen closed path of length $n$. Said differently, $\lambda_{n,m}$ is the distribution on paths of length $n-m$ obtained by pushing $\lambda_n$ forward under the prefix map.
In particular, if $x$ has law $\lambda_n$, then $\widehat{x}$ has law $\lambda_{n, \log n}$. We define $\nu_{n,m}$ in the same way, using $\nu_n$ in place of $\lambda_n$. By the Markov property, $\nu_{n,m} = \nu_{n-m}$.
\begin{proposition} \label{P:RN-der}
With notation as above, $\lambda_{n, m}$ is absolutely continuous with respect to $\nu_{n, m}$, and moreover
$$\sup_{\gamma \in \Omega^{n-m}} \left| \frac{d \lambda_{n, m}}{d \nu_{n, m}}(\gamma) - 1 \right| \to 0$$
as $\min \{ m, n \} \to \infty$.
\end{proposition}
\begin{proof}
Given a path $\gamma = e_1 \cdot \ldots \cdot e_{n-m}$ with starting vertex $v_i$ and end vertex $v_j$ we have
$$\lambda_{n, m}(\gamma) = \frac{\#\{\textup{paths of length }m\textup{ from }v_j\textup{ to }v_i\}}{\# \{\textup{closed paths of length }n \}}$$
$$ = \frac{ \textbf{e}_j^T M^m \textbf{e}_i }{\textup{Tr }M^n}$$
where $\textbf{e}_i$ is the $i^{th}$ basis vector.
Now, recall that we have fixed a right eigenvector $\textbf{v}$ for $M$ of eigenvalue $\lambda$, and a left eigenvector $\textbf{u}$ of the same eigenvalue, normalized so that
$\textbf{u}^T \textbf{v} = 1$.
Since $M$ is irreducible and aperiodic, by the Perron-Frobenius Theorem we have
$$\lim_{n \to \infty} \frac{M^n}{\lambda^n} = \textbf{v} \textbf{u}^T$$
and in particular
$$\lim_{n \to \infty} \frac{\textbf{e}_i^T M^n \textbf{e}_j}{\lambda^n} = \textbf{e}_i^T \textbf{v} \textbf{u}^T \textbf{e}_j = v_i u_j.$$
As before the measure $(\pi_i)$ where $\pi_i = u_i v_i$ is stationary for the Markov chain defined as $q_{ij} = \frac{m_{ij} v_j}{\lambda v_i}$, so
we consider this Markov chain with the stationary measure as starting distribution.
Let $\nu_{n, m} = \nu_{n-m}$ be the pushforward of the Markov measure on the set of paths of length $n-m$. Then
$$\nu_{n, m}(\gamma) = \frac{\pi_i v_j}{v_i \lambda^{n-m}} = \frac{u_i v_j}{\lambda^{n-m}}.$$
Hence
$$\frac{d \lambda_{n, m}}{d \nu_{n, m}}(\gamma) = \frac{ \textbf{e}_j^T M^m \textbf{e}_i }{\textup{Tr }M^n} \frac{\lambda^{n-m}}{u_i v_j} = \frac{ \textbf{e}_j^T M^m \textbf{e}_i }{\lambda^m u_i v_j} \frac{\lambda^n}{\textup{Tr }M^n} \to 1,$$
as $\min\{m,n\} \to \infty$.
\end{proof}
We conclude this section by promoting the CLT for displacement from the Markov chain (Theorem \ref{T:disp}) to the counting measure $\lambda_n$.
\begin{theorem} \label{T:disp2}
For any $a, b \in \mathbb{R}$ with $a < b$ one has
$$\lambda_n \left( x \ : \ \frac{d(\mathrm{ev}(x)z, z) - n L }{\sigma \sqrt{n}} \in [a, b] \right) \to N_{a, b}$$
as $n \to \infty$.
\end{theorem}
\begin{proof}
Since $\log n/\sqrt{n} \to 0$, Theorem \ref{T:disp} (together with Lemma \ref{L:sum}) implies
$$\nu_{n-\log n} \left( x \ : \ \frac{d(\mathrm{ev}(x)z, z) - n L }{\sigma \sqrt{n}} \in [a, b] \right) \to N_{a, b}.$$
Now, since $\nu_{n - \log n} = \nu_{n, \log n}$ and by Proposition \ref{P:RN-der} and Lemma \ref{L:TV} we get
$$\Vert \nu_{n, \log n} - \lambda_{n, \log n} \Vert_{TV} \to 0$$
hence
$$\lambda_{n, \log n} \left( x \ : \ \frac{d(\mathrm{ev}(x)z, z) - n L }{\sigma \sqrt{n}} \in [a, b] \right) \to N_{a, b}.$$
Moreover, by the definition of $\widehat{x}$,
$$\lambda_n \left( x \ : \ \frac{d(\mathrm{ev}(\widehat{x})z, z) - n L }{\sigma \sqrt{n}} \in [a, b] \right) = \lambda_{n, \log n} \left( x \ : \ \frac{d(\mathrm{ev}(x)z, z) - n L }{\sigma \sqrt{n}} \in [a, b] \right). $$
Finally, note the since the orbit map $G \to \mathbb{H}^2$ is Lipschitz, we have
$$|d(\mathrm{ev}(x)z, z) - d(\mathrm{ev}(\widehat{x}) z, z)| \leq C \log n$$
where $C$ is the Lipschitz constant. Then using Lemma \ref{L:sum}
\begin{align*}
\lim_{n \to \infty} \lambda_n \left( x \ : \ \frac{d(\mathrm{ev}(x) z, z) - n L }{\sigma \sqrt{n}} \in [a, b] \right)
&= \lim_{n \to \infty} \lambda_n \left( x \ : \ \frac{d(\mathrm{ev}(\widehat{x}) z, z) - n L }{\sigma \sqrt{n}} \in [a, b] \right) \\
&= N_{a, b}
\end{align*}
which completes the proof.
\end{proof}
\section{The Gromov product}
The remaining step of our proof is to turn the statement about displacement (Theorem \ref{T:disp2}) into a statement about translation length. This is done by controlling the Gromov product.
We begin with the following easy computation. Recall that $\Omega^n$ is the set of all paths of length $n$ and $\mathcal{C}_n \subset \Omega^n$ is the subset of closed paths.
\begin{lemma} \label{L:Rn}
There is a constant $D \ge0$ such that
\[
1 \le \frac {\# (\Omega^n)} {\# (\mathcal{C}_n)} \le D
\]
\end{lemma}
\begin{proof}
We know that $\# (\mathcal{C}_n) = \textup{Tr }M^n = \lambda^n + \sum_i {\lambda_i^n}$, where $\lambda$ is the Perron--Frobenius eigenvalue of $M$ and $|\lambda_i|< \lambda$.
Also, $\# (\Omega^n) = \Vert M^n \Vert_1 \le D \lambda^n$ for some $D \ge 1$.
\end{proof}
Next, we will see that the Gromov product of a random element and its inverse grows slowly in word length. This is our key estimate for relating displacement to translation length.
\begin{proposition} \label{P:gr}
For any $\epsilon >0$,
$$\lambda_n \big ( x \ : \ (\mathrm{ev}(x)z, \mathrm{ev}(x)^{-1}z)_z \leq \epsilon \sqrt{n} \big )\to 1,$$
as $n \to \infty$.
\end{proposition}
\begin{proof}
Fix $\epsilon > 0$. Define
$$A = \big \{ x \in \Omega^\star \ : \ (\mathrm{ev}(x)z, \mathrm{ev}(x)^{-1}z)_z \geq \epsilon \sqrt{|x|} \big \},$$
where $|x|$ denotes the length of $x$. By \cite[Section 6]{GTT} for any vertex $v$ of $\Gamma$ one has
$$\mathbb{P}_v\left( (\mathrm{ev}(x_n) z, \mathrm{ev}(x_n)^{-1}z)_z \geq \epsilon \sqrt{n} \right )\to 0,$$
where $\mathbb{P}_v$ is the distribution of the Markov chain starting at vertex $v$, and $x_n$ is the $n^{th}$ step. (Lemma 6.2 of \cite{GTT} explicitly gives this statement where $v$ is the ``initial vertex'' of the directed graph, however the same argument gives the more general result for all vertices.)
Next, consider any path $p \in \Omega_v^n$ of length $n$ starting at $v$. If $v$ is the $l^{th}$ vertex of $\Gamma$ and the terminal endpoint of $p$ is the $k^{th}$ vertex, then the $n^{th}$ step measure of $p$ is
$
\mathbb{P}^n_v(p) = \frac{1}{\lambda^n} \frac{v_k}{v_l}.
$
(Here, $\mathbb{P}^n_v$ is the measure supported on $\Omega_v^n$ which is the pushforward of $\mathbb{P}_v$ under the prefix map $\Omega_v \to \Omega_v^n$.)
Hence, as in \cite[Lemma 3.4]{GTT}), there is a constant $c>1$, depending only on the adjacency matrix $M$, such that for any subset $B \subset \bigcup_{n\ge1}\Omega_v^n$
\[
\frac{1}{c} \cdot \frac{\# (B \cap \Omega_v^n)}{\# \Omega_v^n} \le \mathbb{P}^n_v(B)\le c \cdot \frac{\# (B \cap \Omega_v^n)}{\# \Omega_v^n}.
\]
In particular, we conclude that
\begin{align} \label{e:growth}
\frac{\# (A \cap \Omega_v^n)}{\# \Omega_v^n} \leq c \cdot \mathbb{P}_v \left( (\mathrm{ev}(x_n) z, \mathrm{ev}(x_n)^{-1}z)_z \geq \epsilon \sqrt{n} \right ) \to 0,
\end{align}
as $n\to \infty$.
Hence, by summing over all vertices $v \in V$
$$\frac{\# (A\cap \Omega^n)}{\# \Omega^n} \to 0.$$
Then by combining this with Lemma \ref{L:Rn},
$$\lambda_n(A) = \frac{\# (A \cap \mathcal{C}_n) }{\# \mathcal{C}_n} \leq \frac{\# (A \cap \Omega^n)}{\# \Omega^n} \cdot \frac{\# \Omega^n}{\# \mathcal{C}_n} \to 0.$$
\end{proof}
\begin{remark}
The main estimate (\ref{e:growth}) in the proof of Proposition \ref{P:gr} can also be obtained via a trick using more recent work of the authors \cite{GTT2}. Using terminology there, one may define a geodesic graph structure $(G,\Gamma)$ by declaring that $v$ be the initial vertex of $\Gamma$. Such a structure may not be surjective, but this is not necessary. Then \cite[Proposition 5.8]{GTT2} precisely gives the required decay result.
\end{remark}
\subsection*{Proof of Theorem \ref{T:main}}
We can now complete the proof of Theorem \ref{T:main}.
\begin{proof}[Proof of Theorem \ref{T:main}]
Let $d_n := \frac{d(z, \mathrm{ev}(x) z) - n L }{\sigma \sqrt{n}}$, $t_n := \frac{\tau( \mathrm{ev}(x) ) - n L }{\sigma \sqrt{n}}$, and $p_n := t_n - d_n = \frac{2 (\mathrm{ev}(x) z, \mathrm{ev}(x)^{-1} z)_z + O(\delta)}{\sigma \sqrt{n}}$.
By the CLT for displacement (Theorem \ref{T:disp}), for any $a < b$
$$\lambda_n(x : d_n \in [a, b]) \to N_{a, b}.$$
Moreover, by decay of Gromov products for any $\epsilon > 0$ (Proposition \ref{P:gr}) we have
$$\lambda_n(x: |p_n| \geq \epsilon) \to 0$$
hence by Lemma \ref{L:sum}
$$\lambda_n \left( x \ : \ \frac{\tau(\mathrm{ev}(x)) - n L }{\sigma \sqrt{n}} \in [a, b] \right) \to N_{a, b}.$$
Finally, by Lemma \ref{L:push} this implies
$$\mu_n \left( \gamma \ : \ \frac{\tau(\gamma) - n L }{\sigma \sqrt{n}} \in [a, b] \right) \to N_{a, b}$$
which completes the proof.
\end{proof}
\section{Generalizations}
While many generalization of Theorem \ref{T:main} are possible, we record the most immediate one here. The proof is the same as the one given above.
\begin{theorem}
Let $G\curvearrowright X$ be any convex cocompact action of a closed orientable surface group on a $\mathrm{CAT}(-1)$ space.
Then the conclusion of Theorem \ref{T:main} holds. The same is true for any convex cocompact free group action on a $\mathrm{CAT}(-1)$ space $X$ as long as the lengths of closed geodesics on $X/G$ are not all contained in $c\Z$ for some $c>0$.
\end{theorem}
The condition on lengths of closed geodesics is known to hold whenever $G\curvearrowright X$ is a discrete action on a $\mathrm{CAT}(-1)$ satisfying at least one of the following:
\begin{itemize}
\item the limit set $\Lambda(G)\subset \partial X$ has an infinite connected component (in particular when $G$ is a surface group) \cite{Bourdon};
\item $X$ is itself a surface (so that $X/G$ is a locally $\mathrm{CAT}(-1)$ surface) \cite{Dal'bo};
\item $X$ is a rank 1 symmetric space \cite{Kim}.
\end{itemize}
|
1,941,325,220,515 | arxiv | \section{Introduction}
In \cite{Ca1}, the author considered the orbit configuration space \(\Conf_n^G(M)\) associated to a free action of a finite group \(G\) on a manifold \(M\),
\[ \Conf_n^G(M) := \{(m_i) \in M^n \mid m_i \ne g m_j \,\forall g \in G\}. \]
The cohomology \(H^i(\Conf_n^G(M); \mathbb{Q})\) is a representation of \(W_n := G^n \rtimes S_n\). We showed that, if we consider all \(n\) at once, then \(H^i(\Conf^G(M); \mathbb{Q})\) forms a module over the category \(\FI_G\) introduced by \cite{SS2}. Furthermore, we proved \cite[Thm 3.1]{Ca1} that this \(\FI_G\)-module is finitely generated, when mild conditions are \(M\) are satisfied (e.g., \(\dim H^*(M; \mathbb{Q}) < \infty\)).
This finite generation implies the following. Given a partition-valued function \(\underline \lambda\) on the irreducible representations of \(G\) with \(\|\underline \lambda\| = n\), let \(L(\underline \lambda)\) be the associated irreducible representation of \(W_n\). Let \(c(G)\) be the set of conjugacy classes of \(G\). Define an \emph{\(\FI_G\) character polynomial} to be a polynomial in \(c(G)\)-labeled cycle-counting functions. This is a class function on \(W_n\). (We will define this and other terminology more precisely in \S2.2).
\begin{thm}[\cite{Ca1}]
Let \(M\) be a connected manifold with a free action of a finite group \(G\). Assume that \(\dim M \ge 2\) and that each \(\dim H^i(M; \mathbb{Q}) < \infty\). Then:\begin{enumerate}
\item The characters of \(H^i(\Conf_n^G(M); \mathbb{C})\) are given by a single character polynomial for all \(n \gg 0\).
\item The multiplicity of each irreducible \(W_n\)-representation in \(H^i(\Conf_n^G(M); \mathbb{C})\) is eventually independent of \(n\), and \(\dim H^i(\Conf_n^G(M); \mathbb{C})\) is given by a single polynomial for all \(n \gg 0\).
\end{enumerate}
\end{thm}
In this paper we consider the case where we replace the manifold \(M\) with a scheme \(X\) over \(\mathbb{Z}[1/N]\), or more generally \(\mathcal{O}_k[1/N]\) for a number field \(k\), with an algebraic action of \(G\). Here the \(\mathbb{C}\)-points \(X(\mathbb{C})\) take the place of \(M\). However, now we can also consider the points \(X(\mathbb{F}_q)\), for \(\mathbb{F}_q\) a residue field of \(\mathcal{O}_k\).
We generalize the results of Farb-Wolfson \cite{FW2} on \(\Conf_n(X)\) to the orbit configuration space \(\Conf^G_n(X)\). The first result is on \emph{\'etale representation stability} in the sense of \cite{FW2}.
\begin{thm}[\bfseries \'Etale representation stability for orbit configuration spaces]
Let \(X\) be a smooth scheme over \(\mathcal{O}_k[1/N]\) with geometrically connected fibers. Let \(G\) be a finite group acting freely on \(X\), such that \(X\) is smoothly compactifiable as a \(G\)-scheme. Let \(K\) be either a number field or an unramified function field over \(\mathcal{O}_k[1/N]\). For each \(i \ge 0\), the \(\Gal(\overline{K}/K)\)-\(\FI_G\)-module \(H^i_{\acute et}(\Conf^G(X)_{/\overline{K}}; \mathbb{Q}_l)\) is finitely generated.
\end{thm}
The next result concerns bounds on \(H^i(\Conf^G_n(X))\) as \(i\) varies, which are necessary to ensure convergence of the point-counts we are interested in.
\begin{thm}[\bfseries Orbit configuration spaces have convegent cohomology]
\thlabel{conf_convg}
Let \(X\) be a connected orientable manifold of dimension at least 2 with \(\dim H^*(M; \mathbb{Q})\) finite-dimensional. Let \(G\) be a finite group acting freely on \(M\). Then for each character polynomial \(P\), the inner product \(|\langle P, H^i(\Conf^G_n(X))\rangle|\) is bounded subexponentially in \(i\) and uniformly in \(n\).
\end{thm}
Finally, we use the Grothendieck-Lefschetz trace formula, along with Theoresm 1.2 and 1.3, to obtain the following result on arithmetic statistics.
\begin{thm}[\bfseries Arithmetic statistics for orbit configuration spaces stabilize]
\thlabel{arith_stats}
Let \(X\) be a smooth quasiprojective scheme over \(\mathcal{O}_k[1/N]\) with geometrically connected fibers. Let \(G\) be a finite group acting freely on \(X\), such that \(X\) is a smoothly compactifiable \(G\)-scheme. Then for any \(\FI_G\) character polynomial \(P\),
\begin{equation} \label{groth}
\lim_{n \to \infty} q^{-n \dim X} \sum_{y \in \UConf_n(X/G)(\mathbb{F}_q)} P(\sigma_y) = \sum_{i = 0}^\infty (-1)^i \tr\left(\Frob_q : \langle H^i(\Conf^G(X(\mathbb{C})); L), P\rangle\right)
\end{equation}
In particular both the limit on the left and the series on the right converge, and they converge to the same limit.
\end{thm}
\subsection*{Gauss Sums}
For the specific case \(G = \mathbb{Z}/d\mathbb{Z}\), the resulting automorphism groups \(W_n = \left(\mathbb{Z}/d\mathbb{Z}\right)^n \wr S_n\) are the so-called \emph{main series of complex reflection groups}. These directly generalize the Weyl groups of type \(BC_n\) of Wilson's paper, for which \(d = 2\). In particular, we can apply \thref{arith_stats} to the action of \(\mathbb{Z}/d\mathbb{Z}\) on \(\mathbb{C}^*\) by rotation. In this case, the sum on the left-hand side of (\ref{groth}) is over \(\UConf_n(\mathbb{G}_m)(\mathbb{F}_q) = \Poly_n(\mathbb{F}_q^*)\), the space of square-free polynomials over \(\mathbb{F}_q\) that do not have 0 as a root. We thus obtain the following result, generalizing the work of Church-Ellenberg-Farb \cite{CEF2} (see \S6 for definitions).
\begin{thm}[\bfseries Gauss sums for \(\Conf^G(\mathbb{C}^*)\) stabilize]
\thlabel{gauss}
For any prime power \(q\), any \(d \mid q-1\), and any character polynomial \(P\) for \(\FI_{\mathbb{Z}/d\mathbb{Z}}\),
\begin{equation} \label{gauss2}
\lim_{n \to \infty} q^{-n} \sum_{f \in \Poly_n(\mathbb{F}_q^*)} P(f) = \sum_{i = 0}^\infty (-1)^i \frac{\langle P^*, H^i(\Conf^{\mathbb{Z}/d\mathbb{Z}}(\mathbb{C}^*); \mathbb{C})\rangle}{q^i}
\end{equation}
In particular both the limit on the left and the series on the right converge, and they converge to the same limit.
\end{thm}
\thref{gauss} essentially says that the \emph{average value} of certain Gauss sums across all polynomials in \(\Poly_n(\mathbb{F}^*_q)\) always converges to the series in \(q^{-1}\) on the right. For example, let \(\chi\) be a character of \(\mathbb{Z}/(q-1)\mathbb{Z}\). Define the character polynomial \(X_i^\chi := \sum_{g \in G} \chi(g) X^g_i\). Then
\begin{gather} \begin{aligned} \label{X1_pcount}
\lim_{n \to \infty} q^{-n} &\sum_{f \in \Poly_n(\mathbb{F}_q^*)} \sum_{\substack{\alpha \ne \beta \in \mathbb{F}_q\\f(\alpha) = f(\beta) = 0}} \chi(\alpha)\chi(\beta)^{-1} \\
&= \sum_i (-1)^i \frac{\langle X_1^{\overline{\chi}} X_1^{\chi}, H^i(\Conf^{\mathbb{Z}/(q-1)\mathbb{Z}}(\mathbb{C}^*); \mathbb{Q}(\zeta_{q-1}))\rangle}{q^i} = -\frac{1}{q} + \frac{5}{q^2} + \cdots
\end{aligned} \end{gather}
That is, the average value of the Gauss sum obtained by applying \(\chi\) to each quotient of pairs of linear factors of \(f\), across all \(f \in \Poly_n(\mathbb{F}_q^*)\), is equal to the series on the right-hand side of (\ref{X1_pcount}) obtained by looking at the inner product of the character polynomial \(X_1^{\overline{\chi}} X_1^{\chi}\) with \(H^i(\Conf^{\mathbb{Z}/(q-1)\mathbb{Z}}(\mathbb{C}^*); \mathbb{Q}(\zeta_{q-1}))\). As another example, suppose \(q\) is odd and let \(\psi = \left(\frac{-}{q^2}\right)\) be the Legendre symbol in \(\mathbb{F}_q^2\), which is \(1\) or \(-1\) according to whether its argument is a square or nonsquare in \(\mathbb{F}_{q^2}\). Then
\begin{gather} \begin{aligned} \label{X2_pcount}
\lim_{n \to \infty} q^{-n} &\sum_{f \in \Poly_n(\mathbb{F}_q^*)} \sum_{\substack{p \mid f \\ \deg(p) = 2}} \psi(\text{root}(p)) \\
&= \sum_i (-1)^i \frac{\langle X_2^{\psi}, H^i(\Conf^{\mathbb{Z}/(q-1)\mathbb{Z}}(\mathbb{C}^*); \mathbb{Q}(\zeta_{q-1}))\rangle}{q^i} = -\frac{1}{q} + \frac{3}{q^2} + \cdots
\end{aligned} \end{gather}
where \(\text{root}(p)\) denotes a root of \(p\), an irreducible degree 2 factor of \(f\), lying in \(\mathbb{F}_{q^2}\); the value \(\psi(\text{root}(p))\) turns out not to depend on the choice of root. Thus, (\ref{X2_pcount}) says that the average value of the Gauss sum obtained by applying \(\psi\) to the \emph{quadratic} factors of \(f\), across all \(f \in \Poly_n(\mathbb{F}^*_q)\), is equal to the series on the right obtained by looking at the inner product of the character polynomial \(X_2^{\psi}\) with \(H^i(\Conf^{\mathbb{Z}/(q-1)\mathbb{Z}}(\mathbb{C}^*); \mathbb{Q}(\zeta_{q-1}))\).
\begin{rem}
Just as in \cite[\S4.3]{CEF2}, it is quite likely that one can compute the left-hand side of (\ref{gauss2}) using twisted L-functions of the form
\[ L(P,s) = \sum_n \sum_{f \in \Poly_n(\mathbb{F}_q^*)} P(f) q^{-ns} \]
or by other analytic methods. Zeev Rudnick sketched such an argument for the case of (\ref{X1_pcount}) in a private communication. However, \thref{gauss} gives a topological interpretation to the left-hand side of (\ref{gauss2}). More to the point, the fact that representation stability holds for orbit configuration spaces is what suggests \thref{gauss} in the first place. In fact, the bridge to topology provided by the Grothendieck-Lefschetz trace formula gives further motivation to study such L-functions. One can often go in the other direction, and prove representation stability by means of counting points over finite fields. Chen \cite{WCh2} has done this for the usual configuration spaces, and it would be interesting to investigate this for orbit configuration spaces as well.
\end{rem}
\subsection*{Related work}
After distributing this paper, we learned of new work of Rolland-Wilson \cite{RW2} investigating the specific case of the type B/C hyperplane arrangement. This corresponds to the case \(d = 2\) in \thref{gauss}. Their proof of the necessary convergence results are by completely different methods. Our proof of the general result \thref{conf_convg} is by the same argument as Farb-Wolfson \cite{FW2}, and we prove the stronger result \thref{convg} for branched covers of \(\mathbb{P}^1\) by a topological argument. Rolland-Wilson's result, \cite[Thm 3.8]{RW2}, is proven using a novel graph-theoretic argument. It would be very interesting to see if their method could be strengthened to prove the stronger polynomial bounds of our \thref{convg}.
\subsection*{Acknowledgements}
I would like to thank Nir Gadish, Joel Specter, Weiyan Chen, Jesse Wolfson, and Sean Howe for helpful conversations. I would like to thank Zeev Rudnick for pointing out mistakes in some calculations in the initial version of this paper, and for other helpful comments. Also, I am very grateful to my advisor, Benson Farb, for all his guidance throughout the process of working on, writing, and revising this paper.
\section{\'Etale homological stability and orbit configuration spaces}
In this section we extend the theory of \'etale representation stability, developed by Farb-Wolfon \cite{FW2} for the category \(\FI\), to the category \(\FI_G\). We then apply this theory to orbit configuration spaces.
\subsection{\'Etale representation stability}
If \(G\) is a group and if \(R\) and \(S\) are sets, define a \emph{\(G\)-map} \((a, (g_i)): R \to S\) to be a pair \(a: R \to S\) and \((g_i) \in G^R\). If \((b, (h_j)): S \to T\) is another \(G\)-map, their composition is \((b \circ a, (g_i \cdot h_{a(i)}))\). Let \(\FI_G\) be the category with objects finite sets and morphisms \(G\)-maps with the function \(a\) injective; in particular, the category \(\FI = \FI_{\text{triv}}\). An \emph{\(\FI_G\)-module} \(V\) is a functor from \(\FI_G\) to \(k\Mod\) for some ring \(k\). We have
\[\Aut_{\FI_G}([n]) = G^n \rtimes S_n =: W_n\]
so that if \(V\) is an \(\FI_G\)-module, then each \(V_n\) is a \(W_n\)-module. An \(\FI_G\)-module is finitely-generated if it has a finite set of elements not contained in any proper submodule.
In \cite{Ca1}, the author developed a theory of representation stability for \(\FI_G\)-modules, extending the theory developed for \(\FI\) by Church-Ellenberg-Farb \cite{CEF} and \(\FI_{\mathbb{Z}/2\mathbb{Z}}\) by Wilson \cite{Wi2}, and building on the work of \cite{SS2} who introduced \(\FI_G\). We thus have the following results about finitely-generated \(\FI_G\)-modules, many of which were also obtained by Gadish \cite{Ga2}.
\begin{thm}[\bfseries Structural properties of finitely-generated \(\FI_G\)-modules]
\thlabel{fig_props}
Let \(G\) be a finite group and let \(V\) be an \(\FI_G\)-module over a field \(k\) of characteristic zero. If \(V\) is finitely-generated then:
\begin{description}
\item [Representation stability (\cite{GL3}):] \(V\) satisfies representation stability in the sense of \cite{CF} for the sequence \(\{V_n\}\) of \(W_n\)-modules.
\item [Isomorphism of trivial isotypics (\cite{Ca1}):] For all sufficiently large \(n\), the map \(V_n \to V_{n+1}\) given by averaging the structure maps induces an isomorphism
\[ V_n^{W_n} \xrightarrow \simeq V_{n+1}^{W_{n+1}} \]
\item [Tensor products (\cite{Ca1}):] If \(W\) is another finitely-generated \(\FI_G\)-module, and \(G\) is finite, then \(V \otimes W\) is finitely-generated.
\item [Character polynomials (\cite{Ca1}):] If \(k\) is a splitting field for \(G\), then there is a character polynomial \(P_V\) so that
\[ \chi_{V_n} = P_V \text{ for } n \gg 0. \]
\end{description}
\end{thm}
Farb-Wolfson define a notion of \emph{\'etale represention stability} for a co-FI scheme \(Z\) over \(\mathbb{Z}[1/N]\). They show that if there is a uniform way of normally compactifying the \(Z_n\), then the base-change maps commute with the induced FI maps. This allows them to pass from knowing that \(H^i(Z(\mathbb{C}))\) is a finitely-generated FI-module to knowing that \(H^i_{\acute et}(Z_{/\overline{\mathbb{F}}_p}; \mathbb{Q}_l)\) is a finitely-generated \emph{\(\Gal(\overline{K}/K)\)-FI-module}. Since Fulton-Macpherson constructed a normal compactification of \(\Conf(X)\), this allows them to conclude that \(\Conf(X)\) satisfies \'etale representation stability.
We want to generalize this to the orbit configuration space \(\Conf^G(X)\). However, to do so, we need to allow for the possibility that \(Z\) is only defined over some finite Galois extension of \(\mathbb{Q}\). For example, the orbit configuration space
\[ \Conf^{\mathbb{Z}/d\mathbb{Z}}_n(\mathbb{G}_m) = \{(x_i) \in \mathbb{G}_m^n \mid x_i \ne \zeta^m x_j\}\]
where \(\zeta\) is a primitive \(d\)-th root of unity and the group \(G = \mathbb{Z}/d\mathbb{Z}\) acts on \(\mathbb{G}_m\) by multiplication by \(\zeta\), is naturally defined over \(\mathbb{Z}[\zeta]\). Thus, rather than reducing modulo primes \(p\), we now have to reduce modulo prime ideals \(\mathfrak{p}\) of \(\mathcal{O}_k[1/N]\). We then have \(\mathcal{O}_k/\mathfrak{p} = \mathbb{F}_{q}\), where \(q = p^f\) with \(f\) the inertia degree of \(\mathfrak{p}\) over \(p\).
We therefore make the following definition, following \cite{FW2}.
\begin{defn}[\bfseries Smoothly compactifiable \(I\)-scheme over \(k\)]
Let \(I\) be a category and \(k\) a Galois number field. A \emph{smooth \(I\)-scheme over \(k\)} is a functor \(Z: I \to \mathbf{Schemes}\) consistings of smooth schemes over \(\mathcal{O}_k[1/N]\) for some \(N\) independent of \(i\). A smooth \(I\)-scheme is \emph{smoothly compactifiable at \(\mathfrak{p} \nmid N\)} if there is a smooth projective \(I\)-scheme \(\overline{Z}\) and a natural transformation \(Z \to \overline{Z}\) so that for all \(i \in I\), \(Z_i \to \overline{Z}_i\) is an open embedding and \(\overline{Z}_i - Z_i\) is a normal crossings divisor with good reduction at \(\mathfrak{p}\).
\end{defn}
We then have the following, generalizing \cite[Thm 2.6]{FW2}:
\begin{thm}[\bfseries Base change for \(I\)-schemes over \(k\)] \thlabel{basechange}
Let \(l\) be a prime, and \(Z: I \to \mathbf{Schemes}\) be a smooth \(I\)-scheme over \(k\) which is smoothly compactifiable at \(\mathfrak{p} \nmid N \cdot l\). Then for all morphisms \(i \to j\) in \(I\), the following diagram of ring homomorphisms commutes:
\[ \begin{diagram}
H^*_{\acute et}(Z_{j/\overline{\mathbb{F}}_q}; \mathbb{Z}_l) & \rTo^{\sim} & H^*_{sing}(Z_j(\mathbb{C}); \mathbb{Z}_l) \\
\dTo && \dTo \\
H^*_{\acute et}(Z_{i/\overline{\mathbb{F}}_q}; \mathbb{Z}_l) & \rTo^{\sim} & H^*_{sing}(Z_i(\mathbb{C}); \mathbb{Z}_l)
\end{diagram}\]
\end{thm}
\begin{proof}
The proof of \cite[Thm 2.6]{FW2} applies verbatim, since as they say, theirs is essentially an \(I\)-scheme version of \cite[Thm 7.7]{EVW}, which is more than general enough to include our extension to number fields. Just replace \(p\) with \(\mathfrak{p}\) and \(\mathbb{F}_p\) with \(\mathbb{F}_q\) everywhere in the proof of \cite[Thm 2.6]{FW2}.
\end{proof}
Now, let \(Z\) be a co-\(\FI_G\) scheme over \(k\) and \(l\) a prime. For each \(i \ge 0\) the \'etale cohomology \(H^i_{\acute et}(Z_{/\overline{k}}, \mathbb{Q}_l)\) is an \(\FI_G\)-module equipped with an action of \(\Gal(\overline{k}/k)\) commuting with the \(\FI_G\) action. Following \cite{FW2}, we call such an object a \emph{\(\Gal(\overline{k}/k)\)-\(FI_G\)-module}. Likewise, if \(\mathfrak{p} \nmid N \cdot l\) is a prime ideal of \(k\), and we put \(\mathbb{F}_q = \mathcal{O}_k/\mathfrak{p}\), then \(H^i_{\acute et}(Z_{\overline{\mathbb{F}}_q}, \mathbb{Q}_l)\) is a \(\Gal(\overline{\mathbb{F}}_q/\mathbb{F}_q)\)-FI-module.
For \(K\) a number field or finite field of characteristic \(p\), a \(\Gal(\overline{K}/K)\)-\(\FI_G\)-module is finitely generated if it has a finite set of element not contained in any proper \(\Gal(\overline{K}/K)\)-FI-module. We can therefore use \thref{basechange} to pass from knowing finite generation of \(H^i(Z(\mathbb{C}))\) as an \(\FI_G\)-module to finite generation of \(H^i(Z_{/K}; \mathbb{Q}_l)\) as a \(\Gal(\overline{K}/K)\)-\(\FI_G\)-module.
\subsection{Stability of orbit configuration spaces}
Let \(X\) be a scheme over \(\mathcal{O}_k[1/N]\) with a free algebraic action of a finite group \(G\). The \emph{orbit configuration space} \(\Conf^G_n(X)\) is defined as the functor of points:
\[ \Conf^G_n(X)(R) = \{(x_i) \in X(R)^n \mid x_i \ne g x_j \; \forall g \in G\}\]
for any \(\mathcal{O}_k[1/N]\)-algebra \(R\). Thus \(\Conf^G_n(X)\) has the structure of a scheme over \(\mathbb{Z}[1/N]\). If \(X\) is smooth, then \(\Conf^G_n(X)\) is smooth.
The group \(W_n := G^n \rtimes S_n\) acts freely on \(\Conf^G_n(X)\), where the copies of \(G\) act on each coordinate and \(S_n\) permutes the coordinates. In fact, \(\Conf^G(X)\) forms a co-\(\FI_G\)-scheme: for an injection \(a: [m] \hookrightarrow [n]\) and \(\vec g \in G^m\), the action on \((x_i) \in \Conf^G_n(X)\) is given by
\[ (\vec g, a) (x_i) = ( g_i \cdot x_{a(i)}) \]
It thus follows that \(H^i(\Conf^G(X))\) is an \(\FI_G\)-module. In \cite[Thm 3.1]{Ca1}, we proved that \(H^i(\Conf^G(X)\) is finitely generated. We would like to use \thref{basechange} to deduce \'etale representation stability for \(\Conf^G(X)\). To do so, we need to know that \(\Conf^G(X)\) is smoothly compactifiable at all but finitely many primes. Luckily, there are general constructions of smooth compactifications for the complement of an arrangement of subvarieties. The basic construction is due to Macpherson-Procesi \cite{MP} in the complex-analytic setting, generalizing Fulton-Macpherson's \cite{FM} compactification of \(\Conf_n(X)\). This work was extended to the scheme-theoretic setting by Hu \cite{Hu} and Li \cite{Li}. Following Li, define the \emph{wonderful compactification} \(\overline{\Conf^G_n}(X)\) to be the closure of the image of the embedding
\begin{equation} \Conf^G_n(X) \hookrightarrow \prod_{\substack{i \ne j \\ g \in G}} \Bl_{\Delta_{i,g,j}}(X^n)
\end{equation}
of the product of the blowups of \(X^n\) along the diagonals \(\Delta_{i,g,j} := \{(x_i) \mid x_i = g x_j\}\).
To carry this out, we first need to know that \(X\) itself can be smoothly compactified in a manner consistent with the action of \(G\). For a smooth scheme \(X\) equipped with an action of a finite group \(G\), say that \(X\) is a \emph{smoothly compactifiable \(G\)-scheme} if there is a scheme \(\overline{X}\) with an action of \(G\) and a \(G\)-equivariant embedding \(X \hookrightarrow \overline{X}\) such that \(\overline{X} - X\) is a normal crossings divisor. Note that this is always possible in characteristic 0 by resolution of singularities, so this assumption is only needed to ensure such a compactification exists in finite characteristic.
We then have the following.
\begin{prop}[\bfseries \(\Conf^G(X)\) is smoothly compactifiable] \thlabel{cpctify}
Let \(X\) be a smooth scheme over \(\mathcal{O}_k[1/N]\) with a free action of a finite group \(G\) so that \(X\) is a smoothly compactifiable \(G\)-scheme. Then \(\overline{\Conf^G_n}(X)\) is a smoothly compactifiable co-\(\FI_G\)-scheme at any prime ideal \(\mathfrak{p} \nmid N\) of \(\mathcal{O}_k\).
\end{prop}
\begin{proof}
First, note that \(X^n\) is a smoothly compactifiable \(G^n\)-scheme, since \(X\) is smoothly compactifiable \(G\)-scheme. We want to use Li's \cite{Li} work on wonderful compactifications, but to do so we need to check that our arrangement satisfies his hypotheses, that the arrangement is a \emph{building set} in the sense of \cite[\S2]{Li}. This follows from the fact that \(G\) acts freely on \(X\), so that diagonals \(\Delta_{i,g,j}\) and \(\Delta_{i,h,j}\) are disjoint for \(g \ne h\).
By definition (3) of \(\overline{\Conf^G_n}(X)\) as the closure in the blowup, we see that \(W_n\) acts on \(\overline{\Conf^G_n}(X)\), and that \(\overline{\Conf^G}(X)\) forms a co-\(\FI_G\)-space. Furthermore, since \(X\) is a scheme over \(R := \mathcal{O}_k[1/N]\), then (3) defines \(\overline{\Conf^G}(X)\) as a scheme over \(R\). There is a natural open embedding \(\Conf^G(X) \hookrightarrow \overline{\Conf^G}(X)\), again just given by (3). To see that the complement \(\overline{\Conf^G}(X) - \Conf^G(X)\) is a normal crossings divisor, it is enough to check this at the geometric fibers over the points of \(R\). This holds by \cite[Thm 1.2]{Li}. Likewise, it is enough to check smoothness of \(\overline{\Conf^G}(X)\) at the geometric fibers over the points of \(R\). This holds by \cite[Thm 1.2]{Li}. Since \(\overline{\Conf^G}(X)\) is smooth and \(\overline{\Conf^G}(X) - \Conf^G(X)\) is a normal crossings divisor relative to \(\Spec R\), in the sense of \cite[Prop 7.7]{EVW}, we conclude that \(\overline{\Conf^G}(X) - \Conf^G(X)\) has good reduction at each prime \(\mathfrak{p}\) of \(R\).
\end{proof}
\begin{thm}[\bfseries \'Etale representation stability for orbit configuration spaces]
\thlabel{etale_repstab}
Let \(X\) be a smooth scheme over \(\mathcal{O}_k[1/N]\) with geometrically connected fibers. Let \(G\) be a finite group acting freely on \(X\), such that \(X\) is smoothly compactifiable as a \(G\)-scheme. Let \(K\) be either a number field or an unramified function field over \(\mathcal{O}_k[1/N]\). For each \(i \ge 0\), the \(\Gal(\overline{K}/K)\)-\(\FI_G\)-module \(H^i_{\acute et}(\Conf^G(X)_{/\overline{K}}; \mathbb{Q}_l)\) is finitely generated.
\end{thm}
\begin{proof}
Again, we follow \cite[Thm 2.8]{FW2}. Since \(X\) has geometrically connected fibers, \(X(\mathbb{C})\) is connected. Since \(X\) is smoothly compactifiable, \(H^*(X; \mathbb{Q})\) is finitely generated. Since \(X(\mathbb{C})\) is a complex manifold, it is orientable and has real dimension at least 2. Thus \(X\) satisfies the hypotheses of \cite[Thm 3.1]{Ca1}, and so \(H^i(X(\mathbb{C}); \mathbb{Q})\) is a finitely generated \(\FI_G\)-module.
If \(K\) is a number field, we conclude immediately from Artin's comparison theorem that \(H^i_{\acute et}(\Conf^G(X)_{/K}; \mathbb{Q}_l)\) is a finitely generated \(\Gal(\overline{K}/K)\)-\(\FI_G\)-module. If \(K\) is a finite field, by \thref{cpctify} \(\Conf^G(X)\) is a smoothly compactifiable co-\(\FI_G\)-scheme, and so by \thref{basechange}, the conclusion likewise follows.
\end{proof}
\section{Convergence}
Recall that the conjugacy classes of \(W_n = G^n \rtimes S_n\) are given by a cycle decomposition of size \(n\) (i.e., the usual conjugacy classes of \(S_n\)), but where each cycle is decorated by a conjugacy class of \(G\), where we write \(c(G)\) for the set of conjugacy classes of \(G\). A \emph{character polynomial for \(\FI_G\)} is a polynomial
\[ P \in k[\{X_i^C \mid C \in c(G)\}] \]
This determines a class function \(P_n : W_n \to k\) for all \(n\), where \(X_i^C\) counts the number of \(C\)-decorated \(i\)-cycles of a given conjugacy class of \(W_n\). Recall from \thref{fig_props} that if \(V\) is a finitely-generated \(\FI_G\)-module, then the characters \(\chi_{V_n}\) are eventually given by a single character polynomial for all large \(n\). Furthermore, for another character polynomial \(P\), we know that the inner product \(\langle P_n, V_n\rangle_{W_n}\) is eventually independent of \(n\). We put
\[ \langle P, V\rangle = \lim_{n \to \infty} \left\langle P, V_n \right\rangle_{W_n} \]
for the limiting multiplicity.
Thus if \(Z\) is a co-\(\FI_G\)-scheme that satisfies \'etale representation stability, there is a stable inner product \(\langle P, H^i_{\acute et}(Z; \mathbb{Q}_l)\rangle\), eventually independent of \(n\). However, for our applications we need to bound how these inner products grow in \(i\). The following proposition is helpful in doing so:
\begin{prop}\thlabel{convg} For any graded \(\FI_G\)-module \(V^*\), the following are equivalent: \begin{enumerate}
\item For each character polynomial \(P\), \(|\langle P_n, V^i_n \rangle|\) is bounded subexponentially in \(i\) and uniformly bounded in \(n\).
\item For every \(a\), the dimension \(\dim (V^i_n)^{W_{n-a}}\) is bounded subexponentially in \(i\) and uniformly bounded in \(n\).
\end{enumerate}
\end{prop}
\begin{proof}
\item This follows from \cite[Defn 3.12]{CEF2}. Their proof applies almost verbatim, we just need to replace \(S_n\) with \(W_n\) and ``cycles'' with ``\(c(G)\)-labeled cycles''.
\end{proof}
If a graded \(\FI_G\)-algebra \(V^*\) satisfies these two equivalent conditions, we say it is \emph{convergent}. \thref{conf_convg} thus states that, under appropriate conditions, the \(\FI_G\) algebra \(H^*(\Conf^G(X); \mathbb{Q})\) is convergent.
\begin{proof}[Proof of \thref{conf_convg}]
Farb-Wolfson \cite[Thm 3.4]{FW2} proved this for \(G = 1\) by using the Leray spectral sequence to reduce to convergence of \(H^*(X^n) \otimes H^*(\Conf(\mathbb{R}^d))\). We will use the same technique to reduce to their result.
Let \(A(G,d)_n\) be the graded commutative \(\mathbb{Q}\)-algebra generated by \(\{e_{a,g,b} \mid 1 \le a \ne b \le n, g \in G\}\), each of degree \(d-1\), modulo the following relations:
\begin{align*}
e_{a,g,b} &= (-1)^d e_{b,g^{-1},a} \\
e_{a,g,b}^2 &= 0 \\
e_{a,g,b} \wedge e_{b,h,c} &= (e_{a,g,b} - e_{b,h,c}) \wedge e_{a,gh,c}
\end{align*}
where the action of \((\vec h, \sigma) \in W_n\) is given by
\[ (\vec h, \sigma) \cdot e_{a,g,b} = e_{\sigma(a), h_a g h_b^{-1}, \sigma(b)} \]
If we let \(A(d)_n = A(1,d)_n\), then there is an embedding \(A(d)_n \hookrightarrow A(G,d)_n\) given by \(e_{a,b} \mapsto e_{a,e,b}\). In \cite[Thm 3.6]{Ca1}, we proved that \(A(G,d)_n\) is spanned by the \(G^n\)-translates of \(A(d)_n\).
In \cite[Thm 3.1]{Ca1}, we proved that \(H^*(\Conf^G_n(X); \mathbb{Q})\) is isomorphic as a graded \(W_n\)-representation, to a subquotient of
\[ H^*(X^n; \mathbb{Q}) \otimes A(G,d)_n. \]
Thus, it suffices to bound
\[ \left(H^*(X^n; \mathbb{Q}) \otimes A(G,d)_n\right)^{W_{n-a}} = \left(\left(H^*(X^n; \mathbb{Q}) \otimes A(G,d)_n\right)^{G^{n-a}}\right)^{S_{n-a}} \]
We know that \(\left(H^*(X^n; \mathbb{Q}) \otimes A(G,d)_n\right)^{G^{n-a}}\) is a subquotient of \(H^*(X^n; \mathbb{Q}) \otimes \left(A(G,d)_n\right)^{G^{n-a}}\) as \(S_n\)-representations. Next, since \(A(G,d)_n\) is spanned by the \(G^n\) translates of \(A(d)_n\), we know that \(\left(A(G,d)_n\right)^{G^{n-a}}\) will be spanned by \(G^a\) translates of \(A(d)_n\). In conclusion,
\begin{equation} \label{convg_calc}
\dim \left(H^*(X^n; \mathbb{Q}) \otimes A(G,d)_n\right)^{W_{n-a}} \le |G|^a \dim \left(H^*(X^n; \mathbb{Q}) \otimes A(d)_n\right)^{S_{n-a}}
\end{equation}
But the right-hand side of (\ref{convg_calc}) is just (a constant times) exactly the term that Farb-Wolfson considered in \cite[Thm 3.4]{FW2} and proved was convergent. Therefore \(H^*(\Conf^G(X); \mathbb{Q})\) is convergent.
\end{proof}
\subsection{Polynomial bounds for branched covers of \(\mathbb{P}^1\)}
Consider the specific case where \(X\) is a surface with punctures and \(X/G\) is a sphere with punctures. For example, take any holomorphic branched cover of the Riemann sphere and remove the branch points. In this case, we can do better than proving that \(H^i(\Conf^G(X))^{W_{n-a}}\) is bounded by a subexponential function in \(i\): we can prove it is actually bounded by a polynomial in \(i\). Moreover, our proof is a straightforward topological one, and avoids relying on the detailed calculations of Farb-Wolfson. This also provides a partial answer to a question of Chen \cite{WCh2}, who proved that for \(X = \mathbb{C}\), \(\langle H^i(\Conf(X)), P\rangle\) is actually given by a quasipolynomial in \(i\), and asked for a topological proof of this fact. We believe that such quasipolynomiality probably exists for a more general class of manifolds. We hope all this provides some motivation for our reproving a special case of \thref{conf_convg} by other means.
So take \(X = \mathcal{S} - \{p_1, \dots, p_b\}\), with a free action of the finite group \(G\), and \(X/G = S^2 - \{q_0, \dots, q_c\} \approx \mathbb{C} - \{q_1, \dots, q_c\}\), where we assume there is at least one branch point, so \(c \ge 0\). To begin, notice that by transfer,
\[H^i(\Conf^G_n(X); \mathbb{Q})^{W_{n-a}} \cong H^i(\Conf^G_n(X)/W_{n-a}; \mathbb{Q}) \]
For any fixed \(a < n\), put \(Y_{n,a} = \Conf^G_n(X)/W_{n-a}\), so that
\[ Y_{n,a} = \left\{\{x_1, \dots, x_{n-a}\}, (z_1, \dots, z_a) \in \UConf_{n-a}(X/G) \times \Conf^G_a(X) \mid x_i \ne \pi(z_j)\right\}\]
There is thus a fiber bundle
\begin{equation} \label{convg_fibr}
\UConf_{n-a}(X/G - \{\pi(z_1), \dots \pi(z_a)\}) \hookrightarrow Y_{n,a} \twoheadrightarrow \Conf^G_a(X)
\end{equation}
and we have the following.
\begin{prop} \thlabel{monodromy}
In the fiber bundle (\ref{convg_fibr}), the monodromy action
\[\pi_1(\Conf^G_a(X)) \curvearrowright H^*(\UConf_{n-a}(X/G - \{\pi(z_1), \dots \pi(z_a)\}) \]
is trivial.
\end{prop}
\begin{proof}
There is an inclusion
\begin{align*}
\pi_1(\Conf^G_a(X), (z_1, \dots, z_a)) &\subset \pi_1(\UConf_a(X/G), \{\pi(z_1), \dots, \pi(z_a)\}) \\
&= \pi_1(\UConf_a(\mathbb{C} - \{q_1, \dots, q_c\}), \{\pi(z_1), \dots, \pi(z_a)\}) \\
&= \text{Mod}(\mathbb{C} - \{q_1, \dots, q_c\}, \{\pi(z_1), \dots, \pi(z_a)\})
\end{align*}
into the mapping class group with marked points, via the well-known isomorphism between braid groups and mapping class groups. That is, the group of homeomorphisms up to isotopy of \(\mathbb{C}\) that fix \(\{q_1, \dots, q_c\}\) pointwise and leave \(\{\pi(z_1), \dots, \pi(z_a)\}\) invariant, but possibly permute them. Thus \(\pi_1(\Conf^G_a(\mathbb{C}^*))\) acts (up to homotopy and conjugacy) by ``point-pushing'' diffeomorphisms on the underlying space \(X/G - \{\pi(z_1), \dots, \pi(z_a)\}\), and the action of \(\pi_1(\Conf^G_a(X))\) on the fiber of (\ref{convg_fibr}) is just the diagonal action.
Now, \(\UConf_{n-a}(\mathbb{C} - \{q_1, \dots, q_c, \pi(z_1), \dots \pi(z_a)\})\) has a normal \(S_{n-a}\)-cover
\[\Conf_{n-a}(\mathbb{C} - \{q_1, \dots, q_c, \pi(z_1), \dots \pi(z_a)\}) \to \UConf_{n-a}(\mathbb{C} - \{q_1, \dots, q_c, \pi(z_1), \dots \pi(z_a)\}) \]
which is a hyperplane arrangement. By transfer,
\begin{gather*}
H^*(\UConf_{n-a}(\mathbb{C} - \{q_1, \dots, q_c, \pi(z_1), \dots \pi(z_a)\}); \mathbb{Q}) \\
\cong \left(H^*(\Conf_{n-a}(\mathbb{C} - \{q_1, \dots, q_c, \pi(z_1), \dots \pi(z_a)\}); \mathbb{Q})\right)^{S_{n-a}}
\end{gather*}
and by Orlik-Solomon \cite{OS}, \(H^*(\Conf_{n-a}(\mathbb{C} - \{q_1, \dots, q_c, z_1^d, \dots z_a^d\}); \mathbb{Q})\) is generated as an algebra by \(H^1\). So it is enough to determine the action of \(\pi_1(\UConf_a(\mathbb{C} - \{q_1, \dots, q_c\}))\) on \(H^1(\Conf_{n-a}(\mathbb{C} - \{q_1, \dots, q_c, \pi(z_1), \dots \pi(z_a)\}); \mathbb{Q})\). But \(H^1\) is spanned by the de Rham cohomology classes
\begin{equation} \label{dR}
\left\{\frac{d(w_i - w_j)}{w_i - w_j} \,\middle|\, 1 \le i < j \le n-a\right\}, \left\{\frac{dw_i}{w_i - q_k} \,\middle|\, \substack{1 \le i \le n-a \\ 1 \le k \le c}\right\}, \left\{\frac{dw_i}{w_i - \pi(z_l)} \,\middle|\, \substack{1 \le i \le n-a \\ 1 \le l \le a}\right\}
\end{equation}
By tracking how points are pushed by a mapping class, we see that a braid in \(\pi_1(\UConf_a(\mathbb{C} - \{q_1, \dots, q_c\}))\) acts on the first two sets of cohomology classes in (\ref{dR}) trivially, and on the third set in (\ref{dR}) by permuting the \(\{\pi(z_1), \dots \pi(z_a)\}\) under the homomorphism \(\pi_1(\UConf_a(\mathbb{C} - \{q_1, \dots, q_c\})) \to S_a\). But \(\pi_1(\Conf_a(X/G))\) is precisely the kernel of this homomorphism, the subgroup of \emph{pure} braids. So the action of \(\pi_1(\Conf^G_a(X))\) on \(H^1(\Conf_{n-a}(\mathbb{C} - \{q_1, \dots, q_c, z_1^d, \dots z_a^d\}))\), and therefore as we have argued, on \(H^*(\UConf_{n-a}(\mathbb{C} - \{q_1, \dots, q_c, \pi(z_1), \dots \pi(z_a)\}); \mathbb{Q})\), is trivial.
\end{proof}
\begin{thm}
Let \(X\) be a surface with punctures, with a free action of a finite group \(G\), such that \(X/G\) is a sphere with punctures. Then for fixed \(a\), the function
\[i \mapsto \dim H^i(\Conf^G_n(\mathbb{C}^*); \mathbb{Q})^{W_{n-a}}\]
is bounded by a polynomial in \(i\) and uniformly bounded in \(n\).
\end{thm}
This is a big improvement on \cite{CEF2}, where they were only able to bound the dimension by \(O(e^{\sqrt{i}})\).
\begin{proof}
Because the action is trivial by \thref{monodromy}, in the Serre spectral sequence for the bundle (\ref{convg_fibr}), the coefficients are constant. Thus the \(E_2\) page of this spectral sequence is just
\begin{align*}
E_2^{p,q} &= H^p(\Conf^G_a(X); H^q(\UConf_{n-a}(X/G - \{\pi(z_1), \dots \pi(z_a)\}); \mathbb{Q})) \\
&= H^p(\Conf^G_a(X)) \otimes H^q(\UConf_{n-a}(X/G - \{\pi(z_1), \dots \pi(z_a)\}); \mathbb{Q}) \implies H^{p+q}(Y_{n,a}; \mathbb{Q})
\end{align*}
Abutment of spectral sequences means that \(H^{i}(Y_{n,a}; \mathbb{Q})\) is some subquotient of the direct sum of the diagonal terms
\[\bigoplus_{p + q = i} H^p(\Conf^G_a(X); \mathbb{Q}) \otimes H^q(\UConf_{n-a}(X/G - \{\pi(z_1), \dots \pi(z_a)\}); \mathbb{Q})\]
Since taking subquotients only decreasing the vector space dimension, we obtain
\begin{align*} \dim H^{i}(Y_{n,a}; \mathbb{Q}) &\le \sum_{p+q = i} \dim H^p(\Conf^G_a(X); \mathbb{Q}) \cdot \dim H^q(\UConf_{n-a}(X/G - \{\pi(z_1), \dots \pi(z_a)\}); \mathbb{Q}) \\
&= O\left(\dim H^i(\UConf_{n-a}(X/G - \{\pi(z_1), \dots \pi(z_a)\}); \mathbb{Q})\right)
\end{align*}
because \(H^p(\Conf^G_a(X); \mathbb{Q}) = 0\) for \(p > 2a\), since it is a \(2a\)-dimensional manifold. So it is enough to show that
\[ \dim H^i(\UConf_{n-a}(\mathbb{C} - \{q_1, \dots, q_c, \pi(z_1), \dots \pi(z_a)\}); \mathbb{Q}) \]
is bounded by a polynomial in \(i\) and is uniformly bounded in \(n\). We can simplify this description by redefining \(n := n - a\) and \(a := c + a\), and picking any \(a\) points, so we are left to analyze \(\dim H^i(\UConf_n(\mathbb{C} \setminus [a]))\), where \([a] = \{1, \dots, a\}\).
Notice that \(\mathbb{C} \setminus [a]\) is a connected orientable manifold of dimension 2 with finite-dimensional homology, so it satisfies the hypotheses of homological stability for unordered configuration spaces \cite[Thm 1]{Chu}. Therefore \(\dim H^i(\UConf_n(\mathbb{C} \setminus [a]); \mathbb{Q})\) is independent of \(n\) for \(n \gg 0\), and thus uniformly bounded in \(n\). We claim that in fact
\[\dim H^i(\UConf_n(\mathbb{C} \setminus [a]); \mathbb{Q}) = O(i^{a-1})\]
Indeed, notice that we have a decomposition:
\begin{equation} \label{decomp}
\UConf_n(\mathbb{C} \setminus [a-1]) = \UConf_n(\mathbb{C} \setminus [a]) \sqcup \UConf_{n-1}(\mathbb{C} \setminus [a])
\end{equation}
This decomposition just says that an unordered collection of \(n\) distinct points in \(\mathbb{C} \setminus [a-1]\) either does not contain the \(a\)-th point, or it does and so is determined by the \(n-1\) other points. \(\UConf_n(\mathbb{C} \setminus [a])\) is an open complex submanifold of \(\UConf_n(\mathbb{C} \setminus [a-1])\), and so there is a Thom-Gysin exact sequence associated to (\ref{decomp}):
\begin{gather} \label{les} \begin{aligned}
\dots &\to H^{i-2}(\UConf_{n-1}(\mathbb{C} \setminus [a]); \mathbb{Q}) \to H^i(\UConf_n(\mathbb{C} \setminus [a-1]); \mathbb{Q}) \to H^i(\UConf_n(\mathbb{C} \setminus [a]); \mathbb{Q}) \\
&\to H^{i-1}(\UConf_{n-1}(\mathbb{C} \setminus [a]); \mathbb{Q}) \to \dots
\end{aligned} \end{gather}
By exactness of (\ref{les}):
\begin{equation} \label{ex_bd}
\dim H^i(\UConf_n(\mathbb{C} \setminus [a]); \mathbb{Q}) \le \dim H^i(\UConf_n(\mathbb{C} \setminus [a-1]); \mathbb{Q}) + \dim H^{i-1}(\UConf_{n-1}(\mathbb{C} \setminus [a]); \mathbb{Q}) \end{equation}
By repeatedly plugging in the third term of (\ref{ex_bd}) back into the left-hand side of (\ref{ex_bd}), we obtain
\begin{gather} \label{ind_bd} \begin{aligned}
\dim H^i(\UConf_n(\mathbb{C} \setminus [a]); \mathbb{Q}) \le &\dim H^i(\UConf_n(\mathbb{C} \setminus [a-1]); \mathbb{Q}) + \dim H^{i-1}(\UConf_{n-1}(\mathbb{C} \setminus [a-1]); \mathbb{Q}) \\
&+ \dots + \dim H^1(\UConf_1 (\mathbb{C} \setminus [a-1]); \mathbb{Q}) \\
&= O\left(i \cdot \dim H^i(\UConf_n(\mathbb{C} \setminus [a-1]); \mathbb{Q})\right)
\end{aligned} \end{gather}
Now we induct on \(a\). Notice that for \(a = 0\), we know \(H^i(\UConf_n(\mathbb{C}); \mathbb{Q}) = 0\) once \(i \ge 2\). Therefore for \(a = 1\), this implies that \(\dim H^i(\UConf_n(\mathbb{C}^*); \mathbb{Q}) = O(1)\). By induction, we can assume that \(\dim H^i(\UConf_n(\mathbb{C} \setminus [a-1]); \mathbb{Q}) = O(i^{a-2})\). The bound (\ref{ind_bd}) then tells us that
\[ \dim H^i(\UConf_n(\mathbb{C} \setminus [a]); \mathbb{Q}) = O\left(i \cdot \dim H^i(\UConf_n(\mathbb{C} \setminus [a-1]); \mathbb{Q})\right) = O(i \cdot i^{a-2}) = O(i^{a-1}) \]
as desired.
\end{proof}
\section{Arithmetic statistics}
Let \(Z\) be a smooth quasiprojective scheme over \(\mathcal{O}_k[1/N]\). Suppose that \(W_n = G^n \rtimes S_n\) acts freely on \(Z\) by automorphisms, and let \(Y = Z/W_n\) be the quotient, which is known to be a scheme.
For any prime \(\mathfrak{p}\) of \(\mathcal{O}_k[1/N]\), we have \(\mathcal{O}_k/\mathfrak{p} \cong \mathbb{F}_q\), where \(q\) is a power of the rational prime \(p\) under \(\mathfrak{p}\). We can then base-change \(Y\) to \(\overline{\mathbb{F}}_q\). The geometric Frobenius \(\Frob_q\) then acts on \(Y_{/\overline{\mathbb{F}}_q}\). The fixed-point set of \(\Frob_q\) is exactly \(Y(\mathbb{F}_q)\).
Fix a prime \(l \ne p\), and let \(L\) be a splitting field for \(G\) over \(\mathbb{Q}_l\). Since all the irreducible representations of \(G\) are defined over \(L\), there is a natural correspondence between finite-dimensional representations of \(W_n\) over \(L\) and finite-dimensional constructible \(L\)-sheaves on \(Y\) that become trivial when pulled back to \(Z\).
Given a representation \(V\) of \(W_n\), let \(\chi_V\) be its character and \(\mathcal{V}\) the associated sheaf on \(Y\). For any point \(y \in Y(\mathbb{F}_q)\), since \(Frob_q\) fixes \(y\), then \(\Frob_q\) acts on the stalk \(\mathcal{V}_y\), which is isomorphic to \(V\). This action determines an element \(\sigma_y \in W_n\) up to conjugacy, so that \(\tr(\Frob_q : \mathcal{V}_y) = \chi_V(\sigma_y)\). The Grothendieck-Lefschetz trace formula says that
\[ \sum_{y \in Y(\mathbb{F}_q)} \tr(\Frob_q: \mathcal{V}_y) = \sum_i (-1)^i \tr\left(\Frob_q : H^i_{\acute et, c}(Y_{/\overline{\mathbb{F}}_q}; \mathcal{V})\right) \]
We then have the following chain of equalities, as in \cite{CEF2} and \cite{FW2}:
\begin{align*}
H^i_{\acute et, c}(Y_{/\overline{\mathbb{F}}_q}; \mathcal{V}) &\cong (H^i_{\acute et, c}(Z; \pi^* \mathcal{V}))^{W_n} & \text{by transfer}\\
&\cong \left(H^i_{\acute et, c}(Z; L) \otimes V\right)^{W_n} & \text{by triviality of pullback} \\
&\cong \left(H^{2\dim Z - i}_{\acute et}(Z; L(-\dim Z))^* \otimes V\right)^{W_n} &\text{by Poincar\'e duality} \\
&\cong \langle H^{2\dim Z - i}_{\acute et}(Z; L(-\dim Z)), V \rangle_{W_n}
\end{align*}
and so we obtain
\begin{equation}
\label{single_grot}
\sum_{y \in Y(\mathbb{F}_q)} \chi_V(\sigma_y) = q^{\dim Z} \sum_i (-1)^i \tr\left( \Frob_q : \langle H^i_{\acute et}(Z; L), V \rangle_{W_n}\right)
\end{equation}
We would like to apply (\ref{single_grot}) to a sequence of schemes \(Z_n\) that form a co-\(\FI_G\)-scheme, and then let \(n \to \infty\). To make this work, we need to know that \(Z\) satisfies \'etale representation stability, and that \(H^*(Z)\) is convergent in the sense of \S3. Following \cite{FW2}, we have the following.
\begin{thm}
\thlabel{fig_stats}
Suppose that \(Z\) is a smooth quasiprojective co-\(\FI_G\)-scheme over \(\mathcal{O}_k[1/N]\) such that \(H^i(Z_{/\overline{\mathbb{F}}_q}; \mathbb{Q}_l)\) is a finitely-generated \(\Gal(\overline{\mathbb{F}}_q/\mathbb{F}_q)\)-\(\FI_G\)-module, and that \(H^*(Z; \mathbb{Q}_l)\) is convergent. Then for any \(\FI_G\) character polynomial \(P\),
\begin{equation}
\lim_{n \to \infty} q^{-\dim Z_n} \sum_{y \in Y_n(\mathbb{F}_q)} P(\sigma_y) = \sum_{i=0}^\infty (-1)^i \tr\left( \Frob_q : \langle H^i_{\acute et}(Z; L), P \rangle\right)
\end{equation}
\end{thm}
\begin{proof}
Since each \(Z_n\) is smooth quasiprojective, we can apply (\ref{single_grot}) to it. By linearity, we can replace a representation \(V\) of \(W_n\) with a virtual representation given by a character polynomial \(P\), so we obtain
\begin{equation}
q^{-\dim Z_n} \sum_{y \in Y_n(\mathbb{F}_q)} P(\sigma_y) = \sum_{i=0}^{2\dim Z_n} (-1)^i \tr\left( \Frob_q : \langle H^i_{\acute et}(Z_n; L), P \rangle_{W_n}\right)
\end{equation}
Call this sum \(A_n\). Furthermore, let
\[ B_n = \sum_{i=0}^{2\dim Z_n} (-1)^i \tr\left( \Frob_q : \langle H^i_{\acute et}(Z; L), P \rangle\right) \]
Our goal is thus to show that \(\lim_{n \to \infty} A_n = \lim_{n \to \infty} B_n\): that is, first of all that both sequences converge, and that their limits are equal.
By our assumption that \(H^*(Z)\) is convergent, we know that there is a function \(F_P(i)\) which is subexponential in \(i\), such that for all \(n\),
\[ |\langle H^i_{\acute et}(Z_n; L), P\rangle_{W_n}| \le F_P(i) \]
and thus in particular (taking \(n\) large enough)
\[ |\langle H^i_{\acute et}(Z; L), P\rangle| \le F_P(i) \]
Furthermore, by Deligne \cite[Thm 1.6]{De}, we know that
\[ \left|\tr\left( \Frob_q : \langle H^i_{\acute et}(Z_n; L), P \rangle\right)\right| \le q^{-i/2} \left|\langle H^i_{\acute et}(Z_n; L), P \rangle\right| \]
We thus have
\begin{align*}
|A_n| &\le \sum_{i=0}^{2\dim Z_n} \left|\tr\left( \Frob_q : \langle H^i_{\acute et}(Z_n; L), P \rangle_{W_n}\right)\right| \\
&\le \sum_{i=0}^{2\dim Z_n} q^{-i/2} \left|\langle H^i_{\acute et}(Z_n; L), P \rangle\right| \\
&\le \sum_{i=0}^{2\dim Z_n} q^{-i/2} F_P(i)
\end{align*}
For exactly the same reason, \(|B_n| \le \sum_{i=0}^{2\dim Z_n} q^{-i/2} F_P(i)\). Since \(F_P(i)\) is subexponential in \(i\), this means that both \(A_n\) and \(B_n\) converge.
It remains to show that \(\lim_{n \to \infty} A_n - B_n = 0\). Let \(N(n,P)\) be the number such that
\[\langle H^i(Z_n), P\rangle_{W_n} = \langle H^i(Z), P\rangle \;\; \text{ for all } i \le N(n,P). \]
We thus have
\begin{align*}
|B_n - A_n| &\le \sum_{i=0}^{2\dim Z_n} q^{-i/2} \left| \langle H^i(Z), P\rangle - \langle H^i(Z_n), P\rangle_{W_n} \right| \\
&= \sum_{i=N(n,P)+1}^{2\dim Z_n} q^{-i/2} \left| \langle H^i(Z), P\rangle - \langle H^i(Z_n), P\rangle_{W_n} \right| \\
&= \sum_{i=N(n,P)+1}^{2\dim Z_n} 2q^{-i/2} F_P(i)
\end{align*}
Since \(N(n,P) \to \infty\) as \(n \to \infty\), and \(F_P(i)\) is subexponential in \(i\), we conclude that \(|B_n - A_n|\) becomes arbitrarily small as \(n \to \infty\).
\end{proof}
We can now apply this to \(\Conf^G_n(X)\) to obtain \thref{arith_stats}.
\begin{proof}[Proof of \thref{arith_stats}]
By \thref{etale_repstab}, \(H^i(\Conf^G(X)_{/\overline{\mathbb{F}}_q}\) is a finitely-generated \(\Gal(\overline{\mathbb{F}}_q/\mathbb{F}_q)\)-\(\FI_G\)-module. By \thref{conf_convg} \(H^*(\Conf^G(X))\) is convergent. We thus conclude by \thref{fig_stats}.
\end{proof}
\section{Point-counts for polynomials}
For special choices of \(X\) in \thref{arith_stats}, we can give an interpretation to the left-hand side of (\ref{groth}) in terms of point-counts of polynomials over \(\mathbb{F}_q\).
\subsection{Complex reflection groups and Gauss sums}
The first example is where \(X = \mathbb{G}_m\) and \(G = \mathbb{Z}/d\mathbb{Z}\) acting by multiplication by a \(d\)-th root of unity. In order for this action to be well-defined, we need to consider \(X\) as a scheme over \(\mathbb{Z}[\zeta_d]\). However, notice that here the action is not free: if we look at the fiber of \(X\) over a prime dividing \(d\), then \(\mathbb{Z}/d\mathbb{Z}\) will act trivially, because these are the primes that ramify in \(\mathbb{Z}[\zeta_d]\). Thus, to satisfy our hypotheses that we have a free action of \(G\), we consider \(X\) as a scheme over \(\mathcal{O}_k[1/d]\), where \(k = \mathbb{Q}(\zeta)\) is the cyclotomic field. Thus the finite fields \(\mathbb{F}_q\) we consider will satisfy \(q \equiv 1 \mod d\), since \(\mathbb{F}_q\) is a residue field of \(\mathbb{Z}[\zeta]\).
In this case, we have
\[ \Conf^{\mathbb{Z}/d\mathbb{Z}}_n(\mathbb{G}_m)(R) = \{(x_i) \in R^n \mid x_i \ne 0, x_i \ne \zeta^k x_j\} \]
so that \(\Conf^G_n(X)\) is the complement of a hyperplane arrangement. The arithmetic and \'etale cohomology of this arrangement was studied by Kisin-Lehrer in \cite{KL}, where they obtained formulas for the equivariant Poincar\'e polynomial of \(\Conf^{\mathbb{Z}/d\mathbb{Z}}_n(\mathbb{G}_m)\).
By Bj\"orner-Ekedahl \cite{BE}, the action of \(\Frob_q\) on \(H^i_{\acute et, c}(Z; \mathbb{Q}_l)\) is given by multiplication by \(q^i\), and thus (again, by Poincar\'e duality)
\[\tr\left(\Frob_q : \langle H^i_{\acute et}(\Conf^G_n(X)), P\rangle\right) = q^{-i} \langle H^i_{\acute et}(\Conf^G_n(X)), P\rangle \]
This lets us compute the right-hand side of (\ref{groth}) explicitly in this case:
\begin{equation} \label{crg_groth}
\lim_{n \to \infty} q^{-n} \sum_{f \in \Poly_n(\mathbb{F}_q^*)} P(y) = \sum_{i=0}^\infty (-1)^i q^{-i} \langle H^i(\Conf^{\mathbb{Z}/d\mathbb{Z}}(\mathbb{C}^*); \mathbb{C}), P \rangle
\end{equation}
since we know by \thref{etale_repstab} that \(H^i_{\acute et}(\Conf^{\mathbb{Z}/d\mathbb{Z}}(\mathbb{G}_m)_{/\overline{\mathbb{F}}_q}; \mathbb{Q}_l) \cong H^i(\Conf^{\mathbb{Z}/d\mathbb{Z}}(\mathbb{C}^*); \mathbb{Q}_l)\).
Notice that \(d \mid q - 1\), so \(\mathbb{Z}/d\mathbb{Z}\) is a quotient of \(\mathbb{Z}/(q-1)\mathbb{Z}\). Thus any representation of \(\mathbb{Z}/d\mathbb{Z} \wr S_n\) lifts to a representation of \(\mathbb{Z}/(q-1)\mathbb{Z} \wr S_n\), so we can always interpret the left-hand side of (\ref{crg_groth}) as a statement about representations of the single group \(\mathbb{Z}/(q-1)\mathbb{Z} \wr S_n\). On the other hand, as remarked in \cite[\S3]{Ca1}, there is a Galois cover
\[\Conf^{\mathbb{Z}/(q-1)\mathbb{Z}}_n(\mathbb{C}^*) \to \Conf^{\mathbb{Z}/d\mathbb{Z}}_n(\mathbb{C}^*)\]
with deck group \((\mathbb{Z}/\frac{q-1}{r}\mathbb{Z})^n\) and so by transfer,
\[ H^i(\Conf^{\mathbb{Z}/d\mathbb{Z}}_n(\mathbb{C}^*); \mathbb{Q}) = \left(H^i(\Conf^{\mathbb{Z}/(q-1)\mathbb{Z}}_n(\mathbb{C}^*); \mathbb{Q})\right)^{(\mathbb{Z}/\frac{q-1}{r}\mathbb{Z})^n} \]
and thus the right-hand side of (\ref{crg_groth}) is the same whether we consider \(V\) as a representation of \(\mathbb{Z}/d\mathbb{Z} \wr S_n\) or \(\mathbb{Z}/(q-1)\mathbb{Z} \wr S_n\). Therefore we lose nothing if we simply assume that in fact \(d = q-1\).
Now we give some number-theoretic meaning to the left-hand side of (\ref{crg_groth}). An element \(f \in \Poly_n(\mathbb{F}_q^*)\) is a polynomial in \(\mathbb{F}_q[T]\) that does not have 0 as a root. The roots of \(f(T)\) are sitting in some extension field of \(\mathbb{F}_q\), and the \(d\)-th roots of those roots possibly in some even higher extension field. The permutation \(\sigma_f\) is the action of \(\Frob_q\) on all these \(d\)-th roots, which permutes the actual roots (think of these as the columns each containing a set of \(d\)-th roots), and then further permutes the \(d\)-th roots cyclically. This precisely gives an element of \(G^n \rtimes S_n\) (up to conjugacy).
Recall that a \emph{Gauss sum} is a certain sum of roots of unity obtained by summing values of a character of the unit group of a finite ring. Now, suppose that \(\chi\) is an irreducible character of \(G\). Write
\[X^\chi_i = \sum_{g \in G} \chi(g) X^g_i.\]
For each \(i\), the \(\{X^\chi_i\}_{\chi \in \widehat{G}}\) have the same span as \(\{X^g_i\}_{g \in G}\), and the \(\{X^\chi_i\}\) are more natural to use here.
Since \(d = q - 1\), there is an isomorphism \(\mathbb{F}^*_q \cong G\), and in fact such a surjection \(\mathbb{F}_{q^k}^* \to G\) for any \(k\). It therefore makes sense to talk about applying \(\chi\) to elements of \(\overline{\mathbb{F}}_q^*\). For a given \(f \in \Poly_n(\mathbb{F}^*_q)\), consider first decomposing \(f\) into irreducibles factors over \(\mathbb{F}_q\), and then each of these into linear factors over \(\overline{\mathbb{F}}_q\); since none are zero, all the roots actually lie in \(\overline{\mathbb{F}}_q^*\). For a given irreducible degree-\(i\) factor \(p\) of \(f\), since \(\Frob_q\) acts transitively on the roots of \(p\) over \(\overline{\mathbb{F}}_q\), and since \(q \equiv 1 \mod d\), then \(\chi\) takes the same value on each of the roots. It is then straightforward to calculate that
\[
X^\chi_i(\sigma_f) = \sum_{\deg(p) = i} \chi(\text{root}(p))
\]
where the sum is taken over all irreducible factors \(p\) of \(f\) of degree \(i\), and \(\text{root}(p)\) denotes any of the roots of \(p\) in \(\overline{\mathbb{F}}_q\). Thus, \(X^\chi_i\) is a Gauss sum of \(\chi\) applied to the roots of degree-\(i\) irreducible factors of \(f\). A general character polynomial is generated as a ring by the \(X^\chi_i\)'s, so this says how to interpret the left-hand-side. Thus, (\ref{crg_groth}) says that the \emph{average value} of any such Gauss sum across all polynomials in \(\Poly_n(\mathbb{F}^*_q)\) always converges to the series in \(q^{-1}\) on the right-hand side of (\ref{crg_groth}). In particular, the decomposition of \(H^1\) and \(H^2\) determined in \cite[\S4.1]{Ca1} allows us to compute the examples (\ref{X1_pcount}) and (\ref{X2_pcount}) from the introduction.
\subsection{Counting solutions to \(f = g^d - t h^d\)}
Consider counting the number of monic polynomials \(f \in \mathbb{F}_q[t]\) with distinct roots of the form \(f = g^d - t h^d\) for \(g, h \in \mathbb{F}_q[t]\). If we assume \(q \equiv 1\) mod d, so that there is a primitive \(d\)-th root of unit \(\zeta \in \mathbb{F}_q\), then we can factor
\begin{align*}
f(t^d) = g^d(t^d) - t^d b^d(t^d) = (a(t^d) - t b(t^d)) (a(t^d) - \zeta t b(t^d)) \cdots (a(t^d) - \zeta^{d-1} b(t^d))
\end{align*}
where each \(a(t^d) - \zeta^i t b(t^d) \in \mathbb{F}_q[t]\). Notice that if \(x\) is a root of \(a(t^d) - t b(t^d)\), then \(\zeta^{n-i} x\) is a root of \(a(t^d) - \zeta^i t b(t^d)\). Thus, \(f(t^d)\) has such a factorization if and only if the \(d\)-th roots of the roots of \(f\) all lie in different Galois orbits. This is therefore equivalent to being able to write \(f = g^d + t h^d\). Also notice that this is equivalent to \(f\) being a norm in the cyclic extension \(\mathbb{F}[\sqrt[d] t] / \mathbb{F}[t]\).
Now, assume further that \(f\) is monic with distinct roots, and that 0 is not a root of \(f\), so that \(f \in \Poly_n(\mathbb{F}_q^*)\). Then as we have seen, we get a group element \(\sigma_f \in W_n\), up to conjugacy. The condition that the \(d\)-th roots of the roots of \(f\) all lie in different Galois orbits precisely says that \(\sigma_f\) is in a conjugacy class where each cycle is decorated with the identity element of \(G\). Let \(\delta_n\) be the indicator function for such conjugacy classes. Then as we have just argued, \(\delta_n(\sigma_f) = 1\) precisely when we can write \(f = g^d + t h^d\).
Then (\ref{single_grot}) tells us that
\begin{gather*}\#\{f \in \Poly_n(\mathbb{F}_q^*) \mid f = g^d + t h^d \text{ for } g,h \in \mathbb{F}_q[t]\} = \sum_{f \in \Poly_n(\mathbb{F}_q^*)} \delta_n(\sigma_f) \\
= q^n \sum_{i = 0}^{2n} (-1)^i q^{-i} \langle H^i(\Conf^{\mathbb{Z}/d\mathbb{Z}}(\mathbb{C}^*)), \delta_n\rangle
\end{gather*}
This formula was investigated in great detail in the case \(d = 2\) by Matei \cite{Mat}, in which case the equation \(f = g^2 - t h^2\) had been studied in a number of previous papers, e.g. \cite{BSSW}.
It is very important to note that while each \(\delta_n\) is a class function of \(W_n\), and thus in the span of character polynomials for \(W_n\), there is \emph{no} character polynomial \(P\) such that \(P_n = \delta_n\). This can be seen, for instance, by the fact that a single character polynomial can only depend on cycles of bounded length, while \(\delta_n\) depends on all the cycles of \(g \in W_n\). Thus we cannot apply \thref{arith_stats} and take the limit as \(n \to \infty\).
Indeed, the calculations in \cite{Mat} make it clear that the result depends on \(n\), although there is a certain kind of stability present in his results. It would be very interesting to carry out the same calculations for general \(d\).
\subsection{Iterated configuration spaces}
Next, take \(X = \Conf_d(\mathbb{A}^1)\) and \(G = S_d\) permuting the indices, which by definition of \(\Conf_d\) is a free action. Thus we are considering \(\Conf^{S_d}_n(\Conf_d(\mathbb{A}^1))\) with the action of \(W_n = S_d \wr S_n\), and quotient \(\UConf_n(\UConf_d(\mathbb{A}^1))\). Here we have
\begin{align*} \Conf^{S_d}_n(\Conf_d(\mathbb{A}^1))(R) = \big\{&[ (x^1_1, \dots, x^1_d), \dots, (x^n_1, \dots, x^n_d)] \in (R^d)^n : \\
& x^i_a \ne x^i_b, \{x^i_1, \dots, x^i_d\} \ne \{x^j_1, \dots, x^j_d\} \big\}
\end{align*}
Thus, \(\Conf^G_n(X)\) is the complement of a linear subspace arrangement: the subspaces are the codimension 1 spaces \(\{x^i_a = x^i_b\}\), and the codimension \(d\) spaces \(\{(x^i_1, \dots, x^i_d) = (x^j_1, \dots, x^j_d)\}\) and all its \(S_d\)-translates. Unfortunately, there are not simple results on the eigenvalues of Frobenius for a linear subspace arrangement, like there are for hyperplanes arrangements, because the different codimensions of the subspaces ``mix up'' the action of Frobenius on different cohomology classes. So all we are able to assert is the content of \thref{arith_stats} for this specific \(X\) and \(G\).
\begin{equation}
\label{iter_groth}
\lim_{n \to \infty} q^{-nd} \sum_{y \in \UConf_n(\UConf_d(\mathbb{F}_q))} P(\sigma_y) = \sum_{i = 0}^\infty (-1)^i \tr\left(\Frob_q : \langle H^i(\Conf^{G}(X); L), P\rangle\right)
\end{equation}
However, we are able to interpret the left-hand side of (8). An element \(\{f_i\} \in \UConf_n(\UConf_d(\mathbb{F}_q))\) is a set \(\{f_1, \dots, f_n\}\) of distinct monic polynomials \(f_i \in \overline{\mathbb{F}}_q[t]\) with \(d\) distinct roots such that \(\Frob_q(f_i) = f_j\). In particular, the product \(f_1 f_2 \cdots f_n \in \mathbb{F}_q[t]\). The element \(\sigma_{\{f_i\}}\) is the action of Frobenius on all the roots of the \(\{f_i\}\), which permute the \(\{f_i\}\) themselves and then further permute each of their roots. This precisely gives an element of \(S_d \wr S_n\), up to conjugacy.
A character polynomial \(P\) for \(S_d \wr S_n\) is a polynomial in \(\{X^\lambda_i \mid 1 \le i \le n, \lambda \vdash d\}\), where \(X^\lambda_i\) counts the number of \(\lambda\)-decorated \(i\)-cycles. For us, this means that we look at the cycle decomposition of how \(\Frob_q\) acts on the \(\{f_j\}\), and on the \(d\) roots of each \(f_j\). This will give us a cycle decomposition of \(n\), the action (up to conjugacy) on the \(\{f_j\}\), and each of the cycles will be labeled by a partition (cycle decomposition) of \(d\), which is the action on the \(d\) roots. \(X_i^{\lambda}\) then counts the number of \(i\)-cycles labeled with the partition \(\lambda\).
That is to say, \(X_i^{\lambda}\) counts the number of \(\{f_{j_1}, \dots, f_{j_i}\} \subset \{f_j\}\) such that Frobenius permutes the \(\{f_{j_1}, \dots, f_{j_i}\}\) cyclically, and that the product \(f_{j_1} \cdots f_{j_i}\), which must therefore lie in \(\mathbb{F}_q[t]\), decompose into irreducible pieces of degrees \(i \lambda_1, \dots, i \lambda_m\).
For example, if \(\lambda\) is a \(d\)-cycle, then \(X_2^{\lambda}\) counts the number of \(\{f_\alpha, f_\beta\} \in \{f_j\}\) such that \(f_\alpha f_\beta\) is an irreducible polynomial in \(\mathbb{F}_q[t]\). We can think of the sum
\[ \sum_{\{f_j\} \in \UConf_n(\UConf_d(\mathbb{F}_q))} X_2^{\lambda}(\sigma_{\{f_j\}})\]
as being over all monic \(F \in \mathbb{F}_q[t]\) with distinct roots of degree \(n \cdot d\), where we sum over all the possible ways of decomposing \(F = f_1 \cdots f_n\). This sum then counts the number of ways of decomposing \(F = G \cdot f_1 \cdots f_{n-2}\), where \(G \in \mathbb{F}_q[t]\) is irreducible of degree \(2d\), and \(f_j \in \overline{\mathbb{F}}_q[t]\), each of degree \(d\), satisfy \(\Frob(f_i) = f_j\). Thus, (\ref{iter_groth}) is saying that the \emph{average} number of such decompositions, across all such \(F\) of degree \(n \cdot d\), stabilizes to the right-hand side of (\ref{iter_groth}) as \(n \to \infty\).
|
1,941,325,220,516 | arxiv | \section{Preliminaries}
Let $\pi=\pi_1(K)$ be the fundamental group of a CW complex $K$. By $u^K:K\to B\pi=K(\pi,1)$ we denote a map that classifies the universal covering $\Wi K$
of $K$. We refer to $u^K$ as a {\em classifying map} for $K$. We note that a map $f:K\to B\pi$ is a classifying map if and only if it induces an isomorphism on the fundamental groups.
\subsection{Inessential manifolds and macroscopic dimension} We recall the following definition of Gromov~\cite{G3}:
\begin{defn}
An $n$-manifold $M$ with fundamental group $\pi$ is called {\it inessential} if its classifying map $u^M:M\to B\pi$ can be deformed into the $(n-1)$-skeleton $B\pi^{(n-1)}$ of a CW-complex structure on $B\pi$ and it is called {\em essential} otherwise.
\end{defn}
Note that for an inessential $n$-manifold $M$ we have $\dim_{mc}\Wi M\le n-1$. Indeed, a lift $\Wi{u^M}:\Wi M\to E\pi^{(n-1)}$ of a classifying
map is a uniformly cobounded proper map to an $(n-1)$-complex.
Generally, if a classifying map $u^M:M\to B\pi$ can be deformed to the $k$-dimensional skeleton, then $\dim_{mc}\Wi M\le k$.
The following was proven in~\cite{BD}:
\begin{prop}[\cite{BD}, Lemma 3.5]\label{ref2}
For an inessential manifold $M$
with a CW complex structure a classifying map $u:M\to B\pi$ can be chosen such that
$$u(M^{(n-1)})\subset B\pi^{(n-2)}.$$
\end{prop}
\subsection{Macroscopically inessential manifolds}
The macroscopic dimension can be used for the description of the macroscopic inessentiality of a manifold.
\begin{thm}[\cite{Dr1}]\label{ref-mc}
Let $M$ be a closed oriented $n$-manifold and let $\Wi u:\Wi M\to E\pi$ be a lift of $u^M:M\to B\pi$.
Then the inequality $\dim_{mc}\Wi M\le n-1$ is equivalent to the following condition:
(*) The map $\Wi u$ can be deformed by a bounded homotopy to $E\pi^{(n-1)}$ for any lift of a proper metric on $B\pi$.
\end{thm}
Note that we may assume that $B\pi$ is a locally finite complex and hence it admits a proper metric.
Also note that $\Wi u$ admits a bounded deformation to a map $f:\Wi M\to E\pi$ if and only if $f$ is in bounded distance to $\Wi u$, i.e. there is $C>0$ such that
$d(f(x),\Wi u(x))< C$ for all $x\in\Wi M$.
Thus, the inequality $\dim_{mc}\Wi M^n\le n-1$ is a macroscopic analog of inessentiality. We call manifolds
$N$ with $\dim_{mc}N<\dim N$ {\em macroscopically inessential}.
There is an analog of Proposition~\ref{ref2}:
\begin{prop}[\cite{Dr1}, Lemma 5.3.]\label{ref3}
For an $n$-dimensional manifold $M$ with a fixed CW complex structure, a classifying map $u:M\to B\pi$ and with macroscopically
inessential universal covering $\Wi M$
any lift $\Wi u:\Wi M\to E\pi$ of $u$ admits a bounded deformation to a proper map $f:\Wi M\to E\pi^{(n-1)}$ with
$f(\Wi M^{(n-1)})\subset E\pi^{(n-2)}.$
\end{prop}
\subsection{Macroscopic dimension and QI-embeddings}
A map $f:X\longrightarrow Y$ between metric spaces is a quasi-isometric embedding (QI-embedding) if for all $x_1, x_2$ in X, there are $\lambda, c >0$, such that $$\frac{1}{\lambda}d_X(x_1,x_2)-c\leq d_Y(f(x_1),f(x_2))\leq\lambda d_X(x_1,x_2)+c.$$
\newline Note that any countable CW complex $B$ admits a metric $d$, called a \emph{weak metric} on $B$, such that the identity map $id: B\longrightarrow (B,d)$ is continuous. It means that $d$ does not necessarily define the CW topology on $B$ but its restriction to any compact set does.
\newline\indent The following theorem, proved in \cite{BD}, characterizes macroscopic dimension of universal coverings of finite CW complexes by continuous QI-embeddings.
\begin{thm}[\cite{BD}, Theorem 2.12]\label{Macroscopic dimension and QI-embeddings}
Suppose that $E\pi$ is given a metric $d$ lifted from a weak metric on $B\pi$ where $\pi=\pi_1(X)$ for a finite CW complex X. Then the inequality $dim_{mc}\Wi X\leq n$ holds true for the universal covering $\Wi X$ of $X$ with the lifted metric if and only if there is a continuous QI-embedding $g:\Wi X\longrightarrow E\pi^{(n)}$.
\end{thm}
It was shown in the proof of the above theorem that given a QI-embedding $g:\Wi X\longrightarrow E\pi^{(n)}$, then it is proper and uniformly cobounded. It follows from the above theorem and Proposition 2.4 that for a closed $n$-manifold $M$, a lift $\Wi u:\Wi M\longrightarrow E\pi$ of $u:M\longrightarrow B\pi$ admits a bounded deformation to a QI-embedding $f:\Wi M\longrightarrow E\pi^{(n-1)}$ with $f(\Wi M^{(n-1)})\subset E\pi^{(n-2)}$.
\subsection{Bordism}
We recall that the group of oriented relative bordisms $\Omega_n(X,Y)$
of the pair $(X,Y)$ consists of the equivalence classes of pairs $(M,f)$
where $M$ is an oriented $n$-manifold with boundary and $f:(M,\partial M)\to (X,Y)$ is a continuous map. Two pairs $(M,f)$ and $(N,g)$ are equivalent if there is a pair $(W,F)$, $F:W\to X$ called a {\em bordism} where $W$ is an orientable $(n+1)$-manifold with boundary such that $\partial W =
M\cup W'\cup N$, $W'\cap M=\partial M$, $W'\cap N=\partial N$, $F|_M=f$, $F|_N=g$, and $F(W')\subset Y$.
In the special case when $X$ is a point, the manifold $W$ is called a bordism between $M$ and $N$.
The following proposition is proven in~\cite{BD}.
\begin{prop}\label{bordism} For any CW complex $K$
there is an isomorphism $$\Omega_n(K,K^{(n-2)})\cong H_n(K,K^{(n-2)}).$$
\end{prop}
In this paper we consider bordisms of open manifolds obtained from an infinite family of bordisms of compact manifolds with boundary
as follows. Let $M$ be an open $n$-manifold with a family of disjoint $n$-dimensional submanifolds with boundary $\{V_\gamma\}$.
Let $U_\gamma$ be a family of stationary on the boundary bordisms between $V_\gamma$ and $N_\gamma$.
Then the manifold $W=A\cup B$ with
$A=(M\setminus\cup_i\Int V_\gamma)\times[0,1]$ and $B=\coprod_\gamma U_\gamma$ with
$A\cap B=\coprod_\gamma\partial V_\gamma\times[0,1]$ is a {\em bordism
defined by the family} $\{U_\gamma\}$. Thus, $W$ is obtained from $M\times[0,1]$ by replacing
the cylinders $V_\gamma\times[0,1]$ by bordisms $U_\gamma$.
We call $W$ a bordism
between manifolds $M=M\times\{0\}$ and $N$ where $\partial W=M\coprod N$.
\newline We consider the product metric on $M\times[0,1]$.
For each $\gamma$ one can define a metric on $U_\gamma$ that extends the metric from $\partial V_\gamma\times[0,1]$. This gives a metric on $W$
such that the inclusion $M\subset W$ is an isometric embedding.
\subsection{Poincare-Lefschtz Duality} The following theorem is classic (\cite{Ha}, p.254).
\begin{thm}[Poincare-Lefschetz duality]
Suppose $M$ is a compact orientable $n$-manifold whose boundary is decomposed as the union of two compact $(n-1)$-dimensional manifolds $A$ and $B$ with a common boundary $\partial A=\partial B=A\cap B$. Then there is an isomorphism $$D:H^k(M,A)\to H_{n-k}(M,B)$$ for all $k$.
\end{thm}
\
\subsection{A Note on Obstruction Theory}
\
Let $(X,Y)$ be a CW-pair and $g$ a map from $Y$ to $Z$. Suppose that $Z$ is path connected and $\pi_1(Z)$ acts trivially on $\pi_n(Z)$ for every $n$. The aim is to extend the map $g$ to a map from $X$ to $Z$ by proceeding inductively over the skeleton of $X$. Suppose $g$ has been extended to a map $\hat{g}$ on the $n$-skeleton $X^{(n)}$, $\hat{g}:X^{(n)}\rightarrow Z$, then the obstruction $[\sigma_{\hat{g}}]$ to extending $\hat{g}|_{X^{(n-1)}}$ to $X^{(n+1)}$ lies in the cohomology group $H^{n+1}(X,Y;\pi_n(Z))$. Obstruction theory says that $\hat{g}|_{X^{(n-1)}}:X^{(n-1)}\rightarrow Z$ extends to a map $X^{(n+1)}\rightarrow Z$ iff $[\sigma_{\hat{g}}]=0$ (See \cite{DK}, ch.7). Here, $[\sigma_{\hat{g}}]$ is generated by the cochain $\sigma_{\hat{g}}\in C^{n+1}(X,Y;\pi_n(Z))$ defined on the $(n+1)$-cells, generators of $C_{n+1}(X,Y)$, by $\sigma_{\hat{g}}(e^{(n+1)}_i)=[\hat{g}\circ\phi_i|_{S^n}:S^n\rightarrow Z]\in\pi_n(Z)$, where $\phi_i:D^{(n+1)}_i\rightarrow X$ is the characteristic map. It easily follows that $\sigma_{\hat{g}}=0$ if and only if $\hat{g}$ extends to a map $X^{(n+1)}\rightarrow Z$.
\
\
\section{Proof of Main theorem}
The condition of residual finiteness in the main theorem comes into the picture in the following lemma:
\begin{lemma}\label{immersion}
Let $M$ be a 4-manifold with a residually finite fundamental group. Then for any $\alpha\in\pi_2(M)$ there is a finite covering $p':M'\to M$ and an immersion $h':S^2\to M'$ such that
$p'_*([h'])=\alpha$ and the inclusion homomorphism of the fundamental groups
$\pi_1(h'(S^2))\to\pi_1(M')$ is trivial.
\end{lemma}
\begin{proof}
Let $h:S^2\to M$ be an immersion without triple points that realizes $\alpha$. Let $\tilde h:S^2\to\Wi M$ be its lift with respect to the universal covering $p:\Wi M\to M$.
Since $\tilde h$ is an immersion, there are finitely many pairs of points $(x^-_1,x^+_1),\dots,(x^-_n,x^+_n)$ in $\tilde h(S^2)$ with $p(x^-_i)=p(x^+_i)$ such that
$p$ restricted to $\tilde h(S^2)\setminus(\cup_{i=1}^n\{x^\pm_i\})$ is 1-to-1. There are $g_i\in\pi=\pi_1(M)$ with $g_i(x_i^-) = x_i^+$. Since $\pi$ is residually finite,
there is an epimorphism $\phi:\pi\to F$ to a finite group such that $\phi(g_i)\ne e$ for all $i$ and $\phi$ restricted on $\{g_1, ..., g_n\}$ is injective. Then the universal covering is factorized as $p=p'\circ q$ with $p':M'\to M$ a finite covering
and $q:\Wi M\to M'$ the projection onto orbit space of the action of $ker\phi$. Since $q$ is $\pi$-equivariant map, $q(x_i^-)\ne q(x_i^+)$ for all $i$. Therfore, $q$ restricted to $\tilde h(S^2)$ is a homeomorphism onto the image. We consider $h'=q\circ\tilde h$. The commutative diagram
$$
\begin{CD}
\pi_1(\tilde h(S^2)) @>>> \pi_1(\Wi M)\\
@Vq|_{\dots}VV @VqVV\\
\pi_1(h'(S^2)) @>>>\pi_1(M')\\
\end{CD}
$$
and the fact that $\pi_1(\Wi M)=0$ imply the required property of $h'$.
\end{proof}
\begin{prop}\label{ker}
Let $\phi:A\to\Z_2$ be a $\pi$-module homomorphism where $A$ is a finitely generated $\pi$-module. Then the kernel $\ker\phi$ is a finitely generated $\pi$ module.
\end{prop}
\begin{proof}
Let $A$ be generated as a $\pi$-module by $S=\{x_1, x_2, ..., x_n\}$ and let $F$ be the free $\pi$-module on the set $S$ and $A$ be the homomorphic image of $F$ . Consider the kernel of the $\pi$-module homomorphism $\phi\circ p:F\longrightarrow \Z_2$, where $p:F\longrightarrow A$. Then $Ker(\phi\circ p)$ is generated as a $\pi$-module by the set $S'=\{x_1^{i_1}, ..., x_n^{i_n}\}, i_j = 1,2$. Hence, $Ker(\phi\circ p)$ is finitely generated. Since $p|_{Ker(\phi\circ p)}:Ker(\phi\circ p)\longrightarrow Ker(\phi)$ is surjective, it follows that $Ker(\phi)$ is finitely generated as a $\pi$-module.
\end{proof}
Let $\nu_M:M\to BSO$ denote
a classifying map for the stable normal bundle of a manifold $M$.
\begin{thm}\label{main} For a closed smooth orientable totally non-spin $4$-manifold $M$ with residually finite fundamental group
the inequality $\dim_{mc}\Wi M\le 3$ for the universal covering $\Wi M$ implies the inequality $\dim_{mc}\Wi M\le 2$.
\end{thm}
\begin{proof} Let $\pi=\pi_1(M)$.
The idea of the proof is to perform surgery on a family $\{V_\gamma\}_{\gamma\in\Gamma}$ of compact submanifolds of $\widetilde M$ for some finite index subgroup $\Gamma\subset\pi$ to obtain a family of stationary on the boundary bordisms $\{(W_\gamma , q_\gamma)\}_{\gamma\in\Gamma}$ between $\{V_\gamma\}$ and $\{N_\gamma\}$ , where $q_\gamma(N_\gamma)\subset E\Gamma^{(2)}.$ Then use obstruction theory to extend the maps ${q_\gamma}$ to a map $q:W \longrightarrow E\Gamma^{(2)}$, where $W=A\cup B$ with $A=(\widetilde M\backslash \bigcup_\gamma\Int V_\gamma) \times [0,1]$ and $B=\coprod_\gamma W_\gamma$ with $A\cap B=\coprod_\gamma \partial V_\gamma \times [0,1]$.
\bigskip
The assumption that $\widetilde M$ is non-spin implies that the homomorphism $$(\nu_{\widetilde M})_*:\pi_2(\widetilde M)\to\pi_2(BSO)=\mathbb{Z}_2$$ is surjective.
Let $ h :S^2\longrightarrow\widetilde M$ be the immersion representing an element $[h]\in\pi_2(\Wi M)$ such that $(\nu_{\widetilde M})_*[ h]\neq 0$.
By Lemma~\ref{immersion} there is a finite index normal subgroup $\Gamma\subset\pi$ and corresponding finite covering $p':M'\to M$ such that
the homomorphism $\pi_1(q\circ h(S^2))\to\pi_1(M')=\Gamma$ is trivial where $q:\Wi M\to M'$ is the universal covering of $M'$. Let $V$ be a regular neighborhood of $qh(S^2)$. Thus $V$ is a 4-manifold with boundary which can be deformed to $qh(S^2)$.
Since the inclusion homomorphism $\pi_1(V)\to\pi_1(M')$ is trivial, there is a section $s:V\to\Wi M$ of the universal covering $q:\Wi M\to M'$.
Let $V_\gamma=\gamma(s(V)),\gamma\in\Gamma$, denote the $\gamma$-translate of $s(V)$. Note that the family $\{V_\gamma\}$ is disjoint.
We may assume that $M'$ has a CW structure with one $4$-dimensional cell. Since $\dim_{mc}\widetilde M\le 3$, then by Proposition~\ref{ref3}, there is a bounded deformation of a lift $\widetilde u:\widetilde M\longrightarrow E\Gamma$ of a classifying map $u:M\longrightarrow B\Gamma$ to a map $f:\widetilde M\longrightarrow E\Gamma^{(3)}$ such that $f(\widetilde M\backslash\coprod_{\gamma\in\Gamma}D_\gamma)\subset E\Gamma^{(2)}$, where $\{D_\gamma\}_{\gamma\in\Gamma}$ are the translates of a section of $q$ over a fixed closed 4-ball $D$ in the 4-dimensional cell of $M'$. We may assume that $D\subset V$ and $D_\gamma\subset V_\gamma$. This can be done while keeping the images $f(D_\gamma)$ uniformly bounded. Clearly, $f$ is a QI-embedding.
\newline \indent Note that the restriction of $f$ to $(D_{\gamma},\partial D_{\gamma})$ defines a zero element in $H_4(E\Gamma,E\Gamma^{(2)})$. Since the family of sets $f(D_\gamma)$ is uniformly bounded, there is $r>0$ such that $f(D_{\gamma})\subset B_r(f(c_{\gamma}))$ where $c_{\gamma}\in D_{\gamma}$ and
$f|_{D_{\gamma}}$ defines a zero element in $H_4(B_r(f(c_{\gamma})), B_r(f(c_{\gamma}))\cap E\Gamma^{(2)}).$
By Proposition~\ref{bordism}, there is a relative bordism
$(W_{\gamma},q_{\gamma})$ of $(D_{\gamma},\partial D_{\gamma})$ to $(N'_{\gamma},S'_{\gamma})$ with $q_{\gamma}(\partial W_{\gamma}\setminus D_{\gamma})\subset E\Gamma^{(2)}$ and $q_\gamma(N'_\gamma)\subset Br(f(c_\gamma))\cap E\Gamma^{(2)}$.
We may assume that the bordism $W'_{\gamma}\subset\partial W_{\gamma}$ of the boundaries $\partial D_{\gamma}\cong S^3$ and $S'_{\gamma}$ is stationary; $W'_{\gamma}\cong\partial D_{\gamma}\times[0,1]$
and $q(x,t)=q(x)=f(x)$ for all $x\in\partial D$ and all $t\in[0,1]$. Applying 1-surgery we may assume that $W_{\gamma}$ is simply connected.
For each $\gamma\in\Gamma$, we can enlarge the bordism $W_\gamma$ by the trivial bordism $(V_\gamma\backslash\Int D_\gamma)\times [0,1]$ to a bordism $\overline{W}_\gamma$ of manifolds with boundaries between $V_\gamma$ and $N_\gamma$. The maps $q_\gamma$ extend by means of $f$ to maps $\overline{q}_\gamma :\overline{W}_\gamma\longrightarrow E\Gamma$ with $\overline{q}_\gamma(V_\gamma\backslash\Int D_\gamma)\times[0,1]\subset E\Gamma^{(2)}$. Thus $(\overline{W}_\gamma,\overline{q}_\gamma)$ is a bordism between $V_\gamma$ and $N_\gamma$ that is stationary on the boundary and $\overline{q}_\gamma(N_\gamma)\subset E\Gamma^{(2)}$. Note that the inclusion $V_\gamma\to\overline{W_\gamma}$ induces an isomorphism of the fundamental groups.
Since the images of $N_\gamma$ are uniformly bounded, there are finite isometric subcomplexes $Z_\gamma, \gamma\in\Gamma$, of $E\Gamma^{(2)}$ and $b>0$ sufficiently large such that $\overline{q}_\gamma(N_\gamma)\subset Z_\gamma$ and $diam( Z_\gamma)<b$. Now consider a finite set of loops $g^\gamma_i:S^1\to Z_\gamma$ that generates
$\pi_1(Z_\gamma)$,
and define $$X_\gamma=Z_\gamma\cup_{\cup g^\gamma_i}(\coprod_i D^2_i)$$ be the space obtained by attaching disks to $Z_\gamma$ along the maps
$g^\gamma_i$. Then $X_\gamma, \gamma\in\Gamma$, are simply connected and isometric. Note that since $\pi_1(E\Gamma^{(2)})=0$, the inclusion map $j_\gamma:Z_\gamma\longrightarrow E\Gamma^{(2)}$ can be extended to a map $\overline{j_\gamma}:X_\gamma\longrightarrow E\Gamma^{(2)}$. Moreover, we can assume that $diam(\overline{j_\gamma}(X_\gamma))<b$.
Consider the induced homomorphism $(\nu_{\overline{W}_\gamma })_*:\pi_2(\overline{W}_\gamma)\to\pi_2(BSO)=\mathbb{Z}_2$. This homomorphism is surjective and every 2-sphere $S$ that generates an element of the kernel has a trivial stable normal bundle. Since $\pi_1(\overline{W}_\gamma)\cong\pi_1(V_\gamma)$ is a finitely generated free group, $\pi_2(\overline{W}_\gamma)$ is a finitely generated $\pi_1(\overline{W}_\gamma)$-module (see~\cite{Ra}). It follows from Proposition~\ref{ker} that the kernel of $(\nu_{\overline{W}_\gamma })_*$ is finitely generated. Hence we can perform 2-surgery on $\overline{W}_\gamma$ to obtain a bordism $(\hat{W}_\gamma,\hat{q}_\gamma)$ between $V_\gamma$ and $N_\gamma$ and a map $\nu_{\hat{W}_\gamma}:\hat{W}_\gamma\longrightarrow BSO$ with $\hat{q}_\gamma=\overline{q}_\gamma=f$ on $\partial\hat{W}_\gamma=\partial\overline{W}_\gamma$ and with a uniform bound for the distance between images of $\hat{q}_\gamma$ and $\overline{q}_\gamma$. Hence, we can assume that $\hat{q}_\gamma(N_\gamma)\subset Z_\gamma$.
Let $i_\gamma:V_\gamma\longrightarrow\hat{W}_\gamma$ denote the inclusion map. Then $(\nu_{\hat{W}_\gamma})_*\circ i_{\gamma*}=(\nu_{V_\gamma})_*$. It follows that $i_{\gamma*}:\pi_2(V_\gamma)\longrightarrow\pi_2(\hat{W}_\gamma)$ is surjective since $(\nu_{V_\gamma})_*$ is surjective and $(\nu_{\hat{W}_\gamma})_*$ is an isomorphism. Thus, by the exact sequence of pairs $(\hat{W}_\gamma,V_\gamma)$, we get $\pi_2(\hat{W}_\gamma,V_\gamma)=0$. By the relative Hurewicz theorem, we get $H_1(\hat{W}_\gamma,V_\gamma)=H_2(\hat{W}_\gamma,V_\gamma)=0$.
Since $\hat{W}_\gamma$ is a smooth 5-manifold with corners, then it can be triangulated and hence it is a CW complex. Therefore, we can use obstruction theory on $(\hat{W}_\gamma, N_\gamma\bigcup\partial V_\gamma\times[0,1])$ to extend the map $i_\gamma\circ\hat{q}_\gamma|_{N_\gamma\bigcup\partial V_\gamma\times[0,1]}:N_\gamma\bigcup\partial V_\gamma\times[0,1]\longrightarrow X_\gamma$ to a map from $\hat{W}_\gamma$ to $X_\gamma$, where $i_\gamma:Z_\gamma\longrightarrow X_\gamma$ is the inclusion map.
Since $X_\gamma$ is simply connected, we can extend the map to the 2-skeleton $\hat W_\gamma^{(2)}$. The primary obstruction for this extension problem lives in $H^3(\hat W_\gamma,N_\gamma\bigcup\partial V_\gamma\times[0,1];\pi_2(X_\gamma))=H^3(\hat W_\gamma,N;\pi_2(X_\gamma))$. By the Poincare-Lefschetz Duality, $H^3(\hat W_\gamma,N;\pi_2(X_\gamma))=H_2(\hat W_\gamma,V_\gamma;\pi_2(X_\gamma))$, $H^3(\hat W_\gamma,N_\gamma)=H_2(\hat W_\gamma, V_\gamma)=0$ and $H^4(\hat W_\gamma,N_\gamma)=H_1(\hat W_\gamma, V_\gamma)=0$. By the Universal Coefficient Formula,
$$H^3(\hat W_\gamma,N_\gamma;\pi_2(X_\gamma))=H^3(\hat W_\gamma,N_\gamma)\otimes\pi_2(X_\gamma)\oplus Tor(H^4(\hat W_\gamma,N_\gamma),\pi_2(X_\gamma))=0.$$
Thus, there is an extension of the map $i_\gamma\circ\hat q_\gamma|_{N_\gamma\bigcup\partial V_\gamma\times[0,1]}$ to $\hat W_\gamma^{(3)}$. The second obstruction lives in the group $H^4(\hat W_\gamma,N_\gamma;\pi_3(X_\gamma)).$
By the Universal Coefficient Formula, $$H^4(\hat W_\gamma,N_\gamma;\pi_3(X_\gamma))=H^4(\hat W_\gamma,N_\gamma)\otimes\pi_3(X_\gamma))=0.$$
And the third obstruction lives in $H^5(\hat W_\gamma,N_\gamma;\pi_4(X_\gamma))= H_0(\hat W_\gamma,V_\gamma;\pi_4(X_\gamma))=0$.
Thus, there is an extension of $i_\gamma\circ\hat{q}_\gamma|_{N_\gamma\bigcup\partial V_\gamma\times[0,1]}:N_\gamma\bigcup\partial V_\gamma\times[0,1]\longrightarrow X_\gamma$ to a map $\tilde q_\gamma:\hat W_\gamma\to X_\gamma$.
Let $\hat{W}=A\cup B$, where $A=(\widetilde M\backslash\cup_\gamma\Int V_\gamma)\times[0,1]$ and $B=\coprod_\gamma\hat{W}_\gamma$ with $A\cap B=\coprod_\gamma\partial V_\gamma\times[0,1]$ and let $i:\widetilde M\longrightarrow \hat{W}$ denote the inclusion map. We may choose a metric on $\hat{W}$ such that $i$ is an isometric embedding.
Then the maps $\cup_\gamma\overline{j_\gamma}\circ \tilde q_\gamma:B\to E\Gamma^{(2)}$ naturally extend by means of $f$ to a QI-embedding $\hat{q}:\hat{W}\longrightarrow E\Gamma^{(2)}$.
The restriction $g=\hat q|_{\Wi M}:\Wi M\to E\Gamma^{(2)}$ is in finite distance to $\tilde u:\Wi M\to E\Gamma$. Thus, by Theorem 2.5, $\dim_{mc}\Wi M\le 2$.
\end{proof}
\begin{cor}
Gromov's Conjecture holds true for 4-manifolds with abelian fundamental groups.
\end{cor}
\begin{proof}
The case where $M$ is spin was proved in~\cite{BD2}, and the case where the universal cover $\Wi M$ is spin was proved in~\cite{Dr1}. Since abelian groups are residually finite, then in view of Theorem~\ref{main}, to complete the proof in the case where $\Wi M$ is non-spin, it suffices to show that a positive scalar curvature 4-manifold with abelian fundamental group is inessential.
Since Gromov's conjecture for a finite index subgroup implies the conjecture for the group~\cite{BD2}, we may assume that $\pi_1(M)$ is free abelian. If $rank(\pi_1(M))\ne 4$, then the manifold $M$ is inessential by definition. If $rank(\pi_1(M))=4$ and $M$ admits an essential map onto the 4-torus $T^4$, then $M$ does not admit a metric with positive scalar curvature~\cite{Sch},~\cite{SY}.
\end{proof}
\bigskip
\bigskip
\noindent\textbf{Question:} Let $\nu:M^4\longrightarrow BSO$ be a classifying map of the stable normal bundle. Does there exist an immersion $h:S^2\longrightarrow M^4$ such that $\nu_\ast[h_\ast(S^2)]\neq0$ in $\pi_2(BSO)$ and the inclusion $h(S^2)\longrightarrow M^4$ induces the trivial homomorphism on the fundamental groups?
\newline
\newline
\textbf{Remark:} In case of a positive answer to the above question, the residually finite condition on the fundamental group in the statement of the main theorem would drop.
|
1,941,325,220,517 | arxiv | \section{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 \citep{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 \citep{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 \citep{Bul72}, bending mode in the self-accelerating branch of Dvali-Gabadadze-Porrati model \citep{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 $T$ \citep{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 \cite{Shami15, Myrza12, Hound12, Alvar13, Barri14, Vacar14, Shari12, Shari14, Yous16a, Yous16b, Sahoo16, Aktas17}. Alvarenga et al. studied the scalar perturbations \citep{Alvar13}, Shabani and Ziaie studied the stability of the model \citep{Shab17, Shab17a}, Sharif and Zubair investigated the energy conditions and stability of power law solutions \citep{Sharif13} in this modified gravity theory. Sharif and Zubair \citep{Sharif12a} and Jamil et al. \citep{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 \citep{Moraes17, Zubair15, Alha16}.
With the advent of recent observations regarding the cosmic anisotropy \citep{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 \citep{Campa06, Grup10, SKT14, Saadeh16, SKT17, Deng18}. In view of this, anisotropic cosmological models that bear a similarity to Bianchi morphology have gained much importance \citep{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 \cite{ 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.\cite{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.
\section{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 \cite{Harko11, Mishra16a}
\begin{equation} \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}
\end{equation}
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, $f(R,T)=R+2\beta T$ \citep{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,
\begin{equation} \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}
\end{equation}
which can also be written as
\begin{equation} \label{eq:3}
R_{ij}-\frac{1}{2}Rg_{ij}=\left[8\pi+2\beta\right]T_{ij} + \Lambda(T) g_{ij}.
\end{equation}
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 \eqref{eq:3} 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
\begin{equation} \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
\end{equation}
\begin{equation} \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
\end{equation}
\begin{equation} \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
\end{equation}
\begin{equation} \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
\end{equation}
\begin{equation} \label{eq:10}
\frac{\dot{B}}{B}+ h\frac{\dot{C}}{C}- (1+h)\frac{\dot{A}}{A}=0.
\end{equation}
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. \cite{skt15} and Mishra et al. \cite{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 \eqref{eq:10} 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{eqnarray}
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{eqnarray}
From the above field equations \eqref{eq:16}-\eqref{eq:18}, we obtain the expressions for pressure, energy density and the string tension density as
\begin{eqnarray}
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{eqnarray}
Consequently, the equation of state parameter $\omega$ and the effective cosmological constant $\Lambda$ can be expressed as
\begin{eqnarray}
\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{eqnarray}
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 \eqref{eq:19}-\eqref{eq:23} 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.
\section{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{eqnarray}
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{eqnarray}
From the field eqns. \eqref{eq:16}-\eqref{eq:18}, the pressure, energy density and string tension density can be obtained as:
\begin{eqnarray}
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{eqnarray}
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.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig1}
\caption{Dynamical evolution of the equation of state parameter for different representative values of the coupling constant $\beta$.}
\end{center}
\end{figure}
From eqs. \eqref{eq:19}- \eqref{eq:20}, we obtain the equation of state parameter $\omega=\frac{p}{\rho}$ and the effective cosmological constant $\Lambda$ respectively as
\begin{eqnarray}
\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{eqnarray}
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.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig2}
\caption{Equation of state parameter as function of the coupling constant at present epoch for a given anisotropic parameter.}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig3}
\caption{Effect of anisotropic parameter on the equation of state.}
\end{center}
\end{figure}
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 $q=-0.598$ \citep{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.
\section{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 \cite{Kapusta94, Aktas07}
\begin{equation}
p_q=\frac{\rho_q}{3}-B_c,
\end{equation}
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{eqnarray}
\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{eqnarray}
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{eqnarray}
\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{eqnarray}
Substituting the expressions for $s_1, s_2$ and $s_3$ in eqs. \eqref{eq:26} and \eqref{eq:27}, the quark matter energy density and quark pressure are obtained as
\begin{eqnarray}
\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{eqnarray}
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.
\section{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.
\section{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.
|
1,941,325,220,518 | arxiv | \section{Introduction}
The pair contact process with diffusion (PCPD) is an interacting particle
system with diffusion, pair annihilation ($2 A \rightarrow 0$), and creation
of particles by pairs ($2 A \rightarrow 3 A$).
The PCPD was first introduced in 1982 by Grassberger~\cite{G1982},
but had remained unnoticed in the statistical physics
community for about 15 years. Since Howard and T{\"a}uber~\cite{HT1997}
(re)introduced the so-called `bosonic' PCPD in 1997,
the PCPD has been captivating statistical physicists and
the effort to understand the critical behavior of the PCPD has continued
until now~\cite{O2000,CHS2001,H2001,PK2002,BC2003,KC2003,JvWDT2004,NP2004,PHP2005a,PHP2005b,dOD2006,PP2006,KK2007,SB2008,PP2009,SB2012}.
It was accepted almost without question
that the upper critical dimensions of the PCPD are 2 and the PCPD
does not belong to the
directed percolation (DP) universality class
in higher dimensions~\cite{OMS2002}.
On the other hand, the one
dimensional PCPD has gained its notorious fame from the beginning because of
its strong corrections to scaling which make it difficult to find the
critical exponents accurately. Consequently, numerical studies have reported
scattered values of critical exponents and, in turn,
as many hypotheses concerning the critical behavior were suggested
as the number of researchers involved (for a review of the early controversy,
see Ref.~\cite{HH2004}). It still remains an open question whether
the one dimensional PCPD belongs to the DP class (DP hypothesis) or
forms a different universality class just like the higher dimensional PCPD
(non-DP hypothesis). In this paper, we will try to settle down this issue
by extensive Monte Carlo simulations.
Since we are mainly interested in the PCPD in one
dimension, by the PCPD in the following is meant the one-dimensional
PCPD if dimensions are not explicitly stated.
In favor of the DP hypothesis, Hinrichsen~\cite{H2006} argued that
the PCPD will eventually crossover to the DP class in the
long time limit. Although the argument is not without imperfection (see
Ref.~\cite{PP2008EPJB} for a critique), it is quite persuasive once
the dynamic exponent of the PCPD is accepted to be smaller than 2.
To confirm the DP hypothesis numerically, Barkema and his
colleagues~\cite{BC2003,SB2008,SB2012} published a series of papers.
Meanwhile, numerical results which seem contradictory
to the DP hypothesis were also published.
The critical behavior of the driven pair contact process with
diffusion~\cite{PHP2005a,PHP2005b} seems to suggest that the PCPD
cannot be described by a field theory with a single component field,
which is not compatible with the DP conjecture~\cite{J1981,G1982}.
Besides, the existence of non-trivial crossover from the PCPD
to the DP~\cite{PP2006,PP2009} also suggests the non-DP nature of the PCPD.
Naturally, these studies have been invoked to support the non-DP hypothesis.
Since the controversy is due to the strong corrections to scaling,
it is necessary to tame the corrections to scaling at our disposal.
To this end, this paper suggests a systematic method to find the leading
correction to scaling from numerical simulations.
Once corrections to scaling are determined, the effective exponents
can be systematically analyzed and one can evaluate the critical
exponents. In this way, this paper will find numerical values
of critical exponents of the PCPD.
The paper is organized as follows: after introducing the model in Sec.~\ref{Sec:model}, we develop a method to calculate the corrections to scaling
and then estimate critical decay exponent $\delta$,
using the knowledge of the corrections
to scaling in Sec.~\ref{Sec:results}. In Sec.~\ref{Sec:z},
we study the finite size scaling as well as microscopic absorbing
phase transitions to find other exponents. For comparison,
microscopic absorbing phase transitions
of a PCPD-like model belonging to the DP class are also investigated.
Section~\ref{Sec:sum} summarizes the work.
\section{\label{Sec:model}Model}
The model we study is the one-dimensional parity-conserving pair
contact process with diffusion first introduced in Ref.~\cite{PHK2001}.
Since the parity conservation does not play any role
in determining the universality class of the PCPD~\cite{PHK2001,PP2009},
we will simply refer to this model as the PCPD.
The model is defined on a one-dimensional lattice with periodic
boundary conditions. The state at each site is either occupied by a
particle ($A$) or empty ($\emptyset$); every site can accommodate at most
one particle. The dynamics can be defined as
\begin{eqnarray}
A\emptyset \stackrel{1/2}{\longleftrightarrow} \emptyset A,\quad
AA \stackrel{p}{\longrightarrow} \emptyset \emptyset,\nonumber\\
AA \emptyset \emptyset\stackrel{\sigma}{\longrightarrow} AAAA,\quad
\emptyset \emptyset AA \stackrel{\sigma}{\longrightarrow} AAAA,
\label{Eq:defmodel}
\end{eqnarray}
where $\sigma = (1-p)/2$.
Strictly speaking, a state is called absorbing if the transition rate from
it to any other states is zero. In the PCPD, a vacant state is the only
absorbing state in the rigorous sense, but the set of states with a single
particle alone is collectively thought of as absorbing because probability flux
from any configuration in this set to other states not in this set
is zero (in mathematics, this set is called closed).
Note that the PCPD in the conventional setting ($2A \rightarrow 0$,
$2A \rightarrow 3A$) also has such two `absorbing' states.
In simulations, we keep a list of the position of every particle for all
time. Assume that there are $N_t$ particles at time $t$. Among $N_t$ particles,
we choose one with equal probability. Let us assume that
the chosen particle is located at site $i$. With probability $\frac{1}{2}$,
one of two nearest neighbors of site $i$ is chosen. Let us call the index of
the chosen site $j$ ($j$ is either $i+1$ or $i-1$; recall that
we are using periodic boundary conditions). If site $j$ is empty, the
particle at site $i$ moves to site $j$. If site $j$ is also occupied,
two particles at both sites ($i$ and $j$) are annihilated with
probability $p$. But with probability $1-p$, the pair will try to
branch another two particles. If a branching event is scheduled, we choose one
of two directions with equal probability. If the direction is determined,
we check if two nearest consecutive sites of the pair
along the chosen direction are both empty. If so, two new particles are placed
there, but otherwise nothing happens and there is no change of configuration.
After the attempt, time increases by $1/N_t$ and the list is appropriately
updated.
\section{\label{Sec:results}Critical Decay Exponent}
\subsection{\label{Sec:CTS}Corrections to scaling}
In this section, the initial condition is set to be the fully occupied state.
During simulations, we measured the (particle) density $\rho(t)$ and the pair
density $\phi(t)$ which are defined as
\begin{equation}
\rho(t) = \frac{1}{L} \sum_i \left \langle s_i(t) \right \rangle ,\quad
\phi(t) = \frac{1}{L} \sum_i \left \langle s_i(t) s_{i+1}(t) \right \rangle,
\label{Eq:rpdef}
\end{equation}
where $L$ is the system size, $s_i(t)$ is a random variable which takes 1
(0) if site $i$ is occupied
(vacant) at time $t$, and $\langle\cdots \rangle$ means the average over all
realizations.
At criticality, the density is expected to show power-law behavior
with corrections to scaling such as
\begin{equation}
\rho(t) = a_1 t^{-\delta} \left ( 1 + c_1 t^{-\chi} + o(t^{-\chi})\right ),
\label{Eq:density_critical}
\end{equation}
where $\delta$ is the critical decay exponent, $\chi$ is the
exponent of the leading correction, and $a_1$, $c_1$ are constants.
Since the two dimensions are believed to be the upper critical dimensions
of the PCPD, we assume that there is no logarithmic correction in the
one dimensional system.
To find the critical exponent $\delta$ from numerical data in a systematic
way, it is customary to analyze an effective exponent defined as
\begin{equation}
-\delta_\textrm{eff}(t) = \frac{ \ln \rho(t)-\ln\rho(t/b) }{\ln b},
\end{equation}
where $b$ is a time independent constant.
At the critical point the asymptotic behavior of the effective exponent
should be
\begin{equation}
-\delta_\textrm{eff}(t) = - \delta - c_1 \frac{b^\chi-1}{\ln b} t^{-\chi} + o(t^{-\chi}).
\label{Eq:deltaeff}
\end{equation}
Thus, if we plot the effective exponent $-\delta_\textrm{eff}$
as a function of $t^{-\chi}$ at criticality with accurate numerical value
of $\chi$, the curve should approach to the critical exponent with finite
slope as $t^{-\chi} \rightarrow 0$.
On the other hand, if the system is in the active (absorbing) phase,
the effective exponent eventually veers up (down) as
$t^{-\chi} \rightarrow 0$. The accurate value of $\chi$, therefore, is
crucial to estimate the accurate value of the critical exponent.
To find $\chi$ without prior knowledge of $\delta$,
we need another quantity which decays as $t^{-\delta}$ in the
asymptotic regime. A natural candidate is the pair density $\phi(t)$.
Although a mean field theory and simulations of higher-dimensional
models~\cite{OMS2002} as well as the one dimensional driven
PCPD~\cite{PHP2005a} have shown that
the pair density does not necessarily follow the same
behavior as the density, numerical studies of the one-dimension PCPD
(for example, see Ref.~\cite{BC2003,PP2006}) show that
$\rho(t) / \phi(t)$ indeed approaches to a finite number as
$t\rightarrow \infty$ at criticality. Hence, it is expected up to the leading
correction that
\begin{equation}
\phi(t) = a_2 t^{-\delta} \left ( 1 + c_2 t^{-\chi} + o(t^{-\chi})\right ),
\label{Eq:rho_rhop}
\end{equation}
where $a_2$ and $c_2$ are constants.
Thus, the ratio $M(t)$ of $\rho(t)$ and $\phi(t)$ eliminates $t^{-\delta}$
dependence.
To describe the behavior of $M(t)$,
we made two assumptions. One is $c_1 \neq c_2$ and the other
is that the second leading correction to scaling is not
larger than $t^{-2\chi}$.
The plausibility of these two assumptions will be discussed
in Sec.~\ref{Sec:discuss}.
With these two assumptions, the ratio $M(t)$ at the
critical point becomes
\begin{equation}
M(t) \equiv \frac{\rho(t)}{\phi(t)} \sim \frac{a_1}{a_2}
\left [ 1 + (c_1-c_2) t^{-\chi}\left (1 + C t^{- \chi} \right )\right ],
\label{Eq:Roft}
\end{equation}
where $C$ is assumed to be nonzero.
This ratio has already been studied in Refs.~\cite{BC2003,SB2008} and,
albeit in a different context, also in Ref.~\cite{PP2006}.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{chieff.eps}
\caption{\label{Fig:chi} (Color online) Plots of $\chi_\textrm{eff}(t;b)$ for $b=10^{3/2}$
(top curve), $b=10^2$ (middle curve), and $b=10^{5/2}$ (bottom curve)
as functions of $t^{-0.37}$. The straight lines are resulting fitting
functions.
}
\end{figure}
Since $\delta$ does not appear in $M(t)$, we can study
$\chi$ directly using the effective exponent
\begin{equation}
\chi_\textrm{eff}(t;b) \equiv
\ln \left ( \frac{M(t/b) - M(t/b^2)}{M(t) - M(t/b)} \right ) / \ln b,
\end{equation}
where $b$ is again a time-independent constant.
From Eq.~\eqref{Eq:Roft}, the asymptotic behavior of $\chi_\textrm{eff}(t)$
at criticality is expected to be
\begin{equation}
\chi_\textrm{eff}(t;b) \sim \chi + C \frac{b^{2\chi} -1 }{\ln b} t^{-\chi}.
\label{Eq:chi_eff}
\end{equation}
Equipped with Eq.~\eqref{Eq:chi_eff} together with the accurately estimated
critical point from Ref.~\cite{PP2009}, we will find $\chi$ from the Monte
Carlo simulations.
To this end, we simulated the system with $L=2^{19}$ for several $p$'s around
the critical point (from 0.180~210 to 0.180~215) up to $t=10^7$.
Since no significant difference is observed up to
$t=10^7$ for the stated range of $p$,
we will only present the results for $p=0.180~214$ which later will be
asserted to be the critical point.
We collected data from 25~000 independent realizations. Then, the effective
exponent $\chi_\text{eff}(t;b)$ is calculated using $b=10^{3/2}$, $10^2$, and
$10^{5/2}$.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{er4.eps}
\caption{\label{Fig:delta_eff} (Color online) Plots of $-\delta_\textrm{eff}$ using
$b=100$ as functions of $t^{-\chi}$ with $\chi = 0.37$
for $p=0.180~213$ (top curve),
$0.180~214$ (middle curve), and $0.180~215$ (bottom curve).}
\end{figure}
To find $\chi$, we take the following procedure.
First, we naively set a value of $\chi$.
Then, we fit $\chi_\textrm{eff}(t;10^{5/2})$ using
Eq.~\eqref{Eq:chi_eff} with $C$ to be the only fitting parameter.
Note that in this step $\chi$ is not a fitting parameter.
We set the fitting range from $t=2\times 10^6$ to $10^7$.
After finding $C$ for given $\chi$, we graphically checked if
Eq.~\eqref{Eq:chi_eff} can fit the other effective exponents calculated
with different $b$'s.
From the above procedure, we observed the best fit when $\chi = 0.37$ is used;
see Fig.~\ref{Fig:chi}. In this case, $C$ is estimated as $\simeq 2.15$.
Since setting $\chi = 0.36$ or $0.38$ does not give a satisfactory fitting for
$\chi_\text{eff}(t;10^2)$ and $\chi_\text{eff}(t;10^{3/2})$,
(see Supplemental Material at [URL] for the results of the procedure
explained above for $\chi = 0.36$ and 0.38), we concluded $\chi = 0.37(1)$.
Here, we have used the convention that
the number in parentheses indicate the error of the
last digit and this convention will be used throughout the paper.
Using $\chi = 0.37$, we will analyze the effective exponent
$\delta_\textrm{eff}$. We present the simulation results at
$p=0.180~213$ (300 runs),
$0.180~214$ (456 runs), and $0.180~215$ (300 runs) with system
size $L=2^{21}$ and with the observation time $t=10^9$.
Up to $t=10^9$, all simulated samples contain at least one pair,
which minimally guarantees that the finite size effect is negligible.
In Fig.~\ref{Fig:delta_eff}, the effective exponents calculated with
$b=100$ are plotted as functions of $t^{-\chi}$ for different $p$'s
(see Supplemental Material at [URL] for a figure depicting
the effective exponents of pair density at different $p$'s,
which looks more or less same as Fig.~\ref{Fig:delta_eff}). The effective
exponent for
$p=0.180~213$ (0.180~215) veers up (down), which signals that the
system is in the active (absorbing) phase.
In fact, the curvature for $p=0.180~215$ is not so conspicuous, but if we
use $b=10$, one can see a clear curvature
for this case (see Supplemental Material at [URL] for the behavior of effective
exponents calculated with $b=10$). Since the graph for
$p=0.180~214$ approaches to the ordinate with finite slope,
we conclude that the critical point is $p_{cr} = 0.180~214(1)$ and
the decay exponent is $\delta = 0.185(10)$.
Our estimation of $\delta$ should be compared with the
DP value, 0.1595.
\subsection{\label{Sec:discuss}Discussion}
While deriving Eq.~\eqref{Eq:chi_eff}, we had to resort to a
few assumptions. First, we assume that the coefficients of the leading
correction to scaling
of the density and the pair density are different ($c_1\neq c_2$).
Second (provided the first assumption is valid), the next leading correction
to scaling is not larger than
$t^{-2\chi}$. We will discuss about the plausibility of
these two assumptions from a numerical perspective.
The assumption of $c_1 \neq c_2$ is actually not built on a firm
ground. We cannot fully exclude the possibility that we actually estimated
the exponent of the next leading correction.
In fact, one can easily come up with two quantities whose coefficients
of the leading correction are exactly the same.
To see this, let
us begin with writing down the time evolution equation of the density.
A straightforward calculation gives
\begin{equation}
\frac{d}{dt} \rho(t) = 2 (1-2 p) \phi(t) - 2 (1-p) \psi(t),
\label{Eq:pcpd}
\end{equation}
where $\psi(t)$ is defined as
\begin{equation}
\psi(t) = \frac{1}{L} \sum_i
\langle s_i s_{i+1} \left [ s_{i+2} + (1 - s_{i+2} ) s_{i+3}
\right ]
\rangle .
\label{Eq:deft}
\end{equation}
We drop the explicit $t$ dependence of $s_i$ in Eq.~\eqref{Eq:deft}
for convenience and
we have assumed that the system has mirror symmetry; that is,
the density of the local configuration $AA\emptyset A$ is assumed to be the same
as that of $A \emptyset AA$.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{ptall.eps}
\caption{\label{Fig:ptratio} (Color online) Log-log plot of $d(t)
\equiv (1-p_{cr})/(1-2p_{cr}) - \phi(t)/\psi(t)$ vs $t$ at $p=p_{cr}$.
A line segment whose slope is $-1$ is for a guide to the eyes.
Inset: Log-linear plot of $\phi(t)/\psi(t)$ vs $t$ at $p=p_{cr}$.
}
\end{figure}
As in Sec.~\ref{Sec:CTS}, let us assume that at the critical point
$\rho(t)$ and $\phi(t)$ behave as Eqs.~\eqref{Eq:density_critical} and
\eqref{Eq:rho_rhop}, respectively, with $\chi$ smaller than 1.
Since Eq.~\eqref{Eq:pcpd} should be consistent with
Eqs.~\eqref{Eq:density_critical}
and \eqref{Eq:rho_rhop}, $\psi(t)$ should behave up to the leading correction as
\begin{equation}
\psi(t) = \frac{1-2p_{cr}}{1-p_{cr}}a_2 t^{-\delta}
\left ( 1 + c_2 t^{-\chi} \right ),
\end{equation}
where $a_2$ and $c_2$ are the same as in Eq.~\eqref{Eq:rho_rhop}.
Actually, any corrections
larger than $t^{-1}$ should vanish on the right-hand side of
Eq.~\eqref{Eq:pcpd}. Hence, if we study the ratio of
$\phi(t)$ and $\psi(t)$, it should approach
to the asymptotic value as
\begin{equation}
\frac{\phi(t)}{\psi(t)} = \frac{1-p_{cr}}{1-2p_{cr}} \left (
1 + c_4 t^{-1} + o(t^{-1}) \right ),
\end{equation}
where $c_4$ is a constant. Note that $c_4$ should be nonzero;
otherwise, Eq.~\eqref{Eq:pcpd} does not hold. Hence
\begin{equation}
d(t) \equiv \frac{1-p_{cr}}{1-2p_{cr}} - \frac{\phi(t)}{\psi(t)}
\end{equation}
should show a $1/t$ behavior in the long time limit at criticality.
In Fig.~\ref{Fig:ptratio}, we depict $d(t)$ vs $t$ on
a double-logarithmic scale.
As expected, $d(t)$ decays as $t^{-1}$ in the long time limit.
As the example of $d(t)$ shows, it is not impossible that the leading
correction to scaling in $\rho$ and $\phi$ happens to be identical.
Although we cannot provide a theoretical argument, we think that
Fig.~\ref{Fig:delta_eff} combined with Eq.~\eqref{Eq:deltaeff} provides
a numerical evidence that the leading
correction of $M(t)$ is same as that of $\rho$.
Note that similar studies in Refs.~\cite{BC2003,SB2008} also implicitly
assume that $M(t)$ indeed contain information about the leading correction
to scaling.
Actually, Fig.~\ref{Fig:chi} also supports the assumption $c_1 \neq c_2$.
To see why, we will investigate what will be the correct asymptotic
behavior of $\chi_\textrm{eff}$ if $c_1=c_2$.
By this assumption, $\rho$ and $\phi$ at criticality should take the form
($\chi_1 < \chi_2<1$)
\begin{eqnarray}
\rho(t) = a_1 \left \{ 1 + c t^{-\chi_1} + f_1t^{-\chi_2} + o(t^{-\chi_2}) \right \},\nonumber\\
\phi(t) = a_2 \left \{ 1 + c t^{-\chi_1} + f_2t^{-\chi_2} + o(t^{-\chi_2}) \right \},
\end{eqnarray}
which gives
\begin{eqnarray}
\frac{a_2}{a_1} M(t) &\approx& \frac{1+c t^{-\chi_1} + f_1 t^{-\chi_2}}{1+c t^{-\chi_1} + f_2 t^{-\chi_2}}\nonumber\\
&\approx& 1 + (f_1-f_2) t^{-\chi_2} ( 1 - c t^{-\chi_1}).
\end{eqnarray}
If this is the case, the procedure of finding $\chi$ explained in
Sec.~\ref{Sec:CTS} would not yield a reasonable conclusion.
We can repeat the above discussion to argue that
the next correction to scaling is not stronger than $t^{-2\chi}$.
Hence, two assumptions we made for developing a method to estimate $\chi$
are supported self-consistently.
\section{\label{Sec:z} Other exponents}
\subsection{\label{Sec:ens} Three ensembles}
In general, survival probability plays an important role in the study
of absorbing phase transitions and the PCPD is no exception.
Conventionally, survival probability at time $t$ is defined as
the probability that the system has avoided falling into any of
absorbing states up to time $t$. Remind that the set of configurations
with a single particle alone is considered absorbing in the study of the PCPD.
The collection of all realizations generated by the dynamic
rules will be called
a C-ensemble (C is intended to mean `conventional') and
the survival probability at time $t$ calculated according to the conventional
definition of absorbing states is denoted by $S_c(t)$.
When we calculate the average of certain quantities over
the C-ensemble, any contribution to these quantities from any realizations
are taken into account with appropriate weight.
Although a C-ensemble is a natural starting point to study absorbing
phase transitions in many cases, the C-ensemble of the PCPD
is an inept guide to the
correct critical behavior especially in higher dimensions (discussion
about the C-ensemble in higher dimensions can be found
in Sec.~\ref{Sec:mabs}).
Even in one dimension, as we will see soon, the C-ensemble is not capable
of capturing the correct critical behavior.
Hence, it is indispensable to introduce a more restricted definition
of ``survival'' which captures the crucial feature of the PCPD.
Since a configuration without any pairs cannot change the number of
particles in the system, such a configuration can be regarded
as tentatively absorbing~\cite{NP2004,KK2007}.
Unlike the conventional absorbing states, however,
a tentatively absorbing state can be resurrected when two
particles are placed
in a row by diffusion. For this reason, we will call a configuration
without pairs a dormant state rather than an absorbing state. According to this
definition, an absorbing state in the conventional sense is one of
dormant states, although it cannot be resurrected any more.
The collection of realizations with dormant states being treated as tentatively
dead will be called the D-ensemble (D is from `dormant') and the survival
probability in the D-ensemble, to be denoted by $S_d(t)$, is defined as the
probability that the system is not in one of dormant states at time $t$.
When we calculate an average of any quantity over the D-ensemble, any
contributions from dormant states are neglected.
In particular, the mean number of particles in the D-ensemble can be
different from that in the C-ensemble because dormant states can have many
particles.
We further introduce another definition of survival by regarding dormant
states in the D-ensemble as truly dead without chance of resurrection.
This is actually equivalent to introducing a global
constraint to the PCPD in such a way that even diffusion
is not allowed once the system does not have any pairs. The PCPD
with this global constraint has now {\em infinitely many absorbing states}
and the survival probability has the conventional meaning.
We will refer to the collection of all realizations generated by dynamic rules
with the global constraint as the G-ensemble (G is from `global constraint')
and survival probability will be denoted by $S_g(t)$. Note that the
ensemble of the quasi-stationary distribution studied in
Ref.~\cite{dOD2006} is in parallel with the G-ensemble because
any state without any pairs is immediately replaced by a randomly chosen
configuration with pairs.
To make the distinction clear among averages of a random variable
over different ensembles, we will denote
an average over the C-ensemble by
$\langle \ldots \rangle_c$, over the D-ensemble by $\Dave{\ldots}$,
and over the G-ensemble by $\Gave{\ldots}$. Likewise, by a mathematical
symbol with one of the subscritps $c$, $d$, and $g$, we denote an
observable in the corresponding ensemble.
When we need to clarify which ensemble is used to calculate
critical exponents, we also add a subscript $c$, $d$, or $g$ to
appropriate mathematical symbols for critical exponents.
If the random variable $N(t)$ ($N_p(t)$) is the number of particles (pairs)
at time $t$ in the system, the survival probability of each
ensemble can be formally written as
\begin{eqnarray}
S_c(t) &=& 1 - \langle \delta_{N(t),0} \rangle_c,\nonumber\\
S_d(t) &=& 1 - \Dave{\delta_{N_p(t),0}} =
1 - \langle \delta_{N_p(t),0} \rangle_c,\nonumber\\
S_g(t) &=& 1 - \Gave{\delta_{N_p(t),0}},
\label{Eq:S}
\end{eqnarray}
where $\delta$ is the Kronecker delta symbol (not the critical exponent).
It is obvious by definition that $S_c(t) \ge S_d(t) \ge S_g(t)$,
$\langle N(t) \rangle_c \ge \Dave{N(t)} \ge \Gave{N(t)}$, and
$\langle N_p(t) \rangle_c = \Dave{N_p(t)} \ge \Gave{N_p(t)}$
if all parameters ($L$ and $p$) as well as the initial condition are the same
for all ensembles.
Since these three ensembles can be different only if
a system has non-zero probability to fall into a dormant state,
the conclusion in Sec.~\ref{Sec:results} does not
depend on which ensemble is used, for $S_c(t) = S_d(t) = S_g(t) =1 $
at finite $t$ in the infinite size limit.
\subsection{\label{Sec:FSS}Finite size scaling}
This section studies the finite size scaling (FSS) to estimate the
dynamic exponent.
At the critical point $p_{cr} = 0.180~214$,
the particle density $\rho$ and the pair density $\phi$ are
expected to follow the scaling form
\begin{equation}
\rho_x(t,L) = t^{-\delta}f_x(t/L^{z_x}),\,\,\,
\phi_x(t,L) = t^{-\delta}g_x(t/L^{z_x}),
\label{Eq:FSS}
\end{equation}
where $x$ is one of $c$, $d$, and $g$ depending on the ensemble under
consideration, $z_x$ is the dynamic exponent of the corresponding
ensemble, and $f_x(y)$ and $g_x(y)$ are scaling
functions which decay exponentially when $y \gg 1$.
Note that $\phi_c(t,L) = \phi_d(t,L)$ though $\rho_c(t,L)$ is not
necessarily equal to $\rho_d(t,L)$.
We first analyze the scaling collapse for $\phi_c(t,L) = \phi_d(t,L)$.
If we plot $\phi_d(t,L) t^\delta$ as a function
of $t/L^z$ at the critical point
with the correct exponents $\delta$ and $z$, it is
expected that curves for different system size will collapse into a
single curve in the scaling regime.
Using $\delta = 0.185$ obtained in Sec.~\ref{Sec:results} and varying
$z$, we observed a best collapse when we use $z = 1.65$;
see Fig.~\ref{Fig:rpp} which depicts a scaling collapse of data from
simulation results with the number of samples for $L=2^{14}$,
$2^{15}, 2^{16}, 2^{17}$, and $2^{18}$ to be $8\times 10^5$, $1.7\times 10^5$,
$8\times 10^4$, $3000$, and $800$, respectively.
If we calculate the average density over survived samples, the density is expected
to saturate to a finite value when $t \gg L^z$ and the saturating density
is expected to show a scaling $L^{-\delta z}$ in the asymptotic regime.
Indeed, as shown in Inset of Fig.~\ref{Fig:rpp},
plots of $\phi_c^S(t,L) L^{\delta z}$ vs $t/L^z$ with the same
exponents as above show a nice scaling collapse.
Here, $\phi_d^S$ is the pair density averaged over survived
samples in the D-ensemble and it is equivalent to $\phi_d^S\equiv \phi_d/S_d$.
In the following, we will add superscript $S$ as in $\phi_d^S$ whenever
we need to clarify that a quantity is averaged over survived samples of the
ensemble in question.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{fssrpp.eps}
\caption{\label{Fig:rpp} (Color online) Log-log plots of $\phi_d (t,L) t^\delta$ vs $t/L^z$
with $\delta = 0.185$ and $z = 1.65$ for $L =2^{14}, 2^{15}, 2^{16}, 2^{17}$,
and $2^{18}$ at criticality. Inset: Scaling collapse of
$\phi_d^S$ with
$\delta = 0.185$ and $z=1.65$ on a double-logarithmic scale.
}
\end{figure}
However, $\rho_c^{S} \equiv \rho_c / S_c$ does not show such a nice scaling
behavior though scaling collapse for $\rho_c$ is nice.
Figure~\ref{Fig:fssrho} shows that $\rho_c t^\delta$ for different values of
$L$ collapses into a single curve when it is drawn as a function of $t/L^z$.
On the other hand, $\rho_c^S L^{\delta z}$ does not collapse
into a single curve when $t/L^z \gg 1$, though in the region of
$t/L^z \le 1$ where $S_c(t) \approx 1$ the scaling collapse is still nice.
This is one of the reasons why we said that the C-ensemble is an awkward
guide to the correct critical behavior. More detailed discussion about the
clumsiness of the C-ensemble is found in Sec.~\ref{Sec:mabs}.
Now we will move on to the FSS in the G-ensemble.
Unlike the D-ensemble,
simulation results for $L=2^{14}, 2^{15}, 2^{16}$, and $2^{17}$ show
a best collapse when we use $z_g=2$ rather than $z_d = 1.65$;
see Fig.~\ref{Fig:sm}.
As we explained before, the G-ensemble was used in
Ref.~\cite{dOD2006} in the context of quasi-stationary state. Our finding
$z_g=2$ for the G-ensemble is indeed consistent with the result
in Ref.~\cite{dOD2006} for slow diffusion.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{fssrho.eps}
\caption{\label{Fig:fssrho} (Color online) Double-logarithmic plots of $\rho_c (t,L) t^\delta$ vs $t/L^z$
with $\delta = 0.185$ and $z = 1.65$ for $L =2^{14}, 2^{15}, 2^{16}$, and
$2^{17}$ at criticality. Inset: Plots of
$\rho_c^S L^{\delta z}$ vs $t/L^z$ with
$\delta = 0.185$ and $z=1.65$ on a double-logarithmic scale.
The saturating value does not show a scaling behavior although
the scaling collapse is nice up to $t \approx L^z$.
}
\end{figure}
\begin{figure}[b]
\includegraphics[width=\columnwidth]{fsssm.eps}
\caption{\label{Fig:sm} (Color online) Double-logarithmic plots of $\phi_g (t,L) t^\delta$ vs $t/L^2$
with $\delta = 0.185$ for $L =2^{14}, 2^{15}, 2^{16}$, and $2^{17}$,
at the critical point. Inset: Scaling collapse for $\phi_g$
using $\delta = 0.185$ and $z= 1.8$ for $L=2^{18}$ and $2^{19}$ on
a double-logarithmic scale.
}
\end{figure}
Since the G-ensemble introduces a global constraint to the PCPD, it might be
possible that the dynamic exponent may differ depending
on ensembles. However, it is impossible that $z_g$ (the dynamics exponent
of the G-ensemble) is larger than $z_d$ (the dynamics exponent
of the D-ensemble), because $\phi_d(t)$ cannot be smaller than $\phi_g(t)$
by definition. To understand the reason why the inequality $z_g > z_d$ is wrong,
let us look at Figs.~\ref{Fig:rpp} and \ref{Fig:sm}.
From these figures, we can estimate the characteristic time scales
as $T_d \approx L^{1.65}$ for the D-ensemble and as $T_g \approx 5 \times 10^{-3} L^2$ for the G-ensemble, assuming
that $z_d=1.65$ and $z_g=2$ are true dynamic exponents of each ensemble.
Up to the system size we simulated, $T_g$ is smaller than $T_d$, but
if $L \gg 4 \times 10^6$, $T_g$ becomes much larger than $T_d$.
If this happens for some $L$, in the regime where $T_d \ll t \ll T_g$
$\phi_d(t)$ decays exponentially while $\phi_g(t)$ decays algebraically.
Hence, at some
time $\phi_g(t)$ becomes larger than
$\phi_d(t)$, which is contradictory to the definition.
Hence, $z_g$ should not be larger than $z_d$.
Indeed, as Inset of Fig.~\ref{Fig:sm} shows, the estimated
value of the dynamic exponent $z_g$
becomes smaller as the system size gets larger.
This observation also suggests that the dynamic exponent obtained
in Ref.~\cite{dOD2006} for small diffusion constant
is not the correct one and investigating systems with larger
size will yield smaller value of $z$ (the largest system size studied
in Ref.~\cite{dOD2006} is 40~960 which is smaller than $2^{16}$).
Since the FSS of the G-ensemble shows an inconsistent results, we
cannot make a firm conclusion concerning the dynamic exponent in this section.
Besides, it is hard to analyze the dynamic exponent systematically within
the FSS scheme.
In the next section, we study microscopic absorbing phase transitions with
a systematic analysis of effective exponents.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{NcNd.eps}
\caption{\label{Fig:n_correct} (Color online) Double-logarithmic plot of $n_c/n_d -1$ vs $t$
at the critical point. The straight line whose slope is $-0.37$ is drawn
for a guide to the eyes.
}
\end{figure}
\subsection{\label{Sec:mabs}Microscopic absorbing phase transition}
Unlike the studies in the previous sections, the system size is now set to
infinity and there are only finitely many particles in the system.
Although macroscopic quantities such as $\rho$ and $\phi$ is zero by definition,
microscopic quantities such as the number of particles, the survival
probability, and so on can exhibit a phase transition. Since
only microscopic quantities are nonzero, this phase transition
is termed as a microscopic absorbing phase transition~\cite{PP2008EPJB}.
Initially, the system has only two particles at two consecutive sites and all other
sites are vacant. During simulations, we measured the number of particles
($N(t)$), the number of pairs ($N_p(t)$), the distance between two most
distant particles ($D_s(t)$), and the distance between two most distant
pairs ($D_p(t)$).
For the C- and D-ensembles, we simulated $10^5$ independent runs
and for the G-ensembles, the number of independent runs is $10^{15}$.
For all cases, the observation time is set to be $10^9$ and the
simulations are run at the critical point $p = 0.180~214$.
For the survival probabilities, we use the same definition as in
Eq.~\eqref{Eq:S}. Likewise, we define average quantities of observables such
that
\begin{eqnarray}
n_x &\equiv& \ave{N(t)},\quad p_x \equiv\ave{N_p(t)},
\label{Eq:dynamic_scaling}
\\
R_x &\equiv& \left (\frac{\ave{D_s(t)^2}}{S_x(t)} \right )^{1/2},\,
R^{P}_x \equiv \left (\frac{\ave{D_p(t)^2}}{S_x(t)} \right )^{1/2},\nonumber
\end{eqnarray}
where $x$ can be one of $c$, $d$, and $g$, depending on which
ensemble is used.
At criticality, we expect power-law behavior
($x \in \{c,d,g\}$)
\begin{eqnarray}
n_x \sim p_x \sim t^{\eta_x}, S_x \sim t^{-\delta_x'},
R_x \sim R^{P}_x \sim t^{1/z_x}
\label{Eq:hdz}
\end{eqnarray}
with critical exponents $\eta_x$, $\delta_x'$, and $z_x$.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{etaC.eps}
\caption{\label{Fig:etaC} (Color online) Plots of effective
exponents $\eta_\text{eff}$ for $n_c$, $n_d$, and $p_c=p_d$
as functions of $t^{-0.37}$ at criticality.
For comparison, the corresponding exponent of the DP class $\eta_\text{DP}
= 0.313$ is also drawn.
}
\end{figure}
Let us begin with the analysis of $n$ and $p$ in the C- and D-ensembles.
As in Sec.~\ref{Sec:results}, we first determine the strength of a
leading correction to scaling. It turns out that $n_c / n_d$ approaches
to 1 as $t\rightarrow \infty$, which reflects that main contribution to
$n_c$ in the long time limit is from configurations with pairs.
In Fig.~\ref{Fig:n_correct}, we depict $n_c/n_d-1$ as a function of $t$
on a double-logarithmic scale. As in Sec.~\ref{Sec:results}, we found
that the leading correction to scaling is $t^{-0.37}$.
Having this leading correction to scaling $t^{-0.37}$ in mind, we depict
effective exponents obtained from $n_c$, $n_d$, and $p_c (=p_d)$
as functions of $t^{-0.37}$ in Fig.~\ref{Fig:etaC}. We use $b=100$ to
calculate the effective exponents. Effective exponents for $n_c$ and
$p_c = p_d$ are hardly
discernible. All effective exponents approach to
$0.285 \pm 0.01$ with a finite slope.
Hence we conclude that $\eta_c = \eta_d = 0.285(10)$
which is different from $\eta_\text{DP} \approx 0.313$.
In Fig.~\ref{Fig:deltaC}, effective exponents calculated from $S_c$ and $S_d$
are depicted as functions of $t^{-0.37}$. From the behavior of the effective
exponents we conclude that $\delta_c' = 0.125(5)$ and $\delta_d'=0.145(5)$
which are different from the corresponding exponent of the
DP class $\delta_\text{DP}' = 0.1595$. Recall that
$S_c \sim t^{-\delta'_c}$ and $S_d \sim t^{-\delta'_d}$.
Also we observe that $\delta' \neq \delta$, which implies that
the PCPD does not have
the time reversal symmetry or rapidity symmetry which is the characteristic
feature of the DP class.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{deltaC.eps}
\caption{\label{Fig:deltaC} (Color online) Plots of effective
exponents $-\delta_\text{eff}'$ for $S_c$ and $S_d$
as functions of $t^{-0.37}$ at criticality.
For comparison, the corresponding exponent of the DP class $-\delta_\text{DP}'
= -0.1595$ is also drawn.
}
\end{figure}
At first sight, it seems surprising that $S_c$ and $S_d$ show different
power-law behavior. However, this discrepancy is actually
the inherent property of the PCPD (and any PCPD-like models such as one studied
in Sec.~\ref{Sec:tp12}), which is easily understandable when considering the
higher dimensional system. Since random walks are not recurrent in more than two
dimensions, that is, it happens with nonzero probability that two random
walkers never encounter each other forever in higher dimensions,
$S_c$ even for $p=1$ (pure annihilation without branching) is nonzero even in
the $t\rightarrow \infty$ limit.
In this regard, $S_c$ suggests that there is no phase transition in higher dimensions
which is incompatible with the mean-field theory.
On the other hand, such non-recurrent realizations in the D-ensemble
become effectively absorbing and do not contribute to $S_d$.
Thus at the critical point, $S_d$ should decay to zero in a power-law fashion
though $S_c$ saturates to a nonzero value.
In other words, the higher dimensional PCPD has the property that
$\delta_c' = 0$ but $\delta_d' \neq 0$.
In one dimension, of course, random walks are recurrent and two walkers should
eventually meet each other. When a dormant state wakes up by diffusion, it restarts the
dynamics effectively from a single pair. However, it again becomes dormant
with high probability in short time, so has little contribution to $S_d$
as well as to $n_d$. This is consistent with the numerical observation that
$n_c/n_d \rightarrow 1$ though $S_c / S_d \rightarrow \infty$.
That is, even in one dimension, the dormant state is effectively absorbing.
However, one should note that this argument does not imply
that isolated particles in a non-dormant state are effectively non-moving.
If so, the PCPD in any dimensions, according
to Hinrichsen's argument~\cite{H2006}, should belong to the DP class
just like the pair contact process without diffusion~\cite{J1993}, which
is not true in two and higher dimensions.
We will further discuss about the C-ensemble after investigating
$R_d$ and $R_d^P$.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{zpC.eps}
\caption{\label{Fig:zC} (Color online) Plots of effective
exponents $1/z_\text{eff}$ for $R_d$ and $R_d^P$
as functions of $t^{-0.37}$ at criticality.
For comparison, the corresponding exponent of the DP class $1/z_\text{DP}
= 0.6325$ is also drawn.
}
\end{figure}
The effective exponents related to $R_d$ and $R_d^P$ are depicted
in Fig.~\ref{Fig:zC} which shows that both
effective exponents approach to the same value
as $t^{-0.37}\rightarrow 0$. Hence we conclude that
$1/z_d = 0.610(5)$ or $z_d = 1.639(14)$ which is consistent with $z$ obtained
from the FSS study in Sec.~\ref{Sec:FSS} for the D-ensemble.
If we plot the effective exponents calculated from $R_c$ and $R_c^P$, we get
$1/z_c = 0.60$ rather than 0.61; see Supplemental Material at [URL].
This difference is simply originated from the fact
that $\delta_c'$ differs from $\delta_d'$.
Since $\delta_d' - \delta_c' \approx 0.02$ and
$\langle D_p(t)^2 \rangle_c =\Dave{ D_p(t)^2}$ by definition,
$1/z_c$ becomes smaller than $1/z_d$ by 0.01;
see Eq.~\eqref{Eq:dynamic_scaling}.
In fact, $z_c$ is an ill-defined quantity.
To clarify the meaning of the above sentence, let us introduce another
averaged quantities,
\begin{equation}
r_x = \frac{\langle D_s(t) \rangle_x}{S_x(t)},\quad
r_x^P = \frac{\langle D_p(t) \rangle_x}{S_x(t)},
\end{equation}
where $x$ is one of $c$, $d$, and $g$.
If the dynamic exponent is well-defined, we expect that
\begin{equation}
r_x\sim R_x \sim r_x^P \sim R_x^P \sim t^{1/z_x}.
\end{equation}
For the D-ensemble the above relation indeed holds, but
for the C-ensemble power-law behavior of $r_c$ differs from $R_c$;
see Supplemental Material at [URL].
Hence, $z_c$ obtained from the definition in
Eq.~\eqref{Eq:dynamic_scaling} cannot properly describe the critical behavior
of the PCPD and we conclude that $z_d = 1.639(14)$ is the correct
dynamic exponent of the PCPD.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{Dhyper.eps}
\caption{\label{Fig:Dhyper} (Color online) Plot of
effective exponent for $p_d/(R_d S_d)$ vs $t^{-0.37}$.
As $t^{-0.37} \rightarrow 0$, the effective exponent approaches
to $-\delta = -0.185$ within error bars, which is consistent with the hyperscaling relation.
}
\end{figure}
\begin{figure}[b]
\includegraphics[width=\columnwidth]{npd.eps}
\caption{\label{Fig:npd} (Color online) Plots of
$n_g$, $p_g$, and $S_g$
as functions of $t$ at criticality on a double-logarithmic scale.
Two line segments with slopes $-0.1$ (upper segment) and $-0.5$ (lower segment)
are drawn for a guide to the eyes. Inset: Plot of
$\chi_\text{eff}$ vs $t^{-0.37}$ on a linear scale.
$\chi_\text{eff}$ is calculated from
$n_g/p_g$ with $b=10^3$.
}
\end{figure}
In Fig.~\ref{Fig:Dhyper}, we plot $\eta_\text{eff} + \delta_\text{eff} -
1/z_\text{eff}$ as a function of $t^{-0.37}$. According to the hyperscaling
relation which is derived from
\begin{equation}
\frac{p_d}{R_d S_d} \sim \phi \sim t^{-\delta},
\end{equation}
where $\phi$ is the pair density in the thermodynamic limit,
the above combination of effective exponents should approach to
$-\delta = -0.185$. Indeed, as Fig.~\ref{Fig:Dhyper} show,
the hyperscaling relation holds.
Now we will analyze the critical behavior within the G-ensemble scheme.
At first, we illustrate the behavior of $n_g$, $p_g$, and $S_g$ with time
in Fig.~\ref{Fig:npd}. Up to $t=10^9$, $n_g$ and $p_g$ decrease with time,
which seems to suggest that $\eta_g$ is negative.
To be more systematic, we also calculated the effective exponents for $\eta_g$,
$\delta_g'$, and $z_g$. To determine the leading correction to scaling,
we analyzed $n_g/p_g$ as in Sec.~\ref{Sec:CTS}. To calculate
$\chi_\text{eff}$, we use $b= 10^3$.
Inset of Fig.~\ref{Fig:npd} depicts the resulting behavior of $\chi_\text{eff}$
as a function of $t^{-0.37}$. Although the behavior of $\chi_\text{eff}$ is
quite noisy, it is still consistent with the previous correction to scaling,
$t^{-0.37}$. Furthermore, as we show in Supplemental Material at [URL],
$n_g/p_g$ shows the similar behavior to $\rho/\phi$ for sufficiently
large $t$, which again supports that $\chi$ indeed is 0.37.
Hence, we will use $\chi = 0.37$ to analyze the effective exponents for
the G-ensemble.
Effective exponents for $-\eta_g$ (note the minus sign)
and $\delta_g'$ as functions of $t^{-0.37}$ are plotted in Fig.~\ref{Fig:eG}
and for $1/z_g$ in Fig.~\ref{Fig:zG}. A naive fitting of the region
$t^{-0.37} \le 0.001$ gives $\eta_g \approx -0.05(5)$, $\delta_g' \approx
0.45(5)$, and $1/z_g = 0.59(2)$ or $z_g = 1.69(6)$.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{etaG.eps}
\caption{\label{Fig:eG} (Color online) Plots of
effective exponents for $-\eta_g$ and $\delta_g'$
calculated from $p_g$ and
$S_g$, respectively, as functions of $t^{-0.37}$ at criticality.
The straight lines are obtained by a linear fitting of the
region $t^{-0.37} \le 0.001$ for the corresponding effective exponents.
}
\end{figure}
As we have seen in Sec.~\ref{Sec:FSS}, the G-ensemble may have larger
corrections to scaling than the D-ensemble. Note that the FSS within
the D-ensemble scheme gives $z\approx 1.65$ which is compatible with
$z_d = 1.639(14)$, whereas the FSS within the G-ensemble
up to $L = 2^{19}$ does not give any conclusion about the value of
the dynamic exponent. Furthermore, the extrapolated values
of $\eta_g$, $\delta_g'$, and $z_g$ are quite far away from the
closest value of the corresponding effective exponents.
Hence, the accuracy of numerical value of $\eta_g$,
$\delta_g'$, and $z_g'$ might be questionable and more elaborated investigation
of the G-ensemble is still desired.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{zG.eps}
\caption{\label{Fig:zG} (Color online) Plots of
$1/z_\text{eff}$ for $R_g$ and $R_g^P$ as functions of $t^{-0.37}$.
The straight lines are obtained by a linear fitting of the
region $t^{-0.37} \le 0.001$ of the data from $R_g$.
}
\end{figure}
\subsection{\label{Sec:tp12}Comparison to a DP model}
In this section, we will investigate how three different ensembles
affect the critical behavior of a PCPD-like model which belongs
to the DP class. By a PCPD-like model is meant a model
in which isolated particles can only diffuse and only multiplets (pairs,
triplets, and so on) can increase/decrease the total number of particles
in the system. To be specific, we study a model with dynamics
\begin{eqnarray}
A\emptyset \stackrel{1/2}{\longleftrightarrow} \emptyset A,\quad
AA \stackrel{p}{\longrightarrow} \emptyset \emptyset,\nonumber\\
AAA \emptyset\stackrel{\sigma}{\longrightarrow} AAAA,\quad
\emptyset AAA \stackrel{\sigma}{\longrightarrow} AAAA,
\label{Eq:tp12}
\end{eqnarray}
where $\sigma =(1-p)/2$. It is well established that in one dimension
this model which will be called \texttt{tp12} following Ref.~\cite{KC2003}
does belong to the DP class. From the critical decay of
the density in the infinite system size limit, we estimated
the critical point to be $p_{cr} = 0.059~0295(5)$ (data not shown).
As in the PCPD, we introduce three different ensembles,
C-, D-, and G-ensembles. For the D-ensemble, we consider two more
cases; in one case, a state without pairs is regarded as dormant (Dp-ensemble)
and in the other case a triplet-less state is considered dormant
(Dt-ensemble). As we will see later, however, these two cases does not
make any difference in terms of critical phenomena.
For the G-ensemble, any states without triplets are regarded as absorbing.
The corresponding survival probabilities are denoted by the same symbols
as in Sec.~\ref{Sec:mabs}. In case the distinction between
the Dp- and Dt-ensembles is necessary, we add superscripts $p$ and $t$
for the Dp-ensemble and Dt-ensemble, respectively.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{nc12.eps}
\caption{\label{Fig:nc12} (Color online) Double-logarithmic plot of $n_c/n_d^p -1$ vs $t$
for the \texttt{tp12}\xspace at the critical point. A line segment whose slope is $-0.33$
is drawn for a guide to the eyes. Inset: Log-log plot of $n_d^p/n_d^t - 1$ as a function of $t$. A line segment whose slope is $-1.33$
is drawn for a guide to the eyes.
}
\end{figure}
In our numerical study,
the number of independent runs for
the C- and D-ensembles is $3\times 10^6$ and that for the G-ensemble is
$2 \times 10^8$. The observation time is set $10^7$ for all ensembles.
During simulations, we measured the number of particles ($N(t)$), pairs
($N_p(t)$), and triples ($N_t(t)$)
as well as the distance between two most distant particles ($D_s(t)$), pairs
($D_p(t)$), and triplets ($D_t(t)$).
Similar to Eq.~\eqref{Eq:dynamic_scaling}, average quantities are defined
as
\begin{eqnarray}
n_x &\equiv& \ave{N(t)},\quad p_x \equiv\ave{N_p(t)},
t_x \equiv \ave{N_t(t)}
\label{Eq:tp12_obs}
\\
R_x &\equiv& \left (\frac{\ave{D_s(t)^2}}{S_x(t)} \right )^{1/2},\,
R^{P}_x \equiv \left (\frac{\ave{D_p(t)^2}}{S_x(t)} \right )^{1/2},\nonumber\\
R^{T}_x &\equiv& \left (\frac{\ave{D_t(t)^2}}{S_x(t)} \right )^{1/2},\nonumber
\end{eqnarray}
where $x$ can be one of $c$, $d$, and $g$, depending on which
ensemble is under consideration.
Critical exponents are denoted by the same symbols as in Eq.~\eqref{Eq:hdz}.
At first, we determine the strength of the leading correction
to scaling by analyzing $n_c/n_d^p - 1$, where $n_d^p$ is the average of
$N(t)$ over the Dp-ensemble. Figure~\ref{Fig:nc12} shows that the leading
correction to the scaling is $t^{-0.33}$. Somewhat interestingly,
the leading corrections to scaling of the \texttt{tp12}\xspace is comparable to that of
the PCPD, though the universality class to which the \texttt{tp12}\xspace belongs
has unequivocally been accepted to be the DP class.
To investigate how different the Dp-ensemble is from the Dt-ensemble,
we also analyzed how $n_d^p/n_d^t-1$ behaves in the long time limit,
where $n_d^t$ is the average of $N(t)$ over the Dt-ensemble.
As illustrated in Inset of Fig.~\ref{Fig:nc12},
we found that $n_d^p/n_d^t - 1 \sim t^{-1.33}$, which shows
that $n_d^p$ and $n_d^t$ are exactly the same up to the leading
correction to scaling. In other words, the difference
between the Dp-ensemble and Dt-ensemble is negligible as far as
the critical behavior is concerned. On this account,
by the D-ensemble is always meant the Dt-ensemble in what follows.
In Fig.~\ref{Fig:h12}, we plot $\eta_\text{eff}$ vs $t^{-0.33}$
for $n_c$, $n_d$, and $t_c = t_d$. As expected, $\eta_\text{eff}$
hovers around $\eta_\text{DP}$. We omit the data
for the mean number of pairs because the relation $n_c \ge p_c \ge t_c$
implies that the
effective exponents related to the mean number of pairs should
show the same asymptotic behavior as in Fig.~\ref{Fig:h12}.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{h12.eps}
\caption{\label{Fig:h12} (Color online) Plots of $\eta_\text{eff}$ vs $t^{-0.33}$
for $n_c$, $n_d$, and $t_c=t_d$. The flat line indicates $\eta_\text{DP}=0.313$.
Although $\eta_\text{eff}$ slightly deviates from $\eta_\text{DP}$ for
large $t$, we think this is due to the error of the critical point.
}
\end{figure}
\begin{figure}[b]
\includegraphics[width=\columnwidth]{d12.eps}
\caption{\label{Fig:d12} (Color online) Plots of $-\delta_\text{eff}'$ vs $t^{-0.33}$
for $S_c$, $S_d$. The flat line indicates $-\delta_\text{DP}'=-0.1595$.
As in the PCPD, $\delta_c'$ is larger than $\delta_d'$.
}
\end{figure}
If we plot the effective exponents for $\delta'$, we again observed that
$\delta_c'$ is larger than $\delta_d'$ just like the PCPD; see
Fig.~\ref{Fig:d12}. That is, the strict inequality
$\delta_c' < \delta_d'$ seems to hold for all absorbing phase
transition models in which an isolated particle cannot change the number
of particles in the system. Also note that the argument of the non-recurrent
nature of random walks in higher dimension is still applicable to the \texttt{tp12}\xspace.
Furthermore, the fact that the relation
$\delta_c' < \delta_d' = \delta_\text{DP}$ holds in the \texttt{tp12}\xspace
supports that the correct critical behavior is described by $\delta_d'$
not $\delta_c'$.
Dynamic exponents are obtained by analyzing $R_d$, $R_d^P$, and $R_d^T$;
see Fig.~\ref{Fig:z12}. As expected, the dynamic exponent of the \texttt{tp12}\xspace
in the D-ensemble is the same as $z_\text{DP}$. Within the C-ensemble,
the dynamic exponent could be erroneously measured to be different from
$z_\text{DP}$ because the survival probability $S_c$ contains misleading
information.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{z12.eps}
\caption{\label{Fig:z12} (Color online) Plots of $1/z_\text{eff}$ vs $t^{-0.33}$
for $R_d$, $R_d^P$, and $R_d^T$.
The flat line indicates $1/z_\text{DP}=0.6325$.
}
\end{figure}
\begin{figure}[b]
\includegraphics[width=\columnwidth]{Gall.eps}
\caption{\label{Fig:Gall} (Color online) Plots of effective exponents for $1/z_g$, $\delta_g'$,
and $\eta_g$ as functions of $t^{-0.25}$. For comparison, corresponding
DP exponents are shown.
}
\end{figure}
Finally, we study the \texttt{tp12}\xspace in the G-ensemble, assuming that
the leading correction to scaling is same as in the D-ensemble.
In Fig.~\ref{Fig:Gall}, we draw three effective exponents for $1/z_g$,
$\delta_g'$ and $\eta_g$ which are calculated from $R_g^T$, $S_g$, and $t_g$,
respectively, as functions of $t^{-0.33}$.
All effective exponents approach to the corresponding DP critical
exponents as $t^{-0.33} \rightarrow 0$. Hence,
we conclude that different ensembles except the misleading C-ensemble does not
affect the critical behavior in the DP class, which reflects the time-reversal
symmetry or rapidity symmetry of the DP in the asymptotic regime.
\section{\label{Sec:sum}Summary}
To summarize,
we studied the critical behavior of the pair contact process with diffusion
and estimated critical exponents by investigating effective exponents.
In the first part of the paper, we analyzed how macroscopic quantities such
as density and pair density decay with time at the critical point. To this end,
we developed a systematic method to calculate the leading correction to
scaling from Monte Carlo simulations, which gives that the leading correction
to scaling of the PCPD is $t^{-0.37}$. With the aid of the
correction to scaling, we estimated the critical decay exponent $\delta$ by
investigating the effective exponents. We found that $\delta = 0.185(10)$ which
should be compared with the corresponding value of the DP class, 0.1595.
In the second part of the paper, we studied the finite size scaling as well as
microscopic absorbing phase transitions to estimate other exponents of the
PCPD. At first, we introduced three different ensembles called
C-, D-, and G-ensembles, respectively. It turned out that the C-ensemble cannot
describe the correct critical behavior. From the finite size scaling analysis,
we have seen that the D-ensemble and G-ensemble up to system size $2^{19}$ do
not give a consistent value of the dynamic exponent $z$. From the analysis
of microscopic absorbing phase transitions, we found that the D-ensemble
gives $\eta = 0.285 (10)$, $\delta'=0.145(10)$, and $z = 1.639(14)$ and
the G-ensemble gives $\eta \approx -0.05(5)$, $\delta' \approx 0.45(5)$, and
$z \approx 1.69(6)$. For comparison, we also studied a PCPD-like model
belonging to the DP class and found that all exponents are consistent
with the DP values regardless of which ensemble (except
the misleading C-ensemble) is under consideration.
All our numerical analyses support that the PCPD does not belong to the
DP class.
\begin{acknowledgments}
This work was supported by the Basic Science Research Program
through the National Research Foundation of Korea
(NRF) funded by the Ministry of Education, Science
and Technology (Grant No. 2011-0014680);
and by the Catholic university of Korea, Research Fund, 2011.
The author acknowledges the hospitality of the Institut f\"ur Theoretische
Physik, Universit\"at zu K\"oln, Germany and support under DFG grant within
SFB 680 {\it Molecular Basis of Evolutionary Innovations}
during the final stages of this work. The computation of this work was
supported by Universit\"at zu K\"oln.
\end{acknowledgments}
\bibliographystyle{apsrev}
|
1,941,325,220,519 | arxiv | \section{Introduction}
Given a $k$-uniform hypergraph (or $k$-graph) $H$, the \emph{Tur\'an function} (or \emph{extremal number}) $\textup{ex}(n,H)$ is the maximum number of edges in an $H$-free $n$-vertex $k$-uniform hypergraph. The \emph{Tur\'an density} of $H$, denote by $\pi(H)$, is the scaled limit
\[
\pi(H)=\lim_{n\to \infty} \frac{\textup{ex}(n,H)}{ \binom{n}{k}}.
\]
Determining these numbers is a central problem in extremal combinatorics. For graphs ($k=2$), this question is well-explored. The Erd\H{o}s-Stone theorem \cite{ErdosStone,ErdosSimonovits} asymptotically determines the Tur\'an density for graphs with chromatic number at least three. For hypergraphs, determining the Tur\'an density is notoriously difficult; very few exact results are known. For example, the Tur\'an density of the innocent looking \emph{tetrahedron} $K_4^3$, the complete $3$-uniform hypergraph on $4$ vertices, is unknown.
In order to get a better understanding of these problems, various different kinds of extrema\-li\-ty such as the generalized Tur\'an function or the minimum codegree threshold have been studied. We~\cite{BalCleLidmain} recently introduced a new type of extremality for hypergraphs and solved the tetrahedron problem asymptotically for this notion. Here, we will systematically study extremal problems regarding this function. \\
Let $G$ be an $n$-vertex $k$-uniform hypergraph. For a vertex set $T\subset V(G)$, the \emph{codegree} of $T$, denoted by $d_G(T)$, is the number of edges in $G$ containing $T$. We drop the index if $G$ is clear from the context.
The \emph{codegree vector} of $G$ is the vector $X \in \mathbb{Z}^{\binom{V(G)}{k-1}}$, where $X(v_1,v_2,\ldots,v_{k-1}) = d_G(v_1,v_2,\ldots,v_{k-1})$ for all $\{v_1,v_2,\ldots,v_{k-1}\} \in \binom{V(G)}{k-1}$.
Finding $\textup{ex}(n,H)$ is equivalent to determining the maximum $\ell_1$-norm of the codegree vector of an $H$-free $n$-vertex $k$-uniform hypergraph. Here, we study maximality with respect to the $\ell_2$-norm of the codegree vector. The \emph{codegree squared sum} $\textup{co}_2(G)$ is the sum of codegrees squared over all $k-1$ sets $T$, i.e.,
\begin{align*}
\textup{co}_2(G)=\sum_{\substack{T \in \binom{[n]}{k-1}} }d_G^2(T).
\end{align*}
In other words, the codegree squared sum is the square of the $\ell_2$-norm of the codegree vector.
\begin{ques}\textup{\cite{BalCleLidmain}}
\label{questionco2}
Given a $k$-uniform hypergraph $H$, what is the maximum $\ell_2$-norm of the codegree vector of a $k$-uniform $H$-free $n$-vertex hypergraph $G$?
\end{ques}
We follow the notation introduced in \cite{BalCleLidmain}. Let $\mathcal{F}$ be a family of $k$-uniform hypergraphs. Denote by $\textup{exco}_2(n,\mathcal{F})$ the maximum codegree squared sum among all $k$-uniform $n$-vertex $\mathcal{F}$-free hypergraphs, and let the codegree squared density $\sigma(\mathcal{F})$ be its scaled limit, i.e.,
\begin{align}
\label{sigmalimit}
\textup{exco}_2(n,\mathcal{F})= \smash{\displaystyle\max_{G \text{ is } \mathcal{F} \text{-free}}} \textup{co}_2(G) \quad \quad \text{ and } \quad \quad
\sigma(\mathcal{F})&=\limsup\limits_{n \rightarrow \infty} \frac{\textup{exco}_2(n,\mathcal{F})}{\binom{n}{k-1}(n-k+1)^2}.
\end{align}
We~\cite{BalCleLidmain} proved general properties of $\sigma$ including the existency of the limit in \eqref{sigmalimit}. In Table~\ref{tab:exactcod2} we present bounds and exact values for the codegree squared density of various hypergraphs.
Table~\ref{tab:pic} provides the definitions with pictures of all hypergraphs included in Table~\ref{tab:exactcod2}. Unless otherwise mentioned, all upper bounds on $\sigma$ in this table were obtained using Razborov's flag algebra machinery \cite{flagsRaz}. It is a standard application of flag algebras to obtain these results; we give a short explanation of it in Section~\ref{flagresults}. Table~\ref{tab:exactcod2} also gives an overview of known results for the Tur\'an density, the minimum codegree threshold and the uniform Tur\'an density. Denote by $\delta(G)$ the minimum $(k-1)$-codegree of a $k$-graph $G$. For a family of $k$-graphs $\mathcal{F}$, the \emph{minimum codegree Tur\'an number} $\textup{ex}_{k-1}(n,\mathcal{F})$ is the maximum $\delta(G)$ over all $\mathcal{F}$-free $k$-graphs $G$ on $n$ vertices. The \emph{minimum codegree threshold},
\[
\pi_{k-1}(\mathcal{F})= \lim_{n \to \infty} \frac{\textup{ex}_{k-1}(n,\mathcal{F})}{ \binom{n}{k-2}},
\]
is its scaled limit. \\
Reiher, R\"{o}dl and Schacht~\cite{MR3764068} recently introduced a variant of the Tur\'an density, where we want to maximize the density of every linear sized subsets of $H$-free hypergraphs. For real numbers $d\in[0,1]$, and $\eta>0$ a $3$-graph $G=(V,E)$ is \emph{$(d,\eta,1)$-dense} if for all $U\subseteq V$ the relation
\[
\left| U^{(3)}\cap E \right| \geq d \binom{|U|}{3}-\eta |V|^3
\]
holds, where $U^{(3)}$ denotes the set of all three element subsets of $U$. The \emph{uniform Tur\'an density} $\pi_u(H)$ of a 3-graph $H$ is defined to be
\begin{align*}
\pi_u(H):&=\sup \{d\in [0,1]: \ \text{ for every } \eta>0 \text{ and } n\in \mathbb{N} \text{ there exists an } \text{$H$-free} \\
& \quad \ \ \text{$(d,\eta,1)$-dense hypergraph } G \text{ with } |V(G)|\geq n\}.
\end{align*}
\begin{theo}
\label{tabletheo}
All bounds presented in Table~\ref{tab:exactcod2} hold.
\end{theo}
Here we collect many known results, and when known results are lacking, we at least mention the `trivial' bounds. Most upper bounds were obtained by a simple application of flag algebras.
In addition to the bounds presented in Table~\ref{tab:exactcod2}, we asymptotically determine the maximum $\ell_2$-norm of $3$-graphs not containing a loose cycle, loose path, matching or star. These problems are not approachable with flag algebra methods due to the fact that their codegree squared extremal number is $o(n^4)$. We use non-computer assisted methods to obtain these results. The discussion about the history of these problems will be deferred to the corresponding sections. Additionally, we provide a non-computer assisted proof determining the exact codegree squared extremal number of $F_5$.
Denote by $S_n$ the complete $3$-partite $3$-graph on $n$ vertices with part sizes $\lfloor n/3\rfloor,\lfloor (n+1)/3\rfloor,$ $\lfloor (n+2)/3\rfloor$.
We~\cite{BalCleLidmain} showed that $S_n$ is the largest $\{F_4,F_5\}$-free $3$-graph in $\ell_2$-norm using a simple double counting argument and the corresponding $\ell_1$-norm result by Bollob\'as~\cite{Bollobascancellative}. Here, we will expand this result for $F_5$-free $3$-graphs, which requires more work than just applying the corresponding $\ell_1$-norm result.
\begin{table}[H]
\footnotesize
\def2.5{1.4}
\setlength{\tabcolsep}{3pt}
\begin{center}
\begin{tabular}{ | l|| c | c | c | c | c | c| c| c| }
\hline
$H$ & $\pi(H)\geq $ & $\pi(H)\leq $ & $\sigma(H)\geq $ & $\sigma(H)\leq $ & $\pi_2(H)\geq$ & $\pi_2(H)\leq$ & $\pi_u(H)\geq$ & $\pi_u(H)\leq$ \\
\hline\hline
\hyperref[K43]{$K_4^3$} &$5/9$\cite{MR177847}&$0.5615$\cite{BaberTuran} & $1/3$\cite{BalCleLidmain} & $1/3$\cite{BalCleLidmain} & $1/2$ \cite{codegreeconj} & 0.529 & $1/2$& 0.529 \\
\hyperref[K53]{$K_5^3$} &$3/4$\cite{Sidorenko1981SystemsOS,MR177847}&
$0.7696$~\cite{flagmatic} & $5/8$\cite{BalCleLidmain} & $5/8$\cite{BalCleLidmain} & $2/3$ \cite{LoMark,codegreeFalgas}& $0.74$ & $2/3$& 0.758
\\
\hyperref[K63]{$K_6^3$} &$0.84$\cite{Sidorenko1981SystemsOS,MR177847}&
$0.8584$\cite{flagmatic}& $0.7348$\cite{BalCleLidmain} &
$0.7536$ & $3/4$ \cite{LoMark,codegreeFalgas}& $0.838$ & $3/4$ & 0.853
\\
\hyperref[F32]{$F_{3,2}$} &$4/9$ \cite{F32Mubayi}&$4/9$~\cite{F32furedi}& $1/4$ & $1/4+10^{-9}$& $1/3$~\cite{codF32falgas}&$1/3$ \cite{codF32falgas} & $0$& $0$~\cite{MR3764068} \\
\hyperref[sec:F33]{$F_{3,3}$} &$3/4$\cite{F32Mubayi}&$3/4$\cite{F32Mubayi}& $5/8$\cite{BalCleLidmain} & $5/8$\cite{BalCleLidmain}& $1/2$& $0.604$ & $1/4$& 0.605 \\
\hyperref[F5]{$F_{5}$} &$2/9$ \cite{Bollobascancellative} &$2/9$ \cite{F5Frankl}& $2/27$ & $2/27$& $0$& $0$ & $0$ & $0$~\cite{MR3764068} \\
\hyperref[Fano]{$\mathbb{F}$} &$3/4$ \cite{VeraSos}&$3/4$ \cite{FanoFuredi}& $5/8$ & $3/4$ & $1/2$\cite{MubayiFano} &$1/2$ \cite{MubayiFano}& 0 & 0~\cite{MR3764068}\\
\hyperref[K43minus]{$K_4^{3-}$} & $2/7$ \cite{K43-extremalfrankl}&
$0.28689$~\cite{flagmatic}& $4/43$ &
$0.09307$& $1/4$ \cite{codegreeconj}&$1/4$\cite{FalgasK4-} & $1/4$~\cite{MR3474967} & $1/4$~\cite{MR3474967} \\
\hyperref[K43minusF32C5]{$K_4^{3-},F_{3,2},C_5$} &$12/49$ \cite{RavryTuran}&$12/49$ \cite{RavryTuran}& $2/27$ & $2/27$ & $1/12$& $0.186$ & $0$ & $0$~\cite{MR3764068} \\
\hyperref[K43-F32]{$K_4^{3-},F_{3,2}$} &$5/18$ \cite{RavryTuran}&$5/18$ \cite{RavryTuran}& $5/54$ & $5/54$& $1/12$ & 0.202 & $0$ & $0$~\cite{MR3764068} \\
\hyperref[K43-C5]{$K_{4}^{3-},C_5$} &$1/4$ \cite{RavryTuran}&
$0.25108$~\cite{RavryTuran}& $1/13$ &
$0.07695$& $1/12$ & $0.204$& 0 & 0.251 \\
\hyperref[F32-J4]{$F_{3,2},J_4$} &$3/8$ \cite{RavryTuran}& $3/8$ \cite{RavryTuran}& $3/16$ & $3/16$& $1/12$& $0.274$ & $0$ & $0$~\cite{MR3764068} \\
\hyperref[F32-J5]{$F_{3,2},J_5$} &$3/8$ \cite{RavryTuran}& $3/8$ \cite{RavryTuran}& $3/16$ & $3/16$ & $1/12$ & $0.28$ & $0$ & $0$~\cite{MR3764068} \\
\hyperref[sec:J4]{$J_4$} &$1/2$ \cite{DaisyBollobas}&
$0.50409$~\cite{flagmatic}& $0.28$ &
$0.2808$& $1/4$& $0.473$ & $1/4$& 0.49 \\
\hyperref[sec:J5]{$J_5$} &$1/2$ \cite{DaisyBollobas} &
$0.64475$& $0.28$ &
$0.44275$& $1/4$& 0.613 & $1/4$& 0.631 \\
\hyperref[sec:C5]{$C_5$} &$2\sqrt{3}-3$ \cite{F32Mubayi}
$0.46829$~\cite{flagmatic}& $0.25194$ &
$0.25311$& $1/3$&$0.3993$ & $1/27$ & 0.402 \\
\hyperref[sec:C5minus]{$C_5^-$} &$1/4$ \cite{F32Mubayi}&
$0.25074$~\cite{flagmatic} & $1/13$ &
$0.07726$ &$0$& $0.136$ & $0$ & $0$~\cite{MR3764068} \\
\hyperref[sec:K_5^=]{$K_5^{=}$} &$0.58656$
$0.60962$&
$0.35794$&
$0.38873$& $1/2$&$0.569$& $1/2$& 0.567 \\
\hyperref[sec:K_5^<]{$K_5^{<}$} &$5/9$
$0.57705$& $1/3$ &
$0.34022$& $1/2$&$0.560$& $1/2$& 0.568 \\
\hyperref[sec:K_5^-]{$K_5^{-}$} &$0.58656$
$0.64209$&
$0.35794$&
$0.41962$& $1/2$& 0.621 & $1/2$& 0.626 \\
\hline
\end{tabular}
\end{center}
\caption[Known bounds]{\label{tab:exactcod2}
This table shows upper and lower bounds on $\sigma$ and displays the known results for $\pi, \pi_2$ and $\pi_u$ for various hypergraphs.
For the definition of the hypergraphs see Table~\ref{tab:3uniformfig}.
Useful approximations: $2/7\approx 0.285714$,
$2\sqrt{3}-3\approx 0.4641016$,
$1/13\approx 0.0769230$,
$4/43\approx 0.0930232$,
$1/12\approx 0.0833333$.
}
\end{table}
\begin{center}
\setlength{\tabcolsep}{1pt}
\renewcommand{2.5}{2.0}
\begin{longtable}{ | l|| c | c | c | }
\hline
$H$ & Edges & Visual 1 & Visual 2 \\
\hline\hline
$K_4^3$
&123, 124, 134, 234
&
\vc{\begin{tikzpicture}\outercycle{5}{4}
\draw (x0) node[unlabeled_vertex]{};\draw (x1) node[unlabeled_vertex]{};\draw (x2) node[unlabeled_vertex]{};\draw (x3) node[unlabeled_vertex]{};
\drawhyperedge{0}{4}
\drawhypervertex{0}{0}
\drawhypervertex{1}{0}
\drawhypervertex{2}{0}
\drawhyperedge{1}{4}
\drawhypervertex{0}{1}
\drawhypervertex{1}{1}
\drawhypervertex{3}{1}
\drawhyperedge{2}{4}
\drawhypervertex{0}{2}
\drawhypervertex{2}{2}
\drawhypervertex{3}{2}
\drawhyperedge{3}{4}
\drawhypervertex{1}{3}
\drawhypervertex{2}{3}
\drawhypervertex{3}{3}
\end{tikzpicture}
}
&
\Kfourthree
\\
\hline
$K_4^{3-} = F_4$
&123, 124, 134
&
\vc{\begin{tikzpicture}\outercycle{5}{4}
\draw (x0) node[unlabeled_vertex]{};\draw (x1) node[unlabeled_vertex]{};\draw (x2) node[unlabeled_vertex]{};\draw (x3) node[unlabeled_vertex]{};
\drawhyperedge{0}{4}
\drawhypervertex{0}{0}
\drawhypervertex{1}{0}
\drawhypervertex{2}{0}
\drawhyperedge{1}{4}
\drawhypervertex{0}{1}
\drawhypervertex{1}{1}
\drawhypervertex{3}{1}
\drawhyperedge{2}{4}
\drawhypervertex{0}{2}
\drawhypervertex{2}{2}
\drawhypervertex{3}{2}
\path (0,0.4) -- (0,-1.1);
\end{tikzpicture}
}
&
\Kfourthreeminus
\\
\hline
$F_{3,2}$ & 123, 145, 245, 345
&
\vc{\begin{tikzpicture}\outercycle{6}{5}
\draw (x0) node[unlabeled_vertex]{};\draw (x1) node[unlabeled_vertex]{};\draw (x2) node[unlabeled_vertex]{};\draw (x3) node[unlabeled_vertex]{};\draw (x4) node[unlabeled_vertex]{};
\drawhyperedge{0}{5}
\drawhypervertex{0}{0}
\drawhypervertex{1}{0}
\drawhypervertex{2}{0}
\drawhyperedge{1}{5}
\drawhypervertex{0}{1}
\drawhypervertex{3}{1}
\drawhypervertex{4}{1}
\drawhyperedge{2}{5}
\drawhypervertex{1}{2}
\drawhypervertex{3}{2}
\drawhypervertex{4}{2}
\drawhyperedge{3}{5}
\drawhypervertex{2}{3}
\drawhypervertex{3}{3}
\drawhypervertex{4}{3}
\end{tikzpicture}
}
&
\vc{
\begin{tikzpicture}
\path (0,1) -- (0,-1);
\draw
(0,-0.7) coordinate(1) node[vtx](a){}
(0,0) coordinate(2) node[vtx](b){}
(0,0.7) coordinate(3) node[vtx](c){}
(1,-0.5) coordinate(4) node[vtx](d){}
(1,0.5) coordinate(5) node[vtx](d){}
;
\draw[hyperedge] (1) to[bend left] (2) to[bend left] (3) to[bend right] (1);
\draw[hyperedge] (1) to[out=0,in=190] (4) to[out=190,in=250] (5) to[out=250,in=0] (1);
\draw[hyperedge] (2) to[out=0,in=140] (4) to[out=140,in=210,looseness=1.5] (5) to[out=210,in=0] (2);
\draw[hyperedge] (3) to[out=0,in=120,looseness=1.2] (4) to[out=120,in=170,looseness=1.2] (5) to[out=170,in=0,looseness=1.2] (3);
\end{tikzpicture}
}
\\
\hline
$J_4$
&
123, 124, 125, 134, 135, 145
&
\vc{\begin{tikzpicture}\outercycle{6}{5}
\draw (x0) node[unlabeled_vertex]{};\draw (x1) node[unlabeled_vertex]{};\draw (x2) node[unlabeled_vertex]{};\draw (x3) node[unlabeled_vertex]{};\draw (x4) node[unlabeled_vertex]{};
\drawhyperedge{0}{5}
\drawhypervertex{0}{0}
\drawhypervertex{1}{0}
\drawhypervertex{2}{0}
\drawhyperedge{1}{5}
\drawhypervertex{0}{1}
\drawhypervertex{1}{1}
\drawhypervertex{3}{1}
\drawhyperedge{2}{5}
\drawhypervertex{0}{2}
\drawhypervertex{1}{2}
\drawhypervertex{4}{2}
\drawhyperedge{3}{5}
\drawhypervertex{0}{3}
\drawhypervertex{2}{3}
\drawhypervertex{3}{3}
\drawhyperedge{4}{5}
\drawhypervertex{0}{4}
\drawhypervertex{2}{4}
\drawhypervertex{4}{4}
\drawhyperedge{5}{5}
\drawhypervertex{0}{5}
\drawhypervertex{3}{5}
\drawhypervertex{4}{5}
\end{tikzpicture}
}
&
\vc{
\begin{tikzpicture}[scale=0.8]
\draw
(0,0) coordinate(1) node[vtx](b){}
(45:1) coordinate(2) node[vtx](c){}
(135:1) coordinate(3) node[vtx](d){}
(225:1) coordinate(4) node[vtx](d){}
(315:1) coordinate(5) node[vtx](d){}
;
\draw[hyperedge] (1) to[out=90,in=190,looseness=1.5] (2) to[out=190,in=-10] (3) to[out=-10,in=90,looseness=1.5] (1);
\draw[hyperedge] (1) to[out=180,in=280,looseness=1.5] (3) to[out=280,in=80] (4) to[out=80,in=180,looseness=1.5] (1);
\draw[hyperedge] (1) to[out=270,in=10] (4) to[out=10,in=170,looseness=1.5] (5) to[out=170,in=270] (1);
\draw[hyperedge] (1) to[out=0,in=100,looseness=1.5] (5) to[out=100,in=260,looseness=1.5] (2) to[out=260,in=0,looseness=1.5] (1);
\draw[hyperedge] (1) to[out=135,in=225,looseness=1.2] (2) to[out=205,in=65,looseness=1.2] (4) to[out=45,in=135,looseness=1.2] (1);
\draw[hyperedge] (1) to[out=45,in=315,looseness=1.2] (3) to[out=335,in=115,looseness=1.2] (5) to[out=135,in=45,looseness=1.2] (1);
\end{tikzpicture}
}
\\
\hline
$J_5$
&
123, 124, 125, 126, 134, 135, 136, 145, 146, 156
&
\vc{\begin{tikzpicture}\outercycle{7}{6}
\draw (x0) node[unlabeled_vertex]{};\draw (x1) node[unlabeled_vertex]{};\draw (x2) node[unlabeled_vertex]{};\draw (x3) node[unlabeled_vertex]{};\draw (x4) node[unlabeled_vertex]{};\draw (x5) node[unlabeled_vertex]{};
\drawhyperedge{0}{6}
\drawhypervertex{0}{0}
\drawhypervertex{1}{0}
\drawhypervertex{2}{0}
\drawhyperedge{1}{6}
\drawhypervertex{0}{1}
\drawhypervertex{1}{1}
\drawhypervertex{3}{1}
\drawhyperedge{2}{6}
\drawhypervertex{0}{2}
\drawhypervertex{1}{2}
\drawhypervertex{4}{2}
\drawhyperedge{3}{6}
\drawhypervertex{0}{3}
\drawhypervertex{1}{3}
\drawhypervertex{5}{3}
\drawhyperedge{4}{6}
\drawhypervertex{0}{4}
\drawhypervertex{2}{4}
\drawhypervertex{3}{4}
\drawhyperedge{5}{6}
\drawhypervertex{0}{5}
\drawhypervertex{2}{5}
\drawhypervertex{4}{5}
\drawhyperedge{6}{6}
\drawhypervertex{0}{6}
\drawhypervertex{2}{6}
\drawhypervertex{5}{6}
\drawhyperedge{7}{6}
\drawhypervertex{0}{7}
\drawhypervertex{3}{7}
\drawhypervertex{4}{7}
\drawhyperedge{8}{6}
\drawhypervertex{0}{8}
\drawhypervertex{3}{8}
\drawhypervertex{5}{8}
\drawhyperedge{9}{6}
\drawhypervertex{0}{9}
\drawhypervertex{4}{9}
\drawhypervertex{5}{9}
\end{tikzpicture}
}
&
\vc{
\begin{tikzpicture}
\draw
(-2,0) coordinate(1) node[vtx](1){}
\foreach \x in {2,...,6}
{
(-144+72*\x:1) coordinate(\x) node[vtx](v\x){}
}
;
\draw[hyperedge,opacity=0.2] (1)--(2)--(3)--(1);
\draw[hyperedge,opacity=0.2] (1)--(2)--(4)--(1);
\draw[hyperedge,opacity=0.2] (1)--(2)--(5)--(1);
\draw[hyperedge,opacity=0.2] (1)--(2)--(6)--(1);
\draw[hyperedge,opacity=0.2] (1)--(3)--(4)--(1);
\draw[hyperedge,opacity=0.2] (1)--(3)--(5)--(1);
\draw[hyperedge,opacity=0.2] (1)--(3)--(6)--(1);
\draw[hyperedge,opacity=0.2] (1)--(4)--(5)--(1);
\draw[hyperedge,opacity=0.2] (1)--(4)--(6)--(1);
\draw[hyperedge,opacity=0.2] (1)--(5)--(6)--(1);
\draw [line width = 0.5pt] (2)--(3)--(4)--(5)--(6)--(2)--(4)--(6)--(3)--(5)--(2) ;
\draw (-2,0) coordinate(1) node[vtx](1){}
\foreach \x in {2,...,6}
{
(-144+72*\x:1) coordinate(\x) node[vtx](v\x){}
};
\end{tikzpicture}
}
\\
\hline
$F_{3,3}$
&
123, 145, 146, 156, 245, 246, 256, 345, 346, 356
&
\vc{\begin{tikzpicture}\outercycle{7}{6}
\draw (x0) node[unlabeled_vertex]{};\draw (x1) node[unlabeled_vertex]{};\draw (x2) node[unlabeled_vertex]{};\draw (x3) node[unlabeled_vertex]{};\draw (x4) node[unlabeled_vertex]{};\draw (x5) node[unlabeled_vertex]{};
\drawhyperedge{0}{6}
\drawhypervertex{0}{0}
\drawhypervertex{1}{0}
\drawhypervertex{2}{0}
\drawhyperedge{1}{6}
\drawhypervertex{0}{1}
\drawhypervertex{3}{1}
\drawhypervertex{4}{1}
\drawhyperedge{2}{6}
\drawhypervertex{0}{2}
\drawhypervertex{3}{2}
\drawhypervertex{5}{2}
\drawhyperedge{3}{6}
\drawhypervertex{0}{3}
\drawhypervertex{4}{3}
\drawhypervertex{5}{3}
\drawhyperedge{4}{6}
\drawhypervertex{1}{4}
\drawhypervertex{3}{4}
\drawhypervertex{4}{4}
\drawhyperedge{5}{6}
\drawhypervertex{1}{5}
\drawhypervertex{3}{5}
\drawhypervertex{5}{5}
\drawhyperedge{6}{6}
\drawhypervertex{1}{6}
\drawhypervertex{4}{6}
\drawhypervertex{5}{6}
\drawhyperedge{7}{6}
\drawhypervertex{2}{7}
\drawhypervertex{3}{7}
\drawhypervertex{4}{7}
\drawhyperedge{8}{6}
\drawhypervertex{2}{8}
\drawhypervertex{3}{8}
\drawhypervertex{5}{8}
\drawhyperedge{9}{6}
\drawhypervertex{2}{9}
\drawhypervertex{4}{9}
\drawhypervertex{5}{9}
\end{tikzpicture}
}
&
\vc{
\begin{tikzpicture}
\draw
(0,-0.7) coordinate(1) node[vtx](a){}
(0,0) coordinate(2) node[vtx](b){}
(0,0.7) coordinate(3) node[vtx](c){}
(1,-0.7) coordinate(4) node[vtx](d){}
(1,0.0) coordinate(5) node[vtx](d){}
(1,0.7) coordinate(6) node[vtx](d){}
;
\draw[hyperedge] (1) to[bend left] (2) to[bend left] (3) to[bend right] (1);
\draw[hyperedge] (1) to[out=0,in=190] (4) to[out=190,in=250] (5) to[out=250,in=0] (1);
\draw[hyperedge] (2) to[out=0,in=140] (4) to[out=140,in=210,looseness=1.5] (5) to[out=210,in=0] (2);
\draw[hyperedge] (3) to[out=-60,in=120,looseness=1.2] (4) to[out=120,in=170,looseness=1.2] (5) to[out=170,in=-60,looseness=1.2] (3);
\draw[hyperedge] (1) to[out=60,in=190] (5) to[out=190,in=250] (6) to[out=250,in=60] (1);
\draw[hyperedge] (2) to[out=0,in=140] (5) to[out=140,in=210,looseness=1.5] (6) to[out=210,in=0] (2);
\draw[hyperedge] (3) to[out=0,in=120,looseness=1.2] (5) to[out=120,in=170,looseness=1.2] (6) to[out=170,in=0,looseness=1.2] (3);
\draw[hyperedge] (1) to[out=10,in=170,looseness=1.5] (4) to[out=170,in=190] (6) to[out=190,in=10] (1);
\draw[hyperedge] (2) to[out=0,in=140] (4) to[out=140,in=210,looseness=1.5] (6) to[out=210,in=0] (2);
\draw[hyperedge] (3) to[out=-10,in=120,looseness=1.2] (4) to[out=120,in=190,looseness=1.2] (6) to[out=190,in=-10,looseness=1.2] (3);
\end{tikzpicture}
}
\\
\hline
$F_5$
&
123, 124, 345
&
\vc{\begin{tikzpicture}\outercycle{6}{5}
\draw (x0) node[unlabeled_vertex]{};\draw (x1) node[unlabeled_vertex]{};\draw (x2) node[unlabeled_vertex]{};\draw (x3) node[unlabeled_vertex]{};\draw (x4) node[unlabeled_vertex]{};
\drawhyperedge{0}{5}
\drawhypervertex{0}{0}
\drawhypervertex{1}{0}
\drawhypervertex{2}{0}
\drawhyperedge{1}{5}
\drawhypervertex{0}{1}
\drawhypervertex{1}{1}
\drawhypervertex{3}{1}
\drawhyperedge{2}{5}
\drawhypervertex{2}{2}
\drawhypervertex{3}{2}
\drawhypervertex{4}{2}
\end{tikzpicture}
}
&
\vc{
\begin{tikzpicture}
\path (0,0.7) -- (0,-0.7);
\draw
(0, 0.5) coordinate(0) node[vtx]{}
(0, -0.5) coordinate(1) node[vtx]{}
(0.7, 0.5) coordinate(2) node[vtx]{}
(0.7,-0.5) coordinate(3) node[vtx]{}
(1.5, 0) coordinate(4) node[vtx]{}
;
\draw[hyperedge] (0) to[out=-10,in=190,looseness=1] (2) to[out=190,in=60,looseness=1] (1) to[out=60,in=-10,looseness=1] (0);
\draw[hyperedge] (1) to[out=10,in=170,looseness=1] (3) to[out=170,in=-60,looseness=1] (0) to[out=-60,in=10,looseness=1] (1);
\draw[hyperedge] (2) to[out=310,in=50,looseness=1] (3) to[out=50,in=180,looseness=1] (4) to[out=180,in=310,looseness=1] (2);
\end{tikzpicture}
}
\\
\hline
$\mathbb{F}$
&
123, 345, 156, 246, 147, 257, 367
&
\vc{\begin{tikzpicture}\outercycle{8}{7}
\draw (x0) node[unlabeled_vertex]{};\draw (x1) node[unlabeled_vertex]{};\draw (x2) node[unlabeled_vertex]{};\draw (x3) node[unlabeled_vertex]{};\draw (x4) node[unlabeled_vertex]{};\draw (x5) node[unlabeled_vertex]{};\draw (x6) node[unlabeled_vertex]{};
\drawhypervertex{0}{0}
\drawhypervertex{1}{0}
\drawhypervertex{2}{0}
\drawhyperedge{1}{7}
\drawhypervertex{0}{1}
\drawhypervertex{3}{1}
\drawhypervertex{6}{1}
\drawhyperedge{2}{7}
\drawhypervertex{0}{2}
\drawhypervertex{4}{2}
\drawhypervertex{5}{2}
\drawhyperedge{3}{7}
\drawhypervertex{1}{3}
\drawhypervertex{3}{3}
\drawhypervertex{5}{3}
\drawhyperedge{4}{7}
\drawhypervertex{1}{4}
\drawhypervertex{4}{4}
\drawhypervertex{6}{4}
\drawhyperedge{5}{7}
\drawhypervertex{2}{5}
\drawhypervertex{3}{5}
\drawhypervertex{4}{5}
\drawhyperedge{6}{7}
\drawhypervertex{2}{6}
\drawhypervertex{5}{6}
\drawhypervertex{6}{6}
\end{tikzpicture}
}
&
\vc{
\begin{tikzpicture}
\path (60:2.2) -- (0,-0.3);
\draw
(60:2) coordinate(1) coordinate(7) node[vtx]{}
(60:1) coordinate(2) coordinate(8) node[vtx]{}
(0,0) coordinate(3) node[vtx]{}
(1,0) coordinate(4) node[vtx]{}
(2,0) coordinate(5) node[vtx]{}
++ (120:1) coordinate(6) node[vtx]{}
(30:1.2) coordinate(X) node[vtx]{}
;
\foreach \i in {1,3,5}{
\pgfmathtruncatemacro{\j}{\i+1}
\pgfmathtruncatemacro{\k}{\i+2}
\pgfmathtruncatemacro{\l}{\i+3}
\draw[hyperedge] (\i) to[bend right=10] (\j) to[bend right=10] (\k) to[bend left=10] (\i) ;
\draw[hyperedge] (\i) to[bend right=10] (X) to[bend right=10] (\l) to[bend left=10] (\i) ;
}
\draw[hyperedge] (2) to[bend right=40] (4) to[bend right=40] (6) to[bend left=80,looseness=2.5] (2) ;
\draw
\foreach \i in {1,2,3,4,5,6,X}{
(\i) node[vtx]{}
}
;
\end{tikzpicture}
}
\\
\hline
$C_5$
&
123, 234, 345, 145, 125
&
\vc{\begin{tikzpicture}\outercycle{6}{5}
\draw (x0) node[unlabeled_vertex]{};\draw (x1) node[unlabeled_vertex]{};\draw (x2) node[unlabeled_vertex]{};\draw (x3) node[unlabeled_vertex]{};\draw (x4) node[unlabeled_vertex]{};
\drawhyperedge{0}{5}
\drawhypervertex{0}{0}
\drawhypervertex{1}{0}
\drawhypervertex{2}{0}
\drawhyperedge{1}{5}
\drawhypervertex{1}{1}
\drawhypervertex{2}{1}
\drawhypervertex{3}{1}
\drawhyperedge{2}{5}
\drawhypervertex{2}{2}
\drawhypervertex{3}{2}
\drawhypervertex{4}{2}
\drawhyperedge{3}{5}
\drawhypervertex{0}{3}
\drawhypervertex{3}{3}
\drawhypervertex{4}{3}
\drawhyperedge{4}{5}
\drawhypervertex{0}{4}
\drawhypervertex{1}{4}
\drawhypervertex{4}{4}
\end{tikzpicture}
}
&
\vc{
\begin{tikzpicture}
\path (0,1.2) -- (0,-1.1);
\draw
\foreach \i in {0,1,...,7}{
(90+72*\i:1) coordinate(\i) node[vtx]{}
}
;
\foreach \i in {0,1,2,3,4}{
\pgfmathtruncatemacro{\j}{\i+1}
\pgfmathtruncatemacro{\k}{\i+2}
\draw[hyperedge] (\i) to[out=230+72*\i,in=270+72*\j,looseness=1] (\j) to[out=270+72*\j,in=310+72*\k,looseness=1] (\k) to[out=310+72*\k,in=230+72*\i,looseness=1] (\i);
}
\end{tikzpicture}
}
\\
\hline
$C_5^-$
&
123, 234, 345, 145
&
\vc{\begin{tikzpicture}\outercycle{6}{5}
\draw (x0) node[unlabeled_vertex]{};\draw (x1) node[unlabeled_vertex]{};\draw (x2) node[unlabeled_vertex]{};\draw (x3) node[unlabeled_vertex]{};\draw (x4) node[unlabeled_vertex]{};
\drawhyperedge{0}{5}
\drawhypervertex{0}{0}
\drawhypervertex{1}{0}
\drawhypervertex{2}{0}
\drawhyperedge{1}{5}
\drawhypervertex{1}{1}
\drawhypervertex{2}{1}
\drawhypervertex{3}{1}
\drawhyperedge{2}{5}
\drawhypervertex{2}{2}
\drawhypervertex{3}{2}
\drawhypervertex{4}{2}
\drawhyperedge{3}{5}
\drawhypervertex{0}{3}
\drawhypervertex{3}{3}
\drawhypervertex{4}{3}
\end{tikzpicture}
}
&
\vc{
\begin{tikzpicture}
\path (0,1.2) -- (0,-1.1);
\draw
\foreach \i in {0,1,...,7}{
(90+72*\i:1) coordinate(\i) node[vtx]{}
}
;
\foreach \i in {0,1,2,3}{
\pgfmathtruncatemacro{\j}{\i+1}
\pgfmathtruncatemacro{\k}{\i+2}
\draw[hyperedge] (\i) to[out=230+72*\i,in=270+72*\j,looseness=1] (\j) to[out=270+72*\j,in=310+72*\k,looseness=1] (\k) to[out=310+72*\k,in=230+72*\i,looseness=1] (\i);
}
\end{tikzpicture}
}
\\
\hline
$\overline{K_5^-}$
&
123
&
\vc{\begin{tikzpicture}\outercycle{6}{5}
\draw (x0) node[unlabeled_vertex]{};\draw (x1) node[unlabeled_vertex]{};\draw (x2) node[unlabeled_vertex]{};\draw (x3) node[unlabeled_vertex]{};\draw (x4) node[unlabeled_vertex]{};
\drawhyperedge{0}{5}
\drawhypervertex{0}{0}
\drawhypervertex{1}{0}
\drawhypervertex{2}{0}
\end{tikzpicture}
}
&
\vc{
\begin{tikzpicture}
\path (0,1.2) -- (0,-1.1);
\draw
\foreach \i in {0,1,...,7}{
(90+72*\i:1) coordinate(\i) node[vtx]{}
}
;
\foreach \i in {0}{
\pgfmathtruncatemacro{\j}{\i+1}
\pgfmathtruncatemacro{\k}{\i+2}
\draw[hyperedge] (\i) to[out=230+72*\i,in=270+72*\j,looseness=1] (\j) to[out=270+72*\j,in=310+72*\k,looseness=1] (\k) to[out=310+72*\k,in=230+72*\i,looseness=1] (\i);
}
\end{tikzpicture}
}
\\
\hline
$\overline{K_5^=}$
&
123, 345
&
\vc{\begin{tikzpicture}\outercycle{6}{5}
\draw (x0) node[unlabeled_vertex]{};\draw (x1) node[unlabeled_vertex]{};\draw (x2) node[unlabeled_vertex]{};\draw (x3) node[unlabeled_vertex]{};\draw (x4) node[unlabeled_vertex]{};
\drawhyperedge{0}{5}
\drawhypervertex{0}{0}
\drawhypervertex{1}{0}
\drawhypervertex{2}{0}
\drawhyperedge{1}{5}
\drawhypervertex{2}{1}
\drawhypervertex{3}{1}
\drawhypervertex{4}{1}
\end{tikzpicture}
}
&
\vc{
\begin{tikzpicture}
\path (0,1.2) -- (0,-1.1);
\draw
\foreach \i in {0,1,...,7}{
(90+72*\i:1) coordinate(\i) node[vtx]{}
}
;
\foreach \i in {0,2}{
\pgfmathtruncatemacro{\j}{\i+1}
\pgfmathtruncatemacro{\k}{\i+2}
\draw[hyperedge] (\i) to[out=230+72*\i,in=270+72*\j,looseness=1] (\j) to[out=270+72*\j,in=310+72*\k,looseness=1] (\k) to[out=310+72*\k,in=230+72*\i,looseness=1] (\i);
}
\end{tikzpicture}
}
\\
\hline
$\overline{K_5^<}$
&
123, 234
&
\vc{\begin{tikzpicture}\outercycle{6}{5}
\draw (x0) node[unlabeled_vertex]{};\draw (x1) node[unlabeled_vertex]{};\draw (x2) node[unlabeled_vertex]{};\draw (x3) node[unlabeled_vertex]{};\draw (x4) node[unlabeled_vertex]{};
\drawhyperedge{0}{5}
\drawhypervertex{0}{0}
\drawhypervertex{1}{0}
\drawhypervertex{2}{0}
\drawhyperedge{1}{5}
\drawhypervertex{1}{1}
\drawhypervertex{2}{1}
\drawhypervertex{3}{1}
\end{tikzpicture}
}
&
\vc{
\begin{tikzpicture}
\path (0,1.2) -- (0,-1.1);
\draw
\foreach \i in {0,1,...,7}{
(90+72*\i:1) coordinate(\i) node[vtx]{}
}
;
\foreach \i in {0,1}{
\pgfmathtruncatemacro{\j}{\i+1}
\pgfmathtruncatemacro{\k}{\i+2}
\draw[hyperedge] (\i) to[out=230+72*\i,in=270+72*\j,looseness=1] (\j) to[out=270+72*\j,in=310+72*\k,looseness=1] (\k) to[out=310+72*\k,in=230+72*\i,looseness=1] (\i);
}
\end{tikzpicture}
}
\\
\hline
\caption{\label{tab:pic}Description of some $3$-uniform hypergraphs.
}
\label{tab:3uniformfig}
\end{longtable}
\end{center}
\begin{theo}
\label{F5exact}
There exists a number $n_0$ such that for all $n\geq n_0$
\begin{align*}
\textup{exco}_2(n,F_5)=\textup{co}_2(S_n).
\end{align*}
Furthermore, $S_n$ is the unique $F_5$-free $3$-graph $G$ on $n$ vertices satisfying $\textup{co}_2(G)=\textup{exco}_2(n,F_{5})$.
\end{theo}
The \emph{loose $s$-path} $P^k_s$ is the $k$-graph with $s$ edges $\{e_1,\ldots, e_s\}$ such that $|e_i\cap e_j|=1$ if $|i-j|=1$ and $e_i\cap e_j=\emptyset$ otherwise. The \emph{loose $s$-cycle} $C^k_s$ is the $k$-uniform hypergraph with $s$ edges $\{e_1,\ldots, e_s\}$ obtained from an $(s-1)$-path $\{e_1,\ldots, e_{s-1}\}$ by adding an edge $e_s$ that shares one vertex with $e_1$, another vertex with $e_{s-1}$ and is disjoint from the other edges.
\begin{theo}\label{codcyclepath}
Let $s\geq 4$. Then,
\begin{align*}
\textup{exco}_2(n,C^3_s)=\left \lfloor \frac{s-1}{2}\right \rfloor n^3(1+o(1)) \quad \quad \text{and} \quad \quad \textup{exco}_2(n,P^3_s)=\left \lfloor \frac{s-1}{2}\right \rfloor n^3(1+o(1)).
\end{align*}
\end{theo}
Denote by $M_s^3$ the \emph{$3$-uniform matching} of size $s$, i.e., the $3$-uniform hypergraph on $3s$ vertices with $s$ pairwise disjoint edges.
\begin{theo}\label{codmatching}
Let $s\geq 2$. Then,
\begin{align*}
\textup{exco}_2(n,M_s^3)=(s-1)n^3(1+o(1)).
\end{align*}
\end{theo}
Denote by $S_s^3$ the \emph{star} with $s$ edges such that the intersection of any pair of edges is exactly the same vertex.
\begin{theo}
\label{codstar}
Let $s\geq 3$. If $s$ is odd, then
\begin{align*}
\textup{exco}_2(n,S_s^3)=s(s-1)n^2 (1+o(1)).
\end{align*}
If $s$ is even, then
\begin{align*}
\textup{exco}_2(n,S_s^3)=\left(s^2-\frac{3}{2}s \right)n^2 (1+o(1)).
\end{align*}
\end{theo}
In this work, we are not intending to duplicate or replace the excellent survey~\cite{Keevashsurvey} by Keevash; our aim is to supplement it with new results and directions.
Our paper is organized as follows; in Section~\ref{prepa} we explain our notation and explain the flag algebra technique briefly.
In Section~\ref{boundsfortable} we present the constructions leading to the bounds on $\sigma,\pi,\pi_u,\pi_2$ in Table~\ref{tab:exactcod2} and state conjectures on some of the non-sharp results.
In Section~\ref{F5section} we present the proof of Theorem~\ref{F5exact}. In Sections~\ref{sec:cycle}, \ref{sec:matching} and \ref{sec:star} we prove Theorems~\ref{codcyclepath}, \ref{codmatching} and \ref{codstar}, respectively. Finally, we present related questions for higher uniformities in Section~\ref{openques}.
\section{Preliminaries}
\label{prepa}
\subsection{Terminology and notation}
Let $H$ be a $3$-uniform hypergraph, $x,y,z\in V(H)$ and $A,B,C\subseteq V(H)$ be pairwise disjoint sets. We will use the following notation throughout the paper.
\begin{itemize}[leftmargin=*]
\item For an edge $e=\{x,y,z\}\in E(H)$ we write $xyz$ for convenience.
\item Denote by $L(x)$ the link graph of $x$, i.e., the graph on $V(H)\setminus \{x\}$ with $ab\in E(L(x))$ iff $abx\in E(H)$.
\item Denote by $L_A(x)=L(x)[A]$ the induced link graph on $A$.
\item Denote by $L_{A,B}(x)$ the subgraph of the link graph of $x$ only containing edges between $A$ and $B$, i.e., $V(L_{A,B}(x))=V(H)\setminus\{x\}$ and $ab\in E(L_{A,B}(x))$ iff $a\in A, b\in B$ and $abx\in E(H)$.
\item Denote by $L^c_{A,B}(x)$ the subgraph of the link graph of $x$ only containing non-edges between $A$ and $B$, i.e., $V(L_{A,B}(x))=V(H)\setminus\{x\}$ and $ab\in E(L^c_{A,B}(x))$ iff $a\in A, b\in B$ and $abx \not\in E(H)$.
\item $e(A,B)$ denotes the number of cross-edges between $A$ and $B$, this means \\ $e(A,B):=|\{xyz\in E(H): x,y\in A, z\in B \}|+ |\{xyz\in E(H): x,y\in B, z\in A \}|.$
\item Denote $e(A,B,C)$ the number of cross-edges between $A,B$ and $C$, i.e.,\\ $e(A,B,C):=|\{abc\in E(H): a\in A, b\in B,c\in C \}|$.
\item Let $e=xyz\in E(H)$ be an edge. Define the weight of $e$ to be
\[
w_H(e)=d(x,y)+d(x,z)+d(y,z).
\]
We drop the index if the hypergraph $H$ is clear from the context.
\item The shadow graph $\delta H$ is the subset of all two element subsets of $V(H)$ that contains all pairs contained in some $e\in E(H)$.
\end{itemize}
\begin{defn}
Let $H$ be a $k$-graph and $t\in \mathbb{N}$. The \emph{blow-up} $H(t)$ of $H$ is the $k$-graph obtained by replacing each vertex $x\in V(H)$ by $t$ vertices $x^1,\ldots,x^t$ and each edge $x_1 \cdots x_k\in E(H)$ by $t^k$ edges $x_1^{a_1}\cdots x_k^{a_k}$ with $1 \leq a_1,\ldots, a_k \leq t$.
\end{defn}
\subsection{The uniform Tur\'an density}
\label{sec:uniformturan}
Recently, Reiher, R\"{o}dl and Schacht~\cite{MR3764068} introduced the uniform Tur\'an density.
They characterised $3$-graphs $H$ satisfying $\pi_u(H)=0$.
\begin{theo}[Reiher, R\"{o}dl and Schacht~\cite{MR3764068}]
\label{uniformturan}
For a $3$-uniform hypergraph $H$, the following are equivalent:
\begin{itemize}
\item[(a)] $\pi_u(H)=0$.
\item[(b)] There is a labeling of the vertex set $V(H)=\{v_1,\ldots,v_f\}$ and there is a three-colouring $\phi: \delta F \rightarrow \{\text{red, blue, green}\}$ of the pairs of vertices covered by hyperedges of $H$ such that every hyperedge $v_iv_jv_k\in E(H)$ with $i<j<k$ satisfies
\begin{align*}
\phi(v_i,v_j)=
\text{red}, \quad \phi(v_i,v_k)=
\text{blue}, \quad \phi(v_j,v_k)=
\text{green}.
\end{align*}
\end{itemize}
\end{theo}
We will use their result to observe that $\pi_u(H)=0$ for $H=F_{3,2}, F_5$ and $\mathbb{F}$ in Section~\ref{boundsfortable}. Reiher, R\"{o}dl and Schacht~\cite{MR3764068} also observed that the uniform Tur\'an density has a jump from $0$ to $1/27$.
\begin{corl}[Reiher, R\"{o}dl and Schacht~\cite{MR3764068}]
\label{uniformturanjump}
If a $3$-graph $H$ satisfies $\pi_u(H)>0$, then $\pi_u(H)\geq \frac{1}{27}$.
\end{corl}
Somewhat surprisingly, it was not easy to prove that $1/27$ is best possible, which was done by Garbe, Kr\'al and Lamaison~\cite{kral}.
\subsection{Flag Algebras}
\label{flagresults}
We use flag algebras to obtain upper bounds on $\pi,\sigma,\pi_2$ and $\pi_u$ for Table~\ref{tab:exactcod2}. The general framework of flag algebras was developed by Razborov~\cite{RazbarovK43}.
Flag algebras has been applied to variety of problems, including problems on $3$-graphs~\cite{BaberTalbot,BaberTuran,RazbarovK43}. By now, it is a standard application to obtain some upper bounds on $\pi$. For $\sigma,\pi_2$ and $\pi_u$ we give a short explanation how upper bounds can be obtained.
For readers familiar with flag algebras, the function counting scaled codegree squared sum can be expressed using flag algebras as follows:
\begin{align}\label{eq:fa}
\lim_{n \to \infty} \frac{\textup{co}_2(G_n)}{\binom{n}{2}(n-2)^2}
=
\left\llbracket
\left(
\vc{
\begin{tikzpicture}
\draw
(0,0) coordinate(1) node[vtxl,label=below:1](a){}
(1,0) coordinate(2) node[vtxl,label=below:2](b){}
(0.5,1) coordinate(3) node[vtx](c){}
;
\draw[hyperedge] (1) to[out=40,in=130] (2) to[out=120,in=280] (3) to[out=260,in=50] (1);
\draw
(1) node[vtxl]{}
(2) node[vtxl]{}
;
\end{tikzpicture}
}
\right)^2
\right\rrbracket_{1,2}
=
\frac{1}{6}
\vc{
\begin{tikzpicture}
\draw
(0,0) coordinate(1) node[vtx](a){}
(1,0) coordinate(2) node[vtx](b){}
(1,1) coordinate(3) node[vtx](c){}
(0,1) coordinate(4) node[vtx](d){}
;
\draw[hyperedge] (1) to[out=30,in=90] (2) to[out=100,in=260] (3) to[out=260,in=30] (1);
\draw[hyperedge] (1) to[out=90,in=150] (2) to[out=150,in=280] (4) to[out=280,in=90] (1);
\end{tikzpicture}
}
+
\frac{1}{2}
\Kfourthreeminus
+
\Kfourthree
,
\end{align}
where $(G_n)_{n \geq 1}$ is a convergent sequence with $G_n$ being an $n$-vertex $3$-graph.
For a well-written explanation of the flag algebra method in the setting of $3$-uniform hypergraphs see \cite{RavryTuran} and for the particular application and an explanation of \eqref{eq:fa} see \cite{BalCleLidmain}.
For the minimum codegree threshold $\pi_2$, we use the formulation from \cite{FalgasK4-}. Suppose we want to show that $\pi_2(H) \leq z$ for some $3$-graph $H$ and $z \in (0,1)$.
We do this by arguing that there is no convergent sequence $(G_n)_{n \geq 1}$ of $H$-free graphs with minimum codegree $z$. Suppose there is such a sequence. Then for any flag $F$ with two labeled vertices
\begin{align}\label{eq:fa2}
\left\llbracket
\left(
\vc{
\begin{tikzpicture}
\draw
(0,0) coordinate(1) node[vtxl,label=below:1](a){}
(1,0) coordinate(2) node[vtxl,label=below:2](b){}
(0.5,1) coordinate(3) node[vtx](c){}
;
\draw[hyperedge] (1) to[out=40,in=130] (2) to[out=120,in=280] (3) to[out=260,in=50] (1);
\draw
(1) node[vtxl]{}
(2) node[vtxl]{}
;
\end{tikzpicture}
}
-z
\right)
\times F
\right\rrbracket_{1,2}
\geq 0.
\end{align}
By combining these inequalities and sum of squares inequalities, one gets a contradiction with the existence of $(G_n)_{n \geq 1}$.
It is not obvious to us how to express in the flag algebras language the conditions for uniform Tur\'an density. Instead of a direct expression, Glebov, Kr\'al', and Volec~\cite{MR3474967} formulated consequences of the conditions that can be expressed using flag algebras.
Suppose graphs in a convergent sequence $(G_n)_{n \geq 1}$ are $(d,\eta,1)$-dense.
The simplest instance is to take two labeled vertices and look at their co-neighborhood $N$. In $N$, the relative density of edges must be at least $d$ if $N$ is sufficiently large. If it is not sufficiently large, then the values in the equation below become all $0$ anyway.
In flag algebras, this can be done as
\begin{align}\label{eq:fau}
\left\llbracket
\left(
(1-d)
\vc{
\begin{tikzpicture}
\draw
(0,0) coordinate(1) node[vtxl,label=below:1](a){}
(1,0) coordinate(2) node[vtxl,label=below:2](b){}
(0,1) coordinate(3) node[vtx](c){}
(0.5,1) coordinate(4) node[vtx](c){}
(1,1) coordinate(5) node[vtx](c){}
;
\draw[hyperedge] (1) to[out=40,in=130] (2) to[out=120,in=280] (3) to[out=260,in=50] (1);
\draw[hyperedge] (1) to[out=40,in=130] (2) to[out=120,in=280] (4) to[out=260,in=50] (1);
\draw[hyperedge] (1) to[out=40,in=130] (2) to[out=120,in=280] (5) to[out=260,in=50] (1);
\draw[hyperedge] (3) to[out=40,in=130] (4) to[out=40,in=130] (5) to[out=90,in=90] (3);
\draw
(1) node[vtxl]{}
(2) node[vtxl]{}
(3) node[vtx](c){}
(4) node[vtx](c){}
(5) node[vtx](c){}
;
\end{tikzpicture}
}
-d
\vc{
\begin{tikzpicture}
\draw
(0,0) coordinate(1) node[vtxl,label=below:1](a){}
(1,0) coordinate(2) node[vtxl,label=below:2](b){}
(0,1) coordinate(3) node[vtx](c){}
(0.5,1) coordinate(4) node[vtx](c){}
(1,1) coordinate(5) node[vtx](c){}
;
\draw[hyperedge] (1) to[out=40,in=130] (2) to[out=120,in=280] (3) to[out=260,in=50] (1);
\draw[hyperedge] (1) to[out=40,in=130] (2) to[out=120,in=280] (4) to[out=260,in=50] (1);
\draw[hyperedge] (1) to[out=40,in=130] (2) to[out=120,in=280] (5) to[out=260,in=50] (1);
\draw
(1) node[vtxl]{}
(2) node[vtxl]{}
(3) node[vtx](c){}
(4) node[vtx](c){}
(5) node[vtx](c){}
;
\end{tikzpicture}
}
\right)
\times F
\right\rrbracket_{1,2}
\geq 0,
\end{align}
where edges with exactly one labeled vertex are not depicted for the sake of readability and $F$ is any flag with two labeled vertices. An analogous equation can be obtained by considering non-co-neighborhood. More generally,
for more labeled vertices, we can decide for each pair if we want to take co-neighborhood or non-co-neighborhood.
The approach by Glebov, Kr\'al', and Volec~\cite{MR3474967} continues by getting an exact solution on the threshold that has density of a particular hypergraph 0. Then to a hypothetical counterexample, one can apply sparsification to obtain an example on the threshold density, which in turn shows that the hypothetical counterexample actually has zero edge density, which is a contradiction.
Since the full method from \cite{MR3474967} is more involved and uniform Tur\'an density is not the main focus of this paper, we opted for a simpler approach. We use \eqref{eq:fau} and its analogue for non-co-neighborhood and obtain an upper bound on the global edge density that is strictly less than $d$, leading to a contradiction.
This simplified approach is not as strong. For $K_4^{3-}$ it gives only $0.259$ while a sharp result 0.25 is known~\cite{MR3474967}.
The calculations which lead to the results in Table~\ref{tab:exactcod2} are computer assisted; we use CSDP~\cite{csdp} to calculate numerical solutions of semidefinite programs and then use SageMath~\cite{sagemath} for rounding these numerical solutions to exact ones.
The data files and programs to reproduce the calculations we developed are available at \url{http://lidicky.name/pub/co2b/}.\\
Next, we will present the constructions which give the lower bounds from Table~\ref{tab:exactcod2}.
\section{Bounds from Table~\ref{tab:exactcod2}}
\label{boundsfortable}
\subsection{\texorpdfstring{$K_4^3$}{TEXT}}
\label{K43}
Tur\'an's tetrahedron problem asks to determine the Tur\'an density of $K_4^3$. The best lower bound $\pi(K_4^3)\geq 5/9$ is obtained by $C_n$, see Figure~\ref{fig:CnBnH5}, the $3$-graph on $n$ vertices with vertex set $V(C_n)=V_1\cup V_2 \cup V_3$ where $||V_i|-|V_j||\leq 1$ for $i\neq j$ and edge set
\begin{align*}
E(C_n)=\{abc: a\in V_1,b\in V_2, c\in V_3 \} \cup \{abc: a,b\in V_1,c\in V_2 \} \\ \cup \ \{abc: a,b\in V_2,c\in V_3 \} \cup \{abc: a,b\in V_3,c\in V_1 \}.
\end{align*}
Brown~\cite{K43brown}, Kostochka~\cite{K43Kostochka}, Fon-der-Flaass~\cite{K43Fonderflaass} and Frohmader~\cite{MR2465761} constructed families of $K_4^3$-free $3$-graphs with the same number of edges. A series of papers~\cite{TetrahedronCaen,ChungLutetrahedron,RazbarovK43} have improved on the upper bound of $\pi(K_4^3)$ culminating in the current best-known bound by Baber~\cite{BaberTuran}
\begin{align*}
\pi(K_4^3)\leq 0.5615.
\end{align*}
We~\cite{BalCleLidmain} solved Tur\'an's tetrahedron problem for the codegree squared density by showing $\sigma(K_4^3)=1/3$, where the lower bound is achieved by $C_n$. \\
For the minimum codegree threshold Czygrinow and Nagle~\cite{MR1829685} provided a construction that shows $\pi_2(K_4^3)\geq 1/2$. Let $T$ be a uniformly at random chosen tournament on $n$ vertices. Define a $3$-graph $G_T$ on $n$ vertices by setting the triple $ijk$ with $i < j < k$ to be an edge of $G_T$ if the ordered pairs $(i,j)$ and $(i,k)$ receive opposite directions in $T$. This $3$-graph is $K_4^3$-free, and has minimum codegree $n/2-o(n)$ with high probability. Using flag algebras we can prove that $\pi_2(K_4^3)\leq 0.529$.
The previous construction was used by R\"{o}dl~\cite{MR837962} to show that for the uniform Tur\'an density $\pi_u(K_4^3)\geq 1/2$. Erd\H{o}s~\cite{MR1083590} suggested that this might be best possible, also see Reiher~\cite{MR4111729}. Using flag algebras as described in Section~\ref{flagresults}, we obtain $\pi_u(K_4^3) \leq 0.531$.
\subsection{\texorpdfstring{$K_5^3$}{TEXT}}
\label{K53}
For the Tur\'an density of $K_5^3$ it is known that
\begin{align*}
\frac{3}{4}\leq \pi(K_5^3)\leq 0.769533,
\end{align*}
where the upper bound is obtained via flag algebras~\cite{flagmatic} and the lower bound is obtained by the balanced complete bipartite 3-graph $B_n$ (see Figure~\ref{fig:CnBnH5}) as observed by Tur\'an~\cite{MR177847}. The following $n$-vertex $3$-graph $H_5$ (presented in \cite{Sidorenko1981SystemsOS}, see Figure~\ref{fig:CnBnH5}) also achieves the lower bound.
The vertex set of $H_5$ is divided into $4$ parts $A_1, A_2, A_3, A_4$ with $||A_j|-|A_i||\leq 1$ for all $1\leq i\leq j\leq 4$ and a triple $e$ is not an edge of $H_5$ iff there is some $j$ ($1\leq j \leq 4$) such that $|e \cap A_{j}|\geq 2$ and $|e\cap A_{j}|+|e\cap A_{j+1}|=3$, where $A_{5}=A_1$.
We~\cite{BalCleLidmain} proved that $\sigma(K_5^3)=5/8$, where the lower bound is obtained by the complete balanced bipartite $3$-graph $B_n$. Notice that $H_5$ does not achieve this lower bound.
For the minimum codegree threshold, we have
\begin{align*}
\frac{2}{3} \leq \pi_2(K_5^3)\leq 0.74,
\end{align*}
where we obtained the upper bound via flag algebras and the lower bound is due to Falgas-Ravry~\cite{codegreeFalgas} and Lo and Markstr\"om~\cite{LoMark} who constructed lower bounds on the codegree threshold for cliques of arbitrary size $s$. Here we present the construction due to Falgas-Ravry. Let $c: E(K_n)\rightarrow [s]$ be a uniformly at random chosen colouring of the edges of the complete graph on $n$ vertices with $s$ colours. Consider the $3$-graph $G$ based on this colouring in the following way: A triple $i,j,k$ with $i < j < k$ forms an edge in $G$ if and only if $c(ij)\neq c(ik)$. This $3$-graph is $K_{s+2}$-free and has minimum codegree $(1-1/s)n-o(n)$ with high probability. This construction can be seen as a generalization of the Czygrinow and Nagle construction from the previous section.
\begin{figure}
\begin{center}
\tikzset{
vtx/.style={inner sep=1.1pt, outer sep=0pt, circle, fill,draw},
hyperedge/.style={fill=pink,opacity=0.5,draw=black},
vtxBig/.style={inner sep=12pt, outer sep=0pt, circle, fill=white,draw},
hyperedge/.style={fill=pink,opacity=0.5,draw=black},
}
\vc{
\begin{tikzpicture}[scale=1.4]
\draw (30:0.9) coordinate(x1) node[vtxBig]{};
\draw (150:0.9) coordinate(x2) node[vtxBig]{};
\draw (270:0.9) coordinate(x3) node[vtxBig]{};
\draw
(30:0.8) coordinate(1)
(150:0.8) coordinate(2)
(270:0.8) coordinate(3)
;
\draw[hyperedge] (1) to[out=210,in=330] (2) to[out=330,in=90] (3) to[out=90,in=210] (1);
\foreach \ashift in {30,150,270}{
\draw
(\ashift:0.8)++(\ashift+60:0.1) coordinate(1)
++(\ashift+60:0.3) coordinate(2)
(\ashift+120:0.8)++(\ashift+120+90:-0.2) coordinate(3
;
\draw[hyperedge] (1) to[bend left] (2) to[bend left=5] (3) to[bend left=5] (1);
}
\draw (30:1) node{$V_1$};
\draw (150:1) node{$V_2$};
\draw (270:1) node{$V_3$};
\draw (0,-1.8) node{$C_n$};
\end{tikzpicture}
}
\hskip 1cm
\vc{
\begin{tikzpicture}[scale=2.0]
\draw (0,0) coordinate(x1) ellipse (0.25cm and 0.7cm);
\draw (1,0) coordinate(x1) ellipse (0.25cm and 0.7cm);
\draw
(0,-0.2) coordinate(1)
(0,-0.4) coordinate(2)
(0,0.3) coordinate(3)
(1,0.2) coordinate(4)
(1,-0.3) coordinate(5)
(1,0.4) coordinate(6)
;
\draw[hyperedge] (1) to[bend left] (2) to[bend left=5] (5) to[bend left=5] (1);
\draw[hyperedge] (4) to[bend left] (6) to[bend left=5] (3) to[bend left=5] (4);
\draw (0,0) node{$A$};
\draw (1,-0) node{$B$};
\draw (0.5,-1) node{$B_n$};
\end{tikzpicture}
}
\hskip 1cm
\vc{
\begin{tikzpicture}[scale=1.4]
\draw (45:0.9) coordinate(x1) node[vtxBig]{};
\draw (135:0.9) coordinate(x2) node[vtxBig]{};
\draw (225:0.9) coordinate(x3) node[vtxBig]{};
\draw (315:0.9) coordinate(x4) node[vtxBig]{};
\foreach \ashift in {45,135,225,315}{
\draw
(\ashift:0.9)
+(\ashift+135-30:0.3) coordinate(2)
+(\ashift+135+30:0.3) coordinate(1)
(\ashift+90:0.9)++(\ashift-45:0.3) coordinate(3
;
\draw[hyperedge] (1) to[bend left] (2) to[bend left=5] (3) to[bend left=5] (1);
\draw
(\ashift:0.9)
+(\ashift+180:0.3) coordinate(1)
+(\ashift+180+120:0.3) coordinate(2)
+(\ashift+180+240:0.3) coordinate(3)
;
\draw[hyperedge] (1) to[bend left] (2) to[bend left] (3) to[bend left] (1);
}
\draw (45:1.2) node{$A_1$};
\draw (135:1.2) node{$A_2$};
\draw (225:1.2) node{$A_3$};
\draw (315:1.2) node{$A_4$};
\draw (0,-1.6) node{$\overline{H_5}$};
\end{tikzpicture}
}
\end{center}
\caption{Illustration of $C_n$, $B_n$ and the complement of $H_5$.}\label{fig:CnBnH5}
\end{figure}
For the uniform Tur\'an density, we have
\begin{align*}
\frac{2}{3}\leq \pi_u(K_5^3) \leq 0.758,
\end{align*}
where the lower bound is obtained from the previously described $3$-graph $G$ and the upper bound from flag algebras.
\subsection{\texorpdfstring{$K_6^3$}{TEXT}}
\label{K63}
For the Tur\'an density of $K_6^3$ we know
\begin{align*}
0.84 \leq \pi(K_6^3)\leq 0.8583903,
\end{align*}
where the upper bound was obtained via flag algebras~\cite{flagmatic} and the lower bound (see \cite{Sidorenko1981SystemsOS}) is obtained by the following $3$-graph $H_{6}$ (see Figure~\ref{fig:G6H6}). Divide the vertex set $[n]$ of $H_6$ into $5$ parts $A_1,\ldots,A_{5}$ with $||A_j|-|A_i||\leq 1$ for all $1\leq i\leq j\leq 5$ and let a triple $e$ not form an edge of $H_{6}$ iff there is some $j$ ($1\leq j \leq 5$) such that
\begin{align*}
|e \cap A_{j}|\geq 2 \quad \quad \text{and} \quad \quad |e\cap A_{j}|+|e\cap A_{j+1}|=3,
\end{align*}
where $A_{6}=A_1$.
For the codegree squared density we have
\begin{align*}
0.7348 \leq\sigma(K_6^3)\leq 0.7535963,
\end{align*}
where we obtained the upper bound by flag algebras and the lower bound is achieved by the following construction from \cite{BalCleLidmain}. Let $G_6$ (see Figure~\ref{fig:G6H6}) be a $3$-graph with a vertex partition $A\cup B_1\cup B_2\cup B_3$ such that $|B_1|=|B_2|=|B_3|=bn$ and $|A|=an$, where $a$ and $b$ are optimized later. The $3$-graph $G_6$ forms a $C_{3bn}$ (see Subsection~\ref{K43}) on $B_1 \cup B_2 \cup B_3$ and further contains all edges intersecting $A$ with exactly $1$ or $2$ vertices. It has codegree squared sum
\begin{align*}
\textup{co}_2(G_6)&= \left(\frac{a^2}{2}(1-a)^2+3ab +3 \frac{b^2}{2}(a+b)^2+3b^2 (a+2b)^2\right) n^4(1+o(1))\\
&= \left(\frac{11}{18}a^4-a^3-\frac{2}{3}a^2+\frac{8}{9}a+\frac{1}{6} \right)n^4(1+o(1)).
\end{align*}
The optimum is one of the roots of $22a^3-27a^2-12a+8=0$. The root is approximately $a=0.412498$, which gives $\textup{co}_2(G_6)= 0.3674 n^4 (1+o(1))$. Note that this $3$-graph with $a=2/5$ is an instance of a family of constructions due to Keevash and Mubayi~\cite{Keevashsurvey} that achieves the best-known lower bound for the Tur\'an density of $K_6^3$.
For the minimum codegree threshold of $K_6^3$ we have
\begin{align*}
\frac{3}{4} \leq \pi_2(K_6^3)\leq 0.838,
\end{align*}
where the upper bound is obtained via flag algebras and the lower bound is due to Falgas-Ravry~\cite{codegreeFalgas} and Lo and Markstr\"om~\cite{LoMark}, see the previous section for the construction by Falgas-Ravry.
For the uniform Tur\'an density, we have
\begin{align*}
\frac{3}{4}\leq \pi_u(K_6^3) \leq 0.853,
\end{align*}
where the lower bound is obtained by the Falgas-Ravry's construction stated in the previous section and the upper bound from flag algebras.
\subsection{$K_t^k$}
Denote by $K_t^k$ the complete $k$-graph on $t$ vertices. Large cliques are studied intensively for all notions of extremality. For an overview of results in $\ell_1$-norm see \cite{MR1341481}, in $\ell_2$-norm see \cite{BalCleLidmain}, and for the minimum codegree threshold see \cite{AllanZhao} and \cite{Sidorenkocode}.
The best-known bounds for the Tur\'an density are
\begin{align*}1-\left(\frac{k-1}{t-1}\right)^{k-1} \leq \pi(K_t^k)\leq 1-\binom{t-1}{k-1}^{-1},
\end{align*}
where the lower bound is due to Sidorenko~\cite{Sidorenko1981SystemsOS} and the upper bound is due to de Caen~\cite{MR734038}.
For $k=3$, Keevash and Mubayi~\cite{Keevashsurvey} constructed a family of $3$-graphs obtaining this lower bound. Their construction for $K_t^3$-free graphs goes as follows. Take a directed graph $G$ on $t-1$ vertices with both in-degree and out-degree equal to one, i.e. $G$ is a vertex disjoint union of directed cycles. Note that cycles of length $2$ in $G$ are allowed but loops are not. Now we construct a $K_t^3$-free hypergraph $H$ on $(t-1)n$ vertices.
Partition $V(H)$ into $A_1,\ldots,A_{t-1}$ of equal sizes corresponding to vertices $v_1,\ldots,v_{t-1}$ of $G$.
A triple $xyz$ is not an edge in $H$ if and only if there exists $A_i$ such that $x,y,z \in A_i$, or there exists $v_iv_j \in E(G)$ with $|\{x,y,z\} \cap A_i|=2$ and $|\{x,y,z\} \cap A_j|=1$.
Notice that $H_5$ from Figure~\ref{fig:CnBnH5} is a case of this construction with $G$ being a directed cycle on four vertices with edges $v_1v_2, v_2v_3, v_3v_4, v_4v_1$.
Other examples are $C_n, B_n$ and $H_6$.
The best-known extremal constructions in $\ell_2$-norm for small $t$ are possibly unbalanced versions of this construction.
\begin{itemize}
\item For $t=4$, $C_n$ is the resulting 3-graph when $G$ is a directed triangle.
\item For $t=5$, $B_n$ is the resulting 3-graph when $G$ is the union of two cycles of length 2.
\item For $t=6$, $G_6$ is the resulting 3-graph when $G$ is formed by the union of a directed triangle with a directed cycle of length 2, up to balancing of the class sizes.
\end{itemize}
This suggests~\cite{BalCleLidmain} that when $G$ is maximizing the number of directed cycles, the resulting $3$-graph $H$ could be extremal in $\ell_2$-norm.
For the minimum codegree threshold of cliques of large size, the best-known bounds, obtained by Lo and Zhao~\cite{AllanZhao}, are
\begin{align*}
1- c_2 \frac{\log t}{t^{k-1}}\leq \pi_2(K_t^k) \leq 1- c_1 \frac{\log t}{t^{k-1}},
\end{align*}
where $c_1,c_2>0$ are constants depending on $k$.
\begin{figure}
\begin{center}
\tikzset{
vtx/.style={inner sep=1.1pt, outer sep=0pt, circle, fill,draw},
hyperedge/.style={fill=pink,opacity=0.5,draw=black},
vtxBig/.style={inner sep=12pt, outer sep=0pt, circle, fill=white,draw},
hyperedge/.style={fill=pink,opacity=0.5,draw=black},
}
\vc{
\begin{tikzpicture}[scale=1.4]
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{
\draw (90-72+72*\x:0.9) coordinate(x\x) node[vtxBig]{};
\draw (90-72+72*\x:1.1) node{$A_{\x}$};
}
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\pgfmathtruncatemacro{\ashift}{(90-72+72*\x)}
\draw
(\ashift:0.9)
+(\ashift+135-30:0.3) coordinate(2)
+(\ashift+135+30:0.3) coordinate(1)
(\ashift+72:0.9)++(\ashift-45:0.3) coordinate(3
;
\draw[hyperedge] (1) to[bend left] (2) to[bend left=5] (3) to[bend left=5] (1);
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(\ashift:0.85)
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\draw[hyperedge] (1) to[bend left] (2) to[bend left] (3) to[bend left] (1);
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\draw (0,-1.6) node{$\overline{H_6}$};
\end{tikzpicture}
}
\hskip 2em
\vc{
\begin{tikzpicture}[scale=1.4]
\draw (30:0.9) coordinate(x1) node[vtxBig]{};
\draw (150:0.9) coordinate(x2) node[vtxBig]{};
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(30:0.8) coordinate(1)
(150:0.8) coordinate(2)
(270:0.8) coordinate(3)
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;
\draw[hyperedge] (1) to[bend left] (2) to[bend left=5] (3) to[bend left=5] (1);
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\foreach \x in {1,2,3}{
\pgfmathtruncatemacro{\ashift}{(30+120*\x)}
\draw
(\ashift:0.85)
+(\ashift+180:0.3) coordinate(1)
+(\ashift+180+120:0.3) coordinate(2)
+(\ashift+180+240:0.3) coordinate(3)
;
\draw[hyperedge] (1) to[bend left] (2) to[bend left] (3) to[bend left] (1);
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\pgfmathtruncatemacro{\ashift}{(30+120*0)}
\draw
+(\ashift+180:0.3) coordinate(1)
+(\ashift+180+120:0.3) coordinate(2)
+(\ashift+180+240:0.3) coordinate(3)
;
\draw[hyperedge] (1) to[bend left] (2) to[bend left] (3) to[bend left] (1);
\draw (30:1.1) node{$B_2$};
\draw (150:1.1) node{$B_1$};
\draw (270:1.1) node{$B_3$};
\draw (0,-0.2) node{$A$};
\draw (0,-1.8) node{$\overline{G_6}$};
\end{tikzpicture}
}
\end{center}
\caption{Illustration of the complement of $H_6$ and $G_6$.}\label{fig:G6H6}
\end{figure}
\tikzset{redge/.style={color=red, line width=1.2pt}}
\tikzset{bedge/.style={color=blue, line width=1.2pt}}
\tikzset{gedge/.style={color=green!80!black, line width=1.2pt}}
\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw
\foreach \x in {1,2,...,5}{
(72*\x+90-72:1) node[vtx](\x){}
(72*\x+90-72:1.3) node{\x}
}
;
\draw[redge](1)--(2) (3)--(4);
\draw[bedge](1)--(3) (1)--(4) (1)--(5) (3)--(5);
\draw[gedge](2)--(3) (2)--(4) (2)--(5) (4)--(5);
\draw(0,-2) node {$F_{3,2}$};
\begin{scope}[xshift=5cm]
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(72*\x+90-72:1) node[vtx](\x){}
(72*\x+90-72:1.3) node{\x}
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;
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\draw[bedge](1)--(3) (1)--(4) (3)--(5);
\draw[gedge](2)--(3) (2)--(4) (4)--(5);
\draw(0,-2) node {$F_{5}$};
\end{scope}
\begin{scope}[xshift=10cm]
\draw
\foreach \x in {1,2,...,5}{
(72*\x+90-72:1) node[vtx](\x){}
(72*\x+90-72:1.3) node{\x}
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(72*1+90-72:1.7) node{$v_3$}
(72*2+90-72:1.7) node{$v_2$}
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\draw(0,-2) node {$C_{5}^-$};
\end{scope}
\end{tikzpicture}
\end{center}
\caption{Colorings for Theorem~\ref{sec:uniformturan}(b).}
\label{fig:colorings}
\end{figure}
\subsection{\texorpdfstring{$F_{3,2}$}{TEXT}}
\label{F32}
F\"{u}redi, Pikhurko and Simonovits~\cite{F32furedi} proved that the Tur\'an density of $F_{3,2}$ is $4/9$ which is achieved by the $3$-graph with vertex set $V = V_1 \cup V_2$, where all edges have two vertices in $V_1$ and one vertex in $V_2$ and $|V_1| = \frac{2}{3}n$.\\
The $F_{3,2}$-free graph $G$ which we conjecture to maximize the codegree square sum has similar structure, but different class sizes: $|V_1|= \lceil \sqrt{1/2}n \rceil$. Clearly, changing the class sizes will not create an $F_{3,2}$, and $\textup{co}_2(G)=1/8 n^4 (1+o(1))$. Thus, $\sigma(F_{3,2})\geq 1/4$. Flag algebras give $\sigma(F_{3,2})\leq 1/4+10^{-9}$.
Here we note that flag algebras give a numerical result of $1/4$ but obtaining an exact solution is tricky since it contains $\sqrt{1/2}$ in addition to rational numbers. Likely, using the stability method one could prove an exact result, however we have not attempted to do it. \\
The minimum codegree threshold of $F_{3,2}$ was determined exactly by Falgas-Ravry, Marchant, Pikhurko and Vaughan~\cite{codF32falgas} who proved that for large enough $n$
\begin{align*}
\textup{ex}_2(n,F_{3,2})=\begin{cases} \left\lfloor \frac{n}{3}\right\rfloor -1 & \text{ if $n$ is congruent to $1$ modulo $3$,} \\
\left\lfloor \frac{n}{3}\right\rfloor & \text{ otherwise.}
\end{cases}
\end{align*}
The lower bound is obtained by the hypergraph with vertex partition $A_1\cup A_2 \cup A_3$, where $||A_i|-|A_j||\leq 1$ for $i,j\in [3]$, and edges $xyz$ for $x,y\in A_i,z\in A_{i+1}$ for $i=1,2,3$, where $A_4:=A_1$.
The uniform Tur\'an density of $F_{3,2}$ is $\pi_u(F_{3,2})=0$ by Theorem~\ref{uniformturan}, because the edges of the shadow graph can be colored in the following way. Color the edges $12,34$ red, the edges $13,14,15,35$ blue, and the edges $23,24,25,45$ green; see Figure~\ref{fig:colorings}, and recall that the edges of $F_{3,2}$ are $123,145,245,345$.
\subsection{\texorpdfstring{$F_{3,3}$}{TEXT}}\label{sec:F33}
Keevash and Mubayi~\cite{MR2912791} and independently Goldwasser and Hansen~\cite{MR3048207} proved that the ba\-lan\-ced, complete, bipartite $3$-graph is extremal in the $\ell_1$-norm for $n\geq 6$. Thus, $\pi(F_{3,3})=3/4$. In \cite{BalCleLidmain} we proved that it is also extremal in the $\ell_2$-norm for $n$ large enough, i.e. $\sigma(F_{3,3})=5/8$. For the minimum codegree threshold we have $1/2 \leq \pi_2(F_{3,3})\leq 0.604$, where the lower bound comes from the balanced, complete, bipartite $3$-graph $B_n$ and the upper bound is obtained via flag algebras. For the uniform Tur\'an density, we have
\begin{align}
\label{inequalityk4minus}
\frac{1}{4}= \pi_u(K_4^{3-}) \leq \pi_u(F_{3,3}) \leq 0.605,
\end{align}
since $K_4^{3-} \subset F_{3,3}$. For $\pi_u(K_4^{3-})$ see Section~\ref{K43minus}.
The upper bound in \eqref{inequalityk4minus} is obtained by the method of flag algebras.
\subsection{\texorpdfstring{$F_5$}{TEXT}}
\label{F5}
Frankl and F\"uredi~\cite{F5Frankl} proved that for $n\geq 3000$ the $F_5$-free $3$-graph with the largest number of edges is $S_n$, thus, $\pi(F_5)=2/9$. The codegree squared density of $F_5$ is $\sigma(F_5)=2/27$ which follows from Theorem~\ref{F5exact}, see Section~\ref{F5section} for more details.
The minimum codegree threshold of $F_5$ is $\pi_2(F_5)=0$.
To be more precise, we have $\textup{ex}_2(n,F_5)\leq 2$, because if there exists an $F_5$-free $3$-graph $G$ with $d(x,y)\geq 3$ for all pairs $xy$, then take any pair $ab$ and two vertices $c,d \in N(a,b)$. Now there exists $e\neq a,b,c,d$ with $e\in N(c,d)$. Thus $a,b,c,d,e$ spans an $F_5$, a contradiction.
The uniform Tur\'an density of $F_5$ is $\pi_u(F_5)=0$ by Theorem~\ref{uniformturan}, because the edges of the shadow graph can be colored in the following way. Color the edges $12,34$ red, the edges $13,14,35$ blue, and the edges $23,24,45$ green; see Figure~\ref{fig:colorings}.
\subsection{Fano plane \texorpdfstring{$\mathbb{F}$}{TEXT}}
\label{Fano}
Denote by $\mathbb{F}$ the Fano plane, i.e., the unique $3$-uniform hypergraph with seven edges on seven vertices in which every pair of vertices is contained in a unique edge. S\'os~\cite{VeraSos} proposed to study the Tur\'an number for the Fano plane and conjectured the complete balanced bipartite $3$-graph $B_n$ to be extremal. This problem was solved asymptotically by Caen and F\"{u}redi \cite{FanoFuredi}. Later, F\"{u}redi and Simonovits \cite{FanoFuredi2} and, independently, Keevash and Sudakov \cite{Fanoplane} determined the extremal hypergraph for large $n$. Recently, Bellmann and Reiher \cite{FanoReiher} solved the question for all $n$. \\
We conjecture that the extremal example in the codegree squared sense also is the complete bipartite graph.
\begin{conj}
There exists $n_0$, such that for all $n\geq n_0$
\begin{align*}
\textup{exco}_2(n,\mathbb{F})= \textup{co}_2(B_n).
\end{align*}
Furthermore, $B_n$ is the unique $\mathbb{F}$-free $3$-graph $H$ on $n$ vertices satisfying $\textup{co}_2(H)=\textup{exco}_2(n,\mathbb{F})$.
\end{conj}
We have $5/8\leq \sigma(\mathbb{F}) \leq 3/4$ as trivial bounds using the complete bipartite $3$-graph as a lower bound and the fact that $\sigma(\mathbb{F})\leq \pi(\mathbb{F})\leq 3/4$.
The minimum codegree threshold was asymptotically determined to be $\pi_2(\mathbb{F})=1/2$ by Mubayi~\cite{MubayiFano}. He conjectured that $\textup{ex}_2(n,F_{3,2})=\lfloor n/2 \rfloor$ for large enough $n$. This conjecture was solved by Keevash~\cite{MR2478542} using an involved quasi-randomness argument. Later, DeBiasio and Jiang~\cite{MR3131883} gave a simplified proof.
We did not use flag algebras for $\mathbb{F}$ because our computers cannot handle the number of $\mathbb{F}$-free $3$-graphs on $7$ vertices.
The uniform Tur\'an density of $\mathbb{F}$ is $\pi_u(\mathbb{F})=0$ by Theorem~\ref{uniformturan}, because the edges of the shadow graph can be colored in the following way. Color the edges $12,34,15,24,14,25,36$ red, the edges $13,35,16,26,17,27,37$ blue, and the edges $23,45,56,46,47,57,67$ green, where the edges of the Fano plane are are $123,345,156,246,147,257,367$.
\subsection{\texorpdfstring{$K_4^{3-}$}{TEXT}}
\label{K43minus}
Denote by $S_6$ the $3$-graph on six vertices with edge set
\begin{align*}
E(S_6)=\{123,234,345,451,512,136,246,356,256,146\}.
\end{align*}
Let $n=6^k$ be a power of $6$ and let $G$ be the iterated blow-up of $S_6$. This $3$-graph $G$ was constructed by Frankl and F\"{u}redi~\cite{K43-extremalfrankl} to give a lower bound on the Tur\'an density of $K_4^{3-}$. Since $G$ is $K_4^{3-}$-free, $\pi(K_4^{3-})\geq 2/7$. The best know upper bound $\pi(K_4^{3-})\leq 0.28689$ is by Vaughan~\cite{flagmatic} using flag algebras.
The $3$-graph $G$ has codegree squared sum
\begin{align*}\textup{co}_2(G)&=\frac{1}{2}n^4 \left(\sum_{i\geq1} \frac{5}{6^i}\left( \frac{2}{6^i}\right)^2 + o(1) \right)=10n^4 \left(\sum_{i\geq1} \frac{1}{216^i} + o(1) \right)\\
&=10n^4 \left(\frac{1}{1-\frac{1}{216}}-1\right) (1+o(1))
=\frac{2}{43}n^4(1+o(1)).
\end{align*}
Thus, $\sigma(K_4^{3-})\geq 4/43$. We conjecture that $G$ is the extremal example in $\ell_2$-norm.
\begin{conj}
\begin{align*}\sigma(K_4^{3-})= \frac{4}{43} \approx 0.0930232558.
\end{align*}
\end{conj}
Flag algebras only give $\sigma(K_4^{3-})\leq 0.09306286$. The minimum codegree threshold of $K_4^{3-}$ was determined to be $\pi_2(K_4^{3-})=1/4$ by Falgas-Ravry, Pikhurko, Vaughan and Volec~\cite{FalgasK4-} solving a conjecture by Nagle~\cite{codegreeconj}. The lower bound is obtained by a construction originally due to Erd\H{o}s and Hajnal~\cite{MR0337636}. Given a tournament $T$ on the vertex set $[n]$, define a $3$-graph $C(T)$ on $[n]$ by taking the edge set to be all triples of vertices inducing a cyclically oriented triangle in $T$. No tournament on $4$ vertices can contain more than two cyclically oriented triangles, hence $C(T)$ is $K_4^{3-}$-free. If $T$ is chosen uniformly at random, then the minimum codegree of $C(T)$ is $n/4-o(n)$ with high probability.
The uniform Tur\'an density of $K_{4}^{3-}$ was determined to be $\pi_u(K_4^{3-})=1/4$ by Glebov, Kr\'al' and Volec~\cite{MR3474967}. The lower bound is also achieved by $C(T)$.
\subsection{\texorpdfstring{\{$K_4^{3-},F_{3,2},C_5$\}}{TEXT}}
\label{K43minusF32C5}
Denote by $H_7$ the $3$-graph on seven vertices with edge set
\begin{align*} E(H_7)=\{124,137,156,235,267,346,457,653,647, 621,542,517,431,327\}.
\end{align*}
Falgas-Ravry and Vaughan~\cite{RavryTuran} proved that $\sigma(K_4^{3-},F_{3,2},C_5)=12/49$, where the lower bound is achieved by the blow-up of $H_7$ on $n$ vertices. However, the codegree squared density is achieved by a different $3$-graph, namely the complete balanced $3$-partite $3$-graph $S_n$. The $3$-graph $S_n$ is $K_4^{3-}$-free, $F_{3,2}$-free and $C_5$-free and it has codegree squared sum $\textup{co}_2(S_n)=1/27 n^4$. Thus, $\sigma(\{K_4^{3-},F_{3,2},C_5\})\geq 2/27$. Flag algebras give $\sigma(\{K_4^{3-},F_{3,2},C_5\})\leq 2/27$, hence,
\begin{align*}
\sigma(\{K_4^{3-},F_{3,2},C_5\})= \frac{2}{27}.
\end{align*}
For the minimum codegree threshold we have
\begin{align*}
\frac{1}{12}\leq\pi_2(\{K_4^{3-},F_{3,2},C_5\})\leq 0.186,
\end{align*}
where the upper bound is obtained by flag algebras and the lower bound by the following construction. Consider the intersection of the Erd\H{o}s-Hajnal~\cite{MR0337636} random tournament construction $C(T)$ with the $3$-graph having vertex partition $A_1\cup A_2 \cup A_3$, where $||A_i|-|A_j||\leq 1$ for $i,j\in [3]$, and edges $xyz$ for $x,y\in A_i,z\in A_{i+1}$ for $i=1,2,3$, where $A_4=A_1$. This $3$-graph has minimum codegree $n/12-o(n)$ with high probability, and is $C_5$, $F_{3,2}$ and $K_4^{3-}$-free.
For the uniform Tur\'an density we have
\begin{align*}
\pi_u(\{K_4^{3-},F_{3,2},C_5\})\leq \pi_u(F_{3,2})=0.
\end{align*}
\subsection{\texorpdfstring{\{$K_4^{3-},F_{3,2}$\}}{TEXT}}
\label{K43-F32}
Let $G$ be the blow-up of $H_6$ on $n$ vertices. This $3$-graph $G$ is $K_4^{3-}$ and $F_{3,2}$-free. Falgas-Ravry and Vaughan~\cite{RavryTuran} proved that $\pi(K_4^{3-},F_{3,2})=5/18$ with the lower bound obtained by $G$. The codegree squared sum of $G$ is
\begin{align*} \textup{co}_2(G)= \binom{6}{2}\left(\frac{n}{6} \right)^2 \left( \frac{n}{3}\right)^2 = \frac{5}{108}n^4.
\end{align*}
Thus, $\sigma(\{K_4^{3-},F_{3,2}\})\geq 5/54$. Flag algebras give $\sigma(\{K_4^{3-},F_{3,2}\})\leq 5/54$, hence
\begin{align*}
\sigma(\{K_4^{3-},F_{3,2}\})=\frac{5}{54}.
\end{align*}
For the minimum codegree threshold we have
\begin{align*}
\frac{1}{12}\leq\pi_2(\{K_4^{3-},F_{3,2},C_5\})\leq \pi_2(\{K_4^{3-},F_{3,2}\} \leq 0.202,
\end{align*}
where the upper bound comes from flag algebras. For the uniform Tur\'an density we have
\begin{align*}
\pi_u(\{K_4^{3-},F_{3,2}\})\leq \pi_u(F_{3,2})=0.
\end{align*}
\subsection{\texorpdfstring{\{$K_4^{3-},C_5$\}}{TEXT}}
\label{K43-C5}
Denote by $G$ be the iterated blow-up of an edge on $n$ vertices. In other words, it is obtained from a complete balanced $3$-partite $3$-graph by inserting a complete balanced $3$-partite $3$-graph in each of the parts iteratively.
The $3$-graph $G$ is $K_4^{3-} $ and $C_5$-free. This construction gives the current best lower bound on the Tur\'an density of $\{K_4^{3-},C_5\}$, see~\cite{RavryTuran}. For the codegree squared density, it gives $\sigma(\{K_4^{3-},C_5\})\geq {1}/{13}$.
We conjecture it to be the extremal example in $\ell_2$-norm.
\begin{conj}
\begin{align*}
\sigma(\{K_4^{3-},C_5\})=\frac{1}{13} \approx 0.0769230769.
\end{align*}
\end{conj}
Flag algebras give $\sigma(\{K_4^{3-},C_5\})\leq 0.07694083$ for the codegree squared density. For the minimum codegree threshold we have
\begin{align*}
\frac{1}{12}\leq\pi_2(\{K_4^{3-},F_{3,2},C_5\})\leq \pi_2(\{K_4^{3-},C_5\} \leq 0.202,
\end{align*}
where the upper bound comes from flag algebras. For the uniform Tur\'an density, we have
\begin{align*}
\frac{1}{27}\leq \pi_u(\{K_4^{3-},C_5\}) \leq 0.251,
\end{align*}
where the upper bound is obtained via the method of flag algebras and the lower bound comes from Theorem~\ref{uniformturan} and Corollary~\ref{uniformturanjump} which can easily be modified to also hold for families of $3$-graphs. We check in Section~\ref{sec:C5} that $C_5$ does not satisfy property (b) from Theorem~\ref{uniformturan}. Since $\pi_u(K_4^{3-})=\frac{1}{4}$, it also does not satisfy property (b).
\subsection{\texorpdfstring{\{$F_{3,2},J_4$\}}{TEXT}}
\label{F32-J4}
Let $G$ be the blow-up of $K_4^3$ on $n$ vertices. This graph $G$ is $F_{3,2}$-free and $J_4$-free. Falgas-Ravry and Vaughan~\cite{RavryTuran} proved that $\pi(\{F_{3,2},J_4\})=3/8$, where the lower bound is achieved by $G$. The $3$-graph $G$ has codegree squared sum $\textup{co}_2(G)= 3/32 n^4(1+o(1))$, thus, $\sigma(\{F_{3,2},J_4\})\geq 3/16$. Flag algebras give $\sigma(\{F_{3,2},J_4\})\leq 3/16$, hence
\begin{align*}
\sigma(F_{3,2},J_4)=\frac{3}{16}.
\end{align*}
For the minimum codegree threshold we have
\begin{align*}
\frac{1}{12}\leq\pi_2(\{K_4^{3-},F_{3,2}\})\leq \pi_2(\{F_{3,2},J_4\} \leq \pi_2(F_{3,2})\leq \frac{1}{3},
\end{align*}
because $K_4^{3-}$ is a subhypergraph of $J_4$. For the uniform Tur\'an density we have
\begin{align*}
\pi_u(\{F_{3,2},J_4\})\leq \pi_u(F_{3,2})=0.
\end{align*}
\subsection{\texorpdfstring{\{$F_{3,2},J_5$\}}{TEXT}}
\label{F32-J5}Falgas-Ravry and Vaughan~\cite{RavryTuran} proved that $\pi(\{F_{3,2},J_5\})=3/8$, where the blow-up of $K_4^3$ on $n$ vertices achieves the lower bound. Since $J_4\subset J_5$, we have
\begin{align*}
\frac{3}{16}\leq \sigma(\{F_{3,2},J_4\}) \leq \sigma(\{F_{3,2},J_5\}) \quad \quad \text{and} \quad \quad \frac{1}{12}\leq \pi_2(\{F_{3,2},J_4\}) \leq \pi_2(\{F_{3,2},J_5\}).
\end{align*}
By flag algebras we have for the codegree squared density $\sigma(\{F_{3,2},J_5\})\leq \frac{3}{16}$ and thus $\sigma(\{F_{3,2},J_5\})=3/16$, and for the minimum codegree threshold $\pi_2(\{F_{3,2},J_5\})\leq 0.28$. For the uniform Tur\'an density we have
\begin{align*}
\pi_u(\{F_{3,2},J_5\})\leq \pi_u(F_{3,2})=0.
\end{align*}
\subsection{\texorpdfstring{$J_4$}{TEXT}}
\label{sec:J4}
Let $G$ be the iterated blow-up of the complement of the Fano plane on $n$ vertices. This $3$-graph is $J_4$-free and conjectured to be an asymptotically sharp example in $\ell_1$-norm, see \cite{DaisyBollobas}. It has codegree squared sum of
\begin{align*}
\textup{co}_2(G)=\frac{48}{343} n^4 \sum \left(\frac{1}{7^4}\right)^i=\frac{7}{50} n^4.
\end{align*}
Thus $\sigma(J_4)\geq 0.28$. We conjecture it to be the extremal example in $\ell_2$-norm.
\begin{conj}
\begin{align*}
\sigma(J_4)=0.28.
\end{align*}
\end{conj}
Flag algebras only give $\sigma(J_4)\leq 0.28079696$. For the minimum codegree threshold and the uniform Tur\'an density of $J_4$ we have
\begin{align*}
\frac{1}{4} \leq \pi_2(K_4^{3-})\leq \pi_2(J_4)\leq 0.473 \quad \quad \quad \text{and} \quad \quad \quad \frac{1}{4} \leq \pi_u(K_4^{3-})\leq \pi_u(J_4)\leq 0.49,
\end{align*}
where the upper bounds are obtained using flag algebras and the lower bound bound holds because $K_4^{3-} \subset J_4$.
\subsection{\texorpdfstring{$J_5$}{TEXT}}
\label{sec:J5}
Since $J_4\subset J_5$, we get the following lower bounds:
\begin{align*}
\frac{1}{2} \leq \pi(J_4)\leq \pi(J_5), \ 0.28 \leq \sigma(J_4)\leq \sigma(J_5), \ \frac{1}{4} \leq \pi_2(J_4)\leq \pi_2(J_5) \quad \text{and} \quad \frac{1}{4} \leq \pi_u(J_4)\leq \pi_u(J_5).
\end{align*}
Flag algebras give us the following upper bounds
\begin{align*}
\pi(J_5)\leq 0.64474184, \quad \quad \sigma(J_5)\leq 0.4427457, \quad \quad \pi_2(J_5)\leq 0.613 \quad \quad \text{and} \quad \quad \pi_u(J_5)\leq 0.631.
\end{align*}
It would be interesting, if one could separate the parameters of $J_4$ and $J_5$ from each other.
\subsection{\texorpdfstring{$C_5$}{TEXT}}
\label{sec:C5}
Denote by $G(n)$ the $3$-graph on $n$ vertices where the vertex set is partitioned into two sets $A$ and $B$ of sizes $bn$ and $(1-b)n$, where $b\in(0,1)$ and $E(G(n))$ consists of all triples $xyz$, where $x,y\in B$ and $z\in A$. This $3$-graph is $C_5$-free. Let $G_{it}$ be the $3$-graph constructed from $G(n)$ by iteratively adding $G(bn)$ inside the class $B$. The $3$-graph $G_{it}$ is also $C_5$-free. This construction gives the current best-known lower bound on the Tur\'an density of $C_5$, see~\cite{F32Mubayi}. It's codegree squared sum can be lower bounded as follows
\begin{align*}
\textup{co}_2(G_{it}) > \left(b^2\frac{(1-b)^2}{2} + b(1-b)b^2 \right) \left( \frac{1}{1-(1-b)^4}\right)n^4 > 0.12597 n^4
\end{align*}
for $b$ of size about $0.7$, thus $\sigma(C_5)\geq 0.25194$. Flag algebras give $\sigma(C_5)\leq 0.25310457$. For the minimum codegree threshold of $C_5$ we have
\begin{align*}
\frac{1}{3}\leq \pi_2(C_5)\leq 0.3993,
\end{align*}
where the upper bound comes from flag algebras and the lower bound from the graph with vertex partition $A_1\cup A_2 \cup A_3$, where $||A_i|-|A_j||\leq 1$ for $i,j\in [3]$, and edges $xyz$ for $x,y\in A_i,z\in A_{i+1}$ for $i=1,2,3$, where $A_4=A_1$.
For the uniform Tur\'an density of $C_5$ we have
\begin{align*}
\frac{1}{27} \leq \pi_u(C_5) \leq 0.402,
\end{align*}
where the upper bound is obtained via flag algebras and the lower bound by the following argument. Let $v_1,v_2,v_3,v_4,v_5$ be the vertices of $C_5$, that is $v_1v_2v_3,v_2v_3,v_4,v_3v_4v_5,v_4v_5v_1,v_5v_1v_2$ are edges. Using Theorem~\ref{uniformturan}, we will show that there exists no vertex ordering satisfying property (b) from Theorem~\ref{uniformturan} and thus $\pi_u(C_5)\neq 0$. Corollary~\ref{uniformturanjump} implies $\pi_u(C_5)\geq 1/27$. Without loss of generality, we can assume that $v_2<v_3$. We perform a case analysis depending on the positioning of $v_4$.\\
\noindent
\underline{Case 1: $v_2<v_3<v_4$:}
Since $v_2v_3v_4$ and $v_3v_4v_5$ are edges, $v_5<v_3<v_4$. Since $v_1v_2v_3$ and $v_2v_3v_4$ are edges, $v_2<v_3<v_1$. Because $v_3v_4v_5$ and $v_4v_5v_1$ are edges, we further have $v_5<v_1<v_4$. Since $v_1v_2v_5$ and $v_1v_2v_3$ are edges, we get $v_2<v_5<v_1$. Thus, $v_2<v_5<v_3<v_1<v_4$, contradicting that $v_5v_1v_2$ and $v_4v_5v_1$ are edges. \\
\noindent
\underline{Case 2: $v_4<v_2<v_3$:}
Since $v_2v_3v_4$ and $v_3v_4v_5$ are edges, $v_4<v_5<v_3$. Since $v_1v_2v_3$ and $v_2v_3v_4$ are edges, $v_1<v_2<v_3$. Because $v_3v_4v_5$ and $v_4v_5v_1$ are edges, we further have $v_4<v_5<v_1$. Thus we have $v_4<v_5<v_1<v_2<v_3$, contradicting that $v_1v_2v_5$ and $v_1v_2v_3$ are edges.\\
\noindent
\underline{Case 3: $v_2<v_4<v_3$:}
Since $v_2v_3v_4$ and $v_3v_4v_5$ are edges, $v_5<v_4<v_3$. Since $v_1v_2v_3$ and $v_2v_3v_4$ are edges, $v_2<v_1<v_3$. Further, since $v_1v_2v_5$ and $v_1v_2v_3$ are edges $v_2<v_1<v_5$. Thus, we have $v_2<v_1<v_5<v_4<v_3$, contradicting that $v_4v_5v_1$ and $v_5v_1v_2$ are edges.
We conclude that there exists no vertex ordering satisfying property (b) from Theorem~\ref{uniformturan}.
\subsection{\texorpdfstring{$C_5^-$}{TEXT}}
\label{sec:C5minus}
Denote by $G$ be the iterated blow-up of an edge on $n$ vertices. The $3$-graph $G$ is $C_5^-$-free. This construction gives the current best-known lower bound on the Tur\'an density of $C_5^-$, see~\cite{F32Mubayi}. For the codegree squared density, it gives $\sigma(C_5^-)\geq {1}/{13}$.
We conjecture it to be the extremal example in $\ell_2$-norm.
\begin{conj}
\begin{align*}
\sigma(C_5^-)=\frac{1}{13} \approx 0.0769230769.
\end{align*}
\end{conj}
Flag algebras give $\sigma(C_5^-)\leq 0.07725405$ for the codegree squared density and $\pi_2(C_5^-)\leq 0.136$ for the minimum codegree threshold. The uniform Tur\'an density of $C_5^-$ is $\pi_u(C_5^-)=0$. This was observed by Reiher, R\"{o}dl and Schacht~\cite{MR3764068} using Theorem~\ref{uniformturan}. For the sake of completeness, we repeat the proof here. Let $v_1,v_2,v_3,v_4,v_5$ form a $C_5^-$, that is $v_1v_2v_3,v_2v_3v_4,v_3v_4v_5,v_4v_5v_1$ are edges. Consider the ordering $v_3<v_2<v_1<v_5<v_4$ of the vertices. The edges of the shadow graph can be colored in the following way. Color the edges $v_3v_2,v_3v_5,v_1v_5$ red, the edges $v_1v_3,v_3v_4,v_1v_4$ blue, and the edges $v_1v_2,v_2v_4,v_4v_5$ green; see Figure~\ref{fig:colorings}. Thus, property (b) of Theorem~\ref{uniformturan} holds.
\subsection{\texorpdfstring{$K_5^=$}{TEXT}}
\label{sec:K_5^=}
Let $K_5^=$ be the $3$-graph on $5$ vertices with $8$ edges, where the two missing edges intersect in exactly one vertex. We have
\begin{align*}
0.58656 \leq \frac{227}{387}\leq \pi(K_5^=)\leq 0.60961944 \quad \text{and} \quad 0.357942 \leq \frac{2909}{8127}\leq \sigma(K_5^=)\leq 0.388725.
\end{align*}
The hypergraph providing both lower bounds is the following. Take the $3$-uniform hypergraph on $n$ vertices with vertex set $V_1\cup V_2 \cup V_3$ such that $||V_i|-|V_j||\leq 1$ for $i\neq j$ and edge set
\begin{align*}
\{abc: a\in V_1,b\in V_2, c\in V_3 \} \cup \{abc: a,b\in V_1,c\in V_2 \} \\ \cup \ \{abc: a,b\in V_2,c\in V_3 \} \cup \{abc: a,b\in V_3,c\in V_1 \}.
\end{align*} Now, place in each of the three classes an iterated blowup of $H_6$, recall $H_6$ was defined in Section~\ref{K43minus}. This $3$-graph has
\begin{align*}
\frac{5}{54}n^3+30 \sum\left( \frac{n}{3} \cdot \frac{1}{6^i}\right)^3 +o(n^3)=\frac{227}{2322}n^3(1+o(1))\geq 0.58656 \binom{n}{3}(1+o(1))
\end{align*}
edges, codegree squared sum of
\begin{align*} \frac{4}{27}n^4 + 3 \frac{1}{2} \left(\frac{n}{3}\right)^4 \sum_{k\geq 1} \frac{5}{6^k} \left(\frac{2}{6^k}+1 \right)^2= \frac{2909}{8127} \frac{n^4}{2},
\end{align*}
and is $K_5^=$-free. We conjecture the lower bounds to be asymptotically sharp.
\begin{conj}
\begin{align*}\pi(K_5^=)=\frac{227}{387} \quad \quad \text{and} \quad \quad \sigma(K_5^=)=\frac{2909}{8127}.
\end{align*}
\end{conj}
For the minimum codegree threshold and uniform Tur\'an density of $K_5^=$, we have
\begin{align*}
\frac{1}{2} \leq \pi(K_4^3)\leq \pi_2(K_5^=)\leq 0.569 \quad \quad \text{and} \quad \quad \frac{1}{2} \leq \pi_u(K_4^3)\leq \pi_u(K_5^=)\leq 0.567,
\end{align*}
where the upper bound is obtained via flag algebras. It would be interesting to decide if those parameters are equal to each other.
\subsection{\texorpdfstring{$K_5^<$}{TEXT}}
\label{sec:K_5^<}
Let $K_5^<$ be the $3$-graph on $5$ vertices with $8$ edges, where the two missing edges intersect in exactly two vertices. Since $K_4^3$ is a subgraph of $K_5^<$, we have
\begin{gather*}
\frac{5}{9} \leq \pi(K_4^3) \leq
\pi(K_5^<), \quad \frac{1}{3} \leq \sigma(K_4^3 ) \leq \sigma(K_5^<), \quad
\frac{1}{2} \leq \pi_2(K_4^3) \leq \pi_2(K_5^<) \\
\text{and} \quad \quad \frac{1}{2} \leq \pi_u(K_4^3) \leq \pi_u(K_5^<).
\end{gather*}
Using flag algebras we obtain
\begin{gather*}
\pi(K_5^<)\leq 0.57704775,\quad \sigma(K_5^<)\leq 0.3402107, \quad \pi_2(K_5^<)\leq 0.560 \quad \text{and} \quad \pi_u(K_5^<)\leq 0.568.
\end{gather*}
We conjecture the lower bounds to be sharp.
\begin{conj}
\begin{align*}
\pi(K_5^<)=\frac{5}{9}, \quad \quad \sigma(K_5^<)=\frac{1}{3}, \quad \quad \pi_2(K_5^<)=\frac{1}{2} \quad \quad \text{and} \quad \quad \pi_u(K_5^<)=\frac{1}{2}.
\end{align*}
\end{conj}
\subsection{\texorpdfstring{$K_5^-$}{TEXT}}
\label{sec:K_5^-}
Let $K_5^-$ be the $3$-graph on $5$ vertices with $9$ edges. We have
\begin{gather*}
\frac{227}{387}\leq \pi(K_5^=)\leq \pi(K_5^-), \quad \frac{2909}{8127} \leq \sigma(K_5^=) \leq \sigma(K_5^-), \quad \frac{1}{2} \leq\pi_2(K_4^3) \leq \pi_2(K_5^-)
\\
\text{and} \quad \quad \frac{1}{2}\leq \pi_u(K_4^3)\leq \pi_u(K_5^-).
\end{gather*}
Using flag algebras we obtain
\begin{align*}
\pi(K_5^-)\leq 0.64208589,\quad \sigma(K_5^-)\leq 0.41961743,\quad \pi_2(K_5^-)\leq 0.621 \quad \text{and} \quad \pi_u(K_5^-)\leq 0.626.
\end{align*}
It would be interesting to decide if those parameters are equal to each other.
\section{Proof of Theorem~\ref{F5exact}}
\label{F5section}
Bollob\'as~\cite{Bollobascancellative} proved that the $\{F_4,F_5\}$-free $3$-graph with the largest number of edges is $S_n$, the complete balanced $3$-partite $3$-graph. His idea was to construct an almost partition of three pairwise disjoint sets with the property that every edge has at most one point from each and then use a counting argument to show that the number of edges is maximized when the classes form a partition and have equal size.
This result of Bollob\'as~\cite{Bollobascancellative} was extended by Frankl and F\"uredi~\cite{F5Frankl} who proved that for $n\geq 3000$ the $F_5$-free $3$-graph with the largest number of edges is $S_n$. Keevash and Mubayi~\cite{MubayiF5} proved a stability result for $F_5$-free graphs in the $\ell_1$-norm using a different method from the one in \cite{Bollobascancellative}. There is a straightforward modification of Bollob\'as' result which we will use to get a stability result for $F_5$-free graphs in the $\ell_2$-norm.
\begin{theo}
\label{F5stability}
For every $\varepsilon > 0$ there exists $\delta>0$ and $n_0\in \mathbb{N}$ such that if $G$ is an $F_5$-free $3$-uniform hypergraph on $n \geq n_0$ vertices with $\textup{co}_2(G)\geq (1-\delta) \frac{1}{27} n^4$, then we can partition $V(H)=A \cup B \cup C$ such that
\begin{align}
\label{stabcondition1}
e(A)+e(B)+e(C)+e(A,B)+e(B,C)+e(A,C)\leq \varepsilon n^3
\end{align}
and
\begin{align}
\label{stabcondition2}
e(A,B,C)\geq \frac{1}{27}n^3 -\varepsilon n^3.
\end{align}
\end{theo}
Note that, unlike the $\ell_1$-norm case, here it is not instantly obvious why \eqref{stabcondition1} and \eqref{stabcondition2} are the same conditions, as having high codegree square sum does not imply the existence of many edges.
\begin{proof}
Let $\varepsilon>0$ and choose a constant $\delta=\delta(\varepsilon)$ such that the following argument holds. Let $G$ be an $n$-vertex $F_5$-free $3$-graph satisfying $\textup{co}_2(G)\geq (1-\delta)1/27n^4$. Choose $v_1,v_2\in V(G)$ such that $d(v_1,v_2)$ is maximum among all pairs of vertices. Denote $A'=N(v_1,v_2)$ and note that there is no hyperedge with two vertices from $A'$ and one vertex from $V(G)\setminus\{v_1,v_2\}$ because $G$ is $F_5$-free, see Figure~\ref{fig:ABC} for an illustration.
Choose $v_3\in A', v_4\in V(G)\setminus (A' \cup \{v_1,v_2\})$ such that $d(v_3,v_4)$ is maximum and denote $B'=N(v_3,v_4)\setminus \{v_1,v_2\}$. We have $A'\cap B'=\emptyset$, because for $x\in A'\cap B'$, $v_3v_4x$ is an edge with two vertices in $A'$ and one in $V(G)\setminus\{v_1,v_2\}$. Again, there are no edges with two vertices from $B'$ and one vertex from $V(G)\setminus\{v_1,v_2,v_3,v_4\}$ because $G$ is $F_5$-free. Next, choose $v_5\in A'\setminus \{v_3\},\ v_6\in B' \setminus \{v_1,v_2,v_3,v_4\}$ such that $d(v_5,v_6)$ is maximum. Set $C'=N(v_5,v_6)\setminus\{v_1,v_2,v_3,v_4\}$. We have $C' \cap A'=\emptyset$ and $C'\cap B'=\emptyset$ as otherwise there is an edge with two vertices in either $A'$ or $B'$ and one in $V(G)\setminus \{v_1,v_2,v_3,v_4\}$, which would give a copy of $F_5$. Set $E:=\{v_1,v_2,v_3,v_4,v_5,v_6\}$. There is no edge with two vertices in $C'$ and one in $V(G)\setminus E$, because if $c_1c_2x$ is such an edge, then $\{v_5,v_6,c_1,c_2,x\}$ induces an $F_5$. Set
\begin{align*}
A:= A' \setminus E, \quad B:= B' \setminus E, \quad D=V(G) \setminus (A\cup B\cup C'), \quad C=C' \cup D=V(G)\setminus(A \cup B)
\end{align*}
and note that $V(G)=A\cup B\cup C' \cup D=A\cup B\cup C$. Denote $a=|A|/n$, $b=|B|/n$, $c'=|C'|/n$ and $d=|D|/n$. Thus, $a+b+c'+d=1$.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw
(-1,-0.2) coordinate (1) node[vtx,label=left:$v_1$]{}
(-1,1.2) coordinate (2) node[vtx,label=left:$v_2$]{}
(0.3,0.5) coordinate (A) node[vtx]{}
(1,0.5) circle(1cm) node {$A'$}
(4,0.5) circle(1cm) node {$B'$}
(2.5,-1.5) circle(1cm) node {$C'$
(1.5,1) coordinate (3) node[vtx,label=left:$v_3$]{}
(2.5,2.5) coordinate (4) node[vtx,label=above:$v_4$]{}
(3.5,1) coordinate (B) node[vtx,label=right:$ $]{}
(1.5,0) coordinate (5) node[vtx,label=left:$v_5$]{}
(3.5,0) coordinate (6) node[vtx,label=right:$v_6$]{}
(2.5,-1.2) coordinate (C) node[vtx,label=right:$ $]{}
;
\draw[hyperedge] (1) to[bend left] (A) to[bend left] (2) to[bend left] (1);
\draw[hyperedge] (3) to[bend left] (B) to[bend left] (4) to[bend left] (3);
\draw[hyperedge] (5) to[bend left] (C) to[bend left] (6) to[bend left] (5);
\end{tikzpicture}
\end{center}
\caption{$A',B',C'$ and $v_1,v_2,v_3,v_4,v_5,v_6$ from the proof of Theorem~\ref{F5stability}.}
\label{fig:ABC}
\end{figure}
\begin{claim}
\label{classsizesofblobs}
We have
\begin{align*}
\left\vert a-\frac{1}{3}\right\vert\leq \frac{\varepsilon}{10} ,\quad \quad \left\vert b-\frac{1}{3}\right\vert\leq \frac{\varepsilon}{10} ,\quad \quad \left\vert c'-\frac{1}{3}\right\vert\leq \frac{\varepsilon}{10} ,\quad \quad d\leq \frac{\varepsilon}{10}.
\end{align*}
\end{claim}
\begin{proof}
Assume, toward a contradiction, that the claim is false.
The only triples with two vertices from either $A$, $B$ or $C'$ which can be edges are those with one vertex in $E$. Therefore, we get
\begin{align*}
d(x,y)\leq 6 \text{ for } x,y\in A, \quad
d(x,y)\leq 6\text{ for } x,y\in B \quad \text{and} \quad
d(x,y)\leq 6 \quad \text{for} \quad x,y\in C'.
\end{align*}
By the choices of the vertices in $E$, we further have
\begin{equation*}
d(x,y)\leq c'n+6 \quad \text{for} \quad x\in A,\ y\in B, \quad d(x,y)\leq bn+6 \quad \text{for} \quad x\in A,\ y\in C' \cup D \setminus E
\end{equation*}
\begin{equation*}
\text{and} \quad d(x,y)\leq an \quad \text{for} \quad x,y \in V(G).
\end{equation*}
Now,
\begin{align}
\nonumber
\label{opti}
\frac{1}{27}(1-\delta)n^4 &\leq \textup{co}_2(G) \leq ab(c'n+6)^2n^2+an(c'n+dn+6)(bn+6)^2 + |E|n(an)^2\\
&+\left( \binom{(b+c'+d)n}{2}-\binom{bn}{2}-\binom{c'n}{2} \right)(an)^2 \nonumber\\
&\leq \left( abc'^2+a(c'+d)b^2 + \left( \frac{(b+c'+d)^2}{2}-\frac{b^2}{2}-\frac{c'^2}{2} \right)a^2 \right)n^4+100n^3 \nonumber \\
&= f(a,b,c',d)n^4 +100n^3,
\end{align}
where $f(w,x,y,z)=wxy^2+w(y+z)x^2 +\left( \frac{(x+y+z)^2}{2}-\frac{x^2}{2}-\frac{y^2}{2} \right)w^2$. Define
\begin{align*}
R=\{(w,x,y,z) \ : \ w,x,y,z \geq 0, \ w+x+y+z=1 \} \subset \mathbb{R}^4.
\end{align*}
The polynomial $f$ has a maximum on $R$ of $1/27$ attained at unique point $w,x,y=1/3, \ z=0$.
This can be checked with a tedious calculation or using a computer, we omit the details. Define the open ball
\begin{align*}
U:=\left\{(w,x,y,z) \ : \ \left\vert w-\frac{1}{3}\right\vert < \frac{\varepsilon}{10} ,\quad \left\vert x-\frac{1}{3}\right\vert <\frac{\varepsilon}{10} ,\quad \left\vert y-\frac{1}{3}\right\vert< \frac{\varepsilon}{10} ,\quad z < \frac{\varepsilon}{10} \right\}.
\end{align*}
The regions $R$ and $R\setminus U$ are closed and bounded, thus also compact on $\mathbb{R}^4$. Therefore, the polynomial $f$ attains a maximum on $R\setminus U$. Set
\begin{align*}
\delta:= \min \left\{\varepsilon,\frac{1}{27}- \max_{(w,x,y,z)\in R \setminus U} f(w,x,y,z) \right\},
\end{align*}
and note that $\delta=\delta(\varepsilon)>0$, since the unique maximum of $f$ on $R$ is also in $U$. The point $(a,b,c',d)\in R\setminus U$ by assumption and therefore $f(a,b,c',d)\leq 1/27-\delta$. Combining this observation with inequality \eqref{opti}, we get
\begin{align*}
\frac{1}{27}(1-\delta)n^4 \leq f(a,b,c',d)n^4 +100n^3 \leq \frac{1}{27}n^4-\delta n^4 +100n^3 < \frac{1}{27}(1-\delta)n^4,
\end{align*}
for $n$ large enough. This is a contradiction and thus Claim~\ref{classsizesofblobs} holds.
\end{proof}
Denote $H=G[A\cup B \cup C']$. Then,
\begin{align*}
\textup{co}_2(H)\geq \textup{co}_2(G)-\frac{6}{10}\varepsilon n^4 \geq (1-\delta)\frac{1}{27}n^4 -\frac{6}{10}\varepsilon n^4 \geq \frac{1}{27}n^4-\varepsilon n^4, \end{align*}
because the number of edges incident to $D$ is at most $(\varepsilon/10) n^3$ by Claim~\ref{classsizesofblobs} and the codegree squared sum of a $3$-graph decreases by at most $6n$ when one edge is removed. The $3$-graph $H$ is $3$-partite. Thus, for each edge $e\in E(H)$, $w_{H}(e)\leq n$. By double-counting, we get a lower bound on the size of $E(H)$,
\begin{align*}
\frac{1}{27}n^4-\varepsilon n^4 \leq \textup{co}_2(H) = \sum_{e\in E(H)} w_{H}(e)\leq |E(H)|n.
\end{align*}
Consider the vertex partition $V(G)=A \cup B \cup C$. For the number of cross-edges, we have
\begin{align*}
e(A,B,C) \geq e(A,B,C')=|E(H)|\geq \frac{1}{27}n^3-\varepsilon n^3
\end{align*}
and thus the partition $V(G)=A \cup B \cup C$ satisfies \eqref{stabcondition2}.
Since there is no edge in $G$ with two vertices in one of the classes $A,B$ or $C'$ and because there are at most $(\varepsilon/10) n^3+6n^2$ edges incident to $D\cup E$ by Claim~\ref{classsizesofblobs}, the partition $V(G)=A\cup B \cup C$ also satisfies
\begin{align*}
& e(A)+e(B)+e(C)+e(A,B)+e(B,C)+e(A,C) \\
&=e(C)+e(B,C)+e(A,C)\leq \frac{\varepsilon}{10} n^3+6n^2 \leq \varepsilon n^3.
\end{align*}
Hence, \eqref{stabcondition1} holds, completing the proof of Theorem~\ref{F5stability}.
\end{proof}
Now, we will prove the exact result for $F_5$-free $3$-graphs under an additional universal minimum-degree-type assumption. Afterwards we will deal with the additional assumption.
\begin{theo}
\label{F5mindegasumption}
There exists $n_0\in \mathbb{N}$ such that for all $n\geq n_0$ the following holds. If $G$ is an $F_5$-free $3$-graph satisfying
\begin{align*}
\textup{co}_2(G)\geq \textup{co}_2(S_n)
\end{align*}
and
\begin{align}
\label{minqassumption2}
q(x):= \sum_{y\in V,y\neq x} d(x,y)^2 + 2\sum_{vw\in E(L(x))}d(x,y) \geq \frac{4}{27}n^3-10n^2=:d(n)
\end{align}
for all $x\in V(G)$, then $G \cong S_n$.
\end{theo}
\begin{proof}
Let $G$ be an $F_5$-free $3$-graph on $n\geq n_0$ vertices satisfying $\textup{co}_2(G)\geq \textup{co}_2(S_n)$.
Choose $\varepsilon=10^{-20}$. Apply Theorem~\ref{F5stability} to $G$ and get a partition $A\cup B \cup C$ of the vertex set such that $e(A,B,C)\geq \frac{1}{27}n^3-\varepsilon n^3$. Among all such partitions we choose one which maximizes $e(A,B,C)$. We start by making an observation about the class sizes.
\begin{claim} We have
\label{classsizes1}
\begin{align*}\frac{n}{3} - \varepsilon^{1/4} n \leq |A|,|B|,|C| \leq \frac{n}{3} + \varepsilon^{1/4} n.
\end{align*}
\end{claim}
\begin{proof}
Without loss of generality, let $|A|\leq |B|\leq |C|$. Assume for contradiction that Claim~\ref{classsizes1} is false, so $|A|< n/3 - \frac{\varepsilon^{1/4}}{2} n$. We have
\begin{align}
e(A,B,C) &\leq |A||B||C|\leq |A| \left(\frac{n-|A|}{2}\right)^2 =
(n-(n-|A|)) \left(\frac{n-|A|}{2}\right)^2 \label{co2setsizes} \nonumber \\
\nonumber &\leq
\left( \frac{n}{3}-\frac{\varepsilon^{1/4}}{2} n\right)\left(\frac{\frac{2n}{3}+\frac{\varepsilon^{1/4}}{2} n}{2}\right)^2
= \left( \frac{n}{3}-\frac{\varepsilon^{1/4}}{2} n\right)\left(\frac{n}{3}+\frac{\varepsilon^{1/4}}{4}n\right)^2 \\
&= \left( \frac{1}{27}-\frac{\varepsilon^{3/4}}{32}- \frac{\varepsilon^{1/2}}{16} \right)n^3 < \frac{1}{27}n^3-\varepsilon n^3,
\end{align}
where the second-to-last inequality holds, because the function $x \rightarrow (n-x)(x/2)^2$ is monotone decreasing for $x>2n/3$. Inequality \eqref{co2setsizes} is in contradiction with $e(A,B,C)\geq \frac{1}{27}n^3-\varepsilon n^3$. Hence, the claim is true.
\end{proof}
Define junk sets $J_A,J_B,J_C$ to be the sets of vertices which are not typical. To be precise,
\begin{align*}
J_A:= \{x\in A: \ |L_{B,C}(x)|\leq |B||C|-\sqrt{\varepsilon}n^2 \}, \\
J_B:= \{x\in B: \ |L_{A,C}(x)|\leq |A||C|-\sqrt{\varepsilon}n^2 \}, \\
J_C:= \{x\in C: \ |L_{A,B}(x)|\leq |A||B|-\sqrt{\varepsilon}n^2 \},
\end{align*}
where $|G| = |E(G)|$.
We have $|J_A|,|J_B|,|J_C|\leq \sqrt{\varepsilon}n$ as otherwise $e(A,B,C)\leq |A||B||C|-\varepsilon n^3$.
\begin{claim}
\label{cleaning1}There is no edge $xyz$ with $z\in V(G)$, $x,y\in A \setminus J_A$ or $x,y\in B \setminus J_B$ or $x,y\in C \setminus J_C$.
\end{claim}
\begin{proof}
Let $xyz$ be an edge with $z\in V(G)$, $x,y\in A \setminus J_A$. We have $|L_{B,C}(x)|,|L_{B,C}(y)|\geq |B||C|-\sqrt{\varepsilon}n^2$ and thus $E(L_{B,C}(x)) \cap E(L_{B,C}(y)) \neq \emptyset$. Let $\{b,c\}\in E(L_{B,C}(x)) \cap E(L_{B,C}(y))$ with $b,c\neq z$. Now, $\{x,y,z,b,c\}$ induces an $F_5$, a contradiction. The remaining parts of the statement follows similarly.
\end{proof}
\begin{claim}
\label{cleaning2}
For $v\in V(G)$, we have
\begin{align*}
|L_A(v)|\leq \sqrt{\varepsilon}n^2,\quad \quad |L_B(v)|\leq \sqrt{\varepsilon}n^2, \quad \quad \text{and} \quad \quad |L_C(v)| \leq \sqrt{\varepsilon}n^2.
\end{align*}
Further,
\begin{itemize}
\item for $a\in A\setminus J_A$, we have $|L_{A,B}(a)|,|L_{A,C}(a)| \leq \sqrt{\varepsilon}n^2$,
\item for $b\in B\setminus J_B$, we have
$|L_{A,B}(b)|,|L_{B,C}(b)| \leq \sqrt{\varepsilon}n^2$,
\item for $c\in C\setminus J_C$, we have
$|L_{B,C}(c)|,|L_{A,C}(c)| \leq \sqrt{\varepsilon}n^2$.
\end{itemize}
\end{claim}
\begin{proof}
Let $v\in V(G)$. We have $|L_A(v)|\leq |J_A||A|\leq \sqrt{\varepsilon}n^2$, because by Claim~\ref{cleaning1} every edge in the link graph $L_A(v)$ needs to be incident to a vertex in $J_A$. Similarly, we get $|L_B(v)|,|L_C(v)|\leq \sqrt{\varepsilon}n^2$. \\
Now, let $a\in A\setminus J_A$. We have $|L_{A,B}(a)| \leq |J_A||B|\leq \sqrt{\varepsilon}n^2$, because by Claim~\ref{cleaning1} every edge in the link graph $L_{A,B}(a)$ needs to be incident to a vertex in $J_A$. Similarly, we get $|L_{A,C}(a)| \leq \sqrt{\varepsilon}n^2$. Further, the other two statements follow by the same reasoning.
\end{proof}
\begin{claim}
\label{cleaning3} Let $xyz$ be an edge with
\begin{align*}
&x\in A\setminus J_A, \ y\in A, \ z\in (B\setminus J_B) \cup (C\setminus J_C) \text{ or } \\[0.5em]
&x\in C\setminus J_C, \ y\in C, \ z\in (A\setminus J_A) \cup (B\setminus J_B) \text{ or } \\[0.5em]
&x\in B\setminus J_B, \ y\in B, \ z\in (A\setminus J_A) \cup (C \setminus J_C).
\end{align*}
Then, $|L_{A,B}(y)|,|L_{A,C}(y)|, |L_{B,C}(y)| \leq 2\sqrt{\varepsilon}n^2$.
\end{claim}
\begin{proof}
Let $xyz$ be an edge with $x\in (A\setminus J_A), y\in A, z\in (B\setminus J_B) \cup (C\setminus J_C)$. Without loss of generality, let $z\in B\setminus J_B$. Since $|L_{B,C}(x)|\geq |B||C|-\sqrt{\varepsilon}n^2$, we have $|L_{B,C}(y)|\leq 2\sqrt{\varepsilon}n^2$, as otherwise $E(L_{B,C}(x)) \cap E(L_{B,C}(y))\neq \emptyset$, allowing us to find an $F_5$. Also, since $|L_{A,C}(z)|\geq |A||C|-\sqrt{\varepsilon}n^2$, we have $|L_{A,C}(y)|\leq 2\sqrt{\varepsilon}n^2$ by the same reasoning. Since we chose the partition $A\cup B \cup C$ such that $e(A,B,C)$ is maximized, we have $|L_{A,B}(y)|\leq |L_{B,C}(y)| \leq 2\sqrt{\varepsilon}n^2$ as otherwise moving $y$ to $C$ increases the number of those edges having one vertex in each set $A,B,C$. The other two statements follow by a similar argument.
\end{proof}
\begin{claim}
\label{cleaning4}
There is no edge $xyz$ with
\begin{align*}
&x\in A\setminus J_A, \ y\in A, \ z\in (B\setminus J_B) \cup (C\setminus J_C) \text{ or } \\[0.5em]
&x\in C\setminus J_C, \ y\in C, \ z\in (A\setminus J_A) \cup (B\setminus J_B) \text{ or } \\[0.5em]
&x\in B\setminus J_B, \ y\in B, \ z\in (A\setminus J_A) \cup (C \setminus J_C).
\end{align*}
\end{claim}
\begin{proof}
Let $xyz\in E(G)$ be an edge with $x\in (A\setminus J_A), y\in A, z\in (B\setminus J_B) \cup (C\setminus J_C)$. By Claim~\ref{cleaning2}, $|L_A(y)|,|L_B(y)|,|L_C(y)|\leq \sqrt{\varepsilon}n^2$ and by Claim~\ref{cleaning3}, $|L_{A,B}(y)|,|L_{A,C}(y)|, |L_{B,C}(y)| \leq 2\sqrt{\varepsilon}n^2$. Therefore, $|L(y)|\leq 9 \sqrt{\varepsilon}n^2$. Our strategy for proving this claim is to upper bound $q(y)$ violating \eqref{minqassumption2}. We have
\begin{align}
\label{mindegreecheck3}
2\sum_{\{b,c\}\in E(L(y))}d(b,c) \leq 2\sum_{\{b,c\}\in E(L(y))}n \leq 18 \sqrt{\varepsilon}n^3\leq \frac{1}{100}n^3.
\end{align}
The number of vertices $v\in V(G)$ satisfying $d(y,v)\geq 9\varepsilon^{1/4}n$ is at most $2\varepsilon^{1/4} n$ as otherwise
$|L(y)|>9 \sqrt{\varepsilon}n^2$.
Using this fact, we get
\begin{align}
\label{mindegreecheck4}
\sum_{v\in V(G)}d(y,v)^2 \leq 2\varepsilon^{1/4} n^3 + n (9\varepsilon^{1/4}n)^2\leq \frac{1}{100} n^3.
\end{align}
Combining \eqref{mindegreecheck3} and \eqref{mindegreecheck4}, we upper bound
\begin{align*}
q(y) &\leq \sum_{v\in V(G)}d(y,v)^2 + 2\sum_{\{b,c\}\in E(L(y))}d(b,c) \leq \frac{2}{100} n^3 < \frac{4}{27}n^3 - 10n^2 ,
\end{align*}
violating \eqref{minqassumption2}. Hence, there cannot exist an edge $xyz$ with $x\in (A\setminus J_A), y\in A, z\in (B\setminus J_B) \cup (C\setminus J_C)$. The remaining two statements of this claim follow by a similar argument.
\end{proof}
\begin{claim}
\label{cleaning5}
The following three statements hold.
\begin{itemize}
\item For $a\in A$ we have $|L_{A,B}(a)|,|L_{A,C}(a)|\leq 2\sqrt{\varepsilon}n^2$.
\item For $b\in B$ we have $|L_{A,B}(b)|,|L_{B,C}(b)|\leq 2\sqrt{\varepsilon}n^2$.
\item For $c\in C$ we have $|L_{A,C}(c)|,|L_{B,C}(c)|\leq 2\sqrt{\varepsilon}n^2$.
\end{itemize}
\end{claim}
\begin{proof}
Let $a\in A$. We have $|L_{A,B}(a)|,|L_{A,C}(a)|\leq 2\sqrt{\varepsilon}n^2$, because by Claim~\ref{cleaning4} every edge in both link graphs need to have one vertex in a junk set. The other two statements follow by a similar argument.
\end{proof}
\begin{claim}
\label{cleaning6}
There is no edge $x_1x_2v$ with $v\in V(G)$ and $x_1,x_2\in A$ or $x_1,x_2\in B$ or $x_1,x_2\in C$.
\end{claim}
\begin{proof}
Without loss of generality, let $x_1x_2v$ be an edge where $x_1,x_2\in A, \ v\in V(G)$. Then, $|L_{B,C}(x_1)| \leq \frac{2}{3} |B||C|$ or $|L_{B,C}(x_2)| \leq \frac{2}{3} |B||C|$, as otherwise $E(L_{B,C}(x_1)) \cap E(L_{B,C}(x_2))\neq \emptyset$ and thus there are two vertices $b\in B,c\in C,b,c\neq v$ such that $\{x_1,x_2,v,b,c\}$ induces an $F_5$ in $G$. Without loss of generality let $|L_{B,C}(x_1)| \leq \frac{2}{3} |B||C|$. By Claim~\ref{cleaning2} and Claim~\ref{cleaning5}, we have
\begin{align*}
|L_A(x_1)|,|L_B(x_1)|,|L_C(x_1)|\leq \sqrt{\varepsilon}n^2 \quad \text{and} \quad |L_{A,B}(x_1)|,|L_{A,C}(x_1)|\leq 2\sqrt{\varepsilon}n^2.
\end{align*}
Our strategy for proving this claim is to upper bound $q(x_1)$ again violating \eqref{minqassumption2}. For all $b\in B\setminus J_B,c\in C\setminus J_C$ we have $d(b,c)\leq |A|\leq n/3 + \varepsilon^{1/4}n$ by Claim~\ref{cleaning4} and Claim~\ref{classsizes1}. Thus, $d(b,c)\leq n/3 + \varepsilon^{1/4}n$ for all but at most $\sqrt{\varepsilon}n^2$ pairs $\{b,c\}$ with $b\in B, c\in C$. Thus,
combining this fact with Claim~\ref{cleaning2}, we have
\begin{align}
\label{mindegreecheck1}
&2\sum_{\{b,c\}\in E(L(x_1))}d(b,c) \leq 14\sqrt{\varepsilon}n^3 + 2 \sum_{\{b,c\}\in E(L_{B,C}(x_1))}d(b,c) \\ \leq& 16 \sqrt{\varepsilon}n^3 + 2 |L_{B,C}(x_1)| \left(\frac{n}{3} + 3\varepsilon^{1/4}n \right) \nonumber \leq 7\varepsilon^{1/4}n^3 + \frac{4}{9}|B||C|n \leq 8\varepsilon^{1/4}n^3 + \frac{4}{81}n^3,
\end{align}
where we used Claim~\ref{classsizes1} in the last inequality.
The number of vertices $y\in A$ satisfying $d(x_1,y)\geq \varepsilon^{1/4}n$ is at most $10\varepsilon^{1/4} n$ as otherwise one of the following three inequalities would hold
\begin{align*}
|L_A(x_1)|>\sqrt{\varepsilon}n^2 \quad \text{or} \quad |L_{A,B}(x_1)|, |L_{A,C}(x_1)|> 2\sqrt{\varepsilon}n^2,
\end{align*}
contradicting Claim~\ref{cleaning2} or Claim~\ref{cleaning5}. Further, the number of vertices $y\in B$ satisfying $d(x_1,y)\geq \frac{n}{3} + 3\varepsilon^{1/4}n$ is at most $4\varepsilon^{1/4} n$ as otherwise
\begin{align*}
|L_B(x_1)|>\sqrt{\varepsilon}n^2 \quad \text{or} \quad|L_{A,B}(x_1)| > 2\sqrt{\varepsilon}n^2,
\end{align*}
contradicting Claim~\ref{cleaning2} or Claim~\ref{cleaning5}. Similarly, we can conclude that the number of vertices $y\in C$ satisfying $d(x_1,y)\geq \frac{n}{3} + 3\varepsilon^{1/4}n$ is at most $4\varepsilon^{1/4} n$. Combining these three facts, we have
\begin{align}
\label{mindegreecheck2}
\sum_{y\in V(G)}d(x_1,y)^2 &\leq 18\varepsilon^{1/4} n^3 +|A| (\varepsilon^{1/4}n)^2+ (|B| + |C|) \left(\frac{n}{3} + 3\varepsilon^{1/4}n \right)^2 \nonumber \\
&\leq 21\varepsilon^{1/4} n^3 + (|B| + |C|) \frac{n^2}{9} \leq \left(\frac{2}{27} + 23 \varepsilon^{1/4} \right)n^3,
\end{align}
where we used Claim~\ref{classsizes1} in the last inequality. Combining \eqref{mindegreecheck1} and \eqref{mindegreecheck2}, we can upper bound $q(x_1)$:
\begin{align*}
q(x_1) &\leq \sum_{y\in V(G)}d(x_1,y)^2 + 2\sum_{\{b,c\}\in E(L(x_1))}d(b,c) \leq \left(\frac{2}{27} + 23 \varepsilon^{1/4} \right)n^3 + \left( 8\varepsilon^{1/4} + \frac{4}{81} \right)n^3 \\
&\leq \left(31 \varepsilon^{1/4} + \frac{10}{81} \right)n^3 < \frac{4}{27}n^3 - 10n^2 ,
\end{align*}
violating \eqref{minqassumption2}. Thus, there is no edge $x_1x_2v$ with $x_1,x_2\in A, v\in B$. The remaining part of this claim follows by a similar argument.
\end{proof}
Now, by Claim~\ref{cleaning6} we have that $G$ only has edges with vertices in all three sets $A,B,C$. Thus,
\begin{align}
\label{classsizes}
\textup{co}_2(S_n) \leq \textup{co}_2(G)\leq |A||B||C|^2+|A||C||B|^2+ |B||C||A|^2=|A||B||C|n.
\end{align}
However, also $|A||B||C|n \leq \textup{co}_2(S_n)$.
Thus, equality must hold in \eqref{classsizes}. Equality only holds iff
\begin{align}
|A|,|B|,|C|\in \left\{\left\lfloor \frac{n}{3} \right\rfloor,\left\lceil \frac{n}{3} \right\rceil \right\}
\end{align}
and all triples with one vertex from each set $A,B,C$ form edges in $G$.
Thus $G\cong S_n$.
\end{proof}
We complete the proof of Theorem~\ref{F5exact} by inductively showing that the additional minimum degree type assumption \eqref{minqassumption2} is not more restrictive. The proof follows the idea of \cite[Theorem~1.7.]{BalCleLidmain}. We repeat the argument here for completeness.
\begin{proof}[Proof of Theorem~\ref{F5exact}]
Let $H$ be a $3$-uniform $F_{5}$-free hypergraph with codegree squared sum at least $\textup{co}_2(H)\geq \textup{co}_2(S_n)$. Set $d(n)=4/27n^3-10n^2$ and note that
\begin{align*}
\textup{co}_2(S_n)-\textup{co}_2(S_{n-1})>d(n).
\end{align*}
If all vertices $x\in V(G)$ satisfy \eqref{minqassumption2}, Theorem~\ref{F5mindegasumption} gives the result. Therefore, we can assume there exists a vertex $x\in V(G)$ not satisfying \eqref{minqassumption2}.
Remove $x$ with $q(x)<d(n)$ to get $G_{n-1}$ with
\begin{align*}
\textup{co}_2(G_{n-1})\geq \textup{co}_2(G_n)-q(x) \geq \textup{co}_2(G_n)-d(n) \geq \textup{co}_2(S_n)-d(n)\geq \textup{co}_2(S_{n-1})+1.
\end{align*}
Repeat this process as long as such a vertex exists. This gives us a sequence of hypergraphs $G_m$ on $m$ vertices with $\textup{co}_2(G_m)\geq \textup{co}_2(S_m)+n-m$. This process stops before we reach a hypergraph on $n_0=n^{1/4}$ vertices, because $\textup{co}_2(G_{n_0})>n-n_0 > \binom{n_0}{2}(n_0-2)^2$ which is not possible.
Let $n'$ be the index of the hypergraph where this process stops. $G_{n'}$ satisfies $q(x)\geq d(n')$ for all $x\in V(G_{n'})$ and $\textup{co}_2(G_{n'})\geq \textup{co}_2(S_{n'})$ where the last inequality is strict if $n>n'$. Applying Theorem~\ref{F5mindegasumption} leads to a contradiction.
\end{proof}
\section{Loose Path and Cycle}
\label{sec:cycle}
Tur\'an problems for loose paths and loose cycles have been studied intensively. In the graph setting, the Erd\H{o}s-Gallai Theorem determines
\begin{align}
\label{ErdosGallai}
\textup{ex}(n,P_s^2)=(s-1)n
\end{align} when $s$ divides $n$. For uniformity $k\geq 4$, F\"{u}redi, Jiang and Seiver \cite{Furedipath} determined $\textup{ex}(n,P^k_s)$ exactly, for $n$ large enough. Kostochka, Mubayi and Verstra\"ete~\cite{KostochkCk} extended this result to the case $k=3$. \\
Cs\'ak\'any and Kahn~\cite{KahnC_3} determined $\textup{ex}(n,C_3^3)$; and for $s\geq3,k\geq 5$, F\"uredi and Jiang \cite{Furedicycle} determined the extremal function for $C_s^k$ and large enough $n$. Kostochka, Mubayi and Vestra\"ete~\cite{KostochkCk} extended this result for $k=3,4$.
In this section we will determine the codegree squared ext\-re\-mal number of $P_s^3$ and $C_s^3$ asymptotically. We will make use of the asymptotic version of the previously mentioned results for these $3$-graphs.
\begin{theo}[see e.g.~\cite{KostochkCk}]
\label{Kostochkapath}
Let $s\geq 4$. Then,
\begin{align*}
\textup{ex}(n,C_s^3)=\left \lfloor \frac{s-1}{2}\right \rfloor \frac{n^2}{2}(1+o(1)) \quad \quad \text{and} \quad \quad \textup{ex}(n,P_s^3)=\left \lfloor \frac{s-1}{2}\right \rfloor \frac{n^2}{2}(1+o(1)).
\end{align*}
\end{theo}
The $n$-vertex $3$-graph containing all $3$-sets intersecting a fixed vertex set of size $\left\lfloor \frac{s-1}{2}\right\rfloor$ achieves the lower bound in Theorem~\ref{codcyclepath} and Theorem~\ref{Kostochkapath}.
In order to prove the upper bound in Theorem~\ref{codcyclepath} we will also use some of the lemmas Kostochka, Mubayi and Vestra\"ete needed to prove their result. Let $H$ be a $3$-uniform hypergraph on $n$ vertices. The hypergraph $H$ is $d$\textit{-full} if for every pair $x,y$ with $d_H(x,y)>0$ we have $d_H(x,y)>d$.
Denote by $\partial H$ the \emph{shadow graph} of $H$, i.e., the graph on $V(H)$ where a pair $\{x,y\}$ forms an edge if $d_H(x,y)>0$.
\begin{lemma}[Lemma 3.1.~in \cite{KostochkCk}]
\label{sashadfull1}
For $d\geq 1$, every $n$-vertex $3$-uniform hypergraph $H$ has a $(d+1)$-full subhypergraph $F$ with
$$ |F|\geq |H|-d|\partial H|.$$
\end{lemma}
\begin{lemma}[Lemma 3.2.~in \cite{KostochkCk}]
\label{sashadfull2}
Let $s\geq 3$ and let $H$ be a nonempty $3s$-full $3$-uniform hyperpraph. Then $C_s^3,P_{s-1}^3\subseteq H$.
\end{lemma}
\begin{lemma}[Lemma 5.1.~in \cite{KostochkCk}]
\label{sashadfull3}
Let $\varepsilon >0$ and $s\geq 4$. Let $H$ be a $3$-uniform hypergraph, and $F\subset E(\partial H)$ with $|F|>\varepsilon n^2$. Suppose that $d_H(f)\geq \lfloor \frac{s-1}{2} \rfloor+1$ for every $f\in F$ and, if $s$ is odd, then in addition, for every $f=xy\in F$ there is $e_f=xy\alpha \in H$ such that $\min\{d_H(x\alpha),d_H(y\alpha)\} \geq 2$ and $\max \{d_H(x\alpha),d_H(y\alpha)\}\geq 3s+1$. Then, for large enough $n$, $H$ contains $P_s^3$ and $C_s^3$.
\end{lemma}
Let $H$ be a $3$-uniform hypergraph. Recall that $w_H(e)=d_H(x,y)+d_H(y,z)+d_H(z,x)$ for $e=xyz\in E(H)$. We denote by $H_{2s}$ the subhypergraph of $H$ only consisting of edges $e$ satisfying $w_H(e)\geq 2n+2s$. Note that every edge $xy$ in the shadow graph $\partial H_{2s}$ has codegree $d_H(x,y)\geq 2s$ in $H$.
\begin{lemma}
\label{badedges}
Let $s\geq4$. Let $H$ be a $3$-uniform hypergraph on $n$ vertices. If $H$ is $P_s^3$- or $C_s^3$-free, then $|E(H_{2s})|=o(n^2)$.
\end{lemma}
\begin{proof}
We will first show that $|E(\partial H_{2s})|=o(n^2)$.
\medskip
\noindent
Case 1: $H$ is $P_s^3$-free.
The shadow graph $\partial H_{2s}$ does not contain a copy of $P_s^2$, because every edge of this path has codegree at least $2s$ in $H$ and thus one could find a $P^3_s$ in $H$. By the Erd\H{o}s-Gallai Theorem, see \eqref{ErdosGallai}, $|E(\partial H_{2s})|=O(n)=o(n^2).$
\medskip
\noindent
Case 2: $H$ is $C_s^3$-free and $s$ is even.
The shadow graph $\partial H_{2s}$ does not contain a copy of $C_s^2$, because every edge of this $s$-cycle has codegree at least $2s$ in $H$ and thus one could find a $C_s^3$ in $H$. Bondy and Simonovits' even cycle theorem~\cite{Bondyevencycle} states that an $n$-vertex graph with no copy of $C_s^2$ contains at most $O(n^{1+2/s})$ edges. Thus, $|\partial H_{2s}|=O(n^{1+2/s})=o(n^2)$.
\medskip
\noindent
Case 3: $H$ is $C_s^3$-free and $s$ is odd.
Suppose $|\partial H_{2s}| > \varepsilon n^2$ for some $\varepsilon>0$. Apply Lemma~\ref{sashadfull3} on $H$ with $F=E(\partial H_{2s}) \subseteq E(\partial H)$. Note that the conditions of Lemma~\ref{sashadfull3} are satisfied: We have $d_H(f)\geq 2s\geq \lfloor \frac{s-1}{2}\rfloor +1$ for $f=xy\in F$. Since $f\in E(\partial H_{2s})$, there exists $e_f=xy\alpha\in E(H_{2s})$. So $w_H(e_f)\geq 2n+2s$ and in particular
\begin{align*}\min\{d_H(x\alpha),d_H(y\alpha)\} \geq w_H(e_f)-2n\geq 2s
\end{align*}
and
\begin{align*}\max\{d_H(x\alpha),d_H(y\alpha)\} \geq \frac{d_H(x\alpha)+d_H(y\alpha)}{2} \geq \frac{w_H(e_f)-n}{2}\geq \frac{n+2s}{2}\geq 3s+1.
\end{align*}
We conclude that there is a copy of $C^3_s$ in $H$, a contradiction. Thus, $|\partial H_{2s}|=o(n^2).$
\medskip
\noindent
We obtained in all three cases $|\partial H_{2s}|=o(n^2)$. By Lemma~\ref{sashadfull1}, $H_{2s}$ has a $3(s+1)$-full subgraph $H'$ with
$$ |E(H')|\geq |E(H_{2s})|-3s|E(\partial H_{2s})|.$$
By Lemma~\ref{sashadfull2}, $H'=\emptyset$ and thus $|E(H_{2s})|\leq 3s|E(\partial H_{2s})|=o(n^2)$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{codcyclepath}]
Let $H$ be a $P_s^3$- or $C_s^3$-free $3$-uniform hypergraph on $n$ vertices. By Lemma~\ref{badedges}, $|E(H_{2s})|=o(n^2)$. Therefore, we get
\begin{align*}\sum_{\{x,y\}\in \binom{[n]}{2}}d_H^2(x,y)&= \sum_{e \in E(H_{2s})}w_H(e)+\sum_{e \in E(H) \setminus E(H_{2s}) }w_H(e) \leq o(n^3)+ (2n+2s)|E(H)|\\
&\leq \left\lceil \frac{s-1}{2}\right\rceil n^3(1+o(1)), \end{align*}
where in the last inequality we used Theorem~\ref{Kostochkapath}
\end{proof}
We raise the problem of determining for $P_s^3$ and $C_s^3$ exactly.
\begin{problem}
Let $s\geq 3$. Determine $\textup{exco}_2(n,P_s^3)$ and $\textup{exco}_2(n,C_s^3)$ as a function of $n$ and $s$.
\end{problem}
For $C_3^3$, we solve this problem.
\begin{theo}
Let $n\geq 6$. Then,
\begin{align*}
\textup{exco}_2(n,C_3^3)= \binom{n-1}{2}(2n-3).
\end{align*}
\end{theo}
\begin{proof}
Let $H$ be a $C_3^3$-free $3$-graph on $n$ vertices.
Cs\'ak\'any and Kahn~\cite{KahnC_3} proved that $|E(H)|\leq \binom{n-1}{2}$.
Assume that there is an edge $e=xyz$ with $w(e)>2n-3$, so one can find distinct vertices $a,b,c$ with $a\in N(x,y), \ b\in N(x,z)$ and $c\in N(y,z)$. Now, $\{xay,ycz,zbx\}$ forms a $C_3^3$. Thus, we can assume that $w(e)\leq 2n-3$ for all edges. This allows us to conclude
\begin{align*}
\textup{co}_2(H)= \sum_{e\in E(H)}w(e) \leq |E(H)|(2n-3)\leq \binom{n-1}{2}(2n-3).
\end{align*}
This proves the upper bound. Now, for the lower bound, consider the 3-graph $H$ containing all $3$-sets intersecting in one fixed vertex. This $3$-graph is $C_3$-free and has codegree squared sum
\begin{align*}
\textup{co}_2(H)= \binom{n-1}{2} 1^2 + (n-1)(n-2)^2 = \binom{n-1}{2}(2n-3).
\end{align*}
\end{proof}
\section{Matchings}
\label{sec:matching}
In this section we prove Theorem~\ref{codmatching}. Denote by $M_s^k$ the $k$-uniform matching of size $s$, i.e., the $k$-uniform hypergraph on $ks$ vertices with $s$ disjoint edges. For graphs, Erd\H{o}s and Gallai \cite{Erdosmatching1} proved that the extremal number for $M_s^2$ is
\begin{align}
\textup{ex}(n,M_s^2)=\max \left \{\binom{2s-1}{2}, (s-1)(n-s+1)+\binom{s-1}{2}\right \}. \label{graphmatching}
\end{align}
Erd\H{o}s \cite{Erdosmatching2} also determined the extremal number in the hypergraph case when the number of vertices is sufficiently large.
\begin{theo}[Erd\H{o}s \cite{Erdosmatching2}]
Let $k\geq 2,s\geq 1$ be integers. Then there exists a constant $c=c(s)$ such that for all $n\geq ck$
\begin{align*}
\textup{ex}(n,M_s^{k})= |E(G(n;k,s-1)|=\sum_{i=1}^{\min\{k,s-1\}} \binom{s-1}{i}\binom{n-s+1}{k-i},
\end{align*}
where $G(n;k,s-1)$ is the $k$-uniform hypergraph with vertices $x_1,\ldots,x_n$ and all edges containing at least one of the vertices $x_1,\ldots,x_{s-1}$.
\end{theo}
We will use the extremal result for matchings in graphs to determine the codegree squared extremal number for the matching asymptotically.
\begin{proof}[Proof of Theorem~\ref{codmatching}]
The lower bound is achieved by $G(n;3,s-1)$.
\begin{align*}
\textup{co}_2(G(n;3,s-1)) &= \binom{s-1}{2} (n-2)^2 + (s-1)(n-s+1)(n-2)^2 + \binom{n-s+1}{2} (s-1)^2 \\ &= (s-1)n^3 (1+o(1)).
\end{align*}
For the upper bound, let $H$ be an $M_s^3$-free $3$-uniform hypergraph. We construct an auxiliary graph $G$ with the same vertex set $V(H)$. A pair $xy$ is an edge in $G$ iff $d_H(x,y)\geq 3s$. $G$ is $M_s^2$-free, because if there is a matching of size $s$ in $G$, then it can be extended to a $3$-uniform matching of size $s$ in $H$. By \eqref{graphmatching} we have $|E(G)|\leq n(s-1) (1+o(1))$.
Now, we can upper bound the codegree squared sum:
\begin{align*}
\textup{co}_2(H)&= \sum_{xy\in E(G)}d^2_H(x,y) + \sum_{xy\notin E(G)}d^2_H(x,y) \leq |E(G)| n^2 + \binom{n}{2}(3s)^2= (s-1)n^3(1+o(1)).
\end{align*}
Thus,
\begin{align*}
\textup{exco}_2(n,M_s^3)=(s-1)n^3 (1+o(1)).
\end{align*}
\end{proof}
\section{Star}
\label{sec:star}
Chung and Frankl \cite{Franklstar} determined the Tur\'an number of a star.
\begin{theo}[Chung, Frankl \cite{Franklstar}]
\label{extremalstar}
Let $s\geq 3$. If $s$ is odd and $n\geq s(s-1)(5s+2)/2$, then
\begin{align*}
\textup{ex}(n,S_s^3)=(n-2s)s(s-1)+ 2\binom{s}{3}.
\end{align*}
If $s$ is even and $n\geq 2s^3-9s+7$, then
\begin{align*}
\textup{ex}(n,S_s^3)=\left(s^2-\frac{3}{2}s \right)n-\frac{1}{2}(2s^3-9s+6).
\end{align*}
\end{theo}
They also determined the extremal example, which depends on the parity of $s$.
\begin{exmp}[Chung, Frankl \cite{Franklstar}]
Their extremal example in the case when $s$ is odd is the following hypergraph $F_o$. The vertex set is the disjoint union of the sets $A,B,C$ with sizes $s,s$ and $n-2s$ respectively. The edge set is
$$ E(F_o)=\{e: |e\cap A| \geq 2, e\cap B=\emptyset \} \cup \{e: |e\cap B| \geq 2, e\cap A=\emptyset \}. $$
\end{exmp}
\begin{exmp}[Chung, Frankl \cite{Franklstar}]
Their extremal example in the even case is constructed as follows. First, consider the auxiliary graph $G$ on $2s-1$ vertices $x_1,\ldots ,x_{s-1},y_1,\ldots ,y_{s-1},z$ with edges being all pairs $\{x_i,y_j\}$ with $i\neq j$, $\{x_i,y_i\}$ for $2i\leq s$ and $\{x_i,z\}, \{y_i,z\}$ for $2i>s$. Now construct $F$ on $n$ vertices with edges being all $3$-sets intersecting $V(G)$ in one edge or containing two edges of $G$. The extremal example in the even case is the hypergraph $F_e$ constructed by adding the edges of the form $x_1y_iz$ to $F$.
\end{exmp}
We will determine the codegree squared extremal number of the star asymptotically by using their result.
\begin{proof}[Proof of Theorem~\ref{codstar}]
The lower bound is achieved by the two extremal examples presented above. We have
\begin{align*}
\textup{co}_2(F_o)= 2 \left (\binom{s}{2}(n-s-2)^2+s(n-2s)(s-1)^2 \right)=s(s-1)n^2 (1+o(1))
\end{align*}
and
\begin{align*} \textup{co}_2(F_e)=\left( s^2-\frac{3}{2}s\right)n^2(1+o(1)).
\end{align*}
For the proof of the upper bound let $H$ be an $n$-vertex $S_s^3$-free $3$-graph. Recall that the weight of an edge $e=xyz\in E(H)$ is defined to be
\[
w_H(e)=d(x,y)+d(x,z)+d(y,z).
\]
Consider the subgraph $H'\subset H$ only consisting of the edges $e\in E(H)$ with $w_H(e)\geq n+4s$. Since $H'$ is a subgraph of $H$, $H'$ is clearly $S_s^3$-free. Further, we claim that $H'$ is $M_{3s}^3$-free and satisfies $d_{H'}(x,y)<2s$ for all $x,y\in V(H)$.
\begin{claim}
$H'$ is $M_{3s}^3$-free.
\end{claim}
\begin{proof}
Assume $H'$ contains a copy $M$ of $M_{3s}^3$. Then
\begin{align*}
\sum_{\substack{\{x,y\}\subset e \\ \text{for some } e\in M }} d(x,y)\geq (n+4s)3s,
\end{align*}
and therefore there exists a vertex $v\in V(H)$ which is contained in at least $3s$ edges of type $\{v,x,y\}$ with $\{x,y\}\subset f$ for some $f \in M$. Thus, $v$ is contained in at least $s$ edges only having $v$ as a pairwise intersection and therefore is the center of a copy of $S_s^3$, a contradiction. We conclude that $H'$ is $M_{3s}^3$-free.
\end{proof}
\begin{claim}
\label{codegreesmall}
$H'$ satisfies $d_{H'}(x,y)<2s$ for all $x,y\in V(H)$.
\end{claim}
\begin{proof}
Assume that there exists $x,y\in V(H')$ such that $d_{H'}(x,y)\geq 2s$. Let $v_1,\ldots,v_{2s}$ be vertices such that $v_ixy\in E(H')$. For each $i$, $d_H(x,v_i)>2s$ or $d_H(y,v_i)>2s$ since $w_H(e)>n+4s$ for all $e\in E(H')$. Thus the number of indices $i\in [2s]$ such that $d_H(x,v_i)>2s$ or the number of indices $i\in [2s]$ such that $d_H(y,v_i)>2s$ is at least $s$. This contradicts that $H$ is $S_{s}^3$-free (see e.g. \cite [Lemma~2.1]{Franklstar}). We conclude that $d_{H'}(x,y)<2s$ for all $x,y\in V(H)$.
\end{proof}
Next, we bound the number of edges of $H'$.
\begin{claim}
\begin{align*}
|E(H')|\leq (12s^2+6s+1)3s.
\end{align*}
\end{claim}
\begin{proof}
Let $e\in E(H')$ be an arbitrary edge. By Claim~\ref{codegreesmall}, the number of edges in $H'$ intersecting $e$ in exactly two elements is at most $6s$. Furthermore, the number of edges in $H'$ intersecting $e$ in exactly one element is at most $12s^2$. To verify this, let $v\in e$ and consider the link graph $L(v)$ in $H'$. By Claim~\ref{codegreesmall}, $L(v)$ has maximum degree at most $2s$. Further, $L(v)$ cannot have a matching of size $s$, as otherwise it forms a star in $H'$ with $v$ being the center. A graph with maximum degree at most $2s$ and no matching of size $s$ has at most $4s^2$ edges. Thus, $e$ is incident to at most $4s^2$ edges in $H'$ intersecting $e$ in $v$. This allows us to conclude that $e$ is incident to at most $12s^2+6s$ edges in $H'$. Since $H'$ is $M_{3s}^3$-free, it can have at most $(12s^2+6s+1)3s$ edges.
\end{proof}
Using the bounds on the edges in $H'$ and $H$, we can get an upper bound for the codegree squared sum of $H$: \begin{align*}
\textup{co}_2(H) &= \sum_{e\in E(H)} w_H(e) = \sum_{e\in E(H')} w_H(e) + \sum_{e\in E(H) \setminus E(H')} w_H(e) \\
&\leq |E(H')| 3n + |E(H)| (n+4s) \leq (12s^2+6s+1)3s (3n)+ |E(H)| (n+4s).
\end{align*}
Thus, by using Theorem~\ref{extremalstar}, if $s$ is odd, then
\begin{align*}
\textup{co}_2(H) \leq s(s-1)n^2(1+o(1))
\end{align*}
and if $s$ is even, then
\begin{align*}
\textup{co}_2(H) \leq \left(s^2-\frac{3}{2}s \right) n^2(1+o(1)).
\end{align*}
\end{proof}
\section{Further Discussion}
\label{openques}
\subsection{Positive minimum codegree}
There are more notions of extremality which could be studied. An interesting variant of the minimum codegree threshold is the positive minimum codegree which very recently was introduced by Halfpap, Lemons and Palmer~\cite{personalComCory}. Given a $3$-graph $G$, the positive codegree $\textup{co}^+(G)$ is the minimum of $d(x,y)$ over all pairs ${x,y}$ with positive codegree, i.e. $d(x,y)>0$. Given a $3$-graph $H$, the \emph{positive minimum codegree} of $H$, denoted by $\textup{co}^+\textup{ex}(n,H)$, is the largest minimum positive co-degree in an $F$-free $n$-vertex graph.
\begin{ques}[Halfpap, Lemons and Palmer~\cite{personalComCory}]
Given a $3$-graph $H$, what is the \emph{positive minimum codegree} of $H$.
\end{ques}
The positive minimum codegree of $K_4^{3-}$ and $F_5$ is obtained by $S_n$, i.e.
\begin{align*}
\textup{co}^+\textup{ex}(n,K_4^{3-})=\textup{co}^+\textup{ex}(n,F_5)=\left\lfloor \frac{n}{3} \right\rfloor.
\end{align*}
This shows that the positive minimum codegree behaves significantly different than the minimum codegree threshold. An interesting property Halfpap, Lemons and Palmer~\cite{personalComCory} proved is that $\textup{co}^+\textup{ex}(n,H) = o(n)$ if and only if $H$ is $3$-partite. This implies that if $H$ satisfies $\textup{co}^+\textup{ex}(n,H)=\Theta(n)$, then in fact $\textup{co}^+\textup{ex}(n,H)\geq \frac{n}{3} (1+o(1))$. This means that the scaled positive minimum codegree jumps from $0$ to $1/3$.
\subsection{Higher Uniformities}
We propose to study $\sigma(H)$ for $k$-uniform hypergraphs with $k\geq 4$. In particular it would be interesting to determine the codegree squared density for $K_5^4$, the $4$-uniform hypergraph on five vertices with five edges. For the Tur\'an density, Giraud~\cite{MR1077144} proved $\pi(K_5^4)\geq 11/16$ using the following construction. Let $A$ be an $n/2 \times n/2$ $\{0,1\}$-matrix. Define the $4$-graph $H(A)$ on $n$ vertices corresponding to the rows and columns of $A$.
The edge set consists of two types of edges: any four vertices with exactly three rows or three columns; and any four vertices of two rows and two columns forming a $2 \times 2$ submatrix with an odd sum.
The best-known upper bound is $\pi(K_5^4)\leq \frac{1753}{2380}$ which is due to Markstr\"{o}m~\cite{MR2548922}.
For the codegree squared density, we get
\begin{align*}
0.484375=\frac{31}{64}\leq \sigma(K_5^4)\leq 0.55241250,
\end{align*}
where the lower bound comes from the same construction and the upper bound
from flag al\-geb\-ras. We conjecture the lower bound to be tight.
\begin{conj}
We have
\begin{align*}
\sigma(K_5^4)=\frac{31}{64}.
\end{align*}
\end{conj}
Another $4$-graph where we can determine the codegree squared density is $K_5^{4=}$, the unique $4$-uniform hypergraph on five vertices with exactly three edges.
\begin{theo}
\label{5vertices3edges}
Let $n$ be an integer divisible by $4$. Then,
\label{codK54=}
\begin{align*}
\textup{exco}_2(n,K_5^{4=})=\frac{n^2}{16} \binom{n}{3}.
\end{align*}
\end{theo}
Gunderson and Semeraro~\cite{K54doppleminusgunderson} presented a construction of a $4$-uniform hypergraph on $n$ vertices, where $n$ is divisible by $4$: every $3$-subset is contained in exactly $n/4$ hyperedges and every $5$ vertices span $0$ or $2$ edges. Denote by $G$ this $4$-uniform hypergraph. Then, $e(G)= n \binom{n}{3}/16$ and $\textup{co}_2(G)= n^2 \binom{n}{3}/16$. Furthermore, they proved that
\begin{align}
\label{Gunderson}
\textup{ex}(n,K_5^{4=})= \frac{n}{16} \binom{n}{3}
\end{align}
for $n$ divisible by $4$ by modifying a double counting idea by de Caen~\cite{MR734038}. Note that Gunderson and Semeraros' result~\cite{K54doppleminusgunderson} also implies the codegree threshold result $\textup{ex}_3(n,K_5^{4=})=\frac{n}{4}$ for $n$ divisible by $4$. Here, we will modify de Caen's argument~\cite{MR734038} to prove Theorem~\ref{5vertices3edges}.
\begin{proof}[Proof of Theorem~\ref{5vertices3edges}]
Let $H$ be a $4$-uniform $n$-vertex $K_5^{4=}$-free hypergraph. Denote $N$ the number of pairs $(e,f)$ with $|e|=|f|=4$, $|e\cap f|=3$, $e\in E(H)$ and $f\notin E(H)$. Then, on one side
\begin{align}
\label{doupecountK54=1}
N=\sum_{e\in E(H)} \sum_{x\not\in e} |\{z\in e: e \cup \{ x \} \setminus \{z\} \not\in E(H)\}| \geq \sum_{e\in E(H)} \sum_{x\not\in e} 3= 3e(H)(n-4),
\end{align}
where the inequality holds because $H$ is $K_5^{4=}$-free.
On the other side,
\begin{align}
\label{doupecountK54=2}
N= \sum_{A\in \binom{[n]}{3}}d_H(A)(n-3-d_H(A)) = (n-3)\sum_{A \in \binom{[n]}{3}} d_H(A) - \textup{co}_2(H)= 4(n-3) e(H)- \textup{co}_2(H).
\end{align}
Combining \eqref{doupecountK54=1} and \eqref{doupecountK54=2}, we get
\begin{align*}
\textup{co}_2(H)\leq 4(n-3)e(H)- 3e(H)(n-4) =
ne(H) \leq \frac{n^2}{16} \binom{n}{3},
\end{align*}
where in the last inequality we used \eqref{Gunderson}.
\end{proof}
We remark that $K_5^{4=}$ is the only non-trivial hypergraph we are aware of which has the property that there is a construction (presented in \cite{K54doppleminusgunderson}) that is extremal in each of $\ell_1$-norm, $\ell_2$-norm and for the codegree threshold.
\subsection{Big Triangle}
Let $T^{(2k)}$ be the $2k$-graph obtained by having three disjoint sets $A,B,C$ of size $k$ and the three edges $A\cup B,\ A \cup C,\ C \cup A$. Frankl~\cite{fattrianglefrankl} proved that $\pi(T^{(2k)})=1/2$. The lower bound comes from the $2k$-graph $G$ where the vertex set is partitioned into two equal sized parts and edges are those $2k$-sets which intersect each part in an odd number of elements. The minimum codegree of this extremal example is $n/2-o(n)$. Mubayi and Zhao~\cite{Mubayicode} observed that in fact $\pi_{2k-1}(T^{(2k)})=1/2$.
The maximum sum of $k$-codegrees squared an $n$-vertex $2k$-uniform $T^{(2k)}$-free graph can have, can be easily calculated. Let $H$ be a $T^{(2k)}$-free $2k$-graph. Then,
\begin{align*}
\sum_{\substack{A\subset V(H) \\ |A|=k}}d(A)^2 &= \sum_{B\in E(H)}\sum_{\substack{A\subset B \\ |A|=k}} d(A)\leq |E(H)| \binom{2k}{k} \frac{1}{2}\binom{n}{k} (1+o(1)) \\
&\leq \frac{1}{2}\binom{n}{2k} \binom{2k}{k} \frac{1}{2}\binom{n}{k}(1+o(1))=\frac{1}{4}\binom{n}{k}^3 (1+o(1)).
\end{align*}
On the other side the $2k$-graph $G$ satisfies
\begin{align*}
\sum_{\substack{A\subset V(G) \\ |A|=k}}d_{G}(A)^2 \geq\binom{n}{k} \left( \frac{\binom{n-k}{k-1}(\frac{n}{2}-2k) }{k} \right)^2 =\frac{1}{4}\binom{n}{k}^3 (1+o(1)).
\end{align*}
This determines asymptotically the maximum $k$-codegree squared sum of an $n$-vertex $2k$-uniform $T^{(2k)}$-free hypergraph. We wonder whether the asymptotic extremal example for the codegree squared sum is also $G$.
\begin{ques}
What is $\sigma(T^{(2k)})$?
\end{ques}
\subsection{Induced Problems}
We propose to study induced problems.
\begin{ques}
Given a family of $k$-graphs $\mathcal{F}$, what is the maximum $\ell_2$-norm a $k$-uniform $n$-vertex hypergraph $G$ can have without containing any $H\in \mathcal{F}$ as an induced subhypergraph?
\end{ques}
Let $\mathcal{F}$ be a family of $k$-uniform hypergraphs. Denote by $\textup{exco}_2^{ind}(n,\mathcal{F})$ the maximum codegree squared sum among all $k$-uniform $n$-vertex hypergraphs not containing any $F\in \mathcal{F}$ as an induced subhypergraph. Here, we will present two examples where we can determine $\textup{exco}_2^{ind}(n,\mathcal{F})$.
Denote by $K_5^{4-}$ the $4$-uniform hypergraph with exactly $4$ edges on $5$ vertices and $E_5^{4}$ the $4$-uniform hypergraph with exactly one edge on five vertices. Observe that the proof of Theorem~\ref{5vertices3edges} actually gives the stronger result
\begin{align*}
\textup{exco}_2^{ind}(n,\{E_5^{4},K_5^{4=}, K_5^{4-}, K_5^4\})=\frac{n^2}{16} \binom{n}{3}
\end{align*}
for $n$ divisible by $4$.
Frankl and F\"uredi~\cite{K43-extremalfrankl} proved that the $3$-graph with the maximum number of edges only containing exactly $0$ or $2$ edges on any $4$ vertices is the $n$-vertex blow-up of $H_6$. As observed in Subsection~\ref{K43-F32}, this $3$-graph has codegree squared sum of $5/108n^4 (1+o(1))$. Using flag algebras for the upper bound, we get
\begin{align*}
\textup{exco}_2^{ind}(n,\{E_4^{3},K_4^{3-},K_4^3\})=\frac{5}{108}n^4 (1+o(1)),
\end{align*}
where $E_4^{3}$ denotes the $3$-graph with exactly one edge on four vertices.
\section*{Acknowledgments}
We thank Emily Heath for useful comments and suggestions.
\bibliographystyle{abbrvurl}
|
1,941,325,220,520 | arxiv | \section*{Introduction}
The Korteweg--de Vries hierarchy (or KdV) is a sequence of pairwise commuting partial differential equations first discovered by A.\! Lenard in 1967. Each equation admits a Lax pair as follows \cite{Lax68}. Let a function $u$ which depends on infinitely many variables indexed by odd integers $\mathbf{t} = (x=t_1,t_3,t_5,\ldots)$. We denote $\mathrm{D}\, u = \partial_x u = u'$. We associate to $u$ the Schrödinger operator $L = \mathrm{D}^2 + u$. Then the KdV hierarchy is defined by the flows
\begin{align*}
\frac{\partial L}{\partial t_{2i-1}} = \left[ \left( P^{2i-1} \right)_+, L\right],
\end{align*}
where $P = L^{1/2}$ is the square root of $L$ and for any pseudo-differential operator $X$, $X_+$ is its purely differential part. The first equation reads $\partial_{t_1} u = u'$ (and we set $x=t_1$), the next few equations read
\begin{align*}
\partial_{t_3} u &= \textstyle \frac{1}{4}u^{(3)} + \frac{3}{4}uu',\\
\partial_{t_5} u &= \textstyle \frac{1}{16}u^{(5)} + \frac{4}{8}uu^{(3)} + \frac{5}{4}u'u'' + \frac{15}{8}u^2u',\\
\partial_{t_7} u &= \textstyle \frac{1}{64}u^{(7)} + \frac{7}{32}uu^{(5)} + \frac{21}{32} u'u^{(4)} + \frac{35}{32}u''u^{(3)} + \frac{35}{32}u^2u^{(3)} + \frac{35}{8}uu'u'' + \frac{35}{32}(u')^3 + \frac{35}{16}u^3u',
\end{align*}
the flow with respect to $t_3$ being the KdV equation. To any solution $u$ we associate a tau function defined, up to a multiplicative constant, by
$$u = -2(\log\tau)''.$$
If $\tau$ is polynomial then $u$ is rational. On the other hand, it was proven by Airault, McKean and Moser \cite{AMcKM77} that there are denumerably many rational solutions of KdV, each one being the orbit of a single function $u_n = \frac{n(n+1)}{x^2}$ under the flows of the hierarchy. In \cite{AM78}, the authors constructed the Adler--Moser polynomials $\theta_n(x=q_1,q_3\ldots, q_{2n-1})$ for $n\geq 0$, defined by the recursion
\begin{align*}
\theta'_{n+1}\theta_{n-1} - \theta_{n+1}\theta_{n-1}' = (2n-1)\theta_n^2.
\end{align*}
An important result of \cite{AM78} (cf. Theorem \ref{thmAM}) is that there exists a unique change of variables that transforms the Adler--Moser polynomials into polynomial tau functions of KdV and that we recover all rational solutions of KdV. But we did not know what this change of variables was.
In this paper, we show that the following change of variables transforms the Adler--Moser polynomials into the polynomial tau functions of KdV: $q_1=t_1=x$, and
\begin{align*}
\sum_{i\geq 2} \frac{q_{2i-1}}{\alpha_{2i-1}} z^{2i-1} = \tanh \left( \sum_{i\geq 2} t_{2i-1} z^{2i-1} \right).
\end{align*}
where $\alpha_{2i-1} = (-1)^{i-1} 3^25^2\ldots (2i-3)^2 (2i-1)$. To do so, we apply this change of variables to the Adler--Moser polynomials to get some polynomials $\tau_n(t_1,t_3,t_5,\ldots)$. Then by seeing them as functions $\tau_n(t_1,t_2,t_3,\ldots)$ of even and odd times we show that they are tau functions of the Kadomstev--Petviasvhili hierarchy (KP). Then since the $\tau_n$'s actually depend only on odd times, they are indeed tau functions of KdV.
\begin{comment}
More precisely, still in \cite{AM78} it was shown that the Adler--Moser polynomials admit a Wronskian representation. That allows us to express our functions as Wronskians $\tau_n(\mathbf{t}) = \Wr(\phi_1,\ldots,\phi_n) := \det( \mathrm{D}^{i-1}\phi_j )_{i,j=1,\ldots, n},$ where the functions $\phi_j(\mathbf{t})$ are defined by the generating series
\begin{align*}
\sum_{j\geq 1} \phi_j z^{2j-1} = \sinh(xz) + \cosh(xz)\tanh \left( \sum_{i\geq 2} t_{2i-1}z^{2i-1} \right).
\end{align*}
Then the procedure to show that a Wronskian is a tau function of KP is well established (see for example \cite{IZ92}, \cite{Caf08}). We define a pseudo-differential operator $S_n(\chi) = \frac{1}{\tau_n}\Wr\left(\chi,\phi_1,\ldots,\phi_n \right)\mathrm{D}^{-n},$ and show that it dresses an operator $\mathcal{L}_n = S_n\mathrm{D} S_n^{-1}$ that satisfies the Lax system of KP:
\begin{align*}
\frac{\partial \mathcal{L}_n}{\partial t_i} = \left[ \left( \mathcal{L}_n^i \right)_+, \mathcal{L}_n \right].
\end{align*}
\end{comment}
It is well known how to compute the polynomial tau functions of KdV without using the Adler--Moser polynomials. For instance, in \cite{Hir04} R.\! Hirota constructs a family of tau functions of KP in terms of Wronskians of the elementary Schur polynomials, which can be reduced to recover the polynomial tau functions of KdV. But the Adler--Moser polynomials reveal a recursive structure in the space of rational solutions of KdV. It would be interesting to investigate how to generalize this to the Drinfeld--Sokolov hierarchies.
\section{The Adler--Moser Polynomials}
Let a set of evolution variables $\mathbf{q} = (x=q_1, q_3, q_5,\ldots)$\footnote{In their original paper \cite{AM78} the authors use the variables $\tau_i$ instead. But since then, the letter $\tau$ has been used rather for the tau functions. Moreover, we use odd indices so that we can later complete the set into the variables of KP.}. We save the variables $\mathbf{t} = (x=t_1, t_3, t_5, \ldots)$ for the tau functions of KdV. The Adler--Moser polynomials form a sequence $\theta_n(q_1,q_3\ldots, q_{2n-1})$ for $n\geq 0$, defined by the following recursion \cite{AM78}: $\theta_0 = 1$, $\theta_1 = x$ and for $n\geq 1$,
\begin{equation}\label{eqDiffRecAM}
\theta'_{n+1}\theta_{n-1} - \theta_{n+1}\theta_{n-1}' = (2n-1)\theta_n^2,
\end{equation}
where $f' = \mathrm{D} f = \partial_x f$. This recursion leaves an integration constant that is chosen to be $q_{2n-1}$ when computing $\theta_{n}$. We can check that $\theta_n$ has degree $d_n = \frac{1}{2}n(n+1)$ in $x$ and $q_{2n-1}$ is actually the coefficient of $x^{d_{n-2}}$ in this polynomial. The first few polynomials read
\begin{align*}
\theta_0 &= x,\\
\theta_1 &= x,\\
\theta_2 &= x^3 + q_3,\\
\theta_3 &= x^6 + 5q_3x^3 + q_5x - 5q_3^2,\\
\theta_4 &=\textstyle x^{10} + 15q_3x^7 + 7q_5x^5 - 35q_3q_5x^2 + 175q_3^3x - \frac{7}{3}q_5^2 + q_7x^3 + q_3q_7.
\end{align*}
In that same article (\cite{AM78}, pp. 17--18), the authors state the following theorem.
\begin{thm}[Adler--Moser, 1978]\label{thmAM}
There exists a unique change of variables $\mathbf{q}\mapsto \mathbf{t}$ that transforms the Adler--Moser polynomials $\theta_n(\mathbf{q})$ into the polynomial tau functions $\tau_n(\mathbf{t})$ of KdV. That is, the rational functions $u_n = -2(\log\tau)''$ define operators $L_n = \mathrm{D}^2 + u_n$ that satisfy the Lax system of KdV:
\begin{align*}
\frac{\partial L_n}{\partial t_{2i-1}} = \left[ \left( L_n^{\frac{2i-1}{2}}\right)_+, L_n \right].
\end{align*}
\end{thm}
\noindent In this article we prove that the desired change of variables is given by: $q_1=t_1=x$, and
\begin{align*}
\sum_{i\geq 2} \frac{q_{2i-1}}{\alpha_{2i-1}} z^{2i-1} = \tanh \left( \sum_{i\geq 2} t_{2i-1} z^{2i-1} \right).
\end{align*}
where $\alpha_{2i-1} = (-1)^{i-1} 3^25^2\ldots (2i-3)^2 (2i-1)$. The latter coefficients $\alpha_{2i-1}$ where already given in Adler and Moser's article. Notice that this change of variables amounts to simply change the choice of the integration constant in the differential recursion \eqref{eqDiffRecAM}. To prove the statement, let us first recall and prove some results stated in \cite{AM78}. These lemmas relate the Adler--Moser polynomials to a Wronskian representation through a multiplicative factor and a simple rescaling of the variables.\\
\indent Let another set of variables $\mathbf{s} = (x=s_1, s_3,s_5,\ldots)$ and functions $\psi_j(\mathbf{s})$, $j\geq 0$, be defined by:
\begin{align}\label{eqPsiGen}
\sum_{j\geq 1} \psi_j z^{2j-1} = \sinh(xz) + \cosh(xz)\sum_{i\geq 2} s_{2i-1}z^{2i-1},
\end{align}
and $\psi_0 = 0$. It readily implies the recursion $\psi_j'' = \psi_{j-1}$, where we use the notations $\psi_j' = \partial_x\psi_j = \mathrm{D}\psi_j$ and $\psi_j^{(i)} = \partial^i_x\psi_j = \mathrm{D}^i\psi_j$.
\begin{lem}
The Wronskians of the functions $\psi_j$, defined by
\begin{align*}
W_n := \mathrm{Wr}(\psi_1,\ldots, \psi_n) = \det\left( \mathrm{D}^{i-1}\psi_j \right)_{i,j=1,\ldots, n},
\end{align*}
satisfy the recursion
\begin{align}\label{eqDiffRecWronsk}
W_{n+1}'W_{n-1} - W_{n+1}W_{n-1}' = W_n^2.
\end{align}
\end{lem}
\begin{proof}
For any smooth function $\chi(\mathbf{s})$ with respect to $x$, denote $W_n(\chi) := \Wr\left( \psi_1,\ldots, \psi_n, \chi \right).$ Then one has Jacobi's identity:
\begin{align}\label{eqJacWronsk}
W_n'(\chi)W_{n+1} - W_n(\chi)W_{n+1}' = W_{n+1}(\chi)W_n.
\end{align}
This can be proven by noticing that the left-hand side is a linear differential operator of order $n+1$ which vanishes for $\chi = \psi_1,\ldots, \psi_n$, as well as for $\psi_{n+1}$. Yet the functions $\psi_j$ being linearly independent, the left-hand side must be proportional to $W_{n+1}(\chi)$. Comparing the highest coefficient, one obtain Equation \eqref{eqJacWronsk}. Now thanks to the relation $\psi_j'' = \psi_{j-1}$ and $\psi_1 = x$, we can compute that for $\chi = 1$,
\begin{align}\label{eqWk(1)}
W_n(1) = (-1)^kW_{n-1}.
\end{align}
\begin{comment}
Indeed, first we have $W_n(1) = \Wr(\psi,\ldots,\psi_n,1) = (-1)^k\Wr(1,\psi_1,\ldots,\psi_n)$. Then writing down the determinants one obtains the following.
\begin{align*}
\Wr(1,\psi_1,\ldots,\psi_n) = \left|\begin{matrix}
1 & 0 & 0 & \cdots & 0\\
x & 1 & 0 & \cdots & 0\\
\psi_2 & \psi_2' & \psi_2'' & \cdots & \psi_2^{(n)}\\
\vdots & & & & \vdots\\
\psi_n & \psi_n' & \psi_n'' & \cdots & \psi_n^{(n)}
\end{matrix}\right| = \left| \begin{matrix}
\psi_2'' & \cdots & \psi_2^{(n)}\\
\vdots & & \vdots\\
\psi_n'' & \cdots & \psi_n^{(n)}
\end{matrix} \right| = \left| \begin{matrix}
\psi_1 & \cdots & \psi_1^{(n-2)}\\
\vdots & & \vdots\\
\psi_{n-1} & \cdots & \psi_{n-1}^{(n-2)}
\end{matrix} \right| = W_{n-1},
\end{align*}
hence Equation \eqref{eqWk(1)} holds true.
\end{comment}
\noindent Then setting $\chi = 1$ in Jacobi's identity \eqref{eqJacWronsk}, one finds Equation \eqref{eqDiffRecWronsk}.
\end{proof}
\noindent Now since $W_0 = \theta_0 = 1$ and $W_1 = \theta_1 = x$, the two sequences of polynomials differ only by a multiplicative factor that can be computed:
\begin{align}\label{eqPropAMvsWronsk}
\theta_n(\mathbf{q}) = \mu_n W_n(\mathbf{s}), & & \mu_n = \prod_{j=1}^k (2k-2j+1)^j.
\end{align}
\begin{lem}
The parameters of the Adler--Moser polynomials $\theta_n (\mathbf{q})$ and those of the Wronskians $W_n(\mathbf{s})$ are related via a rescaling:
\begin{align}\label{eqRescale}
s_{2i-1} = \frac{q_{2i-1}}{\alpha_{2i-1}}, & & \alpha_{2i-1} = (-1)^{i-1} 3^25^2\ldots (2i-3)^2 (2i-1)
\end{align}
\end{lem}
\begin{proof}
It all has to do with the choice of the normalization in the recursions \eqref{eqDiffRecAM} and \eqref{eqDiffRecWronsk}. For $\theta_n$, the choice is such that $q_{2n-1}$ is the coefficient of $x^{d_{n-2}}$. Moreover if $\theta_n$ is a solution of \eqref{eqDiffRecAM}, so is $\theta_n + c\theta_{n-2}$ so that the normalization can be expressed as
\begin{align*}
\theta_n = \mathring{\theta}_n + q_{2n-1}\theta_{n-2},
\end{align*}
where $\mathring{\theta}_n := \left. \theta_n \right|_{q_{2n-1} = 0}.$ Similarly, define $\mathring{W}_n := \left. W_n \right|_{s_{2n-1} = 0}.$ Since $\psi_n = \mathring{\psi_n} + s_{2n-1}$, then by Equation \eqref{eqWk(1)},
\begin{align*}
W_n = \mathring{W_n} + s_{2n-1}\Wr(\psi_1,\ldots,\psi_n,1) = \mathring{W_n} + (-1)^{k-1}s_{2n-1}W_{n-2}.
\end{align*}
Comparing the last two equations and using Equation \eqref{eqPropAMvsWronsk} one obtains the result.
\end{proof}
\section{A Change of Variables to the Polynomial Tau Functions KdV}
\noindent The aim of this section is to prove the following theorem.
\begin{thm}\label{thmChangeVar}
The following change of variables transforms the Adler--Moser polynomials into the polynomial tau functions of KdV: $q_1 = t_1 = x$, and
\begin{align}\label{eqChangeVar}
\sum_{i\geq 2} \frac{q_{2i-1}}{\alpha_{2i-1}} z^{2i-1} = \tanh \left( \sum_{i\geq 2} t_{2i-1} z^{2i-1} \right).
\end{align}
where $\alpha_{2i-1} = (-1)^{i-1} 3^25^2\ldots (2i-3)^2 (2i-1)$.
\end{thm}
\noindent As a matter of fact, this change of variables does not affect the first few variables except for a rescaling. Here are the first variables $q_{2i-1}$ in terms of the $t_{2i-1}$'s: $s_1 = t_1 = x$, and then
\begin{comment}
\begin{align*}
q_1 &= t_1, & ,\\
q_3 &= -3\cdot t_3, & q_9 &=\textstyle 99255\cdot (t_9 - \frac{1}{3}t_3),\\
q_5 &= 45\cdot t_5, & q_{11} &=\textstyle -9823275\cdot (t_{11} - t_3^2t_5),\\
q_7 &= -1575\cdot t_7, & q_{13} &=\textstyle 1404728325\cdot (t_{13} - t_3^2t_7 - t_3t_5^2).
\end{align*}
\end{comment}
\begin{align*}
q_3 &= -3\cdot t_3, & q_7 &= -1575\cdot t_7, & q_{11} &= -9823275\cdot \left(t_{11} - t_3^2t_5 \right),\\
q_5 &= 45\cdot t_5, & q_9 &=\textstyle 99255\cdot \left(t_9 - \frac{1}{3}t_3 \right), & q_{13} &= 1404728325\cdot \left(t_{13} - t_3^2t_7 - t_3t_5^2 \right).
\end{align*}
\noindent Notice that the change of variables is homogeneous for the grading $\deg q_i = \deg t_i = \deg s_i = i$, as proven by Adler and Moser \cite{AM78}. Under this change of variables, we obtain the following polynomial tau functions of KdV:
\begin{align*}
\tau_0 &= 1,\\
\tau_1 &= x,\\
\tau_2 &= x^3 - 3t_3,\\
\tau_3 &= x^6 - 15t_3x^3 - 45t_3^2 + 45t_5x,\\
\tau_4 &= x^{10} -45t_3x^7 + 315t_5x^5 + 4725t_3t_5x^2 - 4725t_3^3x - 4725t_5^2 - 1475t_7x^3 + 4725t_3t_7.
\end{align*}
Then the polynomials $\tau_n/\mu_n$ (cf. Equation \eqref{eqPropAMvsWronsk}) correspond to the tau functions computed via the Wronskians of the elementary Schur polynomials following Hirota \cite{Hir04}.
To prove Theorem \ref{thmChangeVar}, we introduce another sequence of functions defined by the generating series.
\begin{equation}\label{eqGenSeriesPhi}
\sum_{j\geq 1} \phi_j z^{2j-1} = \sinh(xz) + \cosh(xz)\tanh \left( \sum_{i\geq 2} t_{2i-1}z^{2i-1} \right).
\end{equation}
It amounts to applying the change of variables of Equation \eqref{eqChangeVar} to the functions $\psi_j$ of Equation \eqref{eqPsiGen}. With these functions, define another sequence of Wronskians:
\begin{equation}\label{eqWronskianPhi}
\tau_n:= \Wr(\phi_1,\ldots,\phi_n),
\end{equation}
In what follows, we prove that these Wronskians are tau functions of the KP hierarchy. Yet because they depend only on odd times, they are tau functions of KdV, which proves Theorem \ref{thmChangeVar}. We use the same approach as in \cite{Caf08} and \cite{IZ92}. First we need the following lemma.
\begin{lem}\label{lemRelationPhi}
The functions $\phi_j$ satisfy the following relation for any integers $i, j\geq 1$:
\begin{equation}
\phi_j^{(2i-1)} - \frac{\partial \phi_j}{\partial t_{2i-1}} = \sum_{k = 1}^{j-i-1} \phi_k a_{j-i-k+1},
\end{equation}
where $\phi_j^{(2i-1)} = \partial_x^{2i-1}\phi_j$. Here the $a_j$'s are functions that do not depend on $x$ and are defined by
$$\sum_{j\geq 2} a_jz^{2j-1} = \tanh\left( \sum_{i\geq 2} t_{2i-1}z^{2i-1} \right).$$
\end{lem}
\begin{proof}
It is a direct calculation: differentiating Equation \eqref{eqGenSeriesPhi} and using a Cauchy product, one obtains
\begin{align*}
\sum_{j\geq 1} \left( \phi_j^{(2i-1)} - \frac{\partial\phi_j}{\partial t_{2i-1}} \right) z^{2j-1} = \sum_{j\geq 1} z^{2j-1}\sum_{k = 0}^{j -i} \phi_k a_{j-i-k+1}.
\end{align*}
Then, noticing that $\phi_0 = 0$ and $a_0 = a_1 = 0$, one gets the correct boundaries in the last sum.
\end{proof}
\begin{rmq}
This relation is to be compared with the one satisfied by the elementary Schur polynomials defined by
\begin{align*}
\exp\left( \sum_{i\geq 1}t_iz^i \right) = \sum_{j\geq 1} P_jz^j.
\end{align*}
These polynomials satisfy the relation $\partial_x^i P_j = \partial_{t_i} P_j.$ Now define $p_j$ to be $P_{2j-1}$ where all even times are set to 0. They satisfy the relation
$$\partial_x^{2i-1} p_j = \partial_{t_{2i-1}} p_j.$$
The last relation allows to prove that their Wronskians $\Wr(p_1,\ldots, p_n)$ are the tau functions of KdV the same way we prove that the Wronskians of the $\phi_j$'s are (see for instance \cite{IZ92}). In particular the Wronskians of the $p_j$'s and those of the $\phi_j$'s coincide.
\end{rmq}
\noindent Now let us introduce all the variables $(x=t_1, t_2, t_3, \ldots)$ of the KP hierarchy (that is, odd and even). To prove that the $\tau_n$'s are tau functions of KP we use Sato's equation, which we state in a equivalent form in the following proposition.
\begin{prop}
Let the following differential operator
\begin{align*}
\Delta_n(\chi) := \frac{\Wr\left( \chi, \phi_1, \ldots, \phi_n \right)}{\Wr\left( \phi_1, \ldots, \phi_n \right)} = \frac{1}{\tau_n}\Wr\left( \chi, \phi_1, \ldots, \phi_n \right),
\end{align*}
for any differentiable function $\chi$ with respect to $x$. The following equation holds for any $i\geq 1$:
\begin{equation}\label{eqPreSato}
\frac{\partial \Delta_n}{\partial t_{i}} = \left( \Delta_n \mathrm{D}^{i} \Delta_n^{-1} \right)_+ \Delta_n - \Delta_n \mathrm{D}^{i}.
\end{equation}
\end{prop}
\begin{proof}
It is sufficient to prove the equality when acting on $\phi_1,\ldots, \phi_n$ which are $n$ linearly independent functions. Yet these functions are solutions of the equation $\Delta_n (\phi_j) = 0,$ so it amounts to proving that
\begin{align*}
\frac{\partial \Delta_n}{\partial t_i} \left(\phi_j\right) + \Delta_n \left( \phi_j^{(i)} \right) = 0.
\end{align*}
If $i = 2\ell$ is even, then we only have to prove that $\Delta_n \left( \phi_j^{(2\ell)} \right) = 0.$ Yet $\phi_j'' = \phi_j$ so that $\phi_j^{(2\ell)} = \phi_{j-\ell}$, or 0 if $j\leq \ell$. Eventually, it amounts to $\Delta_n\left(\phi_{j-\ell} \right) = 0$ which holds true. If $i = 2\ell-1$ is odd, by Lemma \ref{lemRelationPhi}, it amounts to proving that
\begin{align*}
\frac{\partial \Delta_n}{\partial t_{2\ell-1}} \left(\phi_j\right) + \Delta_n \frac{\partial}{\partial t_{2\ell-1}}\left(\phi_j\right) + \sum_{k = 1}^{j-\ell-1} \Delta_n \left( \phi_k a_{j-\ell-k+1} \right) = 0.
\end{align*}
Yet the functions $a_j$ do not depend on $x$, so the last sum vanishes. And for the other terms it amounts to
\begin{align*}
\frac{\partial \Delta_n}{\partial t_{2\ell-1}} \left(\phi_j\right)+ \Delta_n \frac{\partial}{\partial t_{2\ell-1}}\left(\phi_j\right) = \frac{\partial}{\partial t_{2\ell-1}}\Delta_n \left(\phi_j \right) = 0.
\end{align*}
\end{proof}
\noindent Now the fact that Equation \eqref{eqPreSato} implies the KP hierarchy via Sato's Equation is a classical result which we sum up in the following proposition.
\begin{prop}
Let the pseudo-differential operator $S_n = \Delta_n\mathrm{D}^{-n}$. Then $S_n$ satisfies Sato's equation
\begin{align}\label{eqSato}
\frac{\partial S_n}{\partial t_i} + \left( S_n\mathrm{D}^i S_n^{-1} \right)_- S_n = 0.
\end{align}
Moreover, the pseudo-differential operator $\mathcal{L}_n = S_n\mathrm{D} S_n^{-1}$ is of the form $\mathcal{L}_n = \mathrm{D} + \sum_{k\geq 1} f_{n,k}\mathrm{D}^{-k}$ and satisfies the Lax system of the KP hierarchy
\begin{equation}\label{eqKPLax}
\frac{\partial \mathcal{L}_n}{\partial t_i} = \left[ \left( \mathcal{L}_n^i \right)_+, \mathcal{L}_n \right].
\end{equation}
\end{prop}
\begin{proof}
Equation \eqref{eqPreSato} on differential operators readily implies Sato's equation \eqref{eqSato} on purely pseudo-differential operators. Then it is well known how to derive the Lax system of KP from Sato's equation. One only has to differentiate the relation $\mathcal{L}_nS_n = S_n\mathrm{D}$ by $t_i$, then use the relation $\left[ \mathcal{L}_n, \mathcal{L}_n^{i}\right] = 0$.
\end{proof}
\noindent Finally, the following proposition states that the dressing operator $S_n$ is indeed related to the Wronskians $\tau_n$ via the usual shift equation, ie, that $\tau_n$ is a tau function of KP.
\begin{prop}\label{propShift}
The dressing operator $S_n$ and the Wronskian $\tau_n$ satisfy the following relation
\begin{equation}\label{eqWaveFunction}
S_n\left( \mathrm{e}^{\xi(\mathbf{t};\lambda)} \right) = \mathrm{e}^{\xi(\mathbf{t};\lambda)}\frac{\tau_n\left( \mathbf{t} - [\lambda^{-1}] \right)}{\tau_n(\mathbf{t})},
\end{equation}
where $\xi(\mathbf{t};\lambda) = \sum_{i\geq 1} t_i\lambda^i$ and
\begin{equation}\label{eqShift}
\tau_n \left( \mathbf{t} - [\lambda^{-1}] \right) := \textstyle \tau_n \left( t_1 - \frac{1}{\lambda}, t_3 - \frac{1}{3\lambda^3}, t_5 - \frac{1}{5\lambda^5}, \ldots \right).
\end{equation}
\end{prop}
\noindent To prove that, we need the following lemma.
\begin{lem}\label{lemTriangRel}
The shift of the functions $\phi_j$ reads the following triangular relation for any $j\geq 1$,
\begin{equation}\label{eqTriangRelPhi}
\phi_j(\mathbf{t} - [\lambda^{-1}]) = \phi_j(\mathbf{t}) - \lambda^{-1}\phi_j'(\mathbf{t}) + \sum_{i = 1}^{j-1} \left( \phi_i(\mathbf{t}) - \lambda^{-1}\phi_i'(\mathbf{t}) \right) b_{j-i}(\mathbf{t}),
\end{equation}
where the $b_j$'s are functions that do not depend on $x$ and are defined by
\begin{equation}\label{eqGenSerB}
\sum_{j\geq 0} b_jz^{2j} = \mathrm{sech} \left( z\lambda^{-1} \right) \left[ 1 - z\lambda^{-1}\tanh \left( z\lambda^{-1}\right) - z\lambda^{-1}\tanh \left( \eta \right) + \tanh\left( z\lambda^{-1}\right) \tanh \left( \eta \right) \right]^{-1},
\end{equation}
with $b_0 = 1$, and
$$\eta(\mathbf{t};z) := \sum_{i\geq 2} t_{2i-1}z^{2i-1}.$$
Here $\mathrm{sech} = 1/\cosh$ and the exponent $-1$ stands for the multiplicative inverse of formal power series.
\end{lem}
\begin{proof}
Using the fact that
$$\tanh^{-1}(z\lambda^{-1}) = z\lambda^{-1} + \sum_{i\geq 2} \frac{z^{2i-1}}{(2i-1)\lambda^{2i-1}}$$
for $z\lambda^{-1}\in (-1,1)$, and applying the sum formulae of hyperbolic functions, one obtains that
\begin{align*}
\sum_{j\geq 1} \phi_j\left( \mathbf{t} - [\lambda^{-1}] \right) z^{2j-1} = \frac{\mathrm{sech}\left( z\lambda^{-1}\right) \sum_{j\geq 1} \left( \phi_j - \lambda^{-1}\phi_j' \right) z^{2j-1} }{1 - z\lambda^{-1}\tanh \left( z\lambda^{-1}\right) - z\lambda^{-1}\tanh \left( \eta \right) + \tanh\left( z\lambda^{-1}\right) \tanh \left( \eta \right)}.
\end{align*}
Moreover, the above denominator has its constant term equal to 1 and only even powers, so is its multiplicative inverse. Therefore, the series $\sum_{j\geq 0} b_j z^{2j}$ in Equation \eqref{eqGenSerB} is well defined and has indeed $b_0 = 1$, hence the triangular relation \eqref{eqTriangRelPhi}.
\end{proof}
\begin{rmq}
As in Lemma \eqref{lemRelationPhi}, this relation is to be compared with the one satisfied by the elementary Schur polynomials, namely,
\begin{align*}
P_j(\mathbf{t} - [\lambda^{-1}]) = P_j(\mathbf{t}) - \lambda^{-1}P_{j-1}(\mathbf{t}).
\end{align*}
\end{rmq}
\noindent We can now prove Proposition \ref{propShift} which concludes the proof of Theorem \ref{thmChangeVar}.
\begin{proof}[Proof of Proposition \ref{propShift}]
We prove an equivalent equation which only uses differential operator:
\begin{align*}
\Delta_n \left( \mathrm{e}^{\xi(\mathbf{t};\lambda)} \right) = \lambda^n\mathrm{e}^{\xi(\mathbf{t};\lambda)} \frac{\tau_n\left( \mathbf{t} - [\lambda^{-1}] \right)}{\tau_n(\mathbf{t})}.
\end{align*}
Thanks to Lemma \ref{lemTriangRel}, we can rewrite the right-hand side as
\begin{align*}
\frac{\lambda^n\mathrm{e}^{\xi(\mathbf{t};\lambda)}}{\tau_n} \left| \begin{matrix}
\phi_1 - \lambda^{-1}\phi_1' & \cdots & \phi_1^{(n-1)} - \lambda^{-1}\phi_1^{(n)}\\
\vdots & & \vdots\\
\phi_n - \lambda^{-1}\phi_n' + \sum_{i=1}^{j-1}\left( \phi_i - \lambda^{-1}\phi_i'\right) b_{j-i} & \cdots & \phi_n^{(n-1)} - \lambda^{-1}\phi_n^{(n)} + \sum_{i=1}^{j-1}\left( \phi_i^{(n-1)} - \lambda^{-1}\phi_i^{(n)}\right) b_{j-i}
\end{matrix} \right|.
\end{align*}
On the other hand, the left-hand side reads
\begin{align*}
\Delta_n\left( \mathrm{e}^{\xi(\mathbf{t};\lambda)} \right) = \frac{1}{\tau_n} \left| \begin{matrix}
\mathrm{e}^{\xi(\mathbf{t};\lambda)} & \lambda\mathrm{e}^{\xi(\mathbf{t};\lambda)} & \cdots & \lambda^n \mathrm{e}^{\xi(\mathbf{t};\lambda)}\\
\phi_1 & \phi_1' & \cdots & \phi_1^{(n)}\\
\vdots & & & \vdots\\
\phi_n & \phi_n' & \cdots & \phi_n^{(n)}
\end{matrix} \right|.
\end{align*}
Using operations on rows and columns, we can easily check that these two expressions are equal.
\end{proof}~\\
\noindent \textbf{Acknowledgements.} The author would like to warmly thank M. Cafasso as well as M. Adler for their help and guidance. Besides, the author would like to express their most sincere thanks to J. Leray for his most useful insight in recognizing the Taylor series expansion of hyperbolic tangent.
|
1,941,325,220,521 | arxiv | \section{\label{sec:qhe}A class of quantum heat engines}
Our quantum heat engines are just two-level quantum systems,
which are the quantum version of the Otto engines~\cite{Arnaud:03}.
(They are readily extendable to
systems of many discrete energy levels.) They could be realised with coherent
macroscopic quantum systems like, for instance, a Bose-Einstein condensate confined
to the bottom two energy levels of a trapping potential.
The exact cyclicity will be enforced to ensure that upon completing
a cycle all the output products of the engines are clearly displayed without any hidden effect.
A cycle of the quantum heat engine has four stages:
\begin{enumerate}
\item {\em Stage 1:} The system has the probability $1 - p_u^{\,(1)}(0)$
in the lower state {\em prior} to some kind of contact
with a heat bath at temperature $T_1$.
After some contact time $\tau_1$, the system achieves a probability
$p_u^{\,(1)}(\tau_1)$ to gain some energy from the heat bath to jump up an energy
gap of $\Delta_1$ to be in the upper state. Only heat is transferred in this stage
to yield a change in the occupation probabilities.
\item {\em Stage 2:} The system is then isolated from the heat bath and undergoes
a quantum adiabatic
expansion to reduce the energy gap from $\Delta_1$ to a smaller
value $\Delta_2$. The probability in the upper state is
maintained throughout (provided the expansion rate
is sufficiently slow according to the quantum adiabatic theorem~\cite{Messiah}),
an amount of work is thus performed {\em by} the system, but no heat is
transferred.\item {\em Stage 3:} The system, with a probability
$p_u^{\,(2)}(0)$ being in the upper state, is brought into some kind of contact
with another heat bath at temperature $T_2$ for some time $\tau_2$ until it
gains a probability $1-p_u^{\,(2)}(\tau_2)$ to release some energy to the bath
and jump down the gap $\Delta_2$ to be in the lower state.
\item {\em Stage 4:} The system is removed from the heat bath
and undergoes a quantum adiabatic contraction to increase the energy
gap from $\Delta_2$ back to the larger value $\Delta_1$.
An amount of work is performed {\em on} the system in this stage.
\end{enumerate}
Note that we need not and have not assigned any temperature to the quantum system; all
the temperatures are properties of the heat baths, which are assumed to be in
the Gibbs state.
However, in the operation above, the absorption and release of energy in
stages 1 and 3 occur neither definitely, nor even deterministically.
Quantum mechanics tells us that they can only happen probabilistically; and
the probabilities that such transitions take place depend on the details of the
interactions with and some intrinsic properties (namely, the temperatures)
of the heat baths.
The cyclicity of the heat engines then puts a constraint on the probabilities,
$p_u^{\,(1)}(0) = p_u^{\,(2)}(\tau_2)$, and
$p_u^{\,(2)}(0) = p_u^{\,(1)}(\tau_1)$.
The net work done by our quantum heat engines in the two quantum adiabatic
passagesin stages 2 and 4 is, from~(\ref{id}), given that $\Delta_1>\Delta_2$,
\begin{eqnarray}
\Delta W
&=& \left({p}_u^{\,(1)}(\tau_1)-{p}_u^{\,(2)}(\tau_2)
\right)\left(\Delta_2-\Delta_1\right).
\label{5}
\end{eqnarray}
If the system is allowed to thermalise with the heat baths in stages 1 and 3,
the thermal equilibrium probabilities, for $i=1,2$,
\begin{eqnarray}
\tilde{p}_u^{\,(i)} &=& 1/\left(1 + \exp({\Delta_i}/{kT_i})\right),
\label{thermal}
\end{eqnarray}
are to be used in~(\ref{5}).
These probabilities are definitely {\em non-zero}.
Thus even in the case of ($T_1\le T_2$)
there {\em exist}, even with a small probability
$\tilde{p}_u^{\,(1)}(1-\tilde{p}_u^{\,(2)})$, cycles in which the transitions described
in stages 1 and 3 above actually take place. (The probability, however, diminishes
exponentially for $n_c$ consecutive cycles, $(\tilde{p}_u^{\,(1)}
(1-\tilde{p}_u^{\,(2)}))^{n_c}$,
in all of which the system performs net work on the environment.)
As a result, there {\em are} certain cycles {\em whose \underline{sole} result is the
absorption of heat from a reservoir and the conversion of this heat into work,
of the amount $(\Delta_1-\Delta_2)$}.
This amounts to a violation of the Kelvin-Planck statement of the second law due to the
explicit probabilistic nature of quantum mechanical processes. This violation,
however, occurs only randomly, with some small probability, and thus is not
exploitable.
However, there exists a sure way to {\em always} extract work, to the amount of
$(\Delta_1 - \Delta_2)$,
in each {\em completable} ``cycle." In order to eliminate the
probabilistic uncertainty,
we prepare the system to be in the lower state prior to stage 1;
make an energy measurement after stage 1 and only let the engine
continue to stage 2 {\em if} the measurement result confirms that the system is
in the upper state; we then make another measurement after stage 3 and only let
the engine continue to stage 4 {\em if} the measurement result confirms that the
system is in the lower state. All this can be carried out even for the case
$T_1\le T_2$ to extract, in a controllable manner, some work which would have
been otherwise prohibited by the second law. This apparent violation of the
second law is analogous to that of Szilard's one-atom engine and is nothing but
the result of an act of Maxwell's daemon~\cite{Demon:90}. Our quantum heat
engines can thus provide a feasible and quantum way to realise Maxwell's daemon
in a different way to Szilard's engine. Indeed, the condition of strict
cyclicity of each engine's cycle is broken here. After each cycle
the measurement apparatus, being a Maxwell's daemon, has already registered the
results which are needed to determine the {\em next} stages of the engine's
operation. In this way, there are extra effects and changes to the
register/memory of the apparatus, even if we assume that the quantum measurement
steps themselves cost no energy and leave no net effect anywhere else. To remove
these remnants in order to restore the strict cyclicity, we would need to bring
the register back to its initial condition by erasing any information obtained
in the cycles, either by resetting its bits or by thermalising the register with
some heat bath. Any which way, extra effects are inevitable, namely an amount
of heat of at least $kT\ln 2$ will be released per bit erased. The second law
is thus saved~\cite{Lloyd:97}. More extensive discussions and debates on these
issues can be found in the literature~\cite{Demon:90}.%
Despite the random quantum
mechanical violation, the second law is upheld on the average.
The work that can be performed {\em by} our engines in thermal equilibrium when
and only when,
as can be seen from~(\ref{5}), ${p}_u^{\,(1)}(\tau_1)>{p}_u^{\,(2)}(\tau_2)$
and thus when and only when, from~(\ref{thermal}),
\begin{eqnarray}
{T_1} &>& T_2\left({\Delta_1}/{\Delta_2}\right).
\label{6}
\end{eqnarray}
The expressions~(\ref{5}) and~(\ref{6}) (which was also derived in a more
specific context in~\cite{Feldmann:2000}) not only confirm the broad validity
of the second law but also refine the law further in specifying how much $T_1$
needs to be larger than $T_2$ before some work can be extracted. In other
words,{\em work cannot be extracted, on the average, even when $T_1$ is greater
than $T_2$ but less than $T_2(\Delta_1/\Delta_2)$}, in contradistinction to the
classical necessary condition that $T_1$ only needs to be larger than $T_2$.
(However, for Carnot's engines, see~\cite{Arnaud:03}, the ratio of successive energy gaps
involved tends to one.)
The efficiency of our engines is~{\footnote{A similar expression was also
obtained through a specific context in~\cite{Scovil:59}.}}
\begin{eqnarray}
\eta_q &=& {\Delta W}/{Q_{\rm in}} = \left(1-{\Delta_2}/
{\Delta_1}\right),
\label{eff}
\end{eqnarray}
which is {\em independent} of heat bath temperatures and is the maximum available
within the law of quantum mechanics. This also serves as the upper bound, with
appropriate $\Delta_1$ and $\Delta_2$, of the efficiency of any heat engine,
including Carnot's engines, because the work performed by our heat engines
through their quantum adiabatic processes is the maximum that can be
extracted~\cite{Landau:80}. The efficiency of Carnot's engines, $\eta_C = 1 -
T_2/T_1$, can be derived from~(\ref{eff}) through an infinite number of infinitesimal quantum
adiabatic processes~\cite{Arnaud:03}.
We present next a modification of
the quantum heat engines which can better the work derivation than that
available from thermal equilibrium.
A modified cycle also has four stages, of which stage 2 and stage 4 remain the same
as described previously, whereas stages 1 and 3 are
replaced respectively by:
\begin{enumerate}
\item {\em Stage 1':} The system is entered to a single-mode cavity which is tuned to
match the energy gap $\Delta_1$ of
the system and which is in thermal equilibrium with a heat bath at temperature $T_1$.
The thermal distribution for the cavity mode is given by~(\ref{thermaldist}) below.
After some time interval
$\tau_1$, the system is removed from the cavity to enter stage 2.
\item {\em Stage 3':}
Like stage 1' above but with temperature $T_2$ and smaller energy gap $\Delta_2$.
After some time $\tau_2$, the system departs the cavity for stage 4.
\end{enumerate}
In each of the cavities so described, the state of the system satisfies the Schr\"odinger equation for a
single two-level atom interacting with a single-mode field which has the
thermal probability $P_n(T_i)$ to find exactly $n$ photons of frequency $\Delta_i/h$
in the cavity at temperature $T_i$,
\begin{eqnarray}
P_n(T_i) &=& \frac{1}{1+\bar{n}_i}\left(\frac{\bar{n}_i}{1+\bar{n}_i}\right)^n,
\label{thermaldist}
\end{eqnarray}
where $\bar{n}_i$ is the thermal average boson number,
$
\bar{n}_i = 1/\left({e^{{\Delta_i}/{kT_i}}-1}\right)\, .
\label{planck}
$
The {\em exact} solution of this Schr\"odinger equation is given in~\cite{Scully:97},
from which we arrive at
\begin{eqnarray}
p_u^{\,(i)}(t) &=& \frac{(1+2\bar{n}_i)\,p_u^{\,(i)}(0)
- \bar{n}_i}{(1+\bar{n}_i)}\sum_{n=0}^\infty P_n(T_i)
\cos^2\left(\Omega_n t\right)\nonumber\\
&& + \frac{\bar{n}_i( 1 - p_u^{\,(i)}(0))}{1+\bar{n}_i},
\label{upperprobability}
\end{eqnarray}
where $\Omega_n = g\sqrt{n+1}$ and $g$ is the coupling constant between the system
and the cavity mode.
From~(\ref{upperprobability}) we can derive the bounds of the probability
$p_u^{\,(i)}(t)$ in terms of the initial probability
and temperature,
\begin{eqnarray}
p_u^{\,(i)}(t) &\in& [p_u^{\,(i)}(0),
{\bar{n}_i( 1 - p_u^{\,(i)}(0))}/{(1+\bar{n}_i)}], \nonumber\\
&& {\mbox{\rm for }} p_u^{\,(i)}(0)\in [0,\bar{n}_i/(1+2\bar{n}_i)];\nonumber\\
p_u^{\,(i)}(t) &\in& [{\bar{n}_i( 1 - p_u^{\,(i)}(0))}/{(1+\bar{n}_i)},
p_u^{\,(i)}(0)], \nonumber\\
&& {\mbox{\rm for }} p_u^{\,(i)}(0) \in [\bar{n}_i/(1+2\bar{n}_i), 1].
\label{9}
\end{eqnarray}
Figure~\ref{fig1} depicts these bounds.
The probability at any subsequent time is only accessible in the bounded, dark
triangular areas, each of which corresponds to a different temperature, on the
two sides of the diagonal.
\begin{figure}
\includegraphics[scale=0.4]{Slide1b}%
\caption{\label{fig1}For the cavity in contact with
a heat sink at $T_2$,
the accessible probability at all times, given a particular initial probability,
is bounded in the (blue) area under the diagonal, with its reflection shown in
gray above the diagonal. The bounded (red) area above the diagonal is for the
cavity in contact with a heat bath at $T_1 = T_2(\Delta_1/\Delta_2)$.}
\end{figure}
For the cavity in contact with the heat bath at $T_1$,
we want to have the exit probability to be greater (the greater, the better) than the initial
probability,
$
p_u^{\,(1)}(\tau_1) > p_u^{\,(1)}(0) = p_u^{\,(2)}(\tau_2),
$
thus we need only to consider the lower area for this
temperature. Reversely, for the cavity in contact with the heat bath at $T_2$,
we want to have the exit probability to be smaller (the smaller, the better) than
the initial probability, $
p_u^{\,(2)}(\tau_2) < p_u^{\,(2)}(0) = p_u^{\,(1)}(\tau_1),
$
hence we need only to consider the upper area for this temperature.
Thus, Figure~\ref{fig1} combines the two heat
baths in which the upper area comes from
$T_1$ and the lower from $T_2$, with $T_1 = T_2(\Delta_1/\Delta_2)$.
The coordinate of a point $A$ in the lower area of Fig.~\ref{fig1} is
$(p_u^{\,(2)}(0), p_u^{\,(2)}(\tau_2))$, representing an exit probability less than
the entry one at the cavity with temperature $T_2$. The corresponding point at temperature
$T_1$ must have, by requirement of cyclicity, the coordinate
$
(p_u^{\,(1)}(0), p_u^{\,(1)}(\tau_1)) = (p_u^{\,(2)}(\tau_2), p_u^{\,(2)}(0)),
$
which thus is the reflection of the point $A$ across the diagonal. However, it is clearly seen
that for $T_1 \le T_2(\Delta_1/\Delta_2)$ this reflection is {\em not} in the accessible
upper area. We then conclude that at these temperatures, even with $T_1$
greater than $T_2$ by a factor $(\Delta_1/\Delta_2)$, the quantum heat engines
{\em cannot} do work {\em on the average}.
Thus, we have once again confirmed the second law that,
{\em on the average}, no process is possible whose \underline{sole} result is the
transfer of heat from a cooler to a hotter body, with or without
a production of work.
But we have also clarified the degrees of coolness and hotness in terms of the quantum
energy gaps involved before such a process is possbile; namely, we must have, as
a {\em necessary} but not sufficient condition, $T_1 > T_2(\Delta_1/\Delta_2)$
as in~(\ref{6}).
The perfect agreement between this specific result derived from the quantum dynamical bounds~(\ref{9})
with the general result~(\ref{6}) derived from statistical mechanics is quite remarkable -- but
is yet to be fully understood and hence deserved further investigation elsewhere.
We speculate that such agreement is not accidental but is a consequence of the Gibbs
distributions assumed for the heat baths in both derivations;
and it should thus be independent of specific details of the quantum dynamics involved.
Fig.~\ref{fig3} combines the cases $T_2$ and
$T_1>T_2 (\Delta_1/\Delta_2)$ in which there is some overlap between the areas above the diagonal and thus makes possible the
production of some work, $\Delta W_{\rm cav}$.
If and when we choose to operate with a point below the thermal equilibrium
line in the blue area (for $T_2$) such that its reflection across the diagonal is above the
thermal equilibrium line in the red area (for $T_1$) as shown
in the figure, we can derive more work than the case of thermal equilibrium,
$\Delta W_{\rm th}$. This is because
the net work done is proportional to the difference in probabilities as shown
in~(\ref{5}). Here $\left|\Delta W_{\rm cav}\right|$ is greater than
$\left|\Delta W_{\rm th}\right|$
because the vertical distance between point $A$ and its reflection in
Fig.~\ref{fig3} is greater than the vertical distance between the two horizontal lines,
which represent the two thermal equilibria.
\begin{figure}
\includegraphics[scale=0.4]{Slide3b}%
\caption{\label{fig3}Similar to Fig.~\ref{fig1} but this time with $T_1>
T_2(\Delta_1/\Delta_2) > T_2$.
The maximum work can be extracted in a single cycle is proportional to the length
of the (yellow) vertical double arrow.}
\end{figure}
In summary, by interpreting work and heat, but without introducing entropy directly, in the
quantum domain and by applying this interpretation to the simplest quantum
systems of two levels, we have not only confirmed the broad validity of the
second law but also been able to clarify and refine its various aspects.
On the one hand, the second law is seen to be valid {\em on the average}
in the broad sense.
On the other hand, explicitly because of the probabilistic nature of quantum mechanics,
there do exist physical processes which can violate certain classical statements of the
second law. (Such violation, while can be observed in some cycles of our
two-level heat engine, may not be
observable in any single cycle for a system which consists of many subsystems,
because the averaging effects over the subsystems already occur in {\em each} cycle
for the composite system.)
However, such violation only occurs randomly and can thus
neither be exploitable nor harnessible.
This confirmation of the second law is in accordance
with the fact that, while we can treat the quantum heat engines purely and entirely
as quantum mechanical systems, we still have to assume the Gibbs distributions {\em for the
heat baths} involved.
Such distributions can only be derived~\cite{Landau:80} with non-quantum-mechanical
assumptions, which ignore any quantum entanglement within the heat baths. Indeed,
it has been shown that~\cite{Lenard:78, Tasaki:00} the law of entropy increase is a
mathematical consequence of the initial states being in such general
equilibrium distributions. This illustrates and highlights the connection between the
second law to the unsolved problems of emergence of classicality, of quantum
measurement and of decoherence, all of which are central to quantum mechanics.
Only until some further progress can be made on these problems, the classicality of the heat
baths will have to be assumed and remained in the assumption of the
Maxwell-Boltzmann-Gibbs thermal equilibrium
distributions.
We have also clarified the degree of temperature
difference~(\ref{6}), in terms of the quantum energy gaps involved, between the
heat baths before any work can be extracted. We have also shown how to extract
more work from the heat baths than otherwise possible with thermal equilibrium.
Note that such an operation is subject to the bounds given in~(\ref{9}), which then,
as can be seen through their depiction in the figures, ensure that we stay within the second law.
Note also that this
is not an operation of Maxwell's daemon because the information about the time intervals
$\tau_1$ and $\tau_2$ is fixed and forms an integrated part of the modified
engines. This information, being common to {\em all} cycles,
need not and should not be erased after each cycle.
Finally, our class of quantum heat engines can readily
offer a feasible way to physically realise Maxwell's daemon, in a way different
to Szilard's engine but also through the acts of quantum
measurement and information erasure.
\begin{acknowledgments}
I would like to acknowledge Alain Aspect for his seminar
which directly prompted this investigation.
I would also like to thank Bryan Dalton, Jean Dalibard, Peter Hannaford, Peter Knight
and Bruce McKellar for helpful discussions; Jacques Arnaud, Michael Nielsen, Bill Wootters
and Wojciech Zurek for email correspondence.
\end{acknowledgments}
|
1,941,325,220,522 | arxiv | \section{Introduction}
Stochastic processes constitute the basis of virtually all dynamics and therefore are of fundamental importance, in particular in condensed matter physics. There the quantities of interest are dynamic correlation functions, for example via the fundamental fluctuation-dissipation theorem~\cite{KuboBuch:1991}, they encode the linear response with respect to an external perturbation. Furthermore, scattering techniques employing neutrons, light, or X-rays, directly probe the dynamics via time-dependent correlation functions~\cite{Hansen:Theory_of_Simple_Liquids}.
Physically one expects that all observables are coupled to their environment consisting of many degrees of freedom such that all dynamic processes are exposed to some damping mechanism. Then correlations should decay at large times and display only trivial behavior. In particular, persistent oscillations, quasi-periodic or intermittent behavior should not occur. The usual scenario is thus that correlation functions approach zero at large times which corresponds to an ergodic behavior. A fingerprint of non-ergodic behavior is thus if correlation functions display a non-trivial long-time limit. In the context of liquid dynamics~\cite{Goetze:Complex_Dynamics}
this non-ergodic behavior is associated with an idealized glass state, where part of the initial correlation remains frozen-in for arbitrarily large times. In a scattering experiment a non-trivial long-time limit yields an elastic line at zero frequencies broadened only by the instrument resolution.
The existence of a long-time limit for correlation functions appears to be non-trivial, in particular, peculiar examples for correlation functions (in the mathematical literature refered to as covariance functions or characteristic functions of a finite measure) are known that neither display a limit, nor are they almost periodic (see ~\cite{Lukacs:Characteristic_Functions}, p. 27, Sec. 2.2).
A common strategy to develop a theory for time-correlation functions is to derive equations of motion encoding the dynamics of the underlying system. For example,
the Zwanzig-Mori projection operator formalism yields formally exact equations of motion where the complexity arising due to the many-body interaction is deferred to a memory kernel~\cite{KuboBuch:1991,Hansen:Theory_of_Simple_Liquids,Goetze:Complex_Dynamics}. Then the question of the existence of a (non-trivial) long-time limit for the correlation function
is a property inherited from the memory kernel.
The goal of this work is to provide rather general conditions on equations of motion such that the long-time limit of the correlation function is guaranteed to exist. Such equations naturally emerge for classical many-particle systems where the time evolution is driven by Newtonian dynamics. In particular, they have been the starting point for the mode-coupling theory of the glass transition~\cite{Goetze:Complex_Dynamics} which yields a series of testable non-trivial predictions of the dynamics as a liquid is driven towards a non-ergodic state.
The paper is organized as follows, in Sec.~\ref{Sec:correlation_functions} basic properties of time correlation functions are recalled, The main contribution of this work is in Sec.~\ref{Sec:Model_equations} where the problem is formulated and the existence of a long-time limit is demonstrated. The connection to the mode-coupling theory of the glass transition is discussed in Sec.~\ref{Sec:MCT}. At the end, conclusions and possible generalizations are provided.
\section{Correlation functions in equilibrium}\label{Sec:correlation_functions}
For complex-valued observables $A(t)$, probability theory provides constraints on the class of admissable functions that correspond to (equilibrium) correlation functions $S(t) = \langle \delta A(t)^* \delta A(0) \rangle$~\cite{Feller:Probability_2} in a stationary stochastic process. Here $\delta A(t) = A(t)- \langle A(t) \rangle$ denotes the fluctuation around its stationary mean. In particular, employing time-translation invariance, one verifies that correlation functions are positive definite in the following sense: Given a finite set of complex numbers $\xi_i$ and real times $t_i$ the quantity $\sum_{i,j} \xi_i^* S(t_i-t_j) \xi_j \geq 0$ is non-negative for any correlation function. By Bochner's theorem, the converse is also true and therefore provides a complete characterization of admissible functions. In equilibrium processes, which we define as stationary stochastic processes such that the time-reversed trajectory is equally likely, correlation functions are furthermore real and symmetric in time, $S(t) = S(t)^* = S(-t)$. By the spectral representation theorem~\cite{Feller:Probability_2}, correlation functions can then be written as characteristic functions of a finite Borel measure $\sigma(\cdot)$
\begin{equation}
S(t) = \int_\mathbb{R} e^{-i \omega t} \sigma(\diff \omega),
\end{equation}
and for equilibrium correlation functions the measure can be chosen to be symmetric. In particular, one infers that correlation functions are bounded by their initial value $|S(t)| \leq S(t=0)=\sigma(\mathbb{R})$ and are uniformly continuous.
It will be useful to define the Fourier-Laplace transform by
\begin{equation}
\hat{S}(z) = \text{i} \int_0^\infty e^{\text{i} z t} S(t) \diff t,
\end{equation}
where the complex frequency $z$ is confined to the upper complex half plane $\mathbb{C}_+ := \{ z \in \mathbb{C} | \text{Im}[z] >0 \}$. The spectral representation theorem implies
\begin{equation}\label{eq:Borel_transform}
\hat{S}(z) = \int \frac{1 }{\omega - z} \sigma(\diff \omega),
\end{equation}
i.e. $\hat{S}(z)$ is the Borel-Stieltjes transform of the finite symmetric measure $\sigma$.
The following properties follow readily from the preceding relation for the Fourier-Laplace transforms of correlation functions in equilibrium
\begin{enumerate}
\item[(1)] $\hat{S}(z)$ is complex analytic for $z\in \mathbb{C}_+$
\item[(2)] $\hat{S}( -z^*) = -\hat{S}(z)^*$
\item[(3)] $\lim_{\eta \to \infty} \eta \Imag[\hat{S}(z = i\eta)]$ is finite
\item[(4)] $\Imag[ \hat{S}(z)] \geq 0$ for $z\in \mathbb{C}_+$
\end{enumerate}
Conversely, the Riesz-Herglotz representation theorem~\footnote{see N.I. Akhiezer, \emph{The classical moment problem} (Oliver \& Boyd, 1965) p. 92-93, Sec. 3.1., Eq. [3.3a] or F.~Gesztesy and E.~Tsekanovskii, \emph{Math. Nach.}, {\bf 218}, (2000), p.65, Sec. 2, Theorem 2.2}
reveals that these four conditions are sufficient for $\hat{S}(z)$ being the Laplace transform of an equilibrium correlation function. Functions with these properties are discussed in the literature under the names of Nevanlinna, Herglotz, and Pick.
\section{Model equations and spectral properties}~\label{Sec:Model_equations}
We consider equations of motion for the real function $S: t \in [0,\infty[ \mapsto S(t) \in \mathbb{R}$
\begin{equation}\label{eq:harmonic_oscillator}
J^{-1} \ddot{S}(t) + \nu \dot{S}(t) + S^{-1} S(t) + \int_0^t M(t-t') \dot{S}(t') \diff t' =0 ,
\end{equation}
with parameters $S> 0, \nu > 0, J >0 $ and initial conditions $S(t=0) = S$, $\dot{S}(t=0) = 0$.
Furthermore, the kernel $M(t)$ is assumed to be an equilibrium correlation function, i.e. by the spectral representation theorem~\cite{Feller:Probability_2}
\begin{equation}\label{eq:M_spectral}
M(t) = \int_{\mathbb{R}} e^{-i \omega t}\mu(\diff \omega),
\end{equation}
where $\mu(.)$ is a finite symmetric Borel measure on the real line with the Borel sets
as measurable sets. In particular such equilibrium correlation functions are real and fulfill $|M(t)| \leq M(0) = \mu(\mathbb{R}) =: M$.
The equation of motion, Eq.~\eqref{eq:harmonic_oscillator}, naturally emerges in statistical physics for dynamic phenomena either as a phenomenological model, or as derived by the Zwanzig-Mori projection operator technique~\cite{KuboBuch:1991,Hansen:Theory_of_Simple_Liquids,Goetze:Complex_Dynamics}, or field-theoretic methods. Despite its formal similarity to an harmonic oscillator equation, the description is not restricted to harmonic interactions. Rather the spirit is that the complex many-particle interactions are
included in $M(t)$, referred to as memory kernel, self-energy, or mass operator in different physics communities.
One derives the short-time expansion of Eq.~\eqref{eq:harmonic_oscillator} for $t\to 0$
\begin{align}\label{eq:short_time}
S(t) =& S - J t^2/2 + J \nu J |t|^3/6 \nonumber \\
&+ J(M+S^{-1} - \nu J \nu) J t^4/24 + o(t^4),
\end{align}
where $o(\cdot)$ denotes the little-$o$ Landau symbol. To path the way for generalizations, we shall display equations such that they are valid also in the case of matrix-valued correlation functions, i.e. if the quantities $S, J, \nu$, etc., do not necessarily commute.
The goal of these notes is to demonstrate the existence of the long-time limit $\lim_{t\to\infty}S(t)$.
\subsection{Representation of solutions}
First we recall that $S(t)$ is uniquely determined and corresponds to an equilibrium correlation function thereby introducing some notation and relations that will turn out useful to discuss the long-time limit. The proof is performed most conveniently in the Fourier-Laplace domain and is basically the easy direction of the representation theorems by Nevanlinna, Herglotz, etc.~\cite{Akhiezer:Classical_Moment_Problem}.
The equations of motion, Eq.~\eqref{eq:harmonic_oscillator}, are readily transformed to
\begin{align}
\hat{S}(z) =& -[ z S^{-1}+ S^{-1}\hat{K}(z)S^{-1} ]^{-1}, \label{eq:first_eom}
\end{align}
with
\begin{align}
\hat{K}(z) =& - [ z J^{-1} + i \nu + \hat{M}(z) ]^{-1}. \label{eq:second_eom}
\end{align}
Since $\hat{M}(z)$ is the Borel-Stieltjes transform of a finite measure, the asymptotic expansion $\hat{M}(z) = - M/z + o(z^{-1})$ holds uniformly in the sector $\delta\leq \arg z \leq \pi-\delta$ for any fixed angle $\delta >0$ ($<\pi/2)$. All asymptotic expansions in the complex frequencies in the following are to be understood in this sense. Then one finds the corresponding asymptotic expansions
\begin{align}
\hat{K}(z) =& - J z^{-1} + i J \nu J z^{-2} + J ( \nu J \nu -M) J z^{-3} \nonumber \\
& + o(z^{-3}), \label{eq:high_frequency_K} \\
\hat{S}(z) =& - S z^{-1} - J z^ {-3} + i J \nu J z^{-4} \nonumber \\
& + J (\nu J \nu-M - S^{-1}) J z^{-5} + o(z^{-5}). \label{eq:high_frequency}
\end{align}
Clearly, the expansions are equivalent to the short-time expansion of Eq.~\eqref{eq:short_time}.
Assuming that $M(t)$ is an equilibrium auto-correlation functions is equivalent to
$\hat{M}(z)$ fulfills properties (1)-(4).
Then for $z\in \mathbb{C}_+$ the imaginary part of the denominator in Eq.~\eqref{eq:second_eom} is positive and the inverse is well defined. Thus properties (1) and (2) follow immediately for $\hat{K}(z)$.
The spectral representation shows that $\hat{M}(z= i \eta) = {\cal O}(\eta^{-1})$ as $\eta\to \infty$, and one infers $\hat{K}(z=i \eta)= i J/\eta + \mathcal{O}(\eta^{-3})$ as $\eta \to \infty$, which includes property (3). For the last property, write
\begin{align}
\Imag[\hat{K}(z)] = \hat{K}^*(z) \left\{ \Imag[z]J^{-1} + \nu + \Imag[\hat{M}(z)] \right\} \hat{K}(z) >0 ,
\end{align}
for $z\in \mathbb{C}_+$,
which shows that also property (4) is fulfilled. Hence $\hat{{K}}(z)$ is again the Fourier-Laplace transform of an equilibrium correlation function.
Repeating the argument for Eq.~\eqref{eq:first_eom} shows that $\hat{S}(z)$ corresponds again to an equilibrium correlation function. Hence by uniqueness of the Laplace transform,
the equations of motion yield correlation functions $S(t)$ provided the kernel $M(t)$ is itself a correlation function. By the spectral representation theorem, there is thus a finite symmetric Borel measure $\sigma(\cdot)$ such that
\begin{equation}
S(t) = \int_{\mathbb{R}} e^{-i \omega t} \sigma(\diff \omega),
\end{equation}
in particular $S(0) = S = \sigma(\mathbb{R})$.
\subsection{The spectral measure}
Generally a spectral measure can be uniquely decomposed
\begin{equation}
\sigma = \sigma_\text{ac} + \sigma_\text{s},
\end{equation}
such that $\sigma_\text{ac}$ is absolutely continuous with respect to the Lebesgue measure (i.e. $\sigma_\text{ac}(N)=0$ for any Borel set $N$ of Lebesgue measure zero), and $\sigma_\text{s}$ is singular with respect to the Lebesgue measure (i.e. there is a set $N$ of Lebesgue measure zero with $\sigma_\text{s}(\mathbb{R}\setminus N) =0$) \cite{Teschl:Schroedinger_operators}.
The goal of this subsection is to show that the singular measure $\sigma_\text{s}$ corresponding to $S(t)$ consists at most of an atom at zero-frequency. The proof relies on the characterization of the minimal support of the singular measure as inferred from the boundary values of $\Imag[\hat{S}(z)]$ towards the real line, see Theorem 3.23 in \cite{Teschl:Schroedinger_operators}, in symbols
\begin{equation}
\text{supp } \sigma_\text{s} = \{ \omega \in \mathbb{R} | \limsup_{\eta \downarrow 0} \Imag[\hat{S}(\omega+ i \eta) ] = \infty \}.
\end{equation}
The first step is to rewrite the equation of motion in the Laplace domain Eqs.~\eqref{eq:first_eom},\eqref{eq:second_eom} as
\begin{equation}
\hat{S}(z) = - S/z + \left[z S^{-1} - z^3 J^{-1} - i z^2 \nu - z^2 \hat{M}(z) \right]^{-1}.
\end{equation}
Next, a partial fraction decomposition shows that
\begin{equation}\label{eq:estimate}
\Imag[z^2 \hat{M}(z)] + \Imag[z] M = \Imag[z] \int \frac{\lambda^2 }{|\lambda- z|^2} \mu(\diff \lambda) \geq 0,
\end{equation}
for $z\in\mathbb{C}_+$, in particular $\Imag[z^2 \hat{M}(z)]$ is non-negative close to the real line except for corrections of order ${\cal O}(\Imag[z])$.
For later purpose we also take the limit $z=\omega+i\eta$ to the real axis and infer the stronger estimate
\begin{align}\label{eq:Radon}
\lefteqn{\lim_{\eta\downarrow 0} \Imag[(\omega+i\eta)^2 \hat{M}(\omega+ i\eta) ] \geq \nonumber} \\
& \geq
\omega^2
\lim_{\eta\downarrow 0} \Imag[\hat{M}(\omega+i\eta)] = \frac{1}{\pi}\omega^2 D\mu(\omega) \geq 0.
\end{align}
where $D\mu(\omega)$ denotes the Radon derivative of the Borel measure $\mu(\cdot)$~\cite{Teschl:Schroedinger_operators}.
Since $-\Imag[w^{-1}] \leq (\Imag[w])^{-1}$ for any $w\in\mathbb{C}$, the estimate
\begin{align}\label{eq:master_estimate}
\lefteqn{\Imag[\hat{S}(\omega+ i\eta)] \leq \nonumber } \\
& \eta S/\omega^2 + [ \omega^2 \nu + \Imag[ (\omega+i \eta)^2\hat{M}(\omega+i \eta)]+
{\cal O}(\eta)]^{-1},
\end{align}
shows that $\limsup_{\eta\downarrow 0} \Imag[\hat{S}(\omega+i\eta)]$ remains finite for $\omega\neq 0$. Thus $\text{supp } \sigma_\text{s} \subset \{0 \}$. The decomposition into a singular continuous part and a pure point spectrum $\sigma_\text{s} = \sigma_\text{sc} + \sigma_\text{pp}$ allows to exclude the singular continuous part $\sigma_\text{sc} = 0$ and leaves at most a pure point contribution at zero frequency $\sigma_\text{pp} = F \delta_0, F\geq 0$, where $\delta_0$ is the Dirac measure concentrated at the origin.
The correlation function $S(t)$ thus can be uniquely represented by
\begin{equation}
S(t) = F + \int_{\mathbb{R}} e^{-i \omega t}\sigma_{\text{ac}}(\diff \omega).
\end{equation}
By the Riemann-Lebesgue lemma, the absolutely continuous part does not contribute at large times, such that the long-time limit is guaranteed to exist and equals
\begin{equation}
\lim_{t\to \infty} S(t) = F = \sigma(\{ 0 \}).
\end{equation}
Quite generally the pure point part at zero frequency can be also obtained from the Fourier-Laplace transform (see G\"otze~\cite{Goetze:Complex_Dynamics}, p. 588, Appendix A, Eq. (A.36)),
\begin{equation}
\lim_{t\to\infty} S(t) = \lim_{\eta \downarrow 0} \eta \int_0^\infty e^{-\eta t} S(t) \diff t = \lim_{\eta \downarrow 0} (-i\eta) \hat{S}(z=i\eta).
\end{equation}
One can show that for arbitrary correlation functions $M(t)$ the long-time average defined by
\begin{align}
\limav_{t\to\infty} M(t) &= \lim_{T\to \infty} \frac{1}{T} \int_0^T M(t) \diff t,
\end{align}
always exists~\cite{Feller:Probability_2,Goetze:Complex_Dynamics}. Clearly it coincides with the long-time limit if existent.
Equivalent representation follow from the spectral representation theorem or from the low-frequency behavior of the Laplace transform
\begin{align}
\limav_{t\to \infty} M(t) & = \lim_{\eta \downarrow 0} (-i\eta) \hat{M}(z=i\eta), \\
\limav_{t\to\infty} M(t) &= \mu(\{ 0 \}),
\end{align}
where $\mu(\cdot)$ again represents the spectral measure associated with $M(t)$.
Performing the low-frequency limit of the equations of motion in the Laplace domain, Eqs.~(\ref{eq:first_eom},\ref{eq:second_eom}) reveals that the long-time limit of the correlation function $S(t)$ can be evaluated from the long-time average of the memory kernel
\begin{equation}\label{eq:long_time_limits}
\lim_{t\to\infty } S(t) = S - [S^{-1} + \limav_{t\to\infty} M(t)]^{-1}.
\end{equation}
Hence, the long-time limit can be readily calculated from the long-time average of the memory kernel without solving for the entire time-dependence.
\section{Implications for mode-coupling theory}\label{Sec:MCT}
The mode-coupling theory of the glass transition as developed by G\"otze and collaborators~\cite{Goetze:Complex_Dynamics} deals with a non-linear integro-differential equation, which encodes a plethora of
non-trivial feature in striking agreement with experiments on the evolution of glassy dynamics. Here we review only a simplified version known as schematic models. In the Newtonian case the starting point is an equation of motion as in Eq.~\eqref{eq:harmonic_oscillator}, yet the memory kernel is given as a non-linear functional of the correlation function itself
\begin{align}
& J^{-1} \ddot{S}(t) + \nu \dot{S}(t) + S^{-1} S(t) + \int_0^t M(t-t') \dot{S}(t') \diff t' =0 , \nonumber \\
& M(t) = {\cal F}[S(t)] = \sum_{n\in \mathbb{N}} v_n S(t)^n.
\end{align}
The functional ${\cal F}[F] = \sum_{n} v_n F^n$ is assumed to be absolutely monotone, i.e. $v_n\geq 0$ and $\sum_n v_n S^n < \infty$. Again, the equations are subject to the
initial conditions $S(t=0) = S>0, \dot{S}(t=0)=0$.
The existence and uniqueness of solutions has been demonstrated by Haussmann~\cite{Haussmann:1990}, moreover he proved that the solutions correspond to correlation functions in the sense of Sec.~\ref{Sec:correlation_functions}. The key observation is that the mode-coupling functional ${\cal F}[\cdot]$ maps correlation functions on correlation functions. Assuming that the long-time limit exists $F= \lim_{t\to\infty} S(t)$ it has been shown that the equations display a covariance property which allows to demonstrate a certain maximum principle~\cite{Houches:1991}. As a consequence the long-time limit can be determined by solving merely an algebraic equation rather than by determining the entire time-dependent solution. Therefore the non-equilibrium state diagram, distinguishing between ergodic 'liquid' states and non-ergodic 'glass' states, can be constructed by a convergent iteration procedure~\cite{Goetze:Complex_Dynamics,Goetze:1995}.
A consequence of the proof in Sec.~\ref{Sec:Model_equations} is that the existence of the long-time limit of the solutions of the mode-coupling theory is guaranteed for $\nu>0$. By Haussmann's proof~\cite{Haussmann:1990} both $S(t)$ and $M(t)$ correspond to a correlation function. Taking therefore Haussmann's $M(t)$ as known, the proof of the previous section shows that $F = \lim_{t\to\infty } S(t)$
indeed exists.
The case of pure relaxational dynamics is obtained essentially by dropping the second time derivative in the equation of motion and specifying only the initial value of the correlation function, -- the initial value of the derivative then follows from the equation of motion. Then it has been shown that the solutions are correlation functions that are also completely monotone~\cite{Goetze:1995,Franosch:2002,Goetze:Complex_Dynamics}, i.e. by Bernstein's theorem~\cite{Feller:Probability_2} they can be represented as a Lebesgue-Stieltjes integral of relaxing exponentials. Then the existence of the long-time limit is guaranteed, in addition to the conclusions inferred for the non-equilibrium state diagram also the long-time behavior has been characterized.
\section{Conclusion}
We have shown that the equation of motion displays unique solutions that correspond to equilibrium correlation functions, provided the memory kernel displays the same properties.
Furthermore the long-time limit of these solutions always exists and can be related to the long-time average of the memory kernel. The key ingredient is to rule out a singular continuous spectrum
in the spectrum associated with the correlation function $S(t)$, which has been achieved by relying on a characterization in terms of the boundary values of the imaginary part of the Laplace transform.
In the context of liquid dynamics the proof corroborates that the mode-coupling theory encompassing the phenomenology of the evolution of structural relaxation is built on mathematically solid grounds. More precisely, we have proven the assumed existence of the long-time limit for Newtonian dynamics which is key to characterize idealized glass states and the associated non-equilibrium state diagram.
The generalizations to matrix-valued quantities as they naturally occur considering several fluctuating variables, respectively their covariance matrix,
is straightforward and the formulation chosen suggests that all equations can also be interpreted for matrices. Then the concept of non-negativity has to be adapted to positive semi-definite for matrices, etc., see e.g. Ref.~\cite{Franosch:2002,Lang:2013}. Matrix-valued correlation functions occur for example considering mixtures of different species~\cite{Hansen:Theory_of_Simple_Liquids,Goetze:Complex_Dynamics,Franosch:2002}.
Several relaxation channels emerge naturally for molecular fluids~\cite{Franosch:1997,Scheidsteger:1997} or in confined geometry~\cite{Lang:2010,Lang:2012,Lang:2013} and yield
a different mathematical structure, nevertheless it appears feasible to demonstrate the existence of the long-time limit also for this case.
The key observation was that the singular spectrum could be ruled out, since in $\nu + \Imag[ \hat{M}(z)] > 0$ the term $\nu>0$ dominates for sufficiently small
$\Imag[z]>0$. Conventionally this $\nu$-term is considered to describe processes faster than the time scales of interest, and one may argue that it constitutes an idealization and should not be present in the microscopic problem. Then one should put $\nu$ to zero and consider these fast processes to be included in $M(t)$, resp. $\hat{M}(z)$. Then Eqs.~\eqref{eq:Radon},\eqref{eq:master_estimate} shows that the conclusions still hold
if $\lim_{\eta\downarrow 0} \Imag[ \hat{M}(z=\omega+ i\eta)]>0$, i.e. if there are no spectral gaps associated with $M(t)$. Then the correlation function $S(t)$ still displays a long-time limit, which can be obtained from Eq.~\eqref{eq:long_time_limits}.
One may wonder what kind of correlation functions can be obtained in general from an equation of motion of the type of a generalized harmonic oscillator, or conversely, which correlation functions can be represented as solutions of generalized harmonic oscillator equations. Clearly, a necessary condition is that the short-time expansion, Eq.~\eqref{eq:short_time}, respectively, the high-frequency expansion, Eq.~\eqref{eq:high_frequency}, holds with parameters $J>0, \nu>0, M>0$. Then, it is shown in Appendix~\ref{Sec:Appendix} that the representations Eqs.~(\ref{eq:first_eom},\ref{eq:second_eom}) in the Fourier-Laplace domain hold, such that $\hat{M}(z)$ satisfies conditions (1)-(3) and $\nu+ \Imag[\hat{M}(z)]>0$ for $z\in \mathbb{C}_+$. Going through the proof again reveals that the long-time limit $S(t)$ still exists provided for each fixed $\Real[z]$, the expression
$\Imag[ i\nu z^2+ z^2 \hat{M}(z)]>0$ remains separated from zero for $\Imag[z] \downarrow 0$. Then the value of the long-limit can be inferred from an equation analogous to Eq.~\eqref{eq:long_time_limits}, where the long-time average is replaced by $\lim_{\eta\downarrow 0} (-i\eta) \hat{M}(z=i\eta)$, the existence of the limit being guaranteed from the one of $\lim_{\eta\downarrow 0} (-i\eta) \hat{S}(z=i\eta)$. In the case that already $\Imag[\hat{M}(z)] \geq 0$ for $z \in \mathbb{C}_+$, the kernel $\hat{M}(z)$ corresponds itself to a correlation function $M(t)$, and $S(t)$ also fulfills the generalized harmonic oscillator equation in the temporal domain, Eq.~\eqref{eq:harmonic_oscillator}.
Last, one may consider also correlation functions for stationary stochastic processes that do not correspond to equilibrium states. These correlation functions are then in general complex-valued but are still Fourier transforms of a measure, albeit not necessarily a symmetric one~\cite{Feller:Probability_2}. The corresponding Laplace transforms still fulfill properties (1),(3),(4), yet property (2) may be violated. In particular the imaginary part $\Imag[\hat{S}](z)$ will not be a symmetric function for frequencies approaching the real line, which reflects that absorption and emission of quanta in a scattering process is no longer equally likely~\cite{Hansen:Theory_of_Simple_Liquids}. Going through the proofs again, one verifies that if we relax the assumption for the kernel $M(t)$ to be a correlation function in a non-equilibrium steady state, then so will be the solutions $S(t)$ of the equations of motion, Eq.~\eqref{eq:harmonic_oscillator}, and again the long-time limit $\lim_{t\to\infty} S(t)$ is guaranteed to exist.
\begin{acknowledgments}
I am indebted to Wolfgang G\"otze and Rolf Schilling for priceless discussions on the mathematical properties of the mode-coupling theory of the glass transition.
This work has been supported by the
Deutsche Forschungsgemeinschaft DFG via the Research Unit FOR1394 ``Nonlinear Response to
Probe Vitrification''.
\end{acknowledgments}
|
1,941,325,220,523 | arxiv | \section{Introduction}
Fission has shown to be a relevant mechanism to learn about a wide spectrum of fundamental nuclear properties, as it is driving by both reaction mechanism and nuclear structure aspects \cite{krappe:2012, schmidt:2016}. The importance of fission for applications is obvious also, including nuclear power production, transmutation of waste, medicine, network calculations in astrophysics.At intermediate and high excitation energy (above about 40 MeV) the process is a particularly appropriate laboratory for probing nuclear dynamics.\\
Fission is a very entangled process though. Accordingly, a robust understanding has not been reached yet, and various interpretations of the same experimental data exist (see Refs.~\cite{lestone:2009, schmitt:2014, mahata:2017} and therein). Even under "clean" conditions, satisfying Bohr's hypothesis about the independence of the decay of the excited system on the way it was produced, inconsistencies are found (see the discussion in Ref.~\cite{schmitt:2010}). Un-ambiguous insight into the process is complex to extract from heavy-ion-induced fission. Indeed, in this approach, the first stage of the reaction - during which the two ions fuse and produce the excited compound nucleus (CN), can notably affect the subsequent decay: In addition to the obvious role of the fusion stage in determining the excitation energy ($E^*$) and angular momentum ($L$) of the CN, the dynamics of the fusion mechanism itself can play a non-negligible role. The evolution of the composite system formed by the projectile and the target on the way to a fully equilibrated compact system depends on the entrance-channel asymmetry $\alpha$=($A_t$-$A_p$)/($A_t$+$A_p$) (see Ref.~\cite{kumar:1994} and therein). Above the Businaro-Gallone point $\alpha_{BG}$ \cite{businaro:1955} fusion takes place in terms of absorption of the projectile by the target (more generally, the lightest by the heaviest reactant). In other words, the heavy partner takes up the nucleons from the light one, and a mono-nuclear CN is formed in a short time. On the other hand, for more symmetric reactions where $\alpha < \alpha_{BG}$, a neck develops between the reactants. Projectile and target first form a di-nuclear system which equilibrates in various degrees of freedom; nucleons are exchanged through the neck, and formation of a compact CN takes longer.
\begin{figure*}[!hbt]
\hspace{-0.1cm}
\includegraphics[width=18.cm, height=6cm,angle=0]{Mnpre_BGEexc_chi_exp_full_new.pdf}
\vspace{-.60cm}
\caption{(Color online) Correlation between experimental neutron pre-scission multiplicities $M_n^{pre}$ with a) $\alpha$/$\alpha_{BG}$ and $E^*$, and with b) $\chi$ and $E^*$. For experimental points, we refer to Table 1. Error bars are omitted for clarity. Lines connect different energy points for a given reaction. The same color is employed for reaction couples leading to the same CN.}
\label{fig1}
\end{figure*}
The possible influence of the fusion stage has to be kept in mind whenever experimental data are used to extract fission time scales. In a seminal work in the 90's, Saxena et al. \cite{saxena:1993} analyzed a wide set of projectile-target combinations, with the goal to unfold the different contributions to the total fusion-fission time. In particular, it was attempted to separate the contribution of the fusion stage (CN formation time, $t_{fo}$) from the time scale of the sole fission process. The discrimination is crucial also for further proper unfolding of the fission time into the time the system needs to overcome the barrier (pre-saddle, $t_{presad}$) and the time it takes to descent to scission (post-saddle, $t_{postsad}$). This unfolding is important for getting insight into properties such as fission barriers, level densities, nuclear viscosity, etc. The critical role of entrance-channel asymmetry, on one side, and CN excitation energy and fissility, on the other side, in extracting the different times was demonstrated in Ref.~\cite{saxena:1993}. A recent study \cite{shareef:2016} pointed the still unresolved puzzle.\\
The un-ambigous extraction of fusion-fission time scales, and thus reliable conclusions on underlying nuclear properties, is a multi-dimensional problem. The fusion process depends on the projectile-target combination and the bombarding energy with respect to the Coulomb barrier ($E_{cm}$/$V_b$). These two features determine the angular momentum of the composite system. As for the fission stage, CN fissility $\chi$, excitation energy and angular momentum all play a role. A separate study of each effect is very challenging, if possible at all. Indeed, in a heavy-ion collision, these variables are usually not independent. To quote some example: Producing a compound nucleus with specific fissility and excitation energy by means of different entrance channels implies, in general, different bombarding energies, and leads to different angular momentum distributions for the CN. On the other hand, fixing $E_{cm}$/$V_b$ and $L$ usually requires varying $E^*$ and $\alpha$/$\alpha_{BG}$. The involved multi-dimensional nature of the challenge is illustrated in fig.~\ref{fig1} where a representative set of experimental neutron pre-scission multiplicities $M_n^{pre}$ is shown, as a function of $\alpha$/$\alpha_{BG}$ and $E^*$ in a), and as a function of $\chi$ and $E^*$ in b). We chose to consider neutron multiplicities here, as they were identified as relevant signatures of nuclear reaction times. Figure~\ref{fig1} demonstrates that, while $E^*$ seems to have the leading role in determining $M_n^{pre}$, entrance-channel asymmetry and CN fissility matter also. We note that, on top of the displayed correlations, the angular momentum can vary as well. Furthermore, the comparison of the two panels makes it clear that an independent variation of each parameter is, in general, not available.\\
In their reference work, Saxena et al. \cite{saxena:1993} attempted to extract the individual contributions, $t_{fo}$, $t_{presad}$ and $t_{postsad}$, to the total fusion-fission time, by means of a systematics analysis of available experimental information on $M_n^{pre}$. Complete separation could not be achieved, but a definite correlation between specific time scales and, either $\chi$ or $E^*$, was observed. Shareef et al. \cite{shareef:2016} recently revisited the issue of ordering parameter, with focus on the significance of $\alpha$/$\alpha_{BG}$. From these and similar analyses \cite{hinde:1992}, it is clear that a unique ordering parameter is impossible to extract due to the multi-dimensional nature of the problem.\\
Since nuclear reaction times cannot be measured directly, model calculations are needed to link them to experimental observables, {\it i.e.} $M_n^{pre}$ in present case. In the aforementioned studies \cite{saxena:1993, hinde:1992, shareef:2016} the fusion stage is not explicitly modeled, and the fission stage is treated within the statistical model. Although the latter does not involve times as such, it is possible to implement the influence of time by some {\it ad hoc} ansatz (see Ref.~\cite{jurado:2007} for a recent discussion).\\
Describing the evolution of the system during the fusion stage, and its subsequent decay by fission, implies involved model calculations. Shape evolution as function of time is to be obtained from the solution of a multi-dimensional equation of motion, for both the (entrance) fusion and (exit) fission stages. Solving this equation is rather complex for each stage on its own already. Further coupling of the two stages is also particularly difficult, both in terms of optimal description of the shape and of the potential energy (we refer to Ref.~\cite{karpov:2017} for a update discussion on this challenge). In the present context and for the present goal, a fully-dynamical description does not exist yet. So far, hybrid models are used. Depending on the aim of the study, the dynamical treatment is restricted to, either the entrance \cite{wilczynska:1993, kumar:2014} or the exit \cite{frobrich:1998, adeev:2005}, stage. The present work is trying to go a step further in this direction: By means of dynamical considerations for both the entrance and exit channels, we attempt to deepen previous studies \cite{saxena:1993, shareef:2016} based on the statistical model.
\section{Method and models}
\subsection{Strategy}
The characteristics of the reaction (projectile, target, bombarding energy) give the CN maximum excitation energy $E^*$ and angular momentum $L$. Though, even with initial $E^*$ and $L$ fixed, different entrance channels have shown to experience different decays \cite{ruckelshausen:1986}, what suggested different patterns for the entrance-channel relaxation in shape, excitation energy and angular momentum. Consequently, the CN formation (equivalently, fusion) time is different. What happens to the system during this time defines the initial conditions for the CN decay calculation, which it may affect more or less sizeably depending on the specificities of the entrance-channel.\\
To address the issue of influence of entrance-channel effects on typical fission observables, the following method is proposed. The fusion dynamics is calculated with the dynamical heavy-ion collision code HICOL. More specifically, the code is used to estimate the CN formation time as depending on the nuclear reaction at work. The fission stage is calculated from a multi-dimensional dynamical stochastic Langevin code. In this framework, pre- and post-saddle dynamics emerge naturally from the solution of the equation of motion, avoiding the ansatz required in statistical models. The dependence on the properties of the compound system (mass $A$, charge $Z$, $E^*$ and $L$), and their evolution with time, are consistently taken into account along the decay. The model is supposed to yield a fair description of the CN decay by fission, provided that a fully equilibrated and compact CN is formed in the entrance channel. Any discrepancy with experiment may thus be ascribed to an entrance-channel effect.\\
Combining the time information from HICOL with the CN-decay stochastic Langevin mode, we try to pin down the entangled influence of entrance-channel asymmetry, fusion time, CN fissility, excitation energy and angular momentum on the experimental observables.\\
The goal of this study is to investigate, and possibly, unfold, entrance- and exit-channel aspects. Though, it is emphasized that the present work does {\it not} provide us with a complete, unified dynamical model of fusion-fission. Dynamical codes are run separately for the entrance (HICOL) and exit (stochastic Langevin) channels. Also, the outcome of the fusion code is not explicitly fed to the subsequent treatment of CN decay; it is used to estimate the magnitude of the fusion time, only. The investigation on the magnitude of entrance-channel effects remains thus qualitative at some level. However, the proposed method permits to re-examine experimental information in a multi-dimensional space, where previous work concentrated on one-dimensional (1D) projections for specific variables of the problem. The study shows that the influence of one or the other variable critically depends on the mass region under consideration. As a consequence, 1D projections can, either magnify or hide, the importance of some variable, depending on the system. That can lead to hazardeous interpretation when extrapolation is made in other mass regions where the dominating factor may have changed. In this context, the multi-dimensional analysis shall help to address the apparent unconsistency in measured neutron pre-scission multiplicities noted recently by Shareef et al. \cite{shareef:2016}.
\subsection{Theoretical framework}\label{model}
As introduced above, we do not embark on a unified model of fusion-fission, from the heavy-ion collision to the formation of cold fission fragments. Instead, we divide the description into two parts. As for the first (fusion) stage, the HICOL \cite{feldmeier:1987} code is employed. The second (fission) stage is modelled with the stochastic Langevin code developed in Omsk \cite{adeev:2005, nadtochy:2014}. The predictions by HICOL are analyzed in terms of CN formation time, while we exploit the outcome of the fission stage in details. The HICOL information is used to interpret the difference, if any, between the predictions of the CN-decay calculation and experiment. The main ideas and ingredients of the aforementioned two theoretical codes are given below; we refer to the quoted references for further details.
\subsubsection{The HICOL code for fusion dynamics}
The HICOL code \cite{feldmeier:1987} gives access to the time evolution of shape relaxation in a heavy-ion collision by means of a deterministic description of the motion. The collision is considered in the configuration space of three shape variables, representing, respectively, the distance between the geometrical centers of the reaction partners, the thickness of the neck connecting them, and the asymmetry in their size (or mass). Classical equations of motion of the Langevin type with inclusion of one-body dissipation in the form of the "wall and window" formula \cite{blocki:1978} are solved numerically in the 3D configuration space. The potential energy landscape is purely macroscopic, and given from the FRLDM liquid drop formula \cite{krappe:1979} accounting for the finite range of nuclear forces. Inertia is obtained in the Werner-Wheeler approximation of incompressible irrotational flow \cite{davies:1976}. HICOL does not contain free parameters and has shown to consistently describe the dynamical evolution of various composite systems formed in nucleus-nucleus collisions in a wide range of impact parameters. In the present work, we use the version of the code as implemented in Ref.~\cite{wilczynska:1993}. In the framework of the present work, it is important to note that particle evaporation during the path to fusion is {\it not} included in HICOL.
\subsubsection{Stochastic Langevin approach to fission dynamics}
The model used to compute the heavy CN decay is based on the stochastic classical approach of nuclear dynamics \cite{abe:1996}. The four-dimensional (4D) Langevin code developed in Omsk, see Refs.~\cite{adeev:2005, nadtochy:2014}, is employed.\\
In the stochastic approach of fission, most relevant degrees of freedom are considered as collective coordinates, and their evolution with time is treated as the motion of Brownian particles, which interact stochastically with a surrounding "heat bath". In the present model, four collective coordinates are considered. Three variables $({\bf q})$ describe the shape of the deforming nucleus, and the fourth one corresponds to the orientation of its angular momentum relative to the symmetry axis. The shape coordinates are related to elongation, neck constriction, and left-right asymmetry, while the projection $K$ of the total angular momentum onto the symmetry axis of the fissioning nucleus is chosen for the fourth so-called tilting coordinate. The evolution of the shape and $K$ coordinates with time is obtained by solving, in parallel, the corresponding Langevin equations of motion, and assuming that the motion in the $K$ direction is over-damped. The driving potential is given by the Helmholtz free energy $F({\bf q},K)=V({\bf q},K) - a({\bf q}) T^2$, with $V({\bf q},K)$ being the potential energy, $a({\bf q})$ the level-density parameter \cite{ignatyuk:1975} and $T$ the temperature of the system. The potential energy $V({\bf q},K)$ is calculated within the framework of the macroscopic FRLDM model \cite{krappe:1979}. The calculation of inertia uses the Werner-Wheeler approximation \cite{davies:1976}, and friction is derived assuming the chaos-weighted one-body dissipation formalism \cite{chaudhuri:2001}, which can be considered as an improved variant of the "wall and window" prescription. De-excitation by evaporation of light particles ($n$, $p$, and $\alpha$) by the compound system prior scission, as well as by the fragments after scission, is taken into account employing the Monte-Carlo approach. Particle-decay width are calculated within the Hauser-Feschbach theory. The 4D model of Omsk has shown able to explain a large variety of observables for fission over a wide range of systems (see Ref.~\cite{nadtochy:2014} and therein) \footnote{The unconsistency pointed in \cite{mahata:2017} was recently solved in the code, and found to have no impact for the concern of the present work.}.\\
We note that all Langevin calculations presented in this work were obtained using the prescription of Ref.~\cite{frobrich:1998} for modelling the CN initial $L$ distribution. The latter was compared to coupled-channel calculations by the CCFULL code \cite{hagino:1999, shrivastava:2017}. The maximum angular momenta could differ by several units, but the most probable $L$'s from Ref.~\cite{frobrich:1998} and CCFULL were found similar. Most important for the concern of the present investigation is the trend as a function of entrance-channel asymmetry ($\alpha$/$\alpha_{BG}$), and its evolution with system size ($\sim \chi$) and bombarding energy ($E_{cm}$/$V_b$), which were observed to be the same for the two theoretical $L$ distributions.\\
\\
We note the consistency of the present framework. Both the models employed for treating the fusion and fission stage are based on macroscopic concepts, classical equations of motion, and similar prescriptions for model ingredients such as potential energy, friction, and inertia. We note also that all necessary information about the dynamical evolution till the CN is formed is available from HICOL, and can in principle be supplied as an input to the stochastic Langevin code. As explained above, this is not done here. As a first step, the strategy is to model the fission decay dynamics under Bohr's hypothesis, and to consider {\it a posteriori} the possible influence of fusion dynamics, guided by HICOL predictions. The main goal is to decouple entrance- and exit-channel dynamical effects.
\section{Results}
The reaction systems studied in this work are summarized in Table 1. Both lowly- and highly-fissile CN are investigated, located either below or above the BG point, from near to above Coulomb barrier energies. We restrict to reactions for which $E^*$ exceeds about 40 MeV, where the models used in this work are valid. The population in angular momentum ranges from about 10 to 80 $\hbar$. For those reactions for which experimental information on the neutron pre-scission multiplicity exists, the corresponding reference is specified in the table. We focus the investigation on $M_n^{pre}$ as it is particularly sensitive to fission time scales, and numerous data exist. Other observables are possible \cite{kumar:2014, kaur:2015}, and provide complementary insight. We note that most of the systems which we consider have been discussed also within a statistical model framework in {\it e.g.} Refs.~\cite{saxena:1993, shareef:2016}.
\begin{table}[h!]
\caption{Properties of the reactions studied in this work.}
\begin{tabular}{c|c|c|c|c|c}
\hline
Reaction &$E_{lab}$&$E^*$ & $\alpha$/$\alpha_{BG}$ & $\chi$ & Expt.\\
&(MeV)&(MeV) & & & ref.\\
\hline
$^{28}$Si+$^{134}$Ba $\to$ $^{162}$Yb&132-165&81-115&0.85&0.614& - \\
\hline
$^{20}$Ne+$^{159}$Tb $\to$ $^{179}$Re&89-205&58-174&0.97&0.648& \cite{cabrera:2003}\\
\hline
$^{16}$O+$^{182}$W $\to$ $^{198}$Pb &72-80&45-53&1.00 & 0.7044& \cite{newton:1988}\\
$^{28}$Si+$^{170}$Er $\to$ $^{198}$Pb &113-139&57-83&0.85 & 0.7044& \cite{mahata:2006}\\
\hline
$^{12}$C+$^{194}$Pt $\to$ $^{206}$Po &80-108&63-92&1.04& 0.716& \cite{golda:2013}\\
$^{22}$Ne+$^{184}$W $\to$ $^{206}$Po &99-127&63-92&0.93&0.716& - \\
\hline
$^{16}$O+$^{197}$Au $\to$ $^{213}$Fr &83-113&52-80 &0.99& 0.7428& \cite{newton:1988}\\
$^{19}$F+$^{194}$Pt $\to$ $^{213}$Fr &82-111&50-80 &0.95 &0.7428& \cite{singh:2012}\\
\hline
$^{12}$C+$^{204}$Pb $\to$ $^{216}$Ra &86-90&57-61&1.02 & 0.7504 & \cite{singh:2008}\\
$^{19}$F+$^{197}$Au $\to$ $^{216}$Ra &93-97&57-61&0.95& 0.7504 & \cite{singh:2008}\\
\hline
$^{11}$B+$^{237}$Np $\to$ $^{248}$Cf &73-96&60-82&1.01&0.825& \cite{saxena:1993}\\
$^{16}$O+$^{232}$Th $\to$ $^{248}$Cf &96-118&60-82&0.97&0.825& \cite{saxena:1993}\\
\hline
\hline
\end{tabular}
\end{table}
\subsection{Overview of the calculated results}\label{illus}
\begin{figure*}[!hbt]
\hspace{-1.4cm}
\includegraphics[width=20cm, height=7.cm, angle=0]{formt_ecmvbBG_Mnpre_LEexc.pdf}
\vspace{-.70cm}
\caption{(Color online) a) Calculated dependence of the formation time $t_{fo}$ on entrance-channel asymmetry $\alpha$/$\alpha_{BG}$ and energy with respect to the Coulomb barrier $E_{cm}$/$V_b$, as obtained from HICOL. b) Calculated dependence of the neutron pre-scission multiplicity $M_n^{pre}$ on CN excitation energy $E^*$ and mean angular momentum $<L>$ as obtained from the stochastic Langevin code. The reactions presented in the plots are listed in Table 1. Lines connect different energy points for a given reaction. The same color is employed for reaction couples leading to the same CN.}
\label{fig2}
\end{figure*}
\begin{figure*}[!hbt]
\hspace{-0.85cm}
\includegraphics[width=16cm, height=6.4cm, angle=0]{tform_L_Eexc_th04.pdf}
\vspace{-.40cm}
\caption{(Color online) a) Difference between the formation times $t_{fo}$ (integrated over $L$) below ($< BG$) and above ($> BG$) the BG asymmetry $\alpha_{BG}$ as function of $E^*$ for the reaction couples leading to a given CN and considered in Fig.~\ref{fig2}. b) Similar to panel a) for the difference between the mean angular momenta $<L>$.}
\label{fig2b}
\end{figure*}
A typical outcome of the calculations performed in this work is shown in Fig.~\ref{fig2}, as obtained, respectively, for the fusion andthe fission stage by the corresponding code. The correlation predicted by HICOL between CN formation time, entrance-channel asymmetry $\alpha$/$\alpha_{BG}$, and center-of-mass energy with respect to the Coulomb barrier $E_{cm}$/$V_b$ is shown in panel a). As introduced above, the projectile-target combination plays a key role in determining the time scale of the entrance-channel dynamics: for a given CN, the projectile-target couple with $\alpha < \alpha_{BG}$ needs more time to fuse than the combination with $\alpha > \alpha_{BG}$. The violence of the collision, in terms of $E_{cm}$/$V_b$, plays some role as well. Panel b) displays the dependence of the neutron pre-scission multiplicity $M_n^{pre}$ on CN initial excitation energy $E^*$ and angular momentum $L$ as anticipated from the stochastic approach to fission. The picture precisely illustrates the features of the reaction mechanism which are accounted for in the dynamical CN-decay model used in this work. That is, the relative importance of $E^*$ and $L$, and its dependence on the CN ($A$, $Z$) composition. The calculated multiplicity is observed to be primarly governed by $E^*$. Though, for lowly-fissile systems, as we will see in more detail below, the angular momentum can play a non-negligible role.\\
A quantitative estimate of the aforementioned differences and related to the entrance channel is given in Fig.~\ref{fig2b}. For reaction couples yielding the same CN, the difference between the formation times $t_{fo}$ for the projectile-target combination located below and above $\alpha_{BG}$ is shown in panel a) as function of $E^*$. Similarly, panel b) displays the difference in mean angular momentum between reactions with $\alpha < \alpha_{BG}$ and $\alpha > \alpha_{BG}$. Based on the left panel, one may expect an enhanced influence of the fusion stage for the couple $^{16}$O+$^{182}$W {\it vs.} $^{28}$Si+$^{170}$Er leading to $^{198}$Pb, as compared to the couple $^{12}$C+$^{204}$Pb {\it vs.} $^{19}$F+$^{197}$Au leading to $^{216}$Ra. The right panel suggests the possible emergence of $L$-driven entrance-channel dependences if the difference in $<L>$ for two reactions of a given couple is large enough. The differences constructed in Fig.~\ref{fig2b} are explored in detail further below.\\
The correlations displayed in Fig.~\ref{fig1} for experiment, and Fig.~\ref{fig2} for theory, and their concomitant analysis, are at the centre of this work. According to the right panel of Fig.~\ref{fig2}, for the same $E^*$ and close-by $L$ values, two reactions forming the same compound are expected to yield similar pre-scission multiplicities, in full accordance with Bohr's hypothesis which the dynamical CN-decay calculation used here is based on. However, Fig.~\ref{fig1} shows that rather distinct $M_n^{pre}$ were measured for several such couples of reactions. Inspection of the left panel of Fig.~\ref{fig2} suggests that two reactions within a couple can be characterized by noticeably different fusion times depending on $E_{cm}$/$V_b$. This difference in $t_{fo}$, as function of $E^*$, is explicitly shown in Fig.~\ref{fig2}a). Remarking that, in experiment, neutrons emitted along the path to fusion can in general not be discriminated from those emitted along the fission stage, the excess of measured $M_n^{pre}$ may be attributed \cite{saxena:1993, shareef:2016, wilczynska:1993, hinde:1992} to a large CN formation time. Remind that HICOL does {\it not} account for evaporation along fusion. Hence, a deficit in the "CN-decay stochastic Langevin based" prediction for $M_n^{pre}$ as compared to the measuremnt may sign emission during the fusion mechanism. As we will demonstrate in this work, while such a direct connection sounds realistic in many cases, the situation can turn out to be more complex in others. The reason is the aforementioned entangled interplay of {\it e.g.} un-matched $E_{cm}$/$V_b$ and/or $L$ variables for a given combination of $\chi$ and $E^*$. The quantitative importance of the mismatch critically depends on the compound nucleus mass and charge, ranging from nearly no consequence to sizeable compensation effects. The latter may lead to apparent inconsistencies \cite{shareef:2016}.\\
The previous paragraph establishes the strategy and ambition of the present work. A specific CN, characterized by {\it e.g.} $\chi$, can be populated at similar $E^*$ by means of different entrance channels. The latter govern the fusion time. For reactions with large values of $t_{fo}$ (basically, comparable to the decay time), light-particles can be emitted along the fusion process. Those are not calculated by HICOL, and are obviously not included in the Bohr's hypothesis framework used here for modelling CN decay. Figure~\ref{fig3} shows the same correlation as in Fig.~\ref{fig1}b), but as expected from the dynamical CN-decay code. Naturally, for a given CN, the theoretical results merge for different entrance channels, and exhibit a smooth and continuous evolution with $E^*$. Evidently, experiment, Fig.~\ref{fig1}b), contradicts the theoretical picture in most cases (deliberately shown). Guided by HICOL predictions, Fig.~\ref{fig2}a), we try to interpret the difference between measurements and Langevin CN-decay calculations. Such an analysis is proposed to reveal, and hopefuly unfold, the possible influence of the first stage of the reaction on the interpretation of the second one.
\begin{figure}[!hbt]
\hspace{-0.45cm}
\includegraphics[width=9.cm, height=6.2cm,angle=0]{Mnpre_chiEexc_th.pdf}
\vspace{-.80cm}
\caption{(Color online) Calculated dependence of the neutron pre-scission multiplicity $M_n^{pre}$ on $\chi$ and $E^*$ as obtained from the stochastic Langevin code for the same set of reactions than in Fig.~\ref{fig1}. Lines connect different energy points for a given reaction. The same color is employed for reaction couples leading to the same CN.}
\label{fig3}
\end{figure}
\subsection{Detailed analysis and comparison with experiment}\label{analysis}
\begin{figure*}[!hbt]
\includegraphics[width=16.cm, height=10.cm, angle=0]{Fig4_Mnpre_Eexc.png}
\vspace{-.50cm}
\caption{(Color online) Neutron pre-scission multiplicity $M_n^{pre}$ as function of excitation energy for various reactions. Each panel corresponds to a given CN: a) $^{216}$Ra, b) $^{248}$Cf, c) $^{213}$Fr, and d) $^{198}$Pb, populated by means of two entrance channels. The experimental $M_n^{pre}$ (open symbols) are compared with dynamical Langevin CN-decay calculations (full symbols). For each couple of reactions, the system with largest (smallest) $\alpha$/$\alpha_{BG}$ is shown in red (blue).}
\label{fig4}
\end{figure*}
Towards the aim of the work, we discuss in detail separately several representative reactions. The neutron pre-scission multiplicity expected from the dynamical Langevin CN-decay calculation for four compound nuclei is shown in Fig.~\ref{fig4} as function of excitation energy. Each panel corresponds to a given CN: a) $^{216}$Ra, b) $^{248}$Cf, c) $^{213}$Fr, and d) $^{198}$Pb, populated by means of two entrance channels. Wherever available, experimental results (open symbols) are overlaid with theory (full symbols). As much as possible, we favored systems for which the two reactions were studied in the same paper, measured and analyzed under identical conditions, in order to minimize the possibility of experimental bias. {\it This is important for the present investigation which relies mostly on the relative difference in $M_n^{pre}$ between two reactions, rather than on absolute values.}\\
Starting with $^{216}$Ra, produced in either $^{12}$C+$^{204}$Pb or $^{19}$F+$^{197}$Au collisions, the calculation exhibits nearly the same $M_n^{pre}$ at overlaping $E^*$. This is due to the very similar $L$ distribution for the two reactions (see Fig.\ref{fig2b}b) where the difference in $<L>$ is very small). While theory explains reasonably well the reaction $^{12}$C+$^{204}$Pb within the experimental error bars, it predicts a slightly too small number of pre-scission neutrons for $^{19}$F+$^{197}$Au. Most noteworthy is that, independent on the description of absolute values, theory does not explain the difference seen in experiment between the two reactions. Having excluded strong $L$ effects, we thus ascribe this difference to the fusion stage. According to our calculations with HICOL, the CN formation time is, for the here-discussed energy range, larger by about a factor of 2 for $^{19}$F+$^{197}$Au ($\alpha$/$\alpha_{BG} < 1$) as compared to $^{12}$C+$^{204}$Pb ($\alpha$/$\alpha_{BG} > 1$) at given $L$. At $E^* \approx$ 61 MeV the difference in $t_{fo}$ varies from 20 to 50 $^.$ 10$^{-22}$ s over the $L$ range populated, see Fig.\ref{fig2b}a). This is comparable to typical neutron emission times for heavy nuclei at corresponding excitation energy. Hence, we propose that the difference of about 0.5 neutron between the two systems is due to emission during the CN formation stage. Singh et al. \cite{singh:2008} suggested some influence of angular momentum, but also clearly mentionned entrance-channel dynamics.\\
Switching to $^{248}$Cf formed in $^{11}$B+$^{237}$Np and $^{16}$O+$^{232}$Th collisions \footnote{Note that in Ref.~\cite{shareef:2016} there is a mistake for this couple of reactions, $^{237}$Np having been replaced with $^{232}$Np. This is certainly due to the typo in the table of Ref.~\cite{saxena:1993} from where it was extracted. Hence, the point corresponding to B+Np in Fig. 4 of Ref.~\cite{shareef:2016} has $\alpha$/$\alpha_{BG}$=0.97 rather than 1.01.}, the calculation in Fig.~\ref{fig4} again suggests no dependence of $M_n^{pre}$ on the entrance channel at similar $E^*$. In that case, however, the calculated initial CN angular momentum distributions show quite some difference for the two projectile-target combinations. Similarly to Ref.~\cite{kumar:1994} we find that $^{16}$O+$^{232}$Th populates up to larger $L$'s (see Fig.\ref{fig2b}b) for $<L>$). The fact that the calculated $M_n^{pre}$ is not sensitive to the angular momentum is explained by the high fissility of the CN: the fate of $^{248}$Cf is not hindered by neutron emission, the fission barrier is much smaller than the neutron binding energy, and the system is commited to fission anyhow \cite{charity:1986}. As a consequence, for $^{248}$Cf like for $^{216}$Ra, the difference in the measured pre-scission multiplicities for the considered projectile-target combinations is classified as an entrance-channel effect. This is supported by the difference in $t_{fo}$ obtained from HICOL and in accordance with previous work \cite{kumar:1994}: the more symmetric reaction $^{16}$O+$^{232}$Th ($\alpha$/$\alpha_{BG} < 1$) has a non-negligible probability for evaporating a neutron during its longer fusion phase as compared to the more asymmetric $^{11}$B+$^{237}$Np ($\alpha$/$\alpha_{BG} > 1$) combination.\\
The compound nucleus $^{213}$Fr considered in Fig.\ref{fig4}c) was synthetized in $^{16}$O+$^{197}$Au and $^{19}$F+$^{198}$Pt collisions. The calculated $M_n^{pre}$ shows negligible dependence on the reaction, as a result of the aforementionned two features, {\it i.e.} similar $L$ distributions, see Fig.\ref{fig2b}b), and a rather high fissility for the CN. The HICOL-predicted difference in formation time is rather small also, see Fig.\ref{fig2b}a), in contrast to the previous cases. In experiment, there is finite difference in $M_n^{pre}$. Most notably is the fact that $^{16}$O+$^{197}$Au ($\alpha$/$\alpha_{BG} \approx 1$) displays a larger $M_n^{pre}$ as compared to $^{19}$F+$^{198}$Pt ($\alpha$/$\alpha_{BG} < 1$). That is, the difference in $\alpha$/$\alpha_{BG}$ (equivalently, in $t_{fo}$) does {\it not} reflect the measured $M_n^{pre}$ values; it is even opposite to expectation \cite{shareef:2016}. In an attempt to understand this observation, we first note that the difference in $t_{fo}$ for the close F+Pt and O+Au couples is much less than for the combinations used in the $^{216}$Ra and $^{248}$Cf cases. Second, the difference in experimental $M_n^{pre}$ is seen to be limited. Third, we remark that the energies involved are sub- and near-barrier, what may imply additional effects \cite{mukarami:1986}. Finally, the two reactions were measured in different experiments and involved rather thick targets. Depending on the treatment of energy loss corrections, and the method for determining the suited excitation energy, the $x$-axis in the two experiments are subject to be shifted one with respect to the other by up to a few MeV. All in all, we think that one shall maybe not over-interpret the confrontation of the F+Pt and O+Au data. The apparent, inconsistent \cite{shareef:2016}, difference in experimental $M_n^{pre}$ between the two entrance channels may be the outcome of a complex interplay between various weak physical effects, on on side, as well as, possible experimental bias and limited accuracy, on the other side.\\
The fourth compound considered in Fig.~\ref{fig4} is the less fissile $^{198}$Pb nucleus which pre-scission neutron multiplicities were measured in $^{16}$O+$^{182}$W and $^{28}$Si+$^{170}$Er fusion-fission reactions \footnote{Note that as compared to Ref.~\cite{shareef:2016} the $E^*$ values which we calculated for $^{28}$Si+$^{170}$Er are slightly higher, in accordance with Newton et al. \cite{newton:1988}, what yields a slightly shifted $E^*$ range.}. Only few, rather scattered, experimental points exist; some are around the barrier, have limited accuracy, and come from different experiments. Though, it is interesting to study this case as it reveals interesting features which the above fissile systems do not permit to study. The calculations were performed over an extended $E^*$ range for $^{16}$O+$^{182}$W in order to allow a more robust comparison with $^{28}$Si+$^{170}$Er. According to the scatter of some experimental points, and within the established precision of the model (see Refs.~\cite{adeev:2005, nadtochy:2014} and therein), the Langevin CN-decay calculation describes reasonably the measurements. A considerably larger formation time is predicted by HICOL for $^{28}$Si+$^{170}$Er ($\alpha$/$\alpha_{BG} < 1$), see Fig.~\ref{fig2b}a). One may thus anticipate a larger pre-scission multiplicity for this reaction. However, the measured $M_n^{pre}$ seems to be larger for $^{16}$O+$^{182}$W ($\alpha$/$\alpha_{BG} \approx 1$) in the overlaping $E^*$ region covered. This observation was left partly un-understood \cite{shareef:2016}. Beside the word of caution we emitted previously, it seems unlikely that experimental bias can alone explain why experimental observation is so far from what expected from the calculated sizeable large $t_{fo}$ value for the more symmetric reaction. We suggest that one reason why the multiplicities of the two reactions come close, and possibly are inversed as compared to expectation based on $t_{fo}$, is the significant influence of angular momentum. This is shown in Fig.~\ref{fig4}d) with the CN-decay calculations studied over an extended $E^*$ region. At given, low excitation energy, the calculated $M_n^{pre}$ is similar for $^{16}$O+$^{182}$W and $^{28}$Si+$^{170}$Er as seen from the close lying full circles and squares. With increasing energy the two curves separate due to an increasing difference in $<L>$ population between the two reactions (see Fig.~\ref{fig2b}b) for $<L>$). Below the bombarding energy leading to $E^* \approx$ 65 MeV, the most probable $L$'s contributing to fission differ by about 10 $\hbar$, while for the points above $E^* \approx$ 65 MeV, the $^{28}$Si+$^{170}$Er reaction populates angular momenta which on average exceed those from $^{16}$O+$^{182}$W by 15 $\hbar$ and more. For a CN of moderate fissility such as $^{198}$Pb, the barrier to overcome and proceed to fission is high at small $L$ (around 12 MeV) and becomes comparable to the neutron binding energy (around 9 MeV) at the highest $L \approx$ 55 $\hbar$ populated here. Hence, the competition between neutron emission and fission is particularly effective at large $L$, and any additional evaporated neutron can definitively prevent the system to go to fission. At the same time, the higher the angular momentum, the larger the excitation energy stored in rotational energy, the lower the intrinsic energy available for either evaporating neutrons or overcoming the fission barrier. For the points leading to $E^* \approx$ 65 MeV, the difference in maximal angular momentum populated in $^{16}$O+$^{182}$W and $^{28}$Si+$^{170}$Er collisions leads to 4 MeV less excitation energy for the latter system. This deficit in available intrinsic energy increases with increasing bombarding energ (equivalently, $L$), approaching the neutron separation energy. The $^{198}$Pb nuclei produced in Si+Er collisions have thus less intrinsic energy than when formed the O+W combination at the same total CN excitation energy $E^*$. Those $^{198}$Pb systems formed in the Si+Er bombardment which finally succeed in overcoming the barrier are therefore those which emit less neutrons. That is at the origin of the progressive deviation of the two calculated curves in Fig.~\ref{fig4}d). Angular momentum effects thus compensate formation time effects in determining the neutron pre-scission multiplicity. This may explain the trend which, despite uncertain at some level, seems to emerge from the experimental points. Nevertheless, it is not excluded that contribution from quasi-fission \cite{berriman:2001}, difficult to discriminate from fusion-fission experimentally, and characterized by lower reaction times, can partly explain the lower multiplicity for $^{28}$Si+$^{170}$Er.
\begin{figure}[!hbt]
\hspace{-1.1cm}
\includegraphics[width=9.6cm, height=12.3cm, angle=0]{fig6.pdf}
\vspace{-.30cm}
\caption{(Color online) Correlation between the CN angular momentum $L$ and the neutron pre-scission multiplicity $M_n^{pre}$ as calculated from the stochastic Langevin CN-decay model. The first and second rows display each two reactions leading to a given CN ($^{248}$Cf and $^{206}$Po, respectively) at similar $E^*$. The last row considers a single reaction, at two different $E^*$. In each casel, the predicted mean $M_n^{pre}$ is indicated also.}
\label{fig5}
\end{figure}
The complex picture that progressively develops with decreasing fissility in the CN-decay is investigated in Fig.~\ref{fig5}. The correlation between neutron pre-scission multiplicity and CN angular momentum calculated from the stochastic Langevin model is shown for systems with decreasing fissility ($^{248}$Cf, $^{198}$Pb, $^{162}$Yb, from top to bottom). The first and second rows each display a combination of reactions producing the same CN at similar excitation energy. The last row considers a single projectile-target combination at two bombarding energies. The calculated mean pre-scission multiplicity (after folding with the $L$ distribution) are indicated also. Confrontation with experiment is not possible for the second and third rows due to missing measurements. Though these systems are considered here as per deepening our investigation of entrance-channel dependence.\\
As discussed earlier, the neutron pre-scission multiplicity predicted by the CN-decay model is seen to be not sensitive to angular momentum for the fissile $^{248}$Cf compound; this is materialized in panels a) and b) of Fig.~\ref{fig5} by the slope of the mean $M_n^{pre}$ as function of $L$ which is nearly horizontal.\\
The observation made by comparing the $^{12}$C+$^{204}$Pb and $^{28}$Ne+$^{184}$W reactions leading to $^{206}$Po resembles the result detailed for $^{198}$Pb above: $M_n^{pre}$ on average decreases with increasing $L$. At given angular momentum, $M_n^{pre}$ is the same for both entrance channels. The difference in the calculated mean multiplicities, indicated in each panel, is then due to the different CN angular momentum population in the two reactions, as obvious from Fig.\ref{fig5}, combined to the crucial influence of $L$ in this mass region. For that particular couple, it leads to a lower pre-scission multiplicity for the fission stage for $^{28}$Ne+$^{184}$W. According to HICOL, the $^{28}$Ne+$^{184}$W ($\alpha$/$\alpha_{BG} < 1$) reaction needs, however, more time than $^{12}$C+$^{204}$Pb ($\alpha$/$\alpha_{BG} > 1$) to produce a fully equilibrated the CN, and neutrons may be emitted during this phase. That would increase the measured pre-scission multiplicity for $^{28}$Ne+$^{184}$W as compared to the prediction from the CN-decay calculation. Depending on the ratio of neutrons emitted, respectively, during the CN formation and CN decay stage, fusion time and angular momentum effects can act in the same or opposite direction in influencing the pre-scission multiplicity for $^{28}$Ne+$^{184}$W. The latter would then be smaller or higher than for $^{12}$C+$^{204}$Pb, and it may be classified as, either consistent or unconsistent \cite{shareef:2016}. The present schematic method does not permit to determine quantitatively which, from the two stages, will dominantly contribute in this particular case. However, it illustrates again that the direction of various effects caused by $\alpha$/$\alpha_{BG}$, $L$, $E_{cm}$/$V_b$, and the magnitude of their interplay, can be complex to un-fold from the sole measurement of neutron pre-scission multiplicities. Nuclei at and below $A \approx$ 180-200 show particularly involved features \cite{gontchar:2004, ryabov:2008, charity:1986}. As in previous studies \cite{gontchar:2004}, we observe large fission times in this region, what can further diminish the relative importance of the fusion time. Most measured neutrons are then accounted for by the CN-decay stage.\\
The last row of Fig.~\ref{fig5} considers the reaction $^{28}$Si+$^{134}$Ba leading to $^{162}$Yb at excitation energies of $E^ * =$ 81 and 115 MeV. In this light system, the fission barrier is larger than the neutron binding energy up to high $L$, and still additional features are anticipated \cite{gontchar:2004, ryabov:2008}. Very different angular momenta are populated at the two bombarding energies, and $M_n^{pre}$ decreases very fast with $L$. In spite of the dominant contribution from the highest partial waves in fission of lowly-fissile systems and which would reduce the number of emitted neutrons similarly to the $^{206}$Po case discussed above, it is $E^*$ which finally determines the multiplicity for this even lighter compound nucleus. The reason is as follows. Due to the large fission barrier, the system has to overcome the saddle rather early along the cascade, when still a sizeable amount of $E^*$ is available. The fission time is indeed found to be reduced as compared to the $A \approx$ 180-200 region, in accordance with the results of Ref.~\cite{gontchar:2004}. It may thus be conjectured that, for such light systems, the relative contribution from the fusion stage may again pick up, and again be more straightforwardly reflected in the measured multiplicity.
\begin{figure}[!hbt]
\hspace{-0.8cm}
\includegraphics[width=9.cm, height=5.8cm,angle=0]{FIG_NETB_L.png}
\vspace{-.50cm}
\caption{(Color online) Neutron pre-scission multiplicity $M_n^{pre}$ as calculated from the stochastic CN-decay model for the $^{20}$Ne+$^{159}$Tb reaction as function of CN excitation energy. The calculation, including the full theoretical $L$ distribution is shown with (red) dots, while calculations from selected $L$ values - 45 and 70 $\hbar$, are displayed with open (blue) squares and (green) triangles, respectively. Experimental data wherever available are shown as well \cite{cabrera:2003}.}
\label{fig6}
\end{figure}
Finally, the effect of angular momentum in lowly-fissile systems is illustrated in Fig.~\ref{fig6} where the calculated pre-scission multiplicity for $^{20}$Ne+$^{159}$Tb leading to $^{179}$Re is shown as function of CN excitation energy. This system is intermediate between the above-discussed Ba and Po cases. In addition to the calculation folded with the entire $L$ distribution (which describes experiment where available \cite{cabrera:2003}), the result obtained in two narrow $L$ windows (45 and 70 $\hbar$) are displayed. Over the considered $E^*$ interval, the maximum CN angular momentum ranges from 60 to 100 $\hbar$. It is observed that, depending on the total $E^*$ excitation energy, the predicted multiplicity is given by a specific $L$ window, and which does not necessarily coincide with the highest populated CN partial waves. This is related to the complex competition between the influence of $E^*$ and $L$ and which is very sensitive to the CN fissility. That can blur the influence of the fusion stage. A proper unfolding of the different contributions is most challenging in the Pb region.
\section{Discussion}\label{discussion}
In this work, we have proposed a schematic method to address the possible critical influence of the fusion stage on the extraction of information about fission dynamics from heavy-ion collision experiments. That implies dealing with the influence of entrance-channel mass asymmetry, bombarding energy, angular momentum and excitation energy imparted to the compound nucleus. The interplay between all these paramaters can be complex, and possibily give rise to apparent inconsistencies \cite{shareef:2016} when a measurement is analyzed as function of one or the other parameter. The reason of this lies probably in the impossibility to determine a unique ordering paramater. In an attempt to"deconstruct" the multi-dimensional nature of the problem, we use a stochastic dynamical fission model, and compare its predictions in terms of neutron pre-scission multiplicity for various reactions. The relative difference in $M_n^{pre}$ between bombardments leading to the same CN, for such a calculation {\it vs.} experimental observation, is then discussed with predictions by the HICOL code for the fusion stage.\\
As emphasized above, in our exploratory investigation, among the various quantities computed by HICOL, we so far analyze only the fusion time, to explain possible discrepancy between experiment and dynamical CN-decay calculations. However, concomitant with the CN formation time, depending on the reaction, HICOL also expects some saturation of the $E^*$ imparted to the CN, which is below the value given by the incident energy and fusion $Q$ value, as well as a hindrance to fusion for the highest $L$'s. The information from HICOL can be fully \cite{wilczynska:1993} or in part \cite{kumar:2014} used to compute the subsequent decay. This is not done here. Exploiting the richness of HICOL as an input to the CN de-excitation is beyond the scope of this work, and can be considered in future. We made few tests and obtained that such an improvement of the method can affect the quantitative result for some reactions. Nevertheless, it won't affect the qualitative output of this work, and the take-home message would remain the same. The main results obtained within the proposed framework are summarized below.\\
Over the range of systems studied, measurements (Fig.~\ref{fig1}) like theory (Fig.~\ref{fig2}) show that $E^*$ plays a leading role in determining the neutron multiplicity. While experimental observations suggest that entrance-channel asymmetry plays the next key role in enhancing $M_n^{pre}$ when projecting data in a specific sub-space, the present work shows that this proposal is not the full story. Projection into other sub-spaces demonstrate namely the possible influence of angular momentum. The importance of either formation time or angular momentum at given $E^*$ critically depends on the fissility of the CN. Suming up, on top of the usually dominating influence of $E^*$, the measured $M_n^{pre}$ value is determned by a delicate balance governed by $\alpha$/$\alpha_{BG}$, $\chi$, and $L$. Depending on the magnitude of $E^*$, and on the $\chi$ region, the competition between $\alpha$/$\alpha_{BG}$ and $L$ is in favor of one of the other parameter. While the former usually dominates in fissile systems, the latter can be hidden behind not-necessarily intuitive observations for CN with masses around and below $\approx$ 200. Thanks to the employed models, the present work allows projecting the puzzle onto various parameter sub-spaces, and this way, highlighting compensation effects. The latter may explain observations classified as un-resolved so far.\\
As noted in introduction, the present study does not pretend to provide a unified model for fusion-fission dynamics. In this sense, the schematic method does not guarantee robust quantitative results. However, it provides a suited framework for a reliable qualitative study. The multi-dimensional analysis permitted us to reveal the influence of the different ordering parameters which were proposed in the past, and which all had to face limitations for specific data. The complexity of the interplay between effects un-folded to some level in this work is proposed as a major reason for apparent inconsistencies.\\
We mention that the puzzle deconstructed in this work may be visible only for thoses events which end with fission, since the population in $L$ affects the fission probability. Observables connected to other competing channels (see Refs.~\cite{kumar:2014, ruckelshausen:1986} and therein) are therefore very useful also to complete the understanding.\\
Finally, the present exploratory investigation suggests that, to un-ambigously unfold the respective role of fusion and fission dynamics, and further un-bias the extraction of nuclear properties, requires still a large set of experimental data, spanning an as wide as possible mass domain.
\section{Conclusions}\label{summary}
In order to address puzzling experimental observations on fission observables from heavy-ion collisions, we combine a dynamical stochastic Langevin model for fission of excited compound nuclei, with predictions by the dynamical deterministic HICOL code for the fusion time. A careful study about the influence of entrance-channel asymmetry, bombarding energy, compound-nucleus fissility, angular momentum and excitation energy, is performed, and confronted to experimental results whenever available. Additional theoretical calculations spanning a wider domain in compound-nucleus are done, in order to reveal the complex picture that develops due to the possibility of compensation effects depending on the size of the system. The work deepens previous analysis based on statistical models by the use of a description of the CN-decay which accounts for the dynamical features of the process and preserve the correlation between all quantities along time evolution in a natural way. Although model-dependent, and qualitative to some extent, the analysis performed in this work suggests that so-far unresolved observations can be understood within the multi-dimensional character of the process.\\
\\
\\
{\bf Acknowledgements}
\\
We thank Pr.~K. Siwek-Wilczynska and Dr. A.~Shrivastava for help in HICOL and CCFULL calculations, respectively. We are very grateful to Dr. A.~Chatterjee for discussion related to experimental and statistical-model analysis, and to Pr. Adeev for careful reading of the manuscript. The work was sponsored by the French-Polish agreements LEA COPIGAL (Project No.~18) and IN2P3-COPIN (Project No.~12-145).
|
1,941,325,220,524 | arxiv | \section{Introduction}
For both safety and regulation issues, the reliability of industrial systems has to be assessed. The considered systems (dams or nuclear power plants, for instance) are complex dynamic hybrid systems, so only simulation methods can be reasonably considered to assess their reliability.
The failure of such a dynamic hybrid system generally corresponds to a physical variable of the system (temperature, pressure, water level) entering a critical region. The simulation of such a system requires to accurately model the trajectory of the physical variables. The evolution of these physical variables are generally determined by simple differential equations derived from the laws of physics. As the physical context giving the differential equations generally depends on the statuses of the multiple components of the systems (on, off, or failed), the differential equations can change whenever there is a change in the components statuses. To encounter for this hybrid interplay between the discrete process of components' statuses and the continuous evolution of the physical variables, we model the evolution of the state of a system by a piecewise deterministic Markov process (PDMP) \cite{davis1993markov,zhang2015numerical,de2015numerical,cloez2017probabilistic}. PDMPs are meant to represent a large class of Markovian processes that do not include diffusion, as such, they benefit from high modeling capacity: they can model many industrial systems. For instance, EDF has recently developed the {PyCATSHOO} toolbox \cite{Pycatshoo2}, which allows the modeling of dynamic hybrid systems, the main option within this toolbox is to evaluate the dependability criteria of the studied systems by Monte Carlo simulations. \\
As the industrial systems are highly reliable, their failure is a rare event, and the naive Monte-Carlo simulation method (MC) is computationally too intensive in this context. The objective of our work is to set up new algorithms to accelerate the reliability assessment of such industrial systems. To do so we want to use faster methods, such as variance reduction methods.
A variance reduction method is a method that yields an unbiased estimator with a smaller variance than the Monte-Carlo estimator. The estimation being more accurate, we need less simulation runs to reach the desired accuracy, thus we save computational time. The variance reduction is generally achieved by altering the simulation process, and then correcting the bias induced by this modification of simulation process by appropriately weighting each simulation. Several acceleration methods for variance reduction can be proposed, such as importance sampling methods or particle filter methods (also called subset simulations). In this article we focus on one example of particle filter method: the interacting particle system (IPS) method \cite{del2005genealogical}.\\
Unlike in importance sampling methods, in the IPS method we keep simulating according to the original system. Another difference is that we do not simulate directly the trajectories on the entire observation duration, the observation duration is subdivided into small intervals of time and we simulate the trajectories sequentially one interval of time after the other one.
These sequential simulations of the trajectories consist alternating between an exploration step and a selection step. During the exploration step we simulate the trajectories on a small time interval, therefore exploring the most probable trajectories on a short horizon of time. Then we apply a selection step on these trajectories replicating the trajectories which seem "close" to failure and giving up the less "promising" trajectories. At the next exploration step, only replicated trajectories are continued, before the next selection, and the next exploration and so on... until each selected trajectory reaches the end of the observation time window. This way the effort of simulation is concentrated on selected trajectories, which have higher chance of becoming a failing trajectory before the end of the observation time window. In the end we get more failing trajectories to fuel our estimation, and, if the selection was well done, the IPS method yields an unbiased estimator with a smaller variance than the MC estimator. \\
When we try to apply the IPS method in order to estimate the failure probability of such reliable complex hybrid systems, the IPS methods turn out to be inefficient. Indeed the estimation provided by the IPS method often has, in this case, a higher variance than the one of the Monte-Carlo estimator. IPS does not perform well in this context because the application case (a reliable complex hybrid system) makes it hard to conduct the exploration steps of the IPS method efficiently. Indeed, an industrial system is often modeled by what we call a "concentrated PDMP", which is a PDMP with low jump rates and concentrated jump kernels on boundaries. The typical jump rate is low because it is the sum of the failure rates of the components in working condition and of the repair rates of the failed components which are all very low. These failure and repair rates are very low because the components are reliable and their repairs are slow. The typical jump kernel is concentrated because most of the probability mass of a jump is concentrated on one (safe) output. Indeed the jump kernel on boundaries model the automatic control mechanisms within the system. During such a control mechanism there is a small probability that some component(s) fail on demand but the most likely output is that the system jumps on the safe state aimed by the control mechanism, consequently the probability mass of the jump kernel is concentrated on this output.
Due to these characteristics of the model there is a high probability that no component failure or repair occurs during the short exploration time, and with a PDMP it means that all the trajectories are likely to follow the same deterministic paths. So when we explore the trajectory space most simulated trajectories end up being the same one, hence limiting our exploration of the trajectory space. To avoid this pitfall, a particular filter was proposed in \cite{whiteley2011monte} that enhances the occurrence of random jumps (failure or repairs) or modifies the occurrence time of the last jump. However the proposed method is limited to a different case of PDMP. This class of PDMP does not include concentrated PDMP, as it contains only PDMP without boundaries which allows continuous jumps kernels, i.e. it does not allow to model automatic control mechanisms in components. Moreover, an a priori bound on the number of jumps in a time interval is required, but in our case we do not have such information.
In order to adapt the IPS to concentrated PDMPS, we propose instead to use an approach based on the memorization method developed in \cite{labeau1996probabilistic}. The idea is to start the exploration by finding the most likely trajectories continuing each batch of replicated trajectories. Then, we condition the rest of the exploration to avoid these trajectories. As a result, the simulated trajectories have much more chances to differ which improves the quality of the exploration and reduce the variance of the estimator.
To correct the bias induced by this modification of the simulation process, we have to modify the weight of each trajectory.
We call our adaptation of the IPS to PDMP the IPS+M for sequential Monte-Carlo sampler with memorization method. \\% When the jump intensity of the original PDMP is low, this re-weighting yields very heterogeneous weights. Consequently, the IPS algorithm triggers a selection at each step, which tends to increase the variance of the estimator. To reduce this phenomenon we also include in the algorithm the possibility to trigger a semi-selection that corrects the heterogeneity of the weights and has less impact on the variance. \\
The rest of the paper paper is organized as follows: Section \ref{sec:model} is dedicated to the presentation of our model of the system, Section \ref{sec:IPS} presents the IPS method and introduces the optimal potential functions for this method,
Section \ref{sec:IPS+} presents the IPS+M method which adapts the IPS algorithm for PDMPs with low jump rates, Section \ref{sec:Memo} explains how to force the differentiation of the trajectories using the memorization method, and finally in Section \ref{sec:Res} we illustrate the better efficiency of the IPS+M method on a toy example.
\section{A model of the system based on a PDMP with discrete jump kernel}
\label{sec:model}
\subsection{The model}
We denote by $Z_t$ the state of the system at time $t$. $Z_t$ is the combination of the physical variables of the system, noted $X_t$, and of the statuses of all the components within the system, noted $M_t$: $Z_t=(X_t,M_t)$. We consider that $X_t\in\mathbb R^d$, and that $M_t\in\mathbb M=\{On,Off,F\}^{N_c}$ where $F$ corresponds to a failed status, and $N_c$ is the number of components in the system. The value of $M_t$ is sometimes referred as the mode of the system. Here we consider only three categories for the status of a component : $On,Off,F$, but it is possible to include more categories as long as the set of the possible modes $\mathbb M$ stays countable.
The process $Z_t$ is piecewise continuous, and each discontinuity is called a jump. Between two jumps there is no change in the components' statuses, and the dynamics of the physical variables can be expressed thanks to an ordinary differential equation derived from the law of physics: $$\frac{d\,X_t}{d t}= F_{M_t} (X_t).$$
We note $\phi_{(x,m)}(t)$ the solution of this equation when $X_0=x$ and $M_0=m$. Then for any time $s>0,\,$ if $T$ is the time separating $s$ from the next jump time, we have $$\forall t\in[0,T) ,\quad Z_{s+t} =\big( X_{s+t},M_s\big)= \big(\phi_{(X_s,M_s)} (t),M_s\big).$$ Similarly, a flow function on the states can be defined. If $ z=(x,m) $, then we define $ \Phi_{z}(t)=\big(\phi_{(x,m)} (t),m\big)$, and so \begin{equation}
\forall t\in[0,T) ,\ Z_{t+s} =\Phi_{Z_s}(t) .\label{eq:flow}
\end{equation}
As the physical variables are often continuous, the jumps are essentially used to model changes in the statuses of the components. These jumps can occur for two reasons.\\ Firstly, a jump can correspond to an automatic control mechanism (See Figure \ref{sautdet}). In a given mode $m$, such mechanism is typically triggered when the physical variables cross some threshold. For each mode $m$, we define an open and connected set $\Omega_m$, so that these thresholds determine its boundary $\partial \Omega_m$. So when $M_t=m$ the value of the physical variables $X_t$ is restricted to the set $\Omega_m\subset \mathbb R^d$. Letting $E_m=\{(x,m), x\in\Omega_m\}$ be the set of the possible states with mode $m$, in terms of state a jump associated to a control mechanism is triggered whenever the state $Z_{s+t}$ hits the boundary of $E_m$. The set of possible states is therefore defined by: $$E\,=\underset{m\in\mathbb{M}}{\bigcup} E_m.$$ \begin{figure}[t]\centering
\includegraphics[width= 0.8\linewidth]{Saut_det.png}
\caption{A jump at boundary.\label{sautdet}}
\end{figure} \\Secondly, jumps can correspond to a spontaneous failure or repair. In such case the jump occurs before $Z_{s+t}$ hits the boundary of $E_m$. \begin{figure}[h] \centering
\includegraphics[width=0.8\linewidth]{Saut_alea.png}
\caption{A spontaneous jump.\label{sautalea}}
\end{figure}The occurrence time of a spontaneous jump is modeled by a jump rate $\lambda(Z_{s+t})$. For instance, when we consider that components fail or are repaired one at a time, this jump rate is the sum of the failure rates of the components in working condition and of the repair rates of the broken components. In a more general case, each transition from a mode $m$ to a mode $m^+$ has its own rate $ \lambda_{m\to m^+}(X_{s+t})$, and we have \begin{equation}
\lambda(Z_{s+t})=\sum_{m^+\in\mathbb M}\lambda_{M_s\to m^+}(X_{s+t})
\label{eq:lambda}
\end{equation}
We define the cumulative jump rate $\Lambda_{z}(t)$ by $\Lambda_{z}(t)=\int_0^t \lambda\big(\Phi_z(u)\big)du$, so that $\forall t\in[0,T) ,\ \Lambda_{Z_s}(t) =\int_0^t \lambda\big(Z_{s+u}\big)du$. Eventually the cumulative distribution function (cdf) of $T$ (the time until the next jump starting form a state $Z_s=z$) takes the form:\begin{equation}
\mathbb P (T\leq t|Z_s=z)=\left\{\begin{array}{cr}
1-\exp\left[-\Lambda_{z}(t)\right] & \mbox{ if }t<t^{*}_z\, ,\\
1& \mbox{ if }t\geq t^{*}_z\, .\\
\end{array}\right.
\label{Tlaw}
\end{equation}
Here $t^{*}_z=\inf\{t>0,\Phi_z(t)\in \partial E_m\}$ is the time until the flow hits the boundary starting from a state $z=(x,m)$. When there is no boundary i.e. $\{s>0,\Phi_z(s)\notin E_m\}=\emptyset$, we take the convention that $t^{*}_z= + \infty\ $. With this definition, the law of $T$ has a continuous part associated to spontaneous jumps (spontaneous failures and repairs) and a discrete part associated to forced jumps (control mechanisms). A possible reference measure for $T$ knowing $Z_s=z$ is then
\begin{align}
\forall B\in \mathcal B(\mathbb R^+),\quad \mu_{z}(B) &= leb\left(B \cap( 0,t^*_z)\right)+\indic{t^*_z<\infty}\, \delta_{t^*_z}(B)\ , \label{measureMu}
\end{align}
where $leb(.)$ corresponds to the Lebesgue measure. \\
If a jump occurs at time $S$, then the distribution of the destination of the jump is expressed by a transition kernel $\mathcal K_{Z_S^{\text{-}}}$ where $Z_S^{\text{-}}$ is the departure state of the jump. If $ \xoverline{ E\,}$ is the closure of $E$, we have $Z_S^{\text{-}}\in\xoverline{ E\,}$. Let $Z_S^+\in E$ be the arrival state, and $\mathscr B(E)$ be the Borelian $\sigma$-algebra on $E$. Then, the law of a jump from a departure state $z^{\text{-}}$ is defined by:\begin{align}
\forall B\in \mathscr B (E),\qquad \mathbb P \left(Z_T^+\in B|Z_T^{\text{-}}=z^{\text{-}}\right) &= \mathcal K_{z^{\text{-}}}(B).
\end{align}
For any departure state $ z^{\text{-}}\in\xoverline{ E\,}, $ we define $x_a(z^{\text{-}},m^+)$ as the
arrival for the vector of physical variables when the transition $m^{\text{-}} \to m^+$ is triggered from a state $z^{\text{-}}$. We define by $z_a(z^{\text{-}},m^+)=\big(x_a(z^{\text{-}},m^+),m^+\big)$ the arrival state when the transition $m^{\text{-}} \to m^+$ is triggered from a state $z^{\text{-}}$. Using this definition, for any departure state $ z^{\text{-}}\in\xoverline{ E\,}, $ we define the measure $ \nu_{z^{\text{-}}}$ on $\big(E,\mathscr B(E)\big)$ by:\begin{equation}
\forall B\in \mathscr B(E),\qquad \nu_{z^{\text{-}}}(B)=\sum_{m^+\in \mathbb M}\delta_{ z_a(z^{\text{-}} ,m^+)}(B).
\end{equation}
The kernel $\mathcal K_{z^{\text{-}}}$ is absolutely continuous with respect to the measure $ \nu_{z^{\text{-}}}$. Denoting by $K_{z^{\text{-}}}$ its density with respect to $ \nu_{z^{\text{-}}}$ we have:
\begin{align}
\mathbb P \left(Z_T^+\in B|Z_T^{\text{-}}=z^{\text{-}}\right) &=\int_B K_{z^{\text{-}}}(z^+)\,d \nu_{z^{\text{-}}}(z^+) \ .\label{Kernel}
\end{align}
Note that we only consider systems for which $\nu_{z^{\text{-}}}$ is discrete, whatever the departure state $z^{\text{-}}\in E$ may be. This hypothesis of discreteness of $\nu_{z^{\text{-}}}$ is mandatory to apply the method presented in Section \ref{sec:IPS+}.
If $\nu_z$ has a Dirac contribution at point $z$, then the kernel must satisfy $K_z(z)=0$ , so it is not possible to jump on the departure state. In some applications the physical variables are continuous, so that $K_{z^{\text{-}}}(z)$ is zero whenever $ x^-\neq x $. In some cases, one might want the model to include renewable and aging components. Then the vector $X_t$ should include the time since the last renewal of such component, and so the vector $X_t$ can be discontinuous at the time of a renewal. In both situations, when $z^{\text{-}}$ is not on the boundary of $E_{m^{\text{-}} }$, the jump kernel has this form:
\begin{align}
\forall z^{\text{-}} \in E, \quad B\in \mathscr B(E),\qquad \mathcal K_{z^{\text{-}}}(B)&=\sum_{m^+\in \mathbb M}\frac{\lambda_{m^{\text{-}} \to m^+}(x^{\text{-}})}{ \lambda(z^{\text{-}})} \delta_{ z_a(z^{\text{-}} ,m^+)}(B), \label{eq:K_noboundary} \\
\mbox{and}\qquad K_{z^{\text{-}}}(z^+)&= \frac{\lambda_{m^{\text{-}} \to m^+}(x^{\text{-}})}{ \lambda(z^{\text{-}})}\indic{x^+=x_a(z^{\text{-}},m^+)} .
\end{align} When $z^{\text{-}}\in\partial E_{m^{\text{-}} }$, a control mechanism is triggered : some components are required to turn on, or to turn off, so that the system reaches a desired state $z_c$. The transition to this state is usually very likely, so when $z^{\text{-}}\in\partial E_{m^{\text{-}} }$, the jump kernel $K_{z^{\text{-}}}$ tends to concentrate all or a big portion of the probability mass on this state $z_c$. In industrial systems there can be components which have a small probability to fail when they are asked to turn on. This phenomenon is referred as a failure on demand. We denote by $\kappa_i(z^-)$ the probability that the $i^{th}$ component fails on demand when the control is triggered. We denote by $fod(i,m^{\text{-}},m)$ the indicator being equal to one if the $i^{th}$ component fails on demand during the transition from $m^{\text{-}}$ to $m$, and to zero otherwise. We also define $ask(z^{\text{-}},z^+)$ as the set gathering the indices of the components supposed to turn on during the control mechanism triggered from $z^{\text{-}}$ to $z^+$. Finally, when $z^{\text{-}}\in\partial E_{m^{\text{-}} }$, we have that:
\begin{align}
K_{z^{\text{-}}}(z^+)=
\indic{x^+=x_a(z^{\text{-}},m^+)}
\prod_{i\in ask(z^{\text{-}},z^+)}
\left(\kappa_i(z^-)\right)^{fod (i,m^{\text{-}},m^+)} \left(1-\kappa_i(z^-)\right)^{(1-fod (i,m^{\text{-}},m^+)) } \label{eq:K_boundary}.
\end{align}\\
To generate $\mathbf Z_t=(Z_s)_{s\in[0,t]}$ a trajectory of the states of the system, one can repeat the following steps: Starting with $s=0$,
\begin{enumerate}
\item Given a starting state $Z_s=z$, generate $T$ the time until the next jump using \eqref{Tlaw},
\item Follow the flow $\Phi$ until $s+T$ using \eqref{eq:flow}, and set the departure state of the next jump as being $Z_T^{\text{-}} =\Phi_z(T)$,
\item generate $Z_{s+T}$ the arrival state of the next jump using $K_{Z_T^{\text{-}}}$
\item repeat starting with $s=s+T$ until one gets a trajectory of size $t$.
\end{enumerate} Defined in this way, the process $Z_t$ is Markovian \cite{davis1984}. \\
\subsection{Law of trajectory}
Let us denote by $n(\mathbf Z_{t})$ the number of jumps in the trajectory $\mathbf Z_{t}$, and by $S_k$ the time of the $k^{th}$ jump in the trajectory $\mathbf Z_{t}$ (with the convention that $S_0=0$ and $S_{n(\mathbf Z_{t})+1}=t$). $T_k=S_{k+1}-S_k$ denotes the time between two consecutive jumps. We define the $\sigma$-algebra $\mathscr S_t$ as the $\sigma$-algebra generated by the sets in
$\underset{\ n\in \mathbb N^* }{\bigcup} \mathscr B\Big(\Big\{\big(z_{k},t_k\big)_{k\leq n}\in(E\times\mathbb R^{*}_\text{+})^{n},\, \overset{n}{\underset{i=0}{\sum}} t_i=t\Big\}\Big)$, where $\mathscr B(.)$ indicates the Borelians of a set. Letting $\Theta_{t} : \mathbf Z_{t}\to \left((Z_k,T_k)\right)_{0\leq k\leq n(\mathbf Z_{t})}$ be the application giving the skeleton of a trajectory, we can define the law of trajectories as an image law through $\Theta_{t}$. We have that for $B\in \mathscr S_t$:
\begin{align}
\mathbb P_{z_o}\Big(\mathbf{Z_t} \in \Theta_t^{-1}( B) \Big) = &\int_{B} \ \prod_{k=0}^{n} \Big(\lambda_{z_{k} }(t_k) \Big)^{\mathbbm{1}_{ t_k< t^*_{z_{k}}}}\exp\Big[-\Lambda_{z_{k}}(t_k)\Big] \prod_{k=1}^{n }K_{z_{k}^-}(z_{k})\nonumber\\
&\quad\times d\delta_{t^*_{n}}(t_n)\ d\nu_{z_{n }^-}(z_n)\ d\mu_{z_{n-1}}(t_{n-1}) \ ...\ d\nu_{z_{ 1}^-}(z_1)\ d\mu_{z_{o}}(t_{0})\ ,
\label{eq:loitraj}
\end{align}
where $z_j^-=\Phi_{z_{j-1}}(t_{j-1})$, and $t^*_{n}=t-\sum_{i=0}^{n-1}t_i$.\\
In the rest of the paper we will denote $\mathbf E_{t}$ the set of trajectories of size $t$ that satisfy \eqref{eq:flow}, and by $\mathscr M(\mathbf E_{t})$ the set of bounded $ (\Theta_t^{-1}(\mathscr S_t), \mathbf E_{t})$-measurable functions.
\subsection{A hybrid reference measure for the trajectories of PDMP}
As they involve hybrid reference measures for the times between jumps, PDMPs are very degenerate processes. As a result, some of their realizations have a strictly positive probability to be simulated, but some other do not. The trajectories that concentrate a part of the probability mass on their own, i.e. that verify $\mathbb P\big(\mathbf Z_{t}=\mathbf z_t\big)>0$, are called the preponderant trajectories. For example, assuming that $\lambda$ is positive, if the trajectory $\mathbf z_t$ involves no jump, it is preponderant because we have $\mathbb P\big(\mathbf Z_{t}=\mathbf z_t\big)=1-\exp\big(-\Lambda_{z_0}(t)\big)>0$. Also, as we consider that the reference measures $\nu_{z^{\text{-}}}$ are discrete, there can be an other type of preponderant trajectories, which are the trajectories that only jump on the boundaries $\partial E_m$. Indeed, such trajectories $\mathbf z_t$ would verify \begin{align}
\mathbb P\big(\mathbf Z_{t}=\mathbf z_t\big)= \prod_{k=0}^{n} \exp\Big[-\Lambda_{z_{s_k}}(t^*_{z_{s_k}})\Big] \prod_{k=1}^{n }K_{z_{s_k}^-}(z_{s_k})>0,
\end{align} where $s_k$ is the time of the $k^{th}$ jump in $\mathbf z_t$. Conversely, some realizations can be considered as negligible as they verify $\mathbb P\big(\mathbf Z_{t}=\mathbf z_t\big)=0$. These trajectories are the ones that involve a spontaneous jump, i.e. a jump starting from the interior of a set $E_m$. \\
This can be better understood by looking at the equation \eqref{eq:loitraj} which shows that the measure $\zeta_{z_0,t}$ defined by :
\begin{align}
\forall B\in \mathscr S_t ,\quad \zeta_{z_0,t} (\Theta^{-1}( B))= & \underset{ \mbox{\hspace{-12ex} } (z_{_k},t_{_k})_{k\leq n}
\in B}{\int\quad d\delta_{t^*_{n}}(t_n)\ d\nu_{z_{n }^-}(z_n)}\ d\mu_{z_{n-1}}(t_{n-1}) \ ...\ d\nu_{z_{ 1}^-}(z_1)\ d\mu_{z_{o} }(t_{0})
\label{eq:zeta}
\end{align}
where $z_j^-=\Phi_{z_{j-1}}(t_{j-1})$, and $t^*_{n}=t-\sum_{i=0}^{n-1}t_i$ is a reference measure for the law of trajectories. This reference measure highlights the fact that the trajectories with no jump or with only jumps on boundaries can concentrate some probability mass on their own, because they refer to a Dirac contribution of the measure $\zeta_{z_0,t}$. Indeed for such trajectories the times between two consecutive jumps verify $t_k=t_{z_{s_k}}^*$ $\forall k < n$, and so, they always refer to the discrete part of the measures $\mu_{z_{k-1}}$ $\forall k < n$, therefore such trajectories are related to the discrete part of $\zeta_{z_0,t}$. Equation \eqref{eq:zeta} also shows that the remaining probability mass is distributed continuously among the trajectories with at least one spontaneous jump. Indeed, in a negligible trajectory, if for example the $k+1^{th}$ jump is a spontaneous jump, then $t_k$ relates to the continuous part of the reference measure $\mu_{z_{k-1}}$.\\
\subsection{Concentrated PDMP}
\label{subseq:ConcentratedPDMP}
For reliability assessment of a highly reliable system one often models the system by a PDMP with low jump rates and concentrated jump kernels on the boundaries. Indeed, the components of the system are often reliable and their repair takes time, hence the low jump rates, and as failures on demand during a control mechanism are unlikely the jump kernels on boundaries are concentrated on one safe arrival state (i.e. the state aimed by the control mechanism). We call this kind of PDMP a concentrated PDMP, mainly because the law of one trajectory concentrates a big part of its probability mass. \\
This trajectory happens to be the trajectory with no failure and no repair. As jump rates are low the probability of not having a spontaneous jump is close to one. For instance at a $k+1$-th jump this probability verifies $$\mathbb P_{z_{s_{k}}} \left(T_k=t^*_{z_{s_{k}}}\right)=\exp\Big[-\Lambda_{z_{s_{k}}}(t^*_{z_{s_{k}}})\Big] \simeq 1.$$ So only jumps on boundaries are likely, and when the process hits a boundary $\partial E_m$, the arrival state aimed by the control mechanism is very likely. Denoting by $z_{s_k}$ this state for a $k$-th jump we have : $$K_{z_{s_k}^-}(z_{s_k})\simeq 1.$$ So if $\mathbf z_t$ is a trajectory with no failure and no repair of reasonable size we have:\begin{align}
\mathbb P\big(\mathbf Z_{t}=\mathbf z_t\big)= \prod_{k=0}^{n} \exp\Big[-\Lambda_{z_{s_k}}(t^*_{z_{s_k}})\Big] \prod_{k=1}^{n }K_{z_{s_k}^-}(z_{s_k}) \simeq 1.
\end{align}
In this article we adapt the IPS method for concentrated PDMPs. It is interesting to make the connection between our work and the modified particle filter developed in \cite{fearnhead2003line} which can be applied to the related context of discrete time processes with discrete spaces.
\subsection{Reliability assessment}
Let $t_f$ be an observation time. We denote by $\mathbf E_{t_f}$ the set of trajectories of size $t_f$ that verify \eqref{eq:flow}, and $h\in \mathscr M(\mathbf E_{t_f})$. The methods presented in this article can be used for the estimation of any quantity $p_h$ defined by: $$p_h=\mathbb E[h(\mathbf Z_{t_f})].$$
We are interested in estimating the probability, noted $p_{\mathscr D}$, that the system fails before the final observation time $t_f$ knowing it was initiated in a state $z_0$. Letting $\mathscr D$ be the set of trajectories of length $t_f$ which pass through the critical region $D \subset E$, we have $$ p_{\mathscr D}=\mathbb P_{z_0}\big(\mathbf Z_{t_f}\in \mathscr D\big)=\mathbb E_{z_0}\big[\indic{\mathscr D}( \mathbf Z_{t_f} )\big]. $$
Although our application relates to the case where $h=\indic{\mathscr D}$, the IPS method will be presented with an arbitrary (bounded) function $h$.
\section{The IPS method}
\label{sec:IPS}
\subsection{A Feynman-Kac model}
For a measure $\eta$ and a bounded measurable function $f$ we note $\eta(f)=\int f\,d\eta$. For a measure $\eta$ and a kernel $V$, $\eta V$ denotes the measure such that $\eta V(h)=\int \int h(y)V(dy|x)d\eta(x)$, and for a bounded measurable function $f$, $V(f)$ is the function such that $V(f)(x)=\int h(y)V(dy|x)$.
Consider a subdivision of the interval $[0,t_f]$ into $n$ sub-intervals of equal lengths, noted $[\tau_k,\tau_{k+1})$, and such that $0=\tau_0<\tau_1<\dots<\tau_{n\,\text{-}1}<\tau_n=t_f$.
Let $V_k $ be the Markovian transition measure extending a trajectory of size $\tau_k$ into a trajectory of size $\tau_{k+1} $, such that \begin{equation}
\forall \mathbf z_{\tau_k}\in \mathbf E_{\tau_{k}},\ B\in\boldsymbol{\mathcal E}_{\tau_{k+1}},\qquad \mathbb P\left(\mathbf Z_{\tau_{k+1}}\in B|\mathbf Z_{\tau_{k}}=\mathbf z_{\tau_{k}}\right) =V_k(B|\mathbf z_{\tau_k}).
\end{equation}
For each $k< n$ we denote $ G_k$ the potential function on $\mathbf E_{\tau_k}$, such that: \begin{align}\forall \mathbf z_k\in\mathbf E_k,\quad G_k(\mathbf z_k)&\geq 0 .\end{align}
These potential functions are the main inputs of the method. The choice of the potential functions is important, because it will ultimately determine the variance of the estimator of $p_h$ provided by the IPS method. A good choice depends on the system and on the target function $h$.
The potential functions are used to define the target probability measures $\tilde{\eta}_k$ for each $k\leq n$ , such that: \begin{align}
\tilde{\eta}_k(d\mathbf z_{\tau_k}) &\propto \prod_{s=0}^k G_s(\mathbf z_{\tau_{s}}) \prod_{s=0}^{k-1} V_{s }( d\mathbf z_{\tau_{s+1}}|\mathbf z_{\tau_{s}} ), \label{eq:target}
\end{align}
or equivalently
\begin{align}
\forall B \in \boldsymbol{\mathcal E_{\tau_{k}}},\qquad \tilde{\eta}_k(B) &=\frac{\mathbb E\left[ \indic{B}(\mathbf Z_{\tau_{k}})\prod_{s=0}^k G_s(\mathbf Z_{\tau_{s}})\right] }{\mathbb E\left[ \prod_{s=0}^k G_s(\mathbf Z_{\tau_{s}}) \right]}. \label{eq:target1}
\end{align}
Originally the IPS method comes from filtering methods. Filtering methods aim at estimating the target measures and, in these methods, the potential functions are chosen so that the $\eta_k$ match the target measures. But in our cases we have no interest in estimating the target measures, we only want to estimate $p_h$. So the potential functions can be chosen more freely. They are used to propose a probabilistic representation of $p_h$ in terms of a selection+mutation dynamics, which makes it possible to build an estimator of $p_h$ with a reduced variance.
We define the propagated target measures $\eta_k$ such that $\eta_0=\tilde{\eta}_0$ and for $k\geq 0$, $\eta_{k+1}=\tilde{\eta}_kV_k$. We have : \begin{align}
\eta_{k+1}( d\mathbf z_{\tau_{k+1}}) &\propto \prod_{s=0}^{k} G_s(\mathbf z_{\tau_{s}}) \prod_{s=0}^{k } V_{s }( d\mathbf z_{\tau_{s+1}}|\mathbf Z_{\tau_{s}} ) \label{eq:targetProp},
\end{align}
or equivalently
\begin{align}
\forall B \in \boldsymbol{\mathcal E_{\tau_{k+1}}},\qquad \eta_{k+1}(B) &=\frac{\mathbb E\left[ \indic{B}(\mathbf Z_{\tau_{k+1}})\prod_{s=0}^k G_s(\mathbf Z_{\tau_{s}})\right] }{\mathbb E\left[ \prod_{s=0}^k G_s(\mathbf Z_{\tau_{s}}) \right]}. \label{eq:targetProp1}
\end{align}
For $k=0$ we consider that $\eta_0=\delta_0$, but the methods would still be valid if we had $\eta_0 \neq \delta_0$.
We define $Q_{k}$ such that for $f\in\mathscr M(\mathbf E_{\tau_{k+1}})$,
$$
Q_k(f)(\mathbf Z_{\tau_k})=\int_{\mathbf E_{\tau_{k+1}}}f(\mathbf z_{\tau_{k+1}})\mathbf V_k(d\mathbf z_{\tau_{k+1}}|\mathbf Z_{\tau_{k}})G_k(\mathbf Z_{\tau_{k}})$$ and set $Q_{k,n}=Q_kQ_{k+1} \dots Q_n$. Let $\Psi_k$ be the application that transforms a measure $\eta$ defined on $\mathbf E_{\tau_k}$ into a measure $\Psi_k(\eta)$ defined on $\mathbf E_{\tau_k}$ as follows: \begin{equation}
\Psi_k(\eta)(f)=\frac{\int G_k(\mathbf z)f(\mathbf z) d\eta(\mathbf z) }{\eta(G_k)}.
\end{equation}
We say that $\Psi_k(\eta)$ gives the selection of $\eta$ through the potential $G_k$. Notice that $\tilde{\eta}_k$ is the selection of $\eta_k$ as $\tilde{\eta}_k=\Psi_k(\eta_k)$. The target distributions can therefore be built according to the following pattern, in which a propagation step follows a selection step:
$$\eta_k\overset{\Psi_k}{\xrightarrow{\mbox{\hspace{1cm}}}} \tilde{\eta}_k\overset{.V_k}{\xrightarrow{\mbox{\hspace{1cm}}}} \eta_{k+1}.$$
We also define the associated unnormalized measures $\tilde{\gamma}_k$ and $\gamma_{k+1}$, such that for $f\in\mathscr M(\mathbf E_{\tau_{k}})$:
\begin{equation}
\tilde{\gamma}_{k}(f) = \mathbb E\left[f(\mathbf Z_{\tau_{k}})\prod_{s=0}^{k} G_s(\mathbf Z_{\tau_{s}}) \right] \quad \mbox{and}\quad \tilde{\eta}_k(f) =\frac{\tilde{\gamma}_{k}(f)}{\tilde{\gamma}_{k}(1)}, \label{eq:unnorm1}
\end{equation} and for $f\in\mathscr M(\mathbf E_{\tau_{k+1}})$:
\begin{equation}
\gamma_{k+1}(f) =\mathbb E\left[f(\mathbf Z_{\tau_{k+1}})\prod_{s=0}^{k} G_s(\mathbf Z_{\tau_{s}}) \right] \quad \mbox{and} \quad \eta_{k+1}(f)=\frac{\gamma_{k+1}(f)}{\gamma_{k+1}(1)}. \label{eq:unnorm2}
\end{equation}
Denoting $f_h(\mathbf Z_{\tau_{n}})=\frac{h( \mathbf Z_{\tau_{n}} )}{\prod_{s=0}^{n-1} G_s(\mathbf Z_{\tau_{s}}) }$, notice that we have:
\begin{equation}
p_h=\gamma_n(f_h)= \eta_n(f_h) \overset{n-1}{\underset{{k=0}}{\prod}} \eta_k\big(G_k \big )\label{eq:ph}.
\end{equation}
\subsection{The IPS algorithm and its estimators}
The IPS method provides an algorithm to generate weighted samples which approximate the probability measures $\eta_{k}$ and $ \tilde{\eta}_k$ respectively for each step $k$. For the sample approximating $ \eta_{k }$, we denote $\mathbf Z_{\tau_k}^j$ the $j^{th}$ trajectory and $ W_{k}^j$ its weight. Respectively, for the sample approximating $ \tilde{\eta}_k$, we denote $\mytilde{\mathbf Z}_{\tau_k}^j$ the $j^{th}$ trajectory and $\mytilde{W}_{k}^j$ its associated weight. For simplicity reasons, in this paper, we consider that the samples all contain $ N$ trajectories, but it is possible to modify the sample size at each step, as illustrated in \cite{lee2018variance}.
The empirical approximations of $\eta_{k}$ and $ \tilde{\eta}_k$ are denoted by $\eta_{k}^N$ and $ \tilde{\eta}_k^N$ and are defined by:
\begin{equation}
\tilde{\eta}_k^{N} = \sum_{i=1}^{N} \,\mytilde W_k^i\, \delta_{ \mytilde{\mathbf Z}_{\tau_k}^i} \quad\mbox{ and }\quad
\eta_k^{N} = \sum_{i=1}^{N} \,W_k^i\, \delta_{ \mathbf Z_{\tau_k}^i}\,.\label{eq:estiEta}
\end{equation}
So for all $k\leq n$ and $f\in \mathscr M( \mathbf E_{\tau_k})$,
\begin{equation}
\tilde{\eta}_k^{N} (f) = \sum_{i=1}^{N} \,\mytilde{W}_k^i\, f \big( \mytilde{\mathbf Z}_{\tau_k}^i \big) \quad\mbox{and}\quad
\eta_k^{N} (f) = \sum_{i=1}^{N} \,W_k^i\, f \big( \mathbf {Z}_{\tau_k}^i \big)\,.\label{eq:estiEta2}
\end{equation}
By plugging these estimations into equations \eqref{eq:unnorm1} and \eqref{eq:unnorm1}, we get estimations for the unnormalized distributions. Denoting by $\tilde{\gamma}_k^{N}$ and $\gamma_k^{N}$ these estimations, for all $k\leq n$ and $f\in \mathscr M( \mathbf E_{\tau_k})$, we have:
\begin{equation}
\tilde{\gamma}_k^{ N } (f) = \tilde{\eta}_k^{N }(f) \prod_{s=0}^{k-1} \eta_s^{N }(G_s)
\quad \mbox{ and } \quad
\gamma_k^{ N } (f) =\eta_k^{N }(f) \prod_{s=0}^{k-1} \eta_s^{N}(G_s).
\label{eq:estigamma}
\end{equation}
Plugging the estimations $\eta_k^N$ into equation \eqref{eq:ph}, we get an estimator $\hat p_h$ of $p_h$ defined by:
\begin{equation}
\hat p_h = \eta_{n}^{N}(f_h) \overset{n-1}{\underset{{k=0}}{\prod}} \eta_k^{N}\big(G_k \big ) . \label{eq:hatp}
\end{equation}\\
\begin{figure}[h]\fbox{
\begin{algorithm}[H]
\textbf{Initialization :} $ k=0, \ \forall j =1..N, \ \mathbf Z_0^j\overset{i.i.d.}{\sim} \eta_0$ and $ W_0^j=\frac 1 N $, and $\mytilde W_{0}^j=\frac{ G_0( \mathbf Z_{0}^j)}{\sum_{s } G_0( \mathbf Z_{0}^s)}$ \\
\While{ $k<n$}{
\textbf{Selection:}\\
Sample $ (\tilde N_k^j)_{j=1..N }\sim Mult\big( N , (\mytilde W_{k}^j)_{j=1..N }\big)$\\
$\forall j:=1.. N ,\ \mytilde W_{k}^j:= \frac 1 { N }$ \\
\textbf{Propagation :} \\
\For{$j:=1.. N$}{ Sample $\mathbf{Z}_{\tau_{k+1}}^j$ from $ \mathbf V_{k+1}(. | \mytilde{\mathbf{Z}}_{\tau_{k}}^j) $\\
set $W_{k+1}^j=\mytilde W_k^j$ }
\For{$i:=1.. N$}{Set $\mytilde W_{k+1}^i=\frac{W_{k+1}^i G_{k+1}( \mathbf Z_{\tau_{k+1}}^i)}{\sum_{j } W_{k+1}^j G_{k+1}( \mathbf Z_{\tau_{k+1}}^j)}$ }
\eIf{$\forall j,\ \mytilde W_{k+1}^j=0$}{$\forall q>k, $ set $ \eta_q^N=\tilde \eta_q^N=0$ and Stop }{$k:=k+1$}
}\end{algorithm}}
\caption{IPS algorithm}
\label{fig:algo1}
\end{figure}
The algorithm builds the samples sequentially, alternating between a selection step and a propagation step. The $k^{th}$ selection step transforms the sample $(\mathbf Z_k^j,W_k^j)_{j\leq N}$ into the sample $(\mytilde{\mathbf Z}_k^j,\mytilde{W}_k^j)_{j\leq N}$. This transformation is done with a multinomial resampling scheme: the $\mytilde{\mathbf Z}_k^j$'s are drawn with replacement from the sample $ (\mathbf Z_k^j)_{j\leq N}$, each trajectory $ \mathbf Z_k^j$ having a probability $ \frac{W_{k}^j G_k( \mathbf Z_{\tau_{k}}^j)}{\sum_{i=1}^N W_{k}^i G_k( \mathbf Z_{\tau_{k}}^i)}$ to be drawn each time. We denote by $A^j_k$ the ancestor index of the $j^{th}$ trajectory in the selected sample, such that $\mytilde{\mathbf Z}_k^j=\mathbf Z_{\tau_{k}}^{A^j_k}$. We let $\tilde N_k^j=card\{i,A^i_k=j\}$ be the number of times the particle $ \mathbf Z_k^j$ is replicated in the sample $(\mytilde{\mathbf Z}_k^j,\mytilde{W}_k^j)_j$, so $ N =\sum_{j=1}^{N } \tilde N_k^j$. After this resampling the weights $\mytilde W_k^j$ are set to $\frac 1 N$. The interest of this selection by resampling is that it discards low potential trajectories and replicates high potential trajectories. So the selected sample focuses on trajectories that will have a greater impact on the estimations of the next distributions once extended. \\
If one specifies potential functions that are not positive, there can be a possibility that at a step $k$ we get $\forall j, G_k(\mathbf Z_{\tau_{k}}^j)=0 , $ and so the probability for resampling cannot be defined. When this is the case, the algorithm stops and we consider that $\forall s\geq k$ the measures $\tilde{\eta}_s^N$ and $ \eta_s^N$ are equal to the null measure.\\
Then the $k^{th}$ propagation step transforms the sample $(\mytilde{\mathbf Z}_k^j,\mytilde{W}_k^j)_{j\leq N}$, into the sample \linebreak $(\mathbf Z_{k+1}^j,W_{k+1}^j)_{j\leq N}$. Each trajectory $ \mathbf Z_{k+1}^j $ is obtained by extending the trajectory $ \mytilde{\mathbf Z}_k^j $ on the interval $[\tau_k,\tau_{k+1})$ using the transition kernel $V_k$. The weights satisfy $W_{k+1}^j=\mytilde{W}_k^j ,\ \forall j$. Then the procedure is iterated until the step $n$. The full algorithm to build the samples is displayed in Figure \ref{fig:algo1}.\\
For the sake of simplicity, we will make the following assumption:
$\exists\,\varepsilon_1,\varepsilon_2\in\mathbb R^+$ such that $\forall \,\mathbf Z_{\tau_k} \in \mathbf E_{\tau_k}$:\begin{equation}
\varepsilon_1>G_k( \mathbf Z_{\tau_k})>\varepsilon_2>0 \label{eq:(G)}\tag{(G)},
\end{equation}
\begin{Theorem}
When (G) is verified
the estimator \eqref{eq:hatp} is unbiased and strongly consistent. \label{consitant}
\end{Theorem}
The proof of theorem \ref{consitant} follows from Theorems 7.4.2 and 7.4.3 in \cite{del2004feynman}.
\begin{Theorem}
When (G) is verified: \begin{equation}\sqrt N (\hat p_h- p_h)\overset{d}{\underset{N\to\infty}{\longrightarrow}} \mathcal N\big(0,\sigma^2_{IPS,G}\big) ,\end{equation}
where
\begin{align}
\sigma^2_{IPS,G}&=\sum_{k=0}^{ n-1} \gamma_k(1)^2 \eta_{k} \Big(\big[ Q_{k,n}(f_h)-\eta_{k}Q_{k,n}(f_h)
\big]^2\Big) \\ &=\sum_{k=0}^{ n-1} \left\{\mathbb E_{z_0} \bigg[ \prod_{i=0}^{k-1} G_i (\mathbf Z_{\tau_i})\bigg] \mathbb E_{z_0} \bigg[ \mathbb E[h(\mathbf Z_{\tau_n})|\mathbf Z_{\tau_{k}} ]^2\, \prod_{s=0}^{k-1} G_s^{-1 }(\textbf Z_{\tau_s})\bigg]- p_h^2\right\}.
\label{eq:var}
\end{align}
\end{Theorem}
A proof of this CLT can be found in \cite{del2004feynman} chapter 9 at the theorem 9.3.1 .
For the estimation of the variance $\sigma^2_{IPS,G}$ we refer the reader to \cite{lee2018variance}.
\subsection{Classical improvements of the IPS method: the SMC method, and its alternative resampling}
We have seen that the resampling steps have the advantage of replicating high potential trajectories and discarding low potential trajectories. However the resampling steps also introduce some additional fluctuations into the estimation (see \eqref{eq:var}). So we would like to trigger them only when it is judicious. Typically, not when the potentials of all the trajectories are similar, as in this case there is not point in discarding or replicating some trajectories over others. In order to avoid pointless resampling, one can trigger the selection step only when the weights are unbalanced. This is done in the Sequential Monte Carlo (SMC) algorithm with adaptive resampling presented in Figure \ref{fig:algo2}. In this algorithm, the heterogeneity of the weights is quantified using the effective sample size. At the $k^{th}$ step the effective sample size is defined by:\begin{equation}
ESS_k=\dfrac{\left(\sum_{j=0}^{N} W_k^j \,G_k(\mathbf Z_{\tau_{k}}^j)\right)^2}{\sum_{i=0}^{N} \left(W_k^i\, G_k(\mathbf Z_{\tau_{k}}^i)\right)^2}.
\end{equation}Its value is between $1$ and $N$ and it measures the homogeneity in the candidate weights $\frac{W_{k}^i G_k( \mathbf Z_{\tau_{k}}^i)}{\sum_{j } W_{k}^j G_k( \mathbf Z_{\tau_{k}}^j)}$: when $ESS_k = N$ the weights are perfectly balanced and are all equal to $\frac 1 {N}$, and conversely when $ESS_k = 1$ all the weights are null except one, which concentrates the totality of the mass. Therefore, one considers the weights are too unbalanced when $ESS_k \leq eN$ where $e\in[0,1]$ is a tuning parameter.
\begin{figure}[h]\fbox{
\begin{algorithm}[H]
\textbf{Initialization :} $ k=0, \ \forall i :=1..N, \mathbf Z_0^i=(z_0)$ and $ W_0^i=\frac 1 N $, and $\mytilde W_{0}^i=\frac{ G_0( \mathbf Z_{0}^i)}{\sum_{j } G_0( \mathbf Z_{0}^j)}$ \\
\While{ $k<n$}{
\textbf{Selection:}\\
\eIf{$ESS_k \leq eN$}{ Sample
$(\tilde N_k^j)_{j=1..N}\sim Mult\big( N, (\mytilde W_{k}^j)_{j=1..N}\big)$ and set $\forall i=1.. n ,\ \mytilde W_{k}^i:= \frac 1 {N}$
}{
\For{$i:=1.. N$}{set $\mytilde{\mathbf Z}_{\tau_{k}}^i:= \mathbf Z_{\tau_{k}}^i$ }
}
\textbf{Propagation :} \\
\For{$i:=1.. N$}{ Sample $\mathbf{Z}_{\tau_{k+1}}^j$ from $ \mathbf V_{k+1}(. | \mytilde{\mathbf{Z}}_{\tau_{k}}^j) $\\
set $W_{k+1}^i=\mytilde W_k^i$ }
\For{$i:=1.. N$}{Set $\mytilde W_{k+1}^i=\frac{W_{k+1}^i G_{k+1}( \mathbf Z_{\tau_{k+1}}^i)}{\sum_{j } W_{k+1}^j G_{k+1}( \mathbf Z_{\tau_{k+1}}^j)}$ }
\eIf{$\forall j,\ \mytilde W_{k+1}^j=0$}{$\forall q>k, $ set $ \eta_q^N=\tilde \eta_q^N=0$ and Stop }{$k:=k+1$}
}
\end{algorithm}}
\caption{SMC algorithm with adaptive resampling steps}
\label{fig:algo2}
\end{figure}
\\
Note that in the presented algorithm one can use alternative strategies to select high potential trajectories. Here, the presented algorithms include a standard multinomial resampling procedure, but one can also use residual resampling or stratified resampling without altering the properties of the estimator. Empirical results suggest that these alternative resampling schemes yield estimations with smaller variances~\cite{douc2005comparison,hol2006resampling}. There are also recent theoretical results on the performance of different resampling schemes \cite{gerber2017negative}.
MCMC steps with invariant distribution $\tilde \eta_k$ can also be included in the algorithm after the resampling step. Some adaptations of the algorithm for parallel implementations have also been studied in \cite{verge2015modele} for instance.\\
\section{Choice of the potential functions}
\label{sec:chooseG}
Note that the variance of $\hat p_h$ depend on the number of subdivisions and on the choice of the potential functions. We display here an important hint on how to select potential functions that yield a small variance. Indeed, the theoretical expressions of the potential functions that minimize the asymptotic variance of the IPS estimator are known \cite{chraibi2018optimal}.
\begin{Theorem}
For $k\geq 1$, let $G^{*}_k $ be defined by:
\begin{align}
G^{*}_k(\mathbf z_{\tau_k}) =
\sqrt{\frac{\mathbb E\Big[ \ \mathbb E\big[ h(\mathbf Z_{\tau_n})\big|\mathbf Z_{\tau_{k+1}}\big]^2\big|\mathbf Z_{\tau_k} = \mathbf z_{\tau_k}\Big]\ \ \ }
{\mathbb E\Big[ \ \mathbb E\big[ \, h(\mathbf Z_{\tau_n}) \big|\mathbf Z_{\tau_k} \big]^2 \big|\mathbf Z_{\tau_{k-1}} = \mathbf z_{\tau_{k-1}}\Big]}}
\label{eq:optiG1}
\end{align}
if $\mathbb E\Big[ \mathbb E\big[ h(\mathbf Z_{\tau_n}) \big|\mathbf Z_{\tau_k} \big]^2 \big|\mathbf Z_{\tau_{k-1}} = \mathbf z_{\tau_{k-1}}\Big] \neq 0 $,
and $ G^{*}_k(\mathbf z_{\tau_k}) = 0 $ otherwise.
For $k=0$, we define
\begin{equation}
\ G^{*}_0(\mathbf z_{\tau_0} )= \sqrt{\, \mathbb E\Big[\, \mathbb E\big[ h(\mathbf Z_{\tau_n})\big|\mathbf Z_{\tau_1}\big]^2\big|\mathbf Z_{\tau_0} = \mathbf z_{\tau_0}\Big]}. \label{eq:optiG2}
\end{equation}
The potential functions minimizing $\sigma^2_{IPS,G}$ are the ones that are proportional to the $ G^*_k $'s $\forall k\leq n$. The optimal variance of the IPS method with $n$ steps is then
\begin{align}
&\sigma^2_{IPS,G^*} =
\mathbb E \left[ \, \mathbb E\big[ h(\mathbf Z_{\tau_n})\big|\mathbf Z_{\tau_0}\big]^2 \right]- p_h^2\nonumber\\
&+\sum_{k=1}^{ n } \left\{\mathbb E \left[\sqrt{\mathbb E\Big[ \ \mathbb E\big[ h(\mathbf Z_{\tau_n})\big|\mathbf Z_{\tau_k}\big]^2\big|\mathbf Z_{\tau_{k-1}}\Big]}\right]^2- p_h^2\right\}.
\label{eq:varOpti}
\end{align}
\end{Theorem}
For the reliability assessment case, the optimal target distributions have the form:
\begin{equation}
\tilde \eta^{*}_k(d\mathbf z_{\tau_k}) \propto \sqrt{ \mathbb E\big[\mathbb E[ \indic{\mathscr D}(\mathbf Z_{\tau_n})|\mathbf Z_{\tau_{k+1}}]^2\big|\mathbf Z_{\tau_k}=\mathbf z_{\tau_k}\big]}\, \prod_{s=1}^k V_s(d\mathbf z_{\tau_s}|\mathbf z_{\tau_{s\,\text{-}1}} ) ,
\end{equation} and the potential functions would be defined by \eqref{eq:optiG1} and \eqref{eq:optiG2}, taking $h=\indic{\mathscr D}$. As we do not have closed-form expressions of the functions $\mathbf z_{\tau_k}\to \sqrt{\mathbb E\big[\mathbb E[ \indic{\mathscr D}(\mathbf Z_{\tau_n})|\mathbf Z_{\tau_{s+1}}]^2|\big|\mathbf Z_{\tau_s}=\mathbf z_{\tau_k}\big]}$, we propose to use instead a parametric approximation of these expectations based on our knowledge of the system and denoted by $U_\alpha(\mathbf Z_{\tau_s})$ so that we take \begin{align}
\tilde \eta_k(d\mathbf Z_{\tau_k}) &\propto U_\alpha(\mathbf Z_{\tau_k})\, \prod_{s=1}^k V_s(d\mathbf Z_{\tau_s}|\mathbf Z_{\tau_{s\,\text{-}1}} )\\
\mbox{and }
\forall s>0, \qquad G_s(\mathbf Z_{\tau_s})&=\frac{U_\alpha(\mathbf Z_{\tau_s})}{U_\alpha(\mathbf Z_{\tau_{s\,\text{-}1}})}\qquad\mbox{and}\qquad G_0(\mathbf Z_{\tau_0})=U_\alpha(\mathbf Z_{\tau_0}).
\end{align}
For a system including similar components in parallel redundancy, we propose to set \linebreak $U_\alpha(\mathbf Z_{\tau_s})=1$ when $\mathbf Z_{\tau_s}$ has already reached the failure region once, and to set $U_\alpha(\mathbf Z_{\tau_s})= \exp\big[-\alpha\, (b(Z_{\tau_s})+1)^2\big]\,L(\tau_s) $ otherwise, where $L$ is a positive function, and $b(Z)$ indicates the number of working components within a state $Z$. Here $\alpha$ is a parameter tuning the strength of the selection.
\section{The IPS+M method for concentrated PDMPs}\label{sec:IPS+}
\subsection{The issue with PDMP modeling a reliable system}\label{subsec:issue}
When it is used on a reliable system and therefore on a concentrated PDMP (see Section \ref{subseq:ConcentratedPDMP}), the IPS method tends to loose in efficiency. This efficiency loss can be attributed to the exploration steps. Remember that an exploration step comes after a selection step: it builds a sample $ (\mathbf Z_{\tau_{k+1}}^j, W_{k+1}^j )_{ j\leq N}$ by extending the trajectories of a selected sample $(\mytilde{\mathbf Z}_{\tau_k}^j,\mytilde W_k^j )_{ j\leq \tilde N}$. This newly built sample $ (\mathbf Z_{\tau_{k+1}}^j, W_{k+1}^j )_{ j\leq N}$ fulfills two goals : 1) It contributes to the empirical approximation $\eta_{k+1}^{N}$ of $\eta_{k+1}$. 2) It is used as a candidate sample for the next selection. But this second goal is often poorly achieved with a concentrated PDMP. Indeed, in order to get a good approximation $\tilde \eta_{k+1}^{N}$ of $\tilde \eta_{k+1}$, it is preferable that the candidate sample to selection $(\mathbf Z_{\tau_{k+1}}^j, W_{k+1}^j )_{ j\leq N}$ contains as many different trajectories as possible, along with high potential trajectories. Unfortunately, with this kind of PDMP, it is generally not the case: the candidate sample often contains several replicates of the same trajectories, and no high potential trajectory. Therefore each distribution $\tilde \eta_k $ is poorly represented, and so is each target distribution $\eta_{k+1}$, which eventually deteriorates the quality of the estimator $\hat p_{\mathscr D}$ or $\hat p_{h}$.\\% As the exploration of the trajectory space is limited, we have to rely on big a sample size in order to explore sufficiently the trajectory space, hence the inefficiency of the method for reliability assessment.\\
To understand why the exploration steps are not likely to generate many different trajectories with a concentrated PDMP, we have to come back at the beginning of the propagation step. At that point, the sample $(\mytilde{\mathbf Z}_{\tau_k}^j,\mytilde W_k^j )_{1\leq j\leq N}$ is naturally clustered because of the previous selection step, each of the clusters containing several replicates of the same trajectories. We can rewrite \eqref{eq:estiEta} in the following way \begin{align}
\tilde{\eta}_k^N &= \sum_{i=1}^{N} \,\mytilde W_k^{i}\, \delta_{ \mytilde{\mathbf Z}_{\tau_k}^{i}} = \frac{1}{N}\sum_{j=1}^{N} \tilde N_k^j\delta_{ \mathbf Z_{\tau_k}^{j}}
\label{eq:etaClusterized}
\end{align}
where $\sum_{j=1}^{N} \tilde N_k^j=N$. In practice many of the $\tilde N_k^j$ are null and only a few are positive and the $N$ resampled trajectories $(\mytilde{\mathbf Z}_{\tau_k}^j)_{j\leq N}$ are concentrated on a few trajectories.
Then, each of the $\tilde N_k^j$ trajectories of the $j$-th cluster is extended by using the same distribution $V_k(.| \mathbf Z_{\tau_k}^{j})$. (For all index $i$ such that $A_{k}^{i}= j$ the trajectory $ \mytilde{\mathbf Z}_{\tau_k}^{i}$ is extended with the kernel $V_k(.| \mathbf Z_{\tau_k}^{j})$).
As the kernel $V_k(.| \mathbf Z_{\tau_k}^{j})$ corresponds to a concentrated PDMP, it is likely to extend all the trajectories of a cluster in the same manner. The trajectory $\mathbf a_{\tau_{k+1}}^{k,j}$ which extends $\mathbf Z_{\tau_k}^{j}$ until $ \tau_{k+1}$ without spontaneous jump or failure concentrates the mass of the kernel $V_k(.| \mathbf Z_{\tau_k}^{j})$. Indeed, at this point we have : \begin{equation}V_k(\mathbf a_{\tau_{k+1}}^{k,j}| \mathbf Z_{\tau_k}^{j})=\mathbb P\big(
\mathbf Z_{\tau_{k+1}} =\mathbf a_{\tau_{k+1}}^{k,j} \big|
{\mathbf Z}_{\tau_{k}} =\mathbf Z_{\tau_k}^{j}\big)
\simeq 1.\end{equation}
Therefore each of the trajectories $\mytilde{\mathbf Z}_{\tau_{k}}^{i}= \mathbf Z _{\tau_{k}}^{j}$ in a cluster tends to be extended in $\mathbf a_{\tau_{k+1}}^{k,j}$. Thus, the trajectories within a cluster are likely to stay clumped together during the propagation, and the propagated sample $ (\mathbf Z_{\tau_{k+1}}^j, W_{k+1}^j )_{ j\leq N}$ is very likely to be clustered too. When the preponderant trajectories $\mathbf a_{\tau_{k+1}}^{k,j}$ have low potential values, the sample is not likely to contain high potential trajectories. Consequently the selection step having no good candidates and too few candidates, it tends to yield an inaccurate estimation of the distributions $\tilde \eta_k$.\\
This situation is typical of reliability assessment. In that context, a well constructed potential function is close to $G^*_k$ wherein $h=\indic{\mathscr D}$. So the potential of a trajectory $G_{k+1}(\mathbf Z_{\tau_{k+1}})$ should be high if its final state $Z_{\tau_{k+1}}$ is more degraded than the state $Z_{\tau_k}$. This generally implies that $\mathbf Z_{\tau_{k+1}} $ includes at least one component failure between $\tau_k$ and $\tau_{k+1}$. As the preponderant trajectories $ \mathbf a_{\tau_{k +1}}^{k,j}$ do not contain failure between $\tau_k$ and $\tau_{k+1}$ they generally are associated with low potential values.
The segment of $\mathbf a_{\tau_{k+1}}^{k,j}$ on $(\tau_k,\tau_{k+1}]$ relates to a Dirac contribution of the measure $\zeta_{Z^j_{\tau_k},\tau_{k+1}-\tau_k}$. We can, therefore, decompose the expected propagation of the trajectory $\mathbf Z_{\tau_k}^{j}$ in this way: \begin{align}
\delta_{ \mathbf Z_{\tau_k}^{j}}V_k(f)
&=f\big(\mathbf a_{\tau_{k+1}}^{k,j}\big)V_k\big(\mathbf a_{\tau_{k+1}}^{k,j}|\mathbf Z_{\tau_{k}}^{j}\big)\ +\
\int_{\mathbf E_{\tau_{k+1}}\backslash\{\mathbf a_{\tau_{k+1}}^{k,j}\}} f\big( \mathbf z_{\tau_{k+1}} \big) V_k\big(d\mathbf z_{\tau_{k+1}}|\mathbf Z_{\tau_k}^{j}\big) ,
\label{eq:clusterProp}
\end{align}
where $ f \in\mathscr M (\mathbf E_{\tau_{k+1}})$. And the expected propagation of $\eta_k^N$ would be:\begin{align}
\tilde \eta_{k}^{\tilde N}V_{k}(f)&= \sum_{j=1}^{N} \frac{\tilde N_k^j}{ N} \ \delta_{\mathbf Z_k^j}V_{k}(f)\nonumber\\
&= \sum_{j=1}^{N} \frac{\tilde N_k^j}{ N} f\big(\mathbf a_{\tau_{k+1}}^{k,j}\big)V_k\big(\mathbf a_{\tau_{k+1}}^{k,j}|\mathbf Z_{\tau_{k}}^{j}\big)\ +\
\frac{\tilde N_k^j}{ N}\int_{\mathbf E_{\tau_{k+1}}\backslash\{\mathbf a_{\tau_{k+1}}^{k,j}\}} f\big( \mathbf z_{\tau_{k+1}} \big) V_k\big(d\mathbf z_{\tau_{k+1}}|\mathbf Z_{\tau_k}^{j}\big)\nonumber\\
&\simeq \sum_{j=1}^{N} \frac{\tilde N_k^j}{ N} f\big(\mathbf a_{\tau_{k+1}}^{k,j}\big). \label{eq:clusteredPropIPS}
\end{align}
\subsection{Modify the propagation of clusters}
In order to diversify the simulated trajectories, and to increase the precision of the estimation, we propose to modify the way we extend the selected trajectories. Here we consider that the size of the propagated sample can differ from the size of the previous selected sample. We now denote $\tilde N_{k}$ the size of the $k^{th}$ selected sample, and $N_{k+1}$ the size of the $k^{th}$ propagated sample, with the convention $N_0=N$. We stressed out, in section \ref{subsec:issue}, that the propagation step aims at providing an estimation of $\eta_{k+1}=\tilde \eta_{k}V_{k}$ using the selected sample $(\mytilde{\mathbf Z}_k^j ,\mytilde W_k^{j})_{j\leq \tilde N_k}$. In other words, the selection step aims at providing a propagated weighted sample $(\mathbf Z_{\tau_{k+1}}^j,W_{k+1}^{j})_{j\leq N_{k+1}}$ to estimate the distribution $\tilde \eta_{k}^{\tilde N_k}V_{k}$ defined by: \begin{align}
\tilde \eta_{k}^{\tilde N_k}V_{k}(f)&=\sum_{j=1}^{\tilde N_k} \mytilde W_k^{j}\ \delta_{ \mytilde{\mathbf Z}_k^j}V_{k}(f) =\sum_{j=1}^{N_k} \frac{\tilde N_k^j}{\tilde N_k} \ \delta_{\mathbf Z_k^j}V_{k}(f)\label{eq:clusteredProp} ,
\end{align} where $f\in\mathscr M(\mathbf E_{\tau_{k+1}})$.
We denote by $\bar V_k$ the Markovian kernel from $\mathbf E_{\tau_{k}}$ to $\mathbf E_{\tau_{k+1}}$ such that, for any trajectory $ \mathbf Z_{\tau_{k}}^{j }$, $\bar V_k(.| \mathbf Z _{\tau_{k}}^{j } )$ is the conditioning of $V_k(.| \mathbf Z _{\tau_{k}}^{j } )$ to $\mathbf E_k\backslash \{\mathbf a_{\tau_{k+1}}^{k,j}\}$. $\bar V_k$'s density verifies:
\begin{align}
\bar V_k(\mathbf z_{\tau_{k+1}}| \mathbf Z _{\tau_{k}}^{j } )
& =\frac{V_k(\mathbf z_{\tau_{k+1}}|\mathbf Z_{\tau_{k}}^{j} )}{1-V_k\big(\mathbf a_{\tau_{k+1}}^{k,j} \big| \mathbf Z _{\tau_{k}}^{j } \big)} \indic{ \mathbf z_{\tau_{k+1}} \neq \mathbf a_{\tau_{k+1}}^{k,j}}.
\end{align}
Using \eqref{eq:clusteredProp} we can decompose $\tilde \eta_{k}^{\tilde N_k}V_{k}$ as follows:
\begin{align}
\tilde \eta_{k}^{\tilde N_k}V_{k}(f)
&=\sum_{j=1}^{N_k} \frac{\tilde N_k^j}{\tilde N_k} \Bigg[V_k\big(\mathbf a_{\tau_{k+1}}^{k,j}|\mathbf Z_{\tau_{k}}^{j}\big)f\big(\mathbf a_{\tau_{k+1}}^{k,j}\big) + \left(1-V_k\big(\mathbf a_{\tau_{k+1}}^{k,j}|\mathbf Z_{\tau_{k}}^{j}\big)\right) \delta_{\mathbf Z_k^j}\bar V_{k}(f)\Bigg]. \label{eq:probaA}
\end{align}
In the original IPS algorithm, the sample approximating $\tilde \eta_{k}^{\tilde N_k}V_{k}$ is built by directly extending each trajectory in the selected sample. When we extend the replicates of a cluster, in average a proportion $V_k(\mathbf a_{\tau_{k+1}}^{k,j}| \mathbf Z_{\tau_k}^{j})$ of the replicates are extended in $\mathbf a_{\tau_{k+1}}^{k,j}$. This proportion of trajectories extended in $\mathbf a_{\tau_{k+1}}^{k,j}$ then serves as an estimation of $V_k(\mathbf a_{\tau_{k+1}}^{k,j}| \mathbf Z_{\tau_k}^{j})$. But it is not necessary to misspend all these replicates to estimate the probability of the preponderant trajectory. If we use equation \eqref{eq:probaA}, we would need to generate the trajectory $\mathbf a_{\tau_{k+1}}^{k,j}$ only once to assess its contribution to the propagation of the cluster. Also, $\mathbf a_{\tau_{k+1}}^{k,j}$ is easy to get. To generate it, it generally suffices to run the simulation process starting from the state $Z_{\tau_k}^j$ until time $\tau_{k+1}$, while setting the jumps rates and the probability of failure on demand to zero.\\
Therefore, for each cluster, we propose to use an additional replicate to generate $\mathbf a_{\tau_{k+1}}^{k,j}$ and compute exactly its contribution. So for any $j\in\{1,\dots N_k\}$, we will extend the selected trajectory $\mathbf Z_k^j$, $N_k^j$ times, where $N_k^j=\tilde N_k^j +\indic{\tilde N_k^j>0}$. We denote $j_i$ the index of the $i^{th}$ replicates of $\mathbf Z_k^j$, and consider the added replicate has index $0$ such that for $i\in \{ 0, \dots , \tilde N^j_k\}$ we have $\mytilde{\mathbf Z}^{j_i}_{\tau_k}=\mathbf Z_k^j$. The additional replicate is deterministically extended to the preponderant trajectory, so we have $ \mathbf Z_{\tau_{k+1 }}^{j_{0}}=\mathbf a_{\tau_{k+1}}^{k,j}$, and we set its weight to $W_{k+1}^{j_0}=V_k(\mathbf a_{\tau_{k+1}}^{k,j}|\mathbf Z_{\tau_{k}}^{j}) \frac{\tilde N_k^j}{\tilde N_k}$, so that it carries all the mass associated to the preponderant trajectories of a cluster.
Then we can use all the remaining $\tilde N_k^j$ trajectories in the cluster to estimate the non preponderant part of the cluster's propagation (the $1^{st}$ term in the right hand side of equation \eqref{eq:clusterProp}). For $i>0$, we condition the extensions to avoid $\mathbf a_{\tau_{k+1}}^{k,j}$ generating the $\mathbf{Z\, }_{\tau_{k+1}}^{j_i}$ according to the kernel $\bar V_k(.| \mathbf{Z\,}_{\tau_{k}}^{j})$ and set $W_{k+1}^{j_i}=\frac{1-V_k(\mathbf a_{\tau_{k+1}}^{k,j}|\mathbf Z_{\tau_{k}}^{j}) }{\tilde N_k^j}\frac{\tilde N_k^j}{\tilde N_k}$. \\
Usually, the simulations of a restricted law are carried out using a rejection algorithm, but in our case a rejection algorithm would perform poorly. The rate of rejection would be too high, as it would be equal to $V_k(\mathbf a_{\tau_{k+1}}^{k,j}|\mathbf Z_{\tau_{k}}^{j}) $ which is typically close to 1. For PDMPs, such simulations, conditioned to avoid a preponderant trajectory, can be efficiently carried out using the memorization method. This method, introduced in \cite{labeau1996probabilistic}, shares similarities with the inverse method. It therefore benefits from not using any rejection, and so it is well suited to our applications. The memorization method is presented in section \ref{sec:Memo}.
The target distributions $\tilde \eta_k $ are still estimated with $ \tilde \eta_k^{\tilde N_k}$, using equation \eqref{eq:etaClusterized}, but for $k=0 $ to $n-1$, $\eta_{k+1}$, the propagation of a target distribution, is now estimated by :
\begin{align}
\eta_{k+1}^{N_{k+1}}
&=\sum_{i=1}^{N_{k+1}}W_{k+1}^{i}\delta_{\mathbf Z_{k+1}^i} = \sum_{\,j=1,\tilde N_k^j>0}^{N_{k }\,} \sum_{i=0}^{\tilde N_k^j}W_{k+1}^{j_i}\delta_{\mathbf Z_{k+1}^{j_i}} \nonumber\\
&= \sum_{\,j=1,\tilde N_k^j>0\,}^{N_{k }}\frac{\tilde N_k^j}{\tilde N_k} \Bigg[V_k\big(\mathbf a_{\tau_{k+1}}^{k,j}|\mathbf Z_{\tau_{k}}^{j}\big)\delta_{\mathbf a_{\tau_{k+1}}^{k,j}} + \frac{\left(1-V_k\big(\mathbf a_{\tau_{k+1}}^{k,j}|\mathbf Z_{\tau_{k}}^{j}\big)\right) }{\tilde N_k^j}\sum_{i=1}^{\tilde N_k^j} \delta_{\mathbf Z_{k+1}^{j_i}} \Bigg]
\end{align}
Let $\tilde{\mathbf N}_k=(N_0,\tilde N_0,N_1,\tilde N_1,\dots,\tilde N_k) $ and $ \mathbf N_k=(N_0,\tilde N_0,N_1,\tilde N_1,\dots,N_k) $
We now note $\tilde{\gamma}_k^{\tilde{\mathbf N}_k}$ and $\gamma_k^{ \mathbf N_k}$ the estimations of the unnormalized distributions, and for all $k\leq n$ and $f\in \mathscr M( \mathbf E_{\tau_k})$, we define them by:
\begin{equation}
\tilde{\gamma}_k^{ \tilde{\mathbf N}_k } (f) = \tilde{\eta}_k^{\tilde N_k }(f) \prod_{s=0}^{k-1} \eta_s^{N_s }(G_s)
\quad \mbox{ and } \quad
\gamma_k^{ \mathbf N_k } (f) =\eta_k^{ N_k }(f) \prod_{s=0}^{k-1} \eta_s^{N_s}(G_s).
\label{eq:estigamma2}
\end{equation}
In the end, $p_h$ is estimated using the equation : \begin{equation}
\hat p_h = \eta_{n}^{N_n}(f_h) \overset{n-1}{\underset{{k=0}}{\prod}} \eta_k^{N_k}\big(G_k \big ) .
\end{equation}
The full modified version of the algorithm is presented in Figure \ref{fig:algo3}. We call this modified version of the IPS algorithm the IPS+M algorithm.\\
Throughout the rest of the paper, the notation $\mathbb E_{_M}$ will indicate that the expectation is associated to the IPS+M method and $\mathbb E$ will still denote the expectation for the original IPS method.
\begin{figure}[ht]
\fbox{\begin{algorithm}[H]
\textbf{Initialization :} $ k=0, \ \forall j =1..N, \mathbf Z_0^j=(z_0)$ and $ W_0^j=\frac 1 {N} $, and $\mytilde W_{0}^j=\frac{ G_0( \mathbf Z_{0}^j)}{\sum_{s } G_0( \mathbf Z_{0}^s)}$ \\
\While{ $k<n$}{\textbf{Selection:}\\
$\tilde N_k=N$, and sample
$ ( \tilde N_k^j)_{j=1..N_k}\sim Mult\big( N, ( \mytilde W_{k}^j)_{j=1..N_k}\big)$\\
$\forall i=1..\tilde N_k ,\ \mytilde W_{k}^i:= \frac 1 {N}$ \\
$\forall i=1..\tilde N_k,\ $ set $ N_{k}^i= \tilde N_k^i+\indic{\tilde N_k^i>0}$ \\
set $ N_{k} :=\sum_{i=1}^{N_k} N_{k}^i$
\textbf{Propagation :} \\
\For{$j:=1.. N_k$
}{\If{$N_k^j>0$}{set $ \mathbf{Z\, }_{\tau_{k+1}}^{j_0}=\mathbf a_{\tau_{k+1}}^{k,j}$
and $W_{k+1}^{j_0}= V_k\big(\mathbf a_{\tau_{k+1}}^{k,j}|\mathbf Z_{\tau_{k}}^{j}\big)\sum_{i=1}^{\tilde N_k^j}\mytilde{W}_{k }^{j_i}$\\
\For{$j=1.. \tilde N_k^j$
}{Sample $ \mathbf{Z\, }_{\tau_{k+1}}^{j_i}$ from $ \bar V_k(.| \mathbf{Z\,}_{\tau_{k}}^{j})$
and set $ W_{k+1}^{j_i}=\frac{(1-V_k\big(\mathbf a_{\tau_{k+1}}^{k,j}|\mathbf Z_{\tau_{k}}^{j}\big))}{ \tilde N_k^j }\sum_{i=1}^{\tilde N_k^j}\mytilde{W}_{k }^{j_i}$
}}
}
\For{$i:=1..N_{k}$
}{ $\mytilde W_{k+1}^i=\frac{W_{k+1}^i G_{k+1}( \mathbf Z_{\tau_{k+1}}^i)}{\sum_{j } W_{k+1}^j G_{k+1}( \mathbf Z_{\tau_{k+1}}^j)}$}
\eIf{$\forall j,\ \mytilde W_{k+1}^j=0$}{$\forall q>k, $ set $ \eta_q^{N_q}=\tilde \eta_q^{\tilde N_q}=0$ and Stop }{$k:=k+1$}}
\end{algorithm}}
\caption{IPS+M algorithm}
\label{fig:algo3}
\end{figure}
\subsection{Convergence properties of the IPS+M estimators}
In this section we show that the estimator $\hat p_h$ obtained with the IPS+M method have the same basic properties as the IPS estimator. With the IPS+M method, $\hat p_h$ converges almost surely to $p_h$, it is unbiased, and it satisfies a CLT. The proofs that we provide in this section follow the reasoning of the proofs in \cite{del2004feynman}. We present how to adjust the original proofs to take into account that the extensions of the trajectories within a cluster are no longer identically distributed. Finally we show that the asymptotic variance of the CLT is reduced with the IPS+M method.\\
\subsubsection{The martingale decomposition of the anticipated biases}
With the IPS+M method, we assumed that $\forall k, \,\tilde N_k=N$. For $p\leq 2n,$ we define $\mathcal F_p$ the filtration associated to the sequence of the $p$ first random samples built with the IPS+M algorithm: $\big(( {\mathbf Z}_{\tau_0}^j)_{j\leq N_0},$ $(\mytilde{\mathbf Z}_{\tau_0}^j)_{j\leq N}, ({\mathbf Z}_{\tau_1}^j)_{j\leq N_1}\,, \dots,\, \big)$. So when $p$ is an even number such that $p=2k$, $\mathcal F_p$ is adapted to the vector $\big(( {\mathbf Z}_{\tau_0}^j)_{j\leq N_0},$ $(\mytilde{\mathbf Z}_{\tau_0}^j)_{j\leq N},\, $ $\dots,\, ({\mathbf Z}_{\tau_k}^j)_{j\leq N_k}\big)$. For an odd number $p=2k+1$, $\mathcal F_p$ is adapted to the vector $\big(( {\mathbf Z}_{\tau_0}^j)_{j\leq N_0},(\mytilde{\mathbf Z}_{\tau_0}^j)_{j\leq N},$ $ \dots,\,({\mathbf Z}_{\tau_k}^j)_{j\leq N_k},(\mytilde{\mathbf Z}_{\tau_k}^j)_{j\leq N}\big)$. For $f\in \mathscr M(\mathbf E_{\tau_n})$ we let $\Gamma^{\mathbf N}_{p,2n} (h) $ be define by
\begin{align}
\Gamma^{\mathbf N}_{2k,2n}(f)&=\gamma_k^{\mathbf N_k}(Q_{k,n}(f))-\gamma_k (Q_{k,n}(f)) \nonumber \\
&=\gamma_k^{\mathbf N_k}(Q_{k,n}(f))- \gamma_n(f)
\end{align}
and
\begin{align}
\Gamma^{\mathbf N}_{2k+1,2n}(f)&=\tilde \gamma_k^{\tilde{\mathbf N}_k}(V_k Q_{k+1,n}(f))-\tilde\gamma_k (V_k Q_{k+1,n}(f)) \nonumber \\
&=\tilde \gamma_k^{\tilde{\mathbf N}_k}(V_k Q_{k+1,n}(f))- \gamma_n(f).
\end{align}
Using a telescopic argument we get
\begin{align}
\Gamma^{\mathbf N}_{p,2n}(f)=&\sum_{k=0}^{ \lfloor \frac{p} 2 \rfloor }
\gamma_k^{\mathbf N_k}\left(Q_{k,n}(f)\right)-\tilde \gamma_{k-1}^{\tilde{\mathbf N}_{k-1}}\left(V_{k-1}Q_{k,n}(f)\right)
\nonumber \\
+&\indic{p>0}\sum_{k=1}^{ \lfloor \frac{p +1}{2}\rfloor } \tilde \gamma_{k-1}^{\tilde{\mathbf N}_{k-1}}\left(V_{k-1}Q_{k,n}(f)\right)- \gamma_{k-1}^{\mathbf N_{k-1}}\left(Q_{k-1,n}(f)\right) ,
\label{eq:Gammatelscop}
\end{align}
with the convention for $k=0,\ \tilde \gamma_{ -1}^{\tilde{\mathbf N}_{ -1}}(V_{-1 }Q_{0,n}(f))=\gamma_n(f) $.\\
Noticing that $\gamma_k^{\mathbf N_k}(1)=\tilde \gamma_{k-1}^{\tilde{\mathbf N}_{k-1}}(1)=\gamma_{k-1}^{\mathbf N_{k-1}}(G_{k-1})$, we can rewritte \eqref{eq:Gammatelscop} as
\begin{align}
\Gamma^{\mathbf N}_{p,2n}(f)=&\sum_{k=0}^{\frac{\lfloor 2p \rfloor }{2}}\gamma_k^{\mathbf N_k}(1) \left(\eta_k^{N_k}(Q_{k,n}(f))- \tilde \eta_{k-1}^{\tilde N_{k-1}}V_{k-1}(Q_{k,n}(f))\right)\nonumber \\
+&\indic{p>0}\sum_{k=1}^{\frac{\lfloor 2p + 1\rfloor }{2} }\tilde \gamma_{k-1}^{\tilde{\mathbf N}_{k-1}}(1)\left( \tilde \eta_{k-1}^{\tilde N_{k-1}}(V_{k-1}Q_{k ,n}(f))- \Psi_{k-1}(\eta_{k-1}^{N_{k-1}})(V_{k-1}Q_{k,n}(f))\right) ,
\end{align}
where for $k=0$, we use the convention $ \gamma_0^{\mathbf N_0}(1)\tilde \eta^{\tilde N_{-1}}_{-1}(V_{-1}Q_{0,n}(f)) =\gamma_n(f)$. The benefit of this decomposition is that it distinguishes the errors associated to the propagation steps and the errors associated to the selection steps. For the propagation steps, using \eqref{eq:clusterProp} we easily get that for any $f\in\mathscr M(\mathbf E_{\tau_{k+1}})$: \begin{equation}
\mathbb E_M\left[ \eta_{k}^{N_k}(f)\ \big|\mathcal F_{2k-1} \right
=\tilde{\eta}_{k-1}^{\tilde N_{k-1}} V_{k-1}(f)
\label{eq:unbiasedProp}.
\end{equation}
For the selection steps, as the resampling schemes are the same ones as for the IPS algorithm, we still have for any $f\in\mathscr M(\mathbf E_{\tau_{k}})$: \begin{equation}
\mathbb E_M\left[ \tilde{\eta}_{k}^{\tilde N_k}(f) \big| \mathcal F_{2k}\right
= \Psi_k(\eta_k^N)(f).
\end{equation}
Thus, each selection step and propagation step is conditionally unbiased.
Note that $\gamma_k^{\mathbf N_k}(1) $ is $\mathcal F_{2k-1}$-measurable and $\gamma_{k}^{\tilde{\mathbf N}_{k}}(1)$ is $\mathcal F_{2k}$-measurable, so, when the samples are generated with the IPS+M algorithm, $(\Gamma^{\mathbf N}_{p,2n}(h))_{p\leq2n} $ is a $\mathcal F_p$-martingale. Therefore, $\hat p_h$ stays unbiased with the IPS+M method, because $$\mathbb E_M[\Gamma^{\mathbf N}_{2n,2n}(f_h)]=\mathbb E_M[\hat p_h-p_h]=0.$$
\subsubsection{Almost sure convergence}
Thanks to this martingale decomposition, we can use the same arguments as in the proof in the chapter 7 in \cite{del2004feynman}. The hypotheses of Theorems 7.4.2 and 7.4.3 \cite[page 239 and 241]{del2004feynman} are satisfied with the IPS+M method too, which yields the following theorem:
\begin{Theorem}
For any $h\in\mathscr M (\mathbf E_{\tau_n}) $, $\hat p_h$ converges almost surely to $p_h$, and, for any $f\in\mathscr M (\mathbf E_{\tau_k})$, $\eta_k^{N_k}(f)$ converges to $\eta_k(f)$ almost surely, $\gamma_k^{N_k}(f)$ converges to $\gamma_k(f)$ almost surely.
\end{Theorem}
\subsubsection{A Central Limit Theorem}
\begin{Theorem}
If the potential functions verify \ref{eq:(G)} and the samples are generated with the IPS+M algorithm ,then we have the following convergence in distribution:
$$
\sqrt{N}(\hat p_h-p_h) \underset{N\to\infty}{\longrightarrow} \mathcal N\big(0,\sigma^2_{M,G}\big),
$$
where
\begin{align}
\sigma^2_{M,G}=\ &\eta_0\left( \Big[Q_{0,n}(f_h)-\eta_0Q_{0,n}(f_h)\Big]^2\right)\nonumber\\
+&\sum_{k=1}^{n } \gamma_k(1)^2\tilde \eta_{k\text{-}1}\left( \big(1-V_{k-1}(\mathbf a_{\tau_{k}}| \mathbf Z_{\tau_{k\text{-}1}}) \big)^2 \bar V_{k\text{-}1}\Big[Q_{k,n}(f_h)- \bar V_{k\text{-}1}Q_{k,n}(f_h) \Big]^2 \right)\nonumber\\
+&\sum_{k=1}^{n }\tilde \gamma_k(1)^2\tilde \eta_{k\text{-}1}\left( \Big[V_{k\text{-}1} Q_{k ,n}(f_h)-\tilde \eta_{k\text{-}1} V_{k\text{-}1} Q_{k ,n}(f_h) \Big]^2\right) .
\end{align}
\end{Theorem}
\begin{proof}
This proof is very similar to what is done in Chapter 9 of \cite{del2004feynman}. In order to prove that $\hat p_h$ satisfies a CLT, we begin by proving that the errors associated to the selection and propagation steps are normally distributed using Lindeberg's theorem.\\
For a sequence of function $(f_k)_{k\leq 2n}$ such that $f_{2k}$ and $f_{2k+1}$ are in $\mathscr M(\mathbf E_{\tau_k})$, we define the sum of errors until the $p^{th}$ selection and propagation by:
\begin{align}
M^{\mathbf N}_{p,2n}(f)=
& \sum_{k=0}^{ \lfloor\frac{ p }{2} \rfloor } \eta_k^{N_k}(f_{2k}) - \tilde \eta_{k-1}^{\tilde N_{k-1}}V_{k-1}(f_{2k}) \nonumber \\
&+\indic{p>0} \sum_{k=1}^{ \lfloor\frac{ p+1 }{2} \rfloor } \tilde \eta_{k-1}^{\tilde N_{k-1}}( f_{2k-1})- \Psi_{k-1}(\eta_{k-1}^{N_{k-1}})(f_{2k-1}) .
\end{align}
For $j \in \{1,\dots N\}$ we let
\begin{equation}
U_{(2k+1)N+j}^{N}(f ) =\frac 1 {\sqrt {N}} \left(f_{2k+1}(\mytilde{\mathbf Z}_k^j) - \Psi_{k}(\eta_k^{N_{k}})(f_{2k+1}) \right).
\end{equation}
For $k\geq0$, $j \in \{1,\dots , N_{k}\}$ and $i \in \{0,\dots , N^j\}$, we consider that the indices $j_i$ are ordered in such way that $j_0> N$ and $j_i< N$ when $i>0$. With such indexing $\forall s \in \{1,\dots, N\}$, $\exists j \in \{1,\dots, N_{k}\}$ and $i \in \{1,\dots, N^j\}$ such that $s=j_i$, and for such $s$ we let
\begin{equation}
U_{2kN+s}^{N}(f ) =\frac{1-V_k\big(\mathbf a_{\tau_{k+1}}^{k,j}|\mathbf Z_{\tau_{k}}^{j}\big)} {\sqrt {N}} \left(f_{2(k+1)}(\mathbf Z_{k+1}^{j_i}) - \bar V_{k}(f_{2(k+1)})(\mathbf Z_k^{j}) \right).
\end{equation}
For $j \in \{1,\dots, N_{0}\}$, let
\begin{equation}
U_{ j}^{N}(f ) =\frac{1} {\sqrt {N}} \left(f_{0}(\mathbf Z_{0}^{j}) - \eta_0(f_{0}) \right).
\end{equation}
Thus,
\begin{equation}
\sqrt N M^{\mathbf N}_{p,2n}(f)=\sum_{k=0}^{(p+1)N}U_{k}^{N}(f).
\end{equation}
Noting $\mathcal P_k^N$ a filtration adapted to the $k$ first trajectories generated in the IPS+M algorithm. Note that we have that $
\mathbb E\left[U_{k}^{N}(f) |\mathcal P_{k-1}^N\right] =0$, and $ \mathbb E\left[U_{k}^{N}(f)^2 |\mathcal P_{k-1}^N\right]<\infty$,
and $ |U_{k}^{N}(f)|<\frac{2}{\sqrt{N}}\underset{k\leq n, \mathbf Z_{\tau_k}\in \mathbf E_{\tau_k} }{\sup }\{|f_{2k}(\mathbf Z_{\tau_{k}})| \wedge|f_{2k+1}(\mathbf Z_{\tau_{k}}) | \}
$, so the Lindeberg condition is clearly satisfied. Then, we have that \begin{align}
\langle \sqrt{N} M^{\mathbf N}_{p,2n}(f)\rangle_p &=\sum_{k=0}^{(p+1)N}\mathbb E\left[U_{k}^{N}(f)^2|\mathcal P_{k-1}^N\right]\nonumber\\
& =\eta_0^N\left( \Big[f_{0}-\eta_0^N(f_{0})\Big]^2\right)\nonumber\\
&\quad +\sum_{k=1}^{\frac{\lfloor 2p \rfloor}{2} } \tilde \eta_{k\text{-}1}^N\left( \big(1-V_{k-1}(\mathbf a_{\tau_{k}}| \mathbf Z_{\tau_{k\text{-}1}}) \big)^2 \bar V_{k\text{-}1}\Big[f_{2k}- \bar V_{k\text{-}1}f_{2k} \Big]^2\Big)\right)\nonumber\\
&\quad +\sum_{k=1}^{\frac{\lfloor 2p +1\rfloor}{2}} \tilde \eta_{k\text{-}1}^N\left( \Big[ f_{2k-1}(\mathbf Z_{\tau_{k\text{-}1}})-\Psi_{k\text{-}1}(\eta_{k\text{-}1}^{N_{k\text{-}1}}) f_{2k-1} \Big]^2\right) . \nonumber\\
\end{align}
As $\eta_{k }^{N_{k }}$ and $\tilde \eta_{k }^{N }$ converge almost surely to $\eta_{k } $ and $\tilde \eta_{k } $, $ \langle \sqrt{N}M^{\mathbf N}_{p,2n}(f)\rangle_n $ converge in probability to
\begin{align}
\sigma_p^2(f)
& =\eta_0\left( \Big[f_0-\eta_0 (f_0)\Big]^2\right)\nonumber\\
&\quad +\sum_{k=1}^{\frac{\lfloor p \rfloor}{2} } \tilde \eta_{k\text{-}1}\left( \big(1-V_{k-1}(\mathbf a_{\tau_{k}}| \mathbf Z_{\tau_{k\text{-}1}}) \big)^2 \bar V_{k\text{-}1}\Big[f_{2k}- \bar V_{k\text{-}1}f_{2k } \Big]^2\Big)\right)\nonumber\\
&\quad +\sum_{k=1}^{\lfloor \frac{ p +1}{2}\rfloor} \tilde \eta_{k\text{-}1}\left( \Big[ f_{2k-1}(\mathbf Z_{\tau_{k\text{-}1}})-\tilde\eta_{k\text{-}1} f_{2k-1} \Big]^2\right) . \nonumber\\
\end{align}
By application of the Lindeberg's theorem for triangular array (see for instance Theorem 4 on page 543 in \cite{shiryaev1996probability}), we get that $\sqrt{N}M^{\mathbf N}_{p,2n}(f)$ converges in law to a centered Gaussian of variance $\sigma_p^2(f)$. As a corollary, if for $ p\neq2k$ we take $f_p=0 $ and for $p=2k$ $f_{2k}=Q_{k,n}(f_h) $, we get that \begin{align*} \sqrt N \bigg( \eta_k^{N_k} Q_{k,n}(f_h) &- \tilde \eta_{k-1}^{N}V_{k-1}Q_{k,n}(f_h)\bigg) \\
&\underset{N\to\infty}{\longrightarrow} \tilde \eta_{k\text{-}1}\bigg( \big(1-V_{k-1}(\mathbf a_{\tau_{k}}| \mathbf Z_{\tau_{k\text{-}1}}) \big)^2 \bar V_{k\text{-}1}\Big[Q_{k,n}(f_h) -\ \bar V_{k\text{-}1}Q_{k,n}(f_h)\Big]^2\Big)\bigg)
\end{align*}
and if for $p\neq 2k-1$ we take $f_p=0$ and for $p=2k-1$ $f_{2k-1}=V_{k-1}Q_{k,n}(f_h) $, we get that
\begin{align*} \sqrt N \bigg(\tilde \eta_{k-1}^{N}( V_{k-1}Q_{k,n}(f_h))&- \Psi_{k-1}(\eta_{k-1}^{N_{k-1}})(V_{k-1}Q_{k,n}(f_h))\bigg)\\
&\underset{N\to\infty}{\longrightarrow}\tilde \eta_{k\text{-}1}\bigg( \Big[ V_{k-1}Q_{k,n}(f_h)-\tilde \eta_{k\text{-}1} V_{k-1}Q_{k,n}(f_h) \Big]^2\bigg) .
\end{align*}
Following from Theorem 4, $\gamma_k^{N_k}(1)^2 \ $ and $\tilde \gamma_k^{N}(1)^2 $ converges almost surely to $\gamma_k (1)^2 $ and $\tilde \gamma_k(1)^2$ , by an application of Slutsky's Lemma, we get that $\sqrt N \Gamma^{\mathbf N}_{2N,2n}(f_h )$ converges in law to a centered Gaussian with variance
\begin{align}
\sigma_{M,G}^2
& =\gamma_0(1)^2\eta_0\left( \Big[Q_{0,n}(f_h)-\eta_0Q_{0,n}(f_h)\Big]^2\right)\nonumber\\
&\quad +\sum_{k=1}^{n } \gamma_k(1)^2 \tilde \eta_{k\text{-}1}\left( \big(1-V_{k-1}(\mathbf a_{\tau_{k}}| \mathbf Z_{\tau_{k\text{-}1}}) \big)^2 \bar V_{k\text{-}1}\Big[Q_{k,n}(f_h)- \bar V_{k\text{-}1}Q_{k,n}(f_h) \Big]^2\Big)\right)\nonumber\\
&\quad +\sum_{k=1}^{n} \tilde \gamma_{k\text{-}1}(1)^2\tilde \eta_{k\text{-}1}\left( \Big[ V_{k\text{-}1} Q_{k ,n}(f_h)-\tilde \eta_{k\text{-}1} V_{k\text{-}1} Q_{k ,n}(f_h) \Big]^2\right) .
\end{align}
\end{proof}
\subsubsection{Variance reduction}
\begin{Theorem}
The variance of the original IPS can be decomposed as follows:
\begin{align}
\sigma^2_{IPS,G}&= \sigma^2_{M,G} + \sum_{k=1}^{n } \gamma_k(1)^2\tilde \eta_{k\text{-}1}\left(v_k(\mathbf Z_{\tau_{k\text{-}1}}) \bar V_{k\text{-}1}\Big(\Big[ Q_{k,n}(f_h)(\mathbf a_{\tau_{k}})- Q_{k,n}(f_h)(\mathbf Z_{\tau_k }) \Big]^2\Big)\right) ,\label{eq:stroumph3}
\end{align}
where $v_k(\mathbf Z_{\tau_{k\text{-}1}})=V_{k\text{-}1}(\mathbf a_{\tau_{k}}| \mathbf Z_{\tau_{k\text{-}1}})\big(1-V_{k\text{-}1}(\mathbf a_{\tau_{k}}| \mathbf Z_{\tau_{k\text{-}1}}) \big) $.
Therefore we have $\sigma_{M,G}^2\leq \sigma_{IPS,G}^2 $.
\end{Theorem}
\begin{proof}
\begin{align}
\sigma^2_{IPS,G}&=\sum_{k=0}^{ n} \gamma_k(1)^2 \eta_{k} \Big(\big[ Q_{k,n}(f_h)-\eta_{k}Q_{k,n}(f_h)
\big]^2\Big) \\
&=\eta_0\left( \Big[Q_{0,n}((f_h)-\eta_0Q_{0,n}((f_h)\Big]^2\right)\nonumber\\
&+\sum_{k=1}^{n } \gamma_k(1)^2 \tilde \eta_{k-1}V_{k-1}\Big(\big[ Q_{k,n}(f_h)-V_{k-1}Q_{k ,n}(f_h)+V_{k-1}Q_{k ,n}(f_h)-\eta_{k}Q_{k,n}(f_h)
\big]^2\Big)\nonumber \\
\textbf{}&=\eta_0\left( \Big[Q_{0,n}((f_h)-\eta_0Q_{0,n}((f_h)\Big]^2\right)\nonumber\\
&+\sum_{k=1}^{n } \gamma_k(1)^2 \tilde \eta_{k-1}V_{k-1}\left( \Big[Q_{k,n}(f_h)- V_{k-1}Q_{k ,n}(f_h) \Big]^2 \right)\nonumber\\
&+\sum_{k=1}^{n }\tilde \gamma_{k-1}(1)^2\tilde \eta_{k-1}\left( \Big[V_{k-1} Q_{k ,n}(f_h)-\tilde \eta_{k-1} V_{k-1} Q_{k,n}(f_h) \Big]^2\right) \label{eq:stroumph1}
\end{align}
Temporarily using the notation $V_{k\text{-}1}(\mathbf a_{\tau_{k}}| \mathbf Z_{\tau_{k\text{-}1}}) =p_k $, for any $f\in\mathscr M(\mathbf E_{\tau_k})$, we get
\begin{align}
V_{k-1}&\left( \Big[f(\mathbf Z_{\tau_k})- V_{k-1}f \Big]^2\right)\nonumber\\
&= V_{k-1}\left( \Big[f(\mathbf Z_{\tau_k})-p_k f(\mathbf a_{\tau_k})- (1-p_k)\bar V_{k-1}(f) \Big]^2\right)\nonumber\\
&= V_{k-1}\left( f(\mathbf Z_{\tau_k})^2
-2p_k f(\mathbf Z_{\tau_k})f(\mathbf a_{\tau_k})
-2 (1-p_k) f(\mathbf Z_{\tau_k}) \bar V_{k-1}f \right.\nonumber\\
&\qquad\quad \left. + \,p_k^2 f(\mathbf a_{\tau_k})^2
+2p_k f(\mathbf a_{\tau_k})\bar V_{k-1}f
+(1-p_k)^2\big(\bar V_{k-1}f\big)^2\right)\nonumber\\
&= p_k f(\mathbf a_{\tau_k})^2
+(1-p_k)\bar V_{k-1}(f^2)\nonumber\\
&\quad -2p_k^2 f(\mathbf a_{\tau_k})^2
-2p_k(1-p_k) f(\mathbf a_{\tau_k}) \bar V_{k-1}(f)\nonumber\\
&\quad -2p_k (1-p_k)f(\mathbf a_{\tau_k}) \bar V_{k-1}f -2 (1-p_k)f^2 \bar V_{k-1}(f)^2 \nonumber\\
& \quad + \,p_k^2 f(\mathbf a_{\tau_k})^2
+2p_k f(\mathbf a_{\tau_k})\bar V_{k-1}f
+(1-p_k)^2 \bar V_{k-1}(f)^2 \nonumber\\
&= p_k (1-p_k)\big[ f(\mathbf a_{\tau_k})^2
-2f(\mathbf a_{\tau_k})\bar V_{k-1}f+\bar V_{k-1}(f^2)\big]\nonumber\\
&\quad+(1-p_k)^2\big(\bar V_{k-1}(f^2)-\bar V_{k-1}(f)^2\big)\nonumber\\
&= p_k (1-p_k)\bar V_{k-1}\Big(\big[f(\mathbf a_{\tau_k}) -
f(\mathbf Z_{\tau_k})\big]^2\Big) +(1-p_k)^2\Big(\bar V_{k-1}(f^2)-\bar V_{k-1}(f)^2\Big).\end{align}
In particular, for $f=Q_{k,n}(f_h)$ we get
\begin{align}
V_{k-1} &\left( \Big[Q_{k,n}(f_h)- V_{k-1}Q_{k,n}(f_h)\Big]^2\right)\nonumber\\
&=p_k (1-p_k)\bar V_{k-1}\Big(\big[Q_k,n(f_h)(\mathbf a_{\tau_k}) -
Q_k,n(f_h)\big]^2\Big) +(1-p_k)^2\Big(\bar V_{k-1}(f^2)-\bar V_{k-1}(f)^2\Big) \label{eq:stroumph2}
\end{align}
Plugging \eqref{eq:stroumph2} into the second line of \eqref{eq:stroumph1}, yields \eqref{eq:stroumph3}.
\end{proof}
\section{Efficient extension of the trajectories using the Memorization method}\label{sec:Memo}
This section presents the memorization method that was first introduced in \cite{labeau1996a}. Remember that we considered that a trajectory $\mathbf a_t$ is preponderant whenever $p_{\mathbf a_t}=\mathbb P(\mathbf Z_t=\mathbf a_t)>0$. Assuming we know such a preponderant trajectory $\mathbf a_t$, the memorization method allows to generate a trajectory $\mathbf Z_t$ which differs from this preponderant trajectory $\mathbf a_t$.
\subsection{Advantage of Memorization over a rejection algorithm}
The interest of the method, compared to a rejection algorithm, is that we generate a trajectory $\mathbf Z_t\neq \mathbf a_t$ in one shot, whereas a rejection algorithm may generate several times the preponderant trajectory $\mathbf a_t$ before generating a trajectory different from $\mathbf a_t$. This is especially interesting when the probability $p_{\mathbf a_t}=\mathbb P(\mathbf Z_t=\mathbf a_t)$ is close to 1, as, with a rejection algorithm, the average number of tries to get a trajectory different from $\mathbf a_t$ would be $\frac{1}{1 - p_{\mathbf a_t}}$ which is then very high. Therefore with a rejection algorithm much computational effort would be wasted generating $\mathbf a_t$ over and over.
\subsection{Remarks on improving the IPS+M}
Note that the IPS+M, greatly unbalances the weights of the propagated samples. Consequently it is useless to consider an algorithm which triggers a resampling step, when the value of on the effective sample size is below a threshold. Indeed, the weights are so unbalanced that the effective sample size would always be very low and would trigger a resampling at each step.
\subsection{Principle of the memorization method}
\subsubsection{Work with the differentiation time}
The key idea of the memorization method is to consider the stopping time $\tau$ defined such that:\begin{equation}\forall s<\tau ,\quad Z_s= a_s \quad \mbox{and}\quad Z_{\tau}\neq a_{\tau}.\end{equation} This time $\tau$ is the time at which the trajectory $\mathbf Z_{t}$ differentiates itself from $\mathbf a_{t}$. So, to generate $\mathbf Z_{t}$ knowing $\tau\leq t$ is equivalent to generate $\mathbf Z_{t}$ knowing it differs from $\mathbf a_t$. In order to simulate a trajectory $\mathbf Z_{t}$ avoiding $\mathbf a_{t}$, one can follow these three steps:
\begin{enumerate}
\item generate $\tau$ knowing $ \tau\leq t$, and set $\mathbf Z_{\tau^{-}}=\mathbf a_{\tau^{-} }$,
\item generate $Z_{\tau}$ knowing $Z_{\tau}\neq a_{\tau}$,
\item generate the rest of the trajectory normally until $t$.
\end{enumerate}
These steps are not difficult to realize, except for the first one.
\subsubsection{Generate $\tau$ knowing $ \tau\leq t$}
To achieve this first step, the authors in \cite{labeau1996a} propose to generate $\tau$ knowing $ \tau\leq t$ by using a method equivalent to the inverse transform sampling method. We present hereafter the theoretical foundation for this method.
We denote by $F$ the cdf of $\tau$ knowing $ \tau\leq t$:
\begin{equation}F(v)=\mathbb P\big(\tau<v | \tau \leq t\big) ,\end{equation}
and we denote by $F^{-1}$ its generalized inverse defined by
\begin{equation}
F^{-1}(x)=\underset{v>0}{\inf}\{v\mid F(v)\geq x \}.
\end{equation}
We also denote by $\tilde F$ the function defined by
\begin{equation}
\tilde F (v)=\mathbb P (\mathbf Z_{v^-}=\mathbf a_{v^-})=\prod_{k=0}^{n(\mathbf a_v)}\exp\Big[-\Lambda_{a_{s_k}}(t_k)\Big] \prod_{k=1}^{n(\mathbf a_v) }\left(K_{a_{s_k}^-}(a_{s_k})\right)^{\indic{t_k>0}}
\end{equation}
where $\Theta_v(\mathbf a_v)=\big((a_{s_k},t_k)\big)_{0\leq k\leq n(\mathbf a_v)}$. Note that $\tilde F$ is discontinuous in each jump times $s_k$ where $K_{z_k^-}(z_k)\neq 1$, so the inverse of $\tilde F$ is not necessarily defined everywhere on $[p_{\mathbf a_t},1]$. For this reason we consider $\tilde F^{-1}$, the generalized inverse of $\tilde F$ defined by\begin{align}
\tilde F^{-1}(x)=\underset{v>0}{\sup}\{v\mid \tilde F(v)\leq x \}.
\end{align} $\tilde F^{-1}$ extends the inverse of $\tilde F$ constantly where it is not defined, this extension being done from the left so that $\tilde F^{-1}$ is right continuous.
The inverse transform sampling method consists in generating $U \sim Unif(0,1)$ and taking $F^{-1}(U)$ as a realization of $\tau\,|\, \tau\leq t$ which is a truncated random variable. The simulation of such random variables is also presented in \cite{devroye1986sample}. Note that the expression of the cdf $F$ can be related to $\tilde F$ , indeed we have:
\begin{equation}
\forall v< t,\quad F(v)=\frac{\mathbb P(\tau<v)}{\mathbb P(\tau \leq t)}=\frac{1-\mathbb P(\tau \geq v)}{1 - \mathbb P(\tau > t)}=\frac{1-\mathbb P (\mathbf Z_{v^-}=\mathbf a_{v^-})}{1- \mathbb P (\mathbf Z_t=\mathbf a_t)} = \frac{1-\tilde F(v)}{1- p_{\mathbf a_t}} .\nonumber
\end{equation}
Consequently we have that
\begin{align}
F^{-1}(U)&=\underset{v>0}{\inf}\{v\mid F(v)\geq U \}\nonumber\\
&=\underset{v>0}{\sup}\big\{v\mid \tilde F(v)\leq 1-U\big(1-p_{\mathbf a_t})\big)\big\},\nonumber\\
&= \tilde F \Big( 1-U\big(1-p_{\mathbf a_t}\big)\Big).
\end{align} Also, as $U$ has uniform distribution on $[0,1]$, $\tilde U=1 -U\big(1-p_{\mathbf a_t})$ is a uniform on $[p_{\mathbf a_t},1]$. Therefore,
sampling with the inverse transform method is equivalent to simulating $\tilde U\sim Unif(p_{\mathbf a_t},1)$ and taking $\tilde F^{-1}(\tilde U )$ as a realization of $\tau\,|\, \tau\leq t$.
Assuming we first generate the trajectory $\mathbf a_{t}$ and generate $\tilde U $ according to a uniform distribution on $(p_{\mathbf a_t},1)$, we now show how to evaluate $\tilde F^{-1}(\tilde U)$. We consider that during the generation of $\mathbf a_{t}$, we computed and memorized $\mathbb P (\mathbf Z_{s_k^-}=\mathbf a_{s_k^-})$ and $\mathbb P (\mathbf Z_{s_k}=\mathbf a_{s_k})$ for each jump in the trajectory, and also for $\mathbb P (\mathbf Z_{t}=\mathbf a_{t})$. Then we distinguish two cases: either there exists $k\leq n(\mathbf a_t)$ such that $\mathbb P (\mathbf Z_{s_{k}^-}=\mathbf a_{s_{k}^-})\geq \tilde U> \mathbb P (\mathbf Z_{s_{k}}=\mathbf a_{s_{k}})<$, either there exists $k\leq n(\mathbf a_t) $ such that $\mathbb P (\mathbf Z_{s_{k} }=\mathbf a_{s_{k} })\geq \tilde U> \mathbb P (\mathbf Z_{s_{k}^1}=\mathbf a_{s_{k+1}^-}) $ where we take the convention that $s_{n(\mathbf a_t)+1}^-=t$. The first case is quite simple as by definition of $\tilde F^{-1}$ we get $\tilde F^{-1}(\tilde U)=s_k$. In the second case, $\tilde F $ being continuous and strictly decreasing on $[s_k,s_{k+1})$, it is inversible on this interval, and $\tilde F^{-1} $ corresponds to $F$'s inverse on $(\tilde F(s_{k+1}^-),\tilde F(s_{k})]$. So $\tilde F^{-1}(\tilde U)\in[s_k,s_{k+1})$ and $\tilde F(\tilde F^{-1}(\tilde U)) =\tilde U$. Notice that
\begin{equation}
\forall v \in[s_k,s_{k+1}),\quad \tilde F(v)=\tilde F(s_{k})\times \exp\left[-\Lambda_{a_{s_k}}(v-s_k)\right].
\end{equation}
So in particular, for $v=\tilde F^{-1}(\tilde U)$, we have :
\begin{equation}
\tilde U=\tilde F(\tilde F^{-1}(\tilde U))=\tilde F(s_{k})\times \exp\left[-\Lambda_{a_{s_k}}(\tilde F^{-1}(\tilde U)-s_k)\right],
\end{equation}
or equivalently
\begin{equation}
\log \left( \frac{\tilde F(s_{k})} {\tilde U}\right) =\int_0^{\tilde F^{-1}(\tilde U)-s_k}\lambda_{a_{s_k}}(u)du.
\end{equation}
To determine $\tilde F^{-1}(\tilde U)$ we look for the value $s$ such that the integral $\int_0^{s}\lambda_{a_{s_k}}(u)du$ is equal to $\log \left( \dfrac{\tilde F(s_{k})} {\tilde U}\right) $ by dichotomy, then we set $\tilde F^{-1}(\tilde U)=s_k+s$.\\
To sum up the generation of a realization of $\tau|\tau\leq t$ we proceed as follow: \begin{enumerate}
\item Generate $\tilde U \sim Unif(p_{\mathbf a_t},1)$, and set $k=0$
\item If $\mathbb P (\mathbf Z_{s_{k}}=\mathbf a_{s_{k}})\geq \tilde U> \mathbb P (\mathbf Z_{s_{k+1}^-}=\mathbf a_{s_{k+1}^-})$, then we find $s\in[0,s_{k+1}-s_k)$ such that $$\log \left( \frac{\tilde F(s_{k})} {\tilde U}\right) =\int_0^{s}\lambda_{a_{s_k}}(u)du ,
$$ and we set $\tau=s_k+s$.
\item If $ \mathbb P (\mathbf Z_{s_{k+1}^-}=\mathbf a_{s_{k+1}^-})\geq \tilde U>\mathbb P (\mathbf Z_{s_{k+1}}=\mathbf a_{s_{k+1}})$, then $\tau=s_{k+1} $
\item If the condition above were not satisfied, set $k=k+1$, if $k\leq n(\mathbf a_s)$ repeat the steps 2 to 4
\end{enumerate}
\subsection{The memorization in the IPS+M algorithm}
In the IPS+M algorithm, assuming we apply the memorization method on an interval $(\tau_{k\,\text{-}1},\tau_k]$ and for the $i^{th}$ cluster, we apply it knowing that $\mathbf Z_{\tau_{k\,\text{-}1}}=\mathbf a^{i}_{k\,\text{-}1} $. Note that this trajectory $\mathbf a^{i}_{k\,\text{-}1}$ is not necessarily preponderant but when we extend it into $\mathbf a^{i}_{k}$, the piece of trajectory $\mathbf a^{i}_{(\tau_{k\,\text{-}1},\tau_k]})$ is preponderant because :
\begin{equation}
\mathbb P(\mathbf Z_{(\tau_{k\,\text{-}1},\tau_k]}=\mathbf a^{i}_{(\tau_{k\,\text{-}1},\tau_k]}\big| \mathbf Z_{\tau_{k\,\text{-}1}}=\mathbf a^{i}_{k\,\text{-}1})=\mathbb P(\mathbf Z_{\tau_k}=\mathbf a^{i}_{\tau_k} \big| \mathbf Z_{\tau_{k\,\text{-}1}}=\mathbf a^{i}_{k\,\text{-}1})>0 .
\end{equation}
So, in the IPS we try to generate trajectories of the cluster that verify $\mathbf Z_{\tau_{k\,\text{-}1}}=\mathbf a^{i}_{k\,\text{-}1}$, but avoid the piece of trajectory $\mathbf a^{i}_{(\tau_{k\,\text{-}1},\tau_k]}$.
\section{Numerical illustrations}
\label{sec:Res}
\renewcommand{\arraystretch}{1.27}
In order to confirm our results numerically, we have applied the IPS method and the IPS+M method to two two-components system.
\subsection{The heated-room system}
The first system is a room heated by two heaters in passive redundancy. Heaters are programmed to maintain the temperature of the room above negative values, turning on when the temperature drops below some positive threshold and turning off when the temperature crosses a high threshold. The second heater can activate only when the first one is failed. The system fails when the temperature falls below zero.
$X_t$ represents the temperature of the room at time $t$. $M_t$ represents the status of the heaters at time $t$.
Heaters can be on, off, or out-of-order, so $\mathbb M=\{ON,OFF,F\}^{2}$. The state of the system is $Z_t=(X_t,M_t)$.
The differential equation rule the temperature can be derived from the physics. $x_e$ is the exterior temperature. $\beta_1$ is the rate of the heat transition with the exterior. $\beta_2$ is the heating power of each heater. The differential equation giving the evolution of the the temperature of the room has the following form: $$\frac{d\,X_t}{dt}=\beta_1 (x_e-X_t)+\beta_2 \mathbbm 1_{M^1_t\, or\, M^2_t=ON}\ .$$
The heaters are programmed to maintain the temperature within an interval\linebreak$(x_{min} , x_{max})$ where $x_e<0<x_{min}$. We consider that the two heaters are in passive redundancy in the sense that: when $X\leq x_{min}$ the second heater activates only if the first one is failed. When a repair of a heater occurs, if $X\leq x_{min}$ and the other heater is failed, then the heater status is set to $ON$, else the heater status is set to $OFF$.
To handle the programming of the heaters, we set \scalebox{0.9}{$\Omega_{m}=(-\infty,x_{max})$} when all the heaters are failed $m=(F,F)$ or when at least one is activated, otherwise we set \scalebox{0.9}{$\Omega_{m}=(x_{min},x_{max})$}. \\
Due to the continuity of the temperature, the reference measure for the Kernel is $\forall B\in\mathscr B (E),$ $\nu_{(x,m)}(B)=\sum_{m^+\in\mathbb M\backslash\{m\}}\delta_{(x,m^+)}(B)$.
On the top boundary in $x_{max}$, heaters turn off with probability 1. On the bottom boundary in $x_{min}$, when a heater is supposed to turn on, there is a probability $\gamma=0.01$ that the heater will fail on demand. So, for instance, if $z^-=\big(x_{min}, (OFF,OFF)\big)$, we have $K_{z^-}\big(x_{min}, (ON,OFF)\big)=1-\gamma$, \linebreak and $K_{z^-}\big(x_{min}, (F,ON)\big)=\gamma(1-\gamma)$, and $K_{z^-}\big(x_{min}, (F,F)\big)=\gamma^2$.\\
Let $j$ be a transition from $m$ to $m^+$. For the spontaneous jumps that happen outside boundaries, if the transition $j$ corresponds to the failure of a heater, then: \linebreak$\lambda^j(x,m)=0.0021+0.00015\times x $
and, if the transition corresponds to a repair, then $\lambda^j(x,m)=0.2\quad\mbox{when}\ M^j=F$. Here the system failure occurs when the temperature of the room falls below zero, so $D=\{(x,m)\in E, x<0\}$. A possible trajectory of the state of this system is plotted in figure \ref{fig:schemetraj}.
The probability of failure $p$ was estimated to $2.71\times 10^{-5}$ thanks to a massive Monte-Carlo of $10^7$ simulations.
\begin{figure}[h]\centering
\includegraphics[width= 0.7\linewidth]{traj2comp.png} \caption{A possible trajectory of the heated-room system\label{fig:schemetraj}}{\footnotesize(the mode is represented with colors)}
\end{figure}
\subsection{Results of the simulations for the heated-room system }
The results of the simulation study for the heated-room system are displayed in table \ref{table1}. Here we have used the potential functions proposed in section \ref{sec:chooseG}. The value of $\alpha $ was set to 1.1. We have tried different values of $\alpha $ between $0.5$ and $1.5$ with a step $0.1$. The value of $\alpha=1.1$ was chosen among our trials as the one yielding the best variance reduction for the IPS method. For the IPS, IPS+M, and MC methods the variances are estimated empirically: we run the methods 100 times and we take the empirical variances of the 100 estimates.
The results highlight that the IPS method is ill-suited to PDMPs, as it yields a higher variance than the MC method. Conversely, our IPS+M method performs well and has overcome the issue of the PDMP. Indeed, in the case $n=10$, it reduces the variance by a factor $2.7$ compared to the MC method, and by a factor $10$ compared to the IPS method.
The IPS+M is about 4 times slower than the IPS method, so, in terms of computational cost, the method is only $2.5$ more efficient than the IPS method on this test case. For a run of $N=10^5$ the IPS+M is about $2.7$ time slower than Monte-carlo method. So in terms of computational cost the IPS+M is slightly more efficient than the Monte-Carlo Method.
\begin{table}[h]
\centering
\begin{tabular}{p{0.3cm}cc|c|c|c|p{0.6cm}@{}}
\cline{4-6}
& & & MC & IPS & IPS+M &\\
\cline{2-6}
& \multicolumn{1}{|c|}{\multirow{2}{*}{ }} & $\hat{p }$ & $\, 2.71\times10^{-5}$ & & &\\
&\multicolumn{1}{|c|}{} & $\,\hat \sigma^2$ & $ 2.90\times10^{-10}$ & & &\\
\cline{2-6}
&\multicolumn{1}{|c|}{\multirow{2}{*}{$n=5$}} & $\hat{p }$ & & $\, 2.86\times10^{-5}$ & $2.70\times10^{-5}$ &\\
&\multicolumn{1}{|c|}{} & $\,\hat \sigma^2$ & & $\, 1.78\times10^{-9}$ & $1.37\times10^{-10}$ &\\
\cline{2-6}
&\multicolumn{1}{|c|}{\multirow{2}{*}{$n=10$}} & $\hat{p }$ & & $\, 2.85\times10^{-5}$ & $ 2.64\times10^{-5}$&\\
&\multicolumn{1}{|c|}{} & $\,\hat \sigma^2$ & & $\, 1.08\times10^{-9}$ & $ 1.07\times10^{-10}$ &\\
\cline{2-6}
\end{tabular}
\caption{ Empirical means and empirical variances on $100$ runs with $N=10^5$ for the MC, the IPS and the IPS+M methods\label{table1}}
\end{table}
\subsection{Remark on the SMC with Memorization}
We have seen that it is possible to improve the IPS method to make it similar to the SMC method. We may, therefore, think that the IPS+M algorithm could be improved by adding adaptive optional re-sampling steps in order to get a SMC+M algorithm. In practice, however, it is not beneficial to add these adaptive optional re-sampling steps. Indeed we have noticed that, as we greatly modify the propagation process, the weights are greatly imbalanced and the effective-sample-size ends up being extremely small, which would trigger the re-sampling each time. Therefore adding adaptive optional re-sampling to the IPS+M has no effect, and in practice the IPS+M methods and the SMC+M methods are the same.
\subsection{A dam system}
The second system models a dam subjected to an incoming water flow. The physical variable of interest is the water level in the dam denoted by $X_t$. The failure of the system occurs when the water level exceeds a security threshold $x_{lim}= 10$ before time $t_f=50$. The initial level is set to $X_0=0$. The water flow is characterized by the input debit $Q=10$. The dam has two evacuation valves with output debit $Q$. Each valve can be either open, close or stuck closed. So $\mathbb M =\{Open, Closed, Stuck closed\}^2$. The valves are programmed in passive redundancy, so if the valves are in functioning order there is always one valve open and one valve closed. Though, the valve can get stuck closed and this happens at random times with exponential distribution with intensity $\lambda= 0.001$. The valves are repaired with a repair rate $\mu=0.1$. When both valves are stuck closed the reservoir of the dam starts filling up according to the equation $\frac{dX_t}{dt}=Q/S $, where $S=10$ is the surface of the reservoir.
\subsection{Results of the simulations for the dam system }
The results of the simulation study for the dam system are displayed in table \ref{table2}. Here we have used the potential functions:
\begin{equation}
\forall k<n,\quad G(\mathbf Z_{\tau_k})= \exp\left[\alpha_1(x_{lim}-X_{\tau_k})+\alpha_2(b(Z_{\tau_k})+1)^2\right].
\end{equation}
The value of $\alpha_1 $ was set to $-0.9$ and the value of $\alpha_2$ was set to $-1$ (these are a priori guesses, we have not tried to use any optimization). For the IPS and the IPS+M methods the variances are estimated empirically: we run the methods 50 times and we take the empirical variances of the 50 estimates. The results are presented in table \ref{table2}.
\begin{table}[ht]
\centering
\begin{tabular}{p{0.3cm}cc|c|c|c|p{0.6cm}@{}}
\cline{4-6}
& & & MC & IPS & IPS+M &\\
\cline{2-6}
& \multicolumn{1}{|c|}{\multirow{2}{*}{ }} & $\hat{p }$ & $\, 1.12 \times10^{-4}$ & & &\\
&\multicolumn{1}{|c|}{} & $\,\hat \sigma^2$ & $ 1.12\times 10^{-9}$ & & &\\
\cline{2-6}
&\multicolumn{1}{|c|}{\multirow{2}{*}{$n=5$}} & $\hat{p }$ & & $\, 1.75 \times10^{-4}$ & $ 1.12 \times10^{-4}$ &\\
&\multicolumn{1}{|c|}{} & $\,\hat \sigma^2$ & & $\, 3.08\times10^{-8}$ & $4.37\times10^{-9}$ &\\
\cline{2-6}
\end{tabular}
\caption{ Empirical means and empirical variances on $50$ runs with $N=10^5$ for the MC, the IPS and the IPS+M methods\label{table2}}
\end{table}
The results highlight that the IPS method is again ill-suited to PDMPs, as it yields a variance 30 times larger than the MC method. Our IPS+M method performs better than the IPS method as the variance is reduced by a factor 7. Yet on this example the IPS+M method has not overcome the issue of the PDMP, as its variance is 3.4 times larger than the variance of the Monte-carlo estimator. In terms of computational cost, on this example the IPS+M method was 3.6 times slower than the IPS, and 11.8 times slower then than the Monte-Carlo method. So the efficiency of the IPS+M is about 40 lower than the Monte-Carlo method. Clearly, the implementation of the IPS+M method requires a careful choice of the form of the potential functions and of their parameters.
\section{Conclusion}
This paper investigates the application of the IPS method to PDMPs.
As the IPS method does not perform well when it is used on a concentrated PDMP, we introduce and analyze the IPS+M method, that is a modified version of the IPS that performs better with concentrated PDMP. The IPS+M method is similar to the IPS but has different propagation steps. Its propagation steps focus on clusters of identical particles rather then on particles individually. For each cluster a memorization method is used to get an empirical approximation of the distribution of the propagated cluster, which allows to greatly improve the accuracy of the method. We have shown that the proposed algorithm yields a strongly consistent estimation, and that this estimation satisfies a TCL. We prove that the asymptotic variance of the IPS+M estimator is always smaller than the asymptotic variance of the IPS estimator. Simulations also confirm these results, showing that the IPS+M can yield a variance reduction when the IPS cannot. In terms of computational cost, our implementations of the IPS+M method give approximately the same efficiency as the Monte-Carlo method in the examples considered in this paper, where the goal is to estimate a probability of the order of $10^{-5}$ for rather simple toy models. The numerical implementations certainly deserve more careful attention. We also believe that there are ways to improve the efficiency of the IPS+M method by finding a better class of potential functions. Another interesting improvement to the IPS+M method would be to propose an estimator of the variance.
We believe that it should be possible to adapt one of the estimators proposed in \cite{lee2018variance} for the IPS method in order to get an estimator of the variance for the IPS+M estimator.
\newpage
\bibliographystyle{plain}
|
1,941,325,220,525 | arxiv | \section{Preliminaries}
\subsection{Introduction}
Twisted generalized Weyl (TGW) algebras were introduced by Mazorchuk and Turowska in 1999 \cite{MazTur1999}.
They are constructed from a commutative unital ring $\K$, an associative unital $\K$-algebra $R$, a collection of elements $t=\{t_i\}_{i\in I}$ from the center of $R$, a collection of commuting $\K$-automorphisms $\si=\{\si_i\}_{i\in I}$ of $R$, and a scalar matrix $\mu=(\mu_{ij})_{i,j\in I}$, by adjoining to $R$ new non-commuting generators $X_i$ and $Y_i$ for $i\in I$, imposing some relations and taking the quotient by a certain radical $\mathcal{I}$ (see Section \ref{sec:background} for the full definition). These algebras are denoted by $\mathcal{A}_\mu(R,\si,t)$.
They are naturally graded by the free abelian group on the index set $I$ and come with a canonical map $R\to\mathcal{A}_\mu(R,\si,t)$ making them $R$-rings. In addition, if $\mu$ is symmetric, they can be equipped with an involution.
The structure and representation theory of TGW algebras have been investigated in several papers.
For example, families of simple weight modules were classified in \cite{MazTur1999},\cite{MazPonTur2003},\cite{Hartwig2006}, Whittaker modules classified in \cite{FutHar2012a},
bounded and unbounded $\ast$-representations studied in \cite{MazTur2002},
generalized Serre relations were found in \cite{Hartwig2010}, and
conditions for a TGW algebra to be a simple ring were given in \cite{HarOin2013}.
Examples of TGW algebras include multiparameter quantized Weyl algebras \cite{MazTur2002}, \cite{Hartwig2006}, \cite{FutHar2012a}, $U(\mathfrak{sl}_2)$, $U_q(\mathfrak{sl_2})$, $Q_{ij}$-CCR (Canonical Commutation Relation) algebras \cite{MazTur2002}, quantized Heisenberg algebras, extended OGZ algebras $\mathcal{U}(r-1,r,r+1)$ \cite{MazPonTur2003}, the Mickelsson-Zhelobenko algebra associated to the pair $(\mathfrak{gl}_n,\mathfrak{gl}_{n-1})$ \cite{MazPonTur2003}, an example related to $\mathfrak{gl}_3$ \cite{Sergeev2001}, and examples attached to any symmetric generalized Cartan matrix \cite{Hartwig2010}.
In addition, any higher rank generalized Weyl algebra (GWA) \cite{Bavula1992} in which $X_i$ and $Y_i$ are not zero-divisors is also an example of a TGW algebra. The GWAs are obtained precisely when the following additional condition holds:
\begin{equation}\label{eq:GWA-consistency}
\si_i(t_j)=t_j,\qquad i\neq j.
\end{equation}
In a GWA one has $X_iX_j=X_jX_i$ and $Y_iY_j=Y_jY_i$ for all $i,j$. These relations need not hold in a general TGW algebra, where instead they are replaced by higher degree Serre-type relations \cite{Hartwig2010}.
A question of particular importance is whether a given input datum $(R,\si,t)$ actually gives rise to a non-trivial TGW algebra. Indeed, it can happen that the relations are contradictory so that $\mathcal{A}_\mu(R,\si,t)=\{0\}$, the algebra with one element \cite[Ex.~2.8]{FutHar2012b}.
This does not occur for higher rank GWAs, as the conditions \eqref{eq:GWA-consistency} implies the algebra is consistent. Thus it becomes important to find a substitute for \eqref{eq:GWA-consistency} which ensures that a given TGW algebra is non-trivial. This question was solved in \cite{FutHar2012b}, in the case when the $t_i$ are not zero-divisors in $R$. The answer is the following.
\begin{thm*}[\cite{FutHar2012b}] Assume that the elements $t_i$ are not zero-divisors in $R$. Then the following set of equations is sufficient for a TGW algebra $\mathcal{A}_{\mu}(R,\si,t)$ to be non-trivial:
\begin{subequations}\label{eq:intro-tgw-consistency}
\begin{align}
\label{eq:intro-tgw-consistency1}
\si_i\si_j(t_it_j) &=\mu_{ij}\mu_{ji}\si_i(t_i)\si_j(t_j),\qquad i\neq j \\
\label{eq:intro-tgw-consistency2}
\si_i\si_k(t_j)t_j &=\si_i(t_j)\si_k(t_j),\qquad i\neq j\neq k\neq i
\end{align}
\end{subequations}
Moreover, if one requires the canonical map $R\to\mathcal{A}_\mu(R,\si,t)$ to be injective, then equations \eqref{eq:intro-tgw-consistency} are necessary.
\end{thm*}
Clearly, any solution to \eqref{eq:GWA-consistency} is also a solution to \eqref{eq:intro-tgw-consistency}, if we take $\mu_{ij}=1$ for all $i,j$. The first equation \eqref{eq:intro-tgw-consistency1} was known already in \cite{MazTur1999},\cite{MazTur2002}. The second equation \eqref{eq:intro-tgw-consistency2} was found in \cite{FutHar2012b} and is independent of the first one.
Hence, constructing examples of TGW algebras is equivalent to finding solutions $(R,\si,t)$ to the consistency equations \eqref{eq:intro-tgw-consistency}. This task is much more difficult than solving the GWA equation \eqref{eq:GWA-consistency}.
In the present paper we construct a new family of solutions to \eqref{eq:intro-tgw-consistency}, parametrized by multiquivers. A multiquiver is a quiver (directed graph) where we allow multiple edges between two vertices.
The motivation for this construction came from several directions. Already in \cite{MazTur1999}, the authors essentially attached a graph to a TGW algebra such that vertices $i$ and $j$ are connected if and only if $\si_i(t_j)\neq t_j$. In a sense, therefore, the complexity of this graph measures the extent to which the TGW algebra differs from a GWA. In \cite{Hartwig2006} this idea was further quantified by attaching multiplicities $(a_{ij},a_{ji})$ to each edge according to the degree of the minimal polynomial of $\si_i$ acting on $t_j$. It was also shown in \cite{Hartwig2006} that this results in a generalized Cartan matrix (GCM), that corresponding Serre-type relations hold, and that any symmetric GCM can occur. No examples of TGW algebras with non-symmetric GCM were known at the time. In Section \ref{sec:theGCM} we show that in fact any GCM can occur.
Another inspiration was the paper by A. Sergeev \cite{Sergeev2001}, in which he shows that certain infinite-dimensional primitive quotients of $U(\mathfrak{gl}_3)$ are TGW algebras. The corresponding simple $\mathfrak{gl}_3$-modules are completely pointed (multiplicity-free), and can be realized as eigenspaces of the Euler operator in simple weight modules over the Weyl algebra. The latter is a special case of the classification of such modules over $\mathfrak{sl}_n$ and $\mathfrak{sp}_{2n}$ given in \cite{BenBriLem1997}. Thus it is natural to ask for a generalization of Sergeev's construction to higher rank simple Lie algebras of types $A$ and $C$. We obtain this in Sections \ref{sec:special-linear-lie-algebras} and \ref{sec:symplectic-lie-algebras}.
We now describe the contents and main results of this article in more detail.
We assume throughout that $\K$ is a field of characteristic zero.
In Section \ref{sec:background} we collect some basic facts about TGW algebras.
A new result about quotients of TGW algebras that we will need is proved in Section \ref{sec:epimorphisms}.
The definition of the new TGW algebras $\mathcal{A}(\Gamma)$ attached to multiquivers $\Gamma$ is given in Section \ref{sec:AGamma}, along with some examples in Section \ref{sec:examples}. The relation to a previous family of TGW algebras studied in \cite{Hartwig2006} is given in Section \ref{sec:symmetric}.
In Section \ref{sec:diffops} we study a canonical representation $\varphi_\Gamma$ of $\mathcal{A}(\Gamma)$ by differential operators. We determine the rank of the kernel of the incidence matrix of $\Gamma$ as the number of connected components of $\Gamma$ in \emph{equilibrium} (Theorem \ref{thm:rank-ker-gamma}), give a description of the centralizer of $R$ in $\mathcal{A}(\Gamma)$ (Lemma \ref{lem:centralizer}) and prove that this centralizer is a maximal commutative subalgebra (Corollary \ref{cor:centralizer}). This is applied to obtain the first main result of the paper, which gives precise conditions under which $\varphi_\Gamma$ is faithful.
{
\renewcommand{\thethm}{\ref{thm:faithfulness}}
\begin{thm}
Let $\Gamma$ be a multiquiver, $\gamma$ be its incidence matrix, $\mathcal{A}(\Gamma)$ the corresponding TGW algebra, and $\varphi_\Gamma$ be the canonical representation by differential operators.
Then the following statements are equivalent:
\begin{enumerate}[{\rm (i)}]
\item $\varphi_\Gamma$ is faithful;
\item $\gamma$ is injective;
\item $R_E$ is a maximal commutative subalgebra of $\mathcal{A}(\Gamma)$;
\item No connected component of $\Gamma$ is in equilibrium.
\end{enumerate}
\end{thm}
\addtocounter{thm}{-1}
}
We are also interested in the question of how surjective $\varphi_\Gamma$ is. We say that it is \emph{locally surjective} if it maps each homogeneous component of $\mathcal{A}(\Gamma)$ onto a homogeneous component of the Weyl algebra. The second main theorem of the paper gives a precise condition for $\varphi_\Gamma$ to be locally surjective.
{
\renewcommand{\thethm}{\ref{thm:local-surjectivity}}
\begin{thm}
Let $\Gamma$ be a multiquiver and $\mathcal{A}(\Gamma)$ be the corresponding TGW algebra. Let $\varphi_\Gamma$ be the canonical representation by differential operators.
Let $\bar\Gamma$ denote the underlying undirected graph of $\Gamma$.
Then the following two statements are equivalent:
\begin{enumerate}[{\rm (i)}]
\item $\varphi_\Gamma$ is locally surjective;
\item $\bar\Gamma$ is acyclic.
\end{enumerate}
\end{thm}
\addtocounter{thm}{-1}
}
The third main result gives substance to the statement that $\varphi_\Gamma$ is ``canonical''. In fact, we prove that the pair $(\mathcal{A}(\Gamma), \varphi_\Gamma)$ is universal in the following sense:
{
\renewcommand{\thethm}{\ref{thm:main}}
\begin{thm}
Let $\mathcal{A}=\TGWA{\mu}{R}{\si}{t}$ be any TGW algebra with index set denoted $V$, such that $R$ is a polynomial algebra $R=R_E=\K[u_e\mid e\in E]$ (for some index set $E$),
and $\mu$ is symmetric. Assume that
\[
\varphi:\mathcal{A}\to A_E(\K)
\]
is a map of $R_E$-rings with involution, where $A_E(\K)$ is the Weyl algebra over $\K$ with index set $E$.
Then $\mathcal{A}$ is consistent and
there exists a multiquiver $\Gamma=(V,E,s,t)$ with vertex set $V$ and edge set $E$ and a map
\[\xi:\mathcal{A}\to\mathcal{A}(\Gamma)\]
of $\Z V$-graded $R_E$-rings with involution
such that the following diagram commutes:
\[
\begin{aligned}
\begin{tikzcd}[ampersand replacement=\&,
column sep=small]
\mathcal{A}
\arrow{rr}{\varphi}
\arrow{d}[swap]{\xi}
\& \& A_E(\K) \\
\mathcal{A}(\Gamma)
\arrow{rru}[swap]{\varphi_{\Gamma}}
\& \&
\end{tikzcd}
\end{aligned}
\]
Moreover, if $\varphi(X_i)\neq 0$ for each $i\in V$, then $\Gamma$ is uniquely determined and $\mu_{ij}=1$ for all $i,j\in V$.
\end{thm}
\addtocounter{thm}{-1}
}
In the final part of the paper we address the relation between TGW algebras and universal enveloping algebras of simple Lie algebras. Section \ref{sec:theGCM} describes a simple algorithm which associates a Dynkin diagram $D(\Gamma)$ to any multiquiver $\Gamma$ in such a way that the GCM of $\mathcal{A}(\Gamma)$ is exactly the one corresponding to $D(\Gamma)$ (Theorem \ref{thm:commutative-diagram}). From this it easily follows that for any (not just symmetric) GCM $C$, there exists a TGW algebra $\mathcal{A}(\Gamma)$ whose associated GCM is $C$. Another consequence is the existence of graded homomorphisms from the enveloping algebra of the positive and negative parts of the Kac-Moody algebra associated to $D(\Gamma)$ into $\mathcal{A}(\Gamma)$ (Theorem \ref{thm:serre-relations}).
In Theorem \ref{thm:primitivity} we show that $\mathcal{A}(\Gamma)$ is a primitive ring, provided $\varphi_\Gamma$ is faithful. This shows that perhaps it is more natural to expect that some primitive quotients of $U(\mathfrak{g})$, rather than $U(\mathfrak{g})$ itself, can be realized as TGW algebras.
Indeed, in Sections \ref{sec:special-linear-lie-algebras} and \ref{sec:symplectic-lie-algebras} we establish that for
$\mathfrak{sl}_{n+1}$ and $\mathfrak{g}=\mathfrak{sp}_{2n}$, any primitive quotient $U(\mathfrak{g})/J$, where $J$ is the annihilator of an infinite-dimensional simple completely pointed $\mathfrak{g}$-module, is graded isomorphic to TGW algebras of the form $\mathcal{A}(\widetilde{A_n})/\langle \mathbb{E}-\la\rangle$ where $\mathbb{E}$ is central and $\la\in\K$, respectively $\mathcal{A}(\widetilde{C_n})$, and $\widetilde{C_n}$ and $\widetilde{A_n}$ are certain multiquivers whose Dynkin diagrams are $A_n$ and $C_n$ respectively. Our results clarify and generalize the type $A_2$ case considered in \cite{Sergeev2001} and give a direct relation between the representation $\varphi_\Gamma$ and the classification of simple completely pointed $\mathfrak{g}$-modules in \cite{BenBriLem1997} using Weyl algebras.
This leads naturally to the following question: When is a primitive quotient $U(\mathfrak{g})/J$ for a simple Lie algebra $\mathfrak{g}$ graded isomorphic to a TGW algebra? To get a complete answer we need to slightly generalize the notion of TGW algebra to allow $\si_i\si_j\neq\si_j\si_i$. This is our final main result:
{
\renewcommand{\thethm}{\ref{thm:primquotient}}
\begin{thm}
Assume that $\K$ is an algebraically closed field of characteristic zero.
Let $\mathfrak g$ be a finite-dimensional simple Lie algebra over $\K$ with
Serre generators $e_i,f_i$, $i=1,\dots,n$ and $J$ be a primitive ideal
of $U(\mathfrak g)$. The following conditions are equivalent:
\begin{enumerate}[{\rm (a)}]
\item There exists a not necessarily abelian TGW algebra $\TGWA{\mu}{R}{\si}{t}$ and a surjective homomorphism
\[\psi: U(\mathfrak g)\to\TGWA{\mu}{R}{\si}{t}\]
with kernel $J$ such that $\psi(e_i)=X_i$, $\psi(f_i)=Y_i$;
\item There exists a simple completely pointed (multiplicity free)
weight $\mathfrak g$-module $M$ such that
\[\operatorname{Ann}_{U(\mathfrak g)} M=J.\]
\end{enumerate}
\end{thm}
\addtocounter{thm}{-1}
}
Lastly, in Section \ref{sec:affine}, we prove that some primitive quotients of enveloping algebras
of affine Lie algebras can also be realized as TGW algebras.
\subsection*{Acknowledgements}
Part of this work was completed during the first author's visit at Stanford University.
The first author is grateful to Daniel Bump, Vyacheslav Futorny, Johan \"Oinert,
Joanna Meinel and Apoorva Khare for interesting discussions related to this paper
and to John Baez for Remark \ref{rem:markov} about Kolmogorov's criterion for detailed balance.
Some results from this paper were reported at the School of Algebra in Salvador, Brazil in 2012.
The second author was supported by NSF grant DMS - 1303301.
\subsection{Notation}
Unless otherwise stated, $\K$ will denote a field of characteristic zero.
Rings and algebras are understood to be associative and unital. Subrings, subalgebras and homomorphisms of rings and algebras are unital.
The set of integers $x$ with $a\le x\le b$ is denoted $\iv{a}{b}$.
\subsection{Background on TGW algebras}
\label{sec:background}
We recall the definition of TGW algebras \cite{MazTur1999,MazTur2002} and some useful properties.
\subsubsection{TGW data}
\begin{dfn}[TGW datum]
Let $I$ be a set.
A \emph{twisted generalized Weyl datum over $\K$ with index set $I$} is a triple $(R,\si,t)$ where
\begin{itemize}
\item $R$ is a unital associative $\K$-algebra, called the \emph{base algebra},
\item $\si=(\si_i)_{i\in I}$ a sequence of commuting $\K$-algebra automorphisms of $R$,
\item $t=(t_i)_{i\in I}$ is a sequence of nonzero central elements of $R$.
\end{itemize}
The cardinality of $I$ is called the \emph{rank} (or \emph{degree}) of $(R,\si,t)$.
\end{dfn}
Let $\Z I$ denote the free abelian group on the set $I$.
For $g=\sum g_i i\in\Z I$ we put $\si_g=\prod \si_i^{g_i}$. Then $g\mapsto\si_g$ defines an action of $\Z I$ on $R$ by $\K$-algebra automorphisms.
\begin{rem}
In Section \ref{sec:primitivequotients}, we will need to remove the assumption that $\si_i$ and $\si_j$ commute.
\end{rem}
\subsubsection{TGW constructions}
\begin{dfn}[TGW construction]
Let $I$ be a set and
\begin{itemize}
\item $(R,\si,t)$ be a TGW datum over $\K$ with index set $I$,
\item $\mu$ be an $I\times I$-matrix
without diagonal, $\mu=(\mu_{ij})_{i\neq j}$,
with $\mu_{ij}\in\K\setminus\{0\}$.
\end{itemize}
The \emph{twisted generalized Weyl construction} associated to $\mu$ and $(R,\si,t)$, denoted $\TGWC{\mu}{R}{\si}{t}$, is defined as the free $R$-ring on the set $\bigcup_{i\in I}\{X_i,Y_i\}$ modulo the two-sided ideal generated by the following elements:
\begin{subequations}\label{eq:tgwarels}
\begin{alignat}{3}
\label{eq:tgwarels1}
X_ir &-\si_i(r)X_i, &\quad Y_ir&-\si_i^{-1}(r)Y_i,
&\quad \text{$\forall r\in R,\, i\in I$,} \\
\label{eq:tgwarels2}
Y_iX_i&-t_i, &\quad X_iY_i&-\si_i(t_i),
&\quad \text{$\forall i\in I$,} \\
\label{eq:tgwarels3}
&&\quad X_iY_j&-\mu_{ij}Y_jX_i,
&\quad \text{$\forall i,j\in I,\, i\neq j$.}
\end{alignat}
\end{subequations}
\end{dfn}
The ring $\TGWC{\mu}{R}{\si}{t}$ has a $\Z I$-gradation $\TGWC{\mu}{R}{\si}{t}=\bigoplus_{d\in\Z I} \TGWC{\mu}{R}{\si}{t}_d$ given by requiring
\begin{equation}\label{eq:TGWA-gradation}
\begin{gathered}
\deg X_i= i,\quad \deg Y_i=(-1)i,\quad \forall i\in I,\\
\deg r=0, \quad \forall r\in R.
\end{gathered}
\end{equation}
\subsubsection{TGW algebras}
Let $\TGWI{\mu}{R}{\si}{t}\subseteq \TGWC{\mu}{R}{\si}{t}$ be
the sum of all graded ideals $J\subseteq \TGWC{\mu}{R}{\si}{t}$
such that \[\TGWC{\mu}{R}{\si}{t}_0\cap J=\{0\},\]
where $\TGWC{\mu}{R}{\si}{t}_0$ denotes the degree zero component with respect to the $\Z I$-gradation \eqref{eq:TGWA-gradation}.
It is easy to see that $\TGWI{\mu}{R}{\si}{t}$ is the unique maximal graded ideal of $\TGWC{\mu}{R}{\si}{t}$ having zero intersection with $\TGWC{\mu}{R}{\si}{t}_0$.
\begin{dfn}[TGW algebra]
The \emph{twisted generalized Weyl algebra} $\TGWA{\mu}{R}{\si}{t}$ associated to $\mu$ and $(R,\si,t)$ is
defined as the quotient $\TGWA{\mu}{R}{\si}{t}:=\TGWC{\mu}{R}{\si}{t} / \TGWI{\mu}{R}{\si}{t}$.
\end{dfn}
Since $\TGWI{\mu}{R}{\si}{t}$ is graded,
$\TGWA{\mu}{R}{\si}{t}$ inherits a $\Z I$-gradation from $\TGWC{\mu}{R}{\si}{t}$.
The images in $\TGWA{\mu}{R}{\si}{t}$ of the elements $X_i, Y_i$ will also be denoted by $X_i, Y_i$.
\subsubsection{Example: The Weyl algebra}\label{sec:WeylAlgebraExample}
The Weyl algebra over $\K$ with index set $I$, denoted $A_I(\K)$, is the $\K$-algebra generated by $\{x_i,y_i\mid i\in I\}$ subject to defining relations
\[[x_i,x_j]=[y_i,y_j]=[y_i,x_j]-\delta_{ij}=0,\quad\forall i,j\in I.\]
There is a $\K$-algebra isomorphism $\TGWA{\mu}{R}{\tau}{u}\to A_I(\K)$ where $\mu_{ij}=1$ for all $i\neq j$, $R=\K[u_i\mid i\in I]$, $\tau=(\tau_i)_{i\in I}$, $\tau_i(u_j)=u_j-\delta_{ij}$, $u=(u_i)_{i\in I}$,
given by $X_i\mapsto x_i$, $Y_i\mapsto y_i$, $u_i\mapsto y_ix_i$.
\subsubsection{Reduced and monic monomials}
A \emph{monic monomial} in a TGW algebra is any product of elements from the set $\bigcup_{i\in I}\{X_i,Y_i\}$.
A \emph{reduced monomial} is a monic monomial of the form
$Y_{i_1}\cdots Y_{i_k} X_{j_1}\cdots X_{j_l}$
where $\{i_1,\ldots,i_k\}\cap \{j_1,\ldots,j_l\}=\emptyset$.
The following statement is easy to check.
\begin{lem}{\cite[Lem.~3.2]{Hartwig2006}}
\label{lem:monomials}
$\TGWA{\mu}{R}{\si}{t}$ is generated as a left (and as a right) $R$-module by the reduced monomials.
\end{lem}
Since a TGW algebra $\TGWA{\mu}{R}{\si}{t}$ is a quotient of an $R$-ring, it is an $R$-ring itself with a natural map $\rho:R\to \TGWA{\mu}{R}{\si}{t}$.
By Lemma \ref{lem:monomials}, the degree zero component of $\TGWA{\mu}{R}{\si}{t}$ (with respect to the $\Z I$-gradation) is equal to the image of $\rho$.
\subsubsection{Regularity and consistency}
\begin{dfn}[Regularity]
A TGW datum $(R,\si,t)$ is called \emph{regular} if $t_i$
is regular (i.e. not a zero-divisor) in $R$ for all $i\in I$.
\end{dfn}
Due to relation \eqref{eq:tgwarels2}, the canonical map $\rho:R\to\TGWC{\mu}{R}{\si}{t}$ is not guaranteed to be injective, and indeed sometimes it is not \cite{FutHar2012b}. It is injective if and only if the map $R\to\TGWA{\mu}{R}{\si}{t}$ is injective.
\begin{dfn}[$\mu$-Consistency]
A TGW datum $(R,\si,t)$ is \emph{$\mu$-consistent} if the canonical map $\rho:R\to \TGWA{\mu}{R}{\si}{t}$ is injective.
\end{dfn}
By abuse of language we sometimes say that the TGW algebra $\TGWA{\mu}{R}{\mu}{t}$ is consistent if $(R,\si,t)$ is $\mu$-consistent.
\begin{thm}{\cite{FutHar2012b}}\label{thm:consistency}
If $(R,\si,t)$ is a regular TGW datum, and $\mu=(\mu_{ij})_{i\neq j}$ with $\mu_{ij}\in\K\setminus\{0\}$, then $(R,\si,t)$ is $\mu$-consistent iff
\begin{subequations}\label{eq:consistency_rels}
\begin{align}\label{eq:consistency_rel1}
\si_i\si_j(t_it_j)&=\mu_{ij}\mu_{ji}\si_i(t_i)\si_j(t_j),\quad\forall i\neq j;\\
\si_i\si_k(t_j)t_j&=\si_i(t_j)\si_k(t_j),\quad \forall i\neq j\neq k\neq i.
\end{align}
\end{subequations}
\end{thm}
That relation \eqref{eq:consistency_rel1} is necessary for consistency of a regular TGW datum was known already in \cite{MazTur1999},\cite{MazTur2002}.
If $(R,\si,t)$ is not regular, sufficient and necessary conditions for $\mu$-consistency are not known.
For consistent TGW algebras one can characterize regularity as follows:
\begin{thm}{\cite[Thm.~4.3]{HarOin2013}}
\label{thm:regularity}
Let $A=\TGWA{\mu}{R}{\si}{t}$ be a consistent TGW algebra. Then the following are equivalent
\begin{enumerate}[{\rm (i)}]
\item $(R,\si,t)$ is regular;
\item Each monic monomial in $A$ is non-zero and generates a free left (and right) $R$-module of rank one;
\item $A$ is \emph{regularly graded}, i.e. for all $g\in\Z I$, there exists a nonzero, regular element in $A_g$;
\item If $a\in A$ is a homogeneous element such that $bac=0$ for some monic monomials $b,c\in A$, then $a=0$.
\end{enumerate}
\end{thm}
\subsubsection{Non-degeneracy of the gradation form}
For a group $G$, any $G$-graded ring $A=\bigoplus_{g\in G}A_g$
can be equipped with a $\Z$-bilinear form $f:A\times A\to A_e$ called the \emph{gradation form}, defined by
\begin{equation}
f(a,b)=\mathfrak{p}_e(ab)
\end{equation}
where $\mathfrak{p}_e$ is the projection $A\to A_e$ along the direct sum $\bigoplus_{g\in G} A_g$, and $e\in G$ is the neutral element.
\begin{thm}{\cite[Cor.~3.3]{HarOin2013}}
\label{thm:nondeg}
The ideal $\TGWI{\mu}{R}{\si}{t}$ is equal to the radical of the gradation form on $\TGWC{\mu}{R}{\si}{t}$ (with respect to the $\Z I$-gradation), and thus the gradation form on $\TGWA{\mu}{R}{\si}{t}$ is non-degenerate.
\end{thm}
\subsubsection{\texorpdfstring{$R$}{R}-rings with involution}
\label{sec:R-ring-with-involution}
We need the concept of an $R$-ring with involution, defined as follows.
\begin{dfn}\label{dfn:Rring_with_involution}
\begin{enumerate}[{\rm (i)}]
\item An \emph{involution} on a ring $A$ is a $\Z$-linear map $\ast:A\to A, a\mapsto a^\ast$ satisfying $(ab)^\ast=b^\ast a^\ast$, $(a^\ast)^\ast=a$ for all $a,b\in A$.
\item Let $R$ be a ring.
An \emph{$R$-ring with involution} is a ring $A$ equipped with a ring homomorphism $\rho_A:R\to A$ and an involution $\ast:A\to A$ such that $\rho_A(r)^\ast=\rho_A(r)$ for all $r\in R$.
\item
If $A$ and $B$ are two $R$-rings with involution, then a \emph{map of $R$-rings with involution} is a ring homomorphism $k:A\to B$ such that $k\circ \rho_A=\rho_B$ and $k(a^\ast)=k(a)^\ast$ for all $a\in A$.
\end{enumerate}
\end{dfn}
Any TGW algebra $A=\TGWA{\mu}{R}{\si}{t}$ with index set $I$ for which $\mu_{ij}=\mu_{ji}$ for all $i,j$, can be equipped with an involution $\ast$ given by $X_i^\ast=Y_i,\, Y_i^\ast=X_i\;\forall i\in I$, $r^\ast=r\;\forall r\in R$. Together with the canonical map $\rho:R\to A$ this turns $A$ into an $R$-ring with involution. In particular we regard the Weyl algebra $A_I(\K)$ as an $R$-ring with involution in this way, where $R=\K[u_i\mid i\in I]$ as in Section \ref{sec:WeylAlgebraExample}.
\subsection{A proposition about epimorphisms of TGW algebras}
\label{sec:epimorphisms}
The following proposition about a quotient construction will be used in Section \ref{sec:special-linear-lie-algebras} and is a variation of \cite[Thm.~4.1]{FutHar2012b}.
Let $A=\mathcal{A}_\mu(R,\si,t)$ be a TGW algebra with index set $I$. We say that an ideal $J$ of $R$ is \emph{$\Z I$-invariant} if $\si_d(J)\subseteq J$ for all $d\in \Z I$. Then $J$ satisfies
\begin{equation}\label{eq:AJAJAJA}
AJ=AJA=JA
\end{equation}
Indeed, \[AJA=\sum_{g,h\in \Z I} A_gJA_h \subseteq \sum_{g,h\in \Z I} \si_g(J)A_gA_h\subseteq \sum_{g,h\in \Z I} JA_hA_g=JA.\]
The inclusion $AJA\subseteq AJ$ is proved similarly. The reverse inclusions are trivial since $A$ is unital.
\begin{prp}\label{prp:quotients}
Let $A=\mathcal{A}_\mu(R,\si,t)$ be a consistent and regular TGW algebra with index set $I$ and let $J$ be a $\Z I$-invariant prime ideal of $R$ such that $t_i\notin J$ for all $i\in I$. Assume that $A_g$ is cyclic as a left (equivalently, right) $R$-module for all $g\in \Z I$.
Let $\bar A=\mathcal{A}_\mu(\bar R, \bar\si, \bar t)$ where
$\bar R=R/J$, $\bar\si=(\bar\si_i)_{i\in I}$, $\bar\si_i(r+J)=\si_i(r)+J$, $\bar t=(t_i+J)_{i\in I}$. Then the kernel of the surjective homomorphism
\begin{equation}
\begin{aligned}
Q_J:A &\to \bar A\\
X_i &\mapsto X_i \\
Y_i &\mapsto Y_i \\
r &\mapsto r+J
\end{aligned}
\end{equation}
is equal to $AJA$.
\end{prp}
\begin{proof}
By \cite[Lem.~3.3]{FutHar2012b}, $\ker Q_J$ is equal to the sum of all graded ideals of $A$ whose intersection with $R$ is contained in $J$. In particular
\begin{equation}\label{eq:RkerQJJ}
R\cap \ker Q_J\subseteq J.
\end{equation}
By \eqref{eq:AJAJAJA}, $(AJA)\cap R=(AJ)\cap R = RJ = J$,
hence $AJA\subseteq \ker Q_J$.
For the converse, let $d\in \Z I$ and $a\in A_d\cap\ker Q_J$.
For $g\in \Z I$, let $u_g$ be a generator for $A_g$ as a left $R$-module.
We have $a=ru_d$ for some $r\in R$. We will show that $r\in J$. First we show that $u_du_{-d}\notin J$.
Let $m\in A_d$ and $m'\in A_{-d}$ be monic monomials
(products of elements from $\{X_i\}_{i\in I}\cup\{Y_i\}_{i\in I}$). By the TGW relations \eqref{eq:tgwarels}, $mm'$ is a product of elements of the form $\si_g(t_i)$. Since $J$ is $\Z I$-invariant, prime, and does not contain $t_i$, it follows that $mm'\notin J$.
On the other hand, $m=su_d$ and $m'=s'u_{-d}$ for some $s,s'\in R$. Thus
$mm'=s\si_d(s')u_du_{-d}$. Since $J$ is prime, it follows that $u_du_{-d}\notin J$.
We have $ru_du_{-d}=au_{-d}\in R\cap \ker Q_J\subseteq J$ by \eqref{eq:RkerQJJ}, which implies that $r\in J$ since $J$ is prime. Thus $a=ru_d\in JA=AJA$. Since $d$ was arbitrary and $Q_J$ is graded, this proves that $\ker Q_J\subseteq AJA$.
\end{proof}
\section{A family of TGW algebras parametrized by multiquivers}
\label{sec:AGamma}
\subsection{Multiquivers}
\label{sec:DirectedGraphs}
\begin{dfn}
A \emph{multiquiver} is a quadruple $\Gamma=(V,E,s,t)$ where
\begin{itemize}
\item $V$ and $E$ are sets,
\item $s,t:E\to (V\times\Z_{>0})\cup\{\emptyset\}$ are two functions,
\end{itemize}
subject to the following two conditions:
\begin{align}
\text{(No loops):}&\quad\text{If $s(e)=(u,m)$ and $t(e)=(v,n)$ then $u\neq v$.} \\
\text{(Locally finiteness):}&\quad\text{For each $v\in V$, $s^{-1}(\{v\}\times\Z_{>0})$ and $t^{-1}(\{v\}\times\Z_{>0})$ are finite sets.}
\end{align}
The elements of $V$ and $E$ are called \emph{vertices} and \emph{edges} respectively.
Let $e\in E$. If $s(e)=(v,n)$ (respectively $t(e)=(v,n)$) for some $v\in V$, $n\in \Z_{>0}$, then $v$ is called the \emph{source} (respectively \emph{target}) of $e$, $v$ is said to be \emph{incident} to $e$ and $n$ is the \emph{outgoing} (respectively \emph{incoming}) \emph{multiplicity} of $e$ at $v$.
If $s(e)=\emptyset$ (respectively $t(e)=\emptyset$) we say that $e$ has no source (respectively no target).
If $e$ has a target but no source, or a source but no target, $e$ is a \emph{connected leaf} (or a \emph{half-edge}).
If $e$ has neither a source nor a target, $e$ is a \emph{disconnected leaf} (or a \emph{loose edge}).
We depict an edge $e$ with $s(e)=(v_1,a)$ and $t(e)=(v_2,b)$ as follows:
\[
\begin{tikzpicture}[baseline=(1.base)]
\node[vertex] (1) at (0,0) [label=below:$v_1$] {};
\node[vertex] (2) at (2cm,0) [label=below:$v_2$] {};
\draw[thick,directed] (1) -- node[above] {$e$} node[auto,very near start] {$a$} node[auto,very near end] {$b$} (2);
\end{tikzpicture}
\]
If $a=1$ or $b=1$ we sometimes omit the corresponding incidence multiplicity from the diagram.
A multiquiver $\Gamma=(V,E,s,t)$ is called a \emph{simple quiver} if $\Gamma$ has no leaves and has at most one edge between any two vertices.
A multiquiver $\Gamma=(V,E,s,t)$ without leaves is called \emph{symmetric} (or \emph{equally valued)} if, in each edge, the outgoing and incoming multiplicities coincide.
\end{dfn}
\begin{rem}\label{rem:simple-quiver}
In a simple quiver $\Gamma=(V,E,s,t)$ one may re-interpret each edge $e\in E$, say $s(e)=(i,v_{ij})$ and $t(e)=(j,v_{ji})$, as a set of two labeled directed edges, one going from $i$ to $j$ with label $v_{ij}$, and the other going from $j$ to $i$ with label $v_{ji}$. In this way we obtain a directed graph (digraph) $Q(\Gamma)$ with edges labeled by positive integers. However, some information is lost; switching the direction of $e$ in $\Gamma$, i.e. replacing it by $e'$ with $s(e')=(j,v_{ji})$, $t(e')=(i,v_{ij})$, yields the same set of two labeled edges in $Q(\Gamma)$.
\end{rem}
\subsection{Incidence matrix}
For a set $X$ we let $\Z X$ denote the free abelian group on $X$. To each multiquiver $\Gamma=(V,E,s,t)$ we associate a group homomorphism $\gamma\in\Hom_\Z(\Z V, \Z E)$ as follows:
\begin{equation}
\label{eq:gamma-definition}
\gamma(v)=\sum_{\substack{e\in E,\, n\in\Z_{>0}\\t(e)=(v,n)}}n\cdot e - \sum_{\substack{e\in E,\, n\in\Z_{>0}\\s(e)=(v,n)}}n\cdot e, \qquad \forall v\in V.
\end{equation}
Since $\Gamma$ is locally finite, $\gamma(v)$ is a finite linear combination of elements of $E$, hence $\gamma(v)\in \Z E$.
We call $\gamma$ the \emph{incidence matrix} of $\Gamma$.
If $V$ and $E$ are nonempty, we can identify
$\ga$ with the matrix $(\ga_{ev})_{e\in E, v\in V}$ given by
\[\gamma(v)=\sum_{e\in E}\gamma_{ev} e,
\qquad \forall v\in V.\]
It satisfies the following two properties:
\begin{gather}
\label{eq:condition_C}
\text{Every column $(\gamma_{ev})_{e\in E}$ has at most finitely many nonzero elements.} \\
\label{eq:condition_M}
\text{Every row $(\gamma_{ev})_{v\in V}$ contains at most one positive and at most one negative element.}
\end{gather}
Conversely, if $A=(a_{ij})_{i\in I,j\in J}$ is any matrix with integer entries satisfying conditions \eqref{eq:condition_C} and \eqref{eq:condition_M} there exists a unique multiquiver $\Gamma_A$ whose incidence matrix is $A$. Namely, $\Gamma_A=(J,I,s,t)$ where for any $i\in I$,
\begin{align*}
s(i)&=
\begin{cases}
(j, |a_{ij}|)& \text{if $a_{ij}<0$ for some (unique) $j\in J$,}\\
\emptyset & \text{otherwise,}
\end{cases}\\
t(i)&=
\begin{cases}
(j,a_{ij})& \text{if $a_{ij}>0$ for some (unique) $j\in J$,}\\
\emptyset & \text{otherwise.}
\end{cases}
\end{align*}
Thus there is a one-to-one correspondence between the set of multiquivers and the set of homomorphisms $\Hom_\Z(\Z V, \Z E)$ whose matrix satisfies
conditions \eqref{eq:condition_C} and \eqref{eq:condition_M}.
Connected leaves correspond to rows with exactly one non-zero element while disconnected leaves correspond to rows with only zeroes.
Two multiquivers are \emph{isomorphic} if their incidence matrices coincide up to permutation of rows and columns.
\subsection{Construction of the TGW algebra \texorpdfstring{$\mathcal{A}(\Gamma)$}{A(Gamma)}}
Let $\Gamma=(V,E,s,t)$ be a multiquiver, and $\gamma=(\gamma_{ev})_{e\in E, v\in V}$ be its incidence matrix. Define
\begin{subequations}
\begin{align}
\mu&=(\mu_{ij})_{i,j\in V}, \quad\mu_{ij}=1, \quad\forall i,j\in V,\\
R_E &=\K[u_e\mid e\in E]\quad\text{(polynomial algebra),}\\
\label{eq:sigma-definition}
\sigma^\Gamma &=(\sigma_v)_{v\in V},\quad \sigma_v(u_e)=u_e-\gamma_{ev},\quad\forall v\in V, e\in E,\\
\label{eq:t-definition}
t^\Gamma &=(t_v)_{v\in V}, \quad t_v = \prod_{\substack{e\in E\\ \text{$v$ incident to $e$}}} u_{ev},
\end{align}
where
\begin{equation}
u_{ev}=
\begin{cases}
u_e(u_e+1)\cdots (u_e+n-1),&\text{if $t(e)=(v,n)$}, \\
(u_e-1)(u_e-2)\cdots (u_e-n),&\text{if $s(e)=(v,n)$}.
\end{cases}
\end{equation}
\end{subequations}
We define $\mathcal{A}(\Gamma)$ to be the twisted generalized Weyl algebra $\mathcal{A}_\mu(R_E,\sigma^\Gamma,t^\Gamma)$ with index set $V$.
$\mathcal{A}(\Gamma)$ is a $\Z V$-graded algebra. Since $\mu$ is symmetric, $\mathcal{A}(\Gamma)$ is also an $R_E$-ring with involution
(see Section \ref{sec:R-ring-with-involution}).
\begin{rem}
In terms of the matrix $\gamma$ we have
\begin{equation}
t_v=\prod_{e\in E} u_{ev}, \qquad
u_{ev}=
\begin{cases}
u_e(u_e+1)\cdots (u_e+\ga_{ev}-1),& \ga_{ev}> 0, \\
1, &\ga_{ev}=0,\\
(u_e-1)(u_e-2)\cdots (u_e-|\ga_{ev}|),& \ga_{ev}<0.
\end{cases}
\end{equation}
\end{rem}
\begin{thm} \label{thm:Gamma-solves-consistency}
For any multiquiver $\Gamma=(V,E,s,t)$, the TGW datum $(R_E,\si^\Gamma,t^\Gamma)$ satisfies the TGW consistency conditions \eqref{eq:consistency_rels} with $\mu_{ij}=1$ for $i,j\in V$.
\end{thm}
\begin{proof}
By \eqref{eq:t-definition} it suffices to check that for each $e\in E$,
\[\si_i(u_{ei})\si_j(u_{ej})=\si_i\si_j(u_{ei}u_{ej}),
\quad \forall i,j\in V, i\neq j,\]
and
\[\si_i\si_k(u_{ej})u_{ej}=\si_i(u_{ej})\si_k(u_{ej}),
\quad \forall i,j,k\in V, i\neq j\neq k\neq i.\]
These identities are easy to verify, using \eqref{eq:sigma-definition} and
Property \eqref{eq:condition_M} of the matrix $(\ga_{ev})_{e\in E,v\in V}$.
\end{proof}
\begin{cor} \label{cor:consistency_of_AGamma}
For any multiquiver $\Gamma$, the TGW algebra $\mathcal{A}(\Gamma)$ is consistent,
i.e. the canonical map $\rho:R_E\to\mathcal{A}(\Gamma)$ is injective.
\end{cor}
\begin{proof}
Immediate by Theorem \ref{thm:Gamma-solves-consistency} and Theorem \ref{thm:consistency}.
\end{proof}
We will henceforth use $\rho$ to identify $R_E$ with its image in $\mathcal{A}(\Gamma)$. By Lemma \ref{lem:monomials}, $\mathcal{A}(\Gamma)_0=R_E$.
\begin{cor}\label{cor:domain}
$\mathcal{A}(\Gamma)$ is a domain.
\end{cor}
\begin{proof}
Since $R_E$ is a domain, this follows from Corollary \ref{cor:consistency_of_AGamma}
and \cite[Prop.~2.9]{FutHar2012a}.
\end{proof}
\begin{rem}
Different multiquivers $\Gamma$ can yield isomorphic algebras $\mathcal{A}(\Gamma)$. However, if $\Gamma_1=(V,E,s_1,t_1)$ and $\Gamma_2=(V,E,s_2,t_2)$ are two multiquivers with the same underlying vertex and edge sets, then $\mathcal{A}(\Gamma_1)$ and $\mathcal{A}(\Gamma_2)$ are isomorphic as $\Z V$-graded $R_E$-rings if and only if $\Gamma_1$ and $\Gamma_2$ are isomorphic.
\end{rem}
\section{Examples}
\label{sec:examples}
We list a number of examples of multiquivers, their incidence matrices and describe the corresponding algebras $\mathcal{A}(\Gamma)$.
\begin{enumerate}[{\rm (1)}]
\item (Trivial case) $V=\emptyset$, $E=\emptyset$.
Then $\Z V=\Z E=\{0\}$, $\ga$ is the zero map $\{0\}\to\{0\}$ and $\mathcal{A}(\Gamma)\simeq \K$.
\item (Laurent polynomial algebra) $V$ arbitrary, $E=\emptyset$.
Then $\ga$ is the zero map $\Z V\to \{0\}$.
\[
\Gamma:\;
\begin{tikzpicture}[baseline=(etc.base)]
\node[vertex] (1) at (0,0) {};
\node[vertex] (2) at (.5,0) {};
\node[vertex] (3) at (1,0) {};
\node[vertex] (4) at (2,0) {};
\node (etc) at (1.5,0) {$\cdots$};
\end{tikzpicture}
\qquad\qquad
\ga=0:\Z V\to \{0\}
\]
We have $t_v=1$ for each $v\in V$ and $\mathcal{A}(\Gamma)\simeq \K[X_v,X_v^{-1}\mid v\in V]$, a Laurent polynomial algebra.
\item (Polynomial algebra) $V=\emptyset$, $E$ arbitrary.
Then $\ga$ is the zero map $\{0\}\to\Z E$.
\[
\Gamma:\;
\begin{tikzpicture}[baseline=(etc.base)]
\node (1) at (0,0) {};
\node (2) at (.5,0) {};
\node (3) at (1,0) {};
\node (4) at (2,0) {};
\node (leaf1) at (0,1) {};
\node (leaf2) at (.5,1) {};
\node (leaf3) at (1,1) {};
\node (leaf4) at (2,1) {};
\node (etc) at (1.5,.5) {$\cdots$};
\draw[leaf] (1) -- (leaf1);
\draw[leaf] (2) -- (leaf2);
\draw[leaf] (3) -- (leaf3);
\draw[leaf] (4) -- (leaf4);
\end{tikzpicture}
\qquad\qquad
\ga=0:\{0\} \to \Z E
\]
We have $\mathcal{A}(\Gamma)\simeq R_E=\K[u_e\mid e\in E]$, a polynomial algebra.
\item (Weyl algebra)
Let $I$ be any set, and $V=E=I$, $s(i)=\emptyset$, $t(i)=(i,1)$ for all $i\in I$. Then $\ga$ is the identity map $\Z I\to \Z I$ and $\mathcal{A}(\Gamma)$ is isomorphic to the Weyl algebra $A_I(\K)$.
\[
\Gamma:\;
\begin{tikzpicture}[baseline=(X.base)]
\node (X) at (0,.4) {};
\node[vertex] (1) at (0,0) {};
\node[vertex] (2) at (.5,0) {};
\node[vertex] (3) at (1,0) {};
\node[vertex] (4) at (2,0) {};
\node (leaf1) at (0,1) {};
\node (leaf2) at (.5,1) {};
\node (leaf3) at (1,1) {};
\node (leaf4) at (2,1) {};
\node (etc) at (1.5,0) {$\cdots$};
\draw[leaf] (leaf1) -- (1);
\draw[leaf] (leaf2) -- (2);
\draw[leaf] (leaf3) -- (3);
\draw[leaf] (leaf4) -- (4);
\end{tikzpicture}
\qquad\qquad
\ga=
\begin{bmatrix}
1 & 0 & \cdots & 0 \\
0 & 1 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & 1
\end{bmatrix}
\]
\item (An example of type $A_2$)
\[
\Gamma:\;
\begin{tikzpicture}[baseline=(X.base)]
\node (X) at (0,-0.1) {};
\node[vertex] (1) at (0,0) [label=below:$v_1$] {};
\node[vertex] (2) at (2cm,0) [label=below:$v_2$] {};
\draw[edge] (1) -- (2);
\end{tikzpicture}
\qquad\qquad
\ga =
\begin{bmatrix}
-1 & 1
\end{bmatrix}
\]
Then $\mathcal{A}(\Gamma)$ is graded isomorphic to the TGW algebra defined by Mazorchuk and Turowska \cite[Ex.~1.3]{MazTur1999}. Therefore, by \cite[Ex.~6.3]{Hartwig2010}, $\mathcal{A}(\Gamma)$ is isomorphic to the algebra with generators $u, X_1,X_2,Y_1,Y_2$ and defining relations
\[
\begin{aligned}
X_1u&=(u+1)X_1, & X_2u&=(u-1)X_2, \\
Y_1u&=(u-1)Y_1, & Y_2u&=(u+1)Y_2, \\
Y_1X_1&=X_2Y_2=u-1, & X_1Y_2&=Y_2X_1,\\
Y_2X_2&=X_1Y_1=u, & X_2Y_1&=Y_1X_2.
\end{aligned}
\quad
\begin{aligned}
X_1^2X_2-2X_1X_2X_1+X_2X_1^2&=0, \\
X_2^2X_1-2X_2X_1X_2+X_1X_2^2&=0, \\
Y_1^2Y_2-2Y_1Y_2Y_1+Y_2Y_1^2&=0, \\
Y_2^2Y_1-2Y_2Y_1Y_2+Y_1Y_2^2&=0.
\end{aligned}
\]
\item (An example related to $\mathfrak{gl}_{n+1})$
\[
\Gamma:\;
\begin{tikzpicture}[baseline=(etc.base)]
\node (start) at (0,0) {};
\node[vertex] (1) at (1,0) [label=below:$1$] {};
\node[vertex] (2) at (2,0) [label=below:$2$] {};
\node[vertex] (3) at (3,0) [label=below:$3$] {};
\node[vertex] (N) at (4,0) [label=below:$n$] {};
\node (end) at (5,0) {};
\node (etc) at (3.5,0) {$\cdots$};
\draw[leaf] (start) -- (1);
\draw[edge] (1) -- (2);
\draw[edge] (2) -- (3);
\draw[leaf] (N) -- (end);
\end{tikzpicture}
\qquad\qquad
\ga=\begin{bmatrix}
1 & & & &\\
-1 & 1 & & &\\
& -1 & \ddots & \\
& & \ddots & \ddots & \\
& & & -1 & 1 \\
& & & & -1
\end{bmatrix}
\]
Then $\mathcal{A}(\Gamma)$ is a graded homomorphic image of $U(\mathfrak{gl_{n+1}})$, under $E_{i,i+1}\mapsto X_i$, $E_{i+1,i}\mapsto Y_i$, $E_{jj}\mapsto u_j$ for $i=1,2,\ldots,n$ and $j=1,2,\ldots,n+1$. This generalizes the $\mathfrak{gl}_3$ case considered by A. Sergeev \cite{Sergeev2001}.
See Section \ref{sec:special-linear-lie-algebras} for more details.
\item (An example related to $\mathfrak{sp}_{2n}$)
\[
\Gamma:\;\;
\begin{tikzpicture}[baseline=(etc.base)]
\node (start) at (0,0) {};
\node[vertex] (1) at (1,0) [label=below:$1$] {};
\node[vertex] (2) at (2,0) [label=below:$2$] {};
\node[vertex] (3) at (3,0) [label=below:$3$] {};
\node[vertex] (N-2) at (4,0) [label=below:$n-2$] {};
\node[vertex] (N-1) at (5,0) [label=below:$n-1$] {};
\node[vertex] (N) at (6,0) [label=below:$n$] {};
\node (etc) at (3.5,0) {$\cdots$};
\draw[leaf] (start) -- (1);
\draw[edge] (1) -- (2);
\draw[edge] (2) -- (3);
\draw[edge] (N-2) -- (N-1);
\draw[edge] (N-1) -- node[auto,very near start] {$1$} node[auto,very near end] {$2$} (N);
\end{tikzpicture}
\qquad\qquad
\ga=\begin{bmatrix}
1 & & & && \\
-1 & 1 & & &&\\
& -1 & \ddots & &&\\
& & \ddots & \ddots &&\\
& & & -1 & 1 & \\
& & & &-1 & 2
\end{bmatrix}
\]
Then $\mathcal{A}(\Gamma)$ is a graded homomorphic image of $U(\mathfrak{sp}_{2n})$. See Section \ref{sec:symplectic-lie-algebras} for more details.
\item (Algebras associated to symmetric GCMs)
Let $\Gamma=(V,E,s,t)$ be a symmetric simple quiver.
Then $\mathcal{A}(\Gamma)$ is graded isomorphic to a quotient of the algebra $\mathcal{T}(C)$ from \cite{Hartwig2010}, where $C$ is the (symmetric) generalized Cartan matrix of the Dynkin diagram associated to $\Gamma$. See Section \ref{sec:symmetric} for details.
\end{enumerate}
\section{Representation by differential operators}
\label{sec:diffops}
\subsection{The homomorphism \texorpdfstring{$\varphi_\Gamma$}{phiGamma}}
\label{sec:homomorphism-varphi}
Let $\Gamma=(V,E,s,t)$ be a multiquiver. In this section we show how the algebras $\mathcal{A}(\Gamma)$ can be naturally represented by differential operators.
Let $A_E(\K)$ be the Weyl algebra with index set $E$.
For $k=\sum_{e\in E} k_e e\in \Z E$, define $z^k\in A_E(\K)$ by
\begin{equation}\label{eq:zk_notation}
z^k=\prod_{e\in E}z_e^{(k_e)},\qquad \text{where }z_e^{(p)}=\begin{cases}x_e^p& p\ge 0,\\ y_e^{-p}& p<0.\end{cases}
\end{equation}
\begin{thm}\label{thm:map_to_Weyl_algebra}
Let $\Gamma=(V,E,s,t)$ be a multiquiver and $\gamma\in\Hom_\Z(\Z V, \Z E)$ be the corresponding group homomorphism \eqref{eq:gamma-definition}.
There exists a map of $R_E$-rings with involution
\begin{subequations}
\label{eq:varphi_def}
\begin{gather}
\varphi_\Gamma:\mathcal{A}(\Gamma)\longrightarrow A_E(\K),\\
\intertext{uniquely determined by}
\label{eq:varphi_def_1}
\varphi(X_v)=z^{\gamma(v)},\quad
\varphi(Y_v)=z^{-\gamma(v)},\quad \forall v\in V \\
\label{eq:varphi_def_2}
\varphi(u_e)=y_ex_e,\quad\forall e\in E.
\end{gather}
\end{subequations}
\end{thm}
\begin{proof}
Put $(R,\si,t)=(R_E,\si^\Gamma,t^\Gamma)$. Let $\mathcal{X}=\cup_{v\in V}\{X_v,Y_v\}$ and define $\varphi:\mathcal{X} \to A_E(\K)$ by \eqref{eq:varphi_def_1}-\eqref{eq:varphi_def_2}.
Extend $\varphi$ uniquely to a homomorphism of $R$-rings $\varphi:F_R(\mathcal{X})\to A_E(\K)$, where $F_R(\mathcal{X})$ is the free $R$-ring on the set $\mathcal{X}$.
Using \eqref{eq:condition_M}, one verifies that the ideal in $F_R(\mathcal{X})$ generated by the elements \eqref{eq:tgwarels} is contained in the kernel of $\varphi$. Thus we get an induced homomorphism of $R$-rings
$\varphi:\TGWC{\mu}{R}{\si}{t}\to A_E(\K)$. Suppose $a\in \TGWC{\mu}{R}{\si}{t}$ is a homogeneous element which lies in the radical of the gradation form $f$ on $\TGWC{\mu}{R}{\si}{t}$. Then in particular $0=f(a,a^\ast)=a\cdot a^\ast$. Hence $\varphi(a)\cdot\varphi(a)^\ast=0$ which implies that $\varphi(a)=0$, since $A_n$ is a domain.
Thus, by Theorem \ref{thm:nondeg}, $\varphi$ induces a homomorphism $\varphi_\Gamma:\TGWA{\mu}{R}{\si}{t}\to A_E(\K)$ which is the required map of $R$-rings with involution.
\end{proof}
\begin{rem}
It is well-known that $R_E\to A_E(\K)$, $u_e\mapsto y_ex_e$, is injective. Using this fact and that $\varphi_\Gamma$ is a map of $R_E$-rings, it follows that the canonical map $\rho:R_E\to \mathcal{A}(\Gamma)$ is also injective. This gives another independent proof that $\mathcal{A}(\Gamma)$ is consistent and that the TGW datum $(R_E,\si^\Gamma,t^\Gamma)$ satisfies the consistency relations \eqref{eq:consistency_rels}.
\end{rem}
\begin{example}
For the first four examples of Section \ref{sec:examples} we have:
\begin{enumerate}[{\rm (1)}]
\item $\varphi_\Gamma$ is the identity map $\K\to \K$. (Since $E=\emptyset$, the Weyl algebra $A_E(\K)$ is just $\K$.)
\item $\varphi_\Gamma$ is the evaluation homomorphism $\K[X_v,X_v^{-1}\mid v\in V]\to \K$ given by $\varphi_\Gamma(X_v)=1$ for each $v\in V$.
\item $\varphi_\Gamma$ is the embedding of $R_E$ into the Weyl algebra $A_E(\K)$ given by $\varphi_\Gamma(u_e)=u_e=y_ex_e$ for each $e\in E$.
\item $\varphi_\Gamma:\mathcal{A}(\Gamma)\simeq A_I(\K)$ is an isomorphism mapping $X_i$ (respectively $Y_i$) to $x_i$ (respectively $y_i$) for all $i\in I$.
\end{enumerate}
\end{example}
\subsection{Faithfulness}
\label{sec:faithfulness}
In this section we give a precise condition, in terms of the graph $\Gamma$, for when the representation $\varphi_\Gamma:\mathcal{A}(\Gamma)\to A_E(\K)$ is faithful.
We need several definitions.
\begin{enumerate}[{\rm (i)}]
\item A \emph{subgraph} $\Gamma'=(V',E',s',t')$ of a multiquiver $\Gamma=(V,E,s,t)$ is a multiquiver such that
$V'\subseteq V$, $E'\subseteq E$ and
\[s'(e)=\begin{cases} s(e),&
\text{if $s(e)=(v,n)$ and $v\in V'$}\\
\emptyset,&\text{otherwise}
\end{cases}\qquad
t'(e)=\begin{cases} t(e),&
\text{if $t(e)=(v,n)$ and $v\in V'$}\\
\emptyset,&\text{otherwise}
\end{cases}
\]
for all $e\in E'$.
\item A multiquiver $\Gamma$ is a \emph{directed cycle} if $V$ and $E$ are finite sets of the same cardinality and there are enumerations $V=\{v_1,v_2,\ldots,v_n\}$ and $E=\{e_1,e_2,\ldots,e_n\}$ such that for all $i\in\iv{1}{n}$ we have $s(e_i)=(v_i,n_i)$ and $t(e_i)=(v_{i+1},m_i)$ for some positive integers $n_i,m_i$ and we put $v_{n+1}=v_1$.
\item Two multiquivers $\Gamma_1$ and $\Gamma_2$ are called \emph{sign-equivalent} if they become isomorphic after changing the direction of some edges. In other words, the incidence matrices $\ga_1$ and $\ga_2$ will coincide (up to permutation of rows and columns) after multiplying some rows in $\ga_1$ or $\ga_2$ by $-1$.
\item A multiquiver $\Gamma=(V,E,s,t)$ is a \emph{not necessarily directed (NND) cycle} if it is sign-equivalent to a directed cycle.
\item If $\Gamma'$ is a subgraph of a multiquiver $\Gamma$, and $\Gamma'$ is a (directed or NND) cycle, we simply say that $\Gamma'$ is a (directed or NND) cycle \emph{in $\Gamma$}.
\item A directed cycle $\Gamma$ is \emph{balanced} if the product of all outgoing multiplicities equals the product of all incoming multiplicities. This is equivalent to $\det(\ga)=0$, where $\ga$ is the incidence matrix of $\Gamma$.
\item An NND cycle $\Gamma$ is \emph{balanced} if it is sign-equivalent to a balanced directed cycle.
\item A connected component of a multiquiver $\Gamma$ is said to be \emph{in equilibrium} if it has finitely many vertices, no leaves and every NND cycle in $\Gamma$ is balanced.
\end{enumerate}
\begin{example}\label{ex:graph1}
This multiquiver has a disconnected vertex $v_1$, a disconnected leaf $e_1$, a connected leaf $e_5$, and one unbalanced cycle. Only the connected component consisting of $v_1$ is in equilibrium.
\[
\Gamma:\;\;\begin{tikzpicture}[baseline=(60:1)]
\node[vertex] (A) at (-1,0) [label=below:$v_1$] {};
\node (1) at (-1.5,.5) {};
\node (2) at (-.5,.5) {};
\node (E) at (3,0) {};
\node[vertex] (B) at (0,0) [label=below:$v_2$] {};
\node[vertex] (C) at (60:2) [label=above:$v_3$] {};
\node[vertex] (D) at (2,0) [label=below:$v_4$] {};
\draw[disconnected leaf] (1) -- node[above] {$e_1$} (2);
\draw[leaf] (E) -- node[above] {$e_5$} (D);
\draw[edge] (B) -- node[above left] {$e_2$} node[auto,very near start] {$2$} node[auto,very near end] {$3$} (C);
\draw[xshift=10pt] (B) -- (C);
\draw[edge] (C) -- node[above right] {$e_3$} node[auto,very near start] {$1$} node[auto,very near end] {$2$} (D);
\draw[edge] (D) -- node[above] {$e_4$} node[auto,very near start] {$1$} node[auto,very near end] {$1$} (B);
\end{tikzpicture}
\qquad
\qquad
\ga=\begin{bmatrix}
0 & 0 & 0 & 0 \\
0 &-2 & 3 & 0 \\
0 & 0 &-1 & 2 \\
0 & 1 & 0 &-1 \\
0 & 0 & 0 & 1
\end{bmatrix}
\]
\end{example}
\begin{example}\label{ex:graph2}
This is a connected multiquiver in equilibrium since it is a balanced NND cycle.
\[
\Gamma:\;\;\begin{tikzpicture}[baseline=(60:1)]
\node[vertex] (A) at (0,0) [label=below left:$v_1$] {};
\node[vertex] (B) at (60:2) [label=above:$v_2$] {};
\node[vertex] (C) at (2,0) [label=below right:$v_3$] {};
\draw[edge] (A) -- node[auto,very near start] {$2$} node[auto,very near end] {$1$} (B);
\draw[edge] (C) -- node[above right,very near start] {$4$} node[above right,very near end] {$1$} (B);
\draw[edge] (C) -- node[auto,very near start] {$2$} node[auto,very near end] {$1$} (A);
\end{tikzpicture}
\qquad
\qquad
\ga=\begin{bmatrix}
-2& 1 & 0 \\
0 & 1 &-4 \\
1 & 0 &-2
\end{bmatrix}
\]
\end{example}
\begin{example}\label{ex:graph3}
This connected multiquiver without leaves is not in equilibrium since it has a non-balanced cycle.
\[
\Gamma:\;\;\begin{tikzpicture}[baseline=(A.base)]
\node[vertex] (A) at (0,0) [label=left:$v_1$] {};
\node[vertex] (B) at (2,0) [label=right:$v_2$] {};
\draw[edge] (A) to[out=45,in=135] node[auto,near start] {$2$} node[auto,near end] {$1$} (B);
\draw[edge] (B) to[out=-135,in=-45] node[auto,near start] {$3$} node[auto,near end] {$4$} (A);
\end{tikzpicture}
\qquad
\qquad
\ga=\begin{bmatrix}
-2& 1 \\
4 &-3 \\
\end{bmatrix}
\]
\end{example}
The next result describes the centralizer of $R_E$ in $\mathcal{A}(\Gamma)$.
\begin{lem}\label{lem:centralizer}
Let $\Gamma=(V,E,s,t)$ be a multiquiver, $\ga:\Z V\to \Z E$ its incidence matrix, $A=\mathcal{A}(\Gamma)$ and $C_A(R_E)=\{a\in A\mid ar=ra\,\forall r\in R_E\}$ be the centralizer of $R_E$ in $A$. Then
\[C_A(R_E)=\bigoplus_{d\in\ker \ga} A_d\]
In particular, $\ga$ is injective if and only if $R_E$ is maximal commutative in $A$.
\end{lem}
\begin{proof}
Let $K$ be the kernel of the group homomorphism $\hat\sigma:\Z V\to \Aut_\K(R_E)$ determined by $\hat\sigma(v)=\sigma_v$ for all $v\in V$. By definition \eqref{eq:sigma-definition} of $\si_v$, we have
\[\hat\sigma(d)(u_e)=u_e-\ga(d)_e\]
where $\ga(d)=\sum_{e\in E}\ga(d)_e e$.
Thus $K=\ker \ga$. By \cite[Thm.~5.1]{HarOin2013} the claim follows.
\end{proof}
The following theorem describes the rank of $\ker \ga$ as a free abelian group, in terms of the graph $\Gamma$. In view of Section \ref{sec:symmetric}, it is parallel to \cite[Thm.~5.8]{HarOin2013}, since if $\Gamma$ is a symmetric simple quiver, then every connected component is in equilibrium.
\begin{thm}\label{thm:rank-ker-gamma}
Let $\Gamma$ be a multiquiver and $\gamma$ be its incidence matrix. Then the rank of the kernel of $\gamma$ is equal to the number of connected components of $\Gamma$ in equilibrium.
\end{thm}
\begin{proof}
Put $K=\ker \ga$. Let $\mathcal{C}(\Gamma)$ be the set of connected components of $\Gamma$, and $\mathcal{C}^{\mathrm{eq}}(\Gamma)\subseteq\mathcal{C}(\Gamma)$ be the subset of components in equilibrium. We have
\[\Z V=\bigoplus_{C\in\mathcal{C}(\Gamma)}\Z V_C\]
where $V_C$ is the vertex set of $C$,
and $\ga$ is block diagonal with respect to this decomposition (since no two vertices belonging to different components are adjacent). Therefore
\[K=\bigoplus_{C\in\mathcal{C}(\Gamma)} K_C\]
where $K_C=K\cap \Z V_C$.
Furthermore, the rank of each subgroup $K_C$ is at most one.
Indeed, assume $K_C\neq\{0\}$ and let $a\in K_C\setminus\{0\}$. Write $a=\sum_{v\in V_C} \lambda_v v$, where $\lambda_v\in \Z$.
Let $e$ be any proper edge in $C$, connecting two vertices $v,w\in V_C$. Since $a\in K$, in particular the coefficient of $e$ in $\ga(a)$ is zero. That is, $0=\ga(a)_e=\la_v \ga_{ev} + \la_w \ga_{ew}$.
Since $C$ is connected, it follows that all coefficients $\la_v$ are nonzero and uniquely determined by any single one of them. Thus the rank of $K_C$ equals one.
Suppose $C\in\mathcal{C}(\Gamma)$ is not in equilibrium.
Let $a=\sum_{v\in V_C}\la_v v\in K_C$.
If $V_C$ is infinite, then $\la_v=0$ for some $v\in V_C$, hence $\la_w=0$ for all $w\in V_C$ since $C$ is connected, hence $a=0$.
If $C$ has a connected leaf $e$, let $v\in V_C$ be the vertex incident to $e$. Then $0=\ga(a)_e=\la_v\ga_{ev}$ which implies $\la_v=0$. As above that implies that $a=0$.
If $C$ has a non-balanced cycle $\Gamma'=(V',E',s',t')$, let $\ga':\Z V'\to \Z E'$ be the incidence matrix of $\Gamma'$. Since $\Gamma'$ is non-balanced, $\det(\ga')\neq 0$. Let $a'=\sum_{v\in V'} \la_v v$. Since $\ga(a)=0$ we get $\ga'(a')=0$ which implies $a'=0$. So $\la_v=0$ for all $v\in V'$. Since $C$ is connected, $\la_v=0$ for all $v\in V$. This shows that $K_C=\{0\}$ for $C$ not in equilibrium.
Suppose $C\in\mathcal{C}(\Gamma)$ is a component that is in equilibrium. We have to show that $K_C$ is nonzero.
Fix any vertex $v_0$ in $C$ and define $\la_{v_0}=1$. For any other vertex $w$ in $C$, pick a path (linear subgraph) $p$ from $v_0$ to $w$. Let $\vec{p}$ be the multiquiver which is sign-equivalent $p$ but every edge is directed forward (from $v_0$ to $w$).
Define $\la_w=O_{\vec{p}}/I_{\vec{p}}$
where $O_{\vec{p}}$ (respectively $I_{\vec{p}}$) is the product of all the outgoing
(respectively incoming) multiplicities in edges in $\vec{p}$. This is independent of the choice of path since all cycles are balanced. Put
$a'=\sum_{v\in V_C} \la_v v \in \Q V_C$. Let $k$ be an integer such that $a=ka'\in \Z V_C$.
We claim that $a\in K_C\setminus\{0\}$. Clearly $a\neq 0$.
Let $e$ be any edge in $C$, between $w_1$ and $w_2$. Without loss of generality we may assume there is a path from $v_0$ to $w_2$ that goes through $w_1$. Then $\la_{w_2}=\la_{w_1}\cdot \frac{m_1}{m_2}$ where $m_i$ is the multiplicity of $e$ at $w_i$. Thus
\[\ga(a)_e=k\la_{w_1}\ga_{e,w_1}+k\la_{w_2}\ga_{e,w_2}=k\la_{w_1}(\ga_{e,w_1}+(m_1/m_2)\ga_{e,w_2})=\pm k\la_{w_1}(m_1-(m_1/m_2)m_2)=0.\]
Since $e$ was arbitrary, this proves that $a\in K_C$.
\end{proof}
\begin{cor}
\label{cor:centralizer}
Let $\Gamma=(V,E,s,t)$ be any multiquiver and $A=\mathcal{A}(\Gamma)$ the associated TGW algebra. Then the centralizer $C_A(R_E)$ is a maximal commutative subalgebra of $A$.
\end{cor}
\begin{proof} From the relations \eqref{eq:tgwarels} one checks that $\si_j(t_j)X_iX_j=\si_j\si_i(t_j)X_jX_i$ for all $i,j\in V$, $i\neq j$. Thus, since $A$ is a domain, if $\si_i(t_j)=t_j$ it follows that $X_i$ and $X_j$ commute. Similarly (or by applying the involution), $Y_i$ and $Y_j$ commute if $\si_i(t_j)=t_j$. This implies that if $C$ and $C'$ are two connected components of $\Gamma$, then $[A_g,A_h]=0$ for any $g\in \Z V_C$, $h\in \Z V_{C'}$. Thus, by \cite[Thm.~5.3]{HarOin2013} and Lemma \ref{lem:centralizer}, it follows that $C_A(R_E)$ is commutative. Therefore, since $R_E$ is commutative, $C_A(R_E)$ is maximal commutative in $A$.
\end{proof}
\begin{rem} \label{rem:markov}
We would like to mention a connection with Markov chains.
Assume that $\Gamma=(V,E,s,t)$ is a connected simple quiver.
As in Remark \ref{rem:simple-quiver}, let $Q(\Gamma)$ be the digraph corresponding to $\Gamma$. An edge $e\in E$ from $i$ to $j$ yields two edges in $Q(\Gamma)$, the one from $i$ to $j$ is labeled by $v_{ij}=|\ga_{ei}|$ and the one from $j$ to $i$ is labeled by $v_{ji}=|\ga_{ej}|$.
If there is no edge between $i$ and $j$ in $\Gamma$, we put $v_{ij}=v_{ji}=0$.
Then $d=\sum_{v\in V} d_v v\in\Z V$ is in $\ker \ga$ if and only if $d_iv_{ij}=d_jv_{ji}$ for all $i,j\in V$. That is, $\ker \ga$ is nonzero if and only if the matrix $(v_{ij})_{i,j\in V}$ is \emph{symmetrizable}.
For each $i\in V$, put $Z_i=\sum_{j\in V}v_{ij}$ and define $p_{ij}=v_{ij}/Z_i$. Then one may view $Q(\Gamma)$ as a discrete-time Markov chain with state space $V$ and transition matrix $P=(p_{ij})_{i,j\in V}$. Thus, at each time step, $p_{ij}$ is the probability that the system will jump from state $i$ to state $j$. Let $\{q_i\}_{i\in V}$ denote the stationary distribution.
The system is \emph{reversible} if and only if $q_ip_{ij}=q_jp_{ji}$ for all $i,j$, which holds if and only if $q=\sum_{v\in V}(q_v/Z_v) v$ is in $\ker \ga_\mathbb{Q}$ where $\ga_{\mathbb{Q}}:\mathbb{Q}V\to\mathbb{Q}E$ is the rational extension of $\ga$. Thus, by Theorem \ref{thm:rank-ker-gamma}, the system is reversible if and only if $\Gamma$ is in equilibrium. The latter means exactly that
$p_{i_1i_2}p_{i_2i_3}\cdots p_{i_ki_1}=p_{i_ki_{k-1}}\cdots p_{i_3i_2}p_{i_2i_1}$ for any directed cycle in $Q(\Gamma)$ with vertices $i_1,i_2,\ldots,i_k$, which is known as \emph{Kolmogorov's criterion for detailed balance}.
\end{rem}
We can now prove the main theorem of this section.
\begin{thm}\label{thm:faithfulness}
Let $\Gamma$ be a multiquiver, $\gamma$ be its incidence matrix, $\mathcal{A}(\Gamma)$ the corresponding TGW algebra, and $\varphi_\Gamma$ be the canonical representation by differential operators \eqref{eq:varphi_def}.
Then the following statements are equivalent:
\begin{enumerate}[{\rm (i)}]
\item $\varphi_\Gamma$ is injective;
\item $\gamma$ is injective;
\item $R_E$ is a maximal commutative subalgebra of $\mathcal{A}(\Gamma)$;
\item No connected component of $\Gamma$ is in equilibrium.
\end{enumerate}
\end{thm}
\begin{proof}
(i)$\Rightarrow$(ii):
Assume that $\gamma(d_1)=\gamma(d_2)$ for some $d_1,d_2\in\Z V$. Pick nonzero elements $a_i\in\mathcal{A}(\Gamma)_{d_i}$ and let $b_i=\varphi_\Gamma(a_i)$ for $i=1,2$. From the definition of $\varphi_\Gamma$ we get $\deg (b_1)=\gamma(d_1)=\gamma(d_2)=\deg(b_2)$. Since each homogeneous component of the Weyl algebra $A_E(\K)$ is a free cyclic $R_E$-module there are nonzero $r_1,r_2\in R_E$ such that $r_1b_1=r_2b_2$. Since $\varphi_\Gamma$ is a map of $R_E$-rings we get $\varphi_\Gamma(r_1a_1)=\varphi_\Gamma(r_2a_2)$. If $\varphi_\Gamma$ is injective this implies $r_1a_1=r_2a_2$, which by Corollary \ref{cor:domain} implies $d_1=d_2$.
(ii)$\Rightarrow$(iii): Follows by Lemma \ref{lem:centralizer}.
(iii)$\Rightarrow$(i):
Suppose $R_E$ is a maximal commutative subalgebra of $\mathcal{A}(\Gamma)$.
That means that $C_A(R_E)=R_E$. Since $\varphi_\Gamma$ is a map of $R_E$-rings, $R_E\cap\ker(\varphi_\Gamma)=0$, it follows by
\cite[Thm.~3.6]{HarOin2013}
that $\ker(\varphi_\Gamma)=0$.
(ii)$\Leftrightarrow$(iv): Immediate by Theorem \ref{thm:rank-ker-gamma}.
\end{proof}
\begin{example}
$\varphi_\Gamma$ is faithful in Example \ref{ex:graph3}, but not in Examples \ref{ex:graph1}-\ref{ex:graph2}.
\end{example}
\subsection{Local surjectivity}\label{sec:local-surjectivity}
Let $\Gamma$ be a multiquiver, $\ga$ its incidence matrix and $\varphi_\Gamma$ the representation from Section \ref{sec:homomorphism-varphi}.
We say that $\varphi_\Gamma$ is \emph{locally surjective} if
\begin{equation}
\varphi_\Gamma(\mathcal{A}(\Gamma)_g)=A_E(\K)_{\ga(g)},\qquad\forall g\in \Z V.
\end{equation}
Note that $\gamma$ (hence $\Gamma$) is recoverable from $\varphi_\Gamma$ since $\gamma(v)\in \Z E$ is the degree of $\varphi_\Gamma(X_v)$ for all $v\in V$.
In this section we show that the presence of (not necessarily directed) cycles in $\Gamma$ is precisely the obstruction to $\Phi_\Gamma$ being locally surjective.
\begin{rem}
The notion of local surjectivity is natural in the following sense.
A group-graded algebra $A=\bigoplus_{g\in G} A_g$ can be viewed as a category with object set $G$ and morphism sets $\Hom(g,h)=A_{hg^{-1}}$. In particular, if $\Gamma=(V,E,s,t)$ is a multiquiver, then $\mathcal{A}(\Gamma)$ is $\Z V$-graded and the Weyl algebra $A_E(\K)$ is $\Z E$-graded and we may regard them as categories in this way. The two maps $\gamma\in\Hom_\Z(\Z V, \Z E)$ and $\varphi_\Gamma:\mathcal{A}(\Gamma)\to A_E(\K)$ are naturally the object and morphism components of a functor $\Phi_\Gamma$ from $\mathcal{A}(\Gamma)$ to $A_E(\K)$, viewed as categories. Then $\varphi_\Gamma$ is locally surjective if and only if $\Phi_\Gamma$ is a \emph{full} functor.
\end{rem}
\begin{lem}\label{lem:alpha_lemma}
Let $E,V$ be sets and $\ga=(\ga_{ev})_{e\in E,v\in V}$ be an integer $E\times V$-matrix satisfying
conditions \eqref{eq:condition_C}-\eqref{eq:condition_M}.
Let $g=\sum_{v\in V} g_vv$ and $h=\sum_{v\in V} h_vv$ be elements of $\Z V$ satisfying
\begin{equation}\label{eq:alpha_lemma_1}
\begin{aligned}
g_ih_i &\ge 0,\quad\forall i\in V,\\
g_ih_j &\le 0,\quad\forall i,j\in V, i\neq j.
\end{aligned}
\end{equation}
Then
\begin{equation}
A_E(\K)_{\ga(g)}A_E(\K)_{\ga(h)}=A_E(\K)_{\ga(g+h)}.
\end{equation}
\end{lem}
\begin{proof}
Since $\ga$ satisfies \eqref{eq:condition_M} we have
\begin{equation}\label{eq:pf_alpha_lemma}
\ga_{ei}\ga_{ej}\le 0,\qquad\forall e\in E, i,j\in V, i\neq j.
\end{equation}
Using \eqref{eq:alpha_lemma_1} and \eqref{eq:pf_alpha_lemma} we have
\[\big(\sum_{i\in V} \ga_{ei}g_i\big) \big(\sum_{i\in V} \ga_{ei}h_i\big)\ge 0.
\qquad\forall e\in E,\]
Hence, putting $\ga(d)_e=\sum_{i\in V} \ga_{ei}d_i$ for $d=\sum_{v\in V} d_v v\in\Z V$, we have
\[z_e^{(\ga(g)_e)}z_e^{(\ga(h)_e)}=z_e^{(\ga(g+h)_e)}.\]
Taking the product over $e\in E$ (at most finitely many factors being $\neq 1$ by \eqref{eq:condition_C}) we get
\[z^{\ga(g)}z^{\ga(h)} = z^{\ga(g+h)}.\]
Since $A_E(\K)_p$ is a free cyclic as a left $R_E$-module with generator $z^{p}$ for each $p\in \Z E$, the claim follows.
\end{proof}
Let $\Gamma=(V,E,s,t)$ be a multiquiver. The \emph{underlying undirected graph}, denoted $\bar \Gamma$, is obtained by forgetting the direction of each edge. Formally, $\bar\Gamma$ can be identified with the sign-equivalence class of multiquivers containing $\Gamma$. We will prove a lemma about cycles in $\bar\Gamma$.
Let $L\subseteq E$ be the set of leaves and put $E^\text{prop}=E\setminus L$. Thus $E^\text{prop}=\{e\in E\mid s(e)\neq\emptyset\neq t(e)\}$. The edges in $E^\text{prop}$ will be called \emph{proper}.
To each total order $<$ on $V$ we associate a parity function
\begin{equation}\label{eq:parity-def}
\mathcal{P}_<:E^\text{prop} \to \{-1,1\},\qquad
\mathcal{P}_<(e)=
\begin{cases}
1,&s_1(e)<t_1(e),\\
-1,&t_1(e)<s_1(e),
\end{cases}
\end{equation}
where $s_1(e)=v$ if $s(e)=(v,n)$ for some $n\in\Z_{>0}$ and similarly for $t_1$.
Equivalently,
\begin{equation}\label{eq:parity-alt}
\mathcal{P}_<(e)=
\begin{cases}
1,& \ga_{ev}<0, \ga_{ew}>0 \\
-1, & \ga_{ev}>0, \ga_{ew}<0
\end{cases}
\quad \text{where $\{v,w\}=\{s_1(e),t_1(e)\}$, $v<w$.}
\end{equation}
\begin{lem}\label{lem:acyclic_equiv}
Let $\Gamma=(V,E,s,t)$ be a multiquiver with $V$ finite and $\bar\Gamma$ be the underlying undirected graph. The following statements are equivalent:
\begin{enumerate}[{\rm (i)}]
\item $\bar\Gamma$ is acyclic;
\item For any function $f:E^\text{prop}\to\{-1,1\}$ there exists a total order $<$ on $V$ such that $f=\mathcal{P}_<$.
\end{enumerate}
\end{lem}
\begin{proof}
(i)$\Rightarrow$(ii):
We can assume that $\Gamma$ is connected. Consider $\Gamma$ as a rooted tree by picking any vertex $j_0\in V$ as the root. We think of a total order on $V$ as a visual representation of $\Gamma$ where $v<w$ means $v$ is placed to the left of $w$.
Any desired parity of the edges connected to the root vertex $j_0$ can be obtained by permuting the child vertices to the left or right of $j_0$.
Then continue inductively in this fashion with each child of the root. Since $V$ is finite this process will eventually end.
(ii)$\Rightarrow$(i): Suppose $\Gamma$ has a not necessarily directed cycle. Without loss of generality we can assume this cycle is all of $\Gamma$.
Let $n=\# V$ and denote the elements of $V$ by $v_1,v_2,\ldots,v_n$ such that there is an edge $e_{ij}$ between $v_i$ and $v_j$ if and only if $j\equiv i+1 \text{(mod $n$)}$. Let $<$ be the total order on $V$ given by $v_1<v_2<\cdots <v_n$.
Let $f:E\to \{-1,1\}$ be the function that differs from $\mathcal{P}_<$ only at the edge $e_{n1}$.
By (ii) there is a total order $\prec$ on $V$ such that $f=\mathcal{P}_\prec$.
Then $\mathcal{P}_\prec(e_{i,i+1})=f(e_{i,i+1})=\mathcal{P}_<(e_{i,i+1})$ which implies that $v_i\prec v_{i+1}$ for each $i\in\{1,2,\ldots,n-1\}$. Hence $v_1\prec v_n$ by transitivity, which implies that $f(e_{n1})=\mathcal{P}_\prec(e_{n1})=\mathcal{P}_<(e_{n1})=-f(e_{n1})$ which is a contradiction.
\end{proof}
The following lemma introduces certain useful polynomials $P_{mn}(u)$.
\begin{lem}\label{lem:Pmn}
For all integers $m,n\in \Z$ there exists a unique polynomial $P_{mn}(u)$ such that the following identity holds in the first Weyl algebra $A_1(\K)=\K\langle x,y\mid yx-xy=1\rangle$:
\begin{equation}
z^{(m)}z^{(n)}=P_{mn}(u) z^{(m+n)}
\end{equation}
where $u=yx$ and we use notation \eqref{eq:zk_notation}.
If $mn\ge 0$ then $P_{mn}(u)=1$ (constant polynomial).
If $mn<0$ then $P_{mn}(u)$ is a product of factors of the form $(u-k)$, $k\in\Z$.
If $m>0, n<0$, then all zeroes of $P_{mn}(u)$ are positive integers.
If $m<0, n>0$, then all zeroes of $P_{mn}(u)$ are non-positive integers.
\end{lem}
\begin{proof} Follows by direct calculation using the presentation of $A_1(\K)$ as a generalized Weyl algebra. For example, $x^4y^2=x^3(u-1)y=(u-4)x^3y=(u-4)x^2(u-1)=(u-4)(u-3)x^2$ so $P_{4,-2}(u)=(u-4)(u-3)$.
\end{proof}
Next we calculate some values of $\varphi_\Gamma$ in terms of the polynomials $P_{mn}(u)$.
If $<$ is a total order on a finite set $I$ and $x_i$ for $i\in I$ are not necessarily commuting variables, we put $\prod_{i\in I}^{<} x_i=x_{i_1}x_{i_2}\cdots x_{i_k}$ if $I=\{i_1,i_2,\ldots,i_k\}$ and $i_1<i_2<\cdots<i_k$.
\begin{lem} \label{lem:varphi-calculation}
Let $\Gamma=(V,E,s,t)$ be a multiquiver, $<$ be a total order on $V$, and $V'$ be a finite subset of $V$. Then
\begin{equation}\label{eq:varphi-calculation}
\varphi_\Gamma(\prod_{v\in V'}^{<} X_v) = \Big(\prod_{\substack{v,w\in V'\\ v<w}} \prod_{\substack{e\in E\\ \{s_1(e),t_1(e)\}=\{v,w\}}} P_{\ga_{ev},\ga_{ew}}(u_e) \Big) z^{\ga(g)}
\end{equation}
where $g=\sum_{v\in V'} v\in \Z V$, $s_1(e)=v$ if $s(e)=(v,n)$ for some $n\in\Z_{>0}$ and similarly for $t_1$, $P_{mn}(u)$ are the polynomials from Lemma \ref{lem:Pmn}, and $\ga_{ev}$ are the entries of the incidence matrix $\ga$.
\end{lem}
\begin{proof} By definition of $\varphi_{\Gamma}$ we have
\[
\varphi_\Gamma(\prod_{v\in V'}^{<} X_v) =\prod_{v\in V'}^{<} z^{\ga(v)}
=\prod_{v\in V'}^{<}\prod_{e\in E} z_e^{(\ga_{ev})} \]
Since $[x_e,x_{e'}]=[x_e,y_{e'}]=[y_e,y_{e'}]=0$ for $e,e'\in E$, $e\neq e'$ we may switch the order of the products to get
\begin{equation}\label{eq:varphi-calc-step}
\prod_{e\in E} \prod_{v\in V'}^{<} z_e^{(\ga_{ev})}.
\end{equation}
If $e$ is an edge between two vertices $v,v'$ in $V'$ with $v<v'$ we get
\[\prod_{v\in V'}^{<} z_e^{(\ga_{ev})} = z_e^{(\ga_{ev})}z_e^{(\ga_{ev'})} = P_{\ga_{ev},\ga_{ev'}}(u_e) z^{(\ga_{ev}+\ga_{ev'})}\]
by Lemma \ref{lem:Pmn}. For the other edges there are at most one $v\in V'$ such that $\ga_{ev}\neq 0$.
Thus \eqref{eq:varphi-calc-step} equals
\[\Big( \prod_{\substack{v,w\in V'\\ v<w}}\prod_{\substack{e\in E\\ \{s_1(e),t_1(e)\}=\{v,w\}}}
P_{\ga_{ev},\ga_{ew}}(u_e) \Big)\cdot \prod_{e\in E} z_e^{\big(\sum_{v\in V'}\ga_{ev}\big)} \]
which finishes the proof.
\end{proof}
We now prove the main result in this section.
\begin{thm}\label{thm:local-surjectivity}
Let $\Gamma=(V,E,s,t)$ be a multiquiver with incidence matrix $\gamma$ and let $\mathcal{A}(\Gamma)$ be the corresponding TGW algebra. Let $\varphi_\Gamma:\mathcal{A}(\Gamma)\to A_E(\K)$ be the representation from Theorem \ref{thm:map_to_Weyl_algebra}. Let $\bar\Gamma$ denote the underlying undirected graph of $\Gamma$.
Then the following two statements are equivalent:
\begin{enumerate}[{\rm (i)}]
\item $\varphi_\Gamma$ is locally surjective;
\item $\bar\Gamma$ is acyclic.
\end{enumerate}
\end{thm}
\begin{proof}
(ii)$\Rightarrow$(i): Assume $\bar\Gamma$ has no cycles. Put $\varphi=\varphi_\Gamma$. We have to prove that
\begin{equation}\label{eq:pfb_1}
\varphi\big(\mathcal{A}(\Gamma)_g\big)=A_E(\K)_{\ga(g)},
\end{equation}
for all $g=\sum_{v\in V} g_v v\in\Z V$.
We first reduce to the case when $g_v\ge 0$ for all $v\in V$.
Decompose $g=g_++g_-$ where $g_+=\sum_{v\in V} (g_+)_v v$ $(g_+)_v=\max(0,g_v)$ and $g_-=g-g_+$. By Lemma \ref{lem:monomials},
\begin{equation}\label{eq:pfb_3}
\varphi(\TGWA{\mu}{R}{\si}{t}_g)=\varphi(\TGWA{\mu}{R}{\si}{t}_{g_-})\varphi(\TGWA{\mu}{R}{\si}{t}_{g_+}).
\end{equation}
Assuming \eqref{eq:pfb_1} holds for $g_+$ and $-g_-$ and using that $\varphi$ is a map of rings with involution, \eqref{eq:pfb_3} and Lemma \ref{lem:alpha_lemma} imply that
\begin{equation}\label{eq:pfb_4}
\varphi(\TGWA{\mu}{R}{\si}{t}_g)=A_E(\K)_{\ga(g_-)}A_E(\K)_{\ga(g_+)}=A_E(\K)_{\ga(g)}.
\end{equation}
Thus we may assume that $g_i\ge 0$ for all $i\in V$.
Next we reduce to the case when $V$ is a finite set and $g_i>0$ for each $i\in V$ by the following argument. Let $V'=\{i\in V\mid g_i\neq 0\}$ and $\gamma'=(\ga_{ev})_{e\in E, v\in V'}$. Clearly $V'$ is a finite set. Let $\Gamma'$ be the multiquiver corresponding to $\gamma'$. In other words, $\Gamma'$ is obtained from $\Gamma$ by removing all vertices $v$ for which $g_v=0$ and turning edges at such $v$, if any, into (possibly disconnected) leaves. The multiquiver $\Gamma'$ has no NND cycles either.
Let $\mathcal{A}(\Gamma')$ be the corresponding TGW algebra with homomorphism $\varphi'=\varphi_{\Gamma'}$.
There is a commutative triangle
\begin{equation}
\begin{aligned}
\begin{tikzcd}[ampersand replacement=\&,
column sep=small]
\mathcal{A}(\Gamma')
\arrow{rr}{\psi}
\arrow{dr}[swap]{\varphi'}
\& \& \mathcal{A}(\Gamma)
\arrow{dl}{\varphi} \\
\& A_E(\K) \&
\end{tikzcd}
\end{aligned}
\end{equation}
where $\psi$ is the inclusion map $\psi(X_v)=X_v, \psi(Y_v)=Y_v$ for all $v\in V'$, and $\psi(r)=r$ for all $r\in R_E$.
Note that $g=\sum_{v\in V} g_v v=\sum_{v\in V'} g_v v\in \Z V'\subseteq \Z V$.
We have $\varphi(\mathcal{A}(\Gamma)_g)=(\varphi\psi)(\mathcal{A}(\Gamma')_g)=\varphi'(\mathcal{A}(\Gamma')_g)$.
Thus we may without loss of generality assume that $V$ is finite, and $g_i>0$ for all $i$.
For notational purposes we may assume that $V=\{1,2,\ldots,m\}$.
Finally, we reduce to the case when $g_i=1$ for all $i\in V$. Since $A_E(\K)_{\ga(g)}=R_E z^{\ga(g)}$, relation \eqref{eq:pfb_1} follows if we prove that
\begin{equation}\label{eq:pfb_5}
z^{\ga(g)}\in \sum_{\pi\in S_m} R_E \varphi(X_{\pi(1)}^{g_{\pi(1)}}\cdots X_{\pi(m)}^{g_{\pi(m)}})
\end{equation}
where $S_m$ denotes the symmetric group.
If some $g_i>1$ we may replace $g_i$ by $1$ and the $i$:th column of $\ga$ by
$(g_i\ga_{ei})_{e\in E}$.
The new matrix $\ga_g$ still satisfies condition \eqref{eq:condition_M} and the identity \eqref{eq:pfb_5} for $\ga_g$ will coincide with the original one.
Therefore we may assume that $g_i=1$ for all $i\in V$.
By Lemma \ref{lem:varphi-calculation}, we have
\begin{equation}\label{eq:pfb_7}
\varphi(X_{\pi(1)}\cdots X_{\pi(m)}) =
\prod_{i<j}\prod_{e\in E} P_{\ga_{e\pi(i)},\ga_{e\pi(j)}}(u_e) z^{\ga(g)}.
\end{equation}
If $\pi(i)$ or $\pi(j)$ is not incident to $e$, then $P_{\ga_{e\pi(i)},\ga_{e\pi(j)}}=1$. If $e$ is a proper edge between $\pi(i)$ and $\pi(j)$, $i<j$, then by \eqref{eq:parity-alt} and Lemma \ref{lem:Pmn},
\[
\mathcal{V}(P_{\ga_{e\pi(i)},\ga_{e\pi(j)}})\subseteq
\begin{cases}
\Z_{\le 0},&\mathcal{P}_\pi(e) = 1 \\
\Z_{>0},&\mathcal{P}_\pi(e) = -1
\end{cases}
\]
where $\mathcal{V}(P)$ denote the set of zeroes in the algebraic closure $\bar\K$ of a polynomial $P\in \K[u]$, and we identified $\pi\in S_m$ with the total order $\prec$ on $V$ given by $\pi(i)\prec \pi(j)$ for all $i<j$.
Therefore, by (ii) and Lemma \ref{lem:acyclic_equiv},
the polynomials $\big\{\prod_{i<j}\prod_{e\in E} P_{\ga_{e\pi(i)},\ga_{e\pi(j)}}(u_e)\big\}_{\pi\in S_m}$
have no common zeroes in $\bar\K^E$.
By the weak Nullstellensatz, the ideal these polynomials generate in $R_E$ contains $1$. Hence
\[z^{\ga(g)}\in \sum_{\pi\in S_m} R_E \varphi_{\Gamma}(X_{\pi(1)}\cdots X_{\pi(m)}).\]
This finishes the proof that $\varphi_\Gamma$ is locally surjective.
(i)$\Rightarrow$(ii):
Assume that $\Gamma$ has a not necessarily directed cycle $\Gamma'=(V',E',s',t')$.
We will prove that $\varphi_\Gamma$ is not locally surjective by showing that
for $g=\sum_{v\in V'} v$ we have
\[\varphi_\Gamma(\mathcal{A}(\Gamma)_g)\subseteq J \cdot A_E(\K)_{\ga(g)}\]
where $J$ is a proper ideal of $R_E$.
By Lemma \ref{lem:monomials},
\[\mathcal{A}(\Gamma)_g=\sum_{\prec} R_E \cdot \prod_{v\in V'}^\prec X_v \]
where we sum over all total orders $\prec$ on $V'$.
Thus, by Lemma \ref{lem:varphi-calculation},
\[\varphi_\Gamma(\mathcal{A}(\Gamma)_g)=\sum_{\prec} R_E\cdot
\Big( \prod_{\substack{v,w\in V'\\ v<w}}\prod_{\substack{e\in E\\ \{s_1(e),t_1(e)\}=\{v,w\}}}
P_{\ga_{ev},\ga_{ew}}(u_e)\Big) z^{\ga(g)}\]
Since $P_{\ga_{ev},\ga_{ew}}=1$ unless $v$ and $w$ are both incident to $E$, we may without loss of generality assume that $\Gamma'=\Gamma$. In particular every edge is proper, and $V$ and $E$ are finite.
We have
\begin{equation}\label{eq:varphi-proof-identity}
\prod_{\substack{v,w\in V\\ v<w}}\prod_{\substack{e\in E\\ \{s_1(e),t_1(e)\}=\{v,w\}}}
P_{\ga_{ev},\ga_{ew}}(u_e) =
\prod_{\substack{e\in E\\ \mathcal{P}_<(e)=1}} P_e^+
\prod_{\substack{e\in E\\ \mathcal{P}_<(e)=-1}} P_e^-
\end{equation}
where
\[P_e^+ = P_{\ga_{e,s_1(e)},\ga_{e,t_1(e)}}(u_e),\qquad
P_e^- = P_{\ga_{e,t_1(e)},\ga_{e,s_1(e)}}(u_e). \]
By Lemma \ref{lem:acyclic_equiv}, there exists a function $f:E\to \{-1,1\}$ such that $f\neq \mathcal{P}_\prec$ for all total orders $\prec$ on $V$. This means that for any total order $\prec$ on $V$, there exists $e_{\prec}\in E$ such that $\mathcal{P}_\prec (e_\prec ) = -f(e_{\prec})$.
Define
\begin{equation}\label{eq:ideal-J}
J=\sum_{\prec} R_E\cdot P_\prec
\end{equation}
where
\[
P_\prec =
\begin{cases}
P_{e_\prec }^+, & \text{if $\mathcal{P}_\prec (e_\prec ) = 1 $} \\
P_{e_\prec }^-, & \text{if $\mathcal{P}_\prec (e_\prec ) = - 1$}
\end{cases}
\]
We claim that $J$ is a proper ideal of $R_E$.
Assume that $\prec_1$ and $\prec_2$ are total orders on $V$ such that $e_{\prec_1}=e_{\prec_2}=e$.
Then $\mathcal{P}_{\prec_1}(e)=-f(e)=\mathcal{P}_{\prec_2}(e)$ and hence $P_{\prec_1}=P_{\prec_2}$.
Therefore $J$ is proper.
\end{proof}
\begin{example}
$\varphi_\Gamma$ is locally surjective in
Examples (6)-(7) of Section \ref{sec:examples}, but not in Examples \ref{ex:graph1}-\ref{ex:graph3}.
\end{example}
\begin{example} We illustrate the proof of the theorem by considering a more detailed example.
Let $V=\{1,2,3\}$, $E=\{a,b,c\}$ and let $\Gamma=(V,E,s,t)$ be the following multiquiver:
\[
\Gamma:\;\;\begin{tikzpicture}[baseline=(60:1)]
\node[vertex] (A) at (0,0) [label=below left:$1$] {};
\node[vertex] (B) at (60:2) [label=above:$2$] {};
\node[vertex] (C) at (0:2) [label=below right:$3$] {};
\draw[edge] (A) -- node[left,very near start] {$2$} node[left] {$a$} node[left,very near end] {$3$} (B);
\draw[edge] (C) -- node[right,very near start] {$2$} node[right] {$b$} node[right,very near end] {$1$} (B);
\draw[edge] (A) -- node[below,very near start] {$1$} node[below] {$c$} node[below,very near end] {$1$} (C);
\end{tikzpicture}
\qquad
\ga=\begin{bmatrix}
-2 & 3 & 0 \\
0 & 1 &-2 \\
-1 & 0 & 1
\end{bmatrix}
\]
Let $\mathcal{A}(\Gamma)$ be the TGW algebra associated to $\Gamma$, and $\varphi_\Gamma:\mathcal{A}(\Gamma)\to A_E(\K)$ be the natural representation by differential operators.
Since $\bar\Gamma$ has a cycle, we expect $\varphi_\Gamma$ to not be locally surjective. We check it using the methods and notation of the proof of Theorem \ref{thm:local-surjectivity}.
By definition of $\varphi_\Gamma$ and relations in the Weyl algebra $A_E(\K)$ we have
\begin{align*}
\varphi_\Gamma(X_1X_2X_3)&=y_a^2y_c\cdot x_a^3x_b\cdot y_b^2x_c \\
&= y_a^2x_a^3\cdot x_by_b^2\cdot y_cx_c \\
&= u_a(u_a+1)(u_b-1)u_c\cdot x_ay_b
\end{align*}
Recall the definition \eqref{eq:parity-def} of the parity function
$\mathcal{P}_{\prec}$ attached to a total order $\prec$ on $V$.
Below we abbreviate the total order $i\prec j\prec k$ by $ijk$ and write
$\mathcal{P}_{ijk}(a,b,c)$ for $\big(\mathcal{P}_{ijk}(a),\mathcal{P}_{ijk}(b),\mathcal{P}_{ijk}(c)\big)$. Similar calculations to the above give
\begin{align*}
\varphi_\Gamma(X_1X_2X_3)&=u_a(u_a+1)\cdot (u_b-1)\cdot u_c \cdot x_ay_b,
& \mathcal{P}_{123}(a,b,c)&=(1,-1,1), \\
\varphi_\Gamma(X_1X_3X_2)&=u_a(u_a+1)\cdot (u_b+1)\cdot u_c\cdot x_ay_b,
& \mathcal{P}_{132}(a,b,c)&=(1,1,1), \\
\varphi_\Gamma(X_2X_1X_3)&=(u_a-2)(u_a-3)\cdot (u_b-1)\cdot u_c\cdot x_ay_b,
& \mathcal{P}_{213}(a,b,c)&=(-1,-1,1), \\
\varphi_\Gamma(X_2X_3X_1)&=(u_a-2)(u_a-3)\cdot (u_b-1)\cdot (u_c-1)\cdot x_ay_b,
& \mathcal{P}_{231}(a,b,c)&=(-1,-1,-1), \\
\varphi_\Gamma(X_3X_1X_2)&=u_a(u_a+1)\cdot (u_b+1)\cdot (u_c-1)\cdot x_ay_b,
& \mathcal{P}_{312}(a,b,c)&=(1,1,-1), \\
\varphi_\Gamma(X_3X_2X_1)&=(u_a-2)(u_a-3)\cdot (u_b+1)\cdot (u_c-1) \cdot x_ay_b,
& \mathcal{P}_{321}(a,b,c)&=(-1,1,-1). \\
\end{align*}
These identities agree with Lemma \ref{lem:varphi-calculation}.
For example, consider the total order $2\prec 1\prec 3$ on $V$.
The right hand side of \eqref{eq:varphi-calculation} is
\[P_{\ga_{a2},\ga_{a1}}(u_a)\cdot P_{\ga_{b2},\ga_{b3}}(u_b)\cdot
P_{\ga_{c1},\ga_{c3}}(u_c)\cdot z^{\ga(g)} \]
Identifying $\Z V$ with $\Z^3$ we have $g=(1,1,1)$ and thus $\ga(g)=a-b\in\Z E$ so $z^{\ga(g)}=x_ay_b$. Substituting the matrix elements of $\ga$ we obtain
\[P_{3,-2}(u_a)\cdot P_{1,-2}(u_b)\cdot P_{-1,1}(u_c)\cdot x_ay_b=
(u_a-2)(u_a-3)\cdot (u_b-1)\cdot u_c\cdot x_ay_b.\]
This also agrees with \eqref{eq:varphi-proof-identity}.
Let $f:E\to \{-1,1\}$ be given by $(f(a),f(b),f(c))=(1,-1,-1)$. One verifies that for any total order $\prec$ on $V$, $f\neq\mathcal{P}_\prec$. For any order $\prec$ let
\[e_\prec=
\begin{cases}
a& \mathcal{P}_\prec(a)=-1 \\
b& \mathcal{P}_\prec(a)=1, \mathcal{P}_\prec(b)=1, \\
c& \text{otherwise}.
\end{cases}
\]
Then $\mathcal{P}_\prec(e_\prec)=-f(e_{\prec})$ for any total order $\prec$ on $V$. The ideal $J\subseteq R_E$ from \eqref{eq:ideal-J} is given by
\[J=R_E(u_a-2)(u_a-3) + R_E(u_b+1) + R_Eu_c\]
which is a proper ideal of $R_E$, and
\[\varphi_\Gamma(X_iX_jX_k)\subseteq J \cdot x_ay_b\]
for $\{i,j,k\}=\{1,2,3\}$. This shows that $\varphi_\Gamma$ is not locally surjective.
\end{example}
\subsection{Universality of the pair \texorpdfstring{$(\mathcal{A}(\Gamma), \varphi_\Gamma)$}{(A(Gamma),phiGamma)}}
In this section we show that the algebras $\mathcal{A}(\Gamma)$ are universal among all TGW algebras (with polynomial base ring) that can be represented by differential operators.
\begin{thm}\label{thm:main}
Let $\mathcal{A}=\TGWA{\mu}{R}{\si}{t}$ be any TGW algebra with index set denoted $V$, such that $R$ is a polynomial algebra $R=R_E=\K[u_e\mid e\in E]$ (for some index set $E$),
and $\mu$ is symmetric. Assume that
\begin{equation}
\varphi:\mathcal{A}\to A_E(\K)
\end{equation}
is a map of $R_E$-rings with involution, where $A_E(\K)$ is the Weyl algebra over $\K$ with index set $E$.
Then $\mathcal{A}$ is consistent and
there exists a multiquiver $\Gamma=(V,E,s,t)$ with vertex set $V$ and edge set $E$ and a map
\[\xi:\mathcal{A}\to\mathcal{A}(\Gamma)\]
of $\Z V$-graded $R_E$-rings with involution
such that the following diagram commutes:
\begin{equation}\label{eq:univ-diagram}
\begin{aligned}
\begin{tikzcd}[ampersand replacement=\&,
column sep=small]
\mathcal{A}
\arrow{rr}{\varphi}
\arrow{d}[swap]{\xi}
\& \& A_E(\K) \\
\mathcal{A}(\Gamma)
\arrow{rru}[swap]{\varphi_{\Gamma}}
\& \&
\end{tikzcd}
\end{aligned}
\end{equation}
Moreover, if $\varphi(X_i)\neq 0$ for each $i\in V$, then $\Gamma$ is uniquely determined and $\mu_{ij}=1$ for all $i,j\in V$.
\end{thm}
\begin{proof}
(a) Since $\varphi$ is a map of $R_E$-rings, $\mathcal{A}$ is consistent.
We will define the graph $\Gamma$ by specifying its incidence matrix
$\gamma=(\gamma_{ev})_{e\in E, v\in V}$.
Let $v\in V$. If $\varphi(X_v)=0$ (or equivalently $\varphi(Y_v)=0$, since $\varphi$ is a map of rings with involution) then we define
$\gamma_{ev}=0$ for all $e\in E$.
Assume $\varphi(X_v)\neq 0$.
For each $i\in V$ we have
\begin{equation}\label{eq:pf_alpha1}
\varphi(X_v)=\sum_{g\in \Z E} z^{(g)}s_{v,g},\quad\forall i\in\iv{1}{m},
\end{equation}
for some $s_{v,g}\in R_E$, only finitely many nonzero and $s_{v,g}\neq 0$ for at least one $g$.
Let $e\in E$ and act by $\ad u_e$ on both sides of \eqref{eq:pf_alpha1} to get
\begin{equation}\label{eq:pf_alpha2}
(u_e-\si_v(u_e))\varphi(X_v)=\sum_{g\in\Z E} g_ez^{(g)}s_{v,g},\quad\forall v\in V,\forall e\in E.
\end{equation}
Multiplying both sides of \eqref{eq:pf_alpha1} by $u_e-\si_v(u_e)$ from the left and subtracting the resulting equation from \eqref{eq:pf_alpha2}, and using the $\Z E$-gradation on $A_E(\K)$, we get
\begin{equation}
\big(g_e-u_e+\si_v(u_e)\big)z^{(g)}s_{v,g}=0,\quad\forall g\in\Z E,\forall v\in V,\forall e\in E.
\end{equation}
So if $s_{v,g}\neq 0$ for some $g\in\Z E$, then, since $A_E(\K)$ is a domain,
$g_e-u_e+\si_v(u_e)=0 \forall e\in E$.
This shows that there exists exactly one $\gamma_v\in\Z E$ such that $s_{v,\gamma_v}\neq 0$. Namely, $\ga_v=\sum_{e\in E} \ga_{ev}e$ where
\begin{equation}\label{eq:pf_alpha3}
\si_v(u_e)=u_e-\ga_{ev},\quad \forall v\in V,\forall e\in E.
\end{equation}
This defines a matrix $\ga=(\ga_{ev})_{e\in E, v\in V}$ with integer entries. From the above it follows that the column $(\ga_{ev})_{e\in E}$ is uniquely determined if $\varphi(X_v)\neq 0$. Since $\ga_v\in \Z E$, condition \eqref{eq:condition_C} holds.
Next we show that $\ga$ satisfies condition \eqref{eq:condition_M}.
For $i\in V$, put $s_i=s_{i,\ga_i}$ if $\varphi(X_i)\neq 0$ and $s_i=0$ otherwise. We have shown that
\begin{equation}
\varphi(X_i)= z^{(\ga_i)} s_i,\quad \varphi(Y_i)=\varphi(X_i)^\ast=s_iz^{(-\ga_i)}.
\end{equation}
Let $i,j\in V$, $i\neq j$, and assume $s_i,s_j\neq 0$. Applying $\varphi$ to the relation $X_iY_j=\mu_{ij}Y_jX_i$ we obtain
\[z^{(\ga_i)}s_is_jz^{(-\ga_j)}=\mu_{ij}s_jz^{(-\ga_j)}z^{(\ga_i)}s_i.\]
Using that $\varphi$ is a map of $R_E$-rings we get
\begin{equation}\label{eq:univ-eq1}
\si_i(s_is_j)z^{(\ga_i)}z^{(-\ga_j)}=\mu_{ij}s_j\si_j^{-1}\si_i(s_i)z^{(-\ga_j)}z^{(\ga_i)}.
\end{equation}
We have
\begin{equation}
z^{(\ga_i)}z^{(-\ga_j)}=P_{ij} z^{(\ga_i-\ga_j)}
\end{equation}
and
\begin{equation}
z^{(-\ga_j)}z^{(\ga_i)}=Q_{ij} z^{(\ga_i-\ga_j)}
\end{equation}
where $P_{ij}=\prod_{e\in E} P_{e;ij}$ and $Q_{ij}=\prod_{e\in E} Q_{e;ij}$ where $P_{e;ij}, Q_{e;ij}\in \K[u_e]$.
Substituting into \eqref{eq:univ-eq1}, canceling $z^{(\ga_i-\ga_j)}$ we obtain:
\begin{equation}\label{eq:univ-eq2}
s_i's_j' P_{ij} = \mu_{ij} \si_i^{-1}(s_j')\si_j^{-1}(s_i') Q_{ij}
\end{equation}
where $s_i'=\si_i(s_i)$ and $s_j'=\si_i(s_j)$.
Suppose for the sake of contradiction that $\gamma$ does not satisfy condition
\eqref{eq:condition_M}. Then there exist $e\in E$, $i,j\in V$, $i\neq j$, such that $\ga_{ei}\ga_{ej}>0$. We consider the case when $\ga_{ei}$ and $\ga_{ej}$ are both negative, the case when both are positive being treated analogously.
Since $z_e^{(\ga_{ei})}z_e^{(-\ga_{ej})}=P_{e;ij}z_e^{(\ga_{ei}-\ga_{ej})}$
and
$z_e^{(-\ga_{ej})}z_e^{(\ga_{ei})}=Q_{e;ij}z_e^{(\ga_{ei}-\ga_{ej})}$,
Lemma \ref{lem:Pmn} then implies that
\begin{equation}\label{eq:univ-key1}
\text{All roots of $P_{e;ij}$ are non-positive and all roots of $Q_{e;ij}$ are positive.}
\end{equation}
Let $S$ be the set of integers $k$ satisfying the following two conditions:
\begin{gather}
\label{eq:univ-div1}
(u_e-k)\big| s_i's_j'P_{ij},\\
\label{eq:univ-div2}
(u_e-k)\not\big| P_{ij}.
\end{gather}
Clearly $S$ is finite, due to \eqref{eq:univ-div1}. The set $S$ is nonempty, since it contains all roots of $Q_{e;ij}$, by \eqref{eq:univ-eq2} and \eqref{eq:univ-key1}.
Let $k_0$ denote the largest element of $S$. In particular $k_0>0$.
From \eqref{eq:univ-div1}-\eqref{eq:univ-div2}, $u_e-k_0$ divides $s_i'$ or $s_j'$. Applying $\si_j^{-1}$ respectively $\si_i^{-1}$, using \eqref{eq:pf_alpha3}, we obtain that
\[(u_e+\ga_{ej}-k_0)\big| \si_j^{-1}(s_i)
\quad\text{or}\quad
(u_e+\ga_{ei}-k_0)\big| \si_i^{-1}(s_j).\]
By \eqref{eq:univ-eq2} we get
\[
(u_e+\ga_{ea}-k_0)\big| s_is_jP'_{ij}
\]
for some $a\in \{i,j\}$.
By \eqref{eq:univ-key1}, $-\ga_{ea}+k_0$ is not a root of $P_{e;ij}$. Therefore $-\ga_{ea}+k_0\in S$ which contradicts the maximality of $k_0$.
This contradiction shows that $\gamma$ satisfies condition \eqref{eq:condition_M}.
In particular, using this together with \eqref{eq:univ-eq1} we have
\begin{equation}
\label{eq:univ-eq3}
[z^{(\ga_i)},z^{(-\ga_j)}]=0\quad\forall i,j\in V, i\neq j.
\end{equation}
By \eqref{eq:univ-eq3} and \eqref{eq:univ-eq1} we get
\begin{equation}\label{eq:univ-eq4}
\si_i(s_is_j)=\mu_{ij}s_j\si_j^{-1}\si_i(s_i)
\qquad\forall i,j\in V, i\neq j.
\end{equation}
Giving $E$ any order, the polynomial ring $R_E$ becomes filtered and the automorphisms $\si_i$ act as the identity on the associated graded algebra. Therefore, taking leading terms on both sides of \eqref{eq:univ-eq4} we get that
\begin{equation}\label{eq:univ-eq5}
\mu_{ij}=1\quad\text{if $s_is_j\neq 0$.}
\end{equation}
Next we construct the map $\xi:\mathcal{A}\to\mathcal{A}(\Gamma)$. Let $\mathcal{X}=\{X_v\mid v\in V\}\cup \{Y_v\mid v\in V\}$ and define $\xi:\mathcal{X}\to\mathcal{A}(\Gamma)$ by
\begin{equation}\label{eq:xi-def}
\xi(X_v)=X_v^\Gamma s_v,\quad \xi(Y_v)=s_v Y_v^\Gamma,\qquad \forall v\in V,
\end{equation}
where $X_v^\Gamma$ and $Y_v^\Gamma$ denote the generators of $\mathcal{A}(\Gamma)$.
Extend $\xi$ uniquely to a map $\xi:F_{R_E}(\mathcal{X})\to\mathcal{A}(\Gamma)$ where $F_{R_E}(\mathcal{X})$ is the free $R_E$-ring on the set $\mathcal{X}$. We must verify that the elements \eqref{eq:tgwarels} are in the kernel of $\xi$. By \eqref{eq:pf_alpha3} and \eqref{eq:sigma-definition} one checks that
\[
\xi(X_v u_e-\si_v(u_e)X_v)=
X_v^\ga s_v u_e- (u_e-\ga_{ev}) X_v^\Gamma s_v=0
,\quad\forall v\in V,\;\forall e\in E,\\
\]
and thus
\begin{equation}
\label{eq:univ-check1}
\xi(X_v r-\si_v(r)X_v)=0,
\qquad\forall v\in V,\;\forall r\in R_E.
\end{equation}
Similarly one verifies that
\begin{equation}
\label{eq:univ-check1b}
\xi(Y_v r-\si_v^{-1}(r)Y_v)=0,
\qquad\forall v\in V,\;\forall r\in R_E.
\end{equation}
Let $i,j\in V$, $i\neq j$. Denoting the generators of $\mathcal{A}(\Gamma)$ by $X_i^\Gamma, Y_i^\ga$ we have
\begin{equation}
\label{eq:univ-check2}
[\xi(X_i),\xi(Y_j)]=[X_i^\Gamma s_i,s_jY_j^\Gamma]=
(\si_i(s_is_j)-s_j\si_j^{-1}\si_i(s_i))X_i^\Gamma Y_j^\Gamma
\end{equation}
which is zero by \eqref{eq:univ-eq4} and \eqref{eq:univ-eq5}.
For $i\in V$, put $t_i^\Gamma=Y_i^\Gamma X_i^\Gamma\in R_E$. Note $t_i$ and $t_i^\Gamma$ are related as follows:
\begin{align*}
t_i&=\varphi(t_i)=\varphi(Y_iX_i)=\varphi(Y_i)\varphi(X_i)=s_iz^{(-\ga_i)}z^{(\ga_i)}s_i=\\
&=s_i\varphi_\Gamma(Y_i^\Gamma)\varphi_\Gamma(X_i^\Gamma)s_i=s_i^2\varphi_\Gamma(t_i^\Gamma)=s_i^2 t_i^\Gamma.
\end{align*}
Therefore
\begin{equation}
\label{eq:univ-check3}
\xi(Y_iX_i-t_i)=\xi(Y_i)\xi(X_i)-\xi(t_i)=s_iY_i^\Gamma X_i^\Gamma s_i - t_i = s_i^2 t_i^\Gamma-t_i=0,
\qquad\forall i\in V.
\end{equation}
Similarly one checks that
\begin{equation}
\label{eq:univ-check4}
\xi(X_iY_i-\si(t_i))=0,\qquad\forall i\in V.
\end{equation}
This shows that $\xi$ descends to a map of $R_E$-rings with involution $\xi:\TGWC{\mu}{R_E}{\si}{t}\to \mathcal{A}(\Gamma)$. As in the last part of the proof of Theorem \ref{thm:map_to_Weyl_algebra}, we may use the gradation form on $\TGWC{\mu}{R_E}{\si}{t}$ and that $\mathcal{A}(\Gamma)$ is domain (by Corollary \ref{cor:domain}) to conclude that $\xi$ descends to a map of $R_E$-rings with involution $\xi:\mathcal{A}\to\mathcal{A}(\Gamma)$. From the definition \eqref{eq:xi-def} of $\xi$ it follows that $\xi$ is a map of $\Z V$-graded algebras and that the diagram \eqref{eq:univ-diagram} is commutative.
\end{proof}
\section{Relation to enveloping algebras}\label{sec:relation-to-enveloping-algebras}
In this section we assume that $\K$ is an algebraically close field of characteristic zero.
\subsection{Primitivity of \texorpdfstring{$\mathcal{A}(\Gamma)$}{A(Gamma)}}
\label{sec:primitivity}
We give a sufficient condition for $\mathcal{A}(\Gamma)$ to be primitive.
\begin{thm}\label{thm:primitivity}
Let $\Gamma$ be a multiquiver such that no connected component of $\Gamma$ is in equilibrium.
Then $\mathcal{A}(\Gamma)$ is a primitive ring.
\end{thm}
\begin{proof}
By Theorem \ref{thm:faithfulness}, the incidence matrix $\ga$ is injective.
Let $E$ and $V$ be the edge and vertex set of $\Gamma$ respectively. Recall the definition $\mathcal{A}(R_E,\si^\Gamma,t^\Gamma)$ of $\mathcal{A}(\Gamma)$ as a TGW algebra.
Consider the maximal ideal $\mathfrak{m}_{1/2}=(u_e-1/2\mid e\in E)$
of $R_E$. For $d\in\Z V$ we have $\si_d(\mathfrak{m}_{1/2})=(u_e-\ga(d)_e-1/2\mid e\in E)$. Thus $\si_d(\mathfrak{m}_{1/2})=\mathfrak{m}_{1/2} \Leftrightarrow \ga(d)=0 \Leftrightarrow d=0$ since $\ga$ is injective. Therefore, by \cite[Prop.~7.2]{MazPonTur2003}, the induced module $\mathcal{A}(\Gamma)\otimes_{R_E} R_E/\mathfrak{m}_{1/2}$ has a unique simple quotient $M$ with the weight space $M_{\mathfrak{m}_{1/2}}\neq 0$. Since $t_i$ is invertible modulo $\mathfrak{m}_{1/2}$, $X_i$ and $Y_i$ act injectivly on $M$ for all $i\in V$, and thus the support of $M$ is
$\Supp(M)=\{\si_d(\mathfrak{m}_{1/2})\mid d\in \Z V\}\simeq \Z V$.
Therefore $\Ann_{R_E}M=\bigcap_{\mathfrak{m}\in\Supp(M)} \mathfrak{m}=\{0\}$.
Futhermore, $M$ is a graded $\mathcal{A}(\Gamma)$-module via $M=\bigoplus_{d\in \Z V} M_{\si_d(\mathfrak{m}_{1/2})}$. Hence $I:=\Ann_{\mathcal{A}(\Gamma)}M$ is a graded ideal of $\mathcal{A}(\Gamma)$. If $I\neq\{0\}$, then $I\cap R_E\neq\{0\}$ since $\mathcal{A}(\Gamma)$ is a TGW algebra. But $I\cap R_E=\Ann_{R_E}M=\{0\}$ as we just saw. This proves that $I=0$ and thus $M$ is a faithful simple $\mathcal{A}(\Gamma)$-module. Hence $\mathcal{A}(\Gamma)$ is primitive.
\end{proof}
\subsection{The generalized Cartan matrix of \texorpdfstring{$\mathcal{A}(\Gamma)$}{A(Gamma)}}
\label{sec:theGCM}
In \cite{Hartwig2010} it was found that one can associate generalized Cartan matrices to TGW data. These matrices do not completely determine the TGW algebra (as they do for Kac-Moody algebras) but they can be used to describe certain relations in the algebra.
Let $(R,\si,t)$ be a TGW datum with index set $I$. For $i,j\in I$, put
\begin{align}
V_{ij}&=\text{span}_\K \{\si_i^k(t_j)\mid k\in\Z\}, \\
p_{ij}(x) &=\text{minimal polynomial for $\si_i$ acting on $V_{ij}$ (if $\dim_\K V_{ij}<\infty$)},\\
a_{ij} &=
\begin{cases}
2,& i=j\\
1-\dim_\K V_{ij}, & i\neq j
\end{cases} \\
C&=(a_{ij})_{i,j\in I}
\end{align}
If $(R,\si,t)$ is \emph{regular} (i.e. $t_i$ is not a zero-divisor in $R$ for each $i\in I$) and \emph{$\K$-finitistic} (i.e. $V_{ij}$ is finite-dimensional for all $i,j\in I$) then $C$ is a generalized Cartan matrix, by \cite[Thm.~4.4]{Hartwig2010}. Moreover, the (not necessarily commuting) generators $X_i, X_j$ (respectively $Y_i,Y_j$) satisfy certain homogenous Serre-type relations expressed using the coefficients of the polynomials $p_{ij}(x)$.
In the following lemma we compute the polynomials $p_{ij}(x)$ and integers $a_{ij}$ explicitly for the TGW algebras $\mathcal{A}(\Gamma)$ associated to a multiquiver $\Gamma$.
\begin{thm}\label{thm:GCM}
Let $\Gamma=(V,E,s,t)$ be a multiquiver, $\ga=(\ga_{ei})_{e\in E, i\in V}$ be its incidence matrix, and $(R_E,\si^\Gamma,t^\Gamma)$ the corresponding TGW datum. Then $(R_E,\si^\Gamma,t^\Gamma)$ is regular and $\K$-finitistic. Moreover, for
$i,j\in V$ with $i\neq j$ we have
\begin{align}
\label{eq:pij_from_gamma}
p_{ij}(x) &= (x-1)^{1-a_{ij}}, \\
\label{eq:GCM_from_gamma}
a_{ij} &= -\sum_{\substack{e\in E \\ \ga_{ei}\neq 0}} |\ga_{ej}|.
\end{align}
\end{thm}
\begin{proof} Fix $i,j\in V$, $i\neq j$. Put
\[ N_{ij}=\sum_{\substack{e\in E\\ \ga_{ei}\neq 0}} |\ga_{ej}|. \]
Consider the difference operator
\[ D_i=\si_i-\Id \]
and let $m$ be a non-negative integer. By definition of $t_j$ we have
\begin{equation}\label{eq:Dynkin-lem-pf0a}
D_i^{m}(t_j) = D_i^m\Big(\prod_{e\in E} u_{ej}\Big).
\end{equation}
Since $u_{ej}$ is a monic polynomial in $u_e$, $D_i(u_e a)=u_e D_i(a)$ if $\ga_{ei}=0$, and $u_{ej}=1$ if $\ga_{ej}=0$, we have
\begin{equation}\label{eq:Dynkin-lem-pf0b}
D_i^m\Big(\prod_{e\in E} u_{ej}\Big) =
\Big(\prod_{e\in E\setminus E_{ij}} u_{ej}\Big)\cdot D_i^m\Big(\prod_{e\in E_{ij}} u_{ej}\Big)
\end{equation}
where $E_{ij}=\big\{e\in E\mid \ga_{ei}\neq 0\neq \ga_{ej}\big\}$ is the set of edges between the vertices $i$ and $j$. Using the twisted Leibniz rule
\begin{equation}\label{eq:Di-Leibniz}
D_i(ab)=D_i(a)\si_i(b)+aD_i(b),\quad\forall a,b\in R_E,
\end{equation}
the second factor in \eqref{eq:Dynkin-lem-pf0b} can be computed as the sum over all possible ways of distributing the $m$ difference operators $D_i$ among the factors $u_{ej}$, $e\in E_{ij}$ (and in addition applying $\si_i$ to some factors).
If $m=1+N_{ij}$ then, by the pigeon hole principle, any such term will contain at least one factor of the form $D_i^n(u_{ej})$ where $n\ge 1+|\ga_{ej}|$. Since $u_{ej}$ is a polynomial in $u_e$ of degree $|\ga_{ej}|$, \eqref{eq:Di-Leibniz} and $D_i(u_e)=-\ga_{ei}$ imply that $D_i^n(u_{ej})=0$. This shows that
\begin{equation}
D_i^{1+N_{ij}}(t_j)=0.
\end{equation}
Since $D_i$ and $\si_i$ commute, we get $D_i^{1+N_{ij}}\big|_{V_{ij}}=0$. Hence the minimal polynomial $p_{ij}(x)$ for $\si_i\big|_{V_{ij}}$ divides $(x-1)^{1+N_{ij}}$.
It remains to be shown that $D_i^{N_{ij}}(t_j)\neq 0$.
By the above argument, $D_i^m(p)=0$ if $p$ is a polynomial in $u_e$ of degree less than $m$. Hence we may replace $u_{ej}$ by its leading term when applying $D_i^{|\ga_{ej}|}$:
\begin{equation}\label{eq:Dynkin-lem-pf0c}
D_i^{|\ga_{ej}|}(u_{ej})=D_i^{|\ga_{ej}|}(u_e^{|\ga_{ej}|}).
\end{equation}
By iterating \eqref{eq:Di-Leibniz} we get
\[
D_i(u_e^m)=m u_e^{m-1}\cdot (-\ga_{ei})+\text{lower terms},\quad\forall e\in E.
\]
Thus
\begin{equation} \label{eq:Dynkin-lem-pf0d}
D_i^m(u_e^m)=m!\cdot(-\ga_{ei})^m.
\end{equation}
Combining \eqref{eq:Dynkin-lem-pf0a}-\eqref{eq:Dynkin-lem-pf0b} and \eqref{eq:Dynkin-lem-pf0c}-\eqref{eq:Dynkin-lem-pf0d} with the fact that there is at most one non-zero way to distribute $N_{ij}$ difference operators $D_i$ among the factors $u_{ej}$, $e\in E_{ij}$, we get
\begin{equation}
D_i^{N_{ij}}(t_j)=
\prod_{e\in E\setminus E_{ij}} u_{ej} \cdot
\prod_{e\in E_{ij}} D_i^{|\ga_{ej}|}(u_{ej}) \\
= \prod_{e\in E\setminus E_{ij}} u_{ej} \cdot
\prod_{e\in E_{ij}} |\ga_{ej}|!\cdot (-\ga_{ei})^{|\ga_{ej}|} \neq 0.
\end{equation}
This proves that $(\si_i-\Id)^m(t_j)\neq 0$ for any $m\le N_{ij}$. Hence $p_{ij}(x)=(x-1)^{1+N_{ij}}$ which finishes the proof.
\end{proof}
Theorem \ref{thm:GCM} leads us to the following procedure.
Let $\Gamma$ be a multiquiver. One can directly associate to $\Gamma$ a Dynkin diagram $D(\Gamma)$ by applying the following steps:
\begin{enumerate}[{\rm (i)}]
\item Remove all leaves;
\item Forget the direction of each edge;
\item For each pair of adjacent vertices $i$ and $j$, replace the collection of edges between them by a single edge, adding multiplicities in the process:
\[
\begin{tikzpicture}[baseline=(X.base)]
\node (X) at (0,0) {};
\node[vertex] (1) at (0,0) [label=left:$i$] {};
\node[vertex] (2) at (2,0) [label=right:$j$] {};
\draw[thick] (1) to[out=60,in=120] node[above,very near start] {$a_1$} node[above,very near end] {$b_1$} (2);
\draw[thick] (1) to[out=30,in=150]
node[below,near start] {$a_2$} node[below] {$\vdots$} node[below, near end] {$b_2$} (2);
\draw[thick] (1) to[out=-60,in=-120] node[below,very near start] {$a_k$} node[below,very near end] {$b_k$} (2);
\end{tikzpicture}
\qquad \longmapsto \qquad
\begin{tikzpicture}[baseline=(X.base)]
\node (X) at (0,0) {};
\node[vertex] (1) at (0,0)[label=left:$i$] {};
\node[vertex] (2) at (2,0) [label=right:$j$] {};
\draw[thick] (1) -- node[auto] {$(a_{ij},a_{ji})$} (2);
\end{tikzpicture}
\]
where $a_{ij}=\sum_{e\in E_{ij}} |\ga_{ei}|=\sum_{m=1}^k a_m$ and
$a_{ji}=\sum_{e\in E_{ij}} |\ga_{ej}|=\sum_{m=1}^k b_m$, where $E_{ij}\subseteq E$ is the set of edges in $\Gamma$ between $i$ and $j$.
\end{enumerate}
As is well-known (see for example \cite{Kac}), Dynkin diagrams are in bijection with generalized Cartan matrices $C=(a_{ij})_{i,j\in I}$ consisting of integers satisfying $a_{ij}\le 0$ for $i\neq j$, $a_{ii}=2$ and $a_{ij}=0$ iff $a_{ji}=0$.
\begin{thm}\label{thm:commutative-diagram}
The following diagram of assignments is commutative.
\[
\begin{tikzcd}[ampersand replacement=\&, column sep=small]
\text{multiquiver} \& \Gamma\arrow[mapsto]{rr} \arrow[mapsto]{d}\&\& D(\Gamma) \arrow[mapsto]{d}\& \text{Dynkin diagram}\\
\text{TGW datum}\& (R_E,\si^\Gamma,t^\Gamma) \arrow[mapsto]{rr}\arrow[mapsto]{d}\&\& C \arrow[mapsto]{d}\& \text{GCM}\\
\text{TGW algebra} \& \mathcal{A}(\Gamma) \&\& \mathfrak{g}(C)\& \text{Kac-Moody algebra}
\end{tikzcd}
\]
\end{thm}
\begin{proof}
The statement follows directly from Theorem \ref{thm:GCM} and the definition of the Dynkin diagram $D(\Gamma)$ associated to $\Gamma$.
\end{proof}
\begin{rem}
In \cite{Hartwig2010}, it was shown that
the image of the map $(R,\si,t)\mapsto C$ from TGW data to GCMs contains all symmetric GCMs. It was further asked if this image contains any non-symmetric GCMs. The above shows that the map is in fact surjective.
Indeed, it is easy to see that the map $D:\Gamma\mapsto D(\Gamma)$ from multiquivers to Dynkin diagrams is surjective (but not injective). Therefore, by Theorem \ref{thm:commutative-diagram}, $(R,\si,t)\mapsto C$ is surjective.
\end{rem}
The following theorem gives a partial connection between the algebras $\mathcal{A}(\Gamma)$ and the Kac-Moody algebras $\mathfrak{g}(C)$.
\begin{thm}\label{thm:serre-relations}
Let $\Gamma$ be a multiquiver, $\mathcal{A}(\Gamma)$ be the corresponding TGW algebra, and $\mathfrak{g}$ be the Kac-Moody algebra associated to $D(\Gamma)$. Let $\mathfrak{n}^+$ (respectively $\mathfrak{n}^-$) be the positive (respectively negative) subalgebra of $\mathfrak{g}$ with generators $e_i$ (respectively $f_i$) $i\in V$.
Then there exist graded $\K$-algebra homomorphisms
\begin{equation}\label{eq:psi_pm}
\begin{aligned}
\psi^+:U(\mathfrak{n}^+) &\to \mathcal{A}(\Gamma),\quad e_i\mapsto X_i,\\
\psi^-:U(\mathfrak{n}^-) &\to \mathcal{A}(\Gamma),\quad f_i\mapsto Y_i.
\end{aligned}
\end{equation}
Moreover, for any $i,j\in V$, $i\neq j$, and $k\in\iv{0}{-a_{ij}}$, the restrictions
\[\psi^+|_{U(\mathfrak{n}^+)_{k\al_i+\al_j}}\qquad\text{and}\qquad
\psi^-|_{U(\mathfrak{n}^-)_{-k\al_i-\al_j}}\]
are injective, where $\{\al_i\}_{i\in V}$ is the set of simple roots of $\mathfrak{g}$.
\end{thm}
\begin{proof}
The statement follows from Theorem \ref{thm:GCM} and \cite[Thm~4.4(b)]{Hartwig2010}.
\end{proof}
\subsection{General and special linear Lie algebras}
\label{sec:special-linear-lie-algebras}
Let $\widetilde{A_n}$ be the following multiquiver:
\[
\widetilde{A_n}:\;
\begin{tikzpicture}[baseline=(etc.base)]
\node (start) at (0,0) {};
\node[vertex] (1) at (1,0) [label=below:$1$] {};
\node[vertex] (2) at (2,0) [label=below:$2$] {};
\node[vertex] (3) at (3,0) [label=below:$3$] {};
\node[vertex] (N) at (4,0) [label=below:$n$] {};
\node (end) at (5,0) {};
\node (etc) at (3.5,0) {$\cdots$};
\draw[leaf] (start) -- (1);
\draw[edge] (1) -- (2);
\draw[edge] (2) -- (3);
\draw[leaf] (N) -- (end);
\end{tikzpicture}
\qquad\qquad
\ga=\begin{bmatrix}
1 & & && \\
-1 & 1 & &&\\
& -1 & \ddots & &\\
& & \ddots & \ddots & \\
& & & -1 & 1 \\
& & & & -1
\end{bmatrix}
\]
We identify $V=\iv{1}{n}$ with the set of simple roots of $\mathfrak{sl}_{n+1}$ and regard the edge set as $E=\iv{1}{n+1}$.
Let $\mathbb{E}=\sum_{i\in E} x_iy_i= \sum_{i\in E} (u_i-1)$ be the Euler operator in the Weyl algebra $A_E(\K)$. Let $A_E(\K)^\mathbb{E}=\{a\in A_E(\K)\mid [\mathbb{E},a]=0\}$ be the invariant subalgebra with respect to the adjoint action of $\mathbb{E}$.
The following lemma is well-known.
\begin{lem}\label{lem:sln+1}
There exists a homomorphism
\[ \pi: U(\mathfrak{gl}_{n+1})\to A_E(\K)\]
given by
\[
\pi(e_i)=x_iy_{i+1},\quad
\pi(f_i)=x_{i+1}y_i,\quad
\pi(E_{jj})=u_j,
\]
for $i\in \iv{1}{n}$ and $j\in\iv{1}{n+1}$, where $e_i,f_i$ are Chevalley generators and $E_{ii}$ are the diagonal matrix units of $\mathfrak{gl}_{n+1}$. Moreover, the image of $\pi$ coincides with $A_E(\K)^\mathbb{E}$.
\end{lem}
The next result says that $\pi$ factors through the canonical representation $\varphi_{\Gamma}$ of the TGW algebra $\mathcal{A}(\Gamma)$ where $\Gamma=\widetilde{A_n}$. For $n=3$ we recover the case considered by A. Sergeev \cite{Sergeev2001}.
We will regard $U(\mathfrak{gl}_{n+1})$ as an $R_E$-ring via
\[R_E\to U(\mathfrak{gl}_{n+1}),\quad u_i\mapsto E_{ii}. \]
\begin{prp}
There exists a homomorphism of $\Z V$-graded $R_E$-rings
\[\psi:U(\mathfrak{gl}_{n+1})\to \mathcal{A}(\widetilde{A_n})\]
given by
\[\psi(e_i)=X_i,\quad \psi(f_i)=Y_i,\quad \psi(E_{jj})=u_j,\]
for $i\in\iv{1}{n}$ and $j\in\iv{1}{n+1}$.
Moreover, we have the following commutative diagram:
\begin{equation}
\begin{tikzcd}[ampersand replacement=\&,
column sep=small]
U(\mathfrak{gl}_{n+1})
\arrow[twoheadrightarrow]{rrd}{\pi}
\arrow[twoheadrightarrow]{d}[swap]{\psi}
\& \& \\
\mathcal{A}(\widetilde{A_n})
\arrow{rr}{\simeq}[swap]{\varphi_{\widetilde{A_n}}}
\& \& A_E(\K)^{\mathbb{E}}
\end{tikzcd}
\end{equation}
\end{prp}
\begin{proof}
The multiquiver $\widetilde{A_n}$ is connected and not in equilibrium since it has leaves. Thus by Theorem \ref{thm:faithfulness}, $\varphi_{\widetilde{A_n}}$ is faithful. Since
$\widetilde{A_n}$ has no cycles, by Theorem \ref{thm:local-surjectivity}, the image of $\varphi_{\widetilde{A_n}}$ equals $\bigoplus_{d\in \Z E} (A_E(\K))_{\ga(d)}$, which coincides with $A_E(\K)^\mathbb{E}$ since $\ga(\Z^n)=\{g=\sum_e g_ee\in \Z E\mid \sum_e g_e=0\}$.
It only remains to note that $\psi$ coincides with $\varphi_{\widetilde{A_n}}^{-1}\circ \pi$ on the generators $e_i, f_i, E_{jj}$.
\end{proof}
For $\la\in\K$, let $\mathcal{A}^\la(\widetilde{A_n})$ denote the quotient algebra $\mathcal{A}(\widetilde{A_n})/\langle \mathbb{E}-\lambda\rangle$. Since $\varphi_{\widetilde{A_n}}$ is faithful and locally surjective, each homogeneous component $\mathcal{A}(\widetilde{A_n})_g$ is a cyclic left $R_E$-module generated by $(\varphi_{\widetilde{A_n}})^{-1}(z^{\ga(g)})$. By Proposition \ref{prp:quotients}, $\mathcal{A}^\la(\widetilde{A_n})$ is isomorphic to the TGW algebra $\mathcal{A}(R_E^\la, \bar \sigma, \bar t)$ where $R_E^\la=R_E/\langle \mathbb{E}-\la\rangle$ and $\bar\sigma_i$ is induced from $\sigma_i$, and $\bar t_i$ is the image of $t_i$. Note that $U(\mathfrak{sl}_{n+1})$ becomes an $R_E^\la$-ring via $\bar u_i-\bar u_{i+1}\mapsto h_i$.
\begin{thm}\label{thm:sln-quotient}
Let $M$ be an infinite-dimensional completely pointed $\mathfrak{sl}_{n+1}$-module and let $J=\Ann_{U(\mathfrak{sl}_{n+1})} M$. Then there exists a $\la\in\K$ such that the primitive quotient $U(\mathfrak{sl}_{n+1})/J$ is isomorphic as $\Z V$-graded $R_E^\la$-rings to $\mathcal{A}^\la(\widetilde{A_n})$.
\end{thm}
\begin{proof}
By \cite{BenBriLem1997}, there exists a simple weight module $W$ over the Weyl algebra $A_E(\K)$ and a scalar $\la\in\K$ such that $M$ is the $\la$-eigenspace in $W$ with respect to the Euler operator $\mathbb{E}$.
Moreover the algebra $B_E^\la$ of twisted differential operators on $\mathbb{P}^n$ given by $B_E^\la=A_E(\K)^\mathbb{E}/\langle\mathbb{E}-\la\rangle$ acts faithfully on $M$.
Indeed, since $B_E^\la$ is a domain, it is sufficient to show
that $\operatorname{Ann}_{R_E}M=0$. Recall from \cite{Mathieu2000} that
there exists the set $S$ of commuting roots such that $S$ spans
the whole weight space and a root vector $X_\beta$ acts injectively
on $M$ for all $\beta\in S$. Let $M_\mu$ be a weight subspace. Then
$$\operatorname{Ann}_{R_E}M \subset\bigcap_{n_\beta
>0}\operatorname{Ann}_{R_E}M_{\mu+\sum_{\beta\in S} n_\beta\beta}=0,$$
where the second equality follows from the condition that $S$ spans the whole space.
Let $\al:U(\mathfrak{sl}_{n+1})\to\End_\K(M)$ denote the representation corresponding to the module $M$.
Then we have the following commutative diagram, where we use $\psi$ and $\pi$ to also denote the restrictions to $U(\mathfrak{sl}_{n+1})$, $\xi_\la, \xi_\la'$ are the canonical projections, and $\bar\varphi_{\widetilde{A_n}}$ is induced from $\varphi_{\widetilde{A_n}}$:
\[
\begin{tikzcd}[ampersand replacement=\&,
column sep=small]
U(\mathfrak{sl}_{n+1})
\arrow{rr}{\al}
\arrow{rd}{\pi}
\arrow{d}{\psi}
\& \& \End_\K(M) \\
\mathcal{A}(\widetilde{A_n})
\arrow{r}{\simeq}[swap]{\varphi_{\widetilde{A_n}}}
\arrow[twoheadrightarrow]{d}{\xi_\la}
\& A_E(\K)^\mathbb{E} \arrow{ur} \arrow[twoheadrightarrow]{dr}{\xi_\la'} \& \\
\mathcal{A}^\la(\widetilde{A_n})
\ar{rr}{\simeq}[swap]{\bar\varphi_{\widetilde{A_n}}}
\& \& A_E(\K)^\mathbb{E}/\langle\mathbb{E}-\la\rangle \ar[hookrightarrow]{uu}
\end{tikzcd}
\]
Let $J_\la = \ker (\xi_\la \circ \psi)$. Then it follows from the above diagram that $J=\ker (\xi_\la'\circ\pi) = J_\la$ and thus that $\mathcal{A}^\la(\widetilde{A_n}) \simeq U(\mathfrak{sl}_n)/J_\la$ as required.
\end{proof}
\subsection{Symplectic Lie algebra}
\label{sec:symplectic-lie-algebras}
Let $\widetilde{C_n}=(V,E,s,t)$ be the following multiquiver:
\[
\widetilde{C_n}:\;\;
\begin{tikzpicture}[baseline=(etc.base)]
\node (start) at (0,0) {};
\node[vertex] (1) at (1,0) [label=below:$1$] {};
\node[vertex] (2) at (2,0) [label=below:$2$] {};
\node[vertex] (3) at (3,0) [label=below:$3$] {};
\node[vertex] (N-2) at (4,0) [label=below:$n-2$] {};
\node[vertex] (N-1) at (5,0) [label=below:$n-1$] {};
\node[vertex] (N) at (6,0) [label=below:$n$] {};
\node (etc) at (3.5,0) {$\cdots$};
\draw[leaf] (start) -- (1);
\draw[edge] (1) -- (2);
\draw[edge] (2) -- (3);
\draw[edge] (N-2) -- (N-1);
\draw[edge] (N-1) -- node[auto,very near start] {$1$} node[auto,very near end] {$2$} (N);
\end{tikzpicture}
\qquad\qquad
\ga=\begin{bmatrix}
1 & & & && \\
-1 & 1 & & &&\\
& -1 & \ddots & &&\\
& & \ddots &\ddots &&\\
& & & -1 & 1 & \\
& & & & -1 & 2
\end{bmatrix}
\]
We identify $V=\iv{1}{n}$ with the set of simple roots of $\mathfrak{sp}_{2n}$ relative to a fixed choice of Borel subalgebra. In particular we regard $\Z V$ as the root lattice.
We also identify the edge set $E$ with $\iv{1}{n}$. Let $A_E(\K)=A_n(\K)$ be the Weyl algebra with index set $E$, and let $\ep$ be the automorphism of $A_n(\K)$ defined by $\ep(x_i)=-x_i$ and $\ep(y_i)=-y_i$ for all $i\in E$. Let $A_n(\K)^\ep=\{a\in A_n(\K)\mid \ep(a)=a\}$ be the fixed-point subalgebra of $A_n(\K)$ with respect to $\ep$.
\begin{lem}\label{lem:sp2n-lemma}
There exists a $\K$-algebra homomorphism
\[\pi:U(\mathfrak{sp}_{2n})\to A_n(\K)\]
given by
\[
\begin{aligned}
\pi(e_i) &= x_iy_{i+1} \\
\pi(e_n) &= \frac{\sqrt{-1}}{2}x_n^2
\end{aligned}
\qquad
\begin{aligned}
\pi(f_i) &= x_{i+1}y_i \\
\pi(f_n) &= \frac{\sqrt{-1}}{2}y_n^2
\end{aligned}
\qquad
\begin{aligned}
\pi(h_i) &= u_i-u_{i+1} \\
\pi(h_n) &= u_n-\frac{1}{2}
\end{aligned}
\]
for $i\in \iv{1}{n-1}$.
Moreover, the image of $\pi$ coincides with the subalgebra $A_n(\K)^\ep$.
\end{lem}
\begin{proof}
This is well-known, see for example \cite{Dixmier}. The normalization involving $\sqrt{-1}$ is chosen to make $\pi$ respect the natural involutions.
\end{proof}
Consider $U(\mathfrak{sp}_{2n})$ as an $R_E$-ring via
\begin{align*}
R_E &\to U(\mathfrak{sp}_{2n}), \\
u_i-u_{i+1} &\mapsto h_i, \\
u_n-\frac{1}{2} &\mapsto h_n,
\end{align*}
and equip $U(\mathfrak{sp}_{2n})$ with the $\K$-linear Cartan involution on $U(\mathfrak{sp}_{2n})$ satisfying $e_i^\ast=f_i$, $f_i^\ast=e_i$, $h_i^\ast=h_i$ for all $i$. Then the map $\pi$ from Lemma \ref{lem:sp2n-lemma} is a map of $R_E$-rings with involution.
\begin{prp}
There exists a map of $\Z V$-graded $R_E$-rings with involution
\[\psi:U(\mathfrak{sp}_{2n})\to \mathcal{A}(\widetilde{C_n})\]
given by
\[
\begin{aligned}
\psi(e_i)&=X_i \\
\psi(e_n)&=\frac{\sqrt{-1}}{2} X_n
\end{aligned}
\qquad
\begin{aligned}
\psi(f_i)&=Y_i\\
\psi(f_n)&=\frac{\sqrt{-1}}{2} Y_n
\end{aligned}
\qquad
\begin{aligned}
\psi(h_i)&=u_i-u_{i+1} \\
\psi(h_n)&=u_n-\frac{1}{2}
\end{aligned}
\]
for $i\in\iv{1}{n-1}$. Moreover, we have the following commutative diagram:
\begin{equation}
\begin{tikzcd}[ampersand replacement=\&,
column sep=small]
U(\mathfrak{sp}_{2n})
\arrow[twoheadrightarrow]{rrd}{\pi}
\arrow[twoheadrightarrow]{d}[swap]{\psi}
\& \& \\
\mathcal{A}(\widetilde{C_n})
\arrow{rr}{\simeq}[swap]{\varphi_{\widetilde{C_n}}}
\& \& A_n(\K)^{\ep}
\end{tikzcd}
\end{equation}
where $\varphi_{\widetilde{C_n}}$ is the canonical representation by differential operators from Section \ref{sec:homomorphism-varphi}.
\end{prp}
\begin{proof}
$\widetilde{C_n}$ is connected and not in equilibrium since it has a leaf. Its underlying undirected graph has no cycles. By Theorem \ref{thm:faithfulness} and \ref{thm:local-surjectivity}, $\varphi_\Gamma$ is faithful and locally surjective. Thus $\varphi_\Gamma$ is an isomorphism $\mathcal{A}(\widetilde{C_n})\simeq \bigoplus_{d\in \Z V} A_n(\K)_{\ga(d)}$.
One checks that $\ga(\Z V)=\{d=\sum_v d_v v\in\Z V\mid \sum_i d_i\in 2\Z\}$.
Thus
$\varphi_{\widetilde{C_n}}:\mathcal{A}(\widetilde{C_n})\simeq A_n(\K)^\ep$. It remains to be noted that $\psi$ coincides with $\varphi_{\widetilde{C_n}}^{-1}\circ \pi$ on the generators $e_i,f_i$ of $U(\mathfrak{sp}_{2n})$.
\end{proof}
A weight module $M=\bigoplus_{\mu\in\mathfrak{h}^\ast}M_\mu$ over a semisimple finite-dimensional Lie algebra $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}$ is called \emph{completely pointed} if $\dim_\K M_{\mu}\le 1$ for all $\mu\in\mathfrak{h}^\ast$.
\begin{thm}\label{thm:sp2n-quotient}
Assume $\K$ is algebraically closed. Let $M$ be an infinite-dimensional simple completely pointed $\mathfrak{sp}_{2n}$-module.
Let $J=\Ann_{U(\mathfrak{sp}_{2n})} M$. Then $J=\ker \psi$, and hence the primitive quotient $U(\mathfrak{sp}_{2n})/J$ is isomorphic to $\mathcal{A}(\widetilde{C_n})$ as $\Z V$-graded $R_E$-rings with involution.
\end{thm}
\begin{proof}
Let $\alpha:U(\mathfrak{sp}_{2n})\to\End_\K(M)$ be the corresponding representation of the enveloping algebra. Thus $J=\ker \alpha$. By \cite{BenBriLem1997}, $M$ can be realized as an $A_n(\K)^\ep$-invariant subspace of an irreducible weight $A_n(\K)$-module.
The argument with commuting roots in the proof of Theorem \ref{thm:sln-quotient} can be used here as well for proving $J=\operatorname{Ann}_{U(\mathfrak{sp}_{2n})}M$. Thus $M$ is faithful as a module over $A_n(\K)^\ep$ and we have the following commutative diagram:
\begin{equation}
\begin{tikzcd}[ampersand replacement=\&,
column sep=small]
U(\mathfrak{sp}_{2n})
\arrow{rr}{\alpha}
\arrow[twoheadrightarrow]{rrd}{\pi}
\arrow[twoheadrightarrow]{d}[swap]{\psi}
\& \& \End_\K(M) \\
\mathcal{A}(\widetilde{C_n})
\arrow{rr}{\simeq}[swap]{\varphi}
\& \& A_n(\K)^{\ep} \ar[hookrightarrow]{u}
\end{tikzcd}
\end{equation}
It follows from this diagram that $J=\ker \pi=\ker \psi$,
which proves the claim.
\end{proof}
\subsection{Primitive quotients of enveloping algebras as TGW algebras}
\label{sec:primitivequotients}
Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra. Motivated by the maps $\psi$ from Sections \ref{sec:symplectic-lie-algebras} and \ref{sec:special-linear-lie-algebras}, we consider the following question: When does there exist a surjective homomorphism from the universal enveloping algebra $U(\mathfrak{g})$ to a TGW algebra $\TGWA{\mu}{R}{\si}{t}$?
After generalizing the notion of a TGW algebra to allow $\si_i\si_j\neq\si_j\si_i$, we give a complete answer to the question in the case when $\psi(e_i)=X_i$, $\psi(f_i)=Y_i$ and $\ker \psi$ is a primitive ideal of $U(\mathfrak{g})$.
Let $(R,\si,t)$ be a TGW datum with index set $I$, except we do not require that $\si_i$ and $\si_j$ commute. We will let $G$ denote the group generated by $\si_i$. The definition of the TGW algebra $\TGWA{\mu}{R}{\si}{t}$ goes through without any problems. When some $t_i$'s are zero-divisors, there might exist a different choice of automorphisms $\si_i'$ which do commute and give the same algebra.
$(R,\si,t)$, and by abuse of language $\TGWA{\mu}{R}{\si}{t}$, will be called \emph{abelian} or \emph{non-abelian} depending on whether $G$ is abelian or not.
The following lemma shows that regular and consistent TGW algebras must be abelian.
\begin{lem} Let $D=(R,\si,t)$ be a not necessarily abelian TGW datum. If $D$ is regular and $\mu$-consistent for some $\mu$, then $D$ is abelian.
\end{lem}
\begin{proof} Straightforward to check using the relations \eqref{eq:tgwarels}.
\end{proof}
We now prove the main theorem of this section. Recall that a module over a finite-dimensional simple Lie algebra is \emph{completely pointed} if it is a multiplicity free weight module.
\begin{thm}\label{thm:primquotient} Let $\mathfrak g$ be a finite-dimensional simple Lie algebra with
Serre generators $e_i,f_i$, $i=1,\dots,n$ and $J$ be a primitive ideal
of $U(\mathfrak g)$. The following conditions are equivalent:
\begin{enumerate}[{\rm (a)}]
\item There exists a not necessarily abelian TGW algebra $\TGWA{\mu}{R}{\si}{t}$ and a surjective homomorphism: $\psi: U(\mathfrak g)\to\TGWA{\mu}{R}{\si}{t}$ with kernel $J$ such
that $\psi(e_i)=X_i$, $\psi(f_i)=Y_i$;
\item There exists a simple completely pointed $\mathfrak g$-module $M$ such that
$\operatorname{Ann}_{U(\mathfrak g)} M=J$.
\end{enumerate}
\end{thm}
\begin{proof} First, let us prove that (a) implies (b). Let $Q$ denote
the root lattice of $\mathfrak g$. The
adjoint action of the Cartan subalgebra $\mathfrak h$ defines
a $Q$-grading of $U(\mathfrak g)$ and it induces a $Q$-grading on $\TGWA{\mu}{R}{\si}{t}$.
Thus, $\psi$ is a homomorphism of graded algebras and therefore
$\psi(U(\mathfrak g)_0)=R_0$. Note that $R_0$ is generated by
$X_iY_i$ and $\psi(\mathfrak h)$. Hence $R_0$ is commutative.
By Duflo's theorem there exists a simple highest weight $\mathfrak g$-module $M$
whose annihilator is $J$. By Proposition 9.6.1 in [Dix] any weight
space $M_\gamma$ is a simple $R_0$-module. Therefore
$\operatorname{dim} M_\gamma=1$, i.e. $M$ is completely pointed.
Now let us prove that (b) implies (a).
Let us assume first that $M$ is infinite-dimensional.
By the classification of completely pointed weight modules \cite{BenBriLem1997}, $\mathfrak g=\mathfrak{sl}_n$ or
$\mathfrak{sp}_{2n}$. Then (a) follows from
Theorem \ref{thm:sln-quotient} and Theorem \ref{thm:sp2n-quotient}.
Thus we may assume that $M$ is finite dimensional.
Let $\Gamma$ denote the set of weights of $M$ and $\mathcal A=\operatorname{End}(M)$.
The homomorphism $\rho:U(\mathfrak g)\to \mathcal A$ is surjective by
the Jacobson density theorem.
Note that $\mathcal A$ has a $Q$-grading $\mathcal A=\bigoplus_{\alpha\in Q} \mathcal A_\alpha$ defined
by
$$\mathcal A_\alpha=\{X\in\mathcal A | X M_\gamma\subset M_{\gamma+\alpha}\forall \gamma\in\Gamma\}.$$
We set $R=\mathcal A_0$, $X_i=\rho(e_i), Y_i=\rho(f_i)$, $t_i=Y_iX_i$ and claim that
$\mathcal A$ is a TGW algebra with all $\mu_{ij}=1$. Since $\mathcal A$
is simple, it
suffices to prove the existence of automorphisms $\sigma_i$ of $R$
such that $X_i r=\sigma_i(r) X_i$ and $Y_i r=\sigma_i^{-1}(r) Y_i$
for all $r\in R$.
Let $E_\gamma$ denote the $\mathfrak h$-invariant projector on
$M_\gamma$. Clearly $E_\gamma$ for all $\gamma\in\Gamma$ is a basis of $R$.
Let $\alpha_i$ denote the simple root (weight of $e_i$).
If $\gamma-\alpha_i\in \Gamma$ we set $\sigma_i(E_\gamma)=E_{\gamma-\alpha_i}$.
If $\gamma-\alpha_i\notin \Gamma$, we set $\sigma_i(E_\gamma)=E_{\gamma+k_\gamma\alpha_i}$
where $k_\gamma$ is the maximal integer $k$ such that
$\gamma+k\alpha_i\in \Gamma$. In the former case $E_\gamma X_i=X_iE_{\gamma-\alpha_i}$
since $X_iM_{\gamma-\alpha_i}=M_\gamma$. In the latter case $E_\gamma X_i=0$ since
$M_\gamma\notin X_iM$ and by construction
$X_iE_{\gamma+k_\gamma\alpha_i}=0$. The proof of $Y_ir=\sigma_i^{-1}(r) Y_i$ is similar. That concludes the proof.
\end{proof}
\begin{rem} \label{rem:abelian}
If $M$ is infinite-dimensional then $\sigma_i$ commute by
our construction of the subalgebra in a Weyl algebra in Theorem \ref{thm:primquotient}.
In the case of finite-dimensional $M$, the choice of
$\sigma_i$ is not unique. The choice used in our proof usually does not imply that the group $G$ generated by $\sigma_i$ is abelian. For example, if $M$ is a minuscule representation the choice of
$\sigma_i$ as in the proof of Theorem \ref{thm:primquotient} gives
$G$ isomorphic to the Weyl group of $\mathfrak g$.
\end{rem}
\subsection{Completely pointed loop modules} \label{sec:affine}
Let $\hat{\mathfrak g}$ denote the affinizaion of $\mathfrak g$. Recall that
$$\hat{\mathfrak g}=\mathfrak g\otimes\K[t,t^{-1}]\oplus \K c\oplus\K t\frac{\partial}{\partial t},$$
where $c$ is the central element. For any $\mathfrak g$-module $M$ let
$\hat M=M\otimes \K[t,t^{-1}]$ denote the corresponding loop module
(with trivial action of $c$). Recall that Serre generators of
$\hat{\mathfrak g}$ are $e_i\otimes 1,i=1,\dots n,e_0\otimes t$ and $f_i\otimes 1,i=1,\dots,n,f_0\otimes t^{-1}$,
where is $e_0$ is the lowest and $f_0$ is the higest vector in the
adjoint representation of $\mathfrak g$.
\begin{thm}\label{affine} Let $M$ be a simple completely pointed
$\mathfrak g$-module, $J=\operatorname{Ann}_{U(\hat{\mathfrak g})}\hat M$.
There exists a TGW algebra $\TGWA{\mu}{R}{\si}{t}$ and a surjective homomorphism: $\psi: U(\hat{\mathfrak g})\to\TGWA{\mu}{R}{\si}{t}$ with kernel $J$ such
then $\psi(e_i\otimes 1)=X_i$, $\psi(f_i\otimes 1)=Y_i$ for
$i=1,\dots n$ and $\psi(e_0\otimes t)=X_0$, $\psi(f_0\otimes t^{-1})=Y_0$.
\end{thm}
\begin{proof} Let $\psi: U(\hat{\mathfrak g})\to\operatorname{End}(\hat M)$
be the natural homomorphism. We claim that $\mathcal A=\operatorname{Im}\psi$ is a TGW
algebra with $R$ generated by $X_iY_i$, $i=0,\dots,n$ and
$t\frac{\partial}{\partial t}$. It is easy to see that
$R$ is isomorphic to $\K[t\frac{\partial}{\partial t}]\otimes \psi (U(\mathfrak g))_0$.
For $i=1,\dots,n$ we define $\sigma_i$ on
$\psi(U(\mathfrak g)_0)$ as in Theorem \ref{thm:primquotient} and
set $\sigma_0=\sigma_1^{-a_1}\cdots\sigma_n^{-a_n}$
where $a_1,\dots,a_n$ are the coefficients in the decomposition of the
highest root of $\mathfrak g$ into linear combination of simple roots.
We set $\sigma_i(t\frac{\partial}{\partial t})=t\frac{\partial}{\partial t}$ for $i=1,\dots,n$,
and $\sigma_0(t\frac{\partial}{\partial t})=t\frac{\partial}{\partial t}+1$.
All relations are clear from the construction and it remains to show
that $\mathcal A$ does not have non-zero ideals with zero
intersection with $R$.
We use the $\mathbb Z$-grading of $\mathcal A$ induced
by the adjoint action of $t\frac{\partial}{\partial t}$. By our
construction $\mathcal A_0=\K[t\frac{\partial}{\partial t}]\otimes \psi (U(\mathfrak g))$.
Every ideal $I$ in $\mathcal A$ is homogeneous. Since
$R\subset \mathcal A_0$ we have $I\cap R=I_0\cap R$ for any ideal
$I\subset\mathcal A$. Let $I$ be ideal of $\mathcal A$ such that
$I\cap R=0$. Then $I_0\cap R=0$. Hence $I_0=0$ as follows easily from Theorem \ref{thm:primquotient}.
Suppose $I_0=0$ but $I_k\neq 0$ for some $k$. There exists $h\in\mathfrak h$ such that
$\psi(h\otimes t^{-k})$ is invertible in $\operatorname{End}(M)$. Then $\psi(h\otimes t^{-k})I_k\neq 0$,
which is a contradiction.
\end{proof}
\section{Appendix: Relation to a previous family of TGW algebras}
\label{sec:symmetric}
The following TGW algebras (along with quantum analogues) were first defined in \cite{Hartwig2010}. We show that modulo a graded ideal, they are special cases of $\mathcal{A}(\Gamma)$.
Let $n$ be a positive integer and $C=(a_{ij})_{i,j=1}^n$ be an $n\times n$ symmetric generalized Cartan matrix (GCM). Define a TGW datum $(R_C,\si_C,t_C)$ as follows:
\begin{gather}
R_C=\K[H_{ij}^{(k)}\mid 1\le i<j\le n,\; k=a_{ij},a_{ij}+2,\ldots,|a_{ij}|] \\
\si_C=(\si_i)_{i=1}^n,\quad \si_r(H_{ij}^{(k)})=
\begin{cases}
H_{ij}^{(k)}+H_{ij}^{(k-2)},& \text{$r=j$ and $k>a_{ij}$,}\\
H_{ij}^{(k)}-H_{ij}^{(k-2)}+H_{ij}^{(k-4)}-\cdots \pm H_{ij}^{(a_{ij})}, & r=i,\\
H_{ij}^{(k)},& \text{otherwise,}
\end{cases} \\
t_C=(t_i)_{i=1}^n,\quad
t_i=H_{i1}H_{i2}\cdots H_{in},\quad
H_{ij}=
\begin{cases}
H_{ij}^{(|a_{ij}|)}, & i<j, \\
1,&i=j,\\
\si_i^{-1}(H_{ji}^{(|a_{ij}|)}), & i>j.
\end{cases}
\end{gather}
Define $\mathcal{T}(C)=\mathcal{A}(R_C,\si_C,t_C)$.
The main point of these algebras is that the GCM associated to $(R_C,\si_C,t_C)$ is exactly $C$ (see \cite{Hartwig2010} for details).
We now define a multiquiver $\Gamma_C=(V,E,s,t)$ associated to $C$. Let
\begin{gather}
V=\{1,2,\ldots,n\},\qquad E=\big\{(i,j)\in V\times V\mid i<j, a_{ij}\neq 0\big\},\\
s((i,j))=(i,|a_{ij}|),\quad t((i,j))=(j,|a_{ij}|)\quad \forall (i,j)\in E.
\end{gather}
Note that $\Gamma_C$ is a symmetric simple quiver. Conversely, one can show that any finite symmetric simple quiver satisfying those two conditions is sign-equivalent to $\Gamma_C$ for some symmetric GCM $C$. In what follows we identify the group $\Z V$ with $\Z^n$ in the obvious way.
\begin{prp}
Let $C$ be a symmetric GCM and $\Gamma_C$ the corresponding multiquiver as defined above.
Then there is a surjective homomorphism of $\Z^n$-graded $\K$-algebras
\[\mathcal{F}:\mathcal{T}(C)\to \mathcal{A}(\Gamma_C).\]
satisfying $\mathcal{F}(X_i)=X_i$ and $\mathcal{F}(Y_i)=Y_i$ for all $i\in V$.
\end{prp}
\begin{proof}
Put $\Gamma=\Gamma_C$. Let $F:R_C\to R_E$ be the unique $\K$-algebra homomorphism determined by
\begin{equation}
F(H_{ij}^{(|a_{ij}|-2k)})=(\si_j-\Id)^k(u_{(i,j),i}),\qquad\forall i<j,\; k=0,1,\ldots,|a_{ij}|.
\end{equation}
By definition, $F\circ (\si_j-\Id) = (\si_j^\Gamma-\Id)\circ F$ on $H_{ij}^{(k)}$. So $F\circ \si_j=\si_j^\Gamma\circ F$ on $H_{ij}^{(k)}$. Then $F\circ \si_i = \si_i^\Gamma\circ F$ on $H_{ij}^{(k)}$ too since $\si_i\si_j=\Id$ on $H_{ij}^{(k)}$. Also, $F\circ\si_r=\si_r^\Gamma\circ F$ on $H_{ij}^{(k)}$ for $r\notin\{i,j\}$ because $\si_r(u_{(i,j)})=u_{(i,j)}$ for $r\notin\{i,j\}$. This shows that $F\circ\si_i = \si_i^\Gamma\circ F$ on $R_C$. Furthermore, by a direct calculation using that $\si_j(u_{(i,j)},j)=u_{(i,j),i}$ one verifies that $F(t_i)=t_i^\Gamma$ for all $i$. This means that $F$ is a morphism of TGW data, as defined in \cite{FutHar2012b}, and that applying the functor $\mathcal{A}$ from \cite{FutHar2012b} we obtain a homomorphism
\[\mathcal{F}:\mathcal{A}(R_C,\si_C,t_C)\to\mathcal{A}(\Gamma)\]
of $\Z^n$-graded algebras.
One checks that $F(H_{ij}^{(2-|a_{ij}|)})$ is a polynomial in $u_{(i,j)}$ of degree $1$ for any $(i,j)\in E$. This proves that $F$ is surjective. By \cite[Cor.~3.2]{FutHar2012b}, $\mathcal{F}$ is also surjective.
\end{proof}
\bibliographystyle{siam}
|
1,941,325,220,526 | arxiv | \section{Introduction}
\label{sec:intro}
Temporal action detection is a popular and fundamental problem for video content understanding.
This task aims to predict the precise temporal boundaries and categories of each action instance in the videos.
Similar to two-stage object detection in images, most temporal action detection methods \cite{Lin:2018:BSN,Lin:2019:BMN, Xu:2020:G-TAD,RTDNet:2021} follow a two-stage strategy: temporal action proposal generation and action proposal classification.
Action classification \cite{Wang:2017:Unet} has achieved convincing performance, but temporal action detection accuracy is still unsatisfactory on mainstream benchmarks.
It indicates that the quality of generated proposals is the bottleneck of the final action detection performance.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figure_mm/intro_v2.pdf}
\caption{
Given an untrimmed video sequence, we aim to generate temporal action instance.
The boundaries can hardly distinguished from background due to cluttered content and action-unrelated scenes.
Our tailor-modified PRSA-Net is designed to selectively contextualize local semantic information instead of pairwise similarity.}
\label{fig:intro}
\end{figure}
Existing proposal generation approaches make sorts of efforts to exploit rich semantic information for the sake of better high-quality proposal generation.
The majority of previous works \cite{Lin:2018:BSN,Lin:2019:BMN,Lin:2020:Fast} embed the temporal representation by stacked temporal convolutions.
\cite{Chao:2018:TAL-Net,Zhao:2017:SSN} extend the temporal segment by a pre-defined ratio to capture contextual content, while G-TAD \cite{Xu:2020:G-TAD} proposes to explicitly model temporal contextual information by graph convolutional networks (GCNs).
More recently, like DETR \cite{DETR2020} in object detection, transformer based methods \cite{RTDNet:2021,activity:2021} are introduced to provide long-range temporal context for proposal generation and action detection.
Despite impressive progress, these aforementioned approaches still confront two challenges that remain to be addressed:
1) limited contextual representation learning and
2) sensitivity to the complicated background.
In the former case, although video snippets contain richer information than a single image such as temporal dynamical content, offering useful cues for generating proposals, much fewer efforts are spent on semantic temporal content modeling.
Therefore, effectively capturing action-matter context is critical for generating proposals.
In the latter case, due to the complicated background in untrimmed videos, \cite{Xu:2020:G-TAD} adopts a similarity-based attention mechanism to selectively extract the most relevant information and relationships.
However, it is sub-optimal to exploit the dependencies only based on computing the pairwise similarity.
Because there exists highly similar content for consistent frames, where the pairwise relations may introduce spurious noise such as cluttered background and inaccurate action-related content.
As illustrated in Fig. \ref{fig:intro}, the action instances surrounded by the background content, such as \textit{video segments} $p_1$, $p_2$, can hardly be discriminated due to the camera motion and progressive action transitions.
It demonstrates that applying traditional pairwise similarity-based relations such as the operation \textit{dot product, Euclidean distance} are insufficient to deliver complete contextual information for proposal generation.
To relieve the above issues, we propose a novel Pyramid Region-based Slot Attention (PRSlot) to contextualize action-matter representation.
It is building upon the recently-emerged slot attention \cite{slotattn:2020}, which learns the object-centric representations for image classification task.
Our well-design PRSlot module takes the snippet-level features as input and maps them to a set of output vectors by aggregating local region context that we refer to as \textit{slots}.
\textbf{First}, our PRSlot module is enhanced by a Region-based Attention (RA) which directly estimates the confidence from inputs and its local region to the slots.
Unlike the similarity-based attention mechanism, our RA restricts the scope of slot interactions to local surroundings and learns the semantic attention directly using an encoder-decoder architecture.
In detail, an encoder is deployed to exploit the snippet-relevant feature map in the local region by zeroing out the position outside of the desired local scope, while a decoder is followed to map the correlation features into the relation score vectors directly.
\textbf{Second}, instead of applying recurrent function to update slots over multiple iterations in original slot attention \cite{slotattn:2020}, our tailor-modified PRSlot module presents a parallel pyramid slot representation updating strategy with multi-scale local regions.
\textbf{Finally}, the complementary action-matter representation generated by the PRSlot modules are used to produce boundaries scores and proposal-level confidence respectively by linear layers.
Based on the above components, we present a novel Pyramid Region-based Slot Attention Network called PRSA-Net to capture abundant semantic representation for high-quality proposal generations.
Experimental results show our PRSA-Net is superior to the state-of-the-art performance on two widely used benchmarks.
The main contributions of this paper are therefore as follows,
\begin{itemize}
\item A newly Region-based Attention is proposed to directly generate relation scores from the slot representations and its surroundings using encoder-decoder manner instead of similarity-based operation.
\item
We propose a novel Pyramid Region-based Slot Attention (PRSlot) module, which incorporates a region-based attention mechanism and a parallel pyramid iteration strategy to effectively capture contextual representations for better boundaries discrimination.
Based on this module, we develop a Pyramid Region-based Slot Attention Network (PRSA-Net) to exploit action-matter information for high-quality proposal generation.
\item We perform extensive experiments on the THUMOS14 and ActivityNet-1.3 benchmarks and evaluate the performances of temporal action proposal generation and detection.
The results show that our proposed approach outperforms other state-of-the-art methods.
Our ablation study also highlights the importance of both architecture components.
\end{itemize}
\section{Related Work}
\label{sec:related}
\textbf{Video Feature Extraction.}
Since video data can be generally treated as a sequence of ordered frames, the most straightforward approach to video feature extraction is to first generate deep features on every frame by CNN backbones with 2D convolutional filters, and then fuse them into snippet or video level features \cite{Karpathy:2014:VidConv}.
Besides the 2D filters, 3D or (2+1)D convolutional filters powered backbones \cite{Tran:2015:3DConv,Carreira:2017:I3D,Qiu:2017:P3D} have been designed to directly extract features from multiple stacked video frames, which are more suitable for motion feature extraction.
Another widely used framework to explicitly incorporate motion clues is the two-stream CNNs \cite{Simonyan:2014:2S-CNN, Feichtenhofer:2016:2S-CNN}, in which one stream with 2D CNN is deployed on the RGB frames, while the other stream extracts features from the optical flows.
In this work, we adopt the Kinetics \cite{Kay:2017:Kinetics} pretrained inflated 3D network (I3D)\cite{Carreira:2017:I3D} as the feature extractor.
We consider multi-modal information such as RGB and optical flow information in the adopted backbone.
Also, for a fair comparison, we also conduct experiments using the TSN \cite{Wang:2016:TSN}, where ResNet \cite{resnet2016} network and BN-Inception \cite{inception2015} network are used as the spatial and temporal network respectively.
\noindent
\textbf{Temporal Action Proposals and Detection.}
Based on the extracted video features from the feature extractor, numbers of recent approaches solve the action detection problems in two steps:
(1) proposal generation;
(2) proposal classification and refinement.
In the first step, the proposals can be generated in a top-down fashion, which is based on preset sliding temporal windows or anchors \cite{Shou:2016:S-CNN, Heilbron:2016:TAP, Gao:2017:CBR, Chao:2018:TAL-Net}.
Or alternatively, the proposals can be generated in a bottom-up fashion, which directly predicts proposal locations based on video frames or snippets features \cite{Escorcia:2016:DAP, Buch:2017:SST, Buch:2017:SS-TAD, Yeung:2016:End, Gao:2017:TURN}.
In such way, actionness probabilities estimated at every time step can be analyzed and the potential start and end positions of the predicted action instances can be connected \cite{Zhao:2017:SSN, Lin:2018:BSN, Gong:2020:TSA, Yuan:2017:SMS, Lin:2020:Fast}.
In the second step, each generated proposal is classified into one of the action categories, whose boundaries could be refined by regression models \cite{Shou:2016:S-CNN, Gao:2017:CBR}.
\noindent
\textbf{Video Context Modeling.}
A most simple way to merge the features from multiple snippets is by directly applying pooling or convolution operations on the snippet features \cite{Shou:2016:S-CNN, Shou:2017:CDC}.
Also, multi-scale pooling or convolution has been widely employed to improve the feature representations \cite{Gao:2017:TURN, Zhao:2017:SSN, Yang:2020:TPN, Gong:2020:TSA}.
Besides the modeling by convolutions, some recent works have attempted to utilize graph convolutional networks (GCNs) or transformer \cite{attention2017} for long-term relationship modeling.
For example, G-TAD \cite{Xu:2020:G-TAD} embeds the temporal context with GCNs, while transformer based method RTDNet \cite{RTDNet:2021} is employed to directly generate proposals without post-process.
More recently, many methods \cite{Zeng:2019:P-GCN,localrich:2021,qtemporal:2021} pay attention to refine the proposals.
In particular, the video features and proposals generated by other proposal generation methods, \textit{e}.\textit{g}., BSN\cite{Lin:2018:BSN} or BMN \cite{Lin:2019:BMN}, are all processed to enhance the proposal-level representations.
However, our proposal generation method only inputs video features and generates proposals by ours.
Apart from the above-mentioned ones, some other approaches may even exploit the context information at object-level \cite{Heilbron:2017:SSC, Wu:2020:CA-RCNN}.
However, those methods are beyond the scope of this paper since we mainly focus on the snippet-level feature learning and proposal generation while not proposal refinement.
\begin{figure*}[t]
\centering
\includegraphics[width=12cm,clip]{figure_mm/arch_v7.pdf}
\caption{The overall architecture of the proposed PRSA-Net.
The snippet-level features of videos are extracted by a CNN network in Feature Extractor.
Then PRSlot modules are employed to contextualize slot representations with $T$ times.
Next, our Boundary Classifier and Proposal Regressor are utilized to generate the predicted proposals.
Finally, post-process is utilized to suppress redundant proposals.
}
\label{fig:arch}
\end{figure*}
\noindent
\textbf{Attention Mechanism.}
The vanilla attention mechanism is proposed by \cite{attention2017} and contributes more to capturing long-range dependencies.
Recently, attention mechanism has been widely applied in the computer vision field, \textit{e}.\textit{g}., image classification ViT \cite{vit2020}, video detection \cite{video2019}, group activity recognition \cite{li2021groupformer}, and object detection \cite{DETR2020}.
ViT \cite{vit2020} is the first work to employ a pure attention mechanism for image classification.
Original Slot Attention \cite{slotattn:2020} is proposed to exploit object-centric representation.
In our work, different from the previous attention mechanism, our proposed region-based attention is estimated directly by high-level correlation features instead of similarity operations, which can concentrate on the local context better.
\section{Methodology}
\label{sec:method}
\subsection{Problem Formulation}
\label{sec:Form}
Assume that an arbitrary untrimmed video $V=\{ v_t \}_{t=1}^T$ contains a collection of $N$ action instances $\Psi = \{\psi_n = (t_{s,n}, t_{e,n}\}_{n=1}^{N}$, where $\psi_n$ refers to the $n$-th action instance and $(t_{s,n}, t_{e,n})$ correspond to its annotated start time, end time.
The aim of temporal action proposal generation task is to predict a set of action proposals $\Phi = \{\phi_w = (t_{s,w}, t_{e,w}, p_w\}_{w=1}^{W}$ as close to the ground truth annotations as possible based on the content of $V$.
Here $p_w$ is the $w$-th proposal confidence.
\subsection{Overall Architecture}
\label{sec:arch}
We propose a Pyramid Region-based Slot Attention Network (PRSA-Net) to generate temporal action proposals precisely.
The pipeline of our PRSA-Net is illustrated in Fig. \ref{fig:arch}.
PRSA-Net consists of three major components: feature extractor, PRSlot modules and dual branch proposal generation head.
The original video is first fed into the feature extractor to produce the snippet-level features $X$ of size $L \times C_{\texttt{input}}$, where $L$ is the snippet length and $C_{\texttt{input}}$ is the feature dimension.
Then, PRSlot modules are employed to enhance action-matter contextual representations.
After this, a dual branch generation head maps the output embedding from the PRSlot modules to final boundaries and confidence scores respectively.
The Boundary classifier is used to detect the boundaries,
while the proposal regressor is utilized to evaluate the proposal-level confidence.
We provide detailed descriptions of PRSA-Net below.
\subsection{Feature Extractor}
Given an untrimmed video $V$, we utilize the Kinetics \cite{Kay:2017:Kinetics} pre-trained I3D \cite{Carreira:2017:I3D} to extract video features.
We consider RGB and optical flow features jointly as they can capture different motion aspects.
Following the implementation of previous methods \cite{Lin:2018:BSN,Xu:2020:G-TAD}, each frame feature is flattened into a $C$-dimensional feature vector and then we group every consecutive $\sigma$ frames into $L = \ceil{T / \sigma}$ snippets.
In this work, these generated video snippets are the minimal units for further feature modeling.
For convenience, we denote the extracted snippet features as $X \in \mathbb{R}^{L \times C}$, where $C$ is the feature dimension and $L$ is the total number of video snippets.
In the end, a 1d convolution is applied to transform the channels into $C_{\texttt{input}}$.
For fair comparisons with \cite{Lin:2018:BSN,Lin:2019:BMN,Xu:2020:G-TAD}, we also conduct experiments based on the Kinetics pre-trained TSN \cite{Wang:2016:TSN} to extract video features.
\subsection{Pyramid Region-based Slot Attention}
\label{sec:CTR}
The original slot attention module is firstly proposed for updating the object-centric representations (\textit{slots}) by similarity-based attention.
Slots produced by slot attention bind to any object in the input embedding and are updated via a learned recurrent function for object-centric learning.
Inspired by this, we propose a tailor-modified Pyramid Region-based Slot Attention (PRSlot) module for capturing action-matter representations by a novel attention mechanism.
\subsubsection{\textbf{Overview.}}
We adopt dense slots to capture boundary-aware representation.
The insight is that the motion content in boundaries context transfers rapidly and can be distinguished from the background.
The slot is initialized with snippet features $X$ and then is updated by the proposed PRSlot module for $T$ Times
For convenience, the $i$-th slot representation refined $t$ times denotes as $u^{(t)}_i$.
In particular, slot features are processed through convolutional layers with stride 1, window size 3 and padding 1, followed by a {ReLU} activation layer, and produce the slot features shaped as $L \times C_{\texttt{embed}}$.
Then, based on the proposed Region-based attention mechanism (\textit{will describe below}), we update our slot representations by the weighted sum of input and then apply {batchnorm} to the output vectors.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{figure_mm/module_v6.pdf}
\caption{Overview of our Pyramid Region-based Slot Attention (PRSlot) module.
For input snippet (slot) features $u_i$, we take $x_i$ as \textit{query} and its local context snippet (slots) feature as \textit{key}.
An encoder-decoder operation is applied to map the correlation features into the local region score vector, and then generate the output slot features through the weighted sum of the input value (V).
}
\label{fig:MSSA}
\end{figure}
\subsubsection{\textbf{Region-based Attention Mechanism.}}
As shown in Fig. \ref{fig:MSSA}, to focus on the crucial local region relations, we design a newly region-based attention mechanism to learn the snippets relationships instead of using the similarity computation (\textit{cosine}, \textit{dot product} or \textit{Euclidean distance}).
Our region-based attention directly estimates the snippets interactions by the content of the snippet and its local surroundings.
Mathematically, given the target input feature $u_i$ and its surrounding features denoted as $\rho(u_i, s)= \{ u_{i-s}, ...,u_{i+s}\}$, where $s$ represents the window size of its local region, we formulate the calculation of the interaction scores as
\begin{equation}
A_i = f_\mathrm{region}(u_i, \rho(u_i, s)).
\label{eq:a-sum}
\end{equation}
Here the output $A_i$ is defined to be the attention vector for the $i$-th snippet (slot), and its length exactly equals $2s+1$, which is the temporal length of the covered surrounding $\rho(u_i, s)$.
$f_\mathrm{region}$ is a series of operations to map the local region features into the interaction confidence scores for the target snippet (slot).
The value of $j$-th element in $A_i$, here denoted as $A_{i,j}$, is expected to indicate the relation score between $j$-th slot and the target slot $i$.
We take every input feature $u_i \in \{u_l\}_{l=0}^{L-1}$ as \textit{query}, and the local context slot of $u_i$ as \textit{key}, our designed region-based attention operation constructs a learnable score vector, which is used to quantitatively measure the importance between the target \textit{query} and \textit{key}.
In detail, this attention mechanism consists of an encoder to embed local contextual information and a decoder to estimate the relevant attention.
Due to the permutation invariant of temporal snippets, we additionally add the position embedding to the input features.
After this, the output slot representation can be described based on the attention over the feature map in the local region.
\noindent
\textbf{Encoder:}
Summarizing the surrounding slot context for the target slot is critical to augment slot representation.
The encoder is utilized to exploit contextual information for the surrounding slots of the target slot.
The detailed implementation is described below.
Upon the input slot features $U \in \mathbb{R}^{C_{\texttt{embed}} \times L}$ (we omit the batch dimension for clarity), we repeat the temporal length $L$ in last dimension for sampling local windows, and then generate a 2D feature map $F \in \mathbb{R}^{C_{\texttt{embed}} \times L \times L}$.
A $1 \times 1$ convolutional layer is adopted to transform the $C_{\texttt{embed}}$-dimension channel into $C_{\texttt{out}}$-dimension channel.
Next, we apply the 2D convolution layer to augment the target slot using the local surrounding context shaped as $2s+1$.
Specifically, the 2D convolutional layer is implemented for target slot feature embedding, where we denote the kernel size as $(2s+1,1)$, padding size $(s,0)$ and stride $(1,1)$, indicating that we capture local context with size $2s+1$ from each column slot.
It generates the feature map $F' \in \mathbb{R}^{C_{\texttt{out}} \times L \times L}$, and we regard this feature map as the high-level correlation matrix.
For this content-based feature map $F'$, we only consider the local contextual information for each target slot.
The process can be formulated as,
\begin{equation}
F''_{i,j}=\left\{
\begin{array}{ccl}
F'_{i,j} \, , & & \qquad {u_i \in \rho(u_j,s)},\\
0 \, , & & \qquad {\mathrm{Otherwise}},\\
\end{array} \right.
\end{equation}
where $F'_{i,j} \in \mathbb{R}^{C_{\texttt{embed}}}$ denotes the slot correlation features between $i$-th slot and $j$-th slot.
\noindent
\textbf{Decoder:}
For the sparse relation features $F''$, a decoder is deployed to map the relation features into the correspond relation scores.
It is noted that we only consider the local surroundings of the target slot and operate scores mapping, which can be formulated as,
\begin{equation}
A_{i,j}=\left\{
\begin{array}{ccl}
f_{\mathrm{region}}^{\mathrm{dec}}(F''_{i,j}) \, ,& & \qquad {u_i \in \rho(u_j,s)},\\
0 \, ,& & \qquad {\mathrm{Otherwise}},\\
\end{array} \right.
\end{equation}
Here $f_{\mathrm{region}}^{\mathrm{dec}}$ represents the decoding functions which is used to map the slots relation features into scores confidence.
In practice, a 2D convolutional layer is applied to distribute the relation features into relation scores, where we set kernel size to $(2s+1,1)$, padding size to $(s,0)$, out channel to $1$ and stride to $(1,1)$.
This convolutional layer is only deployed for the local slots $\rho(u_i,s)$ of target slot $i$ and formulate a slot confidence vector.
Finally, a $\mathrm{Softmax}$ operation is followed to normalize the score matrix.
In this way, each interaction score $A_{i,j}$ is actually dependent on the content of $u_j$ and $\rho(u_j,s)$.
\subsubsection{Iteration and update strategy.}
The original slot attention updates representations via recurrent function at each iteration $t=1...T$.
However, the recurrent function (\textit{e}.\textit{g}. GRU) is time-consuming and achieves limited performance boost.
It will be demonstrated in ablation study.
In order to tackle the issues and make the PRSlot specialize in the temporal action-matter representation exploiting, we develop a parallel pyramid iteration strategy for slot representations updating.
Due to the variants of video duration, we apply a variety of local regions to exploit the slot representation completely.
In detail, $|S|$ region-based attention using different scale surroundings $s \in S$ are deployed in parallel, where $S$ is a collection of snippet region size and $|S|$ denotes the carnality of the set.
In the end, we fuse the slot attention by element sum-wise and aggregate the input values to their assigned slots.
\subsection{Dual branch head}
Next, slot embedding provided by PRSlot modules serves as the latent action-aware representations for the proposal estimation.\\
\textbf{Boundary Classifier.}
A lightweight 1d convolution layer is introduced to transform the input channels into 2 for (\textit{start, end}) detection and a non-linear $\mathsf{sigmoid}$ function is followed to form the start/end probabilities $\mathcal{P}^s$/$\mathcal{P}^e$ separately.\\
\textbf{Proposal Confidence Regressor.}
Following the conventional proposal regression method \cite{Lin:2019:BMN}, we use pre-defined dense anchors to generate densely distributed proposals shaped as $D \times L$ and then apply 1DAlignLayer \cite{Xu:2020:G-TAD}
to extract the proposal-level features for corresponding proposal anchors shaped as $D \times L \times C_{\texttt{out}}$.
Finally, 2 FC layers are used to predict the proposal-level completeness maps $M^{com}$ and classification map $M^{cls}$.
\subsection{Training and Inference}
\label{sec:train}
\textbf{Label Assignment.}
We first generate temporal boundary label $G_s$ and $G_e$ followed by BSN \cite{Lin:2018:BSN}.
Next, we generate the dense proposal label map $G^c \in \mathbb{R}^{D \times L}$.
As described in \cite{Lin:2019:BMN}, for a proposal $G^c_{i,j}$ with start frame $i$ and end frame $i+j$, we calculate the intersection over union (IoU) with all ground-truth and denote the maximum IoU as the value of $G^c_{i,j}$.\\
\textbf{Training.}
We define the following loss function to train our proposed model
\begin{equation}
\mathcal{L}=\mathcal{L}_{{b}}+\mathcal{L}_{p}+ \lambda \mathcal{L}_{norm}
\label{eq:loss}
\end{equation}
Here the boundary classification loss $\mathcal{L}_{b}$ is a weighted binary logistic regression loss used to determine the starting and ending scores $\mathcal{P}^s \in \{ \mathcal{P}_l^s \}_{l=1}^L$ or $\mathcal{P}^e \in \{ \mathcal{P}_l^e \}_{l=1}^L$.
$ \mathcal{L}_{norm}$ is the regularization term for network weights, we set $\lambda=0.0002$.
We also construct the proposal confidence losses $\mathcal{L}_{p}$ which is the combination of cross-entropy loss and mean square error loss to optimize dense proposals confidence scores $M^{com}$ and $M^{cls}$.
It can be formulated as,
\begin{equation}
\mathcal{L}_p=\mathcal{L}_{cls}(M^{cls},G^c)+\lambda_{c} \mathcal{L}_{com}(M^{com},G^c)
\end{equation}
where $\mathcal{L}_{cls}$ is binary logistic regression loss and $\mathcal{L}_{com}$ is the MSE loss, usually we set the balance term $\lambda_c=10$.\\
\textbf{Inference.}
Based on the outputs of the boundaries classifier, we select the valid starting snippets from $\{\mathcal{P}^s_{l}\}_{l=1}^L$ by two conditions:
(1) $\mathcal{P}^{s}_{l-1}<\mathcal{P}^{s}_{l}; \mathcal{P}^{s}_{l}>\mathcal{P}^{s}_{l+1}$;
(2) $\mathcal{P}^s_{l}>0.5 \times \max_{n=1}^L\{\mathcal{P}_n^s\}$.
We apply the same rule for recording ending snippets.
Then, we combine these valid starting and ending snippets and obtain the candidate proposals denoted as $\Phi = \{\phi_w = (t_{s,w}, t_{e,w}, p_w, M_w \}_{w=1}^{W}\}$, where $\mathcal{P}_w=\mathcal{P}^{s}_{t_{s,w}} \cdot \mathcal{P}^{e}_{t_{s,w}}$ and proposal-level confidence $M_w=M^{cls}_{t_{s,w},t_{e,w}} \cdot M^{com}_{t_{s,w},t_{e,w}}$.
Finally, we fuse the proposal confidence scores and output the predicted proposals $\Phi = \{\phi_w = (t_{s,w}, t_{e,w}, S_w \}_{w=1}^{W}\}$, here $S_w=p_w \times M_w$.\\
\textbf{Post-Processing.}
The non-maximum suppression (NMS) algorithm \cite{neubeck:2006:NMS} is adopted to suppress redundant proposals with high overlaps.
Also, we use the Soft-NMS \cite{softnms2017} algorithm to suppress the proposals and report the performance for a fair comparison with previous methods.
\section{Experiments}
\label{sec:expm}
\subsection{Datasets and Evaluation Metrics}
\textbf{THUMOS14 \cite{Idrees:2017:THUMOS}.} THUMOS14 dataset contains respectively 200 videos in the validation for training and 213 videos in the test set for inference.
It has a total number of 20 classes, and each video has around 15 action instances on average.\\
\textbf{ActivityNet-1.3 \cite{Caba:2015:ActivityNet}.}
ActivityNet-1.3 dataset contains 19994 untrimmed videos labeled in a wider range of 200 action categories with a lower 1.5 action instances per video on average.
These videos are split into the training, validation, and test set by the ratio of 2:1:1.
We evaluate on the validation set of ActivityNet-1.3.\\
\textbf{Evaluation metric.}
To assess the performances of action proposal generation, we report the average recall (AR) under different intersections over union thresholds (tIoUs) with various average number of proposals (AN) for each video.
Following the conventions, we adopt the tIoUs of $\{0.5:0.05:1.0\}$ on THUMOS14 and $\{0.5:0.05:0.95\}$ on ActivityNet-1.3.
To evaluate the quality of our generated proposals, we also evaluate the performances of action detection using the mean average precision (mAP) at different tIoUs.
Following the official evaluation API, the tIoUs of $\{0.3, 0.4, 0.5, 0.6, 0.7\}$ are used for THUMOS14, and $\{0.5, 0.75, 0.95\}$ are used for ActivityNet-1.3.
\subsection{Experimental Setup}
Following the conventional setting, we extracted the 2048-dimensional video features by two-stream I3D \cite{Carreira:2017:I3D} pre-trained on Kinetics \cite{Kay:2017:Kinetics} on THUMOS14.
Besides, for a fair comparison, we also conduct experiments based on the two-stream network TSN \cite{Wang:2016:TSN} backbone, where ResNet network \cite{resnet2016} and BN-Inception network \cite{inception2015} are applied as the spatial and temporal network respectively.
We also set $C_{\texttt{input}}=256$.
On ActivityNet-1.3, the video features were extracted by the pre-trained model provided in \cite{Xiong:2016:CUHK, Caba:2015:ActivityNet}.
In data preparation, we set the snippet interval $\sigma$ to 4 on THUMOS14 and 16 on ActivityNet-1.3.
Then we cropped each video feature sequence with the overlapped windows of stride of $100$ and length $L = 250$ on THUMOS14.
While on ActivityNet-1.3, we set the temporal length to $L = 100$ by linear interpolation.
Also, we enumerate all possible anchors where the max durations is $D=64$ on THUMOS14 and $D=100$ on ActivityNet-1.3.
For the PRSlot module, we set the embed dimension $C_{\texttt{embed}}=256$, $C_{\texttt{out}}=256$, and the local region scale $S=\{4, 8\}$.
In post-processing, we set the NMS threshold $\theta$ to 0.65 and 0.45 on THUMOS14 and ActivityNet-1.3 respectively.
During the training, it was optimized on one NVIDIA TESLA V100 GPU with batchsize 16 on ActivityNet1.3 and 8 on THUMOS14.
We adopted the Adam optimizer \cite{Kingma:2019:Adam} for network optimization.
For THUMOS14, the models were tuned for 10 epochs with the learning rate set to $2 \times 10^{-4}$.
For ActivityNet-1.3, we trained our models by setting the learning rate to $10^{-3}$ in the first 7 epochs and decaying it to $10^{-4}$ in the last 3 epochs.
\begin{table}[t]
\centering
\caption{Comparison of the action proposal generation performances with state-of-the-arts on THUMOS14 in terms of AR@AN(\%).}
\scalebox{0.7}{
\begin{tabular}{p{4.0cm}p{2.0cm}<{\centering}p{1.3cm}<{\centering}p{1.3cm}<{\centering}p{1.3cm}<{\centering}p{1.3cm}<{\centering}}
\toprule
Method &Backbone & @50 & @100 & @200 & @500\\
\midrule
MGG \cite{Liu:2019:MGG} &TSN & 39.93 & 47.75 & 54.65 & 61.36\\
BSN \cite{Lin:2018:BSN} + SNMS &TSN & 37.46 & 46.06 & 53.21 & 60.64\\
BMN \cite{Lin:2019:BMN} + SNMS &TSN & 39.36 & 47.72 & 54.70 & 62.07\\
BC-GNN \cite{BCGNN2020} + NMS &TSN & 41.15& 50.35 & 56.23 & 61.45\\
BU-TAL \cite{Zhao:2020:Bottom} &I3D & 44.23 & 50.67 & 55.74 & -\\
BSN++ \cite{bsn++2020} +SNMS &TSN &42.44&49.84&57.61&65.17\\
RTD-Net \cite{RTDNet:2021} &I3D &41.52 &49.32 &56.41 &62.91 \\
\hline
Ours + NMS &TSN &{47.49} & {55.14} & {60.18} & 63.53\\
Ours + SNMS &TSN & 44.11 & 52.52 & 59.19 & {65.12}\\
Ours + NMS &I3D & \textbf{49.06} & \textbf{56.12} & \textbf{61.30} & 63.20\\
Ours + SNMS &I3D &45.81 &53.13 &59.32 &\textbf{66.32} \\
\bottomrule
\end{tabular}
}
\label{tab:THUMOS14-proposal}
\end{table}
\subsection{Temporal Proposal Generation}
\begin{table}[t]
\vspace{-0.1cm}
\centering
\caption{Comparison of the action proposal generation performances with state-of-the-arts on ActivityNet-1.3 in terms of AR@AN(\%) and AUC.}
\scalebox{0.62}{
\begin{tabular}{p{2.2cm}<{\centering}p{2.2cm}<{\centering}p{2.2cm}<{\centering}p{2.2cm}<{\centering}p{2.2cm}<{\centering}p{2.2cm}<{\centering}p{2.2cm}<{\centering}p{2.2cm}<{\centering}}
\toprule
Metric &
BSN \cite{Lin:2018:BSN} \quad &
MGG \cite{Liu:2019:MGG} \quad &
BMN \cite{Lin:2019:BMN} \quad &
BC-GNN \cite{BCGNN2020} \quad &
BU-TAL \cite{Zhao:2020:Bottom} \quad &
RTD-Net \cite{RTDNet:2021} \quad &
Ours\\
\midrule
AR@1 & 32.17 & - & - & - & - & 33.05 &\textbf{35.37} \\
AR@100 & 74.16 & 74.54 & 75.01&76.73 & 75.27 &73.21& \textbf{76.90}\\
AUC & 66.17 & 66.43 & 67.10&68.05 & 66.51 &65.78& \textbf{69.21}\\
\bottomrule
\end{tabular}
}
\vspace{-0.3cm}
\label{tab:acnet-proposal}
\end{table}
\subsubsection{Comparisons with State-of-the-Arts.}
We compare our PRSA-Net with other state-of-the-art methods on two backbones for a fair comparison.
The results on THUMOS14 are summarized in Table \ref{tab:THUMOS14-proposal}.
It can be observed that our proposed PRSA-Net outperforms all of the aforementioned methods by a large margin with either NMS or Soft-NMS used in post-processing.
For instance, when using TSN as feature extractor, our method respectively improve the AR@50 from 42.44\% to 47.49\%, AR@100 from 49.84\% to 55.14\%, and AR@200 from 57.61\% to 60.18\%.
It is worth noting that our model with I3D backbone outperforms all previous methods by a large margin (+ 4.83\% AR@50 and +5.45\% AR@100).
With the well-design PRSlot and its region-based attention, our method establishes new state of the art performance on THUMOS14.
The results on ActivityNet-1.3 are displayed in Table \ref{tab:acnet-proposal}.
Our method shows better performances than the previous best results under both AR@100 and AUC scores.
In particular, the AR@1 achieves 2.32\% performance boost in this well-studied benchmark.
\subsection{Ablation Studies}
\label{sec:ablation}
We conduct extensive experiments on THUMOUS14 using the I3D \cite{Wang:2016:TSN} backbone and NMS for post-processing to investigate the effectiveness and proper settings of the different components of our proposed models.\\
\begin{table}[t]
\centering
\caption{Comparisons of different combinations in our model. Evaluated on THUMOS14 in terms of AR@AN(\%). \texttt{SA} and \texttt{RA} denote the implementation of similarity-based and our region-based attention separately. \texttt{original} and \texttt{ours} denote the original and our parallel pyramid strategy respectively.}
\scalebox{0.8}{
\begin{tabular}{p{2.0cm}<{\centering}p{2.0cm}<{\centering}|p{2.0cm}<{\centering}p{2.0cm}<{\centering}|p{1.2cm}<{\centering}p{1.2cm}<{\centering}p{1.2cm}<{\centering}}
\hline
\multicolumn{2}{c|}{{Attention Mechanism}} &\multicolumn{2}{c|}{{Iteration Strategy}}& \multicolumn{3}{c}{AR@AN (\textit{testing set)}} \\
$\texttt{SA}$ & $\texttt{RA}$ & \texttt{original}&\texttt{ours}&@50 & @100 & @200 \\
\hline
\cmark & & \cmark & & 35.6 &43.1 &50.9 \\
\cmark & & &\cmark & 43.5 & 52.3 & 56.5\\
& \cmark & \cmark& & 42.9 & 52.8 & 55.6\\
& \cmark & &\cmark &\textbf{49.1}&\textbf{56.1}&\textbf{61.3}\\
\bottomrule
\end{tabular}
}
\label{tab:clue-combinations}
\end{table}
\begin{table}[t]
\centering
\caption{Comparisons of different local region scale choices in region-based attention. Evaluated on THUMOS14 in terms of AR@AN(\%).}
\scalebox{0.8}{
\begin{tabular}{p{4cm}<{\centering}p{1.7cm}<{\centering}p{1.7cm}<{\centering}p{1.7cm}<{\centering}p{1.7cm}<{\centering}}
\toprule
Local Region Scale&@50 &@100 &@200 &@500 \\
\midrule
2 &47.2 &54.3 &59.4 &62.9\\
4 &47.8 &55.4 &59.9 &63.2 \\
8 &48.8 &56.0 &60.6 &63.0\\
12 &49.0&56.1 &60.5 &62.7\\
\hline
2, 4 &48.6 &55.7 &60.4 &63.0\\
4, 8 &\textbf{49.1} &\textbf{56.1} &61.3 &63.2\\
2, 4, 8 &{49.1} &{56.1} &60.2 &63.2\\
4, 8, 12 &48.7&55.8&\textbf{61.5}&\textbf{63.4}\\
\bottomrule
\end{tabular}
}
\label{tab:scale-choice}
\end{table}
\vspace{-0.3cm}
\begin{table}[t]
\centering
\caption{Different setting choices for the number of pyramid iterations.
Evaluated on THUMOS14 in terms of AR@AN (\%)}
\scalebox{0.7}{
\begin{tabular}{p{4cm}<{\centering}p{2cm}<{\centering}p{2cm}<{\centering}p{2cm}<{\centering}p{2cm}<{\centering}}
\toprule
Iteration times & @50 &@100&@200&@500 \\
\midrule
1 &48.4&55.3&60.1&61.6 \\
2 &\textbf{49.1}&\textbf{56.1}&61.3&\textbf{63.2} \\
3 &48.9&55.7&\textbf{61.9}&62.3 \\
\bottomrule
\end{tabular}
}
\label{tab:blocks}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figure_mm/quality_res_v3.pdf}
\caption{Some action proposal examples predicted by our PRSA-Net on THUMOS14.}
\label{fig:quality_result}
\vspace{-0.1cm}
\end{figure}
\noindent
\textbf{Variants of our PRSA-Net.}
To measure the importance of our proposed region-based attention and parallel pyramid iteration strategy, we conduct the ablation study with the following variants.
Baseline: we replace the PRSlot with the original slot attention, which includes the similarity-based attention and recurrent iteration strategy.
More baseline details can be found in our Supplementary materials.
We also adopt different combinations of our PRSlot architecture components.
Table \ref{tab:clue-combinations} reports the detailed proposal generation results on THUMOS14.
The \cmark \, symbols stand for \textit{with} the corresponding components or strategies.
We can find that our proposed region-based attention (\textit{the last row}) improves the AR by more than 6\%, indicating the effectiveness of our designed PRSlot module.
Additionally, our parallel pyramid iteration strategy also contributes to performance boosts.
We attribute performance differences to variations in architecture design and iteration schemes.
\noindent
\textbf{Scale of local region.}
Next, we investigate the effect of the local region size.
The results on THUMOS14 are displayed in Table \ref{tab:scale-choice}.
The first column of the table lists the corresponding choices of $S$ in a parallel pyramid strategy.
We can find that (1) a single scale with large $s = 12$ can perform already quite well; and
(2) the multi-scale regions of $\{4, 8\}$ and $\{4, 8, 12\}$ alternatively reach the best results under different AN.
These results demonstrate the effectiveness of our parallel pyramid strategy for complementary context representation modeling.
To balance the performance and model simplicity, we implement our models with $\{4, 8\}$.\\
\textbf{Sensitivity to iteration times.}
We also report the results of the sensitivity of our model to the different settings for iteration times in Table \ref{tab:blocks}.
We find that using $2$ iterations can reach the best results while performance slightly degrades at using $3$ times.
\subsection{Qualitative Results}
In Fig. \ref{fig:quality_result}, we visualize some action detection results generated by our PRSA-Net on THUMOS14.
We can find that our model can successfully detect the action instances in these examples with precise action boundaries.
For the untrimmed video with multiple action instances, the foreground actions and background scenes can be well distinguished.
\subsection{Action Detection with our proposals}
\begin{table}[t]
\small
\centering
\caption{Performance comparisons of temporal action detection on THUMOS14 in terms of mAP@tIoUs(\%).}
\scalebox{0.7}{
\begin{tabular}{p{3cm}<{\centering}p{2cm}<{\centering}p{2cm}<{\centering}p{1.5cm}<{\centering}p{1.5cm}<{\centering}p{1.5cm}<{\centering}p{1.5cm}<{\centering}p{1.5cm}<{\centering}}
\toprule
Method &Backbone &Classifier & 0.7 & 0.6 & 0.5 &0.4 &0.3\\
\midrule
BSN \cite{Lin:2018:BSN} &TSN &UNet &20.0 &28.4 &36.9 &45.0 &53.5\\
MGG \cite{Liu:2019:MGG} &TSN &UNet &21.3 &29.5 &37.4 &46.8 &53.9\\
BMN \cite{Lin:2019:BMN} &TSN &UNet &20.5 &29.7 &38.8 &47.4 &56.0\\
G-TAD \cite{Xu:2020:G-TAD} &TSN &UNet &23.4 &30.8 &40.2 &47.6 &54.5\\
BU-TAL et al. \cite{Zhao:2020:Bottom} &I3D &UNet &28.5 &38.0 &45.4 &50.7 &53.9\\
BSN++ \cite{bsn++2020} &TSN &UNet &22.8&31.9&41.3&49.5&59.9\\
BC-GNN \cite{BCGNN2020} &TSN &UNet &23.1&31.2 &40.4 &49.1 &57.1\\
RTD-Net \cite{RTDNet:2021} &I3D &UNet &25.0 &36.4 &45.1 &53.1 &58.5 \\
\hline
Ours &TSN &UNet &{28.8} &{39.2} &{51.1} &{58.9} &{65.4}\\
Ours &I3D &UNet &\textbf{30.9} &\textbf{44.0} &\textbf{55.0} &\textbf{64.4} &\textbf{69.1}\\
\hline
BSN \cite{Lin:2018:BSN} &TSN &PGCN &- &- &49.1 &57.8 &63.6\\
G-TAD \cite{Xu:2020:G-TAD} &TSN &PGCN &22.9 &37.6 &51.6 &60.4 &66.4 \\
RTD-Net \cite{RTDNet:2021} &I3D &PGCN &23.7 &38.8 &51.9 &62.3 &68.3 \\
\hline
Ours &I3D &PGCN &\textbf{28.4}&\textbf{47.3}&\textbf{58.7}&\textbf{73.2}&\textbf{76.3} \\
\bottomrule
\end{tabular}
}
\label{tab:THUMOS14-localization}
\end{table}
\begin{table}[t]
\small
\centering
\caption{Comparison with state-of-the-arts detection methods on ActivityNet-1.3 in terms of mAP@tIoUs(\%).}
\scalebox{0.8}{
\begin{tabular}{p{2cm}<{\centering}p{2cm}<{\centering}p{2cm}<{\centering}p{2cm}<{\centering}p{2cm}<{\centering}}
\toprule
Method & 0.5 & 0.75 & 0.95 & Average\\
\midrule
TAL-Net \cite{Chao:2018:TAL-Net} &38.23 &18.30 &1.30 &20.22\\
BSN \cite{Lin:2018:BSN} &46.45 &29.96 &8.02 &30.03\\
P-GCN \cite{Zeng:2019:P-GCN} &48.26 &33.16 &3.27 &31.11\\
BMN \cite{Lin:2019:BMN} &50.07 &34.78 &8.29 &33.85\\
G-TAD \cite{Xu:2020:G-TAD} &50.36 &34.60 &9.02 &34.09\\
BSN++ \cite{bsn++2020} &51.27 &35.70 &8.33 &34.88\\
BC-GNN \cite{BCGNN2020} &50.56 &35.35 &{9.71} &34.68\\
RTD-Net \cite{RTDNet:2021} &47.21 &30.68 &8.61 &30.83 \\
\hline
Ours &\textbf{52.37} &\textbf{37.18} &\textbf{9.78} &\textbf{36.26}\\
\bottomrule
\end{tabular}
}
\vspace{-0.5cm}
\label{tab:acnet-localization}
\end{table}
When evaluating the quality of our proposals, we follow the conventional two-stage action detection works \cite{Xu:2020:G-TAD,Lin:2019:BMN,Lin:2018:BSN}.
Therefore, we classify the candidate proposals using external classifiers.
We use the video classifier in UntrimmedNet \cite{Wang:2017:Unet} to assign the video-level action classes on THUMOS14, while on ActivityNet1.3, we adopt the video-level classification results from \cite{Xiong:2016:CUHK} to assign the action labels to detect the action class.
Furthermore, we also introduce the proposal-level classifier P-GCN \cite{Zeng:2019:P-GCN} to predict action labels for every candidate proposals.
We evaluate the final action detection performances and make comparisons with the state-of-the-art.
The results on THUMOS14 are summarized in Table \ref{tab:THUMOS14-localization}.
Compared with the other state-of-the-art methods, our approach achieves significant improvements under all tIoU settings.
Especially, the mAP at the typical IoU = 0.5 was boosted from 45.4\% to 55.0\%, reaching a considerable improvement ratio of 21.1\%.
Also, when proposal-level classifier P-GCN is applied to classify our generated proposals following the same implementations in \cite{Xu:2020:G-TAD,RTDNet:2021}, the performance can be boosted rapidly and achieve 58.7\% [email protected].
Table \ref{tab:acnet-localization} shows our action detection results on the validation set of ActivityNet-1.3 compared with previous works.
Again, our method outperforms most other methods by a large margin in almost all cases, including the average mAPs over different tIoUs.
These experiments demonstrate that the proposals generated by our PRSA-Net are able to boost the action detection performance better.
\section{Conclusion}
\label{sec:end}
In this paper, we propose a novel Pyramid Region-based Slot Attention Network (PRSA-Net) for temporal action proposal generation.
Specifically, a Pyramid Region-based Slot Attention (PRSlot) module is introduced to capture action-aware representations, which is enhanced by the proposed region-based attention and parallel pyramid iteration strategy.
The experiments show the advantages of our method both in action proposals and action detection performance.
\bibliographystyle{splncs04}
|
1,941,325,220,527 | arxiv | \section{\label{sec:conclusion} Conclusion}
\vspace{-1em}
A new method for generating electromagnetic waves using the permanent magnet's static magnetic flux has been introduced. By using reluctance modulation, the direction of the magnetic flux and the location of the stored magnetic energy have been changed to create a time-variant field. A method for evaluating a ULF transmitter's performance has also been implemented and used to assess the proposed transmitter. It has been shown that the prototype transmitter produces a time-variant field with a modulation depth of 50 percent. While we have not tried to minimize the size and weight of the transmitter, it has realistic dimensions and weight. We also analyzed the power consumption of the transmitter and the calculated results. The calculations show that we can generate 1 fT of time-variant magnetic flux at 1 km using a magnet volume of $10\,cm^3$.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
\vspace{-1em}
\section{\label{sec:introduction} Introduction/Background}
\vspace{-1em}
The emphasis is primarily on increasing data rate, which leads to the use of higher frequencies and wider bandwidths in modern communication technology research and innovations. However, because of physical constraints, increasing frequency and bandwidth in many areas of technology cannot necessarily be beneficial. Communication under seawater or other challenging RF environment require very low-frequency, VLF, or ultra-low-frequency, ULF signals to penetrate lossy media that block high-frequency signals. Also, new developments in neuroscience have shown the potentials of ULF and VLF electromagnetic, EM, waves to treat neurological conditions such as Alzheimer's disease, amyotrophic lateral sclerosis, persistent vegetative diseases, epilepsy, stroke-related illness, tinnitus, multiple sclerosis, schizophrenia, and traumatic brain injury. The main challenge is that most of VLF and ULF generators are large and power-hungry, which make them impractical or hard to use in many applications. In this paper, we present a new approach for generating EM waves in a compact and low-power fashion.
At first, radio wave technology was developed within the VLF ranges. Because of the broad spectrum of radiated waves and the problem of spark-gap oscillators (invented by Hertz) interference, William Crookes proposed using sinusoidal sources in resonance structures (then called syntony) to minimize the transmitter and receiver bandwidth in 1892 \cite{Crookes92}. It started a race to develop a continuous wave, CW, sinusoidal wave generator to replace the spark-gap sources for RF applications. Innovative structures were proposed by several researchers (Elihu Thomson, Nikola Tesla, Reginald Fessenden, and many others). Finally, the spark-gap oscillators were replaced by the Alexanderson alternator (a mechanical structure based on a rotating permanent magnet) in 1904. Surprisingly, many variants of Alexanderson alternator have been suggested after more than a century \cite{Fawole17,Strachen18,Burch18,Gong18,Golk18,Prasad18,Bickford19}, in response to a DARPA call for ELF and VLF sources in recent years. Such mechanical generators (mechtenna), however, still have the same shortcomings as the original design, such as large size, massive power consumption, hard to modulate and transmit information, synchronization, noise, vibration, and durability problem of a mechanical structure. There have also been other versions of mechanical vibration proposed to generate EM waves in VLF ranges as well \cite{Trouton92,Datskos12,Madan17,Wang18,Scott19}.
In 1961, on the other hand, an analysis of EM radiation from the acoustically-driven ferromagnetic yttrium iron garnet sphere (YIG) introduced the concept of acoustic resonance as strain powered (SP) antenna. Recent studies have shown that in a device with smaller physical dimensions than the EM wavelength, multiferroic antennas can take advantage of acoustic resonance to reduce antenna size \cite{Nan08,Nan17,Domann17}. As a contrast to rotating permanent magnets, strain-coupled piezoelectric and magnetostrictive composites are thus used in magnetostrictive materials to control magnetic spin states \cite{Ueno03,Xu19,Schneider19}. Although this technique removes the necessary inertial force in mechtenna, it faces challenges due to the matching rigidity between piezoelectric and magnetostrictive (i.e., low energy transfer from piezoelectric to magnetostrictive) and sequentially inefficient power transfer to electromagnetic radiation. Besides, making this structure into bulk is also a challenge. As an alternative technique, we intend to use magnetic material to manipulate the magnetic flux of a permanent magnet.
This idea is to alter the reluctance of the flux path to make the magnetic flux time-variant by pushing it to take an alternative path. Our concept is based on `variable material' rather than `variable structure' as in mechanical rotation. We take advantage of a permanent magnet, which is equivalent to a lossless electromagnet with the winding of the superconductor, which produces a static magnetic flux without dissipating power. Meanwhile, we alternate the direction of flux between free space and a medium with high permeability. The permeability of the magnetic material near the permanent magnet varies by adjusting the current through a control coil, depending on the B-H curve of the magnetic material \cite{Strachen18}.
There are many papers published in the last three years on ULF antennas; however, most of them have not evaluated their work with a concrete criterion. Therefore the performances of these proposed antennas are difficult to assess and compare. We consider a permanent magnet's magnetic flux to be a suitable reference to evaluate the performance of any ULF transmitter. Hence, from now on, we believe the field produced by a rotating magnet to be a reference to assess the field generated by any technique with the same volume magnet. In this way, we calibrate the receiving device (searching coil or any other type of magnetometer), especially if we can rotate the magnet to the generator's operating frequency. Also, we suggest calculating the leakage of the windings around the ferrite cores independently of the permanent magnet to be able to distinguish between the permanent magnet's contributions and the entire field of windings.
In this research, the magnetic flux per volume of selected published designs is compared in \textcolor{cyan}{Table~\ref{tab:table1}} to give a better estimate of the performance of our design. Note that most articles do not provide details about the antenna's total volume, and the information is limited to the size of the main radiating element. The objective of this comparison is to determine the minimum volume needed to reach a field strength of 1 fT at 1 km.
As shown in \textcolor{cyan}{Table~\ref{tab:table1}}, the results for the radiator volume of $1\,cm^3$ ($\Delta$B/Vrad) show that the rotating magnet has the maximum magnetic flux, as expected. Without any modulation, the rotating magnet generates a magnetic flux of about $200 \times 10^{-3}\, fT/cm^3$, whereas any designs aimed at modulating magnet rotation reduced its efficiency significantly. Furthermore, our proposed design and the best multiferroic antenna design in the literature can generate $98 \times 10^{-3}$ and $13.3 \times 10^{-3}\, fT/cm^3$ magnetic fluxes corresponding to 49\% and 13\% efficiency of these antennas, respectively. The results show at the time of publication that the proposed design has the best chance to compete with a rotating magnet with considerably lower power consumption and smaller size.
\vspace{-1em}
\begin{table*}
\small
\caption{\label{tab:table1} Comparison of different low-frequency antennas for use in underwate and underground communication.}
\resizebox{0.97\textwidth}{!}{%
\begin{ruledtabular}
\begin{tabular}{l|l|l|l|l|l|l|l|l}
{\bf Method} & {\bf Ref} & {\bf Modulation \footnote{EMR: Electrically Modulated Reluctance, DAM: Direct Antenna Modulation}} & \begin{tabular}[c]{@{}l@{}}{\bm{ $V_{rad}$}\footnote{The volume of the central radiator}}\\{\bf $\left (cm^3 \right)$}\end{tabular} & \begin{tabular}[c]{@{}l@{}}{\bm{$V_{Antenna}/ D$} \footnote{This column describes the total size of the antenna and the largest antenna dimension extracted from the literature, where applicable.}}\\{\bf $\left (cm^3 / cm \right)$}\end{tabular} & {\bf Freq (Hz)} & {\bf $\Delta B$} & \begin{tabular}[c]{@{}l@{}}{\bf $\Delta B (fT)$}\\{\bf $at 1 km$}\end{tabular} & \begin{tabular}[c]{@{}l@{}}{\bf $\Delta B/ V_{rad}$ }\\{$\left (fT/ cm^3 \right) \times 10^{-3}$}\\{\bf $at 1 km$}\end{tabular} \\
\hline
{} & ${}_{\cite{Gong18}}$ & - & \textsc{100} & -/10 & 30 & 1 pT at 264.8 m & 18.5 & 185.00 \\ \cline{2-9}
{} & ${}_{\cite{Scott19}}$ & - & $100$ & - & 100 & 1800 nT at 2.03 m & 15.06 & 150.60 \\ \cline{2-9}
\multirow{-3}{*}{\begin{tabular}[c]{@{}l@{}}{\bf Rotating}\\ {\bf Magnet}\end{tabular}} & ${}_{\cite{Burch18}}$ & - & 3 & 3/1.6 & 500 & 800 fT at 100 m & 0.8000 & 266.67 \footnote{This value shows a higher value than the theory, which may be due to the magnetometer's error.} \\ \hline
{} & ${}_{\cite{Fawole17}}$ & Electromechanically & 8.4 & 58.5/15.6 & 22 & 600 nT at 1 m & 0.6000 & 71.43 \\ \cline{2-9}
{} & ${}_{\cite{Strachen18}}$ & EMR & 3.62 & 353/8 & 150 & 100 nT at 0.3 m & 0.0027 & 0.75 \\ \cline{2-9}
\multirow{-3}{*}{\begin{tabular}[c]{@{}l@{}}{\bf Modulated}\\ {\bf Magnet}\\ {\bf Rotation}\end{tabular}} & ${}_{\cite{Golk18}}$ & Mechanical Shutter & - & - & 960 & 1.3 nT at 1 m & 0.0013 & - \\ \hline
{\bf Pendulum Array} & ${}_{\cite{Prasad19}}$ & DAM & 29.91 & -/13.4 & 1030 & 79.4 fT at 20 m & 0.0006 & 0.02\\ \hline
{\bf Piezoelectric} & ${}_{\cite{Kemp19}}$ & DAM & 18.9 & -/9.4 & 35500 & - & - & - \\ \hline
{} & ${}_{\cite{Xu19}}$ & DAM & 6.4 & -/25 & 28000 & 16 nT at 0.4 m & 0.0010 & 0.16\\ \cline{2-9}
\multirow{-2}{*}{\bf Multiferroic} & ${}_{\cite{Schneider19}}$ & DAM & 1 & -/18 & 10 & 6.05 nT at 1.3 m & 0.0133 & 13.30 \\ \hline
{\bf Motionless} & - & EMR \& DAM & 3 & 280/14 & 430 & 170 nT at 1.2 m & 0.2940 & 98
\end{tabular}
\end{ruledtabular}}
\end{table*}
\section*{References}
\vspace{-1em}
\section{\label{sec:measurement} Measured Results}
\vspace{-1em}
Assessing the performance of the prototyped transmitter (\textcolor{cyan}{Fig.~\ref{fig:fig4(a))}}) is a significant challenge due to the lack of a reliable and calibrated magnetometer. As a result, we use the magnetic field of a rotating permanent magnet as a reference. We also used a low-noise audio amplifier connected to an air-core search coil as a receiver. Besides, the magnet used in the transmitter and the one used as the rotating magnet are identical. If the measurement setup is the same (the transmitter replaces the rotating magnet while the relative location to the search coil is the same), we can assess our transmitter accurately. \textcolor{cyan}{Figure~\ref{fig:fig4(bc)}} shows the permanent magnet plastic case, which connects to a Dremel 4000 rotary tool (35000 rpm) through its main shaft (see the inset of \textcolor{cyan}{Fig.~\ref{fig:fig4(bc)}}). A metal shaft is in place to secure the other end of the plastic enclosure to a solid fixture when it rotates. We were able to rotate the magnet up to 25800 rpm (equivalent to 430Hz). \textcolor{cyan}{Figure~\ref{fig:fig4(c)}} shows the rotating magnet and the search coil with the distance $R = 1.2\, m$.
\begin{figure}
\centering
\subfigure[]
{
\includegraphics[clip, trim=2.3cm 20.5cm 10.5cm 2.5cm,width=0.9\columnwidth]{Fig/Fig4_a.pdf}
\label{fig:fig4(a))}
}
\subfigure[]
{
\includegraphics[clip, trim=2.5cm 20.5cm 10.5cm 2cm,width=0.9\columnwidth,height=4.5cm]{Fig/Fig4_bc.pdf}\label{fig:fig4(bc)}
}
\subfigure[]
{
\includegraphics[clip, trim=2.5cm 23.5cm 12.5cm 2.7cm,width=0.9\columnwidth]{Fig/Fig4_d.pdf}
\label{fig:fig4(c)}
}
\caption{\textit{a}) Photograph of the prototyped transmitter \textit{b}) photograph of the permanent magnet plastic case. The inset shows the photograph of magnet and Dremel 4000 rotary tool \textit{c}) Maxwell model of the magnet rotation setup.}
\end{figure}
\textcolor{cyan}{Figure~\ref{fig:fig5(a)}} displays the measured signal at the output of the low-noise audio amplifier connected to the search coil as the rotary tool rotates the magnet. The distorted waveform is due to the non-linearity of the detection circuitry (audio amplifier). Next, we remove the rotary system and replace it with the proposed transmitter. We used a signal generator to feed the proposed transmitter via a buffer amplifier with a 430 Hz sinusoidal waveform. In addition to the voltage waveform on the audio amplifier output, \textcolor{cyan}{Fig.~\ref{fig:fig5(b)}} displays the input current waveform. Comparing the two voltage waveforms in \textcolor{cyan}{Fig.~\ref{fig:fig5}} is reasonable by maintaining the same method of receiving and measuring for both cases. Notice that the rotating magnet switches its field polarity per half a cycle (swinging between $+B(R)$ and $-B(R)$ or $2\Delta B_{max}$) while the proposed transmitter can open and close the entire magnetic flux of the magnet (swinging between 0 and $+B(R)$ or $\Delta B_{t}$) at its peak. Therefore the rotating magnet produces twice as much a time-varying magnetic flux as our proposed transmitter produces at its ideal performance. Besides, the magnetic flux maximum $\Delta B_{max}$ is equal to its static value for a given permanent magnet, due to the low frequency (quasi-static). Simply, a rotating magnet's time-variant magnetic flux is equal to $B_{max}\, cos \omega t $. From now on, we compare the transmitter's measured time-variant flux with the permanent magnet's static flux at the same point, and we call it modulation depth.
\begin{equation}
Modulation\;depth = \frac{\Delta B_{t}}{\Delta B_{max}} \times 100 \quad \left (\% \right)
\label{eq:four}
\end{equation}
\begin{figure}
\centering
\subfigure[]
{
\includegraphics[clip, trim=2.5cm 21.3cm 11.8cm 2.5cm,width=0.9\columnwidth]{Fig/Fig5_a.pdf}
\label{fig:fig5(a)}
}
\\
\subfigure[]
{
\includegraphics[clip, trim=2.5cm 21cm 11.5cm 2.5cm,width=0.9\columnwidth]{Fig/Fig5_b.pdf}
\label{fig:fig5(b)}
}
\caption{Measured magnetic flux of \textit{a}) rotating permanent magnet and \textit{b}) proposed transmitter.}
\label{fig:fig5}
\end{figure}
Measuring the total power required to generate a time-varying magnetic flux at a given distance is a crucial factor in evaluating the transmitter's performance. Based on the measured signal shown in \textcolor{cyan}{Fig.~\ref{fig:fig5(b)}}, the sinusoidal voltage applied to the control coil is 0.95 V, and the current is 0.6 A, which results in an average power of 0.285 W, while the rotary device needs 60 W to rotate the magnet. The modulation depth of the proposed transmitter and the rotary device can be compared with the measured input power in mind. The measured flux, shown in \textcolor{cyan}{Fig.~\ref{fig:fig5}}, used to calculate the modulation depth of 51\%. Note that the maximum modulation depth for the transmitter is 100\%, while the magnet's modulation depth is 200\%. \textcolor{cyan}{Figure~\ref{fig:fig6}} also shows the measured modulation depth of the transmitter versus the input power.
\begin{figure}
\centering
\includegraphics[clip, trim=5cm 20.3cm 8.5cm 1.5cm,width=0.9\columnwidth]{Fig/Fig6.pdf}
\caption{The modulation depth as a function of the input power of the transmitter.}
\label{fig:fig6}
\end{figure}
One approach to verifying the measurement method is to measure the magnitude of the magnetic flux at various distances for a given sinusoidal drive current. \textcolor{cyan}{Figure~\ref{fig:fig7}} shows the output voltage of the receiver vs. {\it R}. The magnetic flux (which is linearly proportional to the output voltage) decays by $1/ R^3$ as expected. Besides, this figure provides a guideline for estimating the magnitude of the time-variant magnetic flux at any distance where measured/simulated data at least at one point in the same direction is available. The theoretical equation \cite{Manteghi17} was used to find the magnetic flux for the rotating magnet and then to determine the coefficient required to convert the obtained voltage to the magnetic flux.
\begin{figure}
\centering
\includegraphics[clip, trim=2.5cm 22.2cm 12.5cm 2.5cm,width=0.9\columnwidth]{Fig/Fig7.pdf}
\caption{The measured field versus range. The data points show each individual measurement, and the line is the result of curve fitting.}
\label{fig:fig7}
\end{figure}
The transmitter will, therefore, generate $0.17\, \mu T$ at 1.2 m. In the same way, the $1/ R^3$ decay of the magnetic field of the antenna allows extrapolating the field at a distance of 1 km, although the magnetic flux of 1 km is too low to measure with our magnetometer. It is estimated that the magnetic flux will be 0.294 fT at 1 km. This study is conducted to determine the magnet volume needed to achieve a field strength of 1 fT at 1 km. The results show that 1 fT can be accomplished at 1 km with a permanent magnet volume of $10\,cm^3$ with a power consumption of less than 0.5 W. Also, the proposed antenna is compared with other current designs in \textcolor{cyan}{Table~\ref{tab:table1}}. The magnetic field generated by volume ($\Delta B/V$) for different designs shows that the rotary magnet systems produce the maximum field with a range of approximately $0.2\, fT/cm^3$. However, this technique has its limitations. The multiferroic transmitter, which generates a magnetic field of approximately $0.013\, fT/cm^3$, is also far from competing with the rotating magnet. The proposed transmitter in this paper can generate a $0.1\, fT/cm^3$ magnetic field, making it a feasible candidate to compete with the rotating magnet.
In terms of bandwidth and data rate, the proposed transmitter does not comply with the fundamental antenna limits. The conventional antenna design approaches depend on the practical and useful Linear Time-Invariant (LTI) systems. For example, a lossless tuned electrically small antenna (ESA) at resonance can be treated as a second-order resonator, where the stored electrical/magnetic energy in its reactive zone exchanges the stored magnetic/electric energy in the reactive lumped element of the antenna’s matching circuit. For example, a 1-meter lossless resonant antenna at 1 kHz ($\lambda = 300 m$) has a minimum Q of $10^{14}$ (bandwidth of $10^{-11}\, Hz$). However, bandwidth can be increased by sacrificing the antenna efficiency that can be achieved only on the receive side, but not on the transmitter. However, it has been shown that the fundamental limits of the antennas do not bound the non-linear and/or time-variant (non-LTI) antennas \cite{Manteghi19}.
For example, a time-variant field can be created while avoiding resonance, if the stored energy in an antenna's reactive near-zone does not transform into another type of energy every half a cycle (first-order system), and time variation is realized by changing the location where the energy is stored. Therefore, the time-variant basis of the proposed structure gives rise to a parametric or non-LTI system that allows us to change the data transfer rate, independently from the antenna quality factor. As a consequence, this non-LTI system results in higher data rates being feasible. Moreover, it has shown that the stored energy frequency can be quickly shifted (FSK) without breaching the fundamental limits \cite{Salehi13}. Therefore, the frequency of the field modulation in the proposed transmitter can be changed from a few hundred hertz to tens of kilohertz without any restriction. Besides, any type of modulation, such as frequency or amplitude modulation, can be applied to the proposed transmitter.
\vspace{-1em}
\section{\label{sec:simulation} Simulation Results}
\vspace{-1em}
We conduct further analysis in the simulation domain after verifying the transmitter's functionality in the measurement domain. We used magnetostatic simulation in the software package, ANSYS Maxwell, to achieve that objective. In this analysis, four different cases have been simulated: 1- an isolated permanent magnet, 2- an open mode transmitter (current ON), 3- a closed mode transmitter (current OFF), and 4- a deep closed mode transmitter (reverse current ON). One can use the case 1 magnetic flux to examine the effects of the electric current and the magnetic film thickness on magnetic flux in case 2 and case 3. Also, the modulation depth is determined by subtracting from case 2 the magnetic flux in case 3 or 4 and dividing the result by case 1 magnetic flux. The simulation results for different cases at $R = 0.88\,m$ are shown in \textcolor{cyan}{Fig.~\ref{fig:fig8}}.
\begin{figure}
\centering
\subfigure[]
{
\includegraphics[clip, trim=2.5cm 21.2cm 12.5cm 2.3cm,width=0.9\columnwidth]{Fig/Fig8_a.pdf}
\label{fig:fig8(a)}
}
\\
\subfigure[]
{
\includegraphics[clip, trim=2.5cm 22.7cm 12.8cm 2.5cm,width=0.89\columnwidth]{Fig/Fig8_b.pdf}
\label{fig:fig8(b)}
}
\subfigure[]
{
\includegraphics[clip, trim=2.7cm 22.7cm 12.9cm 2.3cm,width=0.85\columnwidth]{Fig/Fig8_c.pdf}
\label{fig:fig8(c)}
}
\caption{Simulation results; \textit{a}) Magnitude of the magnetic flux at 0.88 m away from the magnet in the magnetostatic solver, \textit{b}) time domain solution of the rotating magnet, and \textit{c}) time domain solution of the transmitter in the transient solver, for two different control currents.}
\label{fig:fig8}
\end{figure}
\textcolor{cyan}{Figure~\ref{fig:fig8(a)}} shows that case 2 (open-mode transmitter) generates 54\% of the flux from an isolated magnet (case 1). This value is essential as we determine the size of the magnet required for a given application. Besides, the modulation depth for case 3 and case 4 is 41\% and 46\%, respectively. Although we used an approximate B-H curve for the Metglas film in the simulation domain, the results are in good agreement with the measured results (51\% modulation depth). Note that the drive current is a balanced sinusoidal in our measurement setup (plus and minus currents); therefore, we compare the measured results with modulation depth in case 4 as 46\%. Next, we analyze the time-domain behavior of the rotating magnet and the proposed transmitter using a transient analysis by ANSYS Maxwell. \textcolor{cyan}{Figure~\ref{fig:fig8(b)}} shows the magnetic flux of the rotating magnet at {\it R} = 0.88 m. As we expected for a quasi-static case, the maximum value of the flux is equal to the magnetic flux of the static magnet at the same distance {\it R} = 0.88 m. The same behavior is observed for the proposed transmitter for two different drive currents.
\begin{figure}
\centering
\includegraphics[clip, trim=2.5cm 20.3cm 12.8cm 2.3cm,width=0.9\columnwidth]{Fig/Fig9.pdf}
\caption{Effect of the number of layers in modulation depth, when 0.5 A and 0.7 A, are applied as input control current.}
\label{fig:fig9}
\end{figure}
We have also analyzed the effect of the magnetic film thickness on the transmitter performance. The Metglas film available comes in a roll, with a thickness of 10 mil (0.0254 mm). The thickness of the magnetic film can, therefore, vary from one layer to an integer number of layers $n \times 10\, mil$. \textcolor{cyan}{Figure~\ref{fig:fig9}} shows the simulation results for a variety of Metglas layers used in the magnetic film for two different drive currents. The optimal number of layers for drive current of 500 mA and 700 mA is 7 and 8, respectively. Therefore, to build the magnetic film, one has to know the drive current in addition to the magnetic material's B-H curve.
\vspace{-1em}
\section{\label{sec:theory} Theoretical Background}
\vspace{-1em}
The traditional way of generating electromagnetic waves is to periodically exchange electric and magnetic energy stored in two distinct parts of the radiating system. Any or both types of stored energy may leak some power as radiation. We thus have a specific amount of radiated power, $P_{r}$, for a maximum amount of stored energy, $W_{max}$, and we can calculate the antenna's quality factor as $Q=\omega W_{max}/ P_{r}$. Fundamental limits of antennas \cite{Chu48,Wheeler47} tie an antenna's quality factor to its electrical size as $Q=1⁄ (ka)^3$, where \textit{a} is the radius of smallest surrounding sphere and $k=2\pi⁄\lambda$ is the wave number. That means the smaller the antenna, the more energy we need for a given radiated power to be stored. Moreover, the quality factor is related to the antenna's instantaneous bandwidth. The simple conclusion shows that we need to store a large amount of energy in the antenna reactive-zone in cases of low frequency or small antennas ($a⁄\lambda \ll 1$), and the instantaneous bandwidth will be small.
Instead of exchanging energy between electric and magnetic forms, a static stored energy (e.g., stored energy in a permanent magnet or an electret) can be moved, vibrated, or rotated without altering its form to generate a time-varying field. This approach differs radically from the traditional radiation systems and is therefore not constrained by resonance limitations. However, it may not be desirable to apply any of these approaches to magnets or electrets by using mechanical movements. We propose to modulate the magnetic energy stored around a magnet by manipulating reluctance to the surroundings. Therefore, the direction of the flux or the position of the stored energy variates in time. The magnetic field thus varies in time.
Let us first look at the magnetic flux density of a uniformly magnetized sphere, as shown in \textcolor{cyan}{Fig.~\ref{fig:fig1}}:
\begin{figure}
\includegraphics[clip, trim=2cm 21.3cm 15cm 2cm,width=0.45\columnwidth]{Fig/Fig1.pdf}
\caption{\label{fig:fig1} Magnetic flux density decays by $\frac{1}{r^3}$.}
\end{figure}
\begin{equation}
B \left (r>a \right)=\frac{\mu_{0}}{4\pi}\left [\frac{-\bf{m}}{r^3}+\frac{3\left(\bf{m}.\bf{r} \right) \bf{r}}{r^5} \right] , \qquad {\bf m}=\frac{4}{3}\pi a^3 \bf{M}
\label{eq:one}
\end{equation}
where {\bf M} (A/m) is the magnetic dipole moment per unit volume of the permanent magnet. As is evident from the closed-form magnetic flux density of spherical magnet, there is a $1⁄ r^3$ decay for $r>a$. One can compute the total magnetic energy stored around the magnet as:
\begin{equation}
W_{m} = \frac{\mu_{0} \left |{\bf m} \right|^2}{12\pi a^3} = \frac{\mu_{0}}{9} V \left |{\bf M} \right|^2
\label{eq:two}
\end{equation}
where $V$ is the volume of the magnet with magnetization {\bf M}. One can compute the total energy stored outside a sphere of radius $r>a$ as:
\begin{equation}
W_{r} = \left (\frac{a}{r} \right)^3 W_{m}
\label{eq:three}
\end{equation}
The above equations indicate that the magnetic energy contained in the radius $r$ sphere and the magnetic flux intensity in the distance $r$ decrease by $1/r^3$. Thus, in order to reduce the size of the transmitter, a high-magnetic flux (requires more sophisticated material) must be modulated when selecting a small $r$. Otherwise, miniaturization must be sacrificed in order to modulate smaller magnetic fluxes at larger $r$.
The first approach is to use a material with a controllable reluctance to create a shielding layer at radius \textit{r}. Ideally, one can alter the shield's reluctance from a small to substantial value. This process allows the stored energy to be temporarily decoupled outside the shield from the magnet, and then allows the magnet to store energy outside the shield again by increasing its reluctance. In its low reluctance mode, the spherical shield closes the field lines that pass it and thus dissipates the $W_{r}$ energy every half cycle. For analytical convenience, we presume that variation of the reluctance does not substantially disrupt the magnetic flux within the shield. There are various constraints, including loss, size, the current required to control the shielding material's reluctance, and saturation level, which dictate the proper values for \textit{r}.
We consider the next approach to be an asymmetric system consisting of a ferrite yoke (as the variable reluctance magnetic material) and a permanent magnet (as the magnetic flux source), as shown in \textcolor{cyan}{Fig.~\ref{fig:fig2}(a)} and\textcolor{cyan}{~\ref{fig:fig2}(b)}. Since the permanent magnet attracts the ferrite yoke, the total energy stored in this system is a function of the distance from the yoke to the magnet. We simulated this structure using ANSYS Maxwell for different materials and ranges and compared the energy of the system with the energy stored in the isolated magnet. The simulation results, as shown in \textcolor{cyan}{Fig.~\ref{fig:fig2}(c)}, suggests that nearly half of the magnet's energy is converted to kinetic energy when the ferrite yoke contacts the magnet, and another half is still stored around the system. The system energy for $D = 1 cm$ is about 90\% of its maximum value, as the simulation results show. One can then move the yoke 1 cm away from the magnet back and forth and modulate the stored energy with a modulation depth of 40\%. We can use a mechanical resonance structure (i.e., a spring and a fixture) to conserve the kinetic energy. We intend to modulate the reluctance to make the stored energy time-variant, rather than a mechanical movement.
\begin{figure}[t]
\begin{center}
\begin{tabular}{ m{0.4\columnwidth} m{0.5\columnwidth} }
\quad \raisebox{-.5\height}{\includegraphics[clip, trim=12cm 7.5cm 12.5cm 7.5cm,width=0.25\columnwidth]{Fig/Fig2_a.pdf}}
&
\quad \raisebox{-.5\height}{\includegraphics[clip, trim=10.7cm 7cm 11cm 7cm,width=0.35\columnwidth]{Fig/Fig2_b.pdf}} \\
\qquad \qquad (a) & \qquad \qquad (b) \\
\multicolumn{2}{c}{\includegraphics[clip, trim=3.9cm 22.6cm 11.6cm 1.8cm,width=0.9\columnwidth]{Fig/Fig2_c.pdf}}\\
\multicolumn{2}{c}{(c)}
\end{tabular}
\caption{The system contains magnet and ferrite yoke. \textit{a}) T mode \textit{b)} N mode \textit{c)} normalized system energy of the system compared to the isolated magnet versus distance D; the legend shows the system‘s mode, the thickness of the ferrite yoke (t), and the type of magnetic material used for ferrite yoke implementation.}
\label{fig:fig2}
\end{center}
\end{figure}
While the spherical shield offers a significant modulation depth (close to 100\%), it is large and three-dimensional. On the other hand, the system with ferrite yoke has much smaller dimensions; however, it cannot provide a sufficiently broad modulation depth. Therefore, we combine the two above methods by putting the magnet on a ferromagnetic film with a proper winding to modulate the magnetic flux by adjusting the reluctance of the film. The design parameters include the ferrite characteristics, in particular, the nonlinearity of its B-H curve, the thickness of the ferromagnetic film, the topology of the structure, and the windings. The design objectives are high magnetic flux, a high modulation depth, small size, and low dissipated power. One of the tasks to achieve these objectives is to utilize the relationship between magnetic flux, {\it {\bf B}}, and magnetic field, {\it {\bf H}}, effectively.
\vspace{-1em}
\section{\label{sec:design} Transmitter Design}
\vspace{-1em}
We designed and prototyped different structures to examine our proposed approach. \textcolor{cyan}{Figure~\ref{fig:fig3(a)}} shows the ANSYS model of one of our designs. The permanent magnet used in this transmitter is a rare-earth Neodymium magnet (N52, $6 \times 1 \times 0.5\, cm$), which is the strongest permanent magnet available in the market. Also, we used seven layers of Metglas sheets 2705M ($B_{s} = 0.77\,T$) with a total thickness of 0.178 mm as the magnetic film. Besides, a 40-turn coil around a c-shape magnetic core made of amorphous AMBC ($B_{s} = 1.56\,T$) with a $2 \times 2\,cm$ cross-section generates the magnetic flux needed to modulate the magnetic film's reluctance. We select a low reluctance core with a reasonably broad cross-section to ensure that the c-shape core works at its linear state. As a result, the current through the control coil generates a magnetic flux in the magnetic film.
\textcolor{cyan}{Figure~\ref{fig:fig3}} shows the flux density on the system for two different values for the control current. \textcolor{cyan}{Figure~\ref{fig:fig3(a)}} shows that small areas of the magnetic film are in saturation when the control current is zero. The saturated film helps to spread magnetic flux in the air and to store magnetic energy around the magnet. The small saturated area shows that the magnetic film operates as a barrier and closes inside the magnetic flux. Next, we apply 0.5 A current to the control coil, and the pattern of magnetic film saturation shifts to \textcolor{cyan}{Fig.\ref{fig:fig3(b)}}, which means the saturated area is larger than the closed mode. In this mode, the magnetic flux spreads more in space, and there is more energy stored around the magnet. We name this state of the system,``Open mode.'' This system's operating modes will differ by adjusting the arrangement of the magnet or the magnetic film. For example, the magnetic film may be saturated by a giant magnet with zero current. The saturated area of the magnetic film can then be reduced by a magnetic flux generated by the control current against the magnet's magnetic flux. In this case, the system's operating modes switch to open and closed mode for zero current and high current, respectively. One can apply a sinusoidal current to the control coil to change the amount of energy stored around the magnet periodically. \textcolor{cyan}{Figure~\ref{fig:fig3(b)}} also shows that the magnetic flux density in the AMBC core is less than 0.13 T, indicating the amorphous AMBC cross-section we have is higher than what we needed to keep it out of saturation. One can use a smaller core to reduce overall system size and weight.
\vspace{-1em}
\begin{figure}
\centering
\subfigure[]
{
\includegraphics[clip, trim=2.5cm 21.8cm 8.5cm 2.8cm,width=0.9\columnwidth]{Fig/Fig3_a_.pdf}
\label{fig:fig3(a)}
}
\\
\subfigure[]
{
\includegraphics[clip, trim=2.5cm 21.8cm 14cm 2.5cm,width=0.9\columnwidth]{Fig/Fig3_b_.pdf}
\label{fig:fig3(b)}
}
\subfigure[]
{
\includegraphics[clip, trim=2.5cm 14.5cm 3cm 2.8cm,width=0.9\columnwidth]{Fig/Fig3_c_.pdf}
\label{fig:fig3(c)}
}
\caption{ \textit{a}) Maxwell model of the prototyped system. Magnetic field distribution of the Metglas \textit{b}) control current is zero and the system is in closed mode \textit{c}) Control current force the Metglas to saturation (Open mode).}
\label{fig:fig3}
\end{figure}
|
1,941,325,220,528 | arxiv | \section{Introduction}
\label{sec:intro}
Confining an equilibrium fluid of particles to length scales on the
order of several particle diameters changes both its static and
dynamic properties. Classical density functional theory (DFT) can
often make reliable predictions concerning the former, but
implications of confinement for dynamics remain challenging to
forecast for even the most basic models. For example, Enskog theory
generalized for inhomogeneous fluids~\cite{Davis1987Kinetictheoryof}
predicts that constant-activity confinement of the hard-sphere (HS)
fluid between hard walls will significantly decrease self-diffusivity
parallel to the
boundaries~\cite{Vanderlick1987Self-diffusioninfluids}. Recent
molecular dynamics simulations of that system, however, show that this prediction is
qualitatively incorrect, i.e., self diffusivity increases with
constant-activity confinement~\cite{Mittal2007Doesconfininghard-sphere}. Hydrodynamic
theories, on the other hand, can predict how the presence of a single
wall~\cite{Happel1973LowReynoldsNumber} or confinement between two
walls~\cite{Faucheux1994ConfinedBrownianmotion,Lin2000Directmeasurementsof,Dufresne2001Browniandynamicsof,Saugey2005Diffusioninpores}
impacts the self diffusion of a single Brownian particle in solvent.
But it is not yet clear how to generalize these approaches to account
for the effects that strongly inhomogeneous static structuring has on
the transport coefficients of dense confined
fluids~\cite{Mittal2007Doesconfininghard-sphere,Goel2008Tuningdensityprofiles,Mittal2008LayeringandPosition}.
Given the absence of a reliable microscopic theory, the development
of
new qualitative or semi-quantitative heuristics for predicting
dynamics of confined fluids would be of considerable practical use.
In this spirit, one productive strategy has been to first exhaustively
simulate the static and dynamic behaviors of simple models of confined
fluids. The idea is that comprehensive study of these systems may reveal
static quantities that strongly correlate with transport
coefficients for a wide variety of confining environments. Knowledge
of these correlations together with reliable predictions of the static
quantities from equilibrium theory would, in turn, lead to indirect predictions
for how confinement modifies dynamics.
Indeed, recent simulation data covering hundreds of state points for
various confined HS, Lennard-Jones, and square-well fluids
point to the existence of a robust, isothermal correlation between the self-diffusion
coefficient $D$ and the excess entropy per particle (over ideal gas) $s^{ex}$
--- a relationship that is approximately independent of the degree of
confinement for a wide range of equilibrium conditions~\cite{Mittal2007Doesconfininghard-sphere,
Goel2008Tuningdensityprofiles, Mittal2006ThermodynamicsPredictsHow,
Mittal2007RelationshipsbetweenSelf-Diffusivity, Mittal2007Confinemententropyand}.
The practical implication is that the mathematical form of the
correlation for a given fluid (obtained from bulk fluid simulations) can
be used together with knowledge of the excess entropy of the confined
system (computed, e.g., via DFT) to make
semi-quantitative predictions for how confinement will modify the
self diffusivity.
As has been discussed
elsewhere~\cite{Mittal2007Doesconfininghard-sphere,Goel2008Tuningdensityprofiles},
this strategy can
successfully predict subtle, confinement induced effects of packing
frustration on relaxation processes, behavior not reflected in other static
quantities traditionally thought to track dynamics, such as the
average density.
Although a fundamental and general derivation that explains the observed
relationship between excess entropy and dynamics
of confined fluids is still lacking, the fact that the two are
connected is not surprising. Excess entropy
is a negative quantity that measures the extent to which static
interparticle correlations reduce the number of microstates available
to the fluid. In fact, $-{s^{\rm{ex}}}$ is often used as a metric for
characterizing the ``amount'' of structural order present in condensed
phase
systems~\cite{Truskett2000Towardsquantificationof,Mittal2006QuantitativeLinkbetween,Errington2006Excess-entropy-basedanomalieswaterlike,Krekelberg2007Howshort-rangeattractions,Lipkowitz2007ReviewsinComputational}.
Since interparticle correlations strongly influence collisional
processes, it makes intuitive sense that macrostate changes
which increase structural order ($-{s^{\rm{ex}}}$) might also tend to reduce
single-particle mobility. Qualitative arguments such as these have
previously been presented to rationalize empirically observed
correlations between excess entropy and transport coefficients of
both bulk~\cite{Rosenfeld1977Relationbetweentransport,Rosenfeld1999quasi-universalscalinglaw,
Dzugutov1996universalscalinglaw} and confined~\cite{Mittal2007Doesconfininghard-sphere,Goel2008Tuningdensityprofiles, Mittal2008LayeringandPosition,Mittal2006ThermodynamicsPredictsHow,Mittal2007RelationshipsbetweenSelf-Diffusivity,Mittal2007Confinemententropyand}
equilibrium fluids. Nonetheless, it is natural to wonder whether excess
entropy is unique in this regard. Perhaps other measures also
accurately forecast the implications of confinement for the dynamics
of dense fluids. For example, does the mobility of inhomogeneous HS
fluids also increase with the average amount of space available for
particle motion? Do some measures of free or available volume
correlate much more strongly with dynamics than others? Can one
quantitatively, or semi-quantitatively, predict self diffusivity of
confined fluids based on knowledge of how confinement affects the free
or available volume of the system?
Here, we put the above questions to stringent tests for a variety of
confined fluid systems. To ascertain the effect of pore geometry on
correlations between dynamics and thermodynamics, we study a
monodisperse hard-sphere fluid confined to smooth hard-wall slit
pores, square channels, and cylindrical pores. We also explore the
effects of boundary interactions in the slit-pore geometry by
examining cases for which square-well (attractive) or
square-shoulder (repulsive) particle-boundary interactions are present. Finally, we investigate the
dynamics and thermodynamics of a confined binary hard-sphere mixture
which can be equilibrated in the fluid state at higher packing
fractions without
crystallizing than the corresponding single-component fluid.
Together, this study represents, to our knowledge, the most
comprehensive exploration of the relationships between static and
dynamic properties in these basic inhomogeneous systems to
date. We calculate
the self-diffusion coefficient [via molecular dynamics simulations],
and excess entropy and various measures of available volume [via
Transition Matrix Monte Carlo (TMMC) simulations and fundamental
measure theory] at approximately
$10^3$ state points. Our results illustrate that predictions of
either excess entropy or
a generalized measure of average available volume from classical
density functional theory can be used together with knowledge of bulk fluid
behavior to semi-quantitatively predict the diffusion coefficient of
confined fluids across a wide range of equilibrium conditions.
\section{Connections between density, available volume, and dynamics:
Preliminary evidence}
\label{connection_preliminary}
The available volume in a configuration of the single-component
HS fluid refers to the
space into which the center
of an additional HS particle of equal size can be inserted without
creating an overlap with existing particles or the boundary.
It might also be considered a coarse measure of the space immediately
available for particle motion in that configuration of the system.
For the bulk equilibrium HS fluid, the ensemble-averaged
fraction of the total volume
that is available (in the above sense) is given by $p_0=\rho/\xi$~\cite{Widom1963SomeTopicsin}, where $\rho$ is number density,
$\xi=\exp[\beta\mu]/\Lambda^3$ is activity, $\mu$ is the chemical
potential, $\beta=[k_{\mathrm{B}}T]^{-1}$, $k_{\mathrm{B}}$ is
the Boltzmann constant, $T$ is the temperature, and $\Lambda$ is the thermal deBroglie wavelength.
It is known that increasing $\rho$ monotonically decreases both $p_0$
and
self diffusivity $D$
for this system; i.e. $d p_0/ d \rho<0$ and $d D / d \rho < 0$ (see,
e.g. \cite{Lipkowitz2007ReviewsinComputational}), which is consistent
with the intuitive idea that these static and dynamic properties
might be linked.
\begin{figure}[h]
\includegraphics{freevolconfine}
\caption{Schematic of an inhomogeneous HS fluid confined between
boundaries in a slit pore geometry.
The walls are placed a distance $H=h+\sigma$ apart,
where $h$ is the length accessible to particle centers.
Dark regions indicate hard spheres and
confining walls. Additional particle centers are excluded from
the grey (overlap) region.
The white region indicates the volume available for inserting an
identical hard sphere of diameter~$\sigma$.
\label{freevolschematic}}
\end{figure}
Are density, available space, and self diffusion connected in the same
qualitative way for inhomogeneous HS fluids? Previous studies have
provided some information useful for addressing this question. For
example, it is known that confining the equilibrium HS fluid between
hard walls (while maintaining fixed $\xi$) significantly increases
the average particle density, i.e. $(\partial {\rho_h} /
\partial h^{-1})_{\xi}>0$, over a wide range of
${\rho_h}$ and $h$~\cite{Vanderlick1987Self-diffusioninfluids,Mittal2007Doesconfininghard-sphere}. Here, ${\rho_h}=N/(Ah)$ is average density, $N$ is the total number of particles,
$A$ is the area of the interface between the fluid and one hard wall,
and $h$ is the center-accessible width of the slit pore [i.e., not
including the ``dead space'' that the particle centers are
excluded from due to their interaction with the boundaries (see
Figure~\ref{freevolschematic})]. The fact that ${\rho_h}$ increases
upon constant-$\xi$
confinement initially appeared consistent
with earlier kinetic theory predictions that $D$ of this system would
show a corresponding
decrease~\cite{Vanderlick1987Self-diffusioninfluids}. However,
recent simulation data have demonstrated that the kinetic theory predictions
were qualitatively
incorrect~\cite{Mittal2007Doesconfininghard-sphere}. That is,
both ${\rho_h}$ and $D$ typically increase upon constant-$\xi$ confinement
[$(\partial D/\partial {\rho_h})_{\xi}>0$, the fluid
gets ``denser'' on average and particles diffuse {\em more rapidly}],
a trend that is the opposite of what might be expected based on the
bulk HS fluid behavior. Interestingly, confined HS fluids show a
different trend when an alternative thermodynamic
constraint
is applied. Specifically, if $h$ is held fixed, then increasing ${\rho_h}$ has the
effect of {\em decreasing} $D$, i.e., $(\partial D/\partial {\rho_h})_{h}<0$]~\cite{Mittal2007Doesconfininghard-sphere,Mittal2006ThermodynamicsPredictsHow}.
This preliminary data illustrates that knowledge of how ${\rho_h}$ changes
is not, in and of itself,
enough to even qualitatively predict the implications of
confinement for the dynamics of a fluid. Indeed, in
Section~\ref{sec:diffpredictions} of this article, we present
extensive numerical evidence for a variety of confined fluid systems
which underscores this point. We also explore whether adopting a
definition of average density based on the total rather than
center-accessible volume of the pore (see
also~\cite{Mittal2007Doesconfininghard-sphere,Mittal2006ThermodynamicsPredictsHow})
improves predictions for how confinement modifies dynamics.
Does available volume show a more reliable correlation to dynamics than
average density?
The fractional available volume in an inhomogeneous fluid is
inherently a local quantity, and it is given by $\po _i
(z)=\rho _i(z) \exp[\beta u^w_\mathrm{i} (z)]/\xi _i$,
where $\rho _i(z)$ and
$u^w_\mathrm{i} (z)$ represent the singlet (one-particle) density and
the wall-particle interaction potential for species $i$,
respectively, evaluated at a distance $z$ from one
wall~\cite{Lebowitz1965ScaledParticleTheory,Widom1978Structureofinterfaces,Hendersen1983StatisticalMechanicsOf}.
The volume averaged quantity can be expressed as
\begin{equation}
\label{eq:po_hshw}
{\overline {p_{\rm{0}}}}_i\equiv \frac{1}{V_\mathrm{c,i}}\int_{V_\mathrm{c,i}} \po_i dV
\end{equation}
where the integral is over the particle-center-accessible volume
$V_\mathrm{c,i}$. For the special case of a single-component
HS fluid confined
between smooth hard walls, Eq.~\eref{eq:po_hshw}
reduces to ${\overline {p_{\rm{0}}}}={\rho_h}/\xi$. Note that since $ \left(\partial {\overline {p_{\rm{0}}}} / \partial h^{-1}\right)_\xi=\xi^{-1} \left(\partial {\rho_h} / \partial
h^{-1}\right)_\xi$, and $\left(\partial {\rho_h} / \partial
h^{-1}\right)_\xi>0$ across a wide range of ${\rho_h}$ and $h$~\cite{Vanderlick1987Self-diffusioninfluids,Mittal2007Doesconfininghard-sphere}, it follows that $\left(\partial
{\overline {p_{\rm{0}}}}/\partial h ^{-1} \right) _\xi>0$ for those conditions. This
increase in the fraction of available space with constant-$\xi$
confinement provides a simple physical
explanation for the counterintuitive observation that
$D$ {\em increases} along the same thermodynamic path.
The density-dependent behavior of the ${\overline {p_{\rm{0}}}}$ under the constraint of
constant $h$ is also qualitatively consistent with the dynamical
trends of the confined HS fluid. In particular, both $\left(\partial {\overline {p_{\rm{0}}}}/\partial
{\rho_h} \right)_h<0$ and $\left(\partial D/\partial
{\rho_h} \right)_h<0$. All of this strongly suggests that
${\overline {p_{\rm{0}}}}$ is a more relevant
static metric for particle mobility than the average particle
density ${\rho_h}$.
In the following
sections, we test the generality of these preliminary observations
by carrying out an extensive quantitative comparison of
the correlations between self diffusivity $D$ and
various static measures (density, excess entropy, and fractional
available space) for single-component and binary HS fluids confined to
a variety of channels with different geometries and particle-boundary
interactions. The results clarify which of these static quantities
reliably predict the implications of confinement for
single-particle dynamics.
\section{Methods}
\label{sec:methods}
We study single-component HS fluids of particles with diameter $\sigma$
both in the bulk and confined to channels with three types of geometries:
(i) quasi-two-dimensional slit pores, (ii) quasi-one-dimensional
square channels, and (iii) cylindrical pores. Specifically, we
consider (i) seven slit pores with thickness $H/\sigma=5, 6, 7, 8, 9,
10,~\mathrm{and}~15$ in the confining $z$ direction (see
Figure~\ref{freevolschematic}) together with periodic boundary conditions in
the $x$ and $y$ directions, (ii) seven
square channels with total $x-y$ cross-sectional dimensions of $H^2/\sigma^2
=25, 36, 49, 64, 81, 100,~\mathrm{and}~225$ together with a
periodic boundary condition in the
$z$ direction, and (iii) six cylindrical channels of total diameter
$H/\sigma=6, 7, 8, 9,
10,~\mathrm{and}~15$ together with a periodic boundary condition
in the axial $z$ direction.
We take the interaction of a particle with a channel wall $u_\mathrm{w} (s)$
in all cases to have a generic square-well form:
\begin{equation}
\label{sw_interaction}
u_\mathrm{SW}(s) = \cases{\infty& $s < \sigma /2$\\
\epsilon_\mathrm{w}& $\sigma/2 \le s < \sigma$\\
0& $s \ge \sigma$\\}
\end{equation}
where $s$ is the
shortest distance between the particle center and the wall of
interest. For all three geometries, we study the case
$\epsilon_\mathrm{w}=0$, i.e. smooth hard boundaries. Additionally, for the
slit pore with
size $H/\sigma=5$, we investigate cases with $\epsilon_\mathrm{w}=2
k_{\mathrm B}T$ (a repulsive shoulder) and
$\epsilon_\mathrm{w}=-2k_{\mathrm B}T$ (an attractive well).
We also consider a binary HS mixture confined between smooth hard
walls in a slit pore of width $H/\sigma_1=5$.
For this system, the particle diameter ratio is taken to be $\sigma_\mathrm{2}/\sigma_\mathrm{1}=1.3$ and the particle masses are
proportional to their volume,
i.e., $m_\mathrm{2}/m_\mathrm{1}=(\sigma_\mathrm{2}/\sigma_\mathrm{1})^3$.
These parameter values closely mimic those examined in
recent experiments of
binary colloidal mixtures under
confinement~\cite{Nugent2007ColloidalGlassTransition}.
To explore dynamic properties of these fluids, we perform
molecular dynamics simulations using an event-driven
algorithm~\cite{Rapaport2004TheArtOf} in the microcanonical ensemble
with $N=4000$ particles for monodisperse HS systems and $N=3200$ for
the binary HS systems. For bulk systems, we use a cubic
simulation cell of volume $V$. For the confined systems, we adopt a
rectangular parallelepiped simulation cell of volume $V=H_x H_y H_z$
with appropriate boundary conditions depending on geometry.
We extract the self diffusivity $D$ by
fitting the long-time ($t >> \sigma_\mathrm{1}
\sqrt{m_\mathrm{1}\beta}$) mean-squared displacement to the
Einstein relation $\big<\Delta
\mathbf{r}^2_{d_p}\big>=2 d_p Dt$, where $\big< \Delta
\mathbf{r}^2_{d_p}\big>$ corresponds to motions in the $d_p$
periodic directions. For the sake of clarity, we reserve the symbol
$D$ for the self diffusivity of fluids under confinement and
$D_{\rm{bulk}}$ for the self diffusivity of the bulk
fluid. To obtain reliable estimates, we
average self diffusivities over four
independent trajectories. For simplification, we report
quantities from this point forward
implicitly non-dimensionalized by appropriate combinations
of the characteristic length scale, taken to be the HS diameter of the smallest
particle in the fluid $\sigma_\mathrm{1}$,
and a characteristic time scale, given by
$\sigma_\mathrm{1} \sqrt{m_\mathrm{1}\beta}$. Thus, all energies are
implicitly per unit $k_{\mathrm B} T$, and $T$ is effectively scaled out of
the problem altogether.
We obtain thermodynamic properties using grand-canonical
transition-matrix Monte Carlo (GC-TMMC) simulation. For pure fluids,
we use an algorithm presented by
Errington~\cite{Errington2003DirectCalculationOf} and for binary
mixtures we employ a strategy developed by Shen and Errington~\cite{Shen2005Determinationoffluid},
wherein one combines a series of semigrand ensemble simulations to
construct the system's free energy over a wide range of densities and
compositions. We conduct GC-TMMC simulations in a standard grand
canonical ensemble where the volume $V$, temperature $T$, and
activities \{$\xi _1,\xi _2$\} are held constant and the particle numbers
\{$N _1, N _2$\} and energy $E$ fluctuate. For notational convenience,
we denote the sets \{$N _1, N _2$\} and \{$\xi _1,\xi _2$\} as
$\mathbf{N}$ and $\boldsymbol{\xi}$, respectively, using conventional
vector notation. The activity of component $i$ is
defined as $\xi _i = \Lambda _i ^{-3}\exp(\mu _i)$, where $\mu _i$
is the chemical potential and $\Lambda _i$ is the thermal de Broglie
wavelength. For the pure component GC-TMMC simulations
we present here,
we set $\xi=1$ and adjust the particle center-accessible volume
$V _\mathrm{c}$ to make the total volume $V \approx 1000$.
Simulations of the binary mixture use $V=125$ and $V=245$ for the
bulk and confined fluids, respectively.
For the bulk and confined simulations, we set $\xi _1=173.7$
and $\xi _2=381.5$.
The key quantity we extract from the GC-TMMC simulations is the
particle number probability distribution ${\Pi} (\mathbf{N})$.
Once we obtain this distribution, we use basic
statistical mechanics principles and histogram
reweighting~\cite{Ferrenberg1988NewMonteCarlo} to evaluate
thermophysical properties over a range of activity values.
First, we use histogram reweighting to deduce
${\Pi} (\mathbf{N})$ at a set of activities
$\boldsymbol{\xi}_\mathrm{new}$ generally different from that of the
GC-TMMC simulation $\boldsymbol{\xi}_\mathrm{sim}$,
\begin{equation}
\label{eq:PiN}
\ln {\Pi} (\mathbf{N};\boldsymbol{\xi}_\mathrm{new}) =
\ln {\Pi} (\mathbf{N};\boldsymbol{ \xi}_\mathrm{sim}) +
\sum_i N_i (\ln \xi _{i, \mathrm{new}} - \ln \xi _{i,
\mathrm{sim}}),
\end{equation}
where it is understood that the probability distributions are not
normalized. We obtain average particle numbers
$\left<\mathbf{N}\right>$ from first-order moments of ${\Pi} (\mathbf{N})$,
\begin{equation}
\label{eq:avgN}
\left<\mathbf{N (\boldsymbol{\xi})}\right> = \sum\mathbf{N}{\Pi}(\mathbf{N;
\boldsymbol{\xi}}) / \sum {\Pi}(\mathbf{N; \boldsymbol{\xi}}).
\end{equation}
We calculate $\mathbf{\rho}$ and $\mathbf{\rho} _\mathrm{h}$ via
normalization of $\left<\mathbf{N}\right>$ by $V$ and $V _\mathrm{c}$,
respectively.
We define excess entropy as the difference
between the fluid's entropy and that of an ideal gas with the same
density profile. The particle number
probability distribution provides the density and composition
dependence of the Helmholtz free energy at a given temperature.
Therefore, we combine knowledge of ${\Pi}(\mathbf{N})$, average excess
configurational energies $U^{\rm{ex}} (\mathbf{N})$, and
particle-number-specific spatial density distributions $\rho
(\mathbf{N,r})$ to obtain the total excess entropy
$S^{\mathrm{ex}}$~\cite{Mittal2006ThermodynamicsPredictsHow,Mittal2007Confinemententropyand,Errington2006Excess-entropy-basedanomalieswaterlike},
\begin{eqnarray}
\fl S^{\mathrm{ex}} (\mathbf{N}) = U^{\rm{ex}}(\mathbf{N}) + \ln {\Pi}
(\mathbf{N})/{\Pi} (\mathbf{0}) \nonumber\\
+ \sum_i \left\{\ln N_i! - N_i \ln \xi_i -
N_i \ln N_i + \int \rho_i (\mathbf{N,r})\ln\rho_i
(\mathbf{N,r})d\mathbf{r}\right\}\label{eq:TMMCexS}
\end{eqnarray}
We also predict the thermodynamic quantities of the bulk
single-component and binary HS fluids using the
Carnahan-Starling~\cite{carnahan:635} and
Boublik-Mansoori-Carnahan-Starling-Leland~\cite{boublik:471,mansoori:1523}
equations of state, respectively. We predict the thermodynamic
properties of confined HS fluids
using a recent modification~\cite{Yu2002Structuresofhard-sphere} of
Rosenfeld's fundamental measure
theory~\cite{Rosenfeld1989Free-energymodelinhomogeneous}. Fundamental
measure theory is a classical DFT of inhomogeneous fluids that
has been shown to accurately predict structure and thermodynamics of
confined HS systems in various restrictive geometries up to very high
densities~\cite{Gonzalez2006Densityfunctionaltheory}. For numerical
evaluation of the DFT for slit and
cylindrical pores, we use Picard iterations on a grid spacing
of $0.005$. We update densities according to,
$[\rho]_{n+1}^{\mathrm{in}}=0.95[\rho]_n^{\mathrm{in}}+0.05[\rho]_n^{\mathrm{out}}$, where
$[\rho]_{n+1}^{\mathrm{in}}~\mathrm{and}~[\rho]_n^{\mathrm{in}}$ are the input density
profiles at the $n+1^{th}$ and $n^{th}$ iterations, respectively,
and $[\rho]_n^{\mathrm{out}}$ is the output density profile at
the $n^{th}$ iteration. We stop Picard iterations when the
relative change in output density profile between two successive
cycles [1 cycle = 20 iterations] becomes less than $10^{-5}$. For
numerical evaluation of the DFT
in the square channel geometry, we use Sandia National Laboratories
Tramonto package~\cite{Sears2003ANewEfficient}.
We adopt a grid of $0.05 \times 0.05$ for $H=5,6$ and a
grid of $0.075 \times 0.075$ for $H>6$.
We stop the minimization algorithm when the relative
or absolute change in the grand potential is less than $10^{-7}$.
\section{Testing structure-property relations for predicting
self diffusivity of confined fluids}
\label{sec:diffpredictions}
In this section, we explore the accuracy of the following strategy
for predicting the self diffusivity of confined HS fluids: (1)
determine the
value of a static quantity~$x$ of a confined fluid believed to be
relevant for dynamics (e.g., its density,
excess entropy, or fractional available volume), and (2) input this
value into the
relationship between self diffusivity and that same static
quantity for the bulk
fluid, ${D_{\rm{bulk}}^x} (x)$,
to estimate the confined fluid self diffusivity, $D$.
Of course, such a strategy can only provide approximate predictions.
While there is a one-to-one relationship for the equilibrium HS fluid
between the
self diffusivity
and any one of the
aforementioned static quantities,{\footnote{This is true if the self diffusivity is
appropriately non-dimensionalized, as we have done here,
to remove the trivial effect of the thermal velocity of the
particles.}
the dynamic properties of the confined fluid generally depend
on a larger number of
variables (e.g., the dimensions of the confining geometry,
the nature of the particle-boundary interactions, and
the chemical potential). Nonetheless, the hope is that one can
discover a static quantity $x$ whose relationship with $D$ is
largely insensitive to the effect of confinement. If so, the
bulk structure-property relationship ${D_{\rm{bulk}}^x}$
can be used to predict $D$
independent of the other parameters of the confined
system. Systematic tests of this idea should give
new insights into the structural properties
that are most relevant for single-particle dynamics of inhomogeneous fluids.
To investigate the accuracy of predictions by
this approach, we
use the ``exact'' results of
molecular simulations to examine the ratio of the bulk fluid
self diffusivity to that of confined fluids with the same value of
$x$, i.e., ${D_{\rm{bulk}}^x}/D$. Since one is often
interested in both $D$ and the effective characteristic time associated with
diffusive motion ($D^{-1}$), we present plots of ${D_{\rm{bulk}}^x}/D$ in this work on a
logarithmic scale, a representation for which
overpredictions and underpredictions of $D$ by
the same factor appear the same distance from unity. We also
present statistics associated with the relative errors of the
predictions for different $x$.
For each $x$ that we consider here,
we analyze roughly $10^3$ state points
of the equilibrium HS fluid confined to the various pore geometries
described in Section~\ref{sec:methods}. This
data, when taken as whole, spans approximately four decades in $D$.
\subsection{Confinement in channels with
smooth hard boundaries}
\label{sec:dbydbulk_neutral}
\begin{figure}[h]
\includegraphics{dbydbulk_all4_new-scaled}
\caption{Ratio of self diffusivity of a bulk HS fluid to that of a
confined HS fluid with the same value of a static quantity $x$,
${D_{\rm{bulk}}^x}/D$, obtained via molecular
simulations. Data are shown for the fluid confined to
slit pore [a, d, g, j], square channel [b, e, h, k], and
cylindrical pore [c, f, i, l] geometries.
The static quantity $x$ is indicated in each of the four panels.
$20\%$ bounds on relative error in confined fluid
self diffusivity ``predictions'' by using
the bulk structure-property relation ${D_{\rm{bulk}}^x}$
are shown by red dash-dotted lines.
The equilibrium fluid states shown here span the density range
$0<{\rho_h}<\rho_0$, where $\rho_0 \approx 0.95$ for (a-c) and
$1<\rho_0<1.25$ for (d-l), depending on pore size. Pores shown have
confining dimensions of $5$ (\opencircle), $6$ (\opensquare), $7$
(\opendiamond), $8$ (+), $9$ (x), $10$ (*), and $15$
(\opentriangle). These dimensions correspond to channel width $H$
for slit pores and square channels and channel diameter $d$ for
cylindrical pores. All channels have smooth hard boundaries.
\label{dbydbulk_all}}
\end{figure}
Here, we examine the ratio ${D_{\rm{bulk}}^x}/D$ for the single-component HS fluid confined to various
geometries by smooth hard boundaries [i.e. $\epsilon _{\mathrm{w}}=0$, see \eref{sw_interaction}].
We begin by testing the predictions that follow from
assuming that $x={\rho_h}$, the
number density averaged over the particle-center-accessible volume of the
pore, is the relevant static metric for dynamics [see
Figure~\ref{dbydbulk_all}(a-c)]. It is immediately clear from the
data that
${\rho_h}$ does not, in itself, provide a good basis for prediction. HS fluids
confined in slit-pore, square-channel, and cylindrical
geometries generally
exhibit a wide range of $D$ for each ${\rho_h}$, with the fastest dynamics
occurring in the smallest pores. In fact, note that
the bulk structure-property relation $D_{\mathrm{bulk}}^{{\rho_h}}$
can underpredict $D$ by nearly a factor of ten for fluids in the most
restrictive geometries. The performance of the bulk structure-property
relation using $x={\rho_h}$
is actually even worse than it appears in
Figure~\ref{dbydbulk_all}(a-c) for the following reason.
The freezing transition occurs at a
density of $0.945$ for the bulk
HS fluid, which provides an upper limit on values of ${\rho_h}$ that
can be used for predictions using $D_{\mathrm{bulk}}^{{\rho_h}}$. However, center-accessible
densities for the equilibrium fluid
in the smallest square-channel and cylindrical pores can reach as high
as ${\rho_h} \approx 1.25$. Thus, the bulk structure-property relation
$D_{\mathrm{bulk}}^{{\rho_h}}$ {\em cannot even make predictions}
for a significant fraction of the equilibrium
state points for highly confined HS fluids.
All of this confirms the
preliminary expectation discussed in
Section~\ref{connection_preliminary} that knowledge of ${\rho_h}$ and bulk
fluid behavior is not enough to predict the self diffusivity
of confined fluids. This conclusion is consistent with the
earlier observations of
Mittal et al.~\cite{Mittal2006ThermodynamicsPredictsHow} concerning
a smaller set of data for the HS fluid confined to slit pores.
In Figure~\ref{dbydbulk_all} (d,g, and, j), we again present
${D_{\rm{bulk}}^x}/D$ for the HS fluid confined to slit pores,
but now ${D_{\rm{bulk}}^x}$ is the corresponding bulk fluid relation between
diffusivity and one of three alternative static properties ($x$):
average density based on total pore volume $\rho={\rho_h} (1-H^{-1})$,
excess entropy per particle ${s^{\rm{ex}}}$, and fraction of available volume
${\overline {p_{\rm{0}}}}$. The data in these plots corresponds to confined fluid states with
packing fractions that vary from the dilute gas to the
freezing transition for pore widths $H \ge 5$. As can be seen, each
of these static measures can
provide semi-quantitative predictions for confined fluid diffusivities
when used together with the corresponding bulk structure-property correlation.
In fact, for $93\%$~($x=\rho$), $97\%$~($x={s^{\rm{ex}}}$), and
$100\%$~($x={\overline {p_{\rm{0}}}}$) of equilibrium state points for these systems,
the predictions provided by ${D_{\rm{bulk}}^x}$ are within $20\%$ of
the ``exact'' MD data for $D$. Note that the
very small fraction of overpredictions based on $\rho$ or ${s^{\rm{ex}}}$ that
exceed $20\%$ relative error correspond
to the high density, low $D$ state points near the freezing
transition.
Based on the slit-pore data, it might be tempting to conclude that
total-volume-based average density~$\rho$ tracks dynamics nearly as
reliably as~${s^{\rm{ex}}}$ and
${\overline {p_{\rm{0}}}}$ for confined fluids. To provide a more stringent test of
this preliminary conclusion, we now examine ${D_{\rm{bulk}}^x}/D$
for HS fluids confined to quasi-one-dimensional square channel and cylindrical pore geometries with
edge dimensions $H \ge 5$ and diameters $d \ge 6$, respectively.
Fluids confined in these geometries
have a significantly higher percentage of particles near the
boundaries than in the corresponding slit pores, and hence the effects of confinement on both structure and
dynamics should be more
pronounced.
Figure~\ref{dbydbulk_all}(e,f) shows that for square channel and
cylindrical geometries, self-diffusivity predictions
based on $D_\mathrm{bulk}^{\rho}$ can be significantly higher
than the actual $D$ of a confined fluid with the same $\rho$.
In fact, for the densest fluid systems studied here,
the bulk structure-property relation $D_{\mathrm{bulk}}^{\rho}$ is
between $2$ and $4$ times larger than $D$, depending on $H$.
Furthermore, Figure~\ref{dbydbulk_all}(h,i,k,l) illustrates that
${D_{\rm{bulk}}^x}$ predictions for $x={s^{\rm{ex}}}$ or $x={\overline {p_{\rm{0}}}}$ are in the
semi-quantitative range for
a larger fraction of state points than those based on $x=\rho$.
Specifically, ${D_{\rm{bulk}}^x}$ is within $20\%$ of $D$ for
$46\%$ ($x=\rho$), $82\%$ ($x={s^{\rm{ex}}}$), and $95\%$ ($x={\overline {p_{\rm{0}}}}$)
of the state points.
The main differences occur for high density, low $D$ state points,
where predictions based on fractional available volume are significantly more
accurate than those based on excess entropy or density.
\begin{figure}[h]
\includegraphics{D_versus_H_oscillations_phi0.4_TMMC-scaled.eps}
\caption{Self diffusivity $D$
of a HS fluid confined in narrow slit pores of
width $H=2-5$ by smooth hard boundaries. The density of the confined
fluid is fixed at $\rho=(6/\pi)0.4$. We compare molecular dynamics
simulation data
(\opencircle) for the confined fluid with the self diffusivity of a
bulk HS fluid, ${D_{\rm{bulk}}^x}$, at the same value of $x=\rho$ (green
dash-dotted line),
$x={s^{\rm{ex}}}$ (red square), and $x={\overline {p_{\rm{0}}}}$ (blue triangle). Solid lines are
shown as a guide to the eye. $D$ and ${s^{\rm{ex}}}$, calculated via molecular dynamics and TMMC
simulations, respectively, are taken
from Fig. 7 of~\cite{Mittal2007Doesconfininghard-sphere}. ${\overline {p_{\rm{0}}}}$ was
calculated via TMMC simulations described in Section~\ref{sec:methods}
of the present work.
\label{D_osscilations}}
\end{figure}
Another relevant test case for comparing which of $\rho$, ${s^{\rm{ex}}}$, or
${\overline {p_{\rm{0}}}}$ most faithfully tracks dynamics
is to vary the degree of confinement while fixing $\rho$, an idea
motivated by an earlier study by Mittal
\etal~\cite{Mittal2007Doesconfininghard-sphere}. In particular,
Mittal \etal demonstrated that $D$ and ${s^{\rm{ex}}}$ of a HS fluid
oscillate in phase when $H$ of the confining slit pore is varied (for $H \le
5$) and $\rho$ is held constant.
The maxima in $D$ (high particle mobility)
and ${s^{\rm{ex}}}$ (weak interparticle correlations) occur
for integer values of $H$, geometries which naturally accommodate
the layering of particles near the boundaries. The minima in
$D$ (low particle mobility) and ${s^{\rm{ex}}}$
(strong interparticle correlations) occur for non-integer values of
$H$, which frustrate this natural layering pattern. Along similar
lines, Goel
\etal~\cite{Goel2008Tuningdensityprofiles} recently demonstrated that
particle-boundary interactions that flatten the density profile of a
confined fluid generally reduce $D$ and ${s^{\rm{ex}}}$, while those which
increase layering can have the opposite effect.
All of this suggests that excess entropy captures some of the subtle
frustration induced effects that layering has on both interparticle
correlations and single-particle
dynamics~\cite{Mittal2007Doesconfininghard-sphere}. Does ${\overline {p_{\rm{0}}}}$ also capture
these effects? A very recent study of Mittal
\etal~\cite{Mittal2008LayeringandPosition} suggests that it might. In
particular, the authors of that study showed that the local fraction of
available volume $p_0(z)$ and the position-dependent self diffusivity
normal to the boundaries $D_{\perp}(z)$ of a confined HS fluid
are highest in regions of high local density $\rho(z)$.
Figure~\ref{D_osscilations} provides a more direct test of this idea.
In particular, it shows the $D$ data of
Mittal~\etal~\cite{Mittal2007Doesconfininghard-sphere} for a HS fluid
confined between hard walls calculated via
molecular dynamics simulations. We have also included on this plot
predictions from the three bulk structure-property relations ${D_{\rm{bulk}}^x}$,
where $x=\rho$, $x={s^{\rm{ex}}}$, and $x={\overline {p_{\rm{0}}}}$. Since $\rho$ is fixed here,
it is evident that $D_{\mathrm{bulk}}^{\rho}$ is not able to
predict the oscillatory trends in the dynamics data.
However, note that
both $D_{\mathrm{bulk}}^{{s^{\rm{ex}}}}$ and $D_{\mathrm{bulk}}^{{\overline {p_{\rm{0}}}}}$
predict the correct oscillatory behavior. In fact, the predictions of
$D_{\mathrm{bulk}}^{{\overline {p_{\rm{0}}}}}$ are virtually quantitative over the entire
range of $H$.
\begin{figure}[h]
\includegraphics{dbydbulk_slit_HSmix-scaled}
\caption{Ratio of self diffusivity of (a) small particles and (b)
large particles of a bulk binary HS fluid mixture to
that of a corresponding confined HS fluid mixture with the same
value of a static quantity $x$, ${D_{\rm{bulk}}^x}/D$, obtained
via molecular simulation. The static quantity used for making
predictions is $x=\rho$ (\opencircle), $x={s^{\rm{ex}}}$ (\opensquare),
and $x={\overline {p_{\rm{0}}}} _i$ (\opentriangle).
Data are shown for the $H=5$ slit-pore
geometry with the total packing fraction in the range
$0.025-0.52$. The
mole fraction of the small spheres is $0.75$. $20\%$ bounds on
relative error in self-diffusivity predictions
are shown by red dash-dotted line.\label{dbydbulk_slitmix}}
\end{figure}
Should we expect $D_{\mathrm{bulk}}^{{\overline {p_{\rm{0}}}}}$ to generally
track the dynamics of dense, confined HS fluids
more accurately than $D_{\mathrm{bulk}}^{{s^{\rm{ex}}}}$?
In other words, what is more relevant for dynamics of inhomogeneous fluids:
available space or available states? We further explore that question
by examining the behavior of the confined HS
mixture discussed in Section~\ref{sec:methods}. Specifically, by
studying this binary fluid mixture in a slit pore with $H=5$, we
are able to probe confined fluid states with packing fraction $\phi$
as high as 0.52 (compared with the highest
packing fraction of $0.46$ for a monodisperse HS fluid confined
in a slit-pore of $H=5$) At $\phi=0.52$, the fluid already exhibits
dynamic signatures of supercooling, e.g., the emergence of a
plateau in the time dependence of the mean-squared displacement.
The corresponding $D$ at this packing fraction
($=0.002$) is an order of magnitude smaller than the smallest
$D$ for the confined monodisperse fluid ($=0.02$) in the slit pore
geometry.
Figure~\ref{dbydbulk_slitmix}(a,b) shows the ratio of the bulk
self diffusivity to confined self diffusivity, ${D_{\rm{bulk}}^x}/D$, for the small
and large particles of the mixture, respectively. Again, the comparisons
are made to the bulk fluid mixture of the same composition and
density ($x=\rho$), excess entropy
($x={s^{\rm{ex}}}$), or fractional available volume of the
corresponding species ($x={\overline {p_{\rm{0}}}}_i$). Note that the
bulk structure-property predictions for self diffusivities of
small and large particles are semi-quantitative (within
$20\%$) for $D>0.1$ based on any of the three
aforementioned static quantities. However, relative
errors in predictions based on $\rho$ or ${s^{\rm{ex}}}$ begin to increase
sharply for $D<0.1$. On the other hand, predictions based on ${\overline {p_{\rm{0}}}}_i$
remain semi-quantitative for all $D>0.02$ (covering three decades in
$D$), with significant
overpredictions occuring only for the densest three state points
investigated. Thus, it appears that, for single-particle dynamics,
fractional available volume is
the most relevant of the three static measures investigated here.
The question of whether there exists an alternative static measure
$x$, such that ${D_{\rm{bulk}}^x}$ tracks $D$ for deeply supercooled mixtures,
is currently an open one. The answer to that question will likely have
important implications for understanding how confinement shifts the glass
transition of fluids.
\subsection{Particle-boundary interactions and a generalized measure
of available volume}
\label{sec:dbydbulk_neutral}
Thus far, we have only considered the geometric (i.e., packing)
consequences of confinement on dynamics.
How does the presence of finite particle-boundary interactions
affect the picture described in the previous section?
We explore the answer to this
question by studying a monodisperse HS fluid confined to a slit pore
geometry by smooth walls with either square-shoulder (repulsive) or
square-well (attractive) particle-boundary interactions
[for details, see Eq.~\eref{sw_interaction}].
\begin{figure}[h]
\includegraphics{dbydbulk_slit_int-scaled}
\caption{Ratio of self diffusivity of a bulk HS fluid to
that of a confined HS fluid with the same
value of a static quantity $x$, ${D_{\rm{bulk}}^x}/D$, obtained
via molecular simulation. Data include systems with (a) square-shoulder (repulsive) and (b)
square-well (attractive) particle-boundary interactions (see
Eq.~\ref{sw_interaction}) with
$H=5$ in the slit pore geometry.
The static quantity used for making predictions is $x=\rho$
(\opencircle), $x={s^{\rm{ex}}}$ (\opensquare), $x={\overline {p_{\rm{0}}}}$ (\opentriangle), and
$x={\overline {p_{\rm{0w}}}}$ (\fulltriangle). $20\%$ bounds on relative error in self-diffusivity predictions
are shown by the red dash-dotted line.
Particle center accessible density for the fluid spans the range $0<{\rho_h}<1.05$.
${\overline {p_{\rm{0}}}}$ overpredicts high-diffusivity state points in panel
\textit{b} by more than $1000\%$ and those data points are off
the scale of the graph. \label{dbydbulk_slitint}}
\end{figure}
Figure~\ref{dbydbulk_slitint}(a and b)
shows the ratio of the bulk to confined self diffusivity, ${D_{\rm{bulk}}^x}/D$,
for these two cases, respectively,
with comparisons between bulk and confined fluids being
made at the same density ($x=\rho$), excess entropy
($x={s^{\rm{ex}}}$), and fractional available volume ($x={\overline {p_{\rm{0}}}}$). Clearly, the fractional available volume ${\overline {p_{\rm{0}}}}$ fails to track
the dynamics of the confined fluid for
both cases presented. However, this should perhaps be expected.
Since the interactions of the particles with the boundaries in these
cases are strongly position-dependent, all free space is not
equally ``available'' to the particles. To account for this
energetic imbalance, we suggest a generalized
available volume (${\overline {p_{\rm{0w}}}}$) that appropriately weighs the local available
space by the Boltzmann factor of the particle-boundary interaction,
\begin{eqnarray}
{\overline {p_{\rm{0w}}}} & = & \frac{\int_{V_\mathrm{c}} \po \exp [-u_\mathrm{w} (s)]dV}{\int_{V_\mathrm{c}}\exp [- u_\mathrm{w}(s)] dV}
= \frac{{\rho_h} (\xi)}{\rho_\mathrm{h}^\mathrm{ig} (\xi)} \nonumber \\
& = & \frac{\xi ^\mathrm{ig}({\rho_h})}{\xi ({\rho_h})}=\exp[-\{\mu({\rho_h})-\mu^\mathrm{ig}({\rho_h})\}]
\label{eq:GFV}
\end{eqnarray}
where, in all cases, the superscript ``ig'' denotes the corresponding
quantity for an ideal gas confined to an identical slit pore. As one
can see from the above equation, since this generalized available volume
inherently relates the thermodynamic state of the confined fluid to
that of an ideal gas, it bears some resemblence to an excess
thermodynamic property.
It is important to point out that this generalized available volume
has several distinguishing
features. (i)~It reduces to ${\overline {p_{\rm{0}}}}$ in the limit of fluids confined
to smooth hard boundaries. Thus, all of the results presented earlier
in this paper for ${\overline {p_{\rm{0}}}}$ will remain unchanged for those systems if
one instead uses ${\overline {p_{\rm{0w}}}}$. (ii)~Unlike ${\overline {p_{\rm{0}}}}$ or density, there is no
arbitrary choice that needs to be made about the volume over which
one should do the averaging. This quantity is the same no matter
whether averaging is carried out over the center-accessible or the
total volume of the fluid. Put differently, this definition removes
any arbitrariness about the effective ``diameter'' of the
fluid-boundary interaction.
(iii)~The quantity ${\overline {p_{\rm{0w}}}}$ can be computed directly
from knowledge of standard thermodynamic and system parameters,
namely $\rho$, $\xi$, and $u_\mathrm{w} (z)$. (iv) It is not limited
to HS fluid systems. In fact, the computation of ${\overline {p_{\rm{0w}}}}$ from Eq.~\eref
{eq:GFV} does not even require information about
the particle-particle interactions, as long as the other thermodynamic
quantities can be measured.
How well does this new generalized measure of available volume track
self diffusivity when finite particle-boundary interactions are
present? Figure~\ref{dbydbulk_slitint} clearly illustrates that
${\overline {p_{\rm{0w}}}}$ corrects for the problems that ${\overline {p_{\rm{0}}}}$ faces in predicting
the dynamics in these cases. In particular, the maximum error
in self-diffusivity predictions based on ${\overline {p_{\rm{0w}}}}$
is $16\%$ across the entire range of packing fractions investigated,
which makes it a more reliable predictor of single-particle dynamics
than either ${s^{\rm{ex}}}$ or $\rho$ for these systems. As was seen earlier,
diffusivity predictions based on the bulk structure-property relation for
${\overline {p_{\rm{0w}}}}$ are considerably more accurate than that for $\rho$ or ${s^{\rm{ex}}}$
when considering high density, low diffusivity state points ($D<0.1$).
\section{Using DFT together with bulk structure-property
relations to predict dynamics of confined fluids}
\label{sec:dbydbulk_DFT}
\begin{figure}[h]
\includegraphics{DFT_TMMC_dbydbulk-scaled}
\caption{Ratio of self diffusivity of a bulk HS fluid to
that of a confined HS fluid with the same
value of a static quantity $x$, ${D_{\rm{bulk}}^x}/D$. Here we show a
comparison of ${D_{\rm{bulk}}^x}/D$ with $x$ obtained from the DFT (lines) or
TMMC (symbols) calculations. (a, b, and c) Data for a
monodisperse HS fluid confined by smooth hard
boundaries to a slit pore [$H=5$], a cylindrical pore [$H=6$],
and a square channel [$H=5$] geometry, respectively.
(d and e) Data for small and large
particles of a binary HS mixture confined in a $H=5$ slit
pore, respectively. (f and g) Data for a monodisperse HS fluid
confined between the repulsive and attractive boundaries discussed
in the text, respectively,
placed at a separation of $H=5$ in slit pore geometry.
The density range for the state points shown are the same as in
Figure~\ref{dbydbulk_all} for [a-c],
Figure~\ref{dbydbulk_slitmix} for [d and e], and
Figure~\ref{dbydbulk_slitint} for [f and g].
$20\%$ bounds on relative error in self-diffusivity predictions
are shown by the red dash-dotted line.\label{dbydbulk_DFT}}
\end{figure}
Above we have shown that knowledge of ${D_{\rm{bulk}}^x}$ for the bulk HS fluid
($x={s^{\rm{ex}}}$ or $x={\overline {p_{\rm{0w}}}}$) together with the value of $x$ in confinement
is enough for semi-quantitative prediction of confined (equilibrium) fluid
diffusivity, $D$, across a wide range of parameter space in these systems.
Thus far, we have used TMMC simulations to determine $x$ for each confined
fluid of interest. That raises the following question. Is classical
DFT accurate enough in its
predictions of $x$ that one can eliminate the step of simulating the
confined fluid altogether? In this section, we take a first step toward addressing this question.
In particular, we
present calculations of the ratio of self diffusivity of a bulk HS
fluid to that of a confined HS fluid with the same value of $x$,
${D_{\rm{bulk}}^x}/D$, where $x={s^{\rm{ex}}}$ or ${\overline {p_{\rm{0w}}}}$. In each case, $x$
of the confined fluid is obtained directly from predictions of
Rosenfeld's fundamental measure
theory~\cite{Yu2002Structuresofhard-sphere}, an accurate DFT for these systems.
Figure~\ref{dbydbulk_DFT} shows relative errors
in self-diffusivity predictions based
on ${s^{\rm{ex}}}$ and
${\overline {p_{\rm{0w}}}}$, comparing cases with knowledge of the ``exact'' value of $x$
in confinement (calculated
via TMMC simulations) and the predicted value of $x$ (calculated via DFT).
Selected cases explored in the previous section
involving the three confining geometries (slit pore,
cylindrical pore, and square channel), small and large particles of
binary mixtures, and finite
fluid-boundary interactions are presented. In all cases, the
${D_{\rm{bulk}}^x}/D$ curves obtained via the two routes (TMMC versus DFT)
are virtually indistinguishable over the entire density range of
equilbrium fluid, a demonstration of the reliability of
DFT for computing the static properties of these systems.
As a result, it is clear that one can use the bulk structure-property
relations discussed above together with predictions of $x$ from DFT to make
semi-quantitative estimates of confined fluid self diffusivity for a
wide variety of hard sphere systems.
\section{Conclusions}
\label{sec:theend}
Fluids trapped in small spaces feature prominently in science and
technology, and understanding their properties is key for a number of
research areas that range from the design of membranes
to the engineering of microfluidic devices.
The static and dynamic properties of these confined fluids
can be very different than those of bulk samples.
While quantitatively accurate theories like DFT are available for
predicting static properties of confined fluids, making even
qualitative predictions for dynamics of inhomogeneous fluids
has long been a challenging endeavor.
In this paper, we demonstrate how semi-quantitative (albeit indirect)
predictions of
dynamics from first principles are still possible, even in the absence
of a theory, once one recognizes that certain relationships between
static and dynamic properties are insensitive to confinement.
This study provides a systematic and quantitative
investigation of such relationships.
In particular, we present a comprehensive
study of the effects of confinement on the correlation between
self diffusivity ($D$) and various thermodynamic measures for confined HS
fluids: particle
center-accessible-volume-based and total-volume-based
average densities (${\rho_h}$ and $\rho$, respectively), excess
entropy (${s^{\rm{ex}}}$), and two average measures of fractional available volume
(${\overline {p_{\rm{0}}}}$~and~${\overline {p_{\rm{0w}}}}$). Our main findings are as follows. The
bulk structure-property correlation, ${D_{\rm{bulk}}^x}$,
based on the first
density measure, $x={\rho_h}$, severely underestimates $D$ when ${\rho_h}$ of
the confined fluid is used as the input. Further, for
dense confined fluids, ${\rho_h}$ is often larger than the freezing
density of the bulk fluid, eliminating altogether
the possibility of using the corresponding
bulk structure-property relation for predictions. Self-diffusivity
predictions based on the relation with
total-volume-based density $x=\rho$ provide a
significant improvement over those involving ${\rho_h}$, substantiating the
earlier idea that $\rho$ might be
considered as a more natural measure of density for predicting
dynamics~\cite{Mittal2006ThermodynamicsPredictsHow}.
However, when one considers a wider
variety of geometries [slit pore, cylindrical,
square channel], confined fluid mixtures, finite
particle-boundary interactions, and a wide range of packing fractions,
one finds that
structure-property relations ${D_{\rm{bulk}}^x}$ based on excess entropy ($x={s^{\rm{ex}}}$)
and a new generalized measure of available volume ($x={\overline {p_{\rm{0w}}}}$)
provide much more accurate
estimates for $D$ than those based on either of ${\rho_h}$ or $\rho$.
Further, self-diffusivity predictions based on ${\overline {p_{\rm{0w}}}}$ are
significantly more accurate than those based on ${s^{\rm{ex}}}$ under
conditions of high packing
fractions (e.g., supercooled fluids) and highly restrictive
confining geometries (e.g., quasi-one-dimensional channels). The generalized
available volume ${\overline {p_{\rm{0w}}}}$ may also be easier to compute based on
experimental quantities than ${s^{\rm{ex}}}$,
since the former is related to average density and chemical potential in a
simple way (see Eq.~\ref{eq:GFV}). Importantly, neither ${\overline {p_{\rm{0w}}}}$ nor
${s^{\rm{ex}}}$ require one to arbitrarily define an averaging volume for the
confined system.
Predictions of $x$ via classical DFT are accurate
enough for inhomogeneous HS fluids that one can use them, together
with the bulk structure-property relation (${D_{\rm{bulk}}^x}$), to make
semi-quantitative estimates of confined fluid diffusivities.
This effectively eliminates the need for simulating the confined fluid
altogether, which might be particularly convenient in applications
where one needs to estimate the dynamics of systems
across a wide array of parameter space. For example, in
the design of microfluidic
systems, one might hope to screen a large range of
possible particle-boundary interactions or confining geometries
against design considerations. A
preliminary application of this idea~\cite{Goel2008Tuningdensityprofiles} is to use DFT to passively tune the transport properties of a confined fluid, in a
controlled way, by modifying the geometry or boundary-particle
interactions of the confined space.
Can the
aforementioned static measures predict the dynamics of fluids with
continuous intermolecular potentials and/or attractive interactions?
As mentioned earlier, recent data from molecular simulations have
shown that there is indeed an
isothermal correlation between the self-diffusion
coefficient $D$ and the excess entropy for a variety of confined
fluids (e.g., Lennard-Jones, square-well, Weeks-Chandler-Andersen),
approximately independent of the degree of
confinement for a wide range of equilibrium
conditions~\cite{Mittal2007RelationshipsbetweenSelf-Diffusivity,Goel2008Tuningdensityprofiles}.
We are currently exploring the viability of using the generalized
available volume for predicting dynamics in these fluids, and we will
report on our findings in a future study.
\ack
T.M.T. acknowledges support of the National
Science Foundation (NSF) under Grant No. CTS-0448721, the Welch
Foundation, the David and
Lucile Packard Foundation, and the Alfred P. Sloan
Foundation. W.P.K. and G.G. acknowledges support from a NSF Graduate Research
Fellowship and a UT ChE department fellowship, respectively. The Texas Advanced
Computing Center (TACC), the Biowulf PC/Linux cluster at the
National Institutes of Health, Bethesda, MD, and the University at
Buffalo Center for Computational Research provided computational
resources for this study.
\section*{References}
|
1,941,325,220,529 | arxiv | \section{Introduction}
Biodiversity has not remained constant nor followed a simple
trajectory through geologic time \citep{raup1982, sepkoski1984,
gilinsky1994, liow2007, alroy08}. Instead, it has been marked by
fluctuations in the richness of taxa, both positive in the case of net
origination, or negative in the case of net extinction. Major events,
such as adaptive radiations and mass extinctions have received special
attention \citep{benton1995, Erwin1998}, but fluctuations of all sizes
are ubiquitous \citep{sepkoski1984, alroy08} and follow a fat-tailed
distribution where large events are more probable compared to, e.g. a
Gaussian distribution. Understanding the fat-tailed nature of these
fluctuations continues to elude paleobiologists and biodiversity
theoreticians.
The fat-tailed distribution of fluctuations in taxon richness inspired
earlier researchers to invoke ideas from complex systems with similar
distributions. Such ideas include the hypotheses that biological
systems self-organize to the brink of critical phase-transitions
\citep{bak1993, sole1997} and that environmental perturbations are
highly non-linear \citep{newman1995}. New data and analyses have not,
however, supported these hypotheses at the scale of the entire
Phanerozoic marine invertebrate fauna \citep{kirchner1998, alroy08}.
Other studies have modeled the mean trend in taxon richness as
tracking a potentially evolving equilibrium \citep{sepkoski1984,
alroy2010, quental2013} and yet ignore the role of stochasticity and
non-equilibrium dynamics in producing observed patterns
\citep{erwin2012, liow2007, jordan2016}. Individual, population, and
local ecosystem scale processes that could produce complex dynamics,
such as escalatory co-evolutionary interactions \citep{vermeij1987},
have not been documented to scale up to global patterns
\citep{madin2006} and indeed should not be expected to do so
\citep{vermeij2008}. Thus, we still lack a theory to describe the
striking fat-tailed nature of fluctuations throughout the Phanerozoic.
Despite the heterogeneity of explanations of Phanerozoic biodiversity,
consensus has emerged on one property of macroevolution: clades
experience different rates of morphological evolution, origination and
extinction \citep{simpson1953, sepkoski1984, gilinsky1994,
rabosky2014}. Here we show that the simple fact of conserved rates
within clades and variable rates across clades is sufficient to
describe pervasive, fat-tailed fluctuations in taxonomic richness
throughout the marine Phanerozoic. This biological mechanism has a
precise correspondence to the non-equilibrial theory from statistical
physics known as ``superstatistics'' \citep{beck2003} which has been
applied across the physical and social sciences \citep{beck2004,
fuentes2009}. We leverage this correspondence to explain the
distribution of fluctuations in the standing richness of marine
invertebrates preserved in the Phanerozoic fossil record. We further
show that the specific mathematical form of this superstatistical
distribution is consistent with niche conservatism
\citep{roy2009range, hopkins2014} and pulsed exploration on an
adaptive landscape by higher taxa \citep{simpson1953,
eldredgeGould1972, newman1985adaptive, hopkins2014}. We
operationally define ``adaptive landscape'' to mean a clade's set of
characteristics, and the fitness they impart to the clade, that
influences its macroevolution. Those characteristics could be
ecological (e.g. substrate preference \citep{bambach1983, bush2007,
hopkins2014}), morphological (e.g. body plan \citep{erwin2012}), or
macroecological (e.g. range size \citep{harnik2011,
foote2008paleobiol}).
\subsection{Superstatistics of fossil biodiversity}
Superstatistics \citep{beck2003} proposes that non-equilibrial systems
can be decomposed into many local sub-systems, each of which attains a
unique dynamic equilibrium. The evolution of these dynamic equilibria
across sub-systems occurs more slowly. This separation in time scales
allows local systems to reach equilibrium while the system as a whole
is not \citep{beck2003}. In the context of macroevolution we propose
that a clade with conserved macroevolutionary rates corresponds to a
sub-system in dynamic equilibrial.
In statistical mechanics, local sub-systems can be defined by a simple
statistical parameter $\beta$ often corresponding to inverse
temperature. In macroevolutionary ``mechanics'' we define the
$\beta_k$ of clade $k$ as the inverse variance of fluctuations $x_k$
in the number of genera within that clade, i.e. fluctuations in the
genus richness. The $\beta_k$ thus represent the inverse variances,
what we term volatilities, of the origination-extinction processes of
genera with clades. The details of this origination-extinction
process, e.g. whether it is linear or subject to a carrying capacity,
are not of central importance to our analysis; so long as fluctuations
can reach a stationary distribution and are observed over
time-averaged intervals in a temporally coarse-grained fossil record
they will be approximately Gaussian (see Supplemental Section
\ref{sec:suppLimitDist}; \citep{grassmann1987}).
We make the hypothesis of dynamic equilibrium within a clade following
MacArthur and Wilson \citep{macWilson} in recognition that while the
identity and exact number of taxa will fluctuate stochastically from
random origination and extinction (taking the place of local
immigration and extinction in island biogeography), the overall
process determining the number of taxa, and by extension, fluctuations
in that number, is in equilibrium. Indeed, the different regions of
adaptive space occupied by different clades can be conceptualized as
islands with unique dynamic equilibria, albeit with macroevolutionary
processes determining the ``colonization'' of adaptive peaks, as
opposed to short timescale biogeographic processes.
The volatility of richness fluctuations will vary across these islands
in adaptive space as an emergent trait of a clade resulting from the
macroevolutionary fitness of the clade and the shape of the
surrounding adaptive landscape. Ultimately, volatility emerges from
the life histories, ecologies, and evolutionary histories that
drive each clade's macroevolutionary fitness and characterize its
occupancy of different regions of an adaptive landscape. We do not
attempt to diagnose which characteristics of different regions account
for volatility differences, but others have found rates of origination
and extinction to depend on larval type \citep{jablonski2008}, body
plan \citep{erwin2012}, body size \citep{harnik2011}, range size
\citep{harnik2011, foote2008paleobiol}, and substrate preference
\citep{hopkins2014}. Not all of these traits would be considered
dimensions of an ecological niche or characteristics of a guild
\citep{bambach1983, bambach2007, bush2007}, but they all point to
different strategies that influence a clade's macroevolutionary
success. These characteristics result from interactions between
heritable traits and environments, which themselves may be
semi-heritable \citep{nicheCons}. Thus different regions of adaptive
space, and the clades occupying them, will experience different
magnitudes of stochastic fluctuations in taxonomic richness. As clades
occasionally split to fill new regions of adaptive space their pulsed
diversification determines the non-equilibrium nature of the entire
biota.
\subsection{Real paleontological data to test superstatistics}
To uncover the superstatistical nature of the marine invertebrate
Phanerozoic fauna we analyzed the distribution of fluctuations in
genus richness (the lowest reliably recorded taxonomic resolution)
using the Paleobiology Database (PBDB; {\tt paleobiodb.org}). We
corrected these raw data for incomplete sampling and bias using a new
approach described in the methods section. Occurrences from the PBDB
were matched to 49 standard time bins all of approximately 11MY
duration following previous publications \citep{alroy08,
alroy2010}. Fluctuations in genus richness were calculated as the
simple difference between bias-corrected richnesses in adjacent
time bins.
To focus attention on the variance of fluctuations we zero-centered
each clade's fluctuation distribution. In this way we focus on
fluctuations about any possible trend toward net diversification or
extinction. Because ``equilibrium'' in the statistical mechanical
sense means a system undergoes coherent, concerted responses to
perturbation, the mean trend line (positive or negative) is of less
interest than deviations from it. We also note that the distributions
of fluctuations for most clades are already very close to a mean of 0
(mean at the family level: $0.038 \pm 0.176 \text{ SD}$), and so
centering has little influence on clade-specific fluctuation
distributions, consistent with the observation that origination is
often roughly equal to extinction \citep{foote2010Chapter}. Following
\citep{fuentes2009} we also ignore all instances of no change
(i.e. zero fluctuation).
We define potentially equilibrial sub-systems based on taxonomic
hierarchies as a full phylogenetic hypothesis for all marine
invertebrates is lacking. Taxa ideally represent groups of organisms
that descend from a common ancestor and share similar ecologically and
evolutionary relevant traits \citep{mayr1965systZool, erwin2007}. Thus
our model assumes that at a given higher taxonomic level, within-taxon
fluctuations in richness are driven by equilibrial processes
characterized by Gaussian distributions. We further assume that new
higher taxa arise due to the emergence of sufficiently novel traits
(be they ecological, morphological, life history, or macroecological)
so that those new taxa occupy a new region of an adaptive
landscape. We lastly assume that different regions of adaptive space
are characterized by different volatilities in origination and
extinction.
To evaluate the optimal taxonomic level for sub-system designation, we
test our superstatistical theory using taxonomic levels from family to
phylum. Additionally, we compare our results to randomized taxonomies
and confirm that the observed fit of superstatistics is not an
artifact of arbitrary classification but instead represents real,
biologically relevant diversification processes within and between
clades. We find that families and orders conform to the assumptions of
our superstatistical model while classes and phyla do not.
\section{Results}
We first evaluate the local equilibria of clades from family level to
phylum. We find that family level fluctuation distributions are well
approximated by Gaussians (Figs. \ref{fig:pk_f} and
\ref{figSupp:pkx_allTaxa}). Three exemplar family-level dynamics are
highlighted in Figure \ref{fig:pk_f} to illustrate how different
volatility equilibria express themselves as actual richness
timeseries. This Gaussian approximation also largely holds for
orders, but classes and phyla increasingly show deviations from
Gaussian with greater kurtosis corresponding to more frequent outliers
at these taxonomic levels (Fig. \ref{figSupp:pkx_allTaxa}).
\begin{figure}[!h]
\centering
\includegraphics[scale=0.9]{fig_pkx-fbeta.pdf}
\caption[Variability in trajectories of within-family fluctuations
in genus richness]{The distributions of within-family fluctuations
in genus richness and across family volatilities. Panel (A) shows
the richness fluctuation trajectories of three exemplar families
and (B) shows an empirical cumulative density function of richness
fluctuations aggregated across all families, highlighting the
exemplar families. To display all families simultaneously we
simply collapse their fluctuation distributions by dividing by
their standard deviations. If families conform to the Gaussian
hypothesis their scaled fluctuations should fall along the
cumulative density line of a normal N(0, 1) distribution, as shown
in (B). We further confirm this normal distribution in the
supplement (Fig. \ref{figSupp:pkx_allTaxa}). In (C) the
distribution of volatilities (inverse variances $\beta_k$) across
all families matches very closely to a Gamma distribution (black
line); exemplar families are again highlighted.}
\label{fig:pk_f}
\end{figure}
To predict the superstatistical behavior of the entire marine
invertebrate Phanerozoic fauna we must integrate over all possible
local equilibria that each clade could experience. The stationary
distribution of $\beta_k$ values describes these possible equilibria,
specifying the probability that a given clade, chosen at random, will
occupy a region of adaptive space characterized by $\beta_k$.
We estimate the distribution of $\beta_k$'s simply as the maximum
likelihood distribution describing the set of volatilities for all
families, orders, classes, or phyla. Phanerozoic marine invertebrate
families clearly follow a Gamma distribution in their $\beta_k$ values
(Fig. \ref{fig:pk_f}). The Gamma distribution also holds for orders but
shows increasing deviations again for classes and especially phyla
(Fig. \ref{figSupp:fbeta_allTaxa}).
Using the observation of within family statistical equilibrium and
Gamma-distributed $\beta_k$ parameters we can calculate, without
further adjusting free parameters, the distributions of family-level
fluctuations for the entire marine Phanerozoic, $P(x)$, as
\begin{equation}
P(x) = \int_0^\infty p_k(x \mid \beta) f(\beta) d\beta \label{eq:PxInt}
\end{equation}
where
$p_k(x \mid \beta) = \sqrt{\frac{\beta}{2\pi}} e^{-\frac{\beta
x^2}{2}}$ is the distribution of fluctuations within a family and
$f(\beta) = \frac{1}{\Gamma(b_1/2)}
\left(\frac{b_1}{2b_0}\right)^{b_1/2} \beta^{(b_1/2) - 1}
\text{exp}\left(-\frac{b_1 \beta}{2 b_0}\right)$ is the stationary
distribution of volatilities in richness fluctuations. The integral in
(\ref{eq:PxInt}) leads to
\begin{equation}
\label{eq:Px}
P(x) = \frac{\Gamma\left(\frac{b_1 +
1}{2}\right)}{\Gamma\left(\frac{b_1}{2}\right)}
\sqrt{\frac{b_0}{\pi b_1}} \left(1 + \frac{b_0
x^2}{b_1}\right)^{-\frac{b_1 + 1}{2}}
\end{equation}
This corresponds to a non-Gaussian, fat-tailed prediction for $P(x)$
which closely matches aggregated family level fluctuations in the
bias-corrected PBDB (Fig. \ref{fig:Px}).
\begin{figure}[!h]
\centering
\includegraphics[scale=1]{fig_Px.pdf}
\caption[Clade-level distribution of richness
fluctuations]{Distribution of fluctuations in genus richness within
different taxonomic groupings of marine invertebrates in the
Paleobiology Database \citep{alroy08} after sampling
correction. The distribution is fat-tailed as compared to the
maximum likelihood estimate of the normal distribution (blue
line). At the family and order level the empirical distribution
of fluctuations are well described by our superstatistical
approach, both when computed from integrating over the
distribution of observed variances (red line) and when fit via
maximum likelihood (95\% confidence interval; red shading in
(A) and (B)).}
\label{fig:Px}
\end{figure}
To quantitatively evaluate how well the superstatistical prediction
matches the family- and order-level data we constructed a 95\%
confidence envelope from bootstrapped maximum likelihood estimates of
$P(x)$. Observed fluctuations for both taxonomic levels fall within
these 95\% confidence envelopes (Fig. \ref{fig:Px}), indicating that
the data do not reject the superstatistical prediction. For further
comparison, we fit a Gaussian distribution to the observed
fluctuations, which corresponds to the equilibrium hypothesis that all
families conform to the same dynamic. Using Akaike Information
Criterion (AIC) we find that observed fluctuations are considerably
better explained by the superstatistical prediction than by the
Gaussian hypothesis ({\small $\Delta$}AIC = 1895.622). Thus, as
expected under the superstatistical hypothesis, the fat-tailed
distribution of fluctuations arise from the superposition of
independent Gaussian statistics of fluctuations within families.
Computing the distribution of aggregated fluctuations using orders
also closely matches the observed data (Fig. \ref{fig:Px}) but as we
further coarsen the taxonomy to classes and phyla we see increasingly
poorer correspondence between data and theory (Fig. \ref{fig:Px}).
We quantify this change in the goodness of fit with the
Kolmogorov-Smirnov statistic (Fig. \ref{fig:dStat}). We can see that
both families and orders have low Kolmogorov-Smirnov statistics, and
in fact order level designation of equilibrial subsystems performs
slightly better than the family level. Classes are substantially worse
and phyla worse yet with the Kolmogorov-Smirnov statistic of phyla
being no different than the null randomized taxonomies described
below.
However, if superstatistical theory explains the data, this worsening
fit with increasing taxonomic scale is expected as the different
classes and phyla should not represent dynamically equilibrial
sub-systems in their fluctuation dynamics. Instead, classes and phyla
aggregate increasingly disparate groups of organisms, and thus
effectively mix their associated Gaussian fluctuations, meaning that
one statistic should no longer be sufficient to describe class- and
phylum-level dynamics. We see this confirmed by the increasing
frequency of outlier fluctuations in within class and phylum level
fluctuation distributions (Fig. \ref{figSupp:pkx_allTaxa}). We can
also see that families and orders represent, on average, 1 to 2
ecospace hypercubes (defined by taxon environment, motility, life
habit, vision, diet, reproduction, and ontogeny \citep{bambach1983,
bambach2007, bush2007}), respectively. In contrast, classes and
phyla represent, on average, 8 to 30 hypercubes, respectively
(Fig. \ref{figSupp:eeSpaceOcc}).
Our analysis indicates that orders and families are evolutionarily
coherent units with all subsumed taxa sharing key ecological
and evolutionary attributes allowing them to reach steady state
diversification independently from other clades at global scale. The
fact that both orders and families conform to theoretical predictions
is consistent with superstatistics. If superstatistics operates at the
order level, then the families subsumed by these orders should
represent random realizations of their order's stationary $\beta_k^{(order)}$
volatility. The sum of Gamma random variables is still Gamma, but with
new parameters, thus the family level distribution of
$\beta_k^{(family)}$ is still Gamma.
\begin{figure}[!h]
\centering
\includegraphics[scale=1]{fig_dStat.pdf}
\caption[Goodness of superstatistical theory fit]{Distribution of
Kolmogorov-Smirnov (KS) statistics from randomly permuting genera
within families (gray shading represents 95\% confidence
interval). Solid colored lines are observed KS statistics at
different taxonomic levels as indicated.}
\label{fig:dStat}
\end{figure}
To further test the evolutionary coherence of families we conducted a
permutation experiment in which genera were randomly reassigned to
families while maintaining the number of genera in each family. For
each permutation, we calculated the superstatistical prediction and
its Kolmogorov-Smirnov statistic. The permutation simulates a null
model in which common evolutionary history is stripped away (genera
are placed in random families) but the total number of observed genera
per family is held constant. Because we ignore all instances of no
change (i.e. 0 fluctuation) we remove any possible large and
artificial gaps in the genus occurrences of these permuted
clades. Controlling for the total number of genera per family is key
because this could be purely an artifact of an arbitrary taxonomic
process \citep{yule1925, capocci2008} and genus richness alone could
be solely responsible for differences in the $\beta_k$ across clades.
We test the possibility that richness is responsible for variation in
$\beta_k$ in two ways. First, we find that the distribution of genus
richnesses within families is not itself distributed Gamma
(Fig. \ref{figSupp:richGamma}), indicating that there is not a simple
equivalence between $\beta_k$ and the richness of family $k$. Second,
we find that the number of genera in a family and that family's
$\beta_k$ value are indeed negatively correlated
(Fig. \ref{figSupp:betaByRich}). A negative correlation between clade
richness and $\beta_k$ is not unexpected because fluctuations are the
sums of the random variables representing genus origination and
extinction events; the more of these random variables in the summation
(i.e. the more genus richness in a clade) the higher the variance of
the summation. Because $\beta_k \equiv 1/\sigma_k^2$ increasing
richness should lead to decreasing $\beta_k$ values. Thus we want to
know if this correlation accounts for all downstream superstatistical
results. The permutation test is specifically designed to determine if
the $\beta_k$ imposed by this correlation with richness are sufficient
to explain the observed superstatistical fit.
Repeating the null permutation of genera in families 500 times yields
a null distribution of Kolmogorov-Smirnov statistics that is far
separated from the observed values at the family and order levels
(Fig. \ref{fig:dStat}) suggesting that the good fit at these levels is
not merely a statistical artifact of classification or the richness of
clades, but carries important biological information. Classes approach
the null and phyla are no different. It should also be noted that the
width of 95\% confidence interval of this null distribution is not far
from the distance between the Kolmogorov-Smirnov statistics of orders
versus families, suggesting that differences of fit between these
taxonomic levels is at least partially accounted for by the randomness
of the sampling distribution of Kolmogorov-Smirnov statistics.
\section{Discussion}
Our analysis makes no assumption that orders and families should
correspond to superstatistical subsystems, but identifies them as the
appropriate level for marine invertebrates. Our study is the first to
demonstrate that complex patterns in the fluctuation of taxon richness
in the fossil record are the result of a simple underlying process
analogous to the statistical mechanisms by which complexity emerges in
large, non-equilibrium physical \citep{beck2004} and social systems
\citep{fuentes2009}. We do so by identifying the biological scale at
which clades conform to locally independent dynamic equilibria in
fluctuations. Equilibrium could result from many processes, including
neutrality \citep{macWilson, hubbell2001}, diversity-dependence
\citep{moen2014, foote2018} and processes that dampen---rather than
exacerbate---fluctuations in complex ecological networks
\citep{berlow2009}. These candidate processes are directly opposed to
the presumption of instability underlying the self-organized
criticality hypothesis for paleo biodiversity \citep{bak1993,
sole1997}.
We show that the distribution describing the evolution to different
equilibria between orders and families is Gamma (Fig. \ref{fig:pk_f}).
A Gamma distribution, while consistent with multiple processes, could
result from evolution of diversification rates across an adaptive
landscape that promotes niche conservatism and pulsed exploration of
niche space \citep{cir1985}. Specifically, if $\beta_k$ values are
associated with a clade's macroevolutionarily-relevant traits, and
those traits evolve via Ornstein-Uhlenbeck-like exploration of an
adaptive landscape, the resulting stationary distribution of $\beta_k$
will be Gamma \citep{cir1985}. For macroevolutionary rates to vary in
a way consistent with the observed superstatistical description of
fluctuations this landscape cannot be flat (i.e. equal fitness
everywhere), but instead must be rugged. Thus, niche conservatism
around local fitness optima in adaptive space interrupted by adaptive
exploration is likely \citep{newman1985adaptive,
gavrilets2004book}. The specifics of how this adaptive landscape is
shaped and is traversed by evolving clades determine the exact form of
the distribution of $\beta_k$ volatilities, in the case of the marine
Phanerozoic resulting in a Gamma distribution. Our work thus motivates
further study of the trait spaces and evolutionary shifts consistent
with Gamma-distributed equilibria in richness fluctuation
volatilities.
We show that the pulsed shift to different equilibria between orders
and the families they subsume is sufficient to explain the
characteristically fat-tailed distribution of richness fluctuations
when the marine Phanerozoic invertebrate fauna is viewed as a whole
macrosystem. Armed with an understanding of the statistical origin of
this diversification pattern we can explore which models of niche
conservatism and pulsed adaptive radiation are consistent with the
statistical behavior of the Phanerozoic. Our statistical theory
provides new motivation for identifying the eco-evolutionary causes of
innovations between lineages and how those innovations are eventually
conserved within lineages. Using the superstatistical prediction as a
theoretical baseline, we can also go on to identify and robustly
examine the mechanisms underlying deviations from statistical
theory. For example, some clades wax and wane systematically, and
possibly non-symmetrically, through time \citep{liow2007,
foote2008paleobiol, quental2013}, a pattern that we cannot explain
with superstatistics alone.
Superstatistics could also be applied to other areas of evolution and
macroecology. For example new phylogenetic models already consider
heterogeneous rates of diversification (e.g., \citep{rabosky2014}) as
expected between different subsystems. The superstatistics of clades
in adaptive landscapes could motivate models that jointly predict
changes in traits and diversification, a research area currently
struggling with model inadequacy \citep{rabosky2017fisse}. This
framework could also provide a new paradigm in modeling the
distributions of richness, abundance, and resource use in non-neutral
communities which can be viewed as emerging from the combination of
locally equilibrium subsystems. Non-neutral models in ecology are
criticized for their over-parameterization \citep{rosindell2011}, yet
a persistent counter argument to neutral theory \citep{hubbell2001} is
the unrealistic assumption of ecological equivalency and poor
prediction of real dynamics \citep{rosindell2011}. If ecosystems are
viewed as the superposition of many individualistically evolving
clades, each exploiting the environment differently and thus obeying a
different set of statistics, then diversity dynamics could be
parsimoniously predicted with superstatistics while incorporating real
biological information on ecological differences between taxa.
Superstatistics is a powerful tool to derive macro-scale predictions
from locally fluctuating sub-systems whose evolution is driven by
interesting, but complex and difficult to model, biological
mechanisms. As such, applications of superstatistics to a wide variety
of patterns in ecological and evolutionary systems are ripe for
exploration.
\section{Methods and Materials}
All data processing and analyses were preformed in R \citep{rcite} and
all code needed to reproduce our study are provided, with added
explanation, in supplemental Appendix A.
\subsection{Paleobiology Database data download and filtering}
Data on individual fossil occurrences and the ecospace characteristics
of Phanerozoic marine invertebrates were downloaded from the
Paleobiology Database (PBDB; \newline\texttt{https://paleobiodb.org}) on 16
November 2018 via the database's API (data retrieval and processing
script available in the supplement). Collections were filtered using
the same approach as Alroy \citep{alroy08} to insure that only well
preserved marine invertebrate occurrences were used in subsequent
analyses. This filtering resulted in 815,222 unique genus-level
occurrences. These were further filtered to exclude those occurrences
without family-level taxonomy and those collections with age estimate
resolutions outside the 11MY time bins proposed by Alroy
\citep{alroy08} resulting in 454,033 occurrences. Time bins were
compiled from {\tt http://fossilworks.org} with a custom script
reproduced in the supplement. The first and last of these time bins,
corresponding to the earliest Cambrian and the latest Cenozoic, were
excluded from analysis because their sampling completeness (see below)
could not be assessed.
\subsection{Correcting for imperfect and potentially biased sampling}
\label{sec:3TP}
We use a new and flexible method to correct for known sampling
incompleteness and biases in publication-based specimen databases
\citep{alroy08, alroy2010}. Incompleteness is inherent in all
biodiversity samples, the fossil record being no exception
\citep{miller1996, foote2016, starrfelt2016, close2018}. In addition
to incompleteness, bias may result from preferential publication of
novel taxa \citep{alroy2010} which exacerbates the difference between
poorly-sampled and well-sampled time periods. We therefore develop a
simple two-step method: we first correct genus richness for incomplete
sampling using the ``three-timer'' correction \citep{alroy08} and then
further correct this three-timer richness estimate by accounting for
any correlation between the number of genera and the number of
publications in a time period.
The three-timer correction estimates the probability of failure to
observe a genus in a given time period $p_t$ as the number of times
any genus is recorded before and after that period but not during,
divided by the number of genera whose occurrence histories span the
period $t$. To calculate the sampling-corrected richness
$\hat{D}_{kt}$ of a clade $k$ in the time period in question, the
observed genera within that clade and time period are divided by
$1 - p_t$ and their occurrences summed:
\begin{equation}
\hat{D}_{kt} = \sum_{j \in k} \frac{I_{jt}}{1 - p_t}
\end{equation}
where $j \in k$ designates genera in clade $k$ and $I_{jt}$ is an
indicator equal to 1 if genus $j$ occurs in time period $t$.
$\hat{D}_{kt}$ is the maximum likelihood estimator of richness in a
simple occupancy through time type model assuming binomial sampling
\citep{royleDorazio}, and in that way mimics other proposed methods
for the fossil record \citep{foote2016, starrfelt2016}. We avoid
parametrically modeling the sampling process through time by instead
taking a sliding window of time bins from the Cambrian to the
Cenozoic. It should be noted that the three-timer correction compares
favorably to other similar methods to account for imperfect detection
\citep{alroy2014}
To eliminate further bias due to preferential publication of novel
taxa \citep{alroy2010} we divide the three-timer-corrected number of
genera per family per time period by the expected number of genera
given publications in that time period. The expected number is
calculated by regressing the log-transformed three-timer-corrected
number of genera on log-transformed number of publications. There is
only a weak trend toward higher richness with more publications
(Fig. \ref{figSupp:divByPub}) meaning that the most important correction
comes from the three timer correction.
Our new method re-scales each genus occurrence from 0 or 1 (absent or
present) to a weighted number continuously ranging between 0 and
1. Because these weighted numbers represent sampling and
bias-corrected {\it occurrences} we can add them arbitrarily,
corresponding to the membership of any given genus in any given higher
taxonomic group. We must, however, choose a taxonomic level at which
to evaluate the relationship between richness and publications; we
choose the level of family because this is the most finely resolved
option.
We opt not to use subsampling methods \citep{miller1996, alroy2010,
kocsis2018} because these approaches would not be advisable for
clades with few genera. However, our new method achieves similar
results to subsampling procedures at the global scale across all
clades. We directly compare our predicted time series of global
fluctuations in genus richness with results derived from rarefaction
and shareholder quorum subsampling (SQS) in Figure
\ref{figSupp:3TPub}. Our method shows very minor differences with
these subsampling-based predictions and any discrepancies do not
impact the statistical distribution of fluctuations
(Fig. \ref{figSupp:3TPub}).
\subsection{Superstatistical methods} \label{sec:numMeth}
We first derive the superstatistical distribution $P(x)$ by fitting
Gaussian distributions to clade-level distributions of fluctuations
$p_k(x)$, extracting the inverse variances $\beta_k$ of those
$p_k(x)$, testing the best function to describe the distribution of
$\beta_k$, and then integrating
$P(x) = \int_{\beta}p_k(x | \beta) f(\beta)$. This process allows no
free parameters to hone the fit of $P(x)$ to the data. However, each
inverse variance must of course be estimated for each clade, making
its good fit to data all the more surprising. To do so we use least
squares instead of maximum likelihood because the asymmetric
fluctuation distributions of small clades were more reliably fit with
curve fitting than with maximum likelihood.
We also estimated $P(x)$ directly from the family-level data using
maximum likelihood to compare the fit of our superstatistical
prediction and that of a simple Gaussian distribution using AIC. To
calculate a likelihood-based confidence interval on our prediction we
bootstrapped the data, subsampling fluctuations with replacement from
all families and fit superstatistics using maximum likelihood to the
aggregated fluctuation distribution of each bootstrap replicate.
\bibliographystyle{science-advances}
|
1,941,325,220,530 | arxiv | \section{Introduction}
The Lawrence-Krammer representation \cite{Lawrence,Krammer,Bigelow2}
$\rho_n$
of the braid group $B_n$ into $SL(p,{\Bbb Z}[t^{\pm 1}q^{\pm 1}])$, with
$p=n(n-1)/2$, has become recently of interest as the first faithful
representation of braid groups. In this paper, we are concerned with
the identification of the image of $\rho_n$ on $B_n$ and certain
types of subgroups thereof. An important
property, unitarizability, is found by Budney \cite{Budney} (see
theorem \reference{TB} below). Our main result is the following.
\begin{theorem}\label{t1}
Assume $q,t$ with $|t|=|q|=1$ are chosen so that $t^aq^b=1$ for
$a,b\in{\Bbb Z}$ implies that $a=b=0$, and the Budney form is definite
at $q,t$. Moreover, assume that $\rho_n$ is irreducible at $q,t$.
Then $\ol{\rho_n(B_n)}= U(p)$ (for $p=n(n-1)/2$).
\end{theorem}
This is analogous to a previous result for the \em{Burau
representation} $\psi_n$ (which we simply call `Burau' below)
in \cite{reiko}.
The irreducibility of $\rho_n$ will be treated extra with lemma
\ref{t1_}. It should be pointed out that it has been proved at
separate places.
There is written account by M.~Zinno \cite{Zinno}, though it was
observed also by others, incl. V.~Jones, R.~Budney and W.~T.~Song.
W.~T.~Song has proved the stronger statement that Budney's form is
the only unitarizing form. (He informed me of this result prior to
my proof of theorem \reference{t1}, although now the theorem
implies this uniqueness property, at least for definite form,
by remark \reference{z2}.) But
it appears all this material is (yet) unpublished. There is closely
related work of I.~Marin \cite{Marin}, which we discuss in \S\ref{S6}.
The proof of lemma \reference{t1_} is provided for completeness and
because of its simplicity compared to other methods.
Our main motivation was again the study of braid representations
of links. The problem to determine conjugacy classes of braid
representations of a given link goes back to the 60s. For some
early work see e.g. \cite{MT}. With the increasing attention given
to braids the problem was studied later e.g. in \cite{Fiedler,%
Morton,Fukunaga}. We apply theorem \reference{t1} to prove
\begin{theorem}\label{t2}
Assume $L$ is a link and $n>b(L)$. Then there exist infinitely many
conjugacy classes of $n$-braid representations of $L$, except if\\
{\def\labeltype{(\alph{type})}%
\mylabel $n\le 3$ or (possibly) \label{t2:a}\\
\mylabel $L$ is a $(n-1,k(n-1))$-torus link $(k\in{\Bbb Z})$. (This includes
the case $k=0$ of the $n-1$-component trivial link.) \label{t2:b}\\
\setcounter{type}{0}%
}
\end{theorem}
The number $b(L)$ is the minimal number of strands of a braid
representation of $L$, and is called \em{braid index} (see
e.g. \cite{Murasugi}).
The case \ref{t2:a} is very well-known from \cite{BM2} to need to be
excluded, but we do not know anything about whether any link of case
\ref{t2:b} is indeed exceptional. Still the theorem almost completely
settles the (in)finiteness for $n>b(L)$. For $n=b(L)$ the situation
is far more complicated; there are certain links also for $n\ge 4$
with a single conjugacy class, e.g. unlinks \cite{BM}, and a further
example due to Ko and Lee. Contrarily, Shinjo has announced to me
(in preparation) that she has, as extension of her work \cite{Shinjo},
found very general families of knots with infinitely many minimal
braid conjugacy classes. It is possible that the decision problem
when finitely many and when infinitely many classes occur for $n=b(L)$
is too complex to have a meaningful answer.
Most of the rest of the paper, until the end of \S\reference{S5},
will be devoted to the proof of theorem \reference{t1}. The proof
is rather Lie-group theoretic, and we will need to bring up some
related material along the way. In \S\reference{S6} we extend
theorem \reference{t1} to denseness of the image of subgroups of
$B_n$. In \S\reference{S7} we discuss theorem \reference{t2}.
\section{Lawrence-Krammer representation and its unitarization}
The $n$-strand \em{braid group} $B_n$ is considered generated by
the Artin \em{standard generators} $\sg_i$ for $i=1,\dots,n-1$
\cite{Artin,Garside}. These are subject to relations of the
type $[\sg_i,\sg_j]=1$ for $|i-j|>1$, which we call
\em{commutativity relations} (the bracket denotes the
commutator) and $\sg_{i+1}\sg_i\sg_{i+1}= \sg_i\sg_{i+1}\sg_i$,
which we call \em{Yang-Baxter} (or shortly YB) \rm{relations}. We
write $[\bt]$ for the \em{exponent sum} of $\bt$, and set $B_{k,l}
\subset B_n$ for $1\le k<l\le n$ to be the subgroup $<\,\sg_k,
\dots,\sg_{l-1}\,>$ (where angle brackets mean `generated by').
The representation $\rho_n$ of $B_n$ can be defined as operating
on a complex vector space $R={\Bbb C}^p$ with $p=n(n-1)/2$ with
basis $\{\,v_{i,j}\,:\,1\le i<j\le n\,\}$ by
\begin{eqn}\label{rf}
\rho_n(\sigma_i) v_{j,k} = \left\{
\begin{array}{lr}
v_{j,k} & i\notin \{j-1,j,k-1,k\}, \\
qv_{i,k} + (q^2-q)v_{i,j} + (1-q)v_{j,k} & i=j-1, \\
v_{j+1,k} & i=j\neq k-1, \\
qv_{j,i} + (1-q)v_{j,k} - (q^2-q)tv_{i,k} & i=k-1\neq j,\\
v_{j,k+1} & i=k,\\
-tq^2v_{j,k} & i=j=k-1.
\end{array}
\right.
\end{eqn}
Here $t$ and $q$ may \em{a priori} be arbitrary non-zero
complex numbers. However, we will choose them always so
that $|t|=|q|=1$. (We will sometimes write $q,t$ explicitly
as parameters of $\rho_n$, with the understanding that
a braid cannot be confused with a complex number.) The
reason is the following result, which is of main importance
below.
\begin{theorem}(Budney \cite{Budney})\label{TB}
The Lawrence-Krammer representation unitarizable if $|q|=|t|=1$.
\end{theorem}
In other words, for
such $t$ and $q$, Budney \cite{Budney} defines a \em{unitarizing}
form $<\,.\,,\,.\,>$ of $\rho_n$ on ${\Bbb C}^p$. This is a sesquilinear
pairing respected by the action of $\rho_n$, in the sense that for all
$\bt\in B_n$ and $x,y\in{\Bbb C}^p$, we have
\[
<\rho_n(\bt) x,\rho_n(\bt) y>\,=\,<x,y>\,.
\]
This feature is analogous to the form of Squier \cite{Squier}
for the Burau representation $\psi_n$.
The \em{(reduced) Burau representation} $\psi_n$ of $B_n$, depending
on a complex parameter $q$ and acting on ${\Bbb C}^{n-1}$, is given by:
\begin{eqnarray*}
& \psi_n(\sg_i)\,=\left[
\begin{array}{*9c}
1 \\
& \ddots & & & & & 0 \\
& & 1 \\
& & & 1 & -q \\
& & & & -q \\
& & & & -1 & 1 \\
& & & & & & 1 \\
& & 0 & & & & & \ddots \\
& & & & & & & & 1 \\
\end{array}\,\right]\quad\mbox{for $1<i<n-1$,} & \\[3mm]
& \psi_n(\sg_1)\,=\left[
\begin{array}{*5c}
-q & 0 \\
-1 & 1 & & \multicolumn{2}{c}{0} \\
& & 1 \\
\multicolumn{2}{c}{0} & & \ddots \\
& & & & 1 \\
\end{array}\,\right],\quad\mbox{and}\quad
\psi_n(\sg_{n-1})\,=\left[
\begin{array}{*5c}
1 & & & \multicolumn{1}{c}{0} \\
& \ddots \\
& & 1 \\
\multicolumn{2}{c}{0} & & 1 & -q \\
& & & 0 & -q \\
\end{array}\,\right]\,,
\end{eqnarray*}
where at position $(i,i)$ there is always the entry $-q$.
We used Squier's form
previously to carry out a study of the image of $\psi_n$ in
\cite{reiko}. Again Budney's form is definite for proper $q$ and
$t$. We became aware of Budney's result only recently, and so
we tried to adapt details we had worked out for $\psi_n$.
At least one serious obstacle was visible in advance. Due to the
quadratic increase in dimension, an argument via a rank estimate
for a simple Lie group is more complicated. Still most simple Lie
group representations have a dimension larger than quadratic (in the
rank), and with a certain amount of extra effort we will be able to
deal with them.
\begin{rem}\label{z2}
Let us here remark that when $<\,.\,,\,.\,>$ is definite, the
subgroup of linear transformations of ${\Bbb C}^n$ respecting the
form is conjugate in $SL(n,{\Bbb C})$ to $U(n)$. Conversely, each such
subgroup determines the respected pairing up to complex conjugation
and multiples. This follows from that facts that the only outer
automorphism of $SU(n)$ is (entry-wise) complex conjugation, and
the centralizer of $SU(n)$ in $SL(n,{\Bbb C})$ are the scalar matrices.
\end{rem}
It will not be necessary to study the form very explicitly here.
We need only the following consequence of the formula in the
proof of Theorem 3 of \cite{Budney}. Below for a complex number
$z=re^{i\th}$ with $r\ge 0$ and $\th\in (-\pi,\pi]$
we set $|z|=r$ and $\arg z=\th$ (when $r\ne 0$).
\begin{prop}\label{xy}
If $q_i,-t_i\to 1$ with $|q_i|=|t_i|=1$ are chosen so that
$\ds\frac{1-q_i}{1-q_i\sqrt{-t_i}}\to 0$, then the Budney
form is definite at $q_i,t_i$ for $i$ large.
\end{prop}
\begin{rem}\label{z1}
The additional condition means that $\arg q_i/\arg -t_i\to 0$.
In other words, when $t_i$ is close to $-1$, one should choose
$q_i$ close to $1$ in a way depending on $t_i$.
It is clear that one can choose such $q_i,t_i$ which are
algebraically independent. This property will be used
in the proof of theorem \ref{t2}, but it will \em{not}
be relevant before \S\reference{S7}.
\end{rem}
The reason why we are interested in the value $t=-1$ is (see
\cite{Krammer}):
\begin{lemma}\label{lsym}
$\rho_n$ turns into the symmetric square of $\psi_n$ for $t=-1$.
\end{lemma}
Then, for $q=1$ we have (the symmetric square of) the permutation
homomorphism $\pi_n$. The homomorphism $\pi_n$ means here the
$n-1$-dimensional (irreducible) representation obtained from the
action of the symmetric group $S_n$ (onto which there is an obvious
homomorphism from $B_n$) on the coordinates of ${\Bbb C}^n$, after
removing the (invariant) space generated by $(1,1,\dots,1)$.
The notion of symmetric square is explained more precisely below.
\begin{warn}\label{w1}
The following should be kept in mind regarding the $t$ variable.
\begin{enumerate}
\item
The convention of \cite{Krammer} for the matrices of the representation
differs from \cite{Bigelow,Budney}; $t$ of former is $-t$ of latter.
We stick with Bigelow-Budney's convention for $t$.
\item
Also, in theorem 4.1, p.483 of \cite{Bigelow} there is a misprint: in
the fourth option $t$ should be $-t$; this is set right on p.782 of
\cite{Budney}, and our \eqref{rf} is a reproduction of latter formula.
\item
Next, $t$ is used often, e.g. in \cite{Jones}, as the variable for
$\psi_n$, but it is here $q$, \em{not} $t$, that via lemma \ref{lsym}
originates from Burau (and what we may call the Burau variable).
Apart from replacing $t$ by $q$, our definition of $\psi_n$ is as
in \cite{Jones}.
\end{enumerate}
\end{warn}
Let us clarify and fix some language. For a (complex)
vector space $V$ with basis $e_1,\dots,e_n$, we can define
the \em{symmetric square} ${\operator@font Sym}\,^2 V$ to be subspace
of $V\otimes V$ spanned by elements
\[
v\odot w=\frac{v\otimes w+w\otimes v}{2}\,.
\]
${\operator@font Sym}\,^2 V$ has the standard basis
\[
\{\,e_i\odot e_j\ :\ i\le j\,\}\,.
\]
When an endomorphism $f$ acts on a vector space $V$, then
it induces an endomorphism on ${\operator@font Sym}\,^2 V$ we write ${\operator@font Sym}\,^2 f$.
So we can talk of ${\operator@font Sym}\,^2 \psi_n$.
We will also need the \em{antisymmetric square} $\Lm^2 V$
generated by
\[
v\wedge w=\frac{v\otimes w-w\otimes v}{2}\,.
\]
Similarly there is a meaning to $\bigwedge^2 f$.
\begin{lemma}\label{t1_}
Assume $q,t$ with $|t|=|q|=1$ are chosen so that $t\ne \pm 1$
and $q$ is, dependingly on $t$, sufficiently close to $1$.
Also, assume the Budney form is definite at $q,t$. Then $\rho_n$
is irreducible at $q,t$.
\end{lemma}
\@ifnextchar[{\@proof}{\@proof[\unskip]}
For $(q,t)=(1,-1)$ the symmetric square ${\operator@font Sym}\,^2\pi_n$ of the
permutation homomorphism acts by permuting the indices in $v_{i,j}$.
Now, when $q=1$, but $t\ne \pm 1$, then the action is similar,
except that $\sg_i$ acts on $v_{i,i+1}$ by multiplying by $-t$.
It is clear then that such an endomorphism has eigenvalues
$\pm 1$ and $-t$, with the eigenspace for $-t$ being
1-dimensional, generated by $v_{i,i+1}$.
Since by unitarity every invariant subspace has an invariant
(orthogonal) complement, it follows that if there is an invariant
subspace $V$, then $V^\vd$ is also invariant, and one of both,
say w.l.o.g. $V$, contains $v_{1,2}$. But then by the way the
action is described, $V=R$.
Thus $\rho_n$ is irreducible for $q=1$, and then for $q$
close to $1$ because irreducibility is an open condition.
\hfill\@mt{\Box}
\begin{rem}\label{z4}
The reason we propose this proof, apart from its simplicity,
is to outline a way in which the algebraic independence
condition on $q,t$ can be circumvented. This condition inavoidably
enters if one likes to return from formal algebra to complex-valued
$q,t$, as in the approaches of \cite{Marin} and \cite{Zinno},
where in latter paper, irreducibility follows from the
identification of $\rho_n$ to a summand of the BWM algebra.
Algebraic independence is `generically' satisfied, but for many
concrete values of $q,t$ it may be false, or at least difficult
to establish. (It is needed for the faithfulness, but
latter is not essential in our arguments, until \S\ref{S7}.)
It is clear from our proofs that the condition `$q$ close
to $1$' can, in both lemma \reference{xy} and \reference{t1_},
be made precise by a slightly more technical calculation.
The clarification for which parameters \em{exactly} $\rho_n$ is
irreducible requires far more effort, and is only subject of
ongoing work. The sole
written reference I received, only \em{a posteriori}, from
I.~Marin, is a very recent Ph D thesis of C.~Levaillant
\cite{Levaillant}. (Allegedly Bigelow has an own, unpublished,
proof.)
\end{rem}
\section{Lie groups}
\subsection{Correspondence between compact
and complex Lie groups\label{oo}}
We start by reviewing a few basic facts from Lie group theory,
which mostly occurred in our treatment of Burau \cite{reiko}.
Let $G$ be a connected compact Lie group with Lie algebra ${\frak g}$.
Compactness implies in particular that $G$ is real,
finite-dimensional and linear reductive.
Linear reductive means for a closed subgroup $G\subset GL(n,{\Bbb C})$
that the number of connected components is finite and $G$ is
closed under conjugated matrix transposition $M\mapsto \ol{M^T}$.
A \em{linear representation} of $G$ is understood as a pair $\rho=
(V,\pi)$ made of a vector space $V$ and a homomorphism $\pi\,:\,
G\,\to\,{\operator@font Aut}\,(V)$. We will often omit $\pi$ and identify $\rho$
with $V$ for simplicity, if unambiguous. A representation is
\em{irreducible} (and will be often called \em{irrep} below)
if it has no non-trivial (i.e.\ proper and non-zero)
invariant subspaces. Linear reductiveness of $G$ implies that
each invariant subspace of a linear representation of $G$ has a
complementary invariant subspace, so that each representation of
$G$ is \em{completely reducible} as direct sum of irreducible
representations.
To $G$ there exists a uniquely determined complex connected linear
reductive Lie group $G_{\Bbb C}$, with \\[1.5mm]
\mylabel ${\frak g}_{\Bbb C}={\frak g}\otimes {\Bbb C}$ is the Lie algebra of $G_{\Bbb C}$, and\\
\mylabel $G\subset G_{\Bbb C}$ as a closed subgroup. \\[1.5mm]
\setcounter{type}{0}%
Then $G_{\Bbb C}$ is called a \em{complexification} of $G$.
If $G$ is simply connected, so is $G_{\Bbb C}$, and then any other
connected complex Lie group with Lie algebra ${\frak g}_{\Bbb C}$ is a covering
of $G_{\Bbb C}$. The real group $G$ is always a maximal compact subgroup
of $G_{\Bbb C}$; we call it the \em{compact real form} of $G_{\Bbb C}$.
Thus we have a one-to-one correspondence between a compact connected
real (simply-connected) Lie group and a (simply-connected) connected
linear reductive complex Lie group.
Under this correspondence to $G=SU(n)$ we have $G_{\Bbb C}=SL(n,{\Bbb C})$.
The groups are connected and simply-connected (for $n\ge 2$).
The correspondence behaves well w.r.t.\ many properties. The
real form $G$ is simple, if and only if $G_{\Bbb C}$ is too. (In
particular, if $G$ is semisimple, so is $G_{\Bbb C}$.) For every
complex representation $\rho=(V,\pi)$ of $G$ (`complex' means
that $V$ is a complex vector space) we have an `extension'
to a representation $\tl \rho=(V,\pi_{\Bbb C})$ of $G_{\Bbb C}$, such that
$\pi_{\Bbb C}$ is an extension of $\pi$ from $G$ to $G_{\Bbb C}$. This
extension is still faithful if $G$ is compact.
\subsection{Symmetric pairs\label{SSP}}
Let $G$ be a Lie group and $\sg$ an involution.
Define
\[
G^\sg\,:=\{\,g\in G\,:\,\sg(g)\,=\,g\,\}\,
\]
to be the $\sg$-invariant subgroup of $G$ and $G^\sg_0$
the connected component of the identity. Then a pair
$(G,H)$ for a closed subgroup $H$ with $G^\sg_0\subset H\subset
G^\sg$ is called a \em{symmetric pair}.
In the case $G=SU(n)$ the symmetric pairs have been classified
by Cartan. See \cite[Chapter IX.4.A, table p.354]{Helgason}.
In this case $H$ is some of $S(U(m)\times U(n-m))$, $Sp(n/2)$
if $n$ is even, or $SO(n)$.
Let us give the corresponding involutions $\sg$ that define the
symmetric pairs (see p.~348 of \cite{Helgason}).
Define $M_{i,j}$ to be the matrix with all entries $0$ except that
at the $(i,j)$-position, which is $1$. Let ${\operator@font diag}\,(x_1,\dots,x_n)=
\sum_{i=1}^nx_iM_{i,i}$ be the diagonal matrix with entries
$x_1,\dots,x_n$, so that $Id_n={\operator@font diag}\,(1,\dots,1)$ (with $n$ entries
`$1$') is the identity matrix.
For $S(U(m)\times U(l))$, with $m+l=n$, the involution $\sg$
is of the form $\sg_{m,l}\,:\,M\mapsto I_{m,l}MI_{m,l}$, where
\[
I_{m,l}\,=\,{\operator@font diag}\,(1,\dots,\inx{1}{m},\inx{-1}{m+1},\dots,-1)\,.
\]
For $n=2n'$ even, $Sp(n')$ respects the involution $\sg_J
\,:\,M\mapsto J^{-1}\bar MJ$, where
\begin{eqn}\label{Jsym}
J\,=\,\left[\begin{array}{c|c}
\rule[-4mm]{0mm}{2.9em}\rx{1mm}0\rx{1mm} & \rx{1mm}-Id_{n'}\rx{1mm}\\
\hline \rule[-4mm]{0mm}{2.9em}\rx{1mm}\phantom{-}Id_{n'}
\rx{1mm} & \rx{1mm}0\rx{1mm}
\end{array}
\right]\,,
\end{eqn}
and $\bar M$ is the complex conjugation (of all entries) of $M$.
For $SO(n)$, the involution $\sg$ is given by $\sg(M)=\bar M$.
These subgroups can be also defined in the standard representation
by the linear transformations that respect a certain (complex)
non-degenerate bilinear form, which is Hermitian, skew-symmetric
or symmetric resp.
All transformations that respect such a form determine, up to
conjugacy, a subgroup of one of the three types.
Analogous three types of subgroups $R(m,n,{\Bbb C})$, $Sp(n/2,{\Bbb C})$ and
$SO(n,{\Bbb C})$ can be defined for $SL(n,{\Bbb C})$. Here $R(m,n,{\Bbb C})$ is
the group of all (complex-)linear unit determinant transformations
of ${\Bbb C}^n$ that leave invariant a subspace of dimension $m$.
(In contrast to the unitary case, there is not necessarily a
complementary invariant subspace!)
We call the three types of groups \em{reducible}, \em{symplectic}
and \em{orthogonal} resp.
We call a representation $V$ of $G$ after one of the types, if
it is contained in a conjugate of a group of the same name.
Orthogonal and symplectic subgroups/representations
will be called also \em{symmetric}, the others \em{asymmetric}.
In the real-complex correspondence we explained, we have that
\em{if the representation $\rho$ of $G$ is symmetric, then so is the
representation $\tl \rho$ of $G_{\Bbb C}$}. This is easily seen by
restricting the respected bilinear form to the reals.
\section{The maximal subgroups\label{mz}}
\begin{conv}\label{conv1}
{}From here, until the conclusion of the proof of theorem
\reference{t1} at the end of \S\reference{S5}, we will
assume, unless we clearly indicate otherwise, that
$q,t$ are fixed unit norm complex numbers that \em{satisfy
the assumptions of theorem \reference{t1}}. In this
situation, we will usually omit explicit reference to
the parameters $q,t$ in the notation.
We may mention here that these conditions are
stronger than we actually need, and were chosen so
as to keep the formulation of theorem \reference{t1} simpler.
We will elaborate on weaker (but slightly more technical)
sufficient conditions in \S\reference{S6}.
\end{conv}
For our approach to theorem \reference{t1}, we will mainly
study the \em{normalization} $\rho_n'\subset SU(p)$. That
is, we consider the (determinant) factorization $U(p)=SU(p)
\times U(1)$ and the projection on the first factor. This
means that we multiply $\rho_n$ by a power of a scalar
$\mu\in{\Bbb C}$ (depending on $q,t$),
\[
\rho'(\bt)\,=\,\mu^{[\bt]}\,\cdot\,\rho_n(\bt)\,,
\]
so that $\rho'(\bt)$ has determinant $1$.
Here $[\bt]$ is the exponent sum of $\bt$, its image under
the homomorphism $B_n\to {\Bbb Z}$ sending all $\sg_i$ to $1$.
This scalar $\mu$ can be calculated as in \eqref{qmu}.
We will prove that
\begin{eqn}\label{d1}
\ol{\rho_n'(B_n)}=SU(p)\,.
\end{eqn}
If then $\ol{\rho_n(B_n)}\ne U(p)$, it must have codimension
one, and so be a collection of components isomorphic to
$SU(p)$. But we have
\begin{eqn}\label{dt}
\dt( \rho_n(\sg_i) )\,=\,-t(-q)^n\,,
\end{eqn}
which is not a root of unity by assumption, so the
projection of $\ol{\rho_n(B_n)}$ onto $U(1)$ is not discrete.
To prove \eqref{d1} we argue indirectly, and assume the contrary.
Since ${\hat H}=\ol{\rho_n'(B_n)}$ is a compact Lie subgroup of $SU(p)$,
we can complexify its connected component ${\hat H}_0$ of the identity,
and obtain a (faithful) representation of a
reductive complex Lie subgroup $\tl H$ of $SL(p,{\Bbb C})$, which is a
proper subgroup by dimension reasons. It is contained in a
maximal proper subgroup which we call $H$. We will
show that $H=SL(p,{\Bbb C})$, and have a contradiction.
\begin{conv}
Let us stipulate that until the end of the proof
of lemma \reference{pp6}, we use $n$ to indicate the dimension
rather than the number of braid strands ($\rho_n$ will not
appear in this scope).
\end{conv}
In the 1950s, Dynkin published a series of seminal papers
in the theory of Lie groups. One of his remarkable achievements
was the classification of maximal subgroups of classical Lie groups
\cite{Dynkin}. (See theorems 1.3, 1.5 and 2.1 in \cite{Dynkin}.)
\begin{theo}\label{prop0}(Dynkin \cite{Dynkin})
A maximal proper subgroup of $SL(n,{\Bbb C})$ is conjugate in $SL(n,{\Bbb C})$ to
\\
\mylabel some symmetric representation, i.e., $SO(n,{\Bbb C})$ or $Sp(n/2,
{\Bbb C})$ when $n$ is even, \em{or} \\ \label{1.}%
\mylabel to $SL(m,{\Bbb C})\otimes SL(m',{\Bbb C})$ with $mm'=n$ and $m,m'\ge 2$
(one which is non-simple irreducible), \em{or} \\
\mylabel to $R(m,n,{\Bbb C})$ (one which is reducible), \em{or} \\
\label{3.}%
\mylabel it is an irreducible representation of a simple Lie group.
\label{4.}%
\setcounter{type}{0}%
\end{theo}
For a non-simple group $H=H_1\times H_2$, one considers ${\Bbb C}^n=
{\Bbb C}^{mm'}\simeq {\Bbb C}^{m}\otimes {\Bbb C}^{m'}$ as a tensor (Kronecker)
product, and $H_1$ resp. $H_2$
acts on ${\Bbb C}^{m}$ resp. ${\Bbb C}^{m'}$.
We call the first 3 types of subgroups \em{orthogonal},
\em{symplectic}, \em{product} and \em{reducible} resp.
We will exclude these types in the next section, before
we get to deal with \reference{4.}.
\section{Excluding symmetric, product and reducible
representations\label{Sym}}
Note that case \reference{1.} in theorem \reference{prop0} is in fact
included in case \reference{4.}. It is singled out because it can
be handled by a more elementary eigenvalue analysis. What theorem
\reference{prop0} actually achieves is also a description of the
maximal subgroups of $SO(n,{\Bbb C})$ and $Sp(n/2,{\Bbb C})$, and one can
decide which of the representations of case \ref{4.} are symmetric.
It will be useful to have an elementary analysis of the eigenvalues
on the various subgroups.
\begin{prop}\label{pp5}
If $\lm\in {\Bbb C}$ is an eigenvalue of an orthogonal or symplectic
matrix $M$ in $SU(n)$, then so is $\bar\lm$.
\end{prop}
\@ifnextchar[{\@proof}{\@proof[\unskip]} Orthogonal matrices are conjugate to real ones, so their
characteristic polynomial has real coefficients. So consider
a symplectic matrix $M$. Then the operator $J$ in \eqref{Jsym}
of the involution $\sg_J$ that $M$ respects (up to conjugacy)
satisfies $J^2=-Id_n$. So $M=J^{-1}\bar MJ$ is equivalent to
$MJ=J\bar M$. Now by assumption there is a vector
$v\in{\Bbb C}^{n}\sm \{0\}$ with $Mv=\lm v$. Then
\[
MJ\bar v=J\bar M\bar v=J\ol{Mv}=J\ol{\lm v}=\bar \lm J\bar v\,,
\]
so $J\bar v\ne 0$ is an eigenvector of $M$ for eigenvalue
$\bar \lm$. \hfill\@mt{\Box}
\begin{lemma}\label{pp5'}
Assume that $H=H_1\otimes H_2\in SL(n_1,{\Bbb C})\otimes SL(n_2,{\Bbb C})$.
Let $\lm_i$ be the eigenvalues of $H_1$ (counting multiplicities)
and $\mu_j$ those of $H_2$. Then the eigenvalues (counting
multiplicities) of $H$ are $\lm_i\mu_j$.
\end{lemma}
\@ifnextchar[{\@proof}{\@proof[\unskip]} $H_{1,2}$ have by Jordan box decomposition lower-triangular
bases $e_i$ and $f_j$. Order the tensor basis $\{e_i\otimes f_j\}$
so that elements with smaller $i+j$ appear first. Then w.r.t. the
so ordered basis, $H=H_1\otimes H_2$ is lower-triangular.
Since in lower-triangular matrices, the eigenvalues appear on
the diagonal, the claim is clear. \hfill\@mt{\Box}
\begin{lemma}\label{pp6}
Assume that for $H=H_1\otimes H_2\in U(n_1)\otimes U(n_2)$ all
eigenvalues $\lm_i$ except exactly one (and single-multi\-pli\-ci\-ty)
eigenvalue $\lm_{i_0}$ satisfy $\lm_i\in\{1,a\}$ for $a\in{\Bbb C}$ with
$a\ne \pm 1$. Furthermore assume that \\
\mylabel $\lm_{i_0}\ne a^2$ and that \\
\mylabel $\lm_{i_0}\ne 1/a$, or $\lm_i=1$ for at least half of all
$\lm_i$.\\
\setcounter{type}{0}%
Then one of $n_1$ or $n_2$ is equal to $1$,
in other words, $H$ is a direct product only in a trivial way.
\end{lemma}
\@ifnextchar[{\@proof}{\@proof[\unskip]} Let $H_1$ have eigenvalues $\mu_j$ and $H_2$ have eigenvalues
$\nu_k$. Since $H_{1,2}$ are diagonalizable, $H$ has eigenvalues
$\lm_i=\mu_j\nu_k$, with all $\mu_j,\nu_k\ne 0\,.$
Let $\lm_{i_0}=\mu_{j_0}\nu_{k_0}$ and assume $n_1,n_2>1$,
where $n_1$ is now the number of $\mu_j$ and $n_2$ that of $\nu_k$.
By choosing $k\ne k_0$ and looking at the set $\{\,\mu_j\nu_k\,\}$
(for the fixed $k$ but varying $j$), we see that $\{\,\mu_j\,\}
\subset \{x,y\}$ and $x/y=a^{\pm 1}$. A similar conclusion applies
to $\{\,\nu_k\,\}$ using $\mu_j$ for some fixed $j\ne j_0$.
Then it is clear that $\lm_{i_0}\in\{a^2,1/a\}$, and we
assumed that the former value is not taken. If $\lm_{i_0}=1/a$,
then (for $a\ne \pm 1$) it follows that exactly one of the
$\mu_j$ is different from all the others, which are equal, and
similarly for $\nu_k$. This implies that the multiplicity
of $\lm_i=a$ is $(n_1-1)(n_2-1)\ge n/2-1$, for the number
$n=n_1n_2$ of eigenvalues $\lm_i$ (with $n_1,n_2>1$). Then,
$\lm_i=1$ occurs $n-(n_1-1)(n_2-1)-1\le n/2$ times, which we
excluded. This gives a contradiction to the assumption
$n_1,n_2>1$. \hfill\@mt{\Box}
Now we apply the preceding lemmas to $\rho_n$ (with
$n$ resuming the meaning of number of braid strands).
\begin{lemma}\label{hhi}
The image of $\rho_n'$ for $n\ge 3$ is not orthogonal or
symplectic, and hence neither is $H$.
\end{lemma}
\@ifnextchar[{\@proof}{\@proof[\unskip]} Assume first that $\rho_n'(\sg_1)\in {\hat H}_0$,
in the notation of \S\reference{mz}.
The eigenvalues of $\rho_n(\sg_1)$ can be easily determined
from the Krammer matrix, and replacing $t$ by $-t$ according to
the warning \reference{w1}. The result is
\begin{eqn}\label{ero}
\Big\{\,-tq^2\,,\ \ \underbrace{-q,\dots,-q}_{\scbox{$n-2$ times}}\,,
\ \underbrace{1,\ \ 1,\ \ \dots,1}_{\shortstack{\scbox{$(n-1)(n-2)/2$}\\
\scbox{times}}}\,\Big\}\,.
\end{eqn}
Let us fix, also for outside this proof, the following notation.
A set of $n$ copies of $k$ will be written $\{k\}^n$, and union
of such sets will be written as a product. Then the above set
can be written as $\{-tq^2\}\{-q\}^{n-2}\{1\}^{(n-1)(n-2)/2}$.
To normalize for the eigenvalues of $\rho_n'(\sg_1)$,
this set has to be multiplied by
\begin{eqn}\label{qmu}
\mu=\det(\rho_n(\sg_1))^{-2/n(n-1)}\,,
\end{eqn}
with \eqref{dt}.
For the chosen $q,t$, none of the resulting numbers is
real (i.e. $\pm 1$). This in particular finishes the cases
$n(n-1)/2$ odd, so $n\ge 4$. But then there is a pre-dominant
occurrence of $\mu$, and the set is not closed under conjugation.
Now let $\rho_n'(\sg_1)\not\in {\hat H}_0$. By the assumption on
$q,t$ in theorem \ref{t1}, no two distinct eigenvalues $\lm_i$
of $\rho_n(\sg_1)$ have a quotient which is a root of unity.
Then one can choose a number $m$ large enough so that all
eigenvalues of $\rho_n(\sg_1^m)$ are as close to $1$ as desired.
(This can be seen for example by looking at the closure of the
infinite cyclic subgroup generated by $(\lm_1,\lm_2,\lm_{3})$ for
the 3 distinct $\lm_i$ within the $3$-dimensional torus $T^3$,
and arguing that this closure, which is formally a Lie subgroup
of $T^3$, cannot have a codimension.) Therefore, by compactness
$\rho'_n(\sg_1^m)\in {\hat H}_0$. Then one argues analogously to
above with $\rho_n'(\sg_1^m)$. \hfill\@mt{\Box}
\begin{lemma}\label{lip}
The image of $\rho_n'$ is not contained in a Kronecker product.
\end{lemma}
\@ifnextchar[{\@proof}{\@proof[\unskip]} Consider first the situation when $\rho_n'(\sg_1)\in {\hat H}_0$.
We want to show that there are no (non-trivial) matrices $H_1, H_2$ with
\begin{eqn}\label{xz}
\rho_n'(\sg_1)(q,t)=H_1(q,t)\otimes H_2(q,t)\,.
\end{eqn}
To rule out \eqref{xz}, we may replace $\rho_n'$ by $\rho_n$,
and consider the eigenvalues of $\rho_n(\sg_1)$.
Using \eqref{ero}, and under the restrictions on $q,t$ of convention
\reference{conv1}, we can apply lemma \reference{pp6}.
It gives the desired conclusion.
Again, for $\rho_n'(\sg_1)\not\in {\hat H}_0$, one argues with
$\sg_1^m$ and replaces $t$ by $t^m$ (to which the
restriction of theorem \reference{t1} applies in the
same way). \hfill\@mt{\Box}
\begin{lemma}\label{lred}
The group $H$ acts irreducibly on ${\Bbb C}^p$.
\end{lemma}
\@ifnextchar[{\@proof}{\@proof[\unskip]} Since, in the notation of \S\reference{mz}, we have
$H\supset \tl H\supset {\hat H}_0$, it is enough to prove that
irreducibility of $\rho'_n$, or simpler of $\rho_n$, is not spoiled
when we restrict ${\hat H}$ to ${\hat H}_0$.
When the Budney form is definite, each $\rho_n$-matrix
diagonalizes. Thus invariant subspaces and irreducibility
will be preserved if we pass to $m$-th powers of any generating set
$\{\tau_i\}$ of $B_n$, provided there are no two distinct eigenvalues
of any $\rho_n(\tau_i)$ which differ (multiplicatively) by a
root of unity.
By the assumption on $q,t$ in theorem \ref{t1}, this condition holds
for $\tau_i=\sg_i$ from \eqref{ero}. Clearly one can choose
$m$ in the last paragraph of the proof of lemma \reference{hhi}
so that the eigenvalues of all $\rho_n(\sg_i^m)$ for
$1\le i\le n-1$ are as close to $1$ as desired,
and then all $\rho_n'(\sg_i^m)\in {\hat H}_0$. \hfill\@mt{\Box}
\section{Rank estimate\label{RE}}
In the quest for what $H$ could be, we are left from the list of
theorem \ref{prop0} only with case \ref{4.}. To deal with this,
in the following it is necessary to appeal to a larger
extent to the Lie theory described, mainly in the appendix,
in \cite{Dynkin}. We will repeat a certain part, though we
would have to refer there for further details.
\begin{conv}
References to
pages, and to equations or statements numbered `$0.\,\cdots$' are
to be understood \em{to Dynkin's paper} (in the translated version).
\end{conv}
The \em{rank} ${\operator@font rk}\, G$ of a simple Lie group $G$ is the maximal dimension
of a torus $G$ contains, or the number of nodes in its Dynkin diagram.
The latter description will be used from the next section on.
Here we have to deal with the torus.
We will recur our rank estimate for $\rho_n$ to the one for $\psi_n$
by means of the important observation in lemma \reference{lsym}.
Let $\bt$ be a \em{fixed} braid in $B_n$. A braid $\bt\in B_k$ for
$k\le n$ can be regarded also as a braid $\bt\in B_{1,k}\subset B_n$.
The following lemma tells us how to determine the eigenvalues of
$\rho_n(\bt)$.
Let for a matrix $M$, by $E={\operator@font Ev}\, M=\{\lm_i\}$ be denoted the
eigenvalues of $M$ (counting multiplicities), and let
${\operator@font Sym}\,^2 E\,=\,\{\,\lm_i\lm_j\,:\,i\le j\,\}$.
\begin{lemma}\label{o1}
\[
{\operator@font Ev}\, \rho_n(\bt)\,=\,\bigl(\,{\operator@font Sym}\,^2 {\operator@font Ev}\,\psi_n(\bt)\,\sm\,
{\operator@font Sym}\,^2 {\operator@font Ev}\,\psi_k(\bt)\,\bigr)\,\cup\,{\operator@font Ev}\, \rho_k(\bt)\,.
\]
For $q,t$ of definite Budney form, the eigenspaces of $\rho_n(\bt)$
of eigenvectors in ${\operator@font Ev}\, \rho_k(\bt)$ correspond to ${\cal E}_k:=\{
\,v_{ij}\,:\,1\le i<j\le k\,\}$.
\end{lemma}
\@ifnextchar[{\@proof}{\@proof[\unskip]} We order the basis ${\cal E}_n$ of ${\Bbb C}^p$ so that ${\cal E}_k$
occur first. It is obvious from the definition \eqref{rf} that
$\rho_n\restr{B_k}$ respects ${\cal E}_k \subset {\cal E}_n$. So the
matrix of $\bt\in B_k\subset B_n$ has the form
\begin{eqn}\label{kn}
\rho_{n}(\bt)\,=\,
\def\kern2mm{\kern2mm}
\left[
\begin{array}{c|c@{\ \ }c@{\ \ }c}
\ry{2.0em}\rho_{k}(\bt) & & 0 & \\[3mm]%
\hline%
& & \ry{1.1em} & \\[-2.8mm]
A & & B & \\[2.3mm]
\end{array}
\right]\ ,
\end{eqn}
where $A,B$ also depend on $\bt$. Thus
\begin{eqn}\label{q1}
{\operator@font Ev}\, \rho_n(\bt)={\operator@font Ev}\, \rho_k(\bt)\cup {\operator@font Ev}\, B(\bt)\,.
\end{eqn}
The next important observation is that by definition
the variable $t$ does not occur in $B$ for $\rho_n(\sg_i)$,
$i=1\,\dots,k-1$. Then the same is also true for
their inverses, and finally thus for $\rho_n(B_k)$.
But since $B$ does not depend on $t$, its eigenvalues can be
determined setting $t=-1$. Then $\rho_k={\operator@font Sym}\,^2\psi_k$
and $\rho_n={\operator@font Sym}\,^2\psi_n$. Thus we have
\begin{eqn}\label{q2}
{\operator@font Ev}\, B(\bt)\cup {\operator@font Sym}\,^2{\operator@font Ev}\,\psi_k(\bt)\,=\,{\operator@font Sym}\,^2{\operator@font Ev}\,\psi_n(\bt)\,.
\end{eqn}
Combining \eqref{q1} and \eqref{q2}, we have the claim. \hfill\@mt{\Box}
Note that \eqref{ero} also follows from this lemma.
Now we apply the lemma on the following elements in $B_n$
that were of central importance also for Burau.
\begin{eqn}\label{q3}
\bt_{n,k}\,=\,\Dl_k^2\,=\,(\sg_1\dots\sg_{k-1})^k\,\in\,B_n\,.
\end{eqn}
\begin{lemma}\label{re}
When $q,t$ are chosen as in theorem \reference{t1}, and the
Budney form is definite, we have
\[
\ol{\rho_n'(\,<\,\bt_{n,2},\dots,\bt_{n,n-1}\,>\,)}\,=\,T_{n-2}\,,
\]
an $n-2$-dimensional torus. Thus in particular ${\operator@font rk}\, H\ge n-2$.
\end{lemma}
\@ifnextchar[{\@proof}{\@proof[\unskip]} We have that $\rho_k(\bt_{n,k})$ are scalars, and
with the notation explained below \eqref{ero}, we have
\[
{\operator@font Ev}\, \psi_n(\bt_k)=\{q^k\}^{k-1}\{1\}^{n-k}\,,
\]
as observed
in \cite{reiko}. If the Budney form is definite on $B_n$,
then so it is on $B_k$, and all matrices are diagonalizable.
The eigenspaces corresponding to the eigenvalues $\{q^k\}^{k-1}$
of $\psi_n$ are spun by $e_1\,\dots\,e_{k-1}$. The claim follows by
a careful look at eigenvalues and eigenspaces. \hfill\@mt{\Box}
\section{The irreps of simple Lie groups\label{S5}}
\subsection{Dynkin diagrams. Weyl's dimension formula}
With \S\reference{RE} we are left to consider
irreps of simple Lie groups. Moreover we know quite
exactly the dimension. It will be more convenient to
look at our rank estimate in lemma \reference{re} from the
point of view of the group, not the number of braid
strands. It says then that for a group of rank $n$, the dimension
of the representation must be $(n'+1)(n'+2)/2$ for some $n'\le n$.
Moreover, this irrep should not have an invariant form
by \S\reference{Sym}. We try next to find out how to obtain
all these irreps.
By the work of Cartan, irreps $\phi$ of a simple Lie group $G$ are
determined by their \em{highest weight} $\Lm$, and latter
is completely described by the property that
\begin{eqn}\label{a_i}
a_i\,=\frac{2(\Lm,\ap_i)}{(\ap_i,\ap_i)}
\end{eqn}
are non-negative integers for all simple roots $\ap_i$ of
the Lie algebra ${\frak g}$ of $G$ (and at least one $a_i$ is positive).
The scalar product $(\,.\,,\,.\,)$
is the one defining the Dynkin diagram: with the normalization
that (in the cases we require below) all simple roots have
length $1$, nodes of the Dynkin diagram depicting orthogonal
vectors are not connected, and connected nodes correspond
to vectors of scalar product $-\myfrac{1}{2}$.
Since the $\ap_i$ correspond to nodes in the Dynkin diagram,
our convention, as in \cite[p.329 top]{Dynkin}, will be to write
$a_i$ at the node for $\ap_i$ in the diagram (but omit zero
entries). We will refer to $a_i$ also as \em{labels} of the nodes.
If $a_i=0$ for all $i$ except exactly one, where $a_i=1$, we
call $\phi$ a \em{basic} representation; it is obviously
associated to the simple root (or node) $\ap_i$ with $a_i=1$.
The following formula calculates the dimension $N(\phi)$
of the irrep $\phi$ corresponding to $\Lm$.
\begin{lemma}(Weyl's formula, Theorem 0.24)
\begin{eqn}\label{We}
N(\phi)\,=\,\prod_{\ap\in\Sg_+}\,\frac{(\Lm+g,\ap)}{(g,\ap)}\,,
\end{eqn}
where $\Sg_+$ is the set of positive roots of ${\frak g}$
and
\begin{eqn}\label{We'}
g=\frac{1}{2}\,\sum_{\bt\in \Sg_+}\,\bt\,.
\end{eqn}
\end{lemma}
Mostly we will appeal to the following consequence.
(We have used the fact in a weaker form already for Burau.)
\begin{lemma}\label{ij}
Assume that one increases the label $a$ of a node in the Dynkin
diagram to $a+1$ ($a\ge 0$). Then the dimension of the
irrep grows a least by a factor of $(a+2)/(a+1)$, in particular
it increases strictly.
\end{lemma}
\@ifnextchar[{\@proof}{\@proof[\unskip]} If $\ap$ is a simple root, then it is known that
\[
\frac{2(g,\ap)}{(\ap,\ap)}\,=\,1\,,
\]
(see (0.141)),
so that $(g,\ap)>0$. Then this is also true for all $\ap\in \Sg_+$,
since they are just sums of (distinct) simple roots. For the same
reason $(\Lm,\ap)$ are just sums of $a_i$, and so non-negative.
Thus increasing some $a_i$ will not decrease any of the (positive)
factors in the product of \eqref{We}. The estimate of increase of
$N(\phi)$ follows from looking at the factor that corresponds to
$\ap_i$.
\hfill\@mt{\Box}
\begin{defi}
Let us say that an irrep $\phi'$ \em{dominates} another
irrep $\phi$, if the labels $a_i'$ of $\phi'$ and
$a_i$ of $\phi$ satisfy $a_i'\ge a_i$ for all $i$.
\end{defi}
A further important tool is the decision which irreps are
asymmetric. This goes back to work of Malcev, and
can be done as explained in Theorem 0.20 p.336
and Remark C.a. on p.254. With the exclusion of symmetry in
\S\reference{Sym}, we are left only with representations of $A_n$,
$D_{2k+1}$ and $E_6$, whose labelings do \em{not} admit a certain
symmetry as shown on Figure \reference{FSym} (which reproduces
Table 3, p.365 in \cite{Dynkin}).
\begin{figure}
\[
\begin{array}{c@{\quad\qquad}c@{\qquad}c}
\diag{8mm}{7}{1}{
\pictranslate{0 0.2}{
\picline{0 0.3}{3 0.3}
\picline{4 0.3}{7 0.3}
\picmultigraphics{3}{0.2 0}{\picfilledcircle{3.3 0.3}{0.02}{}}
\picmultigraphics{4}{1 0}{\picfilledcircle{0 0.3}{0.07}{}}
\picmultigraphics{4}{1 0}{\picfilledcircle{4 0.3}{0.07}{}}
\picputtext{0 0.8}{$a_1$}
\picputtext{7 0.8}{$a_1$}
\picputtext{1 0.8}{$a_2$}
\picputtext{6 0.8}{$a_2$}
\picputtext{2 0.8}{$a_3$}
\picputtext{5 0.8}{$a_3$}
\picputtext{3 0.8}{$a_4$}
\picputtext{4 0.8}{$a_4$}
}
}
&
\diag{8mm}{3.5}{2.5}{
\pictranslate{0 0.55}{
\picline{0 0.7}{2 0.7}
\picline{3.5 0.7 3 q 2 : +}{3 0.7}
\picline{3.5 0.7 3 q 2 : -}{3 0.7}
\picmultigraphics{3}{0.2 0}{\picfilledcircle{2.3 0.7}{0.02}{}}
\picmultigraphics{4}{1 0}{\picfilledcircle{0 0.7}{0.07}{}}
\picfilledcircle{3.5 0.7 3 q 2 : -}{0.07}{}
\picfilledcircle{3.5 0.7 3 q 2 : +}{0.07}{}
\picputtext{3.4 1.14 3 q 2 : +}{$a_1$}
\picputtext{3 0.7 3 q 2 : -}{$a_1$}
\picputtext{2.8 1.2}{$a_2$}
\picputtext{2 1.2}{$a_{n-3}$}
\picputtext{1 1.2}{$a_{n-2}$}
\picputtext{0 1.2}{$a_{n-1}$}
}
}
&
\diag{8mm}{4}{2}{
\pictranslate{0 -0.2}{
\picline{0 1.7}{4 1.7}
\picline{2 1.7}{2 0.7}
\picmultigraphics{5}{1 0}{\picfilledcircle{0 1.7}{0.07}{}}
\picfilledcircle{2 0.7}{0.07}{}
\picputtext{0 2}{$a_1$}
\picputtext{4 2}{$a_1$}
\picputtext{1 2}{$a_2$}
\picputtext{3 2}{$a_2$}
\picputtext{2 2}{$a_3$}
\picputtext{2 0.23}{$a_4$}
}
} \\[1.1cm]
A_n & D_{2k+1} & E_6
\end{array}
\]
\caption{The marking of symmetric irreps (i.e. which admit
an invariant symmetric or antisymmetric form).\label{FSym}}
\end{figure}
\subsection{$A_{n}$}
\def\Alabel#1#2#3{
\diag{8mm}{#1}{1}{
\picline{0 0.3}{#2 0.3}
\picline{#2 1 + 0.3}{#2 1.5 + 0.3}
\picmultigraphics{3}{0.2 0}{\picfilledcircle{#2 .3 + 0.3}{0.02}{}}
\picmultigraphics{#2}{1 0}{\picfilledcircle{0 0.3}{0.07}{}}
\picfilledcircle{#2 1.5 + 0.3}{0.07}{}
#3
}
}
\def\Alabelone#1#2#3#4{\Alabel{#1}{#2}{
\picputtext{#3 0.7}{$#4$}}
}
\def\Alabeltwo#1#2#3#4#5#6{\Alabel{#1}{#2}{
\picputtext{#3 0.7}{$#4$}
\picputtext{#5 0.7}{$#6$}}
}
\def\Afivelabelone#1#2{
\diag{8mm}{4}{1}{
\picline{0 0.3}{4 0.3}
\picmultigraphics{5}{1 0}{\picfilledcircle{0 0.3}{0.07}{}}
\picputtext{#1 0.7}{$#2$}
}
}
\def\Afivelabeltwo#1#2#3#4{
\diag{8mm}{4}{1}{
\picline{0 0.3}{4 0.3}
\picmultigraphics{5}{1 0}{\picfilledcircle{0 0.3}{0.07}{}}
\picputtext{#1 0.7}{$#2$}
\picputtext{#3 0.7}{$#4$}
}
}
Let $\pi_1$ be the elementary representation
\[
\Alabelone{4.5}{3}{0}{1}
\]
of $A_n$ as $SL(n+1,{\Bbb C})$. Our assumption was that
$H\ne \pi_1$, so we will discuss the other possibilities.
First we look at the dimension of the basic representations in Table
30, p.378. These representations were written as $\pi_k$ in (0.92).
For symmetry reasons it makes sense to consider only $n\ge 2k-1$.
If $k\ge 3$, then the only case of
\[
N(\pi_k)\,=\,{n+1\choose k}\,\le\,\frac{(n+1)(n+2)}{2}
\]
is that of $k=3, n=5$. This representation
\[
\Afivelabelone{2}{1}
\]
is symmetric, and it has improper dimension 20. So we must consider
(to avoid symmetry) representations dominating
\begin{eqn}\label{111}
\Afivelabeltwo{2}{1}{0}{1}
\qquad\mbox{and}\qquad
\Afivelabeltwo{2}{1}{1}{1}
\end{eqn}
But latter two dimensions are by lemma \reference{ij}
at least $2\cdot 20=40>21$. (The exact dimensions are
$105$ and $210$, resp.)
The representations
\[
\Alabelone{4.5}{3}{1}{2}
\qquad\mbox{and}\qquad
\Alabeltwo{4.5}{3}{1}{1}{0}{1}
\]
have dimensions
\[
\frac{(n+2)(n+1)^2n}{12}\qquad\mbox{and}
\qquad\frac{(n+2)(n+1)n}{3}\,.
\]
They are too large for $n\ge 3$. The dimension of
\[
\Alabelone{4.5}{3}{0}{2}
\]
is precisely $\frac{(n+1)(n+2)}{2}$, so that the first node
label cannot be increased beyond 2.
There remain the irreps
\[
\Alabelone{4.5}{3}{0}{2}
\qquad\mbox{and}\qquad
\Alabelone{4.5}{3}{1}{1
\]
These are ${\operator@font Sym}\,^2\pi_1$ and $\bigwedge^2\pi_1$.
This can be seen as follows: the inclusions are present
by p.342 l.-8 in the first case, and p.347 l.11
in the second case. Equality follows by calculating dimensions
of the irreps using (0.148).
So it remains to rule out the symmetric and antisymmetric
square of the elementary representation $\pi_1$.
For symmetry reasons ${\operator@font Sym}\,^2\pi_1$ must be considered for
$n\ge 2$, and $\bigwedge^2\pi_1$ for $n\ge 4$ only.
By comparing dimensions, it is clear that
the number $m$ of strands of $\rho_m$ is $m=n+2\ge 4$
in the first, and $m=n+1\ge 5$ in the second case.
We will have to
count again eigenvalues, and use the following suggestive
\begin{lemma}\label{yu}
If $M$ has eigenvalues $\lm_i$ (counting multiplicities)
then the eigenvalues of ${\operator@font Sym}\,^2 M$ are $\{\,\lm_i\lm_j\,:\,i
\le j\,\}$ and those of $\bigwedge ^2 M$ are $\{\,\lm_i\lm_j
\,:\,i< j\,\}$.
\end{lemma}
\@ifnextchar[{\@proof}{\@proof[\unskip]} Similar to lemma \reference{pp5'} (only this time with
bases $e_i\odot e_j$ resp. $e_i\wedge e_j$). \hfill\@mt{\Box}
We can thus finish the $A_n$ series with the following
\begin{lemma}
$\rho_n'(\sg_1^m)$ is not a symmetric square for $n\ge 4$
and not an antisymmetric square for $n\ge 5$.
\end{lemma}
\@ifnextchar[{\@proof}{\@proof[\unskip]} It is clear that one can argue with $\rho_n$ instead.
Consider again $q=-1$. Then all eigenvalues but exactly one are $1$.
Let $\lm_i$ be the ($n-1$ resp. $n$) eigenvalues of a potential
matrix, whose (anti)symmetric square is $\rho_n(\sg_1)$.
For symmetric square $n-1$ of the $p=n(n-1)/2$ eigenvalues
must be $\lm_i^2$, and the rest $\lm_i\lm_j$ for $i<j$.
So $\lm_i^2=1$ for all but at most one $i$.
If at least three $\lm_i=\pm 1$ occur, different signs
are easily ruled out. The option $n=4$, $\lm_1=1$,
$\lm_2=-1$ and $\lm_3\ne \pm 1$ is also easy to exclude.
Clearly not all $\lm_i=\pm 1$, so all but
exactly one are (with the same sign). But then again
we must have all $\lm_i\lm_j=1$ for $i\ne j$, which is impossible.
For antisymmetric square the eigenvalues $\lm_i\lm_j=1$ for
$i\ne j$ are $1$ except one. Let us permute indices $j$ of
$\lm_j$ so that the exceptional one $-t$ occurs as $\lm_1\lm_2$.
Then all $\lm_3\lm_j$ are equal, so either all $\lm_j$
are equal, or equal except one ($\lm_3$). Neither option is
possible. \hfill\@mt{\Box}
\subsection{$D_{2k+1}$}
Let $n=2k+1\ge 5$ be the rank.
{}From Figure \reference{FSym} it is evident that
in order the irrep to lack symmetry, we need to label
non-trivially some of the extreme nodes labeled in figure
\reference{FSym} as $a_1$.
They correspond to basic representations called \em{spinor
representations} (see p.351). According to Table 30 page 378, their
dimension is $2^{n-1}$. We call the nodes $a_1$ thus below
\em{spinor nodes}.
Now $2^{n-1}\le (n+1)(n+2)/2$ only if $n=5$.
In this case the only nodes with associated basic
representations of dimension at most $(n+1)(n+2)/2=21$
are the spinor node, and the other extreme node. So we are
left with considering representations dominating
\begin{eqn}\label{D5}
\def\Dfivelabel#1#2#3{
\diag{8mm}{2.5}{2}{
\pictranslate{0 0.3}{
\picline{0 0.7}{2 0.7}
\picline{2.5 0.7 3 q 2 : +}{2 0.7}
\picline{2.5 0.7 3 q 2 : -}{2 0.7}
\picmultigraphics{3}{1 0}{\picfilledcircle{0 0.7}{0.07}{}}
\picfilledcircle{2.5 0.7 3 q 2 : -}{0.07}{}
\picfilledcircle{2.5 0.7 3 q 2 : +}{0.07}{}
\picputtext{2.15 0.7 3 q 2 : +}{$#1$}
\picputtext{#2 1.2}{$#3$}
}
}
}
\Dfivelabel{1}{0}{1}
\qquad\mbox{and}\qquad
\Dfivelabel{2}{0}{\ }
\end{eqn}
But by the estimate in lemma \reference{ij} their dimensions
are at least $3\cdot 16/2=24$. The exact computation using the
formula (0.150) gives $N(\phi)=144$ and $126$ resp.
\subsection{$E_6$}
In case of $H$ being an $E_6$ representation, the possible
dimensions are $(n+1)(n+2)/2$ for $n\le 6$. A look at the
dimensions of the basic representations of $E_6$ in Table 30,
p.378 shows that only the one marked 27 still fits this bound,
and since dimensions do not match, we must try the irrep
\begin{eqn}\label{E62}
\diag{8mm}{4}{2}{
\picline{0 1.5}{4 1.5}
\picline{2 1.5}{2 0.5}
\picmultigraphics{5}{1 0}{\picfilledcircle{0 1.5}{0.07}{}}
\picfilledcircle{2 0.5}{0.07}{}
\picputtext{0 1.0}{$2$}
}\es.
\end{eqn}
It (and dominating representations) can be excluded from the
estimate in lemma \reference{ij}.
Originally we were aware, though, just of the strict increase
property in the lemma.
So we tried to compute the dimension of the irreps of $E_6$ using the
formula (0.153). But this revealed that the preceding calculation of
the numbers involved in the formula lacks several explanations and
has many errors. So here we provide a correction.
We consider an irrep of $E_6$ highest weight $\Lm$ with $a_i$
in \eqref{a_i} corresponding to nodes of the Dynkin diagram thus:
\[
\diag{8mm}{4}{1.8}{
\pictranslate{0 -0.4}{
\picline{0 1.7}{4 1.7}
\picline{2 1.7}{2 0.7}
\picmultigraphics{5}{1 0}{\picfilledcircle{0 1.7}{0.07}{}}
\picfilledcircle{2 0.7}{0.07}{}
\picputtext{0 2}{$a_1$}
\picputtext{4 2}{$a_5$}
\picputtext{1 2}{$a_2$}
\picputtext{3 2}{$a_4$}
\picputtext{2 2}{$a_3$}
\picputtext{1.7 0.53}{$a_6$}
}
}\ .
\]
First, on p.354, l.-11, $\lm_5$ should be $\lm_6$. On p.355, l.8,
one should refer to (0.95) and (0.96) instead of (0.108).
On (0.138') of page 355, way may clarify that if we normalize
the scalar product so that $||\ap_i||=1$, then $K=\myfrac{1}{12}$.
Then, on p.378, Table 29, the left node $\ap_6$ should be labeled
as $\ap_6=\lm_4+\lm_5+\lm_6+\lm$. The positive roots
$\Sg_+(E_6)$ can be then obtained from the list of roots
in (0.133) by choosing therein $q>p$ (in the first shape) and
the $+$ in either $\pm$ signs (for the second and third shapes).
The formulas on p.358, l.8-10, are almost entirely wrong. The
quantities $g_k$ and $g$ on the right are not properly explained,
but it is suggestive that the decomposition, similar to (0.145'),
\[
g=\sum_{i=1}^6 g_i\lm_i+g_0\lm
\]
for the element $g$ of \eqref{We'} or (0.140) is meant, where we
replaced the
$g$ of p.358, l.8-10 by $g_0$ to avoid confusion. (So $g$ is a vector
for us, given in (0.140), and $g_0$ is a scalar.) We have to assume
similarly to (0.146') that
\[
\sum_{i=1}^6 g_i=0\,,
\]
in order to have formula (0.153) working properly, and we should
(with our convention) replace $\myfrac{1}{2}\,g$ by $\myfrac{1}{2}
\,g_0$ therein. The formulas on p.358, l.8-10 should read then
\[
\begin{array}{l@{\qquad}l}
\ds l_k\,=\,l_6+\sum_{i=k}^5a_i, & \ds g_k=\frac{7-2k}{2}
\mbox{\quad for $k\le 5$;}
\\[5mm]
\ds l_6\,=\,-\frac{1}{6}\,\sum_{i=1}^5\,i\,a_i\,, &
\ds g_6=-\frac{5}{2}\,;
\\[5mm]\pagebreak
l=a_1+2a_2+3a_3+2a_4+a_5+2a_6\,, & g_0=11\,.
\end{array}
\]
To make the presentation more self-contained, we rewrite
the dimension formula (0.153) here. With
\[
m_k=l_k+g_k\,,\mbox{\quad and\quad}\,m=l+g_0\,,
\]
we have
\[
N(\phi)\,=\,\frac{m}{g}\,\prod_{1\le p<q\le 6}\frac{m_p-m_q}{g_p-g_q}
\,\prod_{1\le p<q<r\le 6}\frac{m_p+m_q+m_r+\myfrac{1}{2}\,\!m}
{g_p+g_q+g_r+\myfrac{1}{2}\,\!g_0}\,.
\]
Regardless of the aforementioned errors, the data for the dimensions
of the basic representations of $E_6$ in Table 30, p.378, are correct.
We then also found the exact dimension of the irrep \eqref{E62}
to be 351. (This, and some of the preceding, exact computations
of dimension will become helpful below.)
With this the proof of theorem \reference{t1} is complete.
\section{Generalizations\label{S6}}
After we found our proof of theorem \reference{t1}, we became
aware of the paper by I.~Marin \cite{Marin}. It turns out that
our theorem \ref{t1} is more or less equivalent to his theorem
B and its corollary in \S5.1 for groups of type $A$. His result
implies (most of) theorem \reference{t1} as follows. In the
terminology of our proof, one can conclude from Zariski density
(instead of going through Dynkin's list) that $\tl H=SL(p,{\Bbb C})$
when $t$ and $q$ are algebraically independent. (So this
is a slightly stronger restriction than ours.) The converse
implication is also quite obvious. Marin's result applies also
to other generalizations of $\rho_n$ (of his types $D$ and $E$),
and uses an entirely different description of the representation.
The proof is quite abstract and consists in looking at the Lie
algebra of our $H$.
In contrast, our proof is more direct and gives some new insight.
For example, since the eigenvalue argument can be carried out on
$\rho_n(\sg_1^2)$ instead of $\rho_n(\sg_1)$, theorem \ref{t1}
holds by replacing $B_n$ by any subgroup which contains the elements
$\bt_{n,k}$ ($k=2,\dots,n-1$) in \eqref{q3}, and on which $\rho_n$ is
irreducible. Here is a further variation, in which we made also
some effort to extract what conditions on $t$ and $q$ are really
needed in our arguments.
\begin{theorem}\label{t11}
Fix an integer $m\ne 0$. Assume $q,t$ with $|t|=|q|=1$ are chosen
so that $t$, $q$, $tq$, $tq^2$ and $tq^n$ are not roots of unity.
Assume the Budney form is definite at $q,t$, and $G\subset B_n$
is a subgroup as specified below, such that $\rho_n$ is irreducible
on $G$ at $q,t$.
\begin{enumerate}
\item If $G$ contains $<\,\sg_{2k-1}^{2m}\,>$ ($k\le n/2$),
and provided $tq^3$ is not a root of unity when $n=4$,
then $\ol{\rho_n(G)}\simeq U(p)$ (for $p=n(n-1)/2$).
\item If for fixed $a\ge 2$ and $l$, the group $G$ contains
$<\,\sg_{ak+l}^{2m}\,>$ (for all $k$ with $1\le ak+l\le n-1$), then
for $n$ large (in a way dependent on $a$, but independent on $q,t,l$
or $G$) we have $\ol{\rho_n(G)}\simeq U(p)$.
\end{enumerate}
\end{theorem}
\@ifnextchar[{\@proof}{\@proof[\unskip]} The condition on $tq^n$ is needed to reduce the problem
from $U(p)$ to $SU(p)$. The eigenvalue argument remains the same: as
long as $t$ is not a root of unity, we can get disposed of symmetry
and Kronecker product by looking at $\rho_n(\sg_1^{2mm'})\in{\hat H}_0$
for proper $m'$. The condition on $q$, $tq$ and $tq^2$ enters in
order to keep lemma \reference{lred} working. With these
restrictions, the condition on $tq^3$ is what remains
from the second listed assumption in lemma \reference{pp6},
which is needed to adapt the argument for lemma \reference{lip}.
(When $n>4$, then \eqref{ero} shows that the eigenvalue $1$
occurs too often.) The need to exclude these quantities from
being roots of unity (rather than just $\pm 1$) comes again from
the possibility that ${\hat H}$ is not connected (i.e. ${\hat H}\ne{\hat H}_0$).
For the second claim a torus (within $H$) of dimension a
positive multiple of the number of braid strands is found
from looking at the action of $\rho_n(\sg_{ak+p}^{2m})$ on
subspaces of $v_{ak+p-1,ak+p}$ (where the condition on $tq^2$
is needed). Such a torus keeps an irrep analysis still
manageable. $E_6$ is relevant only for finitely many
$n$, and $D_{2k+1}$ needs (in order to prevent symmetry)
a spinor node marked, with an exponential increase in
dimension. Finally for $A_n$ one remarks that any other
labeling than the ones we studied would give a dimension
of the irrep, which is a polynomial in $n$ of degree $>2$.
Thus only finitely many $n$ would be relevant.
In the case $a=2$ of the first claim, we have for a rank-$n$-%
group an irrep of dimension $n'(n'-1)/2$ for $n'\le 2n+1$;
in particular the dimension is at most $(2n+1)n$.
A similar but slightly more involved discussion in cases,
as for the proof of theorem \reference{t1}, shows that in fact
under this weaker condition, still no irreps occur.
To conclude this it is helpful to use the dimension formulas
given in the proof of theorem \reference{t1} (rather than just
the rough estimates). We give just a few details.
For $E_6$ the only new irrep fitting the dimension bound is
\[
\diag{8mm}{4}{2}{
\picline{0 1.7}{4 1.7}
\picline{2 1.7}{2 0.7}
\picmultigraphics{5}{1 0}{\picfilledcircle{0 1.7}{0.07}{}}
\picfilledcircle{2 0.7}{0.07}{}
\picputtext{1.66 0.7}{$1$}
}\es,
\]
but it is symmetric.
For $D_{2k+1}$, after applying suitably lemma \reference{ij},
the only new possibility is the irrep of $A_7$ obtained from
the diagram on the right of \eqref{D5} by adding two nodes on the left.
But from the latter dimension calculated below \eqref{D5},
we conclude that the dimension is $>105$.
For $A_n$, the discussion is slightly lengthier. First, by using
the dimension formulas for the basic representations in Table 30,
and lemma \reference{ij}, one sees that one only needs to look
at representations where only the leftmost 3 nodes may obtain
a non-trivial label. These are discussed case-by-case.
Most of the options were studied already in the proof of theorem
\reference{t1}. We give a little information on the remaining ones,
by noticing that for the irreps
\[
\Alabeltwo{4.5}{3}{2}{1}{1}{1}
\qquad\mbox{and}\qquad
\Alabeltwo{4.5}{3}{2}{1}{0}{1
\]
(which occur in the extended treatment of \eqref{111}),
the dimensions are $(n+1)\cdot{n+2\choose 4}$ and
$3{n+2\choose 4}$, resp. (for $n\ge 5$). The irreps
\[
\Alabelone{4.5}{3}{0}{3}\es=\es {\operator@font Sym}\,^3\pi_1
\qquad\mbox{and}\qquad\es
\Alabelone{4.5}{3}{2}{1}\es=\es\bigwedge\nolimits^3\pi_1\,,
\]
are again (for small $n$, where the dimension estimate fails)
most conveniently ruled out by an eigenvalue argument.
\hfill\@mt{\Box}
The following consequence was motivated by a similar result in
\cite{Marin}. Our advantage is that our restrictions of $q,t$ are
weaker, and more explicit. (We do not appeal to the result
of Crisp-Paris either.)
\begin{corr}\label{t12}
Let $n\ge 3$.
Assume $q,t$ with $|t|=|q|=1$ are chosen so that $t$, $q$, $tq$,
$tq^2$, $tq^3$ (latter only for $n=4$), and $tq^n$ are not roots
of unity, and the Budney form is
definite and $\rho_n$ is irreducible at $q,t$. Let $m\ne 0$ be
any integer. Then
we have
\[
\ol{\rho_n(<\,\sg_{k}^{2m}\,:\,1\le k\le n-1\,>)}\simeq U(p)\,.
\]
\end{corr}
\@ifnextchar[{\@proof}{\@proof[\unskip]} The argument in the second paragraph of the
proof of lemma \reference{lred} with $\tau_i=\sg_i$
explains why the irreducibility of $\rho_n$ implies
the one of its restriction to the specified subgroup.
\hfill\@mt{\Box}
With such an argument one can treat also the \em{Hilden subgroup}
\cite{Hilden} $H_{2n}\subset B_{2n}$; from the presentation in
\cite[\S 5]{Tawn} one exhibits it to contain the elements
$\sg_{2i-1}$ ($1\le i\le n$).
For irreducibility one needs a few extra arguments, which we provide.
\begin{prop}
Let $n>2$ be even. For $q,t$ as in lemma \reference{t1_},
$\rho_n$ is irreducible on $H_{n}$.
\end{prop}
\@ifnextchar[{\@proof}{\@proof[\unskip]} We repeat the proof of lemma \reference{t1_}, until before
the conclusion $V=R$. Now, for $q=1$, we have $R=V_1\oplus V_2$,
where $V_1$ is (linearly) generated by $v_{2i-1,2i}$, $i=1,\dots,
n/2$, and $V_2$ by all the other $v_{i,j}$. It is easily observed
that $V_k$ for $k=1,2$ are irreducible over $H_{n}$.
If now $\rho_n(t,q)$ are reducible for $q$ converging to
$1$, then by orthogonal approximation (see \cite{reiko}) the
irrep decomposition of $\rho_n(t,q)$ is of the form
$R=V_1(t,q)\oplus V_2(t,q)$, where $V_k(t,q)\to V_k(t,1)=
V_k$ for $q\to 1$ in the sense that there are bases that
converge vector-wise; in particular $\dim V_k(t,q)=\dim V_k$.
Now again $-tq^2$ is a unique eigenvalue of
$\rho_n(t,q)(\sg_{2i-1})$ with eigenspace spun by
$v_{2i-1,2i}$. Since the matrices of $\rho_n(\sg_{2i-1})$
are conjugate, we see that some $V_k(t,q)$ must contain all
$v_{2i-1,2i}$, $i=1,\dots,n/2$, and so $V_1$. By convergence
we can have only $V_1\subset V_1(t,q)$, and by dimension
reasons $V_1=V_1(t,q)$. But it is direct to verify that for
$q\ne 1$, $V_1$ is not an invariant subspace of $\rho_n(t,q)$.
\hfill\@mt{\Box}
It should be remarked that \em{not} necessarily the same
$q,t$ as in lemma \reference{t1_} would do, and that the
above indirect argument spoils our control on how $q$
must be to $1$, the way we had it in remark \ref{z4}.
Still it seems not worthwhile to enter into technical
calculations in order to have this shortcoming removed,
and lemma \reference{t1_} remains at least qualitatively
true.
R.~Budney has observed irreducibility of $\rho_n\raisebox{-1mm}
{$\big|_{H_{n}}$}$ previously, at least for small $n$, but it is
the lack of written record that motivated us to supply the preceding
proposition.
\begin{corr}
Let $n\ge 4$ be even.
Assume $q,t$ with $|t|=|q|=1$ are chosen so that $t$, $q$, $tq$,
$tq^2$, $tq^3$ (if $n=4$), and $tq^n$ are not roots of unity, $t$
is close to $-1$, and $q$ is close to $1$ depending on $t$. Then
we have $\ol{\rho_n (H_{n})}\simeq U(p)$\,. \hfill\@mt{\Box}
\end{corr}
There seems no principal obstacle to apply our approach
to more general Artin groups, if more explicit (matrix)
descriptions of the representations are available.
\section{Non-conjugate braids\label{S7}}
The final section is devoted to the proof of Theorem \ref{t2}.
Theorem \ref{t2} was obtained first by Shinjo for knots $L$. However,
her method cannot be pleasantly applied to links, and this was our
motivation for a different approach in \cite{reiko}. We extended
Shinjo's result, showing theorem \reference{t2} when an $n-1$-braid
representation of $L$ has a non-scalar Burau matrix. One could hope
to further remove braids in the Burau kernel (which exists at least
for $n\ge 5$ \cite{Bigelow,LP}), replacing $\psi_n$ by the faithful
representation $\rho_n$. This was the origin for our interest in
$\rho_n$ in this paper. In contrast, the faithfulness of $\rho_n$
was not essential for theorem \ref{t1}. (Our approach there was
set out to apply also for many values of $q,t$ which are not
algebraically independent, and thus for which $\rho_n$ may not
be faithful.)
We should now choose some parameters $q,t$ for
which $\rho_n$ is faithful. They will have to be close to
$(1,-1)$ in the way that will get clear below, but apart from
that they should be kept fixed.
Throughout this section, $\bt\in B_{n-1}$ is a fixed non-central
braid representation of the link $L$. It will turn out very
helpful to take advantage of our work in \cite{reiko} and assume,
by having dealt with the other cases, that $\psi_{n-1}(\bt)$
is a scalar matrix. We write as
\[
C\,:=\,\{\,\ap\bt\ap^{-1}\,:\,\ap\in B_{n-1}\,\}
\]
the conjugacy class of $\bt$ in $B_{n-1}$. An element in
$C$ will typically be written as $\bt'$. Such a $\bt'$ will
be regarded also as element of $B_n$ using the inclusion
$B_{n-1}\simeq B_{1,n-1}\subset B_n$.
It is known that the center of $B_n$ is generated by the
full twist braid $\Dl^2=(\sg_1\dots \sg_{n-1})^n$.
\begin{lemma}\label{l23}
Assume for $\gm\in B_n$, that $\rho_n(\gm)$ is scalar. Then
$\gm$ is a power of the full twist braid.
\end{lemma}
\@ifnextchar[{\@proof}{\@proof[\unskip]} Scalar matrices are central, and by the faithfulness
of $\rho_n$, so must be then $\gm$. \hfill\@mt{\Box}
A \em{linear function} $f$ defined on the set $M(p,{\Bbb C})$ of
complex $p\times p$ matrices $M=(m_{ij})$ is an expression of the form
\begin{eqn}\label{lz}
f(M)\,=\,\sum_{i=1}^p\,\sum_{j=1}^p\,a_{ij}m_{ij}\,,
\end{eqn}
for fixed $a_{ij}\in{\Bbb C}$. We call $f$ a \em{trace multiple}
if $a_{ij}=0$ and $a_{ii}=a_{jj}$ for $1\le i<j\le p$.
It is well-known that central matrices in $SU(p)$ are scalar and
that the trace is a conjugacy invariant. We showed in \cite{reiko}
that, apart from these trivial cases, there are no linear functions
of matrices invariant on a conjugacy class.
\begin{prop}\label{prop2}(\cite{reiko})
Assume that $f\,:\,M(p,{\Bbb C})\to {\Bbb C}$ is a linear
function, which is not a trace multiple. Let $X$ be a non-central
element in $SU(p)$. Then $f$ is not constant on the conjugacy
class of $X$ in $SU(p)$ (considered as a subset of $M(p,{\Bbb C})$).
\end{prop}
\begin{lemma}\label{Q2}
Assume that $\psi_{n-1}(\bt)$ is a scalar matrix. Then
${\operator@font tr}\, \rho_{n}(\bt'\sg_{n-1})$ for $\bt'\in C$ can be expressed
as a linear function of $\rho_{n-1}(\bt')$ for $-t,q$ close
to $1$. Moreover, this linear function is not a trace multiple.
\end{lemma}
\@ifnextchar[{\@proof}{\@proof[\unskip]} We assume that $q,-t$ are chosen close to $1$,
so that $\rho_{n}$ is unitary.
We note from \eqref{rf} that $\rho_{n}\restr{B_{n-1}}$ preserves
the subset ${\cal V}_{n-1}=\{\,v_{ij}\,:\,1\le i<j\le n-1\,\}$.
By unitarity, the vectors $v_{i,n}$ for $1\le i<n$ can be
modified to $\tl v_{i,n}$, so that $\rho_{n}\restr{B_{n-1}}$
acts invariantly on the linear span ${\cal V}'$ of $\tl v_{i,n}$.
We denote by $\tl\rho_n$ the restriction of $\rho_{n}$ (regarded
as a representation of $B_{n-1}$) to this space ${\cal V}'$.
Since we are interested in evaluating the trace, we have the
freedom to change basis. In the basis of ${\cal V}={\cal V}_{n-1}\cup
{\cal V}'$ then we have the form \eqref{kn} (for $k=n-1$) with $B=0$.
Next we look at the matrix $A$ in \eqref{kn}. As in lemma \ref{o1}
and its proof, we noticed that the eigenvalues of $\tl\rho_{n}(\bt)$
do not depend on $t$ (although $\tl\rho_{n}(\bt)$ itself would).
So let $t=-1$. In this case we use lemma \ref{lsym} and the standard
fact (see e.g. Note 5.7 and above Example 3.2 in \cite{Jones})
that $\psi_n\restr{B_{n-1}}$ is the sum of $\psi_{n-1}$ and a
trivial representation $\tau_{n-1}$, to conclude that
\begin{eqn}\label{zq}
\tl\rho_n=\psi_{n-1}\oplus \tau_{n-1}\,.
\end{eqn}
By multiplying the matrix of $\tl v_{i,n}$ by a proper unitary
matrix independent on $t$, we can assume that a basis of ${\cal V}'$
is chosen w.l.o.g. so that the direct sum in \eqref{zq} is visible
in the block $A$ of \eqref{kn} for $t=-1$.
Now we assumed that $\psi_{n-1}(\bt)$ is scalar, so all its
eigenvalues are the same. But lemma \reference{o1} argued
that they do not depend on $t$, so they will be the same
also when $t\ne -1$. Since still $\tl\rho_{n}$ is unitary
for the chosen $q,t$, we see that $\tl\rho_{n}(\bt)$ is a
diagonal matrix independent on $t$ for such $q,t$, and it is
the same matrix $A=\psi_{n-1}(\bt')$ for all $\bt'\in C$.
This means that, in the basis ${\cal V}$, the only entries of
$\rho_n(\bt')$ that vary with $\bt'\in C$ are those in
the block $\rho_{n-1}(\bt')$ in \eqref{kn}. By writing
$\rho_{n}(\sg_{n-1})$ in the same basis ${\cal V}$, we can
then express ${\operator@font tr}\, \rho_{n}(\bt'\sg_{n-1})$ as a linear
combination of entries of $\rho_{n-1}(\bt')$, with coefficients
$a_{ij}$ in \eqref{lz} depending continuously on $q,t$.
(They will involve the entries of $\rho_{n}(\sg_{n-1})$ and
the scalar in $A$, which is up to sign a certain power of $q$.)
To show that this linear combination is not a trace multiple
on $\rho_{n-1}(\bt')$ when $\bt'$ ranges over $C$, it suffices,
by continuity, to look at $q=-t=1$. Then the action of
$\sg_i$ is this of permuting the subscripts $i$ and $i+1$ of
the (basis) elements $v_{i,j}$ in the formula \eqref{rf}. Clearly
$\sg_{n-1}$, exchanging subscripts $n-1$ and $n$, does not
fix (or take to multiples of themselves) all such elements
with $1\le i< j\le n-1$. \hfill\@mt{\Box}
Theorem \ref{t2} follows by combining the previous three
statements, theorem \ref{t1}, and the result in \cite{reiko}.
\@ifnextchar[{\@proof}{\@proof[\unskip]}[of theorem \reference{t2}]
Let $\bt\in B_{n-1}$ be a braid representation of $L$ as a
non-central braid. Then, by using lemma \ref{l23} (and remark
\ref{z1}), we have that $\rho_{n-1}(\bt)$ is not scalar for proper
$q,t$ of definite form.
If $\psi_{n-1}(\bt)$ is not scalar, then
the claim follows from the work in \cite{reiko}. So assume that
$\psi_{n-1}(\bt)$ is scalar. Then, regarding $B_{n-1}\simeq
B_{1,n-1} \subset B_n$, the map $B_{n-1}\to {\Bbb C}$ given by
\begin{eqn}\label{ok}
\bt'\mapsto {\operator@font tr}\,\rho_n(\bt'\sg_{n-1})
\end{eqn}
is for these $q,t$ linear but not a trace multiple by lemma \ref{Q2}.
By theorem \ref{t1}, the closure of the $\rho_{n-1}$-image of the
conjugacy class $C$ of $\bt$ is a $SU(p')$-conjugacy class $D$ with
$p'=(n-1)(n-2)/2$. Since \eqref{ok} is a linear function on $C$,
it can be extended to such a function on $D$, and in a unique way.
This extension is not constant by proposition \ref{prop2}, and $D$
is a connected set. Then this set $D$ cannot contain a dense subset
$C$ on which a continuous map \eqref{ok} takes a finite (or even
discrete) value range.
\hfill\@mt{\Box}
Thus we prove in fact a bit more; e.g. for proper $q,t$ the set of
$|{\operator@font tr}\,\rho_n|$ or $\arg{\operator@font tr}\, \rho_n$ on $n$-braid representations
of $L$ has a closure that contains an interval.
\begin{rem}\label{z3}
{}From the perspective of Markov's theorem, it seems more important
to construct \em{irreducible} braids, i.e. such which are \em{not}
conjugate to $\gm\sg_{n-1}^{\pm 1}$ for $\gm\in B_{n-1}$. The examples
in the proof of theorem \reference{t2} can be easily modified by
exchange moves (see \cite{BM}) to ones which at least \em{may}
be potentially irreducible. (Lemma \reference{Q2} needs a
slight adaptation.) But this promises no real advance, as long
as one cannot \em{prove} irreducibility. No decent general
technique exists to establish this property for non-minimal strand
braid representations, except the arguments in \cite{Morton},
which apply in very special cases.
\end{rem}
\noindent{\bf Acknowledgment.}
I would wish to thank to R. Shinjo, S.~Bigelow, R.~Budney, W.~T.~Song,
and I.~Marin
for some helpful comments, references and discussions.
{\small
\let\old@bibitem\bibitem
\def\bibitem[#1]{\old@bibitem}
|
1,941,325,220,531 | arxiv | \section{Introduction}
The calculation of gravitational waveforms from the inspiral and merger
of binary black holes is a great current challenge to computational
relativity with important relevance to LIGO. A three dimensional
computer code is now being constructed by the Binary Black Hole Grand
Challenge Alliance to solve this problem by Cauchy evolution of initial
data for two black holes \cite{science}. The most difficult problems
with the development of this code are inaccuracy and instability at the
boundaries of the Cauchy grid. Matching to an exterior characteristic
evolution is one method being pursued to handle the outer Cauchy
boundary and to extract the waveform at null infinity. This has been
shown to be a highly accurate and efficient approach in the treatment
of 3-dimensional nonlinear scalar waves \cite{bishop1996,jcp}. For the
purpose of extending this approach to full general relativity, a
3-dimensional characteristic code for the gravitational field has been
developed and fully calibrated to perform with second order accuracy
and robust stability in highly nonlinear regimes \cite{cce,highp}. A
module for matching Cauchy and characteristic gravitational evolution
codes across a worldtube interface has also been written \cite{manual}
and is now in the testing stage. In this paper, we present results
which show that Cauchy-characteristic matching can also solve the very
difficult inner boundary condition necessary for Cauchy evolution of
black holes. Notably, we have used a characteristic code to achieve the
first successful treatment of a distorted black hole moving on a
3-dimensional grid with apparently unlimited long term stability.
The conventional strategy for avoiding the topological and strong field
difficulties in the Cauchy evolution of black holes has been to excise
the region interior to an apparent horizon, as initially suggested by
W. Unruh \cite{thornburg1987}. In a recent work \cite{excise}, we
proposed an alternative version of this strategy in which the black
hole region is evolved by a characteristic evolution based upon ingoing
null cones. These null cones are truncated at an inner boundary
consisting of a trapped or marginally trapped surface, and matched at
their outer boundary to the inner boundary of a Cauchy evolution. In
turn, the outer boundary of the Cauchy evolution is matched to an
exterior characteristic evolution extending to (compactified) infinity.
The potential advantages over a purely Cauchy approach to the inner
boundary were discussed in that work and the global strategy was
successfully implemented for spherically symmetric self-gravitating
scalar waves evolving in a black hole spacetime.
We present here the successful implementation of a characteristic
treatment of an inner black hole boundary for fully 3-dimensional
simulations containing gravitational radiation. We show that the
ingoing characteristic approach is able to locate the black hole and to
track it stably as it moves on the numerical grid. For a report on
recent progress in tackling the same problem by Cauchy evolution see
Ref.~\cite{losalam}. The boundary data for the characteristic initial
value problem is posed on a worldtube and on an ingoing null cone
emanating from the initial slice of the worldtube. A main goal of this
study is to develop new methods which will allow a combination of
characteristic and Cauchy evolution to tackle the computational problem
of the inspiral of two black holes. In this new approach, the two
evolutions are matched across a worldtube, with the Cauchy domain
supplying the boundary values for characteristic evolution and vice
versa. In treating this problem, there are major computational
advantages in posing the Cauchy evolution in a frame which is
co-rotating with the orbiting black holes. Indeed, such a description
may be necessary in order to keep the numerical grid from being
intrinsically twisted and strangled. In this co-orbiting formulation of
the binary black hole problem, the Cauchy evolution requires inner
boundary conditions in two regions approximating the two disjoint
apparent horizons and also an outer boundary condition on a worldtube.
Far enough outside the outer worldtube the coordinate rotation would go
superluminal. Previous work has shown that an outgoing characteristic
code can routinely handle such superluminal gauge flows in the
exterior. Our present results indicate that an ingoing characteristic
code can just as effectively handle the inner boundaries of multiple
black holes.
Consistent worldtube data must satisfy conservation conditions
\cite{cce,highp} which correspond to a hyperbolic version of the
standard constraint equations for Cauchy data. In the present work,
since we are not matching to a Cauchy evolution, we generate this data
from an analytic solution. Specifically, we set Schwarzschild data at
the worldtube. In ingoing Kerr-Schild coordinates ${\hat x}^{\alpha}$,
the Schwarzschild metric takes the form \cite{kersch}
\begin{equation}
ds^2 = -d{\hat t}^2 +d{\hat x}^2+d{\hat y}^2+d{\hat z}^2
+{2m\over {\hat r}} (d{\hat t} +
{{\hat x}d{\hat x}+{\hat y}d{\hat y}+{\hat z}d{\hat z}
\over {\hat r}})^2,
\label{eq:kersch}
\end{equation}
where ${\hat r}^2 ={\hat x}^2 +{\hat y}^2+{\hat z}^2$ and
$k_{\mu}=-\partial_{\mu}(\hat t+\hat r)$ is the ingoing
degenerate null vector.
In order to determine data for a moving black hole, we introduce
an $x^{\alpha}$ coordinate system which is either rotating, boosted
or wobbling with respect to the ${\hat x}^{\alpha}$ coordinates. In the
$x^{\alpha}$ frame, the null coordinates are centered at $r=0$ and the
worldtube is located at $r=R$, where $r^2=x^2+y^2+z^2$. As a result, the
worldtube is stationary with respect to the null grid but the black
hole moves.
The initial value problem also requires data on the initial null
hypersurface. In our formulation, this null data is free of
constraints, other than continuity conditions at the worldtube. Thus we
can introduce an arbitrary pulse of radiation in the data to describe a
distorted Schwarzschild black hole. We can also pose initial null data
by setting to zero the component of the Weyl tensor intrinsic to the
null hypersurface. On a spherically symmetric null hypersurface, the
Weyl data for the Schwarzschild space-time is exactly zero. On a null
hypersurface offset from the center of spherical symmetry, this Weyl
data for a Schwarzschild spacetime is not zero (and it is not possible
to express it in simple analytical form). In this case the choice of
vanishing Weyl data introduces an amount of gravitational radiation
which depends on the size of the offset. The details of setting up the
worldtube-nullcone data are presented in Section \ref{sec:data}.
The worldtube boundary values, as prescribed in this paper, satisfy the
conservation conditions when the spacetime is exactly Schwarzschild,
e.g. a Schwarzschild black hole in either rotating or boosted
coordinates. But in either the distorted or wobbling case, when
gravitational radiation is contained in the initial null data, the
extraction module overdetermines the metric and its normal derivatives
at the worldtube in terms of their Schwarzschild values. As a result,
the reflection of the wave off the worldtube can lead to a violation of
the Bianchi identities. The mismatch between radiation impinging on the
Schwarzschild worldtube introduces an unphysical sheet source of
gravitational radiation on the worldtube which is not necessarily
consistent with energy and angular momentum conservation. Although
this obscures the physical interpretation of the results for those
cases, it is remarkable that the stability of the evolution is not
affected and that the system behaves in accord with the principles of
black hole dynamics, as described in Sec. \ref{sec:results}. In a more
physical implementation, the conservation conditions would be enforced
either (i) directly, by using them to properly evolve the boundary
conditions up the worldtube, or (ii) by matching the interior evolution
across the worldtube to an exterior Cauchy evolution.
Our chief goal here is to demonstrate that the region of spacetime
interior to the world tube can be evolved {\em stably} and {\em
accurately} by means of a characteristic evolution algorithm. The
interior region is truncated in the vicinity of an apparent horizon.
The long term stability we observe indicates a surprising robustness of
the worldtube-nullcone boundary value problem.
In the case of a Schwarzschild spacetime described in a frame rotating
about the center of spherical symmetry, the location of the apparent
horizon is known analytically, as well as the transformation to null
coordinates and the null metric. Thus this case provides an important
test bed to calibrate numerical accuracy. Long term stability and
second order convergence to the analytic values have been confirmed. In
this purely rotating case in which the worldtube is a stationary
boundary, when we superimpose a pulse of radiation on the initial
Schwarzschild null data we find that the surface area of the resulting
distorted black hole grows in time but eventually reaches an
equilibrium value consistent with the Schwarzschild boundary conditions
on the worldtube. In the offset case, the Schwarzschild boundary moves
periodically but the marginally trapped surface associated with the
black hole again reaches equilibrium with it, confirming that the motion
of the boundary is ``pure gauge''.
When the null cones are not spherically symmetric a computational
approach is necessary to find the ``trapping boundary'', where a
marginally trapped surface is located on each ingoing null
hypersurface. The analogous problem in Cauchy evolution is the
excision of a black hole interior by locating the apparent horizon.
There is an extensive literature on marginally trapped surface (MTS)
finders on spacelike hypersurfaces
\cite{nak84,cooky90,tod91,kem91,seidelsuen1992,thorn96,huq,annin,baum96,gundl}.
In Section \ref{sec:find}, we present two methods for use on null
hypersurfaces. Computational design and performance is discussed in
Sec. \ref{sec:comp}.
\section{Determination of the data}
\label{sec:data}
The worldtube data for the null evolution of the interior region is
obtained by recasting the Cartesian form of the Kerr-Schild metric Eq.
(\ref{eq:kersch}) for a Schwarzschild black hole into nonstationary
ingoing null coordinates. However, the Schwarzschild metric can only be
expressed in an analytically manageable null coordinate form when the
null cones are centered to be spherically symmetric. Thus, in the
offset case, numerical techniques must be used to carry out the
transformation from Cartesian to null coordinates in order to provide
the worldtube data. Fortunately, the transformation need only be
implemented in the neighborhood of the world tube, This allows the
solution for the ingoing null geodesics to be carried out by means of a
Taylor expansion in affine parameter, up to the required order of
accuracy. In this way, the transformation can be formulated as a hybrid
analytical-numerical scheme, with the null properties built in
analytically.
Such a scheme for extracting null worldtube data from Cauchy data in a
general spacetime has been implemented as an {\em extraction} module
\cite{manual}, which is part of the computational procedure to match
Cauchy and characteristic evolutions across an interface. We use this
module here to obtain the required worldtube data for null evolution by
extraction from the ``3+1'' form of the Schwarzschild metric which has
been recast into nonstationary coordinates by a nontrivial choice of
lapse and shift.
The initial value problem is completed by giving data on an initial
null hypersurface. In ingoing null coordinates, with $v$ labeling the
null hypersurfaces, with $x^A =(\theta,\phi)$ labeling the angles along
the null rays and with $r$ labeling the surface area measured along the
null rays, the metric takes the Bondi-Sachs form \cite{excise}
\begin{equation}
ds^2=\left(e^{2\beta} {V \over r} +r^2h_{AB}U^AU^B\right)dv^2
+2e^{2\beta}dvdr -2r^2 h_{AB}U^Bdvdx^A +r^2h_{AB}dx^Adx^B,
\label{eq:vmet}
\end{equation}
where $det(h_{AB})= det(q_{AB})= q$, with $q_{AB}$ a unit sphere
metric. The inverses of these 2-dimensional metrics are denoted by
$h^{AB}$ and $q^{AB}$. We express $q_{AB}$ in terms of a complex dyad
$q_A$ (satisfying $q^Aq_A=0$, $q^A\bar q_A=2$, $q^A=q^{AB}q_B$, with
$q^{AB}q_{BC}=\delta^A_C$ and $ q_{AB} =\frac{1}{2}\left(q_A \bar
q_B+\bar q_Aq_B\right)$). $h_{AB}$ can then be represented by its dyad
component $J=h_{AB}q^Aq^B /2$, with the spherically symmetric case
characterized by $J=0$. The full nonlinear $h_{AB}$ is uniquely
determined by $J$, since the determinant condition implies that the
remaining dyad component $K=h_{AB}q^A \bar q^B /2$ satisfies
$1=K^2-J\bar J$. We also introduce the spin-weighted field $U=U^Aq_A$,
as well as the (complex differential) eth operators $\eth$ and $\bar
\eth$. Refer to {}~\cite{cce,eth} for further details.
The complete null data can be specified freely in terms of either the
metric quantity $J$ or the Weyl tensor component
$C_{\alpha\beta\gamma\delta}n^{\alpha} n^{\gamma} m^{\beta} m^{\delta}$
(corresponding to $\Psi_4$ in Newman-Penrose terminology \cite{np}),
where the null vector $n^{\alpha}$ and the complex spacelike vector
$m^{\alpha}$ span the tangent space to the null hypersurface. On a
spherically symmetric null hypersurface, the Weyl data for the
Schwarzschild space-time is exactly zero but since there are no
constraints on the null data, we can freely add a radiation pulse of
any desired shape.
In the case of rotating coordinates, as described in Sec.
\ref{sec:rot}, or boosted coordinates, as described in Sec.
\ref{sec:boost}, the initial null cone is spherically symmetric and
Schwarzschild null data can be determined analytically. However, it is
not possible to express Schwarzschild null data in simple analytical
form on a null hypersurface which is not spherically symmetric.
Instead, we choose initial data for an offset wobbling black hole by
setting the Weyl data to zero on the initial (nonsymmetric) null
hypersurface. The relevant components of the Riemann tensor are
\begin{eqnarray}
R_{rABr} &=& {1\over 2} g_{AB,rr}
-{1\over 2}g^{vr} g_{vr,r}g_{AB,r}
-{1\over 4}g^{CD}g_{AC,r} g_{BD,r}
\nonumber \\
&=& {r^2\over 2} h_{AB,rr}
+r h_{AB,r}
-r\beta_{,r}(rh_{AB,r} +2h_{AB})
-{r^2\over 4}h^{CD}h_{AC,r} h_{BD,r}.
\end{eqnarray}
Here $g^{AB}R_{rABr}=0$ by virtue of the hypersurface equation for
$\beta$ \cite{highp},
\begin{equation}
\beta_{,r} =- \frac{1}{16}rh^{AB}_{,r}h_{AB,r}
= {r\over 8}\big ( J_{,r} \bar J_{,r} -K_{,r}^2 \big ) .
\label{eq:gur}
\end{equation}
The requirement of vanishing Weyl data is equivalent to
$q^{A}q^{B}R_{rABr}=0$, which gives
\begin{equation}
(r^2 J_{,r})_{,r}
-2\beta_{,r}(r^2 J)_{,r}
-\frac{r^2}{2} J \big ( J_{,r} \bar J_{,r}
-K_{,r}^2 \big )
=0 .
\label{eq:zero}
\end{equation}
Combining Eq's. (\ref{eq:gur}) and (\ref{eq:zero}), we then obtain
\begin{equation}
r^2 (r^2 J_{,r})_{,r}
-2\beta_{,r} (r^4 J)_{,r} =0 .
\label{eq:init}
\end{equation}
With $\beta_{,r}$ determined from Eq. (\ref{eq:gur}), Eq. (\ref{eq:init})
is a second order radial differential equation for the initial data
$J$, which may be solved in terms of boundary values for $J$ and
$J_{,r}$ on the worldtube.
The outer worldtube is located at $(x^2+y^2+z^2)^{1/2}=R$, in terms
of coordinates $x^{\alpha} =(t,x,y,z)$ moving with respect to static Kerr-Schild
coordinates ${\hat x}^{\alpha}$. The boundary data on the worldtube is
obtained by first transforming the metric (\ref{eq:kersch}) in ``3+1''
form to the $x^{\alpha}$ frame and then applying the {\em extraction module},
which determines numerically the transformation from a Cartesian to a
null coordinate system in the neighborhood of the worldtube. This
module supplies the boundary values of the null metric quantities $J$,
$\beta$, $U$ and $V$ on the worldtube.
As a check on the extraction module, we examine the rotating
case where the null Schwarzschild metric can be found
analytically. We relate the static coordinate frame $\hat x^{\alpha}$ to the
rotating one,
$x^{\alpha}$ by $t={\hat t}$,
$x={\hat x}\cos \omega (t + r) - {\hat y}\sin \omega (t + r)$,
$y={\hat x}\sin \omega (t + r) + {\hat y}\cos \omega (t + r)$ and
$z={\hat z}$.
In this transformation, the angles $\theta, \phi$
(associated in the standard way with the Cartesian coordinates $x^i$) remain constant along the generators of the null cones emanating from
the world tube. Therefore, the null metric can be easily obtained by
the simple transformation $v = t+r$, $r=\sqrt{(x^2 +y^2 +z^2)}$,
$\theta = \cos^{-1}(z/r)$ and $\phi=\tan^{-1}(y/x)$, leading to
\begin{equation}
ds^2 = \bigg( -1 + {2m\over{r}} + r^2 \omega^2 \sin^2\theta \bigg) dv^2
+ 2 dv dr - 2 r^2 \omega^2 \sin^2\theta du d\phi
+ r^2 (d\theta^2 + \sin^2\theta d\phi^2 ).
\end{equation}
This identifies the spin weighted versions of the variables appearing
in the null metric (\ref{eq:vmet}) as
\begin{eqnarray}
J &=& 0 \: , \: \beta = 0 \\
U &=& i \, \omega \, \sin\theta \, e^{i \phi} \\
V &=& -r + 2m.
\
\label{eq:rot}
\end{eqnarray}
We used this transformation to check the accuracy of the worldtube data
extracted from the Schwarzschild metric in ``3+1'' Cauchy form with
mass $m=.25$. The code was run using the extraction radius $R = 3$.
The Cauchy grid was a Cartesian cube with range $x^i \in [-4, 4]$ and
the range of the characteristic grid was $r \in [0, 4]$. We confirmed
convergence of the numerical error to zero at a second order rate with
grid size for several values of $\omega$ $( 0.8, 0.1, 0.001)$.
\section{Finding the trapping boundary}
\label{sec:find}
The excision of a region inside the black hole is necessary for
numerical evolution, for otherwise contact with the singular
region of spacetime would result. The boundary of the excised region
must either be trapped or marginally trapped in order to insure that it
does not contain points that can causally influence the radiated
waveform. For a slice ${\cal S}$ of an ingoing null hypersurface ${\cal
N}_v$, described in null coordinates by $r=R(v,x^A)$, the divergence of
the outgoing null normals is \cite{excise}
\begin{equation}
{r^2 e^{2\beta}\over 2}\Theta_l = -V
-{1\over \sqrt{q}}[\sqrt{q}(e^{2\beta}h^{AB}R_{,B}-r^2 U^A)]_{,A}
-r(r^{-1}e^{2\beta}h^{AB})_{,r}R_{,A}R_{,B}
+r^2 U^A_{,r}R_{,A}.
\label{eq:diverg}
\end{equation}
This is to be evaluated on ${\cal S}$ after all partial derivatives are
taken; e.g. $r_{,A}=0$. The slice will be marginally trapped if $\Theta_l=0$.
Finding a marginally trapped slice on a converging ingoing null
hypersurface is a 2 dimensional elliptic problem, which entails setting
the right hand side of equation (\ref{eq:diverg}) to $0$ and solving
for $R(v,x^A)$, at a fixed advanced time $v$. It is easier to find
trapped surfaces. In fact, the largest $r=const$ slice of ${\cal N}_v$
that satisfies the algebraic inequality $Q\le 0$, where
\begin{equation}
Q =-V+{r^2\over \sqrt{q}}(\sqrt{q} U^A)_{,A}.
\label{eq:qmarg}
\end{equation}
is either trapped or marginally trapped \cite{excise}. We call this
slice the ``$Q$-boundary''. A comparison of Eq's. (\ref{eq:diverg})
and (\ref{eq:qmarg}) shows that the $Q$-boundary is marginally trapped
when the slice $Q=0$ is an $r=const$ slice. However, the gauge freedom
in the choice of a surface area coordinate ($r$ is a scalar density)
allows any slice to be regauged as an $r=const$ slice. So there is a
gauge in which the $Q$-boundary and the trapping boundary coincide. But
finding this gauge is tantamount to solving the elliptic problem for a
marginally trapped slice.
This presents us with two possible strategies for locating the inner
boundary, both of which ensure that the excised portion of spacetime
cannot causally effect the exterior spacetime: (I) Use the trapping
boundary or (II) use the Q-boundary. Strategy (I) makes the most
efficient use of the spacetime points but a 2D elliptic equation must be
solved. Strategy (II) involves only some simple algebra so it is very
efficient computationally. We will pursue both strategies here and
compare their merits.
In implementing strategy (I), we have tried two methods for finding the
trapping boundary. One is a standard approach to the solution of the
elliptic equation $\Theta_l=0$ by solving the parabolic equation
\begin{equation}
\partial_{\lambda} R = -\Theta_l
\label{eq:parab}
\end{equation}
in terms of a relaxation parameter $\lambda$. At large $\lambda$, the
solution relaxes to the location of the trapping boundary provided that
the procedure is stable.
Another method of finding the trapping boundary is by a minimization
procedure. In the case of a Cauchy hypersurface, this approach was
introduced in \cite{annin}and further developed in \cite{baum96} using
spectral methods. Here we use a finite difference version of the
minimization approach and combine it with an approach based upon
gradient flows proposed in \cite{tod91}. We combine these approaches by
characterizing the trapping boundary on a converging ingoing null cone
${\cal N}_v$ as a marginally trapped slice ${\cal S}_m$ which minimizes
the functional
\begin{equation}
{\cal T}({\cal S}) = {1\over 8}\oint_{{\cal S}} \Theta_n^2 \Theta_l^2 dS
\end{equation}
where $\Theta_n$ is the divergence of the ingoing null vector
$n^{\alpha}$ tangent to ${\cal N}_v$. We normalize $n^a$ and $l^a$
(both normal to ${\cal S}$) by $n^{\alpha}l_{\alpha}=-1$. With this
setup, ${\cal T}$ is invariant under changes in the extensions of the
the null normals (local boosts) that retain $n^a l_a=-1$. For a
standard sphere of radius $R$ in Minkowski space, $ {\cal T} =8\pi/R^2$
so that the minimum on a Minkowski light cone occurs at infinity. This
has the advantage of biasing the search away from the caustic tip of
the light cone when looking for nontrivial minima.
We perform a variation of the form $\delta x^{\alpha} = F n^{\alpha}
\delta \lambda$, which deforms the slice along the generators of ${\cal
N}_v$. In order to find a flow $F$ which leads toward the minimum
consider the variation
\begin{equation}
\delta {\cal T}({\cal S}) = \oint_{{\cal S}}T_{\alpha} \delta x^{\alpha} dS,
\label{eq:calt}
\end{equation}
which serves to define $T_{\alpha}$. We choose $F=-T_{\alpha} n^{\alpha}$. Then
\begin{equation}
\delta {\cal T}({\cal S}) =
-\oint_{{\cal S}}(T_{\alpha} n^{\alpha})^2\delta \lambda dS,
\end{equation}
so that the variation will lead in the direction of the minimum at
${\cal T}=0$.
The main problem then reduces to calculating $T_{\alpha} n^{\alpha}$.
In ingoing null coordinates on ${\cal N}_v$, we describe ${\cal S}$ by
$r=R(x^A)$ and its variation by $r=R(x^A)+\delta R(x^A)$, with $\delta
v=\delta x^A =0$. Choosing the extension
$n_{\alpha}=-g_{vr}v_{,\alpha}$, we have $\Theta_n = -2/r$ so that, for
any slice of ${\cal N}_v$, $\delta \Theta_n = 2\delta r/r^2$.
The variation of terms $f(r,x^A)$ not explicitly dependent on $R$ is
calculated using $\delta f(r,x^A) = f_{,r}\delta R$. Also, since
$dS=r^2 d\Omega$, in terms of the solid angle element in the $x^A$
coordinates, we have $\delta dS=2 dS \delta R/r$. As a result, the
contributions from $\delta \Theta_n$ and from $\delta dS$ cancel in the
variation of Eq. (\ref{eq:calt}) so that
\begin{equation}
\delta {\cal T}({\cal S})=\oint_{{\cal S}}\Theta_l \delta\Theta_l d\Omega,
\end{equation}
From equation ({\ref{eq:diverg}),
\begin{equation}
\delta \Theta_l = \Theta_{l,r}\delta R
+{2 e^{-2\beta} \over r^2}
\{ -{1\over \sqrt{q}}[\sqrt{q}(e^{2\beta}h^{AB}\delta R_{,B}]_{,A}
-2r(r^{-1}e^{2\beta}h^{AB})_{,r}R_{,A}\delta R_{,B}
+r^2 U^A_{,r}\delta R_{,A} \}
\label{eq:deltamess}
\end{equation}
For any vector field $V^A(r,x^B)$ on ${\cal S}$, we have
\begin{equation}
\oint_{\cal S} D_A V^A d\Omega =0,
\end{equation}
where $D_A V^A =(\sqrt{q}V^A)_{,A}/\sqrt{q} +V^A_{,r}R_{,A}$. This
allows us to eliminate terms in $\delta {\cal T}$ containing $\delta
R_{,A}$ by integrating over ${\cal S}$. We obtain
\begin{equation}
\delta {\cal T}({\cal S}) =
\oint_{{\cal S}}\{\Theta_l \Theta_{l,r} +\Psi\}\delta R d\Omega
\end{equation}
where
\begin{eqnarray}
\Psi &=& [{2\over r^2}(\Theta_lh^{AB})_{,r}
-{4\over r^2}(\Theta_l e^{-2\beta})_{,r}
e^{2\beta}h^{AB}]_{,r}R_{,A}R_{,B}
- [2\Theta_l e^{-2\beta}U^A_{,r}]_{,r}R_{,A}
-[2\Theta_l e^{-2\beta}U^A_{,r}]_{,A} \nonumber \\
&-& [{2\over r^2}(\Theta_l e^{-2\beta})_{,A}
e^{2\beta}h^{AB}]_{,r}R_{,B}
+[({2\over r^2}\Theta_l h^{AB})_{,r}R_{,A}
-({4\over r}\Theta_l e^{-2\beta})_{,r}{ e^{2\beta} \over r}
h^{AB}R_{,A}
-{2\over r^2}(\Theta_l e^{-2\beta})_{,A}e^{2\beta}h^{AB}]_{,B}.
\label{eq:Psi}
\end{eqnarray}
Thus, in order to find the trapping boundary, we follow the variational
path determined by $\delta R=-F \delta \lambda$ with
\begin{equation}
F=(\Theta_l \Theta_{l,r} +\Psi)/R^2.
\label{eq:flow}
\end{equation}
As a check on the stability of this minimization scheme, suppose ${\cal
S}_m$ is a trapping boundary located at $r=R_m(x^A)$. Then, on ${\cal
S}_m$, $\Theta_l=0$ and $F=\Psi/R_m^2$. But the operator
$(\partial_A+R_{m,A}\partial_r)$ also annihilates $\Theta_l$ on ${\cal
S}_m$. As a result, direct substitution to eliminate the $\partial_r$
derivatives in equation (\ref{eq:Psi}) gives $\Psi =0$ on ${\cal
S}_m$. Thus $\delta R$ also vanishes and ${\cal S}_m$ is an invariant
slice with respect to the variational scheme.
We have:
(1) A marginally trapped surface is a zero of the positive functional
${\cal T}$;
(2) The effect of the flow $F$ on ${\cal T}$ is everywhere negative or
zero;
(3) A marginally trapped surface is stationary under the flow $F$.
Thus a marginally trapped surface must be stable with respect to the
flow $F$ except in the degenerate case, corresponding to neutral
stability, where a continuum of such surfaces exist in ${\cal N}_v$.
Although this degenerate case is possible it would be improbable to
encounter in an evolution based upon a reasonably behaved foliation.
For an interesting discussion of the wild behavior possible in general
for marginally trapped surfaces see \cite{eard}.
In order to implement either of the above two finders as computational
algorithms, we represent the geometric quantities involved as
spin-weighted fields in stereographic coordinates. The spin-weighted
expressions necessary to determine $\Theta_l$ and $F$ are given in
Appendix \ref{app:a}.
\section{Computational design and performance}
\label{sec:comp}
All numerical algorithms have been based upon explicit finite
difference methods. The metric functions are discretized and placed on
a finite 3 dimensional grid, with $N_r$ radial points and $N_{\xi}^2$
angular points, whose outer boundary is a spherical worldtube. The
spherical coordinates are patched by two overlapping stereographic
grids and angular derivatives of tensors are handled by a computational
version of the $\eth$ formalism \cite{eth}.
In a general null evolution, data at the worldtube would be extracted
from a Cauchy evolution . In this example, since we are not matching to
a Cauchy evolution, we extract data at the worldtube from an
analytical Cauchy solution. The characteristic evolution is carried
out using a code described and calibrated in \cite{highp}, transformed
into an ingoing null code according to the procedure presented in
\cite{excise}.
Inside the grid there is a black hole whose interior is partially
excised at an inner boundary, the ``hole'', which is taken to be
either a marginally trapped surface or the Q-boundary. We need to
evolve only those grid points which are outside a discrete version of
the hole. At the same time, we also need to allow the hole to move
through the grid. In order to accomplish this we use a 3 dimensional
mask function. Each grid point is assigned the value 1 by the mask if
it either neighbors or is exterior to the boundary. All other grid
points are masked to zero. The metric functions are evolved at each
point with mask value 1 using data only from other points with a mask
value 1 (i.e. points outside or neighboring the hole use data only from
other points outside or neighboring the hole). In case they are needed
(see below), values of the metric functions at grid points which are
nearest neighbors just inside the hole are extrapolated radially
inward using points exterior to them with mask 1. Other interior points
are ignored.
After all the metric functions have been evolved, we locate the hole at
the new time. We then recompute the mask function and continue. If the
boundary moves out, we simply throw away the data which we just
evolved. If the boundary moves in, we have data at the new point which
was obtained by extrapolation. Using this approach saves us from having
to figure out if we have any points which were previously in the hole
but are now outside. Such points automatically have data. It should be
noted that we can safely assume that the black hole boundary will never
move more than a grid point during any iteration. If it did, it would
violate the Courant-Friedrichs-Lewy condition, which is built into the
characteristic code to insure stability.
The procedure for locating the trapping boundary is fairly simple.
Basically, we use the previous position of the horizon ($R$) as a guess
for its current location (if this is the first iteration, we use the
position of the Q-boundary for the approximate location of the trapping
boundary). Then, if we are finding the boundary using the parabolic
relaxation technique based upon Eq.(\ref{eq:parab}), we compute
$\Theta_l$ and then let $R = R_{old} - \Theta_l \delta \lambda$, and
repeat until $\| \Theta_l \|_2$ is less than some threshold. If
instead, we are using the minimization procedure, we compute $F$ and
then let $R = R_{old} - F \delta \lambda$, and repeat until $\| F \|_2$
is less than some threshold. The values of the stopping threshold and
$\delta \lambda$ are parameters. The threshold can be safely set to be
approximately $\Delta r$, (where $\Delta r$ is the spacing between
radial grid points) . The value of $\delta \lambda$ should be small
enough so that $F \delta \lambda < \Delta r$ but not so small that
finding the horizon requires a large number of iterations. We have
found it sufficient to choose a fixed $\delta \lambda$ at the start of
the calculation, however it is possible to design a scheme in which
$\delta \lambda$ is self-tuning and changes its value to speed
convergence of each attempt to locate the horizon.
A stability analysis of the explicit scheme (first order in time,
second order in space) used to solve Eq. (\ref{eq:parab}) shows that
$\delta \lambda$ must scale as $\Delta \xi^2$ (where $\Delta \xi$ is
the spacing between angular points). This requirement would suggest
that the proposed method is computationally expensive. However, our
results show that the use of the horizon finder introduces a negligible
overhead when dealing with long evolutions. With the aforementioned
strategy of using the position of the Q-boundary as the initial guess,
the finder may need many iterations to home in on the trapped surface,
but it takes just a few iterations thereafter to track the surface.
It is worth noting that, if instead of using the Q-boundary to
determine the initial guess one uses a more {\em educated} guess for
the location of the surface, the number of steps can be dramatically
decreased. This can be easily done in the case of the boosted black
hole (see Sec. \ref{sec:rot}). Using the expression for $r$ obtained in
that case, one can set as initial guess $R=2m/(\cos \theta \sinh \alpha
+\cosh \alpha)$, reducing the number of iterations on the first
hypersurface from several hundred (using the Q-boundary) to less
than $10$ (for values of $\alpha \leq 0.5$).
In the case of a wobbling black hole (see Sec. \ref{sec:wobbling}), we
do not know the analytic expression for the marginally trapped surface
even at the first hypersurface so that we use the Q-boundary as our
initial guess. Table \ref{wobperf} shows the number of iterations made by the horizon
finder for different values of the offset $b$ and frequency $\omega$
(for a characteristic grid having $45 \times 21^2$ points covering the
space from $r=0$ to $r=4$). After the first ``find'' the number of
iterations necessary to track the hole is small; and since the finder
solves a $N_{\xi}^2$ problem (as opposed to an $N_r \times N_{\xi}^2$
problem), it does not add an appreciable computational time to the
overall numerical evolution.
In comparison, we obtain decidedly inferior efficiency in locating and
tracking the hole by the minimization procedure using the flow given in
Eq. (\ref{eq:flow}). Stability analysis of the finite difference method
shows that $\delta \lambda$ must scale as $\Delta \xi^4$ in this case.
Thus, although this minimization approach is attractive, it is not
practical using finite difference techniques. It may be possible to
improve the minimization algorithm by using pseudo-spectral techniques
\cite{nak84,gundl} to calculate the flow but we have not explored this
possibility in the present work.
\section{Results}
\label{sec:results}
Here we present some results of code runs for various initial
conditions. We describe the physical behavior of the black hole in
terms of the surface area of its marginally trapped surface. This
surface area gives a measure of the energy of the radiation fields
introduced in the initial null data. For the pure Schwarzschild case,
the marginally trapped surfaces have area $A_s=16\pi M_s^2$, in terms
of the mass $M_s$ of the Schwarzschild black hole. More generally, the
surface area of a marginally trapped surface determines its Hawking
mass $M_h$ \cite{hawkm} according to $A=16\pi M_h^2$. Thus, on an
ingoing null hypersurface ${\cal N}_v$, the function $\Delta(v) = M_s
-M_h(v)$ provides a measure of the energy between the marginally
trapped surface and the worldtube.
If a spacetime satisfies some suitable version of cosmic censorship,
such as asymptotic predictability, and settles down to a Kerr black
hole, then the area of any marginally trapped surface must be less than
the area of the final black hole \cite{hawkell}. In the present
context, we do not have a global asymptotically flat solution so these
results are not immediately applicable. However, if the black hole
settles down to equilibrium with the Schwarzschild boundary condition
on the worldtube, then at late advanced times we would expect
$\Delta(v)\rightarrow 0$.
\subsection{Rotating Schwarzschild black holes}
\label{sec:rot}
Our first runs are for a Schwarzschild spacetime described in
coordinates rotating about the center of spherical symmetry, as
described by the null data in Eq. (\ref{eq:rot}). In this case, the
evolved metric is known analytically and the marginally trapped surface
is fixed at the horizon at $r=2m$, so that convergence to the exact
results can be checked. Our results confirm that the numerically evolved
spacetime is accurate to second order in grid size. As expected, the
horizon finder converges to the known location of the spherically
symmetric marginally trapped surface.
\subsection{Boosted Schwarzschild black holes}
\label{sec:boost}
A boosted Schwarzschild black hole provides a test of the ability to
track the motion of a black hole and to calculate its surface area.
Let ${\hat x}^{\alpha}$ be ingoing Eddington-Finkelstein coordinates.
Define boosted coordinates $x^{\alpha}$ by ${\hat x}=x$, ${\hat y}=y$,
${\hat z}=z\cosh\alpha - t\sinh \alpha$ and
${\hat t}=t\cosh\alpha -z\sinh\alpha$.
We locate an initial incoming null hypersurface ${\cal N}_0$ at $t=-(x^2
+y^2 +z^2)^{1/2}$. Initial Schwarzschild null data corresponds to
setting the Weyl data to zero at this initial time. Schwarzschild
data at the extraction worldtube $x^2 +y^2 +z^2 = R^2$ is provided by
transforming the metric to the $x^{\alpha}$ coordinates.
Let $(\hat v, \hat r, \hat \theta, \hat \phi)$ be standard ingoing null
coordinates associated with the Cartesian Eddington-Finkelstein
coordinates, and let $(v,r,\theta,\phi)$ be the null coordinates
associated with the Cartesian coordinates $x^{\alpha}$ by the
extraction module. We synchronize them so that $v=\hat v =0$ on ${\cal
N}_0$, which is the only null hypersurface common to the $v$ and $\hat
v$ foliations. Then the boost transformation implies that $\phi =\hat
\phi$ and, on ${\cal N}_0$, that
\begin{equation}
\cos \hat \theta ={\cos \theta \cosh \alpha +\sinh \alpha
\over \cosh \alpha +\cos \theta \sinh \alpha },
\end{equation}
with inverse
\begin{equation}
\cos \theta ={\cos \hat \theta \cosh \alpha -\sinh \alpha
\over \cosh \alpha -\cos \hat \theta \sinh \alpha },
\end{equation}
Calculation of the Jacobian of the angular transformation gives
$\hat r= r(\cos \theta \sinh \alpha +\cosh \alpha)
=r/(\cosh \alpha-\cos \hat \theta \sinh \alpha )$ on the initial
null hypersurface.
In order to find the initial null data in the boosted null frame we must
also relate $v$ and $\hat v$. Near ${\cal N}_0$, we set
$v=(t+r)+O({\hat v}^2)$. Then by carrying out the transformation to
leading order in $\hat v$ we obtain $v=\hat v/(\cosh\alpha
-\sinh\alpha\cos\hat \theta)$. This is enough to determine that
initially $J=0$ and $\beta =0$.
To the next order, the null condition
$g^{\alpha\beta}v_{,\alpha}v_{,\beta}=0$ implies $v=\hat v/(\cosh\alpha
-\sinh\alpha\cos\hat \theta)+ \kappa {\hat v}^2 +O({\hat v}^3)$ where
\begin{equation}
\kappa_{,\hat r}=-{\sin^2 \hat \theta\sinh^2 \alpha \over
2{\hat r}^2(\cosh \alpha -\sinh \alpha \cos\hat \theta)^3}.
\label{eq:kappar}
\end{equation}
The extraction routine is based upon the gauge condition that
$v=t$ on the worldtube, so that $v_{,t}=1$ at $r=R$. This fixes
the integration constant in Eq. (\ref{eq:kappar}) and gives
\begin{equation}
\kappa={\sin^2 \hat \theta\sinh^2 \alpha \over
2 \hat r(\cosh \alpha -\sinh \alpha \cos\hat \theta)^3}.
\label{eq:kappa}
\end{equation}
Similarly, the condition $g^{ab}v_{,a}\theta_{,b}=0$ that $\theta$ be
constant along the null rays implies
\begin{equation}
\cos \theta ={\cos \hat \theta \cosh \alpha -\sinh \alpha
\over \cosh \alpha -\cos \hat \theta \sinh \alpha }
+\gamma \hat v +O({\hat v}^2),
\label{eq:cos}
\end{equation}
where
\begin{equation}
\gamma_{,\hat r}=-{\sinh \alpha \sin^2 \hat \theta
\over
{\hat r}^2(\cosh \alpha -\sinh \alpha \cos\hat \theta)^3}.
\label{eq:gammar}
\end{equation}
Here the gauge condition built into the extraction routine is
that $\theta_{,t}=0$ on the worldtube. With this boundary condition,
the integral of Eq. (\ref{eq:gammar}) gives
\begin{equation}
\gamma={\sinh \alpha \sin^2 \hat \theta
\over
{\hat r}(\cosh \alpha -\sinh \alpha \cos\hat \theta)^3}.
\label{eq:gamma}
\end{equation}
The $\hat v$ dependence of $r$ can now be obtained from
the defining equation of a surface area coordinate
$r^4 q=\det(g_{AB})$, where $q$ is the determinant of the
unit sphere metric corresponding to the $x^A$ coordinates.
This gives $(r^4 q)_{,\hat v}=-r^4 qg_{AB}{g^{AB}}_{,\hat v}$,
where
\begin{equation}
{g^{AB}}_{,\hat v}=2{x^A}_{,\hat a}{x^B}_{,\hat b \hat v}g^{\hat a \hat b}.
\end{equation}
A straightforward calculation on the initial null cone $v=0$ gives
\begin{equation}
r_{,\hat v} =-{r^3 \over 2 \sin^2 \theta}
(\gamma \gamma_{,\hat r}
+{\hat r}^{-2}(\cos \theta)_{,\hat \theta}\gamma_{,\hat \theta}),
\end{equation}
which, using Eq's. (\ref{eq:cos}) and (\ref{eq:gamma}), reduces to
\begin{equation}
r_{,\hat v} = \sinh \alpha \cos \theta.
\end{equation}
This determines the Jacobian of the transformation between the the
stationary and boosted null frames at $v=0$. Carrying out the
transformation of the metric gives the initial null data for a boosted
Schwarzschild black hole: $J=\beta=U=0$ and
$V=-r+2m(\cosh\alpha+\sinh\alpha\cos\theta)^{-3}$, where $m$ is the
Schwarzschild mass.
After the hole has moved so that it is no longer centered about the
vertex of the null cones, the null metric still has some simple
properties at the poles ($\theta=0$ and $\theta=\pi$) due to the
axisymmetry of the system; e.g. $J=0$ at the poles. This allows an
analytic transformation between null and Kerr-Schild coordinates along
the ingoing polar null geodesics $\pm z=-(t-T) +R$, which lie on the
null foliation and leave the worldtube at $t=T$. Along these polar
rays, $v=T+R$ and the radial null coordinate is given by $r=|z|=-(t-T)
+R$. This allows us to reexpress the location of the poles of the
horizon $\pm \hat z =2m$ analytically in null coordinates as
\begin{equation}
r={2m \pm v\sinh \alpha \over
\cosh \alpha \pm \sinh \alpha}.
\label{eq:poles}
\end{equation}
Since $v=0$ on the initial null cone, the pole of the horizon
hits the vertex of the null cone after retarded time
$v= 2m/\sinh\alpha$.
As a test of the evolution and finder, the surface area of the boosted
event horizon should remain constant. We observed that this is the case
throughout the evolution, modulo the first order error introduced by
the horizon finder. We have confirmed
that the surface area converges to the value determined by the
Schwarzschild mass as the grid spacing is refined. We have also
checked that the poles of the horizon move in accord with
(\ref{eq:poles}), so that the pole which travels inward moves slightly
faster than the pole moving outward.
The algorithm performs the evolution and tracks the motion of the horizon
stably, as long as the CFL condition is satisfied. Figure
\ref{fig:boostmove} shows a 2-D cut (at $y=0$) of the horizon
displaying the position of the hole at three different
times.~\cite{animations}
\subsection{Distorted black holes}
Here we study the approach to equilibrium of a distorted black hole
with Schwarzschild boundary conditions on a stationary worldtube.
First consider the case where the worldtube is static. We introduce a
gravitational wave pulse, with compact support, on the first
hypersurface by
\begin{equation}
J(v=0,r,x^A) = \left\{ \begin{array}{ll}
\displaystyle{\lambda \left(1 - \frac{R_a}{r} \right)^4 \,
\left(1 - \frac{R_b}{r} \right)^4 \;
\sqrt{\frac{4 \pi}{2 l + 1}} \; {}_{2} Y_{l,m}}
& \mbox{if $r \in [R_a,R_b]$} \\
& \\
0 & \mbox{otherwise,}
\end{array}
\right.
\end{equation}
where ${}_{2}Y_{l,m}$ is the spin two spherical harmonic, ${R_a}=1.5$,
${R_b}=3$ and the amplitude factor $\lambda=45$.
As the evolution proceeds, the pulse gets reflected by the outer
boundary and eventually falls into the hole. Our results confirm the
expected behavior of a black hole approaching equilibrium. Figure
\ref{fig:disteq} shows the behavior of $M_h(v)$. The surface area
increases monotonically and approaches the value $16\pi M_s^2$
determined by the Schwarzschild mass of the exterior.
We also introduced a pulse on the initial hypersurface in the case
where the worldtube rotates (thus inducing a shift of its world lines
with respect to the static Schwarzschild streamlines). We observed
that at any given time, this does not result in any change in the
location of the boundary. Hence, a rotating world tube does not affect
the behavior of the Hawking mass confirming that the rotation is a pure
gauge effect.
\subsection{A Wobbling Black Hole}
\label{sec:wobbling}
Beginning with the Schwarzschild metric in Kerr-Schild coordinates
$\hat x^{\alpha}$, we introduce the coordinates of an offset, rotating frame
$x^{\alpha}$ by $t={\hat t}$, $x=({\hat x}+b)\cos \omega t -{\hat y}\sin
\omega t$, $y=({\hat x}+b)\sin \omega t+{\hat y}\cos \omega t$ and
$z={\hat z}$. In this frame, we use the metric and its derivatives on
the world tube $x^2 +y^2 +z^2= R^2$ to provide the boundary values for
a characteristic initial value problem based upon a family of ingoing
null hypersurfaces. Although the Schwarzschild metric is static, the
worldtube wobbles relative to the static Killing trajectories.
The initial value problem is completed by posing null data determined
by setting the Weyl data to zero on the initial null hypersurface. On a
non-spherically symmetric null hypersurface, the Schwarzschild Weyl
data is no longer zero (and it is not possible to express it in simple
analytical form). Thus our choice of vanishing Weyl data introduces an
amount of gravitational radiation which depends on the size of the
offset.
The resulting spacetime is neither spherically symmetric nor static.
Relative to the worldtube, it describes a black hole wobbling and
emitting gravitational radiation. Relative to the static Schwarzschild
symmetry, the worldtube wobbles but the black hole still moves and
radiates. This physical picture is confirmed in Sec. \ref{sec:results} by the
behavior of the surface area of the marginally trapped surface. The
results demonstrate that the region of spacetime interior to the world
tube can be evolved {\em stably} by means of a characteristic evolution
algorithm, when this interior region is truncated in the vicinity of a
trapped region. This is illustrated in Fig. \ref{fig:stabil}, which displays
the maximum values of the norms of $J$ and $U$ over the entire grid vs. time.
After an initial transient stage, they settle into a stationary state without
any sign of instability whatsoever.
Figure \ref{fig:wobbmove} displays a $z=0$ cut of the trapped surface
at different times, showing the ability to track the movement of the
hole by the horizon finder . As the evolution proceeds, the horizon
``wobbles'' through the computational grid with period $T = 2 \pi /
\omega$. We have evolved up to $2000 M$ confirming this
behavior.~\cite{animations}
The accuracy of the numerical evolution in the region exterior to the
horizon is negligibly affected by the choice of using either the
Q-boundary or marginally trapped surface as the inner boundary. This
is illustrated, for the wobbling case, in Fig. \ref{fig:causality},
where we plot the values of $J$ vs time at points outside the inner
boundary, as obtained by both methods. The numerical values have a
negligible difference. However, evolution with the Q-boundary is
somewhat superior with respect to performance since no elliptic solver
or other iteration procedure is required.
The area of the marginally trapped surface again approaches equilibrium
with the Schwarzschild exterior. This is illustrated in
Fig. \ref{fig:wobbarea}, where the surface monotnically increases and
approaches a constant value (which converges to $16 \pi M_s^2$ in first
order). The usefulness of the Hawking mass as a measure of energy is
supported by the observation that $\Delta$ remains positive.
\section{Conclusion}
Our results display many interesting aspects of black hole physics,
although their physical understanding is not completely clear and would
require a deeper study of the surface sources induced on the worldtube.
The most important accomplishment of this work is that characteristic
evolution is now ready to supply both the inner and outer boundary
condition for the Cauchy evolution of black holes as soon as
Cauchy-characteristic-matching is achieved.
\begin{center}
{\bf ACKNOWLEDGMENTS}
\end{center}
\vspace{0.3cm}
This work has been supported by NSF PHY 9510895 to the University of
Pittsburgh and by the Binary Black Hole Grand Challenge Alliance, NSF
PHY/ASC 9318152 (ARPA supplemented). Computer time for this project has
been provided by the Pittsburgh Supercomputing Center under grant
PHY860023P and by the National Center for Supercomputing Applications
under grant PHY970009N to Robert Marsa. We thank R. A. Matzner
and K. P. Tod for helpful comments.
|
1,941,325,220,532 | arxiv | \section{Basics}
Let $\Om\subset\Rn$ be open and non-empty.
\begin{definition}\label{d1} A map $L\in L(\Dop)$ is called a Hadamard operator if it admits all monomials as eigenvectors. The set of Hadamard operators we denote by $\cM(\Om)$.
\end{definition}
Examples of Hadamard operators are the \em Euler operators, \rm these are operators of the form $P(\theta)$ where $P$ is a polynomial and $\theta_j=x_j\frac{\partial}{\partial x_j}$. They are in $\cM(\Om)$ for every open $\Om$ and they are the only ones with this property (see Theorem \ref{t9}).
From \cite{Vdprime} we take the following definitions.
\begin{definition}\label{d3} $T\in \OH$ if for any $k$ there are finitely many functions $t_\beta$ such that $(1+|x|^2)^{k/2}t_\beta\in L_\infty(\Rn)$ and such that $T=\sum_\beta \theta^\beta t_\beta$.
\end{definition}
We compare $\OH$ with the space $\OC$ of L. Schwartz, which may be defined by any of the following equivalent properties (see \cite[\S 5, Th\'eor\`eme IX]{LS}):
\begin{enumerate}
\item For any $k$ there are finitely many functions $t_\beta$ such that $(1+|x|^2)^{k/2}t_\beta\in L_\infty(\Rn)$ and such that $T=\sum_\beta \partial^\beta t_\beta$.
\item For any $\chi\in\Di$, $T*\chi$ is a rapidly decreasing continuous function.
\end{enumerate}
We have $\Oc\subset \Oh$ and $\Oc\neq\Oh$. An example is $T=e^{-ix}$ which is in $\Oh$ but not in $\Oc$ (see \cite[Section 3]{Vdprime}).
\begin{definition}\label{d2} By $\DH$ we denote the set of all $T\in\OH$ the support of which has positive distance to all coordinate hyperplanes.
\end{definition}
From \cite[Section 3]{Vdprime} we cite: If $T\in\OC$ and the support of $T$ has positive distance to all coordinate hyperplanes, then $T\in\DH$.
The main result of \cite{Vdprime} is the following theorem. Here $\sigma(x)=\prod_j \frac{x_j}{|x_j|}$ for $x\in\NZ$.
\begin{theorem} $L\in\cM(\Rn)$ if and only if there is a distribution $T\in\DH$, such that $L(S) = S\star T$ for all $S\in\Dip$. The eigenvalues are $m_\alpha=T_x\Big(\frac{\sigma(x)}{x^{\alpha+\be}}\Big)$.
\end{theorem}
\section{Extension of the operator $M$}
For $L\in\cM(\Om)$ we set $M=L^*\in L(\Do)$ and we study $L$ by means of properties of $M$. Due to reflexivity we have $L=M^*$. For $\vp\in\Do$ we set $\psi=M(\vp)$ and obtain the following characterization:
\begin{lemma}\label{l1} Let $L\in L(\Dop)$, then $L\in\cM(\Om)$ with eigenvalues $m_\alpha$ if and only if the following holds
\begin{equation}\label{eq1}
\int\xi^\alpha (m_\alpha \vp(\xi) - \psi(\xi)) d\xi =0
\end{equation}
for all $\alpha\in\N_0^d,\,\vp \in\Do$.
\end{lemma}
\Proof The condition is equivalent to
$$\int (m_\alpha\xi^\alpha) \vp(\xi) d\xi =\int \xi^\alpha (M\vp)(\xi) d\xi= \int L(\xi^\alpha) \vp(\xi) d\xi$$
for all $\vp\in\Do$. That is $m_\alpha \xi^\alpha = L(\xi^\alpha)$ for all $\alpha\in\N_0$. \qed
Let now the family $m_\alpha$, $\alpha\in \N_0^d$, of real numbers be given.
\begin{definition}\label{d5} We set
$$D(\tM)=\{\vp\in\Di\,:\,\ex \psi\in\Di\text{ such that } (\ref{eq1}) \text{ holds }\fa \alpha\in\N_0^d\}$$
and
$$\Gamma(\tM)=\{(\vp,\psi)\in\Di\times\Di\,:\,\vp,\,\psi\text{ fulfill }(\ref{eq1}) \fa \alpha\in\N_0^d\}.$$
\end{definition}
\begin{lemma}\label{l2} (1) $D(\tM)$ is a linear space, closed under dilations.\\
(2) $\Gamma(\tM)$ is the graph of a linear map $\tM:D(\tM)\to\Di$ which commutes with dilations.
\end{lemma}
\Proof We show only that $\Gamma(\tM)$ is a graph, the rest is obvious. Because of linearity of $\Gamma(\tM)$ it suffices to assume $\vp=0$. Then (\ref{eq1}) says that $\int\xi^\alpha \psi(\xi) d\xi=0$ for all $\alpha\in\N_0^d$, hence $\psi=0$. \qed
For open $\Om$ we set $\tOm=\bigcup_{a\in\NZ}a\Om$. By definition $\tOm$ is invariant under dilations. We obtain:
\begin{lemma}\label{l6} 1. If $\Do\subset D(\tM)$ then $\cD(\tOm)\subset D(\tM)$.\\
2. If $\tM\cD(\Om)\subset \cD(\Om)$ then $\tM\cD(\tOm)\subset \cD(\tOm)$ and $\tM \in L(\cD\,'(\tOm))$.
\end{lemma}
\Proof By Lemma \ref{l2} we have $\cD(a\Om)\subset D(\tM)$ for all $a\in\NZ$. If $\vp\in\cD(\tOm)$ then there are finitely many $a_j\in\NZ$ such that $\vp\in\cD(\bigcup_j a_j\Om)$. Hence there are $\vp_j\in\cD(a_j\Om)$ such that $\vp=\sum_j\vp_j$. Since $D(\tM)$ is a linear space 1. is proved. 2. follows from the fact that $\tM (a\Om)\subset a\Om$ for all $a\in\NZ$ and the above decomposition, since $\tM$ is a linear map. The continuity of $\tM:\cD(\tOm)\to \cD(\tOm)$ follows from de Wilde's Theorem (or Grothendieck's closed graph theorem). \qed
\begin{lemma}\label{l4} $\NZ\subset \tOm$ for all non-empty open $\Om\subset\Rn$.
\end{lemma}
\Proof Let $x_0\in\Om\cap\NZ$. Then $\NZ=\{ax_0\,:\,a\in\NZ\}\subset\tOm$. \qed
We will later study the following two special cases:
\begin{remark}\label{r2} 1. If\, $0\in\Om$ then $\tOm=\Rn$.\\ 2. If $\Om\subset \NZ$ then $\tOm=\NZ$.
\end{remark}
\section{Representation}
In this whole section we assume that $L\in\cM(\Om)$, $M=L^*$ and $m_\alpha$, $\alpha\in\N_0^d$, the family of eigenvalues. In this case $\Do\subset D(\tM)$ and $M\vp=\tM\vp$ for $\vp\in\Do)$.
We will use the following notations:
For $\eps>0$ we set $$W_\eps=\{x\in\Rn\,:\, \min_j |x_j|\ge\eps\}.$$
For $r=(r_1,\dots,r_d)$, where all $r_j\ge 0$, we set $B_r=\{x\in\R^d:|x_j|\le r_j\text{ for all }j\}$ and for $s\ge 0$ we set $B_s:=B_{s\be}=\{x\in\Rn\,:\,|x|_\infty\le s\}$.\\
For $r=(r_1,\dots,r_d)\in\Rn$ and $s\in\R$ we set $r+s=r+s\be=(r_1+s,\dots,r_d+s)$.
Since $\tM$ commutes with dilations we have
$$\tM_\xi(\vp(\eta\xi))[x]=(\tM\vp)(\eta x)$$
for all $\vp\in\cD(\tOm)$ and $\eta\in\NZ$.
For $\vp\in\cD(\tOm)$ we define
$$T\vp = (\tM\vp)(\be).$$
Then $T\in\cD\,'(\tOm)$ and for all $\eta\in\NZ$ and $\vp\in\cD(\tOm)$ we have $\vp(\eta\,\cdot) \in\cD(\tOm)$ and
\begin{equation}\label{eq2} (\tM\vp)(\eta)=T_\xi \vp(\eta\xi).
\end{equation}
We will study the properties of $T$. First we will investigate the consequences of the fact that $\eta\mapsto T_\xi \vp(\eta\xi)\in\Di$ for all $\vp\in\cD(\tOm)$. For that we modify the proof of Lemma 2.1 in \cite{Vdprime}.
\begin{lemma}\label{l10} If $T\in\cD\,'(\tOm)$ and $\eta\mapsto T_\xi \vp(\eta\xi)\in\Di$ for all $\vp\in\cD(\tOm)$ then $\supp T\subset W_\eps$ for some $\eps>0$.
\end{lemma}
\Proof
For $s\in\R_+^d$ we set $$K_s=\prod_j[\frac{s_j}{2},s_j]$$
and we set $K_1=K_{(1,\dots,1)}$.
Since, by Lemma \ref{l6}, the map $\tM$ is continuous there is $r>0$ such that $\tM(\cD(K_1))\subset \cD(B_r)$ and this implies that $T_\xi\vp(\eta\xi)=0$ for any $\vp\in\cD(K_1), \eta\in\NZ$ and $|\eta|_\infty>r$.
We set $\eps=1/r$ and assume that $K_s\subset \{x\,:\, x_j<\eps\}$. Then $|1/s|_\infty>r$. For $\vp\in\cD(K_s)$ we set $\psi(\xi)=\vp(\xi s)$. Then $\psi\in\cD(K_1)$ and
$$T\vp =T_x(\psi(x/s))=0.$$
We have shown that $T|_{\{x\in\R_+^d\,:\,x_j<\eps\}}=0$. Repeating this in an analogous way for all `quadrants' and all relevant half-spaces, we obtain $T|_{\NZ\setminus W_\eps}=0$.
For the support of $T$ as a distribution in $\cD(\tOm)$ we obtain
$$\supp T \subset (W_\eps\cap\tOm)\cup (\Rn\setminus\NZ)\cap\tOm.$$
So $T$ decomposes into two distribution $T_1$ with support in $W_\eps$ and $T_2$ with support in $\tOm\setminus\NZ$. It is well known (see \cite{LSI}) that $T_2$ is locally a finite sum of distributions of the form
$$\vp\mapsto S_{\xi'}\frac{\partial^\nu}{\partial \xi_j^\nu}\vp(\xi_1,\dots,\xi_{j-1},0,\xi_{j+1},\dots,\xi_d)$$
where $\xi'=(\xi_1,\dots,\xi_{j-1},\xi_{j+1},\dots,\xi_d)$ and $S$ a distribution in these variables. This implies that $S_x\vp(xy)$ does not depend on $y_j$. Therefore $T_2$ must be $0$. \qed
We have shown:
\begin{proposition}\label{p2} There is a distribution $T\in\cD\,'(\tOm)$ with support in $W_\eps$ for some $\eps>0$ such that $(\tM\vp)(y)=T_x\vp(xy)$ for all $\vp\in\cD(\tOm)$ and $y\in\NZ$. This holds, in particular, for $\vp\in\cD(\NZ)$ and for $\vp\in\cD(\Om)$. In the latter case this means $(M\vp)(y)=T_x\vp(xy)$.
\end{proposition}
Next we want to get information on the support of $T$. We need a preparatory lemma.
We will use the following notation: For $M,N\subset\Rn$ we set $M^c:=\Rn\setminus M$ and $V_*(M,N)=\{\eta\in\NZ\,:\,\eta M\subset N\}$, $V_*(M)=V_*(M,M)$.
\begin{lemma}\label{r3} Let $M,N\subset\Rn$ then $V_*(M^c,N^c)=\{1/y\,:\,y\in V_*(N,M)\}$.
\end{lemma}
\Proof The statement is symmetric, hence we need only to show one implication. Let $y\in V_*(N,M)$ then $M=y N\,\dot{\cup}\,y N_0$ where $N_0\subset N^c$, hence $\frac{1}{y}\,M=N\,\dot{\cup}\,N_0$, and this implies $\frac{1}{y}\,M^c = (N\,\dot{\cup}\,N_0)^c\subset N^c$. \qed
The following example shows that for the sets $V(M,N)=\{\eta\in\Rn\,:\,\eta M\subset N\}$ there is no such simple relation. For our theory we are only interested in $V_*(M,N)$.
\begin{example}\label{ex1} Let $\Om=\Rn\setminus \{0\}$ then $V(\Om)=\NZ$ while $V(\Om^c)=V(\{0\})=\Rn$.
\end{example}
For $M\subset\NZ$ we set $1/M=\{1/y\,:\, y\in M\}$.
\begin{proposition}\label{l7} If $T\in\cD\,'(\tOm)$ such that $T_x\vp(x \cdot)\in\Di$ for all $\vp\in \cD(\tOm)$, then the following are equivalent:
\begin{enumerate}
\item $T_x\vp(x \cdot)\in\Do$ for all $\vp\in \Do$.
\item For every $\om\ssubset\Om$ there is $L\ssubset\Om$ such that $\supp T\subset V_*(L^c,\om^c)$.
\item $(1/\supp T) K\ssubset\Om$ for every compact $K\subset \Om$.
\end{enumerate}
\end{proposition}
\Proof We first remark that, by Lemma \ref{l10} and the assumption, $\supp T\subset\NZ$.
1.$\Rightarrow$ 2.\; Let $\om$ be an open set such that $\om\ssubset\Om$. Then there is a compact set $L\subset\Om$ such that $T_x\vp(xy)=0$ for all $\vp\in\cD(\om)$ and $y\not\in L$. If $y\in\NZ$ then
$$\{\vp(\cdot\,y)\,:\,\vp\in\cD(\om)\}=\cD\Big(\frac{1}{y}\om\Big).$$
Therefore
$$\supp T\cap\frac{1}{y}\om=\emptyset \text{ for all }y\in\NZ, y\not\in L.$$
We have shown
$$\supp T \subset \bigcap_{y\in \NZ\cap L^c}\frac{1}{y}\, \om^c= V(L^c,\om^c).$$
The last equality comes from \cite[Lemma 1.4]{Vconv}. Since $\supp T\subset\NZ$ we can replace $V(L^c,\om^c)$ with $V_*(L^c,\om^c)$.
2.$\Rightarrow$ 1.\; Let $\om\subset\Om$ be compact and $\vp\in\cD(\om)$. We choose $L$ according to the assumption and assume that $\supp T\subset V_*(L^c,\om^c)$. If $x\in V_*(L^c,\om^c)$ and $y\in L^c$ then $xy\in \om^c$. Since $x\mapsto xy$ is continuous and $\om^c$ open we have $xy\in\om^c$, hence $\vp(xy)=0$, for $x$ in a neighborhood of $\supp T$. This shows that $T_x\vp(xy)=0$ for all $y\in L^c$.
2.$\Leftrightarrow$ 3.\, In 2. we might assume $\om$ and $L$ to be compact. So by Lemma \ref{r3} it is equivalent to:
For every compact $K\subset\Om$ there is a compact $L\subset\Om$ such that $1/\sup M\subset V_*(K,L)$. The equivalence to 3. is now obvious. \qed
From Proposition \ref{l7}, (3.) we get the following:
\begin{corollary}\label{c1} $1/\supp T\subset V_*(\Om)$.
\end{corollary}
In Proposition \ref{l7}, we cannot replace 3. with $1/\supp T\subset V_*(\Om)$ as the following example shows. See, however, Proposition \ref{l9}.
\begin{example}\label{ex4} Let $\Om=]0,1[$ and $T\vp = \int_1^\infty \vp(x) e^{-x} dx$. Then $\vp\mapsto T_x\vp(x\,\cdot)$ maps $\Di$ into $\Di$ (cf. \cite[Proposition 2.5]{Vdprime}). $1/\supp T =]0,1]=V_*(\Om)$. For $\vp\in\Do$, $\vp\ge 0$, $\vp\neq 0$ we obtain
$$T_x\vp(xy) = \frac{1}{y} \int_y \vp(x) e^{-x/y} dx$$
which is $>0$ near $0$, hence $T_x\vp(x\,\cdot)\not\in\Do$.
\end{example}
An important special case is the following.
\begin{theorem}\label{t9} If $\Om\ssubset Q_+$ then all Hadamard operators on $\Dop$ are Euler operators.
\end{theorem}
\Proof Let $L\in\cM(\Om)$, then by Proposition \ref{p2} and Corollary \ref{c1} there is $T\in\Dip$ with $1/\supp T\subset V_*(\Om)$ such that $(M\vp)(y)=T_x\vp(xy)$. Since $V(\Om)=\{\be\}$ (see e.g. \cite[Example 5.3]{Vhad}) we have $\supp T =\{\be\}$ or $T=0$.
Therefore $T=\sum_\alpha b_\alpha \delta_\be^{(\alpha)}$ hence $(M\vp)(y)=\sum_\alpha b_\alpha (-1)^{|\alpha|} y^\alpha \vp^{(\alpha)}(y)$ and therefore $(LS)(\vp)=S(M\vp)=\sum_\alpha b_\alpha (-1)^{|\alpha|} S_y(y^\alpha \vp^{(\alpha)}(y))$ for all $\vp\in\Do$. This means that $L(S)=(\sum_\alpha c_\alpha \theta^\alpha) S$ with suitable constants $c_\alpha$. \qed
\begin{definition}\label{d6} For open $\Om\subset\Rn$ we define: $T\in\DHO$ if $T\in\cD\,'(\tOm)$ and
\begin{enumerate}
\item $T_x\vp(x,\,\cdot)\in \cD(\tOm)$ for all $\vp\in\cD(\tOm)$,
\item $(1/\supp T) K\ssubset\Om$ for every compact $K\subset \Om$.
\end{enumerate}
\end{definition}
We have shown one of our main results:
\begin{theorem}\label{t4} If $L\in\cM(\Om)$ then there is $T\in\DHO$ such that
$L(S)=S\star T$ for all $S\in\Dop$.
\end{theorem}
It remains to give a closer description of the distributions $T$ appearing in Definition \ref{d6}. This can be done it in the two important cases mentioned in Remark \ref{r2}. In the case of $0\in\Om$ it follows from \cite{Vdprime}.
\section{Case of $0\in\Om$}
Let $\Om\subset\Rn$ be open with $0\in\Om$, then $\tOm=\Rn$ (see Remark \ref{r2}). Let $L\in\cM(\Om)$. By Theorem \ref{t4} there is $T\in\DHO$ such that
$L(S)=S\star T$ for all $S\in\Dop$. Condition 1. in Definition \ref{d6} then means that $T\in\DH$, that is, $T\in\OH$ and $\supp T\subset W_\eps$ for some $\eps>0$. We have proved one implication of the following theorem.
\begin{theorem}\label{t5} If $0\in\Om$, then
$L\in\cM(\Om)$ if and only if
there is a distribution $T\in\DH$ with $(1/\supp T) K\ssubset\Om$ for every compact $K\subset \Om$, such that $L(S)=S\star T$ for all $S\in\Dop$.
\end{theorem}
\Proof It remains to show that under the given conditions $M_T:\vp\mapsto T_x\vp(x\,\cdot)$ is the transpose of an operator in $\cM(\Om)$. Clearly $M_T\in L(\Do)$. We have to verify the condition in Lemma \ref{l1} and this follows from the proof of \cite[Theorem 4.2]{Vdprime}. \qed
For the open unit ball and similar sets the we obtain (cf. \cite[Theorem 4.4]{Vhad}).
\begin{example}\label{ex2} Let $\Om\subset\Rn$ be a bounded open set with the following properties:\\
1. If $x\in\Om$ and $|y_j|\le|x_j|$ for all $j$ then also $y\in\Om$.\\
2. $\Om$ is invariant under permutations of the variables.\\
3. $t\,\overline{\Om}\subset \Om$ for all $0<t<1$.\\
Then $\DHO=\{T\in\OH\,:\,\supp T\subset W_1\}$.
\end{example}
\Proof We may choose $\om=\eps\Om$, $L=\delta\Om$. Then
$$V(L^c,\om^c)=V(\delta \Om^c,\eps \Om^c)=\frac{\eps}{\delta} V(\Om^c).$$
By \cite[Theorem 4.4]{Vhad} we have $V(\Om)=Q:=\{x\,:\,|x|_\infty\le1\}$ hence $V_*(\Om^c)=1/V_*(\Om)= W_1$. This implies
$$V_*(L^c,\om^c) =\frac{\eps}{\delta} V_*(\Om^c)= \frac{\eps}{\delta} W_1.$$
So we obtain $T\in\DHO$ if and only if for every $0<\eps<1$ there is $0<\delta<1$ such that $\supp T\subset (\eps/\delta) W_1$ and this is the case if and only if $\supp T\subset W_1$. \qed
\begin{theorem}\label{t6} If $\Om$ is the open unit ball for $\ell_p$, $0<p\le+\infty$, then
$L\in\cM(\Om)$ if and only if
there is a distribution $T\in\OH$, $\supp T\subset W_1$, such that $L(S)=S\star T$ for all $S\in\Dop$.
\end{theorem}
\section{Case of $\Om\subset\NZ$}
Let $\Om\subset\NZ$ be open, then $\tOm=\NZ$ (see Remark \ref{r2}). Let $L\in\cM(\Om)$. By Theorem \ref{t4} there is $T\in\DHO$ such that
$L(S)=S\star T$ for all $S\in\Dop$. In particular, $T\in\cD\,'_H(\NZ)$.
\begin{lemma}\label{l8} $\cD\,'_H(\NZ) = \cE'(\NZ)$
\end{lemma}
\Proof Assume that $T\in\cD\,'_H(\NZ)$ that is $\supp T\subset W_\eps$ for some $\eps>0$ and $(\tM\vp)(y)=T_x\vp(xy)$ for all $\vp\in\cD(\NZ)$ and $y\in\NZ$.
We set $\check{\vp}(x)=\vp(1/x)$ and $\check{T}\vp = T\check{\vp}$ for $\vp\in\cD(\NZ)$. Then $\check{T}\in \cD\,'(\NZ)$. We obtain for $\vp\in\cD(\NZ)$ and $y\in\NZ$
$$\check{T}_x\vp(xy)=T_x\vp\Big(\frac{1}{x}\,y\Big)=T_x\check{\vp}\Big(x\,\frac{1}{y}\Big).$$
$T_x\check{\vp}(x\,\frac{1}{y})\in\cD(\NZ)$ as a function of $y$. By Proposition \ref{p2} there is $\delta>0$ such that $\supp \check{T}\subset W_\delta$.
We set $r=1/\delta$ and $B_r=\{x\,:\,|x|_\infty \le r\}$. Assume that $\supp \vp\in\NZ\setminus B_r$, then $\supp\check{\vp}\in\NZ\setminus W_\delta$. Therefore $T\vp=\check{T}\check{\vp}=0$.
We have shown that $\supp T \subset W_\eps\cap B_r$, hence compact. The other implication is obvious. \qed
We refer to Corollary \ref{c1} and show for $\Om\subset\NZ$:
\begin{proposition}\label{l9} $T\in\DHO$ if and only if $T\in\cE'(\NZ)$ and $1/\supp T\subset V_*(\Om)$.
\end{proposition}
\Proof The `only if' part follows from the Definition \ref{d6} and Corollary \ref{c1}. For the reverse direction we note that, by Lemma \ref{l8}, $T\in\cD\,'_H(\NZ)$.
Since $\supp T\subset \NZ$ is compact, also $1/\supp T$ is compact. Therefore $(1/\supp T) K$ is compact for any compact $K\subset\Om$ and, by assumption, it is contained in $\Om$. Therefore $T\in\DHO$. \qed
So we have shown one implication of the following theorem.
\begin{theorem}\label{t7} If $\Om\subset\NZ$, then
$L\in\cM(\Om)$ if and only if
there is a distribution $T\in\cE'(\NZ)$ with $1/\supp T\subset V_*(\Om)$, such that $L(S)=S\star T$ for all $S\in\Dop$.
\end{theorem}
\Proof It remains to show that under the given conditions $M_T:\vp\mapsto T_x\vp(x\,\cdot)$ is the transpose of an operator in $\cM(\Om)$. Clearly $M_T\in L(\Do)$. We have to verify the condition in Lemma \ref{l1} and this follows from the proof of \cite[Theorem 4.1]{Vdprime}. \qed
\section{Final remarks}
If $d=1$ the the cases in Theorems \ref{t5} and \ref{t7} cover all possibilities, hence we have a complete characterization.
\begin{theorem}\label{t8} If $\Om\subset \R$ is open then
$L\in\cM(\Om)$ if and only if
there is a distribution\\
$T\in\Dh$ with $(1/\supp T) K\ssubset\Om$ for every compact $K\subset \Om$\\ if $0\in\Om$
or\\
$T\in\cE'(\NZ)$ with $\supp T\subset \NZ$ and $1/\supp T\subset V_*(\Om)$\\ if $\Om\subset \R_*$\\
such that
$L(S)=S\star T$ for all $S\in\Dop$.
\end{theorem}
An interesting example in higher dimensions is $\Om=\Rn\setminus\{0\}$:
\begin{example}\label{ex3} In this case we have also
$\cD\,'_H(\Rn\setminus\{0\}) = \cE'(\NZ)$ and the proof is like in the proof of Lemma \ref{l8} except we habe to replace the pointwise reciprocal by the reflection at the euclidian unit sphere $x\mapsto x/|x|^2$.
\end{example}
If, for instance, $L\in\cM(\Om)$, $\Om\subset\R^2$ and $0\not\in\Om$, $e_1,e_2\in\Om$ where $e_j$ are the unit vectors, then $\tOm=\R^2\setminus \{0\}$ and we have for the representing distribution $T$ the condition $T\in\cE'(\NZ)$ plus the support condition.
|
1,941,325,220,533 | arxiv | \section{Introduction}
Atomic fundamental parameters (FPs) are the basis for the description of light-matter interaction. In the X-ray range the atomic FPs include mass attenuation coefficients, photionization cross-sections, fluorescence yields, Coster-Kronig (CK) factors as well as emission line energies, widths and relative transition probabilities. The atomic FPs are element and in general photon energy (sub)shell specific. The determination of the atomic FPs requires suitable and well-defined samples as well as dedicated experiments to be conducted for their accurate determination such as transmission or fluorescence experiments with (preferably calibrated) energy- or wavelength dispersive spectrometers. Nowadays, different technological developments allow for more accurate and reliable experiments for the determination of atomic FPs. These advances are in the field of sample preparation techniques,
X-ray detection instrumentation, where improved sensitivity, signal-to-noise ratio, energy resolution, dynamic range and linear response behavior are offered, and the availability of tunable monochromatic X-ray radiation of high spectral purity. These improved capabilities can be expected to lead to more consistent data between different experiments barring experimental artefacts and inconsistent data treatment. Thus, an independent validation of experimental and theoretical results (or interpolation thereof) reported in databases such as \cite{Bearden,Deslattes,Krause1979,Elam1,xraylib,CAMPBELL2003} is required. With respect to the large uncertainties reported or interpolation of experimental and / or theoretical data, this endeavor can be even be considered as mandatory in order to have reliable and consistent data in future experiments. An improved and accurate understanding of X-ray interaction with matter will be beneficial for fundamental sciences, quantitative XRF analysis and calculations of atomic systems to test limits and needs of single-particle approaches and perturbative approaches.
In this respect a substantial part of the research activities of Prof. Jean-Claude Dousse was directed to an accurate determination of different atomic FPs using reflection-type or transmission-type high energy-resolution crystal-based spectrometers \cite{Hoszowska96,Szlachetko13} and stand-alone laboratory based X-ray sources, synchrotron radiation or charged particles for the excitation of the X-ray emission processes to be studied. Examples of this work include the study of linewidths of X-ray emission processes \cite{Hoszowska1994}, radiative Auger transitions \cite{Herren96}, X-ray resonant Raman scattering \cite{Szlachetko2006}, CK factors \cite{Cao2009,Cao2010}, hypersatellite X-ray transitions \cite{Hoszowska2010,Maillard2018}, two-electron one-photon transitions \cite{Hoszoswka2011}, atomic level widths \cite{Fennane2013} and off-resonant X-ray spectroscopy \cite{Blachucki2014}. In this work, an independent investigation of the CK factors of the Gd L shell using two different detection schemes, a radiometrically calibrated silicon drift detector (SDD) and a full cylinder von Hamos spectrometer, is reported as a further illustration of the valuable contributions of wavelength dispersive spectrometry to an accurate determination of atomic FPs.
\section{Coster-Kronig transition factors}
The CK factors are included in the Sherman equation \cite{Sherman1955} which is the theoretical basis for quantitative X-ray fluorescence (XRF) since it describes the basics of the production and emission of XRF photons from thin one-elemental foils. Indeed, it can be considered as the chronological order of the physical processes occurring: from the number and energy of the monochromatic incident X-ray photons, the incidence angle on the sample surface, the thickness and density of the considered one-elemental foil and the XRF production factor $\sigma$ for a selected electronic subshell it is possible to calculate the number of produced XRF photons produced after an (usually dipole allowed) electronic transition
to the subshell where the vacancy was created to minimize the total energy of the system (atom). When the solid angle of detection $\Omega/4\pi$ and the detection efficiency $\epsilon$ are known the quantitative amount of the isotropically emitted XRF photons which is detected can be calculated. The XRF production factor $\sigma$ is hereby defined as the product of the incident photon energy dependent photoionization cross section of inner shell electron, the fluorescence yield, which provides the probability for a radiative relaxation, the relative transition probability in case only fluorescence photons originating from the electronic transition between selected subshells from different electronic levels is of interest, and the CK factor(s) for all but the innermost subshell of an electronic level. The CK factor takes into account the relative transition probability for a non-radiative transition of the vacancy created from an inner to an outer subshell of an electronic level such that the principal quantum number of the vacancy does not change \cite{COSTER1935}. A CK transition must be energetically allowed in the sense that the energy difference between the subshells between which the vacancy is transferred is large enough to promote an outer electron to the continuum. This intrashell transition, which can also be described as an Auger transition within the same electronic level, leads hence to an increase in the number of vacancies created in the outer subshells before the electronic transition during which an Auger electron or XRF photon will be emitted takes place. Such multiply ionized atoms can be encountered as well in plasma physics and astrophysics. But this consideration is also important if the number of atoms contributing to the XRF signal is used for quantification purposes by using a fundamental parameter based approach \cite{deBoer89} or the reference-free X-ray spectrometry (XRS) method \cite{Beckhoff2008} but present as well a very sensitive probe to a correct modelling of initial- and final-state wavefunctions and electron binding energies \cite{Bambynek72,Chen81}.
For the definition of the CK factors the description of Jitschin et al. is used \cite{W.Jitschin1985}. When considering the L subshells and using an incident photon energy above the L$_1$ ionization threshold, a possible transfer of vacancies from the L$_1$ and L$_2$ subshells to higher L subshells is possible. The fluorescence production factor for each L subshell is then written as follows:
\begin{align}
\sigma_{\text{L}_3}(E_0) &= \omega_{\text{L}_3}\left[ \tau_{\text{L}_3}(E_0) + f_{2,3} \tau_{\text{L}_2}(E_0) + (f_{1,3}+f_{1,2}f_{2,3}) \tau_{\text{L}_1}(E_0)\right] \label{eq:sigma_L_3}\\
\sigma_{\text{L}_2}(E_0) &= \omega_{\text{L}_2}\left[ \tau_{\text{L}_2}(E_0) + f_{1,2} \tau_{\text{L}_1}(E_0)\right] \label{eq:sigma_L_2}\\
\sigma_{\text{L}_1}(E_0) &= \omega_{\text{L}_1}\left[ \tau_{\text{L}_1}(E_0) \right] \label{eq:sigma_L_1}
\end{align}
where $E_0$ is the incident photon energy, $\omega_{\text{L}_i}$ is the fluorescence yield, $\tau_{\text{L}_i}(E_0)$ the photoionization cross section of the subshell L$_i$ and $f_{i,j}$ the CK factor (with $i=1,2$ or $3$ and depending on the value of $i$, $j=2$ or $3$ ). For incident photon energies below the ionization threshold of subshell L$_i$, the corresponding CK factor $f_{i,j}$ is equal to zero. By using tunable monochromatic X-ray sources, readily available at synchrotron radiation facilities, it is thus possible to turn the CK transitions selectively on or off by a changing the energy of the incident photons \cite{W.Jitschin1985}, which enables determining individually the different CK factors while respecting the boundary conditions that $f_{i,j} \leq 1$ and $f_{1,3}+f_{1,2} \leq 1$.
The CK factors were determined in the past by means of coincidence measurements, which are restricted to the $f_{2,3}$ factor, using radionuclides \cite{Bambynek72} or simultaneously photoinduced K and L XRF \cite{Santra04,Dunford06} or by means of a selective excitation by detecting Auger electrons or XRF photons using a tunable excitation source \cite{W.Jitschin1985,Sorensen91}. The former approach is suitable for obtaining results with low uncertainties but restricted in terms of elements for which the CK factors can be investigated, the latter approach is affected by larger errors which are mainly connected to the an incomplete knowledge on the photoionization cross-section of the different subshells \cite{Papp}. Still the access to all relevant subshell transition yields is granted. The trade-off between Auger and XRF resides in the complexity of the spectra generated and, hence, the availability of a priori information required for a proper separation of the contributions originating from the different L subshells. For XRF based works, mostly energy-dispersive detectors were used \cite{W.Jitschin1985,Stotzel1992,Barrea2004,M.Kolbe2012,M.Kolbe2015,Menesguen18,Menesguen20} but wavelength-dispersive spectrometers for measurements on selected X-ray emission lines were used as well \cite{Cao2009,Cao2010}.
\section{Experimental}
The reference-free XRF and X-ray emission spectroscopy (XES) experiments for the determination of the fundamental parameters of interest here were carried out at PTB’s four-crystal-monochromator (FCM) beamline \cite{Krumrey1998}. It provides monochromatized radiation between 1.75 keV and 10.5 keV by means of either four InSb(111) or Si(111) crystals. The contribution of higher harmonics in the spectral range applied is well below $10^{-5}$ \cite{Krumrey1998}. The incident photon energy was varied from 7.2$\,$keV (below the Gd-L$_3$ edge to exclude sample contamination) up to 9.0$\,$keV (above the Gd-L$_1$ edge) in steps of 50$\,$eV using the Si(111) crystals for monochromatizing the X-ray radiation originating from a bending magnet. The beam diameter in the focal plane of the beamline, where the measurements were performed, was about 300 $\mu m$.
A thin Gd coating with a thickness of 250$\,$nm on the top of a 500$\,$nm thick silicon nitride membrane acting as support material was used as sample. The Gd layer was coated with a thin 40$\,$nm thick Al layer in order to protect it from oxidation. The thickness selected for the Gd layer ensures a good transmittance in the photon energy regime of the Gd-L absorption edges and emission lines, i.e., the attenuation is of the order of 10$\%$
when considering the incidence and detection angles used.
From the transmission experiments it was possible to determine sample specfic mass attenuation in order to not rely on theoretical dat
. In addition, good measurement statistics for the fluorescence and emission experiments were granted, and secondary excitation effects and background contributions from the Al layer and silicon nitride membrane are also negligible.
Furthermore, it should be avoided that the photoionization process involves electrons that are more tightly bound to the atom than those in the electronic level considered for the calculation of the XRF photon rate. Otherwise cascade effects (transition of the vacancy created from inner to outer electronic shells) and possible secondary fluorescence effects do not need to be considered. This condition is usually granted by the large energy difference between the ionization thresholds of the K shell and the different L subshells. A different factor, which can inherently not be avoided for all elements, is possible secondary fluorescence once incident photon energies above the L$_2$ or L$_1$ ionization threshold are used, resulting in the emission of XRF photons with sufficient energy to create a vacancy in the L$_3$ and / or the L$_2$ subshell. For Gd, this is the case for the L$\gamma$ emission lines for which the line energies and relative transition probabilities were established in recent works \cite{Wansleben, Menesguen}: the L$\gamma_2$ (L$_1$N$_2$) and L$\gamma_3$ (L$_1$N$_3$) emission lines resulting from a vacancy in the L$_1$ subshell have energies above the L$_2$ ionization threshold with a relative transition probability of 0.166(17) and 0.091(9), respectively and the the L$\gamma_1$ (L$_2$N$_4$) emission line resulting from a vacancy in the L$_2$ subshell has an energy above the L$_3$ ionization threshold with a relative transition probability of 0.154(15). However, due to the relatively low transition probabilities of the involved emission lines and the low sample thickness the expected secondary enhancement effect is only in the order of one permille.
\subsection{Transmission measurements}
The determination of the CK factors $f_{i,j}$ and the fluorescence yield $\omega_{\text{L}_i}$ depends on the absolute knowledge of the incident photon energy dependence of subshell photoionization cross-sections $\tau_{\text{L}_3}(E_0)$ and $\tau_{\text{L}_2}(E_0)$. To this respect transmission measurements with the sample were performed using a diode downstream of the sample location at different incident photon energies. To correct for the substrate and cap layer, a bare substrate with the same thin Al cap layer was measured as well. A subsequent subtraction of inelastic and elastic scattering cross-sections as reported by Ebel \cite{Elam1} and using a 5th order polynomial interpolation following the approach described in ref. \cite{Unterumsberger2018} allowed obtaining the product $\tau_{\text{L}_i}(E_0) \rho d$ (Fig. \ref{fig:cross-sections}). With this experiment no tabulated or theoretical values for the photon energy dependent mass attenuation coefficients at the relevant incident photon energies and exact information on the density and thickness of the foil are needed and thus unknown uncertainty contributions are avoided
The relative error on uncertainty of the product $\tau_{\text{L}_i}(E_0) \rho d$ is estimated to be 3$\%$ for the L3 subshell and 5$\%$ for the L2 and L1 subshells.
\begin{figure}[!h]
\centering
\includegraphics[width=8cm]{cross-section_dependencies.pdf}
\caption{Incident photon energy dependence of the product of the mass attenuation, respectively photoionization cross section with the mass deposition as obtained for the 250$\,$nm thick Gd film used. In the selection of the film thickness care must be taken that the transmittance remains within a range of 20$\%$ to 80$\%$ for the sake of precision of the measurement.}
\label{fig:cross-sections}
\end{figure}
\subsection{Reference-free XRF}
For the experiments, an in-house developed ultrahigh vacuum chamber dedicated to reference-free XRS was used \cite{J.Lubeck2013}. The incident photon flux was determined using a thin transmission diode upstream and a calibrated photodiode \cite{A.Gottwald2006} downstream of the sample position. The thin transmission diode allowed monitoring flux variations during each measurement, while a cross-referencing to the calibrated photodiode for each incident photon energy used allowed to determine the absolute photon flux for each measurement with a relative uncertainty of 1.5$\%$. A second, non-calibrated diode positioned downstream of the sample position was used for the transmission measurements.
\begin{figure}[!h]
\centering
\includegraphics[width=13cm]{Gd_Lbeta_sum_Entfaltung_SDD_paper.pdf}
\caption{Exemplary Gd L XRF spectrum measured with a radiometrically calibrated SDD at an excitation energy higher than the L$_1$ subshell ionization threshold (9 keV). The knowledge of the detector's response function permits for a deconvolution of the XRF spectrum using only physically relevant contributions.}
\label{fig:sdd_entfaltung}
\end{figure}
The sample was positioned with its surface plane intersecting the pivot point of the goniometer installed within the ultrahigh vacuum chamber and oriented such that the incidence angle of the monochromatized synchrotron radiation and the detection angle of the XRF radiation were each 45$^\circ$ (Fig. \ref{fig:sdd_entfaltung}). The experimental chamber is positioned such that the pivot point of the goniometer is located on the axis along which the synchrotron radiation propagates. The fluorescence radiation emitted by the sample is detected by means of a radiometrically calibrated silicon drift detector (SDD) \cite{Scholze2005,F.Scholze2009}, which is placed behind a well-characterized aperture in a well-defined distance from the sample.
The diameter of the aperture was determined by an optical microscope, and defined together with the sample-detector distance the solid angle of detection with a relative uncertainty of 0.7$\%$. Since the detector is positioned within the polarization plane and perpendicular to the propagation direction of the incident synchrotron radiation, scattering contributions in the recorded spectra are minimized. The detector response functions as well as the detection efficiency of the SDD were known \cite{Scholze2005,F.Scholze2009}, the latter with a relative uncertainty of 1.5$\%$. The intensity of the fluorescence lines of interest were derived from the recorded spectra by a spectral deconvolution using physically motivated detector response functions. The natural Lorentzian broadening of the fluorescence lines could be neglected here due to the resolving power of the detector, which is about 2 orders of magnitude larger.
For the determination of the different CK factors corresponding to the electronic transitions between the different Gd L subshells a similar strategy as described in refs. \cite{M.Kolbe2012,M.Kolbe2015} was followed: XRF spectra were recorded with the SDD while varying the incident photon energies from beneath the L$_3$ ionization threshold to above the L$_1$ ionization threshold and were subsequently deconvolved and normalised to the incident photon flux, detector live time, solid angle of detection and energy-dependent detector efficiency to extract the absolute emission rate of XRF photons appertaining to an electronic transition to the L$_1$, L$_2$ and L$_3$ subshell respectively. For this purpose, the spectra recorded at incident photon energies between the L$_3$ and L$_2$ ionization thresholds were analyzed first since only fluorescence lines appertaining to electronic transitions to the L$_3$ subshell contribute to the recorded spectra. Thus, the relative transition probabilities of the different Gd L$_3$ emission lines that could be resolved by the SDD were accurately determined by means of a spectral deconvolution of each spectrum recorded when considering the contributions of of each individual fluorescence line and of relevant background contributions. By using the information from the spectral deconvolution for different incident photon energies the consistency of the established relative transition probabilities could be verified against possible incident photon energy dependent background or scattering contributions. The relative transition probabilites were then used as a fixed input parameter for incident photon energies above the L$_3$ ionization threshold: the emission lines appertaining to an electronic transition to the L$_3$ subshell were not allowed to vary individually in the spectral deconvolution, only the total net intensity of the ensemble of these emission lines was determined in the following after the spectral deconvolution and data normalization. The same procedure was applied to the XRF fluorescence lines appertaining to a transition to the L$_2$ subshell and L$_1$ subshell for incident photon energies between the L$_2$ and L$_1$ ionization thresholds
This approach allowed reducing the number of free parameters in the deconvolution procedure such that its numerical stability was improved and the contribution of the XRF from each subshell to the measured spectrum could be established in a robust manner (Fig. \ref{fig:sdd_entfaltung}). Finally, the strategy described using sets with fixed relative intensities for the emission lines connected to an electronic transition to the different subshells allowed deriving the respective absolute detected count rates with uncertainties of 1.5 \% by the spectral deconvolution \cite{M.Kolbe2012}. Note that only the determination of the fluorescence yields necessitates the determination of absolute fluorescence emission rates for each subshell, the CK factors can be obtained from relative measurements as shown in the XES experiments. Moreover, and as discussed in ref. \cite{M.Kolbe2012}, it is assumed that the polarization of the incident radiation does not lead to anisotropic emission of the L shell emission lines beyond the uncertainties associated with the detection and deconvolution.
\subsection{X-ray emission spectroscopy}
In an independent experiment, which followed a similar concept, a wavelength dispersive spectrometer in a full-cylinder von Hamos geometry (as introduced in ref. \cite{Anklamm2014}) was made use of for the determination of the CK factors (but not the L subshell fluorescence yields). The spectrometer used \cite{Holfelder2022} is characterized by a medium to high energy resolution such that a good separation of the different Gd emission lines is ensured. In this respect the spectrometer was already applied to the determination of the energy and width of the Gd L emission lines as well as the relative intensity ratios of the L emission lines from the different L subshells \cite{Wansleben}. The spectrometer is attached to a measurement chamber similar to the one described in ref. \cite{J.Lubeck2013} and additionally equipped with an SDD. The sample orientation was similar as during the reference-free XRF experiments, i.e., the incidence angle on the sample surface was 45$^\circ$ and the dispersion axis of the spectrometer was perpendicular to the propagation direction and contained within the polarization plane of the incident synchrotron radiation. The dispersion axis is defined as the axis of the full cylinder highly annealed pyrolithic graphite (HAPG) crystal used.
The SDD was positioned mid-way between the incidence direction of the synchrotron radiation and the detection direction of the von Hamos spectrometer.
\begin{figure}[!h]
\centering
\includegraphics[width=13cm]{Gd_Lbeta_sum_Entafltung_paper.pdf}
\caption{Exemplary Gd L$\beta$ emission spectrum measured with the von Hamos spectrometer at an excitation energy higher than the L$_1$ subshell ionization threshold (9 keV). The fit is a convolution of experimentally determined response functions and Lorentzians and a linear background.}
\label{fig:vH_entfaltung}
\end{figure}
The spectrometer is operated in a slitless geometry, which makes the use of micro-focusing optics necessary, and equipped with two full-cylinder HAPG crystals with a bending radius of 50$\,$mm and a CCD camera as a detector \cite{Holfelder2022}. The incident monochromatized synchrotron radiation was therefore focused by means of a polycapillary half-lens resulting in a 60$\,\mu$m FWHM focus as determined with the knife-edge method \cite{R.Unterumsberger2012} at the sample position.
In the presented experiment only one optic was used as this configuration provided a good enough resolving power of $E/\Delta E$ = 1000 to resolve the emission lines of interest.
Moreover, due to the small radius of curvature of the crystals and the width of the crystals and size of the CCD chip, the advantage of a large spectral energy window of over 600$\,$eV at a single position of the HAPG crystal and the CCD camera can be exploited. The energy window mentioned is monitored in a scanning-free mode due to the cylindrical crystal shape and the position-sensitivity of the detector used.
The CK factors were determined by solely measuring a part of the the Gd L$\beta$ emission spectrum which includes emission lines originating from ionization of all three L subshells: L$_2$M$_4$ (L$\beta_1$), L$_3$N$_1$ (L$\beta_6$) and L$_3$N$_{4,5}$ (L$\beta_{2,15}$). The emission lines L$_1$M$_2$ (L$\beta_4$), L$_1$M$_3$ (L$\beta_3$) are also included in the recorded spectra but not required for the determination of the CK factors. In addition, a satellite emission on the low-energy side of the L$\beta_{2,15}$ is observed, which is due to exchange and spin-orbit interactions between the 4$d$ and 4$f$ electronic levels and discussed in a previous work \cite{Wansleben}. As in the reference-free XRF experiments, the different X-ray emission lines monitored enable the direct measurement of changes in emitted intensity of the emission lines associated with an electronic transition to the L$_3$ or L$_2$ subshell when increasing the incident photon energy above the L$_2$ and L$_1$ ionization thresholds. The deviations from the expected energy-dependent photoionization cross-sections of the L$_3$, respectively L$_2$ subshell are directly connected to the CK transitions. Since all the emission lines are monitored at once for each incident photon energy a more accurate normalization to the incident photon flux is enabled than compared to sequential measurements of the emission lines.
In contrast to the reference-free XRF measurements, the energy dependent transmission of the polycapillary optics needs to be taken into account when normalizing the data acquired at the different incident photon energies to the incident photon flux. Therefore, the intensity of the Ti K$\alpha$ emission lines from a metallic Ti foil was measured using the SDD with and without the polycapillary inserted in the incident beam path. This procedure is conducted for all incident photon energies before and after the acquisition of the Gd L$\beta$ emission spectrum with the von Hamos spectrometer in order to ascertain the stability and reproducibility of the alignment of the polycapillary optics. Fig. \ref{fig:trans_cap} shows the derived transmission of the capillary. This approach was found to be more stable and less affected by experimental noise than a comparable measurement with a diode, which needs to be positioned at an adequate distance of the capillary optics to avoid both depletion and illuminating an area larger than the diode area when focusing the X-ray beam. Finally, incident X-ray photon intensity variations during the XES measurements were monitored with the same transmission diode positioned upstream of both the sample and the polycapillary optics as in the reference-free XRF experiments. By referencing the transmission diode to an absolutely calibrated photodiode for each photon energy used, the absolute photon flux on the sample could be determined for each measurement. The relative uncertainty on the incident photon flux is about 3.1$\%$ and is determined by the counting statistics during the determination of the transmission of the polycapillary optics (relative contribution of 2.7$\%$) and the diode measurements (relative contribution of 1.5$\%$, as for the reference-free XRF measurements). Note, that an absolute determination of the incident photon flux is not strictly required for the determination of the CK factors. However, the use of radiometrically calibrated diode allows to alleviate possible incident photon energy dependent changes in sensitivity and thus for an improved accuracy of the results provided.
\begin{figure}[!h]
\centering
\includegraphics[width=8cm]{Kapillare_Trans_Ti_Mess_paper.pdf}
\caption{The derived transmission of the capillary as a function of all applied incident energies. }
\label{fig:trans_cap}
\end{figure}
To extract the respective line intensities the measured Gd L$\beta$ spectra are deconvolved using the experimentally determined spectrometer's response function as well as Lorentzians that model the natural line shape of the individual emission lines. The response functions are measured prior to the Gd L$\beta$ emission by introducing a 4 $\mu$m-thick gold foil as scattering target and tuning the monochromator energy to the expected emission line energies of the Gd L$\beta$ spectrum. The choice of Au was motivated by the absence of X-ray emission lines in the energy range monitored and the large scattering cross-sections of high-Z elements in the X-ray energy range considered, which enabled the use of thin foils in view of optimizing the inelastically versus elastically scattered radiation. This measurement using elastically scattered X-ray photons is necessary as the response function slightly changes as a function of the emission energy \cite{Anklamm2014,Holfelder2022}. The respective calibration procedure is described in more details in ref. \cite{Wansleben}. In addition to the X-ray emission lines, a linear background is included in the deconvolution algorithm. An exemplary fit of the recorded spectra is displayed in Fig. \ref{fig:vH_entfaltung} showing the Gd L$\beta$ emission lines with the incident energy higher than the L$_1$ subshell threshold (9 keV). Using the residuals the relative uncertainty on the deconvolution of the L$\beta_{2,15}$ (including the satellite transition) and the L$\beta_{1}$ lines, which are the most intense lines in the recorded spectra, is estimated to be about 7$\%$ while the counting statistics is not a relevant contribution in the uncertainty budget.
\section{Results \& Discussion}
The CK factors were determined on the basis of the Sherman equation for quantification of the elemental mass
\begin{equation}
\rho d=\frac{{N_{\text{L}_i}(E_0)}\sin(\theta) M(E_0,E_{\text{L}_i})}{I_0 \sigma_{\text{L}_i}(E_0) \Omega/(4\pi) \epsilon(E_{\text{L}_i}) }
\label{eq:Sherman}
\end{equation}
where $\rho$ is the mass density of the sample, $d$ the sample thickness, $\theta$ the incidence angle of the synchrotron radiation on the sample, $M(E_0,E_{\text{L}_i})$ the correction factor for attenuation of incident and emitted X-rays within the sample (as defined in ref. \cite{M.Kolbe2015}, less than 10$\%$ for the energies considered in the present experiments), $N_{\text{L}_i}(E_0)$ the detected count rate of the fluorescence (e.g., for the L$_3$ subshell the ensemble of emission lines appertaining to an electronic transition to the L$_3$ subshell) using an incident photon flux $I_0$, and $\epsilon(E_{\text{L}_i})$ the detection efficiency of the detector used for the emission lines considered. In the case of XES measurements where only selected emission lines are monitored $N_{\text{L}_i}(E_0)$ would need to be corrected for the relative transition probability.
\begin{figure}[!h]
\centering
\includegraphics[width=16cm]{Gd_L_tau_floureszenz_paper.pdf}
\caption{Comparison of the subshell photoionisation cross section $\tau$ and the normalized count rate of a respective emission line: (a) L$\beta_{2,15}$ (L$_3$N$_{4,5}$ transition) for the L$_3$ subshell and (b) L$\beta_1$ (L$_2$M$_4$ transition) for the L$_2$ subshell. The jumps in the count rate when tuning the photon energy over the next subshell threshold indicates the CK transitions. }
\label{fig:Gd_L_tau_floureszenz}
\end{figure}
Since the mass deposition $\rho d$ is invariant, the CK factors can be determined by using different incident photon energies above and below the respective L subshell ionization thresholds and inserting equations \ref{eq:sigma_L_3} and \ref{eq:sigma_L_2} into \ref{eq:Sherman} assuming that $\sigma_{\text{L}_i} M(E_0,E_{\text{L}_i}) /\omega_{\text{L}_i}$ is proportional to the measured count rates normalized to the incident flux $I_{\text{L}_i}(E_0)=N_{\text{L}_i}(E_0)/I_0$. In Fig. \ref{fig:Gd_L_tau_floureszenz}, the intensity jump at the ionization thresholds for the emission lines monitored is due to vacancies created by a CK transition in addition to the photoionization process. Thus, taking into account the incident photon energy dependency of the individual subshell photoionization cross sections, a pair of equations can be established that can be solved for the CK factors,
\begin{align}
f_{23} &= \frac{\tau_{\text{L}_3}(E_B)}{\tau_{\text{L}_2}(E_B)} \left [ \frac{\tau_{\text{L}_3} (E_A)}{I_{\text{L}_3}(E_A) } \frac{I_{\text{L}_3}(E_B)}{\tau_{\text{L}_3}(E_B)} - 1 \right ]
\label{eq:f23} \\
f_{12} &= \frac{\tau_{\text{L}_B}(E_C)}{\tau_{\text{L}_C}(E_C)} \left [ \frac{\tau_{\text{L}_2} (E_B)}{I_{\text{L}_2}(E_B) } \frac{I_{\text{L}_2}(E_C)}{\tau_{\text{L}_2}(E_C)} - 1 \right ]
\label{eq:f12} \\
f_{13} &= \frac{\tau_{\text{L}_3}(E_C)}{\tau_{\text{L}_1}(E_C)} \left [ \frac{\tau_{\text{L}_3} (E_A)}{I_{\text{L}_3}(E_A) } \frac{I_{\text{L}_3}(E_C)}{\tau_{\text{L}_3}(E_C)} - \left (1 + f_{23} \frac{\tau_{\text{L}_2}(E_C)}{\tau_{\text{L}_3}(E_C)} \right ) \right ] - f_{12}f_{23},
\label{eq:f13}
\end{align}
when using incident photon energies such that $E_{\text{L}_3} < E_A < E_{\text{L}_2} < E_B < E_{\text{L}_1} < E_C$. The uncertainties then depend mainly on the relative contributions of the photoionization cross-section $\tau_{\text{L}_i}$ and the deconvolution result $I_{\text{L}_i}$ or the ensemble of emission lines appertaining to an electronic transtion to subshell L$_i$ for the reference-free XRF experiment, respectively for the XES results by using $I(\text{L}\beta_{2,15})$ for $I_{\text{L}_3}$ and $I(\text{L}\beta_{1})$ for $I_{\text{L}_2}$. Additionally, for each CK factor multiple different excitation energies were used that fulfill the above written condition. Hence, for each CK factor multiple results are obtained which allowed for a consistency check of the CK factors determined, for example whether the incident photon energy dependent trend for the subshell photoionization cross-sections $\tau_{\text{L}_i}(E_0)$ is correctly extrapolated. The individual results for each CK factor scattered on a range of 6$\%$ to 8$\%$ which is well within the uncertainty for, both reference-free XRF and XES. The final result reported for each CK factor is an average over these multiple results. This procedure leads to more robust results and provides a consistency check of the experimental results, which is important for assessing whether an incident photon energy dependence attributable to the subshell photoionization cross-sections used may be present. The results obtained for the XES and reference-free XRF measurements are summarized and compared to literature values in Table \ref{tab:CKs}.
\begin{table}[htbp]
\centering
\caption{L subshell Coster Kronig factors}
\begin{tabular}{|c|c|c|c|}
\hline
& $f_{2,3}$ & $f_{1,2}$ & $f_{1,3}$ \\
\hline
Present work (RF-XRF) & 0.235(59) & 0.169(96) & 0.197(121) \\
\hline
Present work (XES) & 0.206(60) & 0.148(43) & 0.193(147) \\
\hline
M\'{e}nesguen et al. (2020) \cite{Menesguen20} & 0.095(20) & 0.053(23) & 0.27(4) \\
\hline
Puri et al. (1993) \cite{S.Puri1993} & 0.16 & 0.216 & 0.334 \\
\hline
X-RayLib (2012) \cite{xraylib} & 0.149 & 0.19 & 0.279 \\
\hline
Krause (1979) \cite{Krause1979} & 0.147 & 0.19 & 0.3 \\
\hline
Papp et al. (1998) \cite{Papp_1998} & & 0.166(20) & 0.287(14) \\
\hline
Douglas (1972) \cite{Douglas_1972}& 0.223(11) & & \\
\hline
Gnade et al. (1981) \cite{Gnade_1981} & 0.157(12) & & \\
\hline
\end{tabular}%
\label{tab:CKs}%
\end{table}%
First it can be noticed that the CK factors extracted from the reference-free XRF and the XES experiments agree well with each other. The uncertainties attributed to the values are quite large due to the error propagation based on the difference of subshell photoionization cross-section ratios. For the CK factor $f_{1,3}$ the relative error of the two other CK factors needs to be included as well, resulting in uncertainties of the order of the extracted value itself. Indeed, these uncertainty budgets are established based on the uncertainties connected to the different factors entering into Eqs. \ref{eq:f23}, \ref{eq:f12} and \ref{eq:f13}.
However, the agreement with other reported CK factors for Gd is not always given. Especially, the comparison with the other reported synchrotron radiation based measurements \cite{Menesguen20} shows a pronounced discrepancy in the sense that especially the CK factors $f_{2,3}$ and $f_{1,2}$ are less than half of the values deduced in the present work. One major aspect to explain this difference is related to the subshell photoionization cross-sections. Indeed, extrapolation of the cross-sections connected to the L$_3$ or L$_2$ beyond the ionization thresholds of the L$_2$ and, respectively the L$_1$ subshells is required. Differences in the data ranges and polynomials used can result in major deviations in the final results. Hence, a critical revision and intercomparison of such aspects as well as of the spectral modelling, semi-empirically versus response function-based, is required in future to assess the origin of such discrepancies.
With regard to other experimentally established values \cite{Papp_1998,Douglas_1972,Gnade_1981}, a better agreement is found for the CK factors $f_{1,2}$ and $f_{1,3}$, where an overlap within the uncertainties of the respective values can be observed, but not necessarily for $f_{2,3}$, where a major inconsistency between the two coincidence measurements was observed and discussed in ref. \cite{Gnade_1981}. With respect to calculations \cite{S.Puri1993} and databases \cite{xraylib,Krause1979}, an agreement within the margin of the uncertainties for the CK factors $f_{1,2}$ and $f_{1,3}$ can be observed. For $f_{2,3}$ the agreement can only be achieved within the combined uncertainty of the presented value and the database value. For databases the consistency of the data used is a measure of the reliability assumed and the uncertainty assigned. For Gd it is for example recommended to use a relative error of 10$\%$ to 20 $\%$ \cite{Krause1979}.
From the reference-free XRF data, the fluorescence yield $\omega_{\text{L}_i}$ of each subshell could also be obtained using the photoionization cross-section $\tau_{\text{L}_i}$ and the deconvolution result $I_{\text{L}_i}$ for the ensemble of emission lines appertaining to an electronic transition to subshell L$_i$
\begin{equation}
\omega_{\text{L}_i}=\frac{N_{\text{L}_i}(E_0)\sin(\theta) M(E_0,E_{\text{L}_i})}{I_0 \Omega/(4\pi) \epsilon(E_{\text{L}_i}) \tau_{\text{L}_i}(E_0) \rho d}
\end{equation}
for i = 1, 2 or 3 with incident photon intensity $I_0$ and energy $E_0$. For the determination of the fluorescence yield $\omega_{\text{L}_i}$ only the incident photon energy where no CK factors contribute shall be used for the different subshells. The product $\tau_{\text{L}_i}(E_0) \rho d$ was established from the transmission measurements. As during the determination of the CK factors, each subshell fluorescence yield $ \omega_i$ is determined for multiple incident photon energies such that a consistency check can be conducted. For each factor the standard deviation of the results obtained is below 1$\%$. Note that for both, the fluorescence yield and the CK factor, the incident photon energy shall not include energies close to the ionization threshold due to the fine structure on the photoionization cross-section. The results obtained are summarized and compared to literature values in Table \ref{tab:fluoyields}.
\begin{table}[htbp]
\centering
\caption{Experimentally determined Gd L subshell fluorescence yields in comparison to available literature sources.}
\begin{tabular}{|c|c|c|c|}
\hline
& $\omega_{\text{L}_3}$ & $\omega_{\text{L}_2}$ & $\omega_{\text{L}_1}$ \\
\hline
Present work (RF-XRF) & 0.159(7) & 0.164(11) & 0.076(7) \\
\hline
M\'{e}nesguen et al. (2020) \cite{Menesguen20} & 0.159(3) & 0.162(4) & 0.099(3) \\
\hline
Puri et al. (1993) \cite{S.Puri1993} & 0.167 & 0.175 & 0.083 \\
\hline
xraylib (2012) \cite{xraylib} & 0.155 & 0.158 & 0.102 \\
\hline
Krause (1979) \cite{Krause1979} & 0.155 & 0.158 & 0.079 \\
\hline
Sahnoune et al. (2016) \cite{Y.Sahnoune2016} & 0.162 & 0.1686 & 0.085\\
\hline
Krishnananda et al. (2016) \cite{Krishnananda} & 0.167(7) & 0.176(8) & 0.089(4) \\
\hline
Kumar et al. (2010) \cite{Kumar_2010} & &0.165(13) & 0.101(9) \\
\hline
Papp et al. (1998) \cite{Papp_1998} & & & 0.101(5) \\
\hline
Douglas (1972) \cite{Douglas_1972} & 0.187(6) & 0.182(8) & \\
\hline
Gnade et al. (1981) \cite{Gnade_1981} & 0.161(19) & 0.159(22) & \\
\hline
\end{tabular}%
\label{tab:fluoyields}%
\end{table}%
The uncertainty for the fluorescence yields is considerably lower than the one for the CK factors. In contrast to the CK factors a good agreement is obtained with the values from different synchrotron radiation experiments \cite{Menesguen20,Krishnananda}. For $\omega_{\text{L}_3}$ and $\omega_{\text{L}_2}$ the comparisons to most other experimental values \cite{Gnade_1981,Papp_1998,Kumar_2010}, theory \cite{S.Puri1993} and compilations \cite{xraylib,Krause1979,Y.Sahnoune2016} match within the error bars with each other. In ref. \cite{Douglas_1972} larger values are reported, which was already noted in ref. \cite{Gnade_1981} where the same type of experiment was performed. The discrepancies were attributed to the detectors used, background contributions and the data treatment performed \cite{Gnade_1981}. For $\omega_{\text{L}_1}$, most of the different values reported do not agree with and are larger than our result. With respect to experimental works the reason can be the subshell photoionization cross-section $\tau_{\text{L}_1}(E_0)$ \cite{Honicke2016} or the XRF fluorescence rate originating from electronic transitions to the L$_1$ subshell extracted from the experimental data. Without exact information on the these aspects, the situation can currently not be assessed in a more accurate manner.
\section{Conclusions}
The CK factors were determined with a reliable uncertainty using monochromatized synchrotron radiation at different photon energies and two different detection schemes, firstly using an energy-dispersive detector and secondly using a wavelength-dispersive detector. In addition, the fluorescence yields are reported on the basis of reference-free XRF analysis. The uncertainty budget is established relying on experimental values established within this work and only require little input from tabulated data, i.e., the cross-sections for inelastic and elastic scattering with very minor contributions to the uncertainty budget. The agreement between the results and the similarity in relative errors show that for the element investigated both setups are equally well suited. Note that the probability of a radiationless transition was investigated indirectly by means of emitted radiation from the sample, which implies also the quite large errors. CK factors are, indeed, intrinsically difficult to derive due to the required accurate knowledge on the subshell photoionizaition cross-section. The equivalency between the results is relevant since the Coster-Kronig factors are a good probe for theory based on the Hartree-Fock method and the requirement to include relativistic effects \cite{Crasemann84}. Note, that theory applies to single atoms, whereas experiments may be affected by considerable solid-state effects.
A wavelength-dispersive detection scheme, which is more complex to operate compared to the use of an energy-dispersive detector, can be beneficial in different aspects. When turning towards the more complex M or even N-shell spectra a correct assignment of the different emission lines becomes more challenging due to the different transitions involved \cite{McGuire72,Chauchan2008}. Wavelength-dispersive setups have demonstrated their suitability for the accurate determination of the line energies, line widths and relative transition probabilities. In addition, the sensitivity to the chemical state of an element of XES allows to monitor changes in line position or relative transition probabilities, for example due to exchange interactions and crystal effects. Investigations on the dependence of CK factors or fluorescence yields on the chemical state would, however, require improving first the accuracy of the results obtainable. The information on the spectral distribution of the emission lines is required for a correct deconvolution of the experimental spectra and discrimination of the emission lines originating from electronic transitions to different subshells. In addition, the assessment of possible secondary excitation due to XRF appertaining to an electron transition to the inner subshells of an electronic level is facilitated. Such complementary investigations are not always readily obtainable with energy-dispersive detectors, which could therefore profit from such complementary investigations. The strength of energy-dispersive detection schemes is that absolute measurements on the emitted fluorescence intensities are enabled. The latter type of experiment is mandatory for the determination of fluorescence yields and cannot be easily conducted using wavelength-dispersive detection setups, as has been pointed out for the full-cylinder von Hamos geometry used in this study for which the effective solid angle defined as the product of the CCD detection efficiency, the integral reflectivity of HAPG and the solid angle of acceptance of the optic needs to be evaluated in this respect \cite{Malzer2018}. Finally, a correct assessment of the many atomic FP requires even further instrumentation development and the complementary use of energy- and wavelength dispersive detection setups.
\section{Acknowledgements}
Parts of this research was performed within the EMPIR project 19ENV08 AEROMET II. This project has received funding from the EMPIR programme co-financed by the Participating States and from the European Union’s Horizon 2020 research and innovation programme. This project has received funding from the ECSEL Joint Undertaking (JU) IT2 under grant agreement No 875999. The JU receives support from the European Union’s Horizon 2020 research and innovation programme and Netherlands, Belgium, Germany, France, Austria, Hungary, United Kingdom, Romania and Israel.
|
1,941,325,220,534 | arxiv | \section{Introduction}
\label{s:intro}
Collisions of heavy nuclei at high energies create a highly dynamical system, which develops some collective behavior over a timescale of order 10\;fm$/c$.
The emission pattern of particles in the final state appears to be strongly correlated to the initial system geometry determined by the overlap region of the colliding nuclei.
In particular, initial asymmetries in the geometry are converted into transverse momentum space anisotropies, referred to as anisotropic flow, as the system evolves~\cite{Heinz:2013th,Luzum:2013yya,Bhalerao:2020ulk}.
When two nuclei collide, the transverse geometry of their overlap region is customarily characterized in the transverse plane by spatial eccentricities.
In polar coordinates $(r,\theta)$, these are defined by~\cite{Alver:2010gr,Teaney:2010vd,Gardim:2011xv}
\begin{align}
\label{eccentricity}
\varepsilon_n^{\bf x} {\rm e}^{{\rm i}n\Phi_n}\equiv
-\frac{\langle r^n {\rm e}^{{\rm i}n\theta}\rangle_{\bf x}}{\langle r^n\rangle_{\bf x}},
\end{align}
where $\langle\cdots\rangle_{\bf x}$ is an average over the transverse plane --- ${\bf x}$ denotes the transverse position vector ---, weighted with the centered entropy or energy density.\footnote{In the paper, we will consider averages weighted with particle number or energy.}
The angle $\Phi_n$ is the $n$-th participant plane angle, which we shall set equal to 0 in the following.
In turn, momentum space anisotropies are mostly characterized by the Fourier coefficients of the transverse momentum distribution of emitted particles~\cite{Voloshin:1994mz}:
\begin{align}
v_n{\rm e}^{{\rm i}n\Psi_n}\equiv
\langle {\rm e}^{{\rm i}n\varphi_\mathbf{p}}\rangle_{\bf p},
\label{vn_def}
\end{align}
where $\varphi_\mathbf{p}$ is the azimuth of a particle transverse momentum ${\bf p}$ and $\Psi_n$ is the $n$-th ``event plane'' angle.
The angular brackets now denote an average over the momentum distribution of particles.
Most of the modern models of the dynamics of the fireball created in high-energy nuclear collisions include a fluid dynamical stage~\cite{Ollitrault:1992bk,Kolb:2003dz,Huovinen:2006jp,Romatschke:2009im,Teaney:2009qa,Romatschke:2017ejr} or a proxy thereof --- possibly preceded by a dynamical ``prehydrodynamic'' evolution~\cite{Schlichting:2019abc} and a subsequent transport ``afterburner'' ---, which describes well a large amount of the experimentally measured anisotropic-flow signals from collisions of heavy nuclei~\cite{Heinz:2013th}.
However, the application of fluid dynamics to smaller systems or in peripheral collisions is more disputed~\cite{Weller:2017tsr,Zhao:2020pty}.
In addition, several approaches still compete for the early dynamics.
A possible description of the system is given by kinetic transport theory, as we do in this paper.
Within that model, it is possible to model (part of) the fireball evolution from the few-collisions regime to the hydrodynamic limit.
In this paper we investigate how the early-time evolution of several observables varies between these two extremes.
In the next section we introduce the tools we employ in this paper, which consist of a numerical transport code on the one hand and an analytical approach to the kinetic Boltzmann equation via a Taylor-series ansatz on the other hand.
Within both approaches we investigate several characteristics of a system of massless particles, in particular its anisotropic flow and spatial eccentricities, focusing on their development at early time (Sect.~\ref{sec:results}).
Finally we summarize our main findings and discuss them in Sect.~\ref{sec:discussion}.
\section{Methods and setup}
\label{s:methods}
In this section we briefly present the two methods used in our study, namely a numerical transport model (Sect.~\ref{subsec:numerical_simulations}) and an analytical approach to the kinetic Boltzmann equation (Sect.~\ref{subsec:analytical_approach}).
Both require as starting point an initial distribution function, which we specify in Sect.~\ref{subsec:distribution_function}.
\subsection{Transport simulations}
\label{subsec:numerical_simulations}
To simulate the expansion of a system of relativistic degrees of freedom, we use on the one hand a numerical code implementing the covariant transport algorithm introduced in Ref.~\cite{Gombeaud:2007ub}.
As in Ref.~\cite{Roch:2020zdl}, to which we refer for further details, we consider massless (test) particles, modeled as hard spheres.
Since we focus on characteristics of the system, namely its anisotropic flow and the evolution of the spatial eccentricities, which are mostly driven by transverse dynamics~---at least at the qualitative level~---, the system is purely two dimensional.
This restriction to the transverse plane allows us to overcome the statistical noise on the observables, in particular in the few collisions limit.
An advantage of the transport algorithm over the analytical calculations of Sect.~\ref{subsec:analytical_approach} is that one can smoothly cover the whole range from free-streaming to the hydrodynamic limit with a single model, by changing the cross section $\sigma$.
To ensure covariance and locality, we make sure that the system remains dilute even in the fluid-dynamical regime:
the parameter $D\equiv n^{-1/2}/\ell_\mathrm{mfp}$, which compares the relative sizes of the typical distance $n^{-1/2}$ between particles and the mean free path given by $\ell_\mathrm{mfp}\equiv 1/(n\sigma)$ --- both estimated in the initial state at the center of the system, where the particle density $n$ is maximum ---, is smaller than 0.1 for all our simulations.
The computation time of the simulations typically grows as $N_{\mathrm{p}}^{3/2}$, where $N_\mathrm{p}$ denotes the number of test particles in the system.
To reduce statistical fluctuations, for every setup (initial geometry, Knudsen number) that we consider we perform $N_\mathrm{iter.}$ iterations of the simulation, which is equivalent to performing a single run with $N_\mathrm{p}\cdot N_\mathrm{iter.}$ test particles but computationally cheaper.
We typically use $N_{\mathrm{p}} = 2\times 10^5$ particles in the initial state and $N_\mathrm{iter.}={\cal O}(10^3)$, so that their product is always larger than $10^8$.
To allow a more accurate comparison with the analytical calculations presented in Sect.~\ref{subsec:analytical_approach}, which only account for the loss term of the kinetic Boltzmann equation, we also performed simulations with a ``$2\to 0$'' collision kernel.
That is, two (test) particles that collide disappear from the system, which obviously violates every conservation law: energy, momentum, particle number.
For those simulations we introduced labels ``active'' and ``passive'' for the test particles.
A collision can then only happen between two ``active'' particles.
After the collision, these are then labeled as ``passive'' for the remainder of the evolution and from that moment on they are no longer taken into account in the computation of any observable.
\subsection{Analytical calculations}
\label{subsec:analytical_approach}
A dilute system of particles undergoing binary collisions can also be described by a single-particle phase space distribution $f(t,\vec{x},\vec{p})$ obeying the kinetic Boltzmann equation with the appropriate $2\to 2$ collision term.
In the regime of large Knudsen numbers, i.e.\ when the particles undergo very few rescatterings, one can expect that $f(t,\vec{x},\vec{p})$ will not depart much from a free-streaming distribution.
This observation underlies a number of (semi-)analytical studies, in particular of anisotropic flow, in the few-collisions limit, using various approximations for the collision kernel~\cite{Heiselberg:1998es,Borghini:2010hy,Romatschke:2018wgi,Kurkela:2018ygx,Borghini:2018xum,Kurkela:2019kip,Kurkela:2020wwb,Kurkela:2021ctp,Ambrus:2021fej}.
Irrespective of any approximation, it was pointed out in Ref.~\cite{Borrell:2021cmh} that one can start from a Taylor expansion of the phase-space distribution at early times
\begin{align}
f(t,\vec{x},\vec{p}) = f^{(0)}(\vec{x},\vec{p}) &+
t\, \partial_t f(t,\vec{x},\vec{p})\big|_0 \cr
& + \frac{t^2}{2} \partial_t^2 f(t,\vec{x},\vec{p})\big|_0 + \cdots,
\label{eq:Taylordg}
\end{align}
where $f^{(0)}$ denotes the initial distribution while the successive time derivatives are evaluated at the initial time $t=0$.
By making use of the relativistic Boltzmann equation, rewritten in the form
\begin{equation}
\partial_t f(t,\vec{x},\vec{p}) =
-\frac{\vec{p}}{E}\cdot\vec{\nabla}_{\!x} f(t,\vec{x},\vec{p}) + {\cal C}[f],
\label{eq:be}
\end{equation}
one can replace every time derivative in Eq.~\eqref{eq:Taylordg}, so that given the form of the collision kernel, the whole evolution of $f(t,\vec{x},\vec{p})$ is governed by the initial distribution and its spatial derivatives~\cite{Borrell:2021cmh}:
\begin{widetext}
\begin{align}
f(t,\vec{x},\vec{p}) = f_{\mathrm{f.s.}}(t,\vec{x},\vec{p}) &+
t\,{\cal C}[f]\big|_0 +
\frac{t^2}{2} \bigg(\!\!-\!\frac{\vec{p}}{E}\cdot\vec{\nabla}_{\!x} {\cal C}[f] +
\partial_t {\cal C}[f]\bigg)_{\!0} \cr
&+\frac{t^3}{3!}\bigg( \frac{\big(\vec{p} \cdot \vec{\nabla}_{\!x}\big)^2}{E^2} {\cal C}[f] -
\frac{\vec{p}}{E}\cdot\vec{\nabla}_{\!x} \partial_t {\cal C}[f] +
\partial_t^2 {\cal C}[f] \bigg)_{\!0} + \mathcal{O} (t^4),
\label{FullExpansion}
\end{align}
\end{widetext}
where $f_{\mathrm{f.s.}}$ is obtained by grouping all terms which do not contain the collision kernel and is therefore the free-streaming distribution with the same initial condition at $t=0$.
Again, the time derivatives of the collision kernel can be re-expressed using the Boltzmann equation, so that only $f^{(0)}$ and its spatial derivatives are involved.
Note that these time derivatives of ${\cal C}[f]$ actually involve increasing powers of the cross section characterizing the rescatterings~\cite{Borrell:2021cmh}, as will be illustrated in the following section.
From this point, one can in principle compute any quantity at early times.
Obviously if more orders in time are included, the result will be closer to the full solution, although it will always depart at later times when only a finite number of powers of $t$ are taken into account.
To match the numerical simulations, we shall make all analytical calculations in 2 dimensions and with massless particles.
A drawback of the analytical approach is that the gain term of the usual $2\to 2$ collision kernel of the Boltzmann equation is difficult to handle.
In contrast to the loss term, the ``particle of interest'' --- that with the same momentum as appears on the left hand side of the Boltzmann equation --- does not enter it directly, and the form of the differential cross section does matter.
To bypass this issue, here we only consider the loss term of the collision integral, i.e.\ a $2 \to 0 $ collision kernel~\cite{Heiselberg:1998es,Borghini:2010hy}
\begin{align}
\label{C[f]_2->0}
{\cal C}_{2\to 0}[f] = -\frac{E}{2}\int\! f({\bf p})f({\bf p}_1)_{} v_{\mathrm{rel.}}
\sigma\,{\rm d}^2\,{\bf p}_1,
\end{align}
with $v_{\rm rel.}$ the M{\o}ller velocity between the two colliding particles, while for brevity we did not denote the time and position variables.
This collision term clearly does not conserve energy, momentum or particle number, but on the other hand we are able to push our calculations to higher order in the total cross section $\sigma$.
As stated above, we also implement a $2\to 0$ collision kernel in the numerical code by ignoring particles that have already undergone a collision.
A remaining important difference between the analytical calculations and the simulations is the order in $\sigma$.
While the analytical calculations contain only a finite amount of orders due to the Taylor expansion, the numerical simulations contain the total cross section to all orders.
\subsection{Initial distribution function}
\label{subsec:distribution_function}
In this paper, similar in that respect to other studies in the literature~\cite{Heiselberg:1998es,Borghini:2010hy,Romatschke:2018wgi,Kurkela:2018ygx,Kurkela:2019kip,Kurkela:2020wwb,Kurkela:2021ctp,Ambrus:2021fej}, we use a simplified semi-realistic geometry for the initial state for both our numerical and analytical investigations.
We assume that the initial phase space distribution $f^{(0)}$ factorizes like
\begin{align}
f^{(0)}({\bf x},{\bf p}_{\mathrm{T}}) = F({\bf x})\, G({\bf p}_\mathrm{T},T({\bf x})),
\label{InitialDistribution}
\end{align}
where the spatial part $F$ governs the geometry, while the momentum part $G$ takes the form of a thermal distribution with a temperature that can depend on position:
\begin{align}
G({\bf p}_\mathrm{T}) = \frac{1}{2\pi T({\bf x})^2}e^{-|{\bf p}_\mathrm{T}|/T({\bf x})}.
\label{eq:momentum_dist}
\end{align}
Note that $G$ is normalized to unity, so that $F({\bf x})$ is actually the particle-number density.
The temperature is determined by the latter via
\begin{align}
T({\bf x}) \propto \sqrt{F({\bf x})},
\label{eq:T_match}
\end{align}
following the equation of state of a perfect gas of massless particles in two dimensions.
This relation implies that the central region will have a higher temperature than the outer ones.
Using polar coordinates in the transverse plane, the profile in position space is taken to be
\begin{align}
F(r,\theta) = \frac{N_{\rm p}\,{\rm e}^{-\frac{r^2}{2R^2}}}{2\pi R^2}
\left[1-\sum_{k=2}^3 \tilde{\varepsilon}_k {\rm e}^{-\frac{r^2}{2R^2}}\left(\frac{r}{R}\right)^{\!\!k}\cos(k\theta)\right],
\label{eq:initial_distribution}
\end{align}
where $N_{\rm p}$ is the total initial number of particles and $R$ a characteristic system size, which sets the typical time scale of the system evolution.
The value of $R$ (6.68\;fm in the calculations) alone is irrelevant for the results, it is only meaningful when combined with $\sigma$ (which in two dimensions has the dimension of a length) and $N_{\rm p}$ to yield a dimensionless quantity like the Knudsen number.
Consistent with the normalization of the momentum distribution $G$, $F$ is normalized to $N_{\rm p}$, so that the initial phase-space distribution is also normalized to the total number of particles, as it should.
The two (real) parameters $\tilde{\varepsilon}_2$ and $\tilde{\varepsilon}_3$ give the degree of asymmetry of the initial geometry: a straightforward integral gives for the eccentricities~\eqref{eccentricity} $\varepsilon_2^{\bf x} = \tilde{\varepsilon}_2/4$ and $\varepsilon_3^{\bf x} = \tilde{\varepsilon}_3/\sqrt{2\pi}$ with $\Phi_2 = \Phi_3 = 0$, where the latter choice was made for the sake of simplicity, without any impact on our results.\footnote{If $\tilde{\varepsilon}_n$ is taken to be complex, then the symmetry-plane angle $\Phi_n$ is the argument of $\tilde{\varepsilon}_n$.}
All results shown in the following were obtained with initial profiles such that only one eccentricity $\tilde{\varepsilon}_2$ or $\tilde{\varepsilon}_3$ is non-zero --- up to numerical fluctuations.
Our standard choice for the non-vanishing $\tilde{\varepsilon}_n$ is that it yields $\varepsilon_n^{\bf x}\simeq 0.15$.
\begin{figure*}[!ht]
\includegraphics[width=0.495\linewidth]{figures/flow_time_scaling_v2.pdf}
\includegraphics[width=0.495\linewidth]{figures/flow_time_scaling_v3.pdf}\vspace{-3mm}
\caption{$v_2$ (left) and $v_3$ (right) as function of $t/R$ for different Knudsen numbers. The green shaded regions mark the interval for the upper end points of the fits with Eq.~\eqref{vn_fit-function}.}
\label{fig:loglog_flow}
\end{figure*}
An important feature is that the initial distribution~\eqref{InitialDistribution} is isotropic in momentum space.
This is a necessary ingredient for the analytical calculations and it is mostly fulfilled by the transport simulations, up to numerical fluctuations that we now discuss.
In the initial state of the transport simulation, $N_\mathrm{p}$ test particles are first sampled from the particle number density~\eqref{eq:initial_distribution}, and their momenta are sampled from the Boltzmann distribution~\eqref{eq:momentum_dist} with the appropriate local temperature.
Due to the finite number of particles, the resulting initial (transverse) momentum distribution, integrated over all positions, is not exactly isotropic, but shows small anisotropic flow coefficients of order $1/\sqrt{2N_\mathrm{p}}$~\cite{Voloshin:1994mz}.
Discarding the initial $v_1$ is easy, by subtracting $1/N_\mathrm{p}$ times the total momentum of the system from the momentum of every particle.\footnote{Strictly speaking, this slightly deforms the momentum distribution~\eqref{eq:momentum_dist}, but only minimally, as shown by the agreement with analytical calculations using Eq.~\eqref{eq:momentum_dist}.}
However, setting the other initial Fourier coefficients to zero is not so easy.
To circumvent the problem, we perform $N_\mathrm{iter.}$ iterations with exactly the same initial geometry, i.e.\ the positions of the $N_\mathrm{p}$ test particles are unchanged, but with different samplings of the momentum distribution.
The results of the $N_\mathrm{iter.}$ simulations with the same geometry are then averaged, reducing fluctuations in particular in the initial state by a factor $\sqrt{N_\mathrm{iter.}}$.
As stated in Sect.~\ref{subsec:numerical_simulations}, this is (at least to a good approximation) equivalent to performing a single simulation with $N_\mathrm{p}\cdot N_\mathrm{iter.}$ test particles, but computationally significantly cheaper.
In our simulations, the initial $v_n$ values after averaging over iterations is close to zero within the expected numerical uncertainty.
Accordingly we shall systematically shift our anisotropic-flow curves to start at 0, adding at $t=0$ an error bar indicating the numerical fluctuation $1/\sqrt{2N_{\rm p}N_{\rm iter.}}$.
We characterize the amount of rescatterings in the evolution via the Knudsen number $\mathrm{Kn}\equiv\ell_\mathrm{mfp}/R$, where the particle mean free path $\ell_\mathrm{mfp}$ is estimated at the center of the system --- where the particle density is largest --- in the initial state.
A small Knudsen number means a large mean number of rescatterings per particle, corresponding to the ``fluid dynamical limit'', while a large Kn means a system with few rescatterings: by construction, Kn is inversely proportional to the cross section $\sigma$.
\section{Results}
\label{sec:results}
In this section we present our results, starting in Sect.~\ref{subsec:flow_coefficients_Kn} with the change in the early-time scaling behavior of anisotropic flow as one goes from the few collisions regime to the hydrodynamic limit.
In the following subsections we focus on the few collisions regime and compare the numerical and analytical approaches for the early time behaviors of the number of rescatterings (Sect.~\ref{subsec:number_particles}), anisotropic flow (Sect.~\ref{subsec:flow_coefficients}), and eventually spatial eccentricities and other geometric quantities (Sec.~\ref{subsec:eccentricities}).
Eventually, we discuss alternative observables to quantify the momentum anisotropies in Sect.~\ref{subsec:alternative_flow_observables}, again across the whole range of Knudsen numbers and comparing with analytical calculations when Kn is large.
\subsection{Onset of anisotropic flow from small to large Knudsen number}
\label{subsec:flow_coefficients_Kn}
Let us first investigate how the early-time development of anisotropic flow --- here quantified via the Fourier harmonics~\eqref{vn_def} of the particle transverse-momentum distribution, as is most often done --- evolves across Knudsen numbers for a fixed initial geometry.
As is customary in transverse flow studies~\cite{Sorge:1996pc,Heinz:2001xi}, ``early times'' are to be understood in comparison to the typical transverse size of the system $R$, say roughly $t/R$ of order 0.1--0.5.
As is well known and will be shown again below, the value of $v_n$ at a fixed final time increases with the number of rescatterings in the system, when going from the free-streaming limit ${\rm Kn}\to\infty$ --- in which no anisotropic flow develops --- to the fluid-dynamical limit ${\rm Kn}\to 0$.
Throughout this paper we are not so much interested in that known behavior, but rather in the onset of $v_n(t)$, namely its dependence on $t$ at early times.
Let us quickly recall the findings in the literature:
In fluid dynamics, numerical simulations~\cite{Heinz:2002rs,Kolb:2002cq,Kolb:2003dz,Teaney:2010vd} or general scaling arguments~\cite{Vredevoogd:2008id} yield $v_2(t) \propto t^2$ and more generally $v_n(t) \propto t^n$ at early times.
In contrast, transport simulations~\cite{Gombeaud:2007ub,Alver:2010dn} or analytical studies~\cite{Heiselberg:1998es,Borghini:2010hy} in the few-rescatterings regime yield the slower growth $v_n(t) \propto t^{n+1}$.
Here we wish to bridge the gap between the two regimes, looking at both elliptic flow $v_2$ and triangular flow $v_3$.
In this subsection, we use the normal $2\to 2$ collision kernel in our simulations.
\begin{figure*}[!t]
\includegraphics[width=0.495\linewidth]{figures/v2_scaling_exponents.pdf}
\includegraphics[width=0.495\linewidth]{figures/v3_scaling_exponents.pdf}\vspace{-3mm}
\caption{Scaling exponent $\gamma_2$ (left) and $\gamma_3$ (right) as a function of the Knudsen number.
The gray lines indicate the 500 realizations of the fit over different time intervals, while the green points correspond to the weighted average~\eqref{eq:weighted_av}.}
\label{fig:gamma}
\end{figure*}
In Fig.~\ref{fig:loglog_flow} we display $v_2(t)$ and $v_3(t)$ at early times, namely for $t\leq 0.7R$ resp.\ $t\leq 0.9R$, where $R$ is the typical system size, for various Knudsen numbers ranging from ${\rm Kn} = 0.02$ to ${\rm Kn} \approx 10$.
The latter amounts to less than 0.1 rescatterings per particle on average, while in the former case the particles rescatter about 25 times on average, corresponding to the fluid-dynamical regime.
As expected, for larger Knudsen numbers the spatial asymmetry is less efficiently translated into a momentum anisotropy.
Note also that for a fixed $t$ and a given Kn the value of $v_3$ is significantly smaller than that of $v_2$, which is why we restrict ourselves to a smaller range in Kn.
Inspired by the known results, we fit the onset of those curves with a power-law ansatz
\begin{equation}
v_n\bigg(\frac{t}{R}\bigg) =
\beta_n\bigg(\frac{t}{R}\bigg)^{\!\!\gamma_n}.
\label{vn_fit-function}
\end{equation}
To account for the small initial flow value due to the finite test particle number, we shifted all curves to zero at $t=0$.
Since we are only interested in the scaling exponent $\gamma_n$ this shift does not influence our result, yet has the benefit that we do not have to introduce an extra offset parameter in the fit routine.
More physical information is contained in the parameters $\beta_n$ and $\gamma_n$ which are functions of the Knudsen number.
In fact, it is somewhat intuitive that $\beta_n$ should increase with decreasing Kn, which is what we indeed found.
A somewhat problematic issue is the size of the interval in $t/R$ over which we should perform the fit.
The lower end point of the fitting interval is always taken at $t=0$.
Regarding the upper end point of the interval, on the one hand it should not be too large to make sure that we are still fitting early-time behavior.
On the other hand, this final end point should not be too small either, otherwise the signal to be fitted is too small and may still be contaminated by numerical noise.
Accordingly, we defined a range of values, represented by the green bands in Fig.~\ref{fig:loglog_flow}, for the maximum time of the fitting interval: $0.3\leq t_{\max}/R\leq 0.55$ for $v_2$ resp.\ $0.4\leq t_{\max}/R\leq 0.65$ for $v_3$.
We randomly chose 500 values for $t_{\max}$,\footnote{We drew $t_{\max}$ from a uniform distribution over the respective range of values.} and used them to perform fits with the power law~\eqref{vn_fit-function} for all values of Kn.
In Fig.~\ref{fig:gamma} we show for $n=2$ and $n=3$ the exponents $\gamma_n$ of these fits, joining by gray lines the values obtained at different Knudsen numbers but with the same value of $t_{\max}$.
We then performed a weighted average of the exponents $\gamma_n$ over the 500 different realizations of the fit interval, giving less weight to the fits with larger uncertainties by using
\begin{equation}
\bar{\gamma}_n = \frac{\sum_j \gamma_{n,j}/\sigma^2_{\gamma_n,j}}{\sum_j 1/\sigma^2_{\gamma_n,j}},
\label{eq:weighted_av}
\end{equation}
where $\sigma^2_{\gamma_n,j}$ is the (squared) uncertainty on $\gamma_{n,j}$ given by the fitting routine for the $j$-th realization, while $j$ runs over the 500 realizations.
Performing the 500 realizations of the fit gives a better insight on the range of possible exponents at each Knudsen number and provides a very conservative error band for the uncertainty on the exponent, which is actually not symmetric about the average value~\eqref{eq:weighted_av}.
Even accounting for the uncertainty bands, $\gamma_2$ (top panel of Fig.~\ref{fig:gamma}) and $\gamma_3$ (bottom panel) show a clear trend, namely a crossover from the hydrodynamic ($\gamma_n = n$) to the few-collisions regime ($\gamma_n = n+1$) over some range in the Knudsen number.
Comparing the horizontal axes of the two plots, we see that the transition takes place over a much smaller Kn-range for $v_3$.
Moreover, the change in the early-time evolution of $v_3$ takes place for a typical Kn for which $v_2$ is already close to its fluid-dynamical behavior.
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/Vn_Kn_Fit_v2_v3.pdf}
\caption{$v_n/\varepsilon_n$ as a function of the Knudsen number and the mean number of rescatterings $N_\mathrm{resc.}$ per particle, where $v_n$ and $N_\mathrm{resc.}$ are estimated at time $t=100R$. The fits were performed with Eqs.~\eqref{eq:v2_Kn}--\eqref{eq:v3_Kn}. The colored bands show the $1\sigma$ fit error.}
\label{fig:vn_vs_Kn}
\end{figure}
The latter finding parallels the already known different behaviors of the ``final'' values of $v_2$ and $v_3$ --- that is, once $v_n(t)$ has reached its maximum value and evolves only little --- with varying Knudsen number:
starting from the free-streaming limit and decreasing Kn, the $v_2$ signal rises before that of $v_3$~\cite{Alver:2010dn,Roch:2020zdl}, as we also show in Fig.~\ref{fig:vn_vs_Kn} for the specific initial conditions used in this paper.
The fits to the points were performed with~\cite{Bhalerao:2005mm}
\begin{equation}
v_2=\frac{v_2^{\mathrm{hydro}}}{1+\mathrm{Kn}/\mathrm{Kn}_0}
\label{eq:v2_Kn}
\end{equation}
and~\cite{Alver:2010dn}
\begin{equation}
v_3=\frac{v_3^{\mathrm{hydro}}(1+B_3\mathrm{Kn})}{1+(A_3+B_3)\mathrm{Kn}+C_3\mathrm{Kn}^2},
\label{eq:v3_Kn}
\end{equation}
where the parameter $v_n^\mathrm{hydro}$ is the value in the ideal fluid-dynamical limit.
That $v_3$ departs from its few-rescatterings behavior at a smaller Kn than $v_2$ can readily be explained at least on the qualitative level:
At fixed system size $R$, a triangular structure, as will give rise to $v_3$, involves a smaller typical ``wavelength'' (in the azimuthal direction) than an elliptic shape.
To resolve this smaller wavelength, a smaller mean free path is needed, i.e.\ a smaller Knudsen number.\footnote{An alternative argument in the language of fluid dynamics was given in Ref.~\cite{Alver:2010dn}.}
As a final remark, note that fitting the slow early-time growth of triangular flow in the few-collisions regime, $v_3(t)\propto t^{\gamma_3}$ with $\gamma_3\approx 4$, appears to be at the limit of what we can do with reasonable uncertainties.
This is why we did not try to investigate the early time behavior of $v_4$, which should grow as $t^4$ in the hydrodynamic limit and slower when going to larger Knudsen numbers.
\subsection{Number of rescatterings}
\label{subsec:number_particles}
In the few-rescatterings regime we have expansion~\eqref{FullExpansion} at our disposal, with which one can compute the early-time behavior of observables.
Keeping only the first non-vanishing contribution from the Taylor expansion to a given observable, in particular $v_n(t)$, one can hardly expect to obtain a non-integer scaling exponent like the $\gamma_n$ found at ``intermediate'' Knudsen numbers in the previous subsection.
However, we can investigate whether summing the contributions from several terms in the Taylor expansion could lead to an effective behavior similar to that of the full transport simulations.
We begin with the number of rescatterings in the system, scaled by twice the number of particles $N_{\rm p}$ in the initial state, where we include the factor 2 since each collision involves two particles.
In analytical calculations, this number of rescatterings per particle $N_{\rm resc.}$ is simply one half of the integral over time of the collision rate, i.e.\ it can be deduced from the integral of the $2\to 0$ collision kernel~\eqref{C[f]_2->0} over phase space:
\begin{align}
N_{\rm resc.}(t) &= \frac{1}{2N_{\rm p}}\!\int_0^t \Gamma(t')\,{\rm d} t' \cr
&= \frac{1}{N_{\rm p}}\!\int_0^t \!\int\! -\mathcal{C}_{2\to 0}[f(t',{\bf x},{\bf p})]\,
\frac{{\rm d}^2{\bf x}\,{\rm d}^2{\bf p}}{E}\,{\rm d} t'.\qquad
\label{Nresc_analytical}
\end{align}
Note that this formula also holds for a system undergoing any kind of binary collisions between two identical particles, despite the apparently restrictive notation $2\to 0$.
Using expansion~\eqref{FullExpansion} one can compute the phase-space distribution at time $t'$ entering Eq.~\eqref{Nresc_analytical}.
As stated in Sect.~\ref{subsec:analytical_approach}, we were able to calculate $f(t,{\bf x},{\bf p})$ at early times only in the $2\to 0$ scenario, which is why we now consider also that model --- in which the total particle number is not conserved --- in the numerical simulations.
On the analytical side, we computed $N_{\rm resc.}$ up to order $t^{11}$, considering terms up to order $\sigma^5$ in the cross section.\footnote{Strictly speaking, at ${\cal O}(t^{11})$ one finds terms up to order $\sigma^{10}$, so our $N_{\rm resc.}(t)$ at order $t^{11}$ is incomplete. Every new order in $\sigma$ means a significant increase in the number of terms to be calculated, which is why we were slightly inconsistent here.}
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/Ncoll.pdf}
\caption{Number of rescatterings per particle for $\mathrm{Kn}\approx 25$ (full) and $\mathrm{Kn}\approx 5$ (dashed) for the $2\rightarrow 0$ collision kernel simulation (green). The analytical results --- computed up to ${\cal O}(t^{11})$ --- are shown at $\mathcal{O}(\sigma)$ (blue) and $\mathcal{O}(\sigma^5)$ (orange) for the corresponding Knudsen numbers.}
\label{fig:Nresc}
\end{figure}
The importance of going beyond linear order in $\sigma$ is visible in Fig.~\ref{fig:Nresc}.
There we show in green the results of simulations at a Knudsen number of order 25 (full line) at a ${\rm Kn}\approx 5$ (dashed line), where the latter are scaled by a factor $1/5$: if $N_{\rm resc.}(t)$ was simply proportional to $\sigma$, the two curves would coincide, which is clearly not the case.
We also display in blue the computed $N_{\rm resc.}(t)$ at order $\sigma$: the curves at ${\rm Kn}\approx 25$ and ${\rm Kn}\approx 5$ do coincide, as they should, and they are reasonably close to the value of numerical simulations at ${\rm Kn}\approx 25$ up to $t/R \approx 0.3$.
The agreement between analytical and numerical results is further increased when going up to order $\sigma^5$ (in orange) for both values of Kn we consider here.
Overall the number of rescatterings can thus be well described in the analytical approach.
This is a necessary ingredient in our study since other observables of more complicated nature, as e.g.\ $v_n(t)$, depend in a way or another on the number of rescatterings.
Note that ${\rm Kn}\approx 5$ means that about 16\% of the initially present particles disappear during the whole system evolution, most of them in the initial stage shown in Fig.~\ref{fig:Nresc}.
Accordingly, the results of $2\to 2$ and $2\to 0$ simulations differ significantly over that time interval, which is why it is necessary to compare the latter with our (equally $2\to 0$) analytical approach.
\subsection{Anisotropic flow coefficients}
\label{subsec:flow_coefficients}
Let us come back to our starting point, namely the behavior of anisotropic flow harmonics $v_n$ at early times, in a system without ``preflow'', i.e.\ $v_n(t=0) = 0$.
As in the case of the number of rescatterings and in contrast to Sect.~\ref{subsec:flow_coefficients_Kn}, here we compute the flow coefficients in the $2\to 0$ scenario, as was done in Refs.~\cite{Heiselberg:1998es,Borghini:2010hy}.
It turns out that at large Kn numerical simulations with the full $2\to 2$ kernel or the $2\to 0$ version yield very similar results for $v_2(t)$ as long as $t\lesssim R$.\footnote{The signal in the $2\to 0$ setup is slightly smaller than in the $2\to 2$ case, and the deviation between both models increases with decreasing Kn, which is due to the faster dilution in the $2\to 0$ system.
For a more in-depth study of the differences between the $2\to 2$ and $2\to 0$ collision kernels, including also the late time behavior of the flow coefficients, we refer to Ref.~\cite{Bachmann:2022cls}.}
Note that it is hardly possible to study the dependence of $v_n$ over the full Kn range as in Figs.~\ref{fig:gamma} und \ref{fig:vn_vs_Kn}, but only at large Kn, since ${\rm Kn} = 1$ means that already about 60--80\% of the initially present particles disappear.
In Fig.~\ref{fig:v2} we show our results for the development of elliptic flow $v_2(t)$ for two values of the Knudsen number.
The outcome of the simulations is shown in green, where the error bar at $t=0$ is the statistical error due to finite particle number in the initial state of the simulation, as described in Sec.~\ref{subsec:numerical_simulations}.
The blue resp.\ orange curve shows $v_2(t)$ as obtained within the analytical approach starting from the Taylor expansion~\eqref{FullExpansion}, including terms up to order $t^{15}$ but keeping only the linear order in $\sigma$ resp.\ terms up to order $\sigma^3$.
In Ref.~\cite{Borrell:2021cmh} we showed that the contributions at ${\cal O}(\sigma)$ show up at odd orders starting from $t^3$, those at ${\cal O}(\sigma^2)$ start at order $t^4$ and show up at even order in $t$, while ${\cal O}(\sigma^3)$-terms again show up in odd powers of $t$ from order $t^5$ on.
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/v2_Kn25.pdf}
\includegraphics[width=\columnwidth]{figures/v2_Kn5.pdf}
\caption{Elliptic flow $v_2$ as a function of $t/R$ for $\mathrm{Kn}\approx 25$ (top) and $\mathrm{Kn}\approx 5$ (bottom). Results from the numerical $2\to 0$ simulations are shown in green and the analytical results are shown to $\mathcal{O}(\sigma)$ (blue) and $\mathcal{O}(\sigma^3)$ (orange). Both analytical curves include terms up to $\mathcal{O}(t^{15})$.}
\label{fig:v2}
\end{figure}
For the calculations at the largest Knudsen number (upper panel of Fig.~\ref{fig:v2}) we find an nice agreement between the numerical and analytical results at order $\sigma$ up to $t/R\approx 0.4$.
The agreement extends even further at $\mathcal{O}(\sigma^3)$, where including these extra terms seems to slow down the growth of $v_2(t)$, such that at $t/R = 1$ the two curves differ by less than 10\% --- which is by far not obvious for a Taylor expansion in $t/R$.
For the five times smaller Knudsen number ${\rm Kn}=5$ (lower panel), the importance of including higher powers of $\sigma$ is even more striking, since it yields a pretty good agreement between the analytical calculations and the simulation up to $t/R\approx 0.6$.
Overall the inclusion of higher orders in the cross section has a major impact on the numerical comparison.
In particular, it is clear that considering enough powers in $t$ and in $\sigma$ allows to reproduce the behavior that was quantified by a Kn-dependent scaling exponent $\gamma_2$ in Sect.~\ref{subsec:flow_coefficients_Kn}.
Note that further orders in time will indeed improve the curve at late times but not visibly at early times and including those terms is extremely costly computationally.
Similarly higher powers in the cross section would guarantee a better result for smaller Knudsen numbers but we are again limited by the computational power.
Turning now to triangular flow $v_3$, one first finds~\cite{Bachmann:2022cls} that numerical simulations with the $2\to 0$ model differ more than those with the $2\to 2$ collision kernel, especially at larger times ($t\gtrsim R$).
Looking back at Fig.~\ref{fig:vn_vs_Kn}, we see that at ${\rm Kn} \approx 5$ or 25 the $v_3$ signal at the end of the evolution is already quite small in $2\to 2$ simulations, and thus it is even smaller in $2\to 0$ calculations stopped at $t=R$.
Accordingly, the noise from numerical fluctuations, in particular the small non-zero $v_3$ in the initial state, starts to play an important role --- especially in the simulations at ${\rm Kn}\approx 25$ which we present below.
As argued in Ref.~\cite{Borrell:2021cmh}, a non-vanishing $v_3$ can lead to a (small) linear rise of $v_3(t)$ at early times, which is indeed what we observe.
In addition, there is an unpleasant feature in the analytical calculations with the $2\to 0$ collision term, namely that they yield vanishing odd flow harmonics at all times, in particular $v_3$, if one considers only terms of $\mathcal{O}(\sigma)$~\cite{Bachmann:2022cls}.
On the other hand, an advantage of the Taylor series approach is that we can go to higher orders in $\sigma$, which we attempted here.
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/v3_Kn25.pdf}
\includegraphics[width=\columnwidth]{figures/v3_Kn5.pdf}
\caption{Triangular flow $v_3$ as a function of $t/R$ for $\mathrm{Kn}\approx 25$ (top) and $\mathrm{Kn}\approx 5$ (bottom). Results from the $2\to 0$ collision kernel are shown in green and the analytical results are shown to $\mathcal{O}(\sigma^2)$ (orange).}
\label{fig:v3}
\end{figure}
With these caveats in mind, we present our results for $v_3(t)$ in Fig.~\ref{fig:v3}, for the same values of the Knudsen number as $N_{\rm resc.}$ and $v_2$.
Here the analytical calculations include terms up to order $t^{11}$ and ${\cal O}(\sigma^2)$.
This means that the leading order at early times is $v_3(t)\propto \sigma^2 (t/R)^5$, i.e.\ a very slow growth.
To remain careful in our statements, let us just notice that the analytical and numerical signals seem to be of the same magnitude.
This came somewhat as a surprise to us, because it means that in a $2\to 0$ scenario $v_3$ grows as $\sigma^2$, not proportional to $\sigma$.
In contrast, the simulations with a $2\to 2$ collision kernel clearly give $v_3$ rather proportional to $\sigma$ --- see for instance Fig.~\ref{fig:vn_vs_Kn} and the successful fit with Eq.~\eqref{eq:v3_Kn} which gives $v_3\propto {\rm Kn}^{-1}$ at large Kn.
Nevertheless, this seems to be the message conveyed by our findings in Fig.~\ref{fig:v3}, together with the ``success'' of the analytical approach in describing the numerical results.
As we shall see in Sect.~\ref{subsec:alternative_flow_observables}, an alternative measure of triangular flow, namely weighted with the particle energy, gives rather different results.
\subsection{Spatial characteristics}
\label{subsec:eccentricities}
Let us now discuss the early-time dependence of various spatial characteristics, for instance the eccentricities $\varepsilon_n^{\bf x}$ or the mean square radius of the expanding system.
A significant difference with anisotropic flow is that most of these characteristics evolve in a free streaming system, in the absence of rescatterings.
Indeed, it was even found in Ref.~\cite{Borrell:2021cmh} that the change in the geometrical ``observables'' caused by particle rescatterings is at times subleading with respect to the free evolution.
Accordingly, we first look at collisionless systems, and check that we can describe them in our analytical approach, i.e.\ taking into account only the term $f_{\mathrm{f.s.}}$ in Eq.~\eqref{FullExpansion}, before we turn to systems with (few) collisions.
Throughout this subsection we come back to simulations with the $2\to 2$ collision kernel, which makes no difference in the free streaming case but is rather crucial when collisions are allowed.
An important class of geometrical quantities, which are correlated with anisotropic flow via rescatterings, are the spatial eccentricities.
Here we consider both eccentricities computed via Eq.~\eqref{eccentricity} with particle-number weighting
\begin{equation}
\label{eccentricity_N}
\varepsilon_n^{\bf x}(t)\equiv
-\frac{\displaystyle\int\!\! f(t,{\bf x},{\bf p})\,r^n{\rm e}^{{\rm i}n\theta}\,{\rm d}^2{\bf x}\,{\rm d}^2{\bf p}}{\displaystyle\int\!\! f(t,{\bf x},{\bf p})\,r^n\,{\rm d}^2{\bf x}\,{\rm d}^2{\bf p}},
\end{equation}
where we used the fact that for our special case of initial geometry~\eqref{eq:initial_distribution} the reaction-plane angle $\Phi_n$ is zero, and energy-weighted eccentricities $\varepsilon_n^{{\rm x},E}$
\begin{equation}
\label{eccentricity_E}
\varepsilon_n^{{\bf x},E}(t)\equiv
-\frac{\displaystyle\int\!\! E\, f(t,{\bf x},{\bf p})\,r^n{\rm e}^{{\rm i}n\theta}\,{\rm d}^2{\bf x}\,{\rm d}^2{\bf p}}{\displaystyle\int\!\! E\, f(t,{\bf x},{\bf p})\,r^n\,{\rm d}^2{\bf x}\,{\rm d}^2{\bf p}},
\end{equation}
with $E\equiv |{\bf p}|$ the energy of a particle with momentum ${\bf p}$.
The behavior of these eccentricities in a free streaming system is readily computed~\cite{Borrell:2021cmh}:
\begin{align}
\label{eq:en_fs}
\varepsilon_n(t) \simeq \frac{\varepsilon_n(0)}{1+a_n(t/R)^2},
\end{align}
where the approximate equality is actually exact in the case $n=2$, while for $n\neq 2$ it holds up to terms of order $(t/R)^4$ and possibly higher (if $n\neq 4$).
For the initial condition(s) we consider in this paper, see Sect.~\ref{subsec:distribution_function}, the coefficient $a_2$ equals $1/2$ if a particle number weight is used, i.e.\ for $\varepsilon_2^{\bf x}(t)$, while it is $3/4$ for $\varepsilon_2^{{\bf x},E}(t)$ with energy weight.
In the third harmonic, $a_3$ takes the respective values $3/4$ for $\varepsilon_3^{\bf x}(t)$ and $9/8$ for $\varepsilon_3^{{\bf x},E}(t)$.
Note that these numerical values are valid only for the initial distribution~\eqref{InitialDistribution}--\eqref{eq:initial_distribution}, while Eq.~\eqref{eq:en_fs} holds for any initial distribution in the freely streaming case.
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/en_free_stream_particle_weight.pdf}
\includegraphics[width=\columnwidth]{figures/en_free_stream_energy_weight.pdf}
\caption{Time evolution of the spatial eccentricities in a collisionless system. The top panel shows particle-number weighted eccentricities $\varepsilon_2^{\bf x}$, $\varepsilon_3^{\bf x}$, the bottom panel eccentricities $\varepsilon_2^{{\bf x},E}$, $\varepsilon_3^{{\bf x},E}$ with energy weighting. The dotted curves are fits with Eq.~\eqref{eq:en_fs} and the parameters given in Table~\ref{tab:en_fs_fit}.}
\label{fig:en_fs}
\end{figure}
For our numerical calculations, we start with an input value of $\varepsilon_n^{\bf x}=0.15$ in the initial state.
Due to the finite number of particles, which is $N_\mathrm{p} = 2\times 10^5$ in this subsection, this value is not reached exactly in the simulations.\footnote{To minimize fluctuations, we have fixed the initial positions for $N_{\rm iter.} = 3.2\times 10^4$ iterations and resampled the momenta only.}
In addition, the energy-weighted initial eccentricities $\varepsilon_2^{{\bf x},E}$ and $\varepsilon_3^{{\bf x},E}$ can differ from each other, since the momentum (and thus energy) distribution depends on the position.
Figure~\ref{fig:en_fs} shows the time dependence of the eccentricities in free-streaming.
The full curves showing the free-streaming evolution can be nicely fitted with Eq.~\eqref{eq:en_fs} resulting in the fit parameters of Tab.~\ref{tab:en_fs_fit}, where we fitted the numerical simulations up to $t/R=0.5$.
\begin{table}[!t]
\centering
\caption{\label{tab:en_fs_fit}Parameters of the fit function Eq.~\eqref{eq:en_fs} for $\varepsilon_n$ with particle-number and energy weight.}
\medskip
\begin{tabular}{ccccc}
\hline\hline
weight & $\varepsilon_2(0)$ & $a_2$ & $\varepsilon_3(0)$ & $a_3$ \\
\hline
particle number & $0.1496$ & $0.5005$ & $0.1499$ & $0.7626$ \\
energy & $0.250$ & $0.745$ & $0.246$ & $1.141$ \\
\hline\hline
\end{tabular}
\end{table}
For particle-number and energy weighting, and for both $n=2$ and $n=3$, the fitted $\varepsilon_n(0)$ coincide with the initial values computed directly at the beginning of the simulations.
In turn, the fitted values of $a_n$ are in good agreement with the analytical prediction: as could be anticipated, the agreement is better for $n=2$, in which case Eq.~\eqref{eq:en_fs} is exact, than for $n=3$, in which case the denominator contains higher (even) powers of $t$.
Note that the energy-weighted eccentricities $\varepsilon_n^{{\bf x},E}$ decrease slightly faster than their particle-number weighted counterparts, which is reflected in the larger values of $a_n$.
In the free-streaming case there is no transfer of spatial asymmetry into a momentum anisotropy and thus no buildup of anisotropic flow.
That is the reason why the eccentricity decreases most slowly in the collisionless case: introducing rescatterings in the system will lead to a faster decrease.
With collisions it is even possible to change the sign of the eccentricity --- as observed in fluid-dynamical simulations~\cite{Kolb:2002cq}, while for the free-streaming case this does not happen.
However, at large Knudsen numbers (${\rm Kn}\approx 25$ or 5) the change in the evolution of eccentricities due to rescatterings is barely visible (see Fig.~\ref{fig:en_collisions}), even at large times.
For reference, we also include the behavior of the eccentricities for a system close to the fluid-dynamical regime, namely at ${\rm Kn}\approx 0.1$.
One sees that sizable deviations from the collisionless behavior only appear for $t\gtrsim R$.
This is consistent with the fact that it takes about a similar time for anisotropic flow to acquire a sizable value across the whole system, and thus to affect the evolution of geometry.
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/en_particle_weight.pdf}
\includegraphics[width=\columnwidth]{figures/en_energy_weight.pdf}
\caption{Time evolution of the spatial eccentricities in a collisionless (full lines) and in an interacting system with few rescatterings (dotted and dashed lines). The top panel shows particle-number weighted eccentricities $\varepsilon_2^{\bf x}$, $\varepsilon_3^{\bf x}$, the bottom panel eccentricities $\varepsilon_2^{{\bf x},E}$, $\varepsilon_3^{{\bf x},E}$ with energy weighting.
For comparison, we also show the behavior in a system in the fluid-dynamical regime (dotted-dashed lines).}
\label{fig:en_collisions}
\end{figure}
To pinpoint the influence of rescatterings, let us investigate the behaviors of the numerator and denominator of the eccentricity separately.
Here we only discuss results relevant for $\varepsilon_2^{\bf x}$ and $\varepsilon_2^{{\bf x},E}$, since the behaviors in the third harmonic (shown in Appendix~\ref{appendix:epsilon3}) are similar.
To see the effect of collisions more easily, we will subtract the free-streaming behaviors.
Let us start with the numerator of the eccentricity, i.e.\ (minus) the average value of $r^2\cos(2\theta)$, where the averaging can be done with particle-number or energy weighting.
This is actually the simpler case, since in the free streaming case --- and in the absence of initial anisotropic flow, as we assume throughout this subsection --- that quantity remains constant over time: any evolution is thus due to rescatterings.
We show this departure from the free-streaming value in Fig.~\ref{fig:rsquaredcos2theta} for simulations with Knudsen number ${\rm Kn} \approx 25$ (top panel) and ${\rm Kn} \approx 5$ (bottom panel).
To obtain dimensionless quantities, we scaled the curves by the initial value of the corresponding average of $r^2\cos(2\theta)$: each curve thus represents the relative change in the average value due to rescatterings.
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/rsquaredcos2theta_fit_Kn25.pdf}
\includegraphics[width=\columnwidth]{figures/rsquaredcos2theta_fit_Kn5.pdf}
\caption{Time evolution of the departure of $-\langle r^2\cos(2\theta)\rangle_{{\bf x},{\bf p}}$ with particle-number (green) and energy (blue) weighting from its free streaming value, for simulations with $\mathrm{Kn}\approx 25$ (top panel) or $\mathrm{Kn}\approx 5$ (bottom panel).}
\label{fig:rsquaredcos2theta}
\end{figure}
One sees that this relative change is of order a few $10^{-4}$ for the system with ${\rm Kn} \approx 25$, and of order $10^{-3}$ when ${\rm Kn} \approx 5$.
The latter signal is thus about 5 times larger than the former, i.e.\ seems to scale with ${\rm Kn}^{-1}$ or equivalently $\sigma$ or the number of rescatterings in the system.
To quantify the behaviors displayed in Fig.~\ref{fig:rsquaredcos2theta}, we tried to fit the curves with a simple power-law ansatz
\begin{equation}
-\!\langle r^2\cos(2\theta)\rangle_{{\bf x},\bf p}(t) + \langle r^2\cos(2\theta)\rangle_{{\bf x},\bf p}^{\rm f.s.} \propto \left(\frac{t}{R}\right)^{\!\!\chi}.
\label{eq:r2cos2thetafit}
\end{equation}
As in Sect.~\ref{subsec:flow_coefficients_Kn}, we perform the fit over a time interval from $t=0$ to some final $t/R$ in the range 0.35--0.7, letting the end point of the interval vary.
The fit results for the exponents lie in the range $\chi = 3.5$--5.5.
This is a rather slow growth, and we know from our fitting of the early-time behavior of $v_3(t)$ in Sect.~\ref{subsec:flow_coefficients_Kn} that it probably points at an exponent $\chi\geq 4$, but we do not want to make any strong claim.
Indeed, the signal of Fig.~\ref{fig:rsquaredcos2theta} is one to two orders of magnitude smaller than that of $v_3(t)$.
With our analytical approach, one finds invoking general principles --- namely particle-number or energy conservation, depending on the weight of the average --- that the contributions from the Taylor expansion~\eqref{FullExpansion} at order $t^2$ and below vanish, leaving $\chi\geq 3$~\cite{Borrell:2021cmh}.
We can further show that the contributions from the loss term of the $2\to 2$ collision kernel vanish at any order, invoking parity arguments in momentum space.
Unfortunately, we were unable to compute the contribution from the gain term of the collision integral, so we cannot give a more accurate prediction than the inequality $\chi\geq 3$, in agreement with the fit values that we find.
Turning now to the denominator of $\varepsilon_2^{\bf x}$, i.e.\ the mean square radius, it differs from the numerator in that it is evolving in a collisionless system:
\begin{equation}
\langle r^2\rangle_{{\bf x},{\bf p}}^{\rm f.s.}(t) =
\langle r^2\rangle_{{\bf x},{\bf p}}(0) +
\langle v^2\rangle_{{\bf x},{\bf p}}(0)\,t^2,
\end{equation}
where $v^2$ denotes the squared (transverse) velocity of the particles --- here simply equal to the squared velocity of light since we consider massless particles.
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/rsquared_fit_Kn25.pdf}
\includegraphics[width=\columnwidth]{figures/rsquared_fit_Kn5.pdf}
\caption{Time evolution of the departure of $\langle r^2\rangle_{{\bf x},{\bf p}}$ with particle-number (green) or energy (blue) weighting from its free streaming behavior, for simulations with $\mathrm{Kn}\approx 25$ (top panel) or $\mathrm{Kn}\approx 5$ (bottom panel).}
\label{fig:rsquared}
\end{figure}
We show in Fig.~\ref{fig:rsquared} the departure $\langle r^2\rangle_{{\bf x},{\bf p}}(t)-\langle r^2\rangle_{{\bf x},{\bf p}}^{\rm f.s.}(t)$ of the mean square radius in a interacting system from this free-streaming evolution, scaled by the initial value of $\langle r^2\rangle_{{\bf x},{\bf p}}$.
The departure is negative, which means that rescatterings slow down the system expansion:
this seems intuitive, since collisions cannot accelerate the outwards motion of particles already traveling at the speed of light.
As in the case of $\langle r^2\cos(2\theta)\rangle_{{\bf x},{\bf p}}$, one sees that the relative deviation of the mean square radius computed with particle weight from its free-streaming behavior (green curves) changes by about a factor of 5 from ${\rm Kn}\approx 25$ to ${\rm Kn}\approx 5$, i.e.\ it seems to scale linearly with the number of rescatterings.
A power-law fit of these curves
\begin{align}
\langle r^2\rangle_{{\bf x},{\bf p}} - \langle r^2\rangle_{{\bf x},{\bf p}}^\mathrm{f.s.} \propto
\left(\frac{t}{R}\right)^{\!\!\chi^\prime}
\label{eq:r2fit}
\end{align}
yield exponents in the range $\chi'=4.1$--4.5, depending on the end point of the time interval chosen for the fit.
In the analytical approach we found $\chi'\geq 3$~\cite{Borrell:2021cmh}, consistent with the numerical finding.
Quite strikingly, the blue curves in Fig.~\ref{fig:rsquared} representing the energy-weighted mean square radius stay constant.
That is, rescatterings in the system do not modify the evolution of the energy-weighted $\langle r^2\rangle_{{\bf x},{\bf p}}$, i.e.\ of the denominator of $\varepsilon_2^{{\bf x},E}(t)$.
We detail in Appendix~\ref{appendix:<Er^2>} how this can actually be shown within our analytical approach~\cite{Borrell:2021cmh}, mostly invoking energy and momentum conservation --- one has however to beware that it holds only for a two-dimensional system of massless particles.
This absence of departure from the free-streaming behavior is specific to the mean square radius: as shown in Fig.~\ref{fig:rcubed} in Appendix~\ref{appendix:epsilon3}, the evolution of the energy-weighted average of the cubed radius does depend on the amount of rescatterings in the system.
Yet the qualitative differences between particle-number weighted and energy-weighted quantities like the moments $\langle r^2\rangle_{{\bf x},{\bf p}}$, $\langle r^3\rangle_{{\bf x},{\bf p}}$ (Figs.~\ref{fig:rsquared} and \ref{fig:rcubed}) show that the transport of energy density in the system does not match one-to-one that of particle-number density.
\subsection{Alternative measures of anisotropic flow}
\label{subsec:alternative_flow_observables}
Motivated by the observation at the end of the previous subsections, we now look at alternative measures of anisotropic collective flow.
We shall compute the latter in both numerical simulations and analytical calculations and check if their early-time behaviors agree with the results obtained for $v_n(t)$ in Sect.~\ref{subsec:flow_coefficients_Kn} and \ref{subsec:flow_coefficients}.
One possibility, which was adopted in a number of recent kinetic theory studies~\cite{Kurkela:2018ygx,Kurkela:2019kip,Kurkela:2020wwb,Kurkela:2021ctp,Ambrus:2021fej}, is to consider the Fourier harmonics of the transverse energy distribution.
Using the terminology of Sect.~\ref{subsec:eccentricities}, these are effectively ``energy-weighted anisotropic flow coefficients''
\begin{equation}
\label{v_n^E_def}
v_n^{E\,}(t) \equiv
\frac{\displaystyle\int\!E_{}f(t,{\bf x},{\bf p})\,\cos[n(\phi_{\bf p}-\Psi_n^E)]\,{\rm d}^2{\bf x}\,{\rm d}^2{\bf p}}%
{\displaystyle\int\!E_{}f(t,{\bf x},{\bf p})\,{\rm d}^2{\bf x}\,{\rm d}^2{\bf p}},
\end{equation}
with $\Psi_n^E$ the $n$-th harmonic event plane.
In contrast, the traditional coefficients~\eqref{vn_def} are ``particle-number weighted'' flow harmonics.
In the following $\Psi_n^E=0$, consistent with the orientation of the initial-state symmetry planes $\Phi_n=0$.
For massless particles, as we consider throughout the paper, with $p_z=0$, the energy-weighted elliptic flow $v_2^E$ actually coincides with a definition that has been used in fluid-dynamical simulations~\cite{Ollitrault:1992bk,Kolb:2000sd},%
\footnote{Note that there is a misprint in the definition of $\varepsilon_2^{\bf p}$ in Eq.~(3.2) of Ref.~\cite{Kolb:2000sd}, as mentioned in Ref.~\cite{Teaney:2009qa}.}
namely (in our two-dimensional setup)
\begin{align}
\varepsilon_2^{\bf p}\equiv\frac{\displaystyle\int\!\big[T^{xx}({\bf x})-T^{yy}({\bf x})\big]_{}{\rm d}^2{\bf x}}%
{\displaystyle\int\!\big[T^{xx}({\bf x})+T^{yy}({\bf x})\big]_{}{\rm d}^2{\bf x}}.
\label{eq:e2p}
\end{align}
Note that the notation emphasizes the similarity to the geometrical eccentricities.
Definition~\eqref{eq:e2p} involves two diagonal components $T^{xx}$ and $T^{yy}$ of the energy-momentum tensor
\begin{align}
T^{\mu\nu}({\bf x}) \equiv
\int\! p^\mu p^\nu f(t,{\bf x},{\bf p})\;\frac{{\rm d}^2{\bf p}}{E}.
\label{eq:Tmunu}
\end{align}
An advantage of Eq.~\eqref{eq:e2p} is that it can also be implemented for a system that is not described as a collection of particles or a particle density distribution, but rather in terms of its energy momentum tensor.
In particular, there is no need to ``particlize'' the system using a Cooper--Frye-like approach.
As in Sect.~\ref{subsec:flow_coefficients_Kn}, let us first look at the behavior of $\varepsilon_2^{\bf p}$ across a large range of Knudsen numbers.
The results of simulations with the $2\to 2$ collision kernel from the few-rescatterings regime (${\rm Kn}\approx 25$) to the fluid-dynamical limit (${\rm Kn}\approx 0.02$) for the onset of $\varepsilon_2^{\bf p}(t) = v_2^E(t)$ is displayed in the top panel of Fig.~\ref{fig:eta_2}.
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/eps2p_scaling.pdf}
\includegraphics[width=\columnwidth]{figures/eps2p_scaling_exponents.pdf}
\caption{Upper panel: Early time dependence of $\varepsilon_2^{\bf p} = v_2^E$ for various Knudsen numbers. Lower panel: Dependence of the scaling exponent $\gamma'_2$ [Eq.~\eqref{eq:eps2p_time}] on the Knudsen number, using the same color code as in Fig.~\ref{fig:gamma}.}
\label{fig:eta_2}
\end{figure}
Similar to $v_2(t)$, we fit the early time development with a power law ansatz
\begin{equation}
\varepsilon_2^{\bf p}\bigg(\frac{t}{R}\bigg) =
\beta'_2\bigg(\frac{t}{R}\bigg)^{\!\!\gamma'_2},
\label{eq:eps2p_time}
\end{equation}
performing 500 different realizations of the fit over varying time intervals.
The change with Kn of the resulting scaling exponents $\gamma'_2$ is shown in the bottom panel of Fig.~\ref{fig:eta_2}.
The dependence of $\gamma'_2$ on the Knudsen number closely parallels that of $\gamma_2$ (Fig.~\ref{fig:gamma}), seemingly ranging from 2 in the fluid-dynamical limit --- as found in earlier hydrodynamical simulations --- to roughly 3 in the few-rescatterings regime.
That is, $\varepsilon_2^{\bf p}(t)$ behaves as $v_2(t)$ at early times
Using the same arguments as in Ref.~\cite{Borrell:2021cmh} (especially Sect.~IV.1), one finds that in the absence of initial flow, the leading contribution to $\varepsilon_2^{\bf p}(t)$ at early times is
\begin{equation}
\label{eps2p(t)_early-t}
\varepsilon_2^{\bf p}(t) \propto \sigma t^3 + \mathcal{O}(t^4),
\end{equation}
where the term at order $t^4$ is actually of order $\sigma^2$.
This is similar to the scaling of $v_2(t)$ and is shown as dotted line in the lower panel of Fig.~\ref{fig:eta_2}.
We computed the proportionality coefficient --- as well as terms up to ${\cal O}(t^9)$ at linear order in $\sigma$ --- for the $2\to 0$ collision kernel.
This is compared to results from numerical simulations, also with the $2\to 0$ kernel, in Fig.~\ref{fig:eps2p_Kn25_NumAnalyt}.
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/eps2p_Kn25.pdf}
\caption{Early time evolution of $\varepsilon_2^{\bf p} = v_2^E$ at $\mathrm{Kn}\approx 25$. The results of the $2\to 0$ simulation are shown in green, the analytical calculation at ${\cal O}(\sigma)$ in blue.}
\label{fig:eps2p_Kn25_NumAnalyt}
\end{figure}
The results from both approaches are in good agreement until $t/R\simeq 0.4$, and we know from our investigations of $v_2(t)$ that we can improve the agreement by including higher orders in $\sigma$ and $t$ in the (already extensive) analytical calculations.
Setting $n=3$ in Eq.~\eqref{v_n^E_def} gives the energy-weighted triangular flow $v_3^E$.
\begin{figure}[!htb]
\centering
\includegraphics[width=\columnwidth]{figures/v3_Kn25_energy_weight.pdf}
\includegraphics[width=\columnwidth]{figures/v3_Kn5_energy_weight.pdf}
\caption{\label{fig:v3_energy}Early time development of energy-weighted triangular flow $v_3^E$ for $\mathrm{Kn}\approx 25$ (top panel) and $\mathrm{Kn}\approx 5$ (bottom panel) from numerical simulations with the $2\to 0$ collision kernel (green) and analytical calculations at $\mathcal{O}(\sigma)$ (blue).}
\end{figure}
We show in Fig.~\ref{fig:v3_energy} results of simulations (green curves) with the $2\to 0$ collision kernel at both $\mathrm{Kn}\approx 25$ (top panel) and $\mathrm{Kn}\approx 5$ (bottom panel).
Comparing with the particle-number weighted $v_3(t)$ in Fig.~\ref{fig:v3}, the $v_3^E(t)$ signal is significantly larger, and in particular less sensitive to noise in the initial state.
Figure~\ref{fig:v3_energy} also shows the results of analytical calculations (blue curve), where in contrast to $v_3$ we already find a non-vanishing $v_3^E$ at $\mathcal{O}(\sigma)$ and including terms up to order $t^{10}$ --- while in the same $2\to 0$ scenario $v_3$ was found to scale as ${\cal O}(\sigma^2)$.
The analytical results and those of simulations are actually in quite good agreement for both Knudsen numbers being considered, especially given the relative size of the initial fluctuations of the numerical signal, which yield a visible linear rise at small $t/R$.
Other differences like the different curvature as $t/R\approx 1$ could certainly be improved by including more orders in $\sigma$ and in $t/R$ in the analytical approach.
As mentioned above, a drawback of the coefficients $v_n$ or $v_n^E$ from the theorist's point of view is that their definition assumes a particle-based model.
For massless particles with $p_z=0$, $v_2^E$ coincides with the ``momentum space eccentricity'' $\varepsilon_2^{\bf p}$, Eq.~\eqref{eq:e2p}, defined in terms of the energy momentum tensor, but the latter has no straightforward generalization to higher harmonics.
A possible measure of anisotropic flow at the ``macroscopic'' level --- i.e.\ a priori usable for any microscopic model --- was proposed in Ref.~\cite{Teaney:2010vd} for hydrodynamical calculations, which has the advantage of being generalizable for any harmonic.
The simplest example is the measure of elliptic flow, via\footnote{We use a different notation from the original one~\cite{Teaney:2010vd} and adapt it to our 2-dimensional setup.}
\begin{align}
\alpha_2^{\bf p} \equiv \frac{\displaystyle\int\!\big[T^{0x}({\bf x})_{}u_x({\bf x}) - T^{0y}({\bf x})_{}u_y({\bf x})\big]\,{\rm d}^2{\bf x}}%
{\displaystyle\int\!T^{00}({\bf x})_{}u_0({\bf x})\,{\rm d}^2{\bf x}},
\label{eq:alpha2p}
\end{align}
where $u^\mu({\bf x})$ is the system flow velocity.
For a collection of particles of a single species as considered in the present paper, the natural choice for that velocity is
\begin{equation}
u^\mu({\bf x}) = \frac{N^{\mu}({\bf x})}{\sqrt{N^{\mu}({\bf x})N_{\mu}({\bf x})}},
\label{eq:umu}
\end{equation}
with
\begin{equation}
N^\mu({\bf x}) \equiv \int\! p^\mu f(t,{\bf x},{\bf p})\;\frac{{\rm d}^2{\bf p}}{E}.
\label{eq:Numu}
\end{equation}
In analytical calculations, implementing the above equations is straightforward.
The numerical implementation of $\alpha_2^{\bf p}$ in the transport algorithm is more challenging than in a hydrodynamic simulation, as we do not have the energy-momentum tensor directly given at each point in space and time.
To calculate the necessary quantities, we divided the system of test particles into $N_{\rm cells} = 40^2$ cells on a rectangular grid.%
\footnote{We have checked that our result is independent of the number of cells chosen in the computation by varying the number of cells from $N_{\rm cells} = 20^2$ to $N_{\rm cells} = 120^2$ and fixing $N_{\rm p} = 5\times 10^5$ particles.
Fluctuations in the signal arise above $N_{\rm cells} = 60^2$, as the number of particles in the individual cells is too small.}
We then computed the sum of the momentum components $p_x$ resp.\ $p_y$ of the particles in each cell, which we denote by ${\sf P}_x$ resp.\ ${\sf P}_y$.
For each cell, we also computed the total energy $E_{\rm tot.}$, the number of particles, and the mean velocity per particle $\bar{\sf v}_x\equiv {\sf P}_x/E_{\rm tot.}$ and $\bar{\sf v}_y\equiv {\sf P}_y/E_{\rm tot.}$.
Then we can approximate the numerator of Eq.~\eqref{eq:alpha2p} by
\begin{align}
\big\langle T^{0x}u_x-T^{0y}u_y \big\rangle_{\textbf{x}} \approx
\frac{1}{N_{\rm cells}}\sum_{i=1}^{N_{\rm cells}}
\frac{{\sf P}_{x,i} \bar{\sf v}_{x,i}-{\sf P}_{y,i} \bar{\sf v}_{y,i}}{\sqrt{1-\bar{\sf v}_{x,i}^2-\bar{\sf v}_{y,i}^2}}
\end{align}
and the denominator by
\begin{align}
\big\langle T^{00}u_0 \big\rangle_{\textbf{x}} \approx
\frac{1}{N_{\rm cells}}\sum_{i=1}^{N_{\rm cells}}
\frac{E_{{\rm tot.},i}}{\sqrt{1-\bar{\sf v}_{x,i}^2-\bar{\sf v}_{y,i}^2}},
\end{align}
where the term in the denominators is due to the normalization of the velocity.
In the analytical approach starting from the Taylor expansion~\eqref{FullExpansion}, a necessary ingredient is the calculation in the free-streaming case.
It turns out that in a collisionless system $\alpha_2^{\bf p}$ is not constant, but grows quadratically with time:
\begin{align}
\alpha_2^{\bf p}(t) \propto t^2 + \mathcal{O}(t^3),
\end{align}
where the proportionality factor of the term in $t^2$ involves the initial spatial eccentricity $\varepsilon_2^{\bf x}$.
This non-constant behavior is somewhat unexpected, since no anisotropic (here elliptic) flow in the usual acceptation of the term develops in a free streaming system.
In turn, collisions modify this behavior at order $t^3$, consistent with the onset of $v_2(t)$ or $\varepsilon_2^{\bf p}(t)$, yet subleading compared to the collisionless evolution.
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{figures/alpha2p_fs.pdf}
\caption{Time dependence of $\alpha_2^{\bf p}$ for simulations (full lines) in a collisionless (green) system or in the few collisions regime (orange), together with the analytical result in the free streaming case (dotted blue line).}
\label{fig:alpha2p_fs}
\end{figure}
In Fig.~\ref{fig:alpha2p_fs} we show the early-time behavior of $\alpha_2^{\bf p}$ for a transport simulation with a moderate number of rescatterings (full orange line) with the $2\to 2$ collision kernel and in the free-streaming case (full green curve), together with the analytical calculation for the collisionless system (dotted blue curve).
As anticipated, rescatterings only lead to a sizable departure from the free streaming behavior when $t/R\approx 0.8$, but their contribution is subleading at early times.
\section{Discussion}
\label{sec:discussion}
In this paper, we have studied the early time development of several quantities describing the asymmetry in the transverse plane, either in position or in momentum space, of a system of massless particles.
For that purpose, we used simulations from a numerical transport code and semi-analytical calculations based on a recently introduced~\cite{Borrell:2021cmh} Taylor expansion of the single-particle phase space density at early times.
Results from both approaches were compared in the few-rescatterings regime where the application of the analytical model makes sense.
We also performed numerical simulations over a large range of Knudsen numbers, to study the early-time behavior of anisotropic flow coefficients from the few-rescatterings regime to the fluid-dynamical limit.
In a system with few rescatterings, we found that the Taylor-expansion based ansatz can indeed capture the numerical findings at early times, namely the scaling behavior $v_n(t) \propto t^{n+1}$ --- which also holds for the energy-weighted anisotropic flow coefficients $v_n^E(t)$ ---, the change in time of the number of rescatterings in the system, or the slow departure of geometrical characteristics like the spatial eccentricities $\varepsilon_n^{\bf x}(t)$ from their free-streaming evolution.
We also showed that including higher-order terms in $t$ and/or in the cross section $\sigma$ improves the quality of the analytical predictions --- say typically until $t/R\approx 0.5$.
However, we should quickly add that any new order in $t$ or in $\sigma$ typically means the appearance of many more terms in the calculations.
It would thus be highly desirable to find resummation schemes if one wishes to push the analytical expansion to larger times.\footnote{For $v_2(t)$ in the $2\to 0$ scenario and with our specific case of initial condition, one can find an analytical expression resumming all order in $t$ at ${\cal O}(\sigma)$~\cite{Bachmann:2022cls}, and it seems feasible also at ${\cal O}(\sigma^2)$, although we did not attempt it.}
Letting the number of rescatterings per particle grow, so that we can also reproduce the hydrodynamic limit, we found that the early-time scaling behavior of $v_n(t)$ changes smoothly from $t^{n+1}$ at large Knudsen number to $t^n$ when Kn is small.
We quantified this change via a non-integer scaling exponent $v_n(t)\propto t^{\gamma_n}$, which we cannot find directly in the Taylor-series approach.
It could be that a double resummation in $t$ and $\sigma$ can indeed provide such a non-analytic behavior, but it could also be an artifact from our fitting ansatz and procedure.
In any case, we view $\gamma_n$ as an effective exponent characterizing the behavior of $v_n(t)$ on times of order 0.1--0.5$R$.
On such a time scale set by the transverse system size, the fluid-dynamical scaling behavior appears as the many-collision limit of kinetic theory.
This convergence in studies of anisotropic flow was found previously (for the $v_n$ values at the end of the system evolution) in Refs.~\cite{Gombeaud:2007ub,Drescher:2007cd}, which use a similar two-dimensional setup as ours, or Ref.~\cite{Ambrus:2021fej} in the case of a longitudinally boost-invariant expansion.\footnote{There the authors emphasize that their kinetic transport calculations in the relaxation time approximation (RTA) only converge towards the hydrodynamic behavior if they consider the large-opacity limit at fixed initialization time $\tau_0$ of the system.}
Instead of defining early times in comparison to $R$ as we did, one could replace the latter by the mean free path $\ell_\mathrm{mfp} = {\rm Kn}\cdot R$.
In that case, we would expect that for times smaller than say about $0.3\ell_\mathrm{mfp}$, kinetic theory yields the behavior $v_n(t)\propto t^{n+1}$ of the few-collisions regime~\cite{Borrell:2021cmh}, even for values of Kn ``in the fluid-dynamical limit'', that is much smaller than 1.
Unfortunately, investigating this ``very early'' behavior in a setup with ${\rm Kn}\ll 1$ is unfeasible with our cascade code if we wish to remain in the dilute regime.
Yet with this alternative definition of ``early times'', depending --- at fixed initial geometry --- on the interaction strength, we would probably find different scaling laws for the flow coefficients according as the system is described by kinetic theory (irrespective of the Knudsen number) or fluid dynamics.
This would be similar to the finding that kinetic theory in the RTA and hydrodynamics have different early-time attractors for the ratio of longitudinal pressure over energy density~\cite{Kurkela:2019set}.
While we cannot claim that our findings are of immediate relevance for heavy-ion phenomenology, although this is not excluded for the case of small systems, yet we think that they may still be of interest, in particular for studies of the initial stages~\cite{Schlichting:2019abc}.
Whatever dynamical model is used for the prehydrodynamic evolution, some amount of anisotropic flow will develop --- unless of course it is assumed that the system is freely streaming ---, and it is not uninteresting to know whether the model is ``hydro-like'' or rather ``few-collisions-like'', or somewhere inbetween.
One could even assume as initial-stage model a set of simple scaling laws like $v_n(t)\propto t^{\gamma_n}$ and similar for other properties of the system, and see how this affects global Bayesian reconstructions of the properties of the medium created in heavy ion collisions.
More on the theoretical side, our results are relevant for searches for the existence of dynamical attractor solutions for systems with transverse dynamics~\cite{Ambrus:2021sjg}: even ``late time attractors'' have to accommodate qualitatively different ``early time'' behaviors.
In that respect, it would be interesting to extend our study to spatially asymmetric setups for which analytical results in the hydrodynamic limit are known~\cite{Csanad:2014dpa}.
\begin{acknowledgments}
We thank Nina Feld, Sören Schlichting and Clemens Werthmann for discussions, as well as M\'at\'e Csan\'ad and Tam\'as Cs\"org\H o for helpful comments on a preliminary version of our results.
The authors acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the CRC-TR 211 'Strong-interaction matter under extreme conditions' - project number 315477589 - TRR 211.
Numerical simulations presented in this work were performed at the Paderborn Center for Parallel Computing (PC$^2$) and the Bielefeld GPU Cluster, and we gratefully acknowledge their support.
\end{acknowledgments}
|
1,941,325,220,535 | arxiv | \section{Introduction} \label{sec:intro}
Detailed observation of molecular clouds in the Milky Way \citep{2010ApJ...723.1019H, 2010ApJ...724..687L, 2012ApJ...745..190L,2014ApJ...782..114E,2016ApJ...831...73V} and studies of external galaxies \citep{2004ApJ...606..271G} have shown that star formation is more accurately predicted by the amount of dense molecular gas rather than by the total amount of molecular gas. The molecules with high dipole moment and high critical density, such as \mbox{{\rm HCN}}, \mbox{{\rm HCO}$^+$}\ are used as dense gas tracers. The pioneering studies by \citet{2004ApJ...606..271G,2004ApJS..152...63G} revealed a tight linear correlation between the luminosity of the \mbox{{\rm HCN}}\ \jj10 transition (as a tracer of the mass of dense molecular gas) and the far-infrared luminosity, indicative of the star formation rate (SFR) for whole galaxies.
Later works extended this relation to resolved structures in other galaxies \citep{2019ApJ...880..127J, 2022ApJ...930..170H} or even to massive clumps in the Milky Way \citep{2005ApJ...635L.173W,2016ApJ...829...59L,2016ApJ...824...29S, 2017A&A...604A..74S}.
Recent studies \citep{2017A&A...599A..98P, 2017A&A...605L...5K,2020ApJ...894..103E,2020ApJ...891...66N} have questioned how well the lowest $J$ transitions of \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ trace the dense gas. These authors report that \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$} \jj10 emission is easily detectable from the diffuse part of the cloud with typical density $\mathrm{100-500\ cm^{-3}}$. The concept of detecting a particular line above or around the critical density ($n_{c}$) is oversimplified \citep{1989RMxAA..18...21E}.
In reality, multilevel excitation effects and radiative trapping effects all tend to lower the effective excitation density ($n_{eff}$), defined as the density needed to detect a molecular line with integrated intensity of 1 K km s$^{-1}$\ \citep{1999ARA&A..37..311E,2015PASP..127..299S}.
For the low-$J$ transitions the effective excitation densities are typically 1-2 orders of magnitude lower than the critical densities. The values of $n_{eff}$ for \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ are $4.5\times 10^{3}\ \mathrm{cm^{-3}}$ and $5.3 \times 10^{2}\ \mathrm{cm^{-3}}$ at 20 K, respectively \citep{2015PASP..127..299S}. For modern surveys with low RMS noise, lines at the detection limits can be produced by gas with density as low as 50-100 \cmv, explaining the emission from extended translucent regions where the density of gas is much lower than the critical densities of \mbox{{\rm HCN}}, \mbox{{\rm HCO}$^+$}\ \citep{2020ApJ...894..103E}.
\citet{2017A&A...599A..98P} found that only a small fraction of the luminosity of \mbox{{\rm HCN}} (18\%) and \mbox{{\rm HCO}$^+$} (16\%) arises from regions with $\mbox{$A_{\rm V}$} > 15$ mag for Orion B. \citet{2017A&A...605L...5K} found that \mbox{{\rm HCN}} (\jj10) mainly traces gas with $\mbox{$A_{\rm V}$} \approx 6$ mag, or $n \approx 870\ \mathrm{cm^{-3}}$ towards Orion A. A substantial fraction (44-100\%) of the total line luminosities of \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$} arises outside the $\mbox{$A_{\rm V}$} > 8$ mag region often associated with star formation for 6 inner Galaxy clouds \citep{2020ApJ...894..103E}.
Most dense cores and YSOs are found at $\mbox{$A_{\rm V}$} \geq 8$ mag
\citep{2010ApJ...723.1019H, 2010ApJ...724..687L, 2012ApJ...745..190L}.
None of the studies described above considered the star forming regions in the outer Galaxy, i.e., clusters having \mbox{Galactocentric }\ distance (\mbox{$R_{G}$}) greater than $\sim 8.2$ kpc. The star-forming regions in the outer Galaxy (low-density regime) behave distinctively from the environment around the solar neighborhood (intermediate density regime) and also from the Central Molecular Zone (CMZ, high-density regime) \citep{2012ARA&A..50..531K}. The physical and chemical conditions in star-forming regions are dependent on the environment, which is in turn a function of Galactocentric radius ($R_{G}$).
The Milky Way and other normal spiral galaxies are known to have a negative gradient of metallicity with \mbox{$R_{G}$}\ \citep{1971ApJ...168..327S}. Many studies have found a radial abundance gradient of oxygen, nitrogen, etc. in the Milky Way \citep{2000MNRAS.311..329D,2017MNRAS.471..987E,2018MNRAS.478.2315E}. Most recently, \citet{2022MNRAS.tmp...13M} report linear radial abundance gradients for $\mathrm{O/H}$, $\mathrm{N/H}$ with slopes of $-0.044 \pm 0.009 \ \rm dex \ kpc^{-1}$, $-0.063 \pm 0.009 \ \rm dex \ kpc^{-1}$, respectively, for $R_G =$ 4.88 to 17 kpc and for $\mathrm{C/H}$ the slope is $-0.077 \pm 0.019 \ \rm dex \ kpc^{-1}$ for $R_G = 6.46$ to $11.6$ kpc.
The efficiency of gas cooling and dust shielding processes decrease at lower metallicity, thereby affecting the condensation and fragmentation of giant molecular clouds (GMCs). Conversely, the lower interstellar radiation field and cosmic-ray fluxes at larger $R_{G}$ decreases the gas heating \citep{2011ApJ...738...27B}. The balance between these processes sets the gas and dust physical conditions may cause different star formation outcomes from those in the CMZ or Solar neighborhood.
While studies of nearby galaxies (LMC, SMC) with low metallicity are used to understand these effects \citep{2020A&A...643A..63G}, the outer Milky Way is much closer to us. The outer Galaxy targets provide a less confusing view of the ISM as they are widely separated and there is no blending of emission. This clear perspective of the outer Galaxy enables studies of individual giant molecular clouds from which global properties of star formation can be estimated.
Thus, studying the star forming regions in the outer Galaxy can bridge the gap between the star formation studies in Galactic and extragalactic environments.
To inform interpretation of extragalactic studies, we investigate the distribution of \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ \jj10 emission over the full area of molecular clouds and range of physical conditions. This work complements the inner Galaxy work \citep{2020ApJ...894..103E} for the outer Galaxy.
\section{Sample} \label{sec:Sample}
We selected 7 targets beyond the solar circle with $\mbox{$R_{G}$} > 9.5 \ \mbox{kpc}$ with diversity in properties such as cloud mass, physical size, number of massive stars, metallicity etc (see Table \ref{tab:cloud details}). We also considered the availability of molecular data (\textsuperscript{12}CO \ and \mbox{$^{13}$CO} \ emission from the Extended Outer Galaxy survey, see \ref{subsec:ancillary_data}), dust emission from the BGPS survey \citep{2013ApJS..208...14G} and FIR emission from Herschel data \citep{2017MNRAS.471.2730M}.
Several distance values are available in the literature based on kinematic or stellar measurements. In order to maintain uniformity we adopt the GAIA EDR3 based distance values of all the targets (except \mbox{S252}) from \cite{2022MNRAS.tmp...13M} and use the same methodology to obtain the distance for \mbox{S252}.
The distance of the Sun from the Galactic Center is taken as 8.2 kpc \citep{2019A&A...625L..10G}. \mbox{$R_{G}$}\ values of our sample range between 9.85 to 14.76 kpc (see Table \ref{tab:cloud details}).
The spectral type(s) (SpT) of the main ionizing star(s) and \mbox{12+log[O/H]} values of each target and their respective references are also given in Table \ref{tab:cloud details}.
We mapped the \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ (\jj10) emission from the parent clouds of 7 \mbox{\ion{H}{2}}\ regions over an area based on \mbox{$^{13}$CO}\ emission (see details in Section \ref{sec:Obs}).
The cloud mass (\mbox{$M_{\rm cloud}$}) is obtained using the CO data described in \S \ref{sec:Obs} and $\mbox{$M_{\rm cloud}$}=\alpha_{\mathrm{CO}}L_{\mathrm{CO}}$ with $\alpha_{\mathrm{CO}}= 4.3\ \mathrm{M_{\odot}\ (K\ km\ s^{-1} pc^{2})^{-1}}$, where $L_{\mathrm{CO}}$ is the luminosity expressed in $\mathrm{K\ km\ s^{-1} pc^{2}}$ \citep{2013ARA&A..51..207B}.
A brief description of each region can be found in \S Appendix \ref{app:target}.
\section{Observations and Data Sets} \label{sec:Obs}
\subsection{\mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ Observations with TRAO }
We mapped the seven outer Galaxy clouds in the HCN (\jj10, 88.631847 GHz) and \mbox{{\rm HCO}$^+$} (\jj10, 89.1885296 GHz) lines simultaneously at Taeduk Radio Astronomy Observatory (TRAO) in January 2020. The TRAO telescope is equipped with the SEcond QUabbin Optical Image Array (SEQUOIA-TRAO), a multi-beam receiver with 16 pixels in a $4 \times 4$ array and operates in 85-115 GHz range. The back-end has a 2G FFT (Fast Fourier Transform) Spectrometer. The instrumental properties are described in \citet{2019JKAS...52..227J}. The TRAO has a main-beam size of 58\arcsec\ at these frequencies and a main-beam efficiency (defined so that $T_{\rm mb} = T_A^*/(f_b \eta_{\rm mb})$, where $T_{\rm mb}$ is the Rayleigh-Jeans main-beam temperature and $f_b$ is the fraction of the beam filled by the source) of 46\% at 89 GHz \citep{2019JKAS...52..227J}.
The individual SEQUOIA beams vary in beam size (efficiency) by only a few arcseconds (few percent). The beams are very circular and beam efficiencies vary less than $3\%$ with elevation angle.
We used the OTF (On-The-Fly) mode to map the clouds. The mapped areas for the individual targets were determined from maps of their $^{13}$CO (\jj10) line emission. We used a threshold value of 5 times the RMS noise in the \mbox{$^{13}$CO}\ map to estimate the mapping area for \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ from the TRAO 14-m telescope.
The \mbox{$A_{\rm V}$}\ values corresponding to the threshold values are listed in Table \ref{tab:cloud details}.
All the targets were mapped in Galactic co-ordinates. The mapped areas for individual targets are listed in Table \ref{tab:cloud details}.
The system temperature was $\sim 190 \rm K$ during \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ observations. We used $\rm SiO$ maser sources at 86 GHz for pointing and focusing the telescope every 3 hours. We smoothed the data with the boxcar function to a velocity resolution of about 0.2 km s$^{-1}$, resulting in an RMS sensitivity of 0.06-0.08 K on the $T_{A}^{*}$ scale for all the targets.
We obtained $\sim 150\ \rm hrs$ of observations in total.
The following steps were done to reduce the data.
We used the OTFTOOL to read the raw on-the-fly data for each map of each tile and then converted them into the CLASS format map after subtracting a first order baseline.
We applied noise weighting and chose the output cell size of $20\arcsec$ in OTFTOOL.
We did not apply any convolution function to make the maps, so the effective beam size is same as the main-beam size.
Further reduction was done in CLASS format of GILDAS\footnote{\url{https://www.iram.fr/IRAMFR/GILDAS/}}, where we chose the velocity range ($v_{sp}$) to get good baselines on each side of the velocity range of significant emission ($v_{win}$). The reduction details are listed in \S Appendix \ref{app:reduction} Table \ref{tab:reduction}.
We subtracted a second-order polynomial baseline and converted those files to FITS format using GREG.
\subsection{Ancillary data available} \label{subsec:ancillary_data}
For this work, we have also used the following data sets :
\begin{enumerate}
\item \textbf{\textsuperscript{12}CO\ and \textsuperscript{13}CO\ (\jj10) Observations:} \textsuperscript{12}CO\ and \mbox{$^{13}$CO}\ data for all the targets are extracted
from the Exeter-FCRAO, Extended Outer Galaxy survey using the Five College Radio Astronomy Observatory (FCRAO) 14-m telescope. The EXFC survey covered \textsuperscript{12}CO\ and \mbox{$^{13}$CO}\ \jj10 emission within two longitude ranges: $55^{\circ}<l<100^{\circ}$ with the Galactic Latitude range $-1.4^{\circ} \le b \le +1.9^{\circ}$ and $135^{\circ}<l<195^{\circ}$ with Galactic Latitude range $-3.6^{\circ}<b<5.6^{\circ}$ \citep{2016ApJ...818..144R}. Our outer Galaxy targets are covered by the latter survey. The angular resolution of \textsuperscript{12}CO\ and \mbox{$^{13}$CO}\ are $45\arcsec$ and $48\arcsec$ respectively and the grid size is $22.5\arcsec$. The spectral resolution is 0.127 km s$^{-1}$.
All of the data have been de-convolved to remove contributions from the antenna error beam and therefore, are in main beam temperature units.
\item \textbf{Millimeter Dust Continuum Emission:} We use 1.1 mm continuum emission data from the Bolocam Galactic Plane Survey (BGPS)\footnote{\citet{https://doi.org/10.26131/irsa482}} using Bolocam on the Caltech Submillimeter Observatory. The BGPS Survey does not cover the outer Galaxy regions uniformly; it covers four targeted regions (IC1396, a region towards the Perseus arm, W3/4/5 and Gem OB1).
The effective resolution of this survey is $33\arcsec$. We use the mask files of the sources from V2.1 table\footnote{\url{https://irsa.ipac.caltech.edu/data/BOLOCAM_GPS/}}, where the pixel values in the mask file are set with the catalog number where the source is above the $2\sigma$ threshold; otherwise the pixel values are zero \citep{2013ApJS..208...14G}.
The Bolocat V2.1 contains the 1.1 mm photometric flux of 8594 sources for the aperture radii of $20\arcsec$, $40\arcsec$, and $60\arcsec$ and it also provides flux densities integrated over the source area.
As this is not a contiguous survey towards the outer Galaxy, not all the targets have BGPS data. The data availability of the targets are indicated by the footnotes in Table \ref{tab:cloud details}.
\item \textbf{Far-Infrared data:}
We have used the column density maps determined from Herschel data \citep{2017MNRAS.471.2730M} and available at the Herschel infrared Galactic Plane (Hi-GAL) survey site table\footnote{\url{http://www.astro.cardiff.ac.uk/research/ViaLactea/}}. These column density maps are derived using the PPMAP procedure \citep{2015MNRAS.454.4282M} on the continuum data in the wavelength range 70-500 $\rm \mu$m from the Hi-GAL survey. The spatial resolution of the maps is $12\arcsec$. Not all the targets in our list are covered by the Hi-GAL survey. Table \ref{tab:cloud details} shows the data availability.
\end{enumerate}
\section{Results and Analysis}
Our goal is to compare the luminosities of \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ \jj10\ transitions to other conventional tracers of dense gas, in particular extinction thresholds and millimeter-wave emission from dust. We estimate the extinction from the \mbox{$^{13}$CO}\ data and relations between gas column density and extinction. For the emission from dust, we use the mask file of the BGPS data, an indicator of the existence of gas of relatively high volume density \citep{2011ApJ...741..110D}. We also examine the ability of HCN, \mbox{{\rm HCO}$^+$}, and \mbox{$^{13}$CO}\ to trace column density, determined from \textit{Herschel} maps.
\subsection{Luminosity calculation based on Extinction criterion} \label{subsec:Av}
This section describes the method of calculating luminosity of the line tracers coming from the dense regions and also from outside the dense region, depending on \mbox{$^{13}$CO}\ column density based extinction(\mbox{$A_{\rm V}$}) map. It consists of two main steps: \textbf{(i)} making molecular hydrogen column density (\mbox{$\rm N(H_{2})$}) maps of each target, convertible into maps of \mbox{$A_{\rm V}$}; and \textbf{(ii)} measuring the luminosity of regions satisfying the \mbox{$A_{\rm V}$}\ criterion.
\subsubsection{Identifying the pixels with $\mbox{$A_{\rm V}$} \geq 8$ mag}
To derive the molecular hydrogen column density maps of each target, we use their \textsuperscript{12}CO\ and \mbox{$^{13}$CO}\ data. We first identify the velocity range by examining the spectrum of \mbox{$^{13}$CO}\ emission averaged over the whole mapped region of each target. The method of making column density maps of \mbox{$^{13}$CO}\ is discussed in detail in \S Appendix \ref{app:H2}. We follow equation C1-C5 from \S Appendix \ref{app:H2} to produce the total column density map of \mbox{$^{13}$CO}\ (N(\mbox{$^{13}$CO})) using \mbox{$^{13}$CO}\ and \textsuperscript{12}CO\ data.
Next, we convolve and regrid the \mbox{$^{13}$CO}\ column density map to match the TRAO map and convert it to a molecular hydrogen column density map using the following equation.
\begin{equation}
\rm{N(H_{2})}=\rm{N(\mbox{$^{13}$CO})} \left[\frac{\rm{H_{2}}}{\rm{CO}} \right] \left[\frac{\rm{CO}}{\mbox{$^{13}$CO}} \right]
\end{equation}
We assume a fractional abundance of 6000 for CO \citep{2017AAS...23021502L} and we apply the $[^{12}$C]/[$^{13}$C] isotopic ratio recently derived by \citet{2020A&A...640A.125J}. This ratio is linearly dependent on the \mbox{Galactocentric }\ distance (\mbox{$R_{G}$}) of the target.
\begin{equation}
\frac{[^{12} \mbox{C}]}{[^{13}\mbox{C}]} = 5.87\times \frac{\mbox{$R_{G}$}}{\mathrm{kpc}} +13.25
\end{equation}
Finally, we use the relation $\mbox{$A_{\rm V}$}= 1.0 \times 10^{-21}\mbox{$\rm N(H_{2})$} $ to convert the molecular hydrogen column density to extinction \citep{2017AAS...23021502L}. This procedure allows us to define the pixels satisfying the criterion: $\mbox{$A_{\rm V}$} \geq 8$ mag.
\subsubsection{The Emission from the Total region and Inside Region with $\mbox{$A_{\rm V}$} \geq 8$ mag}
Next we calculate the average spectrum for two regions:
(1) including all pixels in the dense region and
(2) all pixels outside the dense region based on the \mbox{$A_{\rm V}$}\ criterion ($1.5 \le \mbox{$A_{\rm V}$} < 8 $ mag). The lower limit on \mbox{$A_{\rm V}$}\ is based on 5 times the RMS noise threshold in the \mbox{$^{13}$CO}\ maps. Because the RMS noise levels are not the same for all the clouds (see Table \ref{tab:cloud details}), we use $\mbox{$A_{\rm V}$} = 1.5$ mag as the lower limit to make the analysis uniform.
We first identify the dense regions with $\mbox{$A_{\rm V}$} \ge 8\ \rm{mag}$ in the extinction map (denoted by \textbf{in}). Figure \ref{fig:panel} shows the integrated intensity map of \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ for all the targets; white contours on each image in Figure \ref{fig:panel} indicate the $\mbox{$A_{\rm V}$} \ge 8$ mag regions. The spectra averaged over pixels that satisfy the criterion $\mbox{$A_{\rm V}$} \ge 8\ \rm{mag}$ (blue line) and the spectra for pixels that do not (denoted \textbf{out} and shown in red line) are shown in Figure \ref{fig:spectrum} (a, b and upper panels of c, d, e, f, g). Figure (\ref{fig:spectrum}c) (\mbox{S212}) does not show any blue line in the upper panel because there is no pixel satisfying the $\mbox{$A_{\rm V}$} \ge 8\ \rm{mag}$ criterion. The average spectral lines from the \textbf{in} region are stronger than those from the \textbf{out} region whenever both spectra are available.
One odd feature appears in Figure \ref{fig:panel} (\mbox{S228}): the \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ agree well with the BGPS, but there is no peak in \mbox{$A_{\rm V}$}\ in that region. Figure (\ref{fig:spectrum}d) does not show strong emission for \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ at the position of $\mbox{$A_{\rm V}$}\ge8$ mag region (upper panel).
This oddity and other information regarding individual targets are described in \S Appendix \ref{app:target}.
\begin{figure*}
\epsscale{1.2}
\plotone{f1.pdf}
\caption{This figure shows the integrated intensity map of (a) \mbox{{\rm HCN}}\ and (b) \mbox{{\rm HCO}$^+$}\ for each target (name of each targets are mentioned in each image). The white contours indicate the $\mbox{$A_{\rm V}$} \ge 8$ mag regions and black contours indicate the BGPS mask regions. There are no black contours in \mbox{S206}\ and \mbox{S208}\ as they lack BGPS data. There is no white contour in \mbox{S212}, as there is no region above $\mbox{$A_{\rm V}$}=8$ mag.}
\label{fig:panel}
\end{figure*}
\begin{figure*}
\gridline{\fig{f2_a.pdf}{0.38\textwidth}{(a)}
\fig{f2_b.pdf}{0.38\textwidth}{(b)}
}
\gridline{\fig{f2_c.pdf}{0.38\textwidth}{(c)}
\fig{f2_d.pdf}{0.38\textwidth}{(d)}
}
\gridline{\fig{f2_e.pdf}{0.38\textwidth}{(e)}
\fig{f2_f.pdf}{0.38\textwidth}{(f)}
}
\gridline{\fig{f2_g.pdf}{0.38\textwidth}{(g)}}
\caption{This is the average spectrum plot for the cloud (a)\mbox{S206}, (b)\mbox{S208}, (c)\mbox{S212}, (d)\mbox{S228}, (e)\mbox{S235}, (f)\mbox{S252}\ and (g)\mbox{S254}\ based on \mbox{$A_{\rm V}$}\ criterion and BGPS mask map. The \textit{left} panel in all images are for \mbox{{\rm HCN}}\ and \textit{right} panels are for \mbox{{\rm HCO}$^+$}.
The blue line indicates the line emission coming from $\mbox{$A_{\rm V}$} \ge8$ mag regions and red line indicates the emission from $1.5 \le \mbox{$A_{\rm V}$} < 8$ mag regions in \textit{top} panel of each sub-figure. In the \textit{bottom} panels of each sub-figures (except (a) and (b)), the blue line indicates emission coming from the BGPS masked region and red line is for emission outside the masked region.
\label{fig:spectrum}}
\end{figure*}
For the calculation of luminosity, we use the main-beam corrected temperature of species $X$ ($T_{X}= T_{A}^{*}/\eta_{mb}$, where $\eta_{mb}=0.46$).
The velocity ranges for \mbox{$^{13}$CO}\ and \mbox{{\rm HCO}$^+$}\ are similar and for \mbox{{\rm HCN}}\ we expand the velocity range to account for the hyperfine structure (around 3-6 km s$^{-1}$\ on each side depending on the target requirements). The velocity ranges are explained in \S Appendix \ref{app:reduction}.
We measure the luminosity of species $X$ arising from the $\mbox{$A_{\rm V}$} \ge 8\ \rm{mag}$ region using the following equation:
\begin{equation}
L_{\rm X, in}= D^2 \int_{v_{\rm sp, l}}^{v_{\rm sp, u}} dv \int d\Omega_{in} T_X(l,b,v),
\end{equation}
where $D$ is the distance of the target from the Sun in pc
and the integrals are taken over the line velocity between $v_{\rm sp, l}$ and $v_{\rm sp, u}$, the lower and upper limits of the line spectral window, and the solid angle that satisfies the extinction criterion.
In practice, we use the average spectrum in Figure \ref{fig:spectrum} to define the integrated intensity
\begin{equation}
I_{\rm X} = \int_{v_{\rm sp, l}}^{v_{\rm sp, u}} T_X dv
\end{equation}
and compute
the luminosity from the following equation:
\begin{equation}
L_{\rm X, in}= D^2 {I_{\rm X}} N_{\rm in} \Omega_{\rm pix},
\end{equation}
where $N_{\rm in}$ is the number of pixels satisfying the \textbf{in} condition and $\Omega_{\rm pix}$ is the solid angle of a pixel.
The uncertainty in luminosity is
\begin{equation}
\sigma_{L} = D^{2} \delta v \sqrt{N_{ch}} \ \sigma_{T_X} \ N_{\rm in} \Omega_{\rm pix},
\end{equation}
where $N_{ch}$ is the number of channels in the line region, $\delta v$ is the channel width, and $\sigma_{T_X}$ is the RMS noise in the baseline of the average spectrum.
To calculate the total luminosity ($L_{\rm X,tot}$), we follow the same method but include all pixels with $\mbox{$A_{\rm V}$}>1.5 \ \rm mag$ for $L_{\rm X,tot}$.
We follow the same procedure for HCN, \mbox{{\rm HCO}$^+$}\ and \mbox{$^{13}$CO}; for \mbox{$^{13}$CO}, we made integrated intensity maps for the specified velocity range using the FITS files from FCRAO and these maps are convolved, resampled, and aligned with the TRAO maps.
We tabulated the values for the fraction of pixels inside the \mbox{$A_{\rm V}$}\ criterion ($N_{\rm in}/N_{\rm tot}$), the log of total line luminosity (Log \ltot), the log of the line luminosity coming from the $\mbox{$A_{\rm V}$} \ge 8 \ \rm mag$ region (Log \lin) and the fractional luminosity (\mbox{$f_{\rm L}$}$=L_{in}/L_{tot}$) in Table \ref{tab:av8} for \mbox{{\rm HCN}}, \mbox{{\rm HCO}$^+$} and \mbox{$^{13}$CO}. Also we have given the statistical parameters (mean, standard deviation and median) for the relevant columns. \mbox{S212}\ has no pixels with $\mbox{$A_{\rm V}$} \ge8 \ \rm mag$.
\begin{figure}
\epsscale{1.25}
\plotone{f3.pdf}
\caption{The fraction of luminosity (\mbox{$f_{\rm L}$}) within \textbf{in} region is plotted for each tracer versus clouds. The \mbox{$f_{\rm L}$}\ in the \textit{left} panel is obtained from $\mbox{$A_{\rm V}$} \ge8$ mag region and in the \textit{right} panel, \mbox{$f_{\rm L}$}\ is from BGPS mask region. The targets are organized with increasing metallicity. Different symbols are used to separate the different tracers. Different color indicates the metallicity of each source.}
\label{fig:fL_vs_cloud}
\end{figure}
Figure \ref{fig:fL_vs_cloud}a shows \mbox{$f_{\rm L}$}\ within the \textbf{in} regions defined by $\mbox{$A_{\rm V}$} \ge8$ mag (left) for each cloud sorted by increasing metallicity.
The values of \mbox{$f_{\rm L}$}\ vary between 0.014 and 0.661 for \mbox{{\rm HCN}}, between 0.009 and 0.608 for \mbox{{\rm HCO}$^+$}, and between 0.018 and 0.570 for \mbox{$^{13}$CO}. The values of \mbox{$f_{\rm L}$}\ for \mbox{{\rm HCN}}\ are higher than \mbox{{\rm HCO}$^+$}\ and \mbox{$^{13}$CO}.
The mean value of the line luminosities coming from the \textbf{in} region is higher by 0.12 for \mbox{{\rm HCN}}\ compared to \mbox{{\rm HCO}$^+$}. For \mbox{$^{13}$CO}, \lin\ is higher but the values of fractional luminosity (\mbox{$f_{\rm L}$}) are lower than those of \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}. In terms of the logarithm of the total line luminosity, the mean is 2.78 for \mbox{$^{13}$CO}, 1.98 for \mbox{{\rm HCO}$^+$}\ and 1.99 for \mbox{{\rm HCN}}. Comparing the two dense gas tracers,
\mbox{{\rm HCO}$^+$}\ and \mbox{{\rm HCN}}\ provide similar total luminosity,
but \mbox{{\rm HCN}}\ shows better correlation with the extinction criterion in terms of \mbox{$f_{\rm L}$}\ followed by \mbox{{\rm HCO}$^+$}\ and \mbox{$^{13}$CO}\ (see Figure \ref{fig:fL_vs_cloud}a).
The correlation between fractional line luminosity (\mbox{$f_{\rm L}$}) arising from the $\mbox{$A_{\rm V}$} \ge 8$ mag region with $\mbox{Log}\ \ltot$ and $\mbox{Log}\ \lin$
(Figure \ref{fig:fL_vs_Luminosity}) shows that the clouds with more material (i.e., higher luminosity) tend to have higher values of \mbox{$f_{\rm L}$}.
\begin{figure}
\epsscale{1.2}
\plotone{f4.pdf}
\caption{The fraction of line luminosity arising above $\mbox{$A_{\rm V}$} \ge 8$ mag is plotted versus line luminosity.
HCN is plotted on the \textit{top}, \mbox{{\rm HCO}$^+$} on the \textit{middle}, and \mbox{$^{13}$CO}\ on the \textit{bottom}.
The total line luminosities ($\mathrm{Log L_{tot}}$) are on the \textit{left} and luminosity from \textbf{in} region ($\mathrm{Log L_{in}}$) on \textit{right}}
\label{fig:fL_vs_Luminosity}
\end{figure}
\subsection{Luminosity calculation from BGPS} \label{subsec:bgps}
In this section, we follow the same basic method but use the dust thermal continuum emission to trace the `dense' gas. We use the data from BGPS to compare the line tracers in those regions. Millimeter continuum dust emission remains an optically thin tracer of the gas column density even in dense molecular clouds and efficiently identifies the molecular clumps that are possible sites of star formation. Because of the limits on sensitivity to large angular scale emission, the BGPS emission traces volume density,
but the characteristic density probed decreases with distance \citep{2011ApJ...741..110D}. For distances in this sample of 2-6 kpc, the mean density traced is 5\ee3 to 1\ee4 \cmv\ (Fig. 12 of \citealt{2011ApJ...741..110D}) with properties of clumps rather than cores or clouds. We use the mask maps of BGPS, downloaded from Bolocat V2.1 to define the \textbf{in} region. As before, we then computed the line luminosities inside and outside the mask regions.
First, we changed the BGPS mask maps to the same resolution and pixel size as the TRAO maps. Next we make the average spectra of the \textbf{in} and \textbf{out} regions and use equation 5 to calculate the luminosity for \mbox{{\rm HCN}}, \mbox{{\rm HCO}$^+$}\ and \mbox{$^{13}$CO}. The total luminosities (\ltot) are the same as those obtained using the extinction condition $\mbox{$A_{\rm V}$} \ge 8 \ \rm mag$ in Section \ref{subsec:Av}.
Among all the targets, 5 of the regions are covered by the BGPS survey. The black contours on the integrated intensity maps of \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ in Figure \ref{fig:panel} show the position of the BGPS \textbf{in} region.
The bottom panels of Figure \ref{fig:spectrum}(c, d, e, f, g) show the average spectra of the luminosities coming from the BGPS \textbf{in} and \textbf{out} regions. All 5 targets show clear detection of \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ from the BGPS mask region, but significant emission is coming from outside the BGPS mask regions.
We present the result of those five targets in Table \ref{tab:bgps} along with the mean, standard deviation and median for relevant columns. We have listed the values for the fraction of pixels inside the BGPS mask region ($N_{\rm{in}}/N_{\rm{tot}}$), the log of the total line luminosity (Log $L_{\rm tot}$), the log of line luminosity arising from the BGPS \textbf{in} region (Log \lin) and the fractional luminosity (\mbox{$f_{\rm L}$}$=L_{in}/L_{tot}$) in Table \ref{tab:bgps} for \mbox{{\rm HCN}}, \mbox{{\rm HCO}$^+$} and \mbox{$^{13}$CO}.
The mean \mbox{$f_{\rm L}$}\ values, listed in Table \ref{tab:bgps}, are higher for \mbox{{\rm HCN}}, followed by \mbox{{\rm HCO}$^+$}\ and \mbox{$^{13}$CO}\ (see Figure \ref{fig:fL_vs_cloud}b).
Dense line tracers (\mbox{{\rm HCN}}, \mbox{{\rm HCO}$^+$}) show better agreement with BGPS emission than they did with the extinction criterion ($\mbox{$A_{\rm V}$} \ge 8 \mathrm{mag}$) for these 5 clouds having BGPS data.
The standard deviation of the \mbox{$f_{\rm L}$}\ values is 0.149 and 0.137 for \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ respectively, considerably lower than the values for the extinction criterion. Figure \ref{fig:fL_vs_Luminosity-bgps} shows the correlation between fractional line luminosity (\mbox{$f_{\rm L}$}) arising from inside the BGPS mask region with $\mbox{Log} \lin$ and total luminosity $\mbox{Log}\ltot$.
The \mbox{$f_{\rm L}$}\ values show no significant dependence on $\mbox{Log}\ \ltot$ and $\mbox{Log}\ \lin$ for any of the 3 tracers.
\begin{figure}
\epsscale{1.2}
\plotone{f5.pdf}
\caption{The fraction of line luminosity arising from the BGPS mask region is plotted versus line luminosity.
HCN is plotted on the \textit{top}, \mbox{{\rm HCO}$^+$} on the \textit{middle}, and \mbox{$^{13}$CO}\ on the \textit{bottom}.
The total line luminosities ($\mathrm{Log L_{tot}}$) are on the \textit{left} and luminosity from \textbf{in} region ($\mathrm{Log L_{in}}$) on \textit{right}.}
\label{fig:fL_vs_Luminosity-bgps}
\end{figure}
\subsection{Conversion of Line Luminosity to Mass of Dense Gas} \label{subsec:Mdense}
In this section, we measure the mass conversion factor for the outer Galaxy targets. The conversion factor is defined as the ratio between the mass of gas and line luminosities ($M=\alpha_{Q} L_{Q}$, where the unit of $\alpha_{Q}$ is $\mathrm{\msun (K \ km \ s^{-1}\ pc^{2})^{-1}}$. This conversion factor is used in extragalactic studies to estimate the mass of dense gas.
The dense gas mass of each source having BGPS data is estimated from its integrated flux density, $S_{1.1}$ (available in BGPS Source catalog table\footnote{\url{https://irsa.ipac.caltech.edu/data/BOLOCAM_GPS/tables/bgps_v2.1.tbl}}, \citealt{2010ApJS..188..123R}) combined with distance ($D$) and dust temperature ($T_{dust}$). The 1.1 mm dust emission is assumed to be optically thin and at a single temperature, and a dust opacity is assumed.
\begin{equation}
\mbox{$M_{\rm dense}$} = \frac{S_{1.1}D^{2}(\rho_{g}/\rho_{d})}{B_{\nu}(T_{dust})\kappa_{dust,1.1}}
\end{equation}
where $B_{\nu}$ is the Planck function evaluated at $\lambda=1.1$ mm, $\kappa_{dust,1.1}=1.14$ $\rm{cm^2 g^{-1}}$ is the dust opacity per gram of dust \citep{2006ApJ...638..293E} and gas-to-dust ratio ($\rho_{g}/\rho_{d}$) is assumed to be 100 \citep{1983QJRAS..24..267H}.
We use the relation after scaling it \citep{2010ApJS..188..123R}
\begin{equation}
\mbox{$M_{\rm dense}$} = 13.1 M_{\odot}\left(\frac{D}{1 \mathrm{ kpc}}\right)^{2}\left(\frac{S_{1.1}}{1 \mathrm{Jy}}\right)
\left[ \frac{e^{(13.0 \mathrm{K}/T_{dust})}-1 }{e^{(13.0/20)}-1} \right]
\end{equation}
For consistency with \citet{2010ApJS..188..123R}, we assume a single dust temperature ($T_{dust}=20$ K) for all sources to estimate the dense gas mass. For the sources with Herschel data, the derived $T_{dust}$ is consistent with this choice. The values of dense mass (\mbox{$M_{\rm dense}$}) are listed in Table \ref{tab:Mdense} in logarithmic scale. The uncertainties include distance uncertainties.
The mass conversion factor \at\ is calculated as the mass of dense gas divided by the total luminosity of the clouds ($\mbox{$A_{\rm V}$} > 1.5$ mag) ($\at = \mbox{$M_{\rm dense}$}/\ltot$) and for \ain\ the luminosity is restricted to the dense region based on BGPS mask ($\ain=\mbox{$M_{\rm dense}$}/\lin$).
In Table \ref{tab:Mdense}, we list the conversion factors along with total line luminosities (\ltot), line luminosities inside the BGPS mask (\lin) in logarithmic scale for \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}. We also estimated the statistical parameters (mean, standard deviation and median) to understand the correlation between \mbox{$M_{\rm dense}$}\ and the conversion factors (\ain, \at).
The mean value of \ain\ in logarithmic scale is $\langle \mathrm{Log \ \ain(\mbox{{\rm HCN}})} \rangle = 1.28\pm 0.08$, translating to $\mbox{$M_{\rm dense}$}= 19 ^{+3.8}_{-3.2} \lin(\mbox{{\rm HCN}})$ and for \at, $\langle \mathrm{Log \ \at(\mbox{{\rm HCN}})} \rangle = 0.93\pm 0.16$, translating to $\mbox{$M_{\rm dense}$}= 8.5^{+3.8}_{-2.6} \ltot(\mbox{{\rm HCN}})$.
Our \at\ value is close to the extragalactic conversion factor ($\alpha_{\rm{HCN(1-0)}} = 10$ \msun$\mathrm{(K\ km\ s^{-1}\ pc^{2})^{-1}}$, \citet{2004ApJS..152...63G}), where luminosity from the whole cloud is considered to obtain mass.
The \ain\ value is similar to the value obtained by \citet{2010ApJS..188..313W} for dense clumps.
The theoretical prediction of $\mathrm{\alpha(HCN)}$ by \citet{2018MNRAS.479.1702O} is $14\pm6$ $\mathrm{\msun(K\ km \ s^{-1}\ pc^{2})^{-1}}$ for $n \geq$1\ee4 \cmv\ which corresponds to the dense gas of the cloud. Including the uncertainty, this theoretical value is consistent with either our \ain(\mbox{{\rm HCN}}) or \at(\mbox{{\rm HCN}}).
Calculations for \mbox{{\rm HCO}$^+$}\ yields $\mathrm{\ain= 20.4^{+4.1}_{-3.4}}$ $\mathrm{\msun(K\ km \ s^{-1}\ pc^{2})^{-1}}$ and $\mathrm{\at= 9.1^{+3.2}_{-2.3}}$ \msun$\mathrm{(K\ km\ s^{-1}\ pc^{2})^{-1}}$, similar to the values for \mbox{{\rm HCN}}.
Variation in $T_{dust}$ by $\pm3$ K produce changes in \ain\ and \at\ within the variation among sources, whereas variation by $5$ K in dust temperature begin to produce changes outside that range. Higher $T_{dust}$ of course produce lower values of \ain, \at; lower $T_{dust}$ gives higher values.
We also estimate the mass of BGPS clumps from the Herschel based column density maps from \citet{2017MNRAS.471.2730M} for \mbox{S228}, \mbox{S252}\ and \mbox{S254}. We use $\mathrm{M_{dense} = \mu_{H_{2}} m_{H} A_{pix} \sum N(H_{2})}$, where mean molecular weight $\mathrm{\mu_{H_{2}}}$ is assumed to be 2.8 \citep{2008A&A...487..993K}, $\mathrm{m_{H}}$ is mass of hydrogen, $\mathrm{A_{pix}}$ is area of a pixel in $\mathrm{cm^{2}}$ and $\mathrm{\sum N(H_{2})}$ is the integrated column density.
The mass values in logarithmic scale are 2.36, 4.09 and 3.72 respectively for 3 targets. The values are higher than BGPS mass measurement by average 0.34 in log scale, which is 2.2 in linear scale. The difference in the values is mainly caused by differing assumptions about the dust opacity. \citet{2017MNRAS.471.2730M} used $\mathrm{\kappa(\lambda)=0.1 (\frac{\lambda}{300})^{-2}}$ $\mathrm{cm^{2}g^{-1}}$, which implies a dust opacity per gram of dust at 1.1 mm of $\mathrm{\kappa_{1.1}}=0.74$ $\mathrm{cm^{2}g^{-1}}$, a value 1.5 times lower than the $\mathrm{\kappa_{dust,1.1}}$ value assumed in the BGPS mass measurement.
BGPS resolves out the extended structure of the clouds, it gives a better measure of mean volume density of the dense regions than Herschel data. Our aim is to measure the L(\mbox{{\rm HCN}}) in the mean high volume density region, not the column density region, for which Herschel works best.
\begin{figure*}
\gridline{\fig{f6_a.pdf}{0.35\textwidth}{(a)}
\fig{f6_b.pdf}{0.35\textwidth}{(b)}
\fig{f6_c.pdf}{0.35\textwidth}{(c)}
}
\caption{Line-to-mass ratio is plotted against \mbox{$A_{\rm V}$}\ based on Herschel dust column density for the region (a) \mbox{S228}, (b) \mbox{S252}\ and (c) \mbox{S254}.
\label{fig:herschel}}
\end{figure*}
\subsection{Line Tracers versus Herschel Column density} \label{subsec:Herschel}
We also explore how well molecular lines trace the dust column density derived from Herschel data. \citet{2017A&A...605L...5K} and \citet{2020MNRAS.497.1972B} studied the relationship between several molecular line emissions and dense gas in Orion A and W49 molecular clouds.
The line-to-mass ratio is defined by $h_{Q}=W(Q)/\mbox{$A_{\rm V}$}$, where $W(Q)=\int T_{mb}dv$ and $\mbox{$A_{\rm V}$} \propto \mathrm{N(H_{2})}$.
This ratio shows the relation between molecular line emission from transition $Q$ and the mass reservoir characterized by \mbox{$A_{\rm V}$}\ \citep{2017A&A...605L...5K}.
The factor $h_{Q}$ is considered as a proxy for the line emissivity, or the efficiency of an emitting transition per \mbox{{\rm H}$_2$}\ molecule \citep{2020MNRAS.497.1972B}.
We plot line-to-mass ratio ($h_{Q}$) versus extinction (\mbox{$A_{\rm V}$}) based on Herschel column density for the 3 targets for which we have data (see Table \ref{tab:cloud details}). We have used the Herschel column density data from \citet{2017MNRAS.471.2730M} for \mbox{S228}, \mbox{S252}, \mbox{S254}\ after regridding and convolving to TRAO maps resolution and pixel size. Figure \ref{fig:herschel} shows how $W(Q)/\mbox{$A_{\rm V}$}$ varies with increasing \mbox{$A_{\rm V}$}\ obtained from dust based column density. We exclude pixels with $\mbox{$A_{\rm V}$} < 2$ mag because of observational uncertainties.
The line emissivity ($h_{Q}$) of \mbox{$^{13}$CO}\ increases up to $\mbox{$A_{\rm V}$} \approx 6-7$ mag and then decreases gradually with increasing \mbox{$A_{\rm V}$}, whereas for \mbox{{\rm HCO}$^+$}\ and \mbox{{\rm HCN}}\ it does not decrease. \mbox{$^{13}$CO}\ traces the diffuse part well but fails in high column density regions, where \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ are more reliable. Both \mbox{$f_{\rm L}$}\ and $W(Q)/\mbox{$A_{\rm V}$}$ indicate that dense gas is traced best by \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ but less well by \mbox{$^{13}$CO}.
Two possible reasons for \mbox{$^{13}$CO}\ falling at high \mbox{$A_{\rm V}$}\ are the following: (i) it becomes optically thick at high column densities or (ii) if dust temperature falls below 20-25 K, \mbox{$^{13}$CO}\ depletes on dust grains \citep{1997ApJ...486..316B}.
Figure \ref{fig:optical_depth} explores the first reason by plotting the spectra of \mbox{$^{13}$CO}\ and \mbox{{\rm HCO}$^+$}\ with increasing \mbox{$A_{\rm V}$}\ for \mbox{S254}. \mbox{{\rm HCO}$^+$}\ continues to grow stronger with increasing column density while \mbox{$^{13}$CO}\ actually becomes somewhat weaker. Although no self-absorption features are prominent, the \mbox{$^{13}$CO}\ line is likely optically thick. A similar effect has been seen in the other 2 targets as well. In principle, the optical depth is corrected for in the LTE model, but that model can fail if the excitation temperatures of the \textsuperscript{12}CO\ and \mbox{$^{13}$CO}\ differ, as we discuss in \S Appendix \ref{app:H2} in more detail. Fortunately, the failure of \mbox{$^{13}$CO}\ to trace column density mostly arise at $\mbox{$A_{\rm V}$} > 8$ mag, so that our method using \mbox{$^{13}$CO}\ to identify the dense regions is still valid.
\begin{figure}
\epsscale{1.2}
\plotone{f7.pdf}
\caption{\mbox{{\rm HCO}$^+$}\ and \mbox{$^{13}$CO}\ spectra are plotted for \mbox{S254}\ with increasing \mbox{$A_{\rm V}$}\ value for \mbox{S254}. \mbox{$^{13}$CO}\ becomes optically thick in higher column densities.}
\label{fig:optical_depth}
\end{figure}
\section{Discussion} \label{sec:discussion}
\subsection{Comparison with Inner Galaxy targets\label{subsec:innercomp}}
A preliminary comparison between the outer and inner Galaxy targets can provide hints about the effects of Galactic environment. In their study of
6 inner Galaxy clouds with \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$} (\jj10), \citet{2020ApJ...894..103E} found a significant amount of luminosity from the diffuse part of the clouds. That is also true for the outer Galaxy clouds, but the fraction of luminosity arising inside the $\mbox{$A_{\rm V}$} \geq 8$ mag or the BGPS region is larger in the outer Galaxy clouds. Also, the behavior of individual tracers differ in both regimes. For inner Galaxy clouds,
the average ratio $\langle \mbox{$f_{\rm L}$}(\mbox{{\rm HCO}$^+$})/\mbox{$f_{\rm L}$}(\mbox{{\rm HCN}}) \rangle = 1.14\pm 0.18$ for $\mbox{$A_{\rm V}$} \ge 8 \ \mathrm{mag}$ criterion compared to $0.83\pm 0.10$ for the outer Galaxy clouds.
For inner Galaxy clouds \mbox{$f_{\rm L}$}(\mbox{{\rm HCO}$^+$}) is greater than \mbox{$f_{\rm L}$}(\mbox{{\rm HCN}}), but for the outer Galaxy the order is opposite.
\begin{figure}
\epsscale{1.2}
\plotone{f8.pdf}
\caption{Variation of conversions factors $\mathrm{Log\ \ain(\mbox{{\rm HCN}})}$ and $\mathrm{Log\ \at(\mbox{{\rm HCN}})}$ for inner and outer Galaxy targets with \mbox{Galactocentric }\ distance(\mbox{$R_{G}$}).}
\label{fig:conversionfactor}
\end{figure}
\citet{2020ApJ...894..103E} also calculated the conversion factor between \mbox{$M_{\rm dense}$}\ measured by millimeter-wave continuum data and luminosity coming from the dense region. For inner Galaxy clouds \ain(\mbox{{\rm HCN}}) is $20^{+16}_{-9}$ $\mathrm{\msun (K \ km \ s^{-1}\ pc^{2})^{-1}}$, whereas it is $19^{+3.8}_{-3.2}$ $\mathrm{\msun (K \ km \ s^{-1}\ pc^{2})^{-1}}$ for outer Galaxy clouds. Considering the whole cloud, the mass-luminosity conversion factor, \at(\mbox{{\rm HCN}}) was $6.2^{+6.6}_{-3.0}$ $\mathrm{\msun (K \ km \ s^{-1}\ pc^{2})^{-1}}$ for inner Galaxy clouds and \at(\mbox{{\rm HCN}}) is $8.5^{+3.8}_{-2.6}$ $\mathrm{\msun (K \ km \ s^{-1}\ pc^{2})^{-1}}$ for outer Galaxy clouds.
From the above values, we can see that \at\ and \ain\ for outer Galaxy clouds are consistent with inner Galaxy values.
Figure \ref{fig:conversionfactor} shows the variation of conversion factors with \mbox{$R_{G}$}\ for both the set of targets.
However, the mean density probed depends on heliocentric distance, and the inner Galaxy clouds are on average farther away. The high outlier in the lower panel of Figure \ref{fig:conversionfactor} is an inner Galaxy cloud that had very strong self-absorption in \mbox{{\rm HCN}}, lowering the line luminosity and raising \at(\mbox{{\rm HCN}}). Bigger samples are needed to understand all the effects.
\subsection{Potential Impact of metallicity on fractional Luminosity?\label{subsec:metallicity}}
\mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ are linear molecules with large dipole moments and similar energy levels, but differing in the elemental compositions ($\mathrm{N}$ versus $\mathrm{O}$). This difference in chemical structure makes them sensitive to the chemistry of the interstellar medium.
Hence we can expect a connection between the abundance ratio of $\mathrm{N/O}$ and $\mathrm{\mbox{{\rm HCN}}/\mbox{{\rm HCO}$^+$}}$ \citep{2017A&A...597A..44B}.
The ratio of total luminosities, $L(\mbox{{\rm HCO}$^+$})/L(\mbox{{\rm HCN}})$, averaged in the logs, is 0.98 for outer Galaxy clouds, while this factor is 0.56 for inner Galaxy targets \citep{2020ApJ...894..103E}. The higher value in the outer Galaxy is consistent with the extragalactic studies on low metallicity systems (1.8-3, LMC, SMC \citep{1997A&A...317..548C, 2020A&A...643A..63G}; 1.1-2.8, IC 10 \citep{2016ApJ...829...94N, 2017A&A...597A..44B, 2018ApJ...862..120K}; 1.1-2.5, M33 \citep{2013A&A...549A..17B, 2017A&A...597A..44B}; 1.2, M31 \citep{2005A&A...429..153B}).
So, in the low metallicity environments \mbox{{\rm HCO}$^+$}\ is as luminous as \mbox{{\rm HCN}}\ but less concentrated in dense regions compared to high metallicity regions.
The \mbox{$f_{\rm L}$}\ values based on \mbox{$A_{\rm V}$}\ analysis increase with the metallicity (Figure \ref{fig:fL_vs_cloud}), but \mbox{$f_{\rm L}$}\ values based on BGPS analysis do not.
A possible reason for this difference is that we use $\mathrm{[\textsuperscript{12}CO]/[\mbox{$^{13}$CO}]}$ ratio in the derivation of molecular hydrogen column density ($\mathrm{\mbox{$\rm N(H_{2})$}}$) map (see \ref{subsec:Av}), which is a function of \mbox{$R_{G}$}\ and indirectly related to metallicity. For BGPS, the same opacity value is used regardless of metallicity, so the metallicity effect is not incorporated there. This study hints at interesting trends with Galactic environment and indicates the need for a larger sample to study it further.
\section{Conclusions} \label{sec:conclusion}
In this paper, we study 7 outer Galaxy star-forming regions with \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ \jj10 transitions obtained from TRAO 14-m telescope and compare these dense line tracers with extinction, 1.1 mm thermal dust continuum emission (BGPS), and dust column density from Herschel. The results are summarized below.
\begin{enumerate}
\item The luminosity coming from the `dense' region based on extinction criteria and BGPS mask maps (indicated as \textbf{in}) is prominent for most of the clouds, but there are significant amounts of luminosity coming from outside the `dense' region (indicated as \textbf{out}) for most of the clouds.
The fraction of the total line luminosity arising from the dense region, as indicated by both the BGPS and extinction maps, is higher in \mbox{{\rm HCN}}, followed by \mbox{{\rm HCO}$^+$}\ and \mbox{$^{13}$CO}. The \mbox{{\rm HCN}}\ emission is generally less extended than the \mbox{{\rm HCO}$^+$}\ emission.
\item In the outer Galaxy, \mbox{{\rm HCO}$^+$}\ is as luminous as \mbox{{\rm HCN}}, but \mbox{{\rm HCO}$^+$}\ is less concentrated in the dense regions; these are the opposite of the trends in the inner Galaxy.
\item Both \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ show better agreement with millimeter continuum emission than they do with the extinction criterion for the clouds having BGPS data. The dense line tracers in \mbox{S212}\ and \mbox{S228}\ do not agree with the extinction criterion.
\item The fraction of line luminosity arising from the dense region based on extinction criterion increases with metallicity, but no such variation for the analysis based on millimeter continuum (Figure \ref{fig:fL_vs_cloud}).
For lower metallicity targets, the fraction of the luminosity of the three tracers (\mbox{{\rm HCN}}, \mbox{{\rm HCO}$^+$}, and \mbox{$^{13}$CO}) arising in high extinction regions are comparable, but all are low.
\item For 5 clouds, we estimate the mass conversion factor (\ain) between dense gas mass (\mbox{$M_{\rm dense}$}) arising from BGPS mask region and the line luminosities of \mbox{{\rm HCN}}, \mbox{{\rm HCO}$^+$}, both within that region (\ain) and for the whole cloud (\at) in \S \ref{subsec:Mdense}.
The \at\ value is consistent with the literature value used in extragalactic studies ($10\ \msun \ (\rm K\ km\ s^{-1}\ pc^{2})^{-1}$, \citet{2004ApJS..152...63G}), while the \ain\ value is consistent with those found in studies of dense Galactic clumps \citep{2010ApJS..188..313W}. Also, the theoretical value obtained by \citet{2018MNRAS.479.1702O} for \mbox{{\rm HCN}}\ is consistent with either our \ain(\mbox{{\rm HCN}}) or \at(\mbox{{\rm HCN}}).
\item We measure the line-to-mass ratio ($W(Q)/\mbox{$A_{\rm V}$}$) for the 3 targets (\mbox{S228}, \mbox{S252}\ and \mbox{S254}) with column densities from Herschel (\S \ref{subsec:Herschel}). The \mbox{$^{13}$CO}\ traces column density well over the range $\mbox{$A_{\rm V}$} = 4-10$ mag, but \mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}\ trace it better for higher \mbox{$A_{\rm V}$}, confirming the view that dense line tracers (\mbox{{\rm HCN}}\ and \mbox{{\rm HCO}$^+$}) are more sensitive to the high column density regions than is \mbox{$^{13}$CO}.
\end{enumerate}
\begin{acknowledgments}
The authors thank the referee for constructive comments which improved the clarity of the paper. We thank the staff of the TRAO for support during the course of these observations. SP thanks DST-INSPIRE Fellowship (No. IF180092) of the Department of Science and Technology to support the Ph.D. program. NJE thanks the Department
of Astronomy at the University of Texas at Austin for ongoing research support. We are grateful to Adam Ginsburg for sharing information on BGPS.
\end{acknowledgments}
\vspace{5mm}
\facilities{TRAO 14-m Telescope, FCRAO, BGPS, \textit{Herschel}}
\software{astropy \citep{2013A&A...558A..33A,2018AJ....156..123A,2022arXiv220614220T},
numpy \citep{2011CSE....13b..22V},
matplotlib \citep{2007CSE.....9...90H},
GILDAS \citep{2018ssdd.confE..11P},
DS9 \citep{2003ASPC..295..489J, 2000ascl.soft03002S}}
\begin{deluxetable*}{l c c l l c c c c c c}[h]
\tablenum{1}
\tabletypesize{\footnotesize}
\tablecaption{Sample of Clouds \label{tab:cloud details}}
\tablewidth{0pt}
\tablehead{
\colhead{Target} & \colhead{$l$} & \colhead{$b$} & \colhead{\mbox{$R_{G}$}} & \colhead{$D$} & \nocolhead{size} & \colhead{\mbox{$M_{\rm cloud}$}} & \colhead{SpT} & \colhead{\mbox{12+log(O/H)}} & \colhead{Map Size} & \colhead{\mbox{$A_{\rm V}$}\tablenotemark{*}}\\
\colhead{} & \colhead{(deg)} & \colhead{(deg)} & \colhead{(kpc)} & \colhead{(kpc)} & \nocolhead{(pc)} & \colhead{(\eten4\ \msun)} & \colhead{(Ref)} & \colhead{(Ref)} & \colhead{($\ \ '\ \times \ \ '$)} & \colhead{(mag)}
}
\startdata
\mbox{S206}\tablenotemark{a} & $150.5886$ & $-0.8570$ & $10.88^{+0.25}_{-0.26}$ & $2.96^{+0.17}_{-0.15}$ & & $18.5$ & O4V (1) & 8.37 (8) & 15 $\times$ 15 & 1.0 \\
\mbox{S208}\tablenotemark{a} & $151.3098$ & $1.9045$ & $11.87^{+0.37}_{-0.34}$ & $4.02^{+0.27}_{-0.25}$ & & $6.8$ & B0V (2) & 8.43 (8) & 10 $\times$ 20 & 0.3 \\
\mbox{S212}\tablenotemark{b} & $155.3375$ & $2.6345$ & $14.76\pm1.30$ & $6.65^{+1.36}_{-1.26}$ & & $19.0$ & O7 (3) & 8.34 (9) & 13 $\times$ 14 & 0.5 \\
\mbox{S228}\tablenotemark{c} & $169.1432$ & $-1.0475$ & $10.72\pm0.19$ & $2.56\pm0.09$ & & $11.6$ & O8V (4) & 8.39 (8) & 11 $\times$ 33 & 0.4 \\
\mbox{S235}\tablenotemark{b} & $173.6682$ & $2.7799$ & $9.85^{+0.16}_{-0.17}$ & $1.66\pm0.07$ & & 41.1 & O9.5V (5) & 8.54 (8) & 30 $\times$ 30 & 1.5 \\
\mbox{S252}\tablenotemark{c} & $189.8418$ & $0.3156$ & $10.09^{+0.12}_{-0.11}$ & $1.92^{+0.12}_{-0.11}$ & & 49.0 & O6.5V,O9.5V,B V (6) & 8.51 (8) & 30 $\times$ 30 & 1.5 \\
\mbox{S254}\tablenotemark{c} & $192.7935$ & $0.0290$ & $10.13\pm0.21$ & $1.96^{+0.12}_{-0.09}$ & & 51.5 &O9.5V,B0.5V (7) & 8.69 (8) & 50 $\times$ 34 & 1.5 \\
\enddata
\tablecomments{(1)\citet{2016ApJS..224....4M}, (2)\citet{moffat1979rotation}, (3)\citet{2014ApJS..211...10S}, (4)\citet{chini1984galactic}, (5)\citet{georgelin1973observations}, (6)\citet{2012MNRAS.424.2486J}, (7)\citet{2011ApJ...738..156O}, (8)\cite{2018PASP..130k4301W}, (9)\cite{2022MNRAS.tmp...13M}}
\textbf{Data availability:} \tablenotetext{a}{\textsuperscript{12}CO, \mbox{$^{13}$CO}} \tablenotetext{b}{\textsuperscript{12}CO, \mbox{$^{13}$CO}, BGPS} \tablenotetext{c}{\textsuperscript{12}CO, \mbox{$^{13}$CO}, BGPS, Herschel}
\tablenotetext{*}{\mbox{$A_{\rm V}$} \ values corresponding to the threshold value of 5 times the RMS noise in the \mbox{$^{13}$CO}\ map.}
\end{deluxetable*}
\begin{deluxetable*}{l c c c c c c c c c c}[h]
\tablenum{2}
\tabletypesize{\footnotesize}
\tablecaption{Line Luminosities versus $\mbox{$A_{\rm V}$} \ge 8 \ \mbox{mag}$ \label{tab:av8}}
\tablewidth{0pt}
\tablehead{
\colhead{Source} & \colhead{$N_{\rm{in}}/N_{\rm tot}$} & \colhead{Log $L_{\rm tot}$} & \colhead{Log $L_{\rm in}$} & \colhead{\mbox{$f_{\rm L}$}} & \colhead{Log $L_{\rm tot}$} & \colhead{Log $L_{\rm in}$} & \colhead{\mbox{$f_{\rm L}$}} & \colhead{Log $L_{\rm tot}$} & \colhead{Log $L_{\rm in}$} & \colhead{\mbox{$f_{\rm L}$}} \\
\colhead{} & \colhead{} & \colhead{\mbox{{\rm HCN}}} & \colhead{\mbox{{\rm HCN}}} & \colhead{\mbox{{\rm HCN}}} & \colhead{\mbox{{\rm HCO}$^+$}} & \colhead{\mbox{{\rm HCO}$^+$}} & \colhead{\mbox{{\rm HCO}$^+$}} & \colhead{\mbox{$^{13}$CO}} & \colhead{\mbox{$^{13}$CO}} & \colhead{\mbox{$^{13}$CO}}
}
\startdata
\mbox{S206} & 0.067 & $1.88^{+0.05}_{-0.05}$ & $1.14^{+0.05}_{-0.05}$ & 0.180(0.008) & $1.84^{+0.05}_{-0.05}$ & $1.05^{+0.05}_{-0.05}$ & 0.164(0.006) & $2.54^{+0.05}_{-0.05}$ & $1.76^{+0.05}_{-0.05}$ & 0.167(0.004)\\
\mbox{S208} & 0.061 & $1.31^{+0.06}_{-0.06}$ & $0.63^{+0.06}_{-0.06}$ & 0.209(0.011) & $1.39^{+0.05}_{-0.06}$ & $0.62^{+0.05}_{-0.06}$ & 0.170(0.004) & $2.04^{+0.05}_{-0.06}$ & $1.18^{+0.07}_{-0.07}$ & 0.138(0.014)\\
\mbox{S212} & \nodata & $1.89^{+0.15}_{-0.21}$ & \nodata & \nodata & $1.95^{+0.15}_{-0.21}$ & \nodata & \nodata & $2.49^{+0.15}_{-0.21}$ & \nodata & \nodata \\
\mbox{S228} & 0.008 & $1.40^{+0.05}_{-0.05}$ & $-0.44^{+0.08}_{-0.09}$ & 0.014(0.003) & $1.29^{+0.04}_{-0.05}$ & $-0.75^{+0.10}_{-0.13}$ & 0.009(0.002) & $2.77^{+0.03}_{-0.03}$ & $1.02^{+0.07}_{-0.08}$ & 0.018(0.003) \\
\mbox{S235} & 0.164 & $2.45^{+0.04}_{-0.04}$ & $2.18^{+0.03}_{-0.04}$ & 0.537(0.011) & $2.43^{+0.03}_{-0.04}$ & $2.10^{+0.03}_{-0.04}$ & 0.471(0.007) & $3.09^{+0.03}_{-0.04}$ & $2.64^{+0.03}_{-0.04}$ & 0.355(0.002) \\
\mbox{S252} & 0.265 & $2.53^{+0.05}_{-0.05}$ & $2.35^{+0.05}_{-0.05}$ & 0.661(0.025) & $2.48^{+0.05}_{-0.05}$ & $2.27^{+0.05}_{-0.05}$ & 0.608(0.010) & $3.32^{+0.05}_{-0.05}$ & $3.08^{+0.05}_{-0.05}$ & 0.570(0.003)\\
\mbox{S254} & 0.107 & $2.50^{+0.05}_{-0.04}$ & $2.16^{+0.05}_{-0.04}$ & 0.459(0.008) & $2.49^{+0.05}_{-0.04}$ & $2.06^{+0.05}_{-0.04}$ & 0.371(0.006) & $3.21^{+0.05}_{-0.04}$ & $2.64^{+0.05}_{-0.04}$ & 0.269(0.002)\\
\tableline
Mean & 0.112 & 1.99 & 1.34 & 0.343 & 1.98 & 1.22 & 0.299 & 2.78 & 2.05 & 0.253\\
Std Dev. & 0.083 & 0.48 & 1.01 & 0.225 & 0.47 & 1.07 & 0.204 & 0.42 & 0.78 & 0.177\\
Median & 0.087 & 1.89 & 1.65 & 0.334 & 1.95 & 1.56 & 0.271 & 2.77 & 2.20 & 0.218\\
\enddata
\tablecomments{1. Units of luminosities are K km s$^{-1}$ pc$^2$.}
\end{deluxetable*}
\begin{deluxetable*}{l c c c c c c c c c c}[h]
\tablenum{3}
\tabletypesize{\footnotesize}
\tablecaption{Line Luminosities versus BGPS Emission }\label{tab:bgps}
\tablewidth{0pt}
\tablehead{
\colhead{Source} & \colhead{$N_{\rm{in}}/N_{\rm tot}$} & \colhead{Log $L_{\rm tot}$} & \colhead{Log $L_{\rm in}$} & \colhead{\mbox{$f_{\rm L}$}} & \colhead{Log $L_{\rm tot}$} & \colhead{Log $L_{\rm in}$} & \colhead{\mbox{$f_{\rm L}$}} & \colhead{Log $L_{\rm tot}$} & \colhead{Log $L_{\rm in}$} & \colhead{\mbox{$f_{\rm L}$}} \\
\colhead{} & \colhead{} & \colhead{\mbox{{\rm HCN}}} & \colhead{\mbox{{\rm HCN}}} & \colhead{\mbox{{\rm HCN}}} & \colhead{\mbox{{\rm HCO}$^+$}} & \colhead{\mbox{{\rm HCO}$^+$}} & \colhead{\mbox{{\rm HCO}$^+$}} & \colhead{\mbox{$^{13}$CO}} & \colhead{\mbox{$^{13}$CO}} & \colhead{\mbox{$^{13}$CO}}
}
\startdata
\mbox{S212} & 0.352 & $1.89^{+0.15}_{-0.21}$ & $1.75^{+0.15}_{-0.21}$ & 0.728(0.045) & $1.95^{+0.15}_{-0.21}$ & $1.80^{+0.15}_{-0.21}$ & 0.717(0.025) & $2.49^{+0.15}_{-0.21}$ & $2.06^{+0.15}_{-0.21}$ & 0.376(0.017) \\
\mbox{S228} & 0.049 & $1.40^{+0.05}_{-0.05}$ & $0.84^{+0.03}_{-0.03}$ & 0.277(0.027) & $1.29^{+0.04}_{-0.05}$ & $0.89^{+0.03}_{-0.03}$ & 0.391(0.034) & $2.77^{+0.03}_{-0.03}$ & $1.60^{+0.04}_{-0.04}$ & 0.068(0.004) \\
\mbox{S235} & 0.118 & $2.45^{+0.04}_{-0.04}$ & $2.08^{+0.03}_{-0.04}$ & 0.433(0.009) & $2.43^{+0.03}_{-0.04}$ & $2.02^{+0.03}_{-0.04}$ & 0.386(0.006) & $3.09^{+0.03}_{-0.04}$ & $2.49^{+0.03}_{-0.04}$ & 0.248(0.001) \\
\mbox{S252} & 0.165 & $2.53^{+0.05}_{-0.05}$ & $2.26^{+0.05}_{-0.05}$ & 0.533(0.014) & $2.48^{+0.05}_{-0.05}$ & $2.18^{+0.05}_{-0.05}$ & 0.496(0.008) & $3.32^{+0.05}_{-0.05}$ & $2.90^{+0.05}_{-0.05}$ & 0.381(0.002) \\
\mbox{S254} & 0.085 & $2.50^{+0.05}_{-0.04}$ & $2.12^{+0.05}_{-0.04}$ & 0.418(0.007) & $2.49^{+0.05}_{-0.04}$ & $2.01^{+0.05}_{-0.04}$ & 0.331(0.005) & $3.21^{+0.05}_{-0.04}$ & $2.53^{+0.05}_{-0.04}$ & 0.208(0.001)\\
\tableline
Mean & 0.154 & 2.15 & 1.81 & 0.478 & 2.13 & 1.78 & 0.464 & 2.97 & 2.32 & 0.256 \\
Std Dev.& 0.106 & 0.44 & 0.51 & 0.149 & 0.46 & 0.46 & 0.137 & 0.31 & 0.50 & 0.116 \\
Median & 0.118 & 2.45 & 2.08 & 0.433 & 2.43 & 2.01 & 0.391 & 3.09 & 2.49 & 0.248 \\
\enddata
\tablecomments{1. Units of luminosities are K km s$^{-1}$ pc$^2$.}
\end{deluxetable*}
\begin{deluxetable*}{l c c c c c c c c c c}[h]
\tablenum{4}
\tabletypesize{\footnotesize}
\tablecaption{\mbox{$M_{\rm dense}$}\ versus Luminosities \label{tab:Mdense}}
\tablewidth{0pt}
\tablehead{
\colhead{Source} & \colhead{$\mathrm{Log\ \mbox{$M_{\rm dense}$}}$} & \colhead{Log $L_{\rm tot}$} & \colhead{Log $L_{\rm in}$} & \colhead{$\mathrm{Log \ \alpha_{tot}}$} & \colhead{$\mathrm{Log \ \alpha_{in}}$} & \colhead{Log $L_{\rm tot}$} & \colhead{Log $L_{\rm in}$} & \colhead{$\mathrm{Log \ \alpha_{tot}}$} & \colhead{$\mathrm{Log \ \alpha_{in}}$} \\
\colhead{} & \colhead{\msun} & \colhead{\mbox{{\rm HCN}}} & \colhead{\mbox{{\rm HCN}}} & \colhead{\mbox{{\rm HCN}}} & \colhead{\mbox{{\rm HCN}}} & \colhead{\mbox{{\rm HCO}$^+$}} & \colhead{\mbox{{\rm HCO}$^+$}} & \colhead{\mbox{{\rm HCO}$^+$}} & \colhead{\mbox{{\rm HCO}$^+$}}
}
\startdata
\mbox{S212} & $3.06^{+0.16}_{-0.25}$ & $1.89^{+0.15}_{-0.21}$ & $1.75^{+0.15}_{-0.21}$ & $1.17\pm 0.09$ & $1.31\pm 0.08$ & $1.95^{+0.15}_{-0.21}$ & $1.80^{+0.15}_{-0.21}$ & $1.11\pm0.08$ & $1.26\pm0.08$\\
\mbox{S228} & $2.19^{+0.07}_{-0.08}$ & $1.40^{+0.05}_{-0.05}$ & $0.84^{+0.03}_{-0.03}$ & $0.80\pm 0.07$ & $1.35\pm 0.06$ & $1.29^{+0.04}_{-0.05}$ & $0.89^{+0.03}_{-0.03}$ & $0.90\pm0.07$ & $1.31\pm0.06$\\
\mbox{S235} & $3.22^{+0.05}_{-0.05}$ & $2.45^{+0.04}_{-0.04}$ & $2.08^{+0.03}_{-0.04}$ & $0.77\pm 0.04$ & $1.13\pm 0.03$ & $2.43^{+0.03}_{-0.04}$ & $2.02^{+0.03}_{-0.04}$ & $0.79\pm0.04$ & $1.20\pm0.03$\\
\mbox{S252} & $3.53^{+0.06}_{-0.07}$ & $2.53^{+0.05}_{-0.05}$ & $2.26^{+0.05}_{-0.05}$ & $1.00\pm 0.04$ & $1.28\pm 0.04$ & $2.48^{+0.05}_{-0.05}$ & $2.18^{+0.05}_{-0.05}$ & $1.05\pm0.04$ & $1.35\pm0.04$\\
\mbox{S254} & $3.43^{+0.06}_{-0.06}$ & $2.50^{+0.05}_{-0.04}$ & $2.12^{+0.05}_{-0.04}$ & $0.93\pm 0.04$ & $1.31\pm 0.04$ & $2.49^{+0.05}_{-0.04}$ & $2.01^{+0.05}_{-0.04}$ & $0.93\pm0.04$ & $1.41\pm0.04$\\
\tableline
Mean & 3.09 & 2.15 & 1.81 & 0.93 & 1.28 & 2.13 &1.78 & 0.96 & 1.31 \\
Std Dev.& 0.53 & 0.44 & 0.51 & 0.16 & 0.08 & 0.46 &0.46 & 0.13 & 0.08\\
Median & 3.22 & 2.45 & 2.08 & 0.93 & 1.31 & 2.43 &2.01 & 0.93 & 1.31\\
\enddata
\tablecomments{1. Units of luminosities are K km s$^{-1}$ pc$^2$.\\
2. Units of conversion factors are $\msun \ (\rm K\ km\ s^{-1}\ pc^{2})^{-1}$}
\end{deluxetable*}
\clearpage
\bibliographystyle{aasjournal}
|
1,941,325,220,536 | arxiv | \section{Introduction}
Observation of neutrinoless double-beta decay ($0 \nu \beta \beta$) would determine the
Majorana nature of the neutrino and would demonstrate that lepton number is not
conserved. Additionally, the measurement of a $0 \nu \beta \beta$\ rate will provide
information about the absolute scale of neutrino mass. The \textsc{Majo\-ra\-na}\
Collaboration~\cite{Aalseth201144} will search for neutrinoless double beta
decay with an array of germanium detectors enriched in the $0 \nu \beta \beta$\ candidate
isotope \nuc{76}{Ge}.
The \textsc{Majo\-ra\-na}\ and GERDA~\cite{Gerda} Collaborations are independently investigating
technologies for deployment of ultra-low-background germanium detector arrays as
research and development toward a future collaborative tonne-scale $0 \nu \beta \beta$\
experiment. As part of this effort, \textsc{Majo\-ra\-na}\ is constructing the \textsc{Dem\-on\-strat\-or}, a germanium
detector array, with the goal of testing a recent claimed observation of
$0 \nu \beta \beta$~\cite{KlapdorKleingrothaus:2006ff} and determining the background rates
achievable with the proposed compact shield design. For a tonne-scale
experiment, \textsc{Majo\-ra\-na}\ and GERDA have a background goal of one count per tonne-year
after analysis cuts in a narrow energy region surrounding the double-beta decay
endpoint at 2039 keV. The energy region of interest (ROI) is expected to be
approximately 4 keV wide, with the actual width depending on detector
resolution. This background goal is approximately 100 times more stringent than
background rates observed in previous low-background germanium
experiments~\cite{HM}, \cite{IGEX}.
\section{The \textsc{Majo\-ra\-na}\ \textsc{Dem\-on\-strat\-or}}
The \textsc{Dem\-on\-strat\-or}\ will contain 40 kg of germanium semiconductor diode detectors, of
which at least 20 kg and up to 30 kg will be enriched to 86\% in \nuc{76}{Ge}.
The \textsc{Dem\-on\-strat\-or}\ will be located at the 4850-foot level of Sanford Underground Research
Facility in Lead, SD. An array of enriched detectors is expected to begin
operation in 2013. Germanium detectors will be housed in two vacuum cryostats
constructed from electroformed copper and cooled with liquid nitrogen. The
detectors will be mounted in \textit{strings}, rigid columns of detectors
suspended from an electroformed copper cold plate. Each cryostat will contain
seven strings of five detectors in a close-packed geometry.
The \textsc{Dem\-on\-strat\-or}\ cryostats will be surrounded by several layers of shielding, shown in
Fig.~1. In order of proximity to the detectors, these layers will consist of
electroformed copper, commercial copper, lead, a radon exclusion volume, an
active muon scintillator veto, and polyethylene neutron moderator. A background
count rate at or below four counts per tonne-year in the \textsc{Dem\-on\-strat\-or}\ ROI would scale
to the background goal of one count per tonne-year in the ROI in a tonne-scale
\nuc{76}{Ge} experiment.
\begin{figure}[h]
\includegraphics[height=0.18\textheight]{4E3_Schubert_Demonstrator}
\caption{%
A drawing of the \textsc{Majo\-ra\-na}\ \textsc{Dem\-on\-strat\-or}, showing insertion of a cryostat module
into the shield.
}
\end{figure}
\section{$0 \nu \beta \beta$\ detection and background identification techniques}
A $0 \nu \beta \beta$\ signal would appear as a peak at the Q-value in the energy spectrum
collected by the \textsc{Dem\-on\-strat\-or}. The sensitivity of the \textsc{Dem\-on\-strat\-or}\ is limited by background
contributions in the ROI; any radiation with energy greater than the Q-value
could potentially create a background in the ROI.
Backgrounds are expected from in-situ cosmogenic flux, long-lived unstable
cosmogenically activated contaminants such as \nuc{68}{Ge} in the detectors and
\nuc{60}{Co} in the copper, and primordial radioimpurities, such as \nuc{40}{K},
\nuc{238}{U}, and \nuc{232}{Th} decay products.
Only materials certified by ongoing assay and simulation campaigns will be used.
Component mass is minimized to reduce contaminants. Close packing of the
detectors enhances effectiveness of a crystal-to-crystal \textit{granularity}
tag.
Measurements in dedicated thermal and electronics test cryostats have refined
the designs of front-end electronics and detector mounts.
\textsc{Majo\-ra\-na}\ has chosen p-type point contact (PPC) germanium detectors for their
superior background rejection capabilities. The low energy threshold achievable
with these detectors should allow identification of certain backgrounds that are
time-correlated with low-energy events and will provide sensitivity to some dark
matter candidates.
Enriched germanium detectors provide an advantageous source-as-detector
configuration with extremely high efficiency for detecting potential $0 \nu \beta \beta$\
decays. Germanium detectors exhibit excellent energy resolution,
approximately 0.2\% FWHM at the $0 \nu \beta \beta$\ Q-value, which allows the definition of
a narrow ROI and minimizes contributions of backgrounds.
Pulse-shape analysis can distinguish an energy deposit occurring in a single
site in a detector, characteristic of the $0 \nu \beta \beta$\ signal, from multi-site
interactions characteristic of gamma backgrounds. This distinction can be made
with with high efficiency for accepting single-site events and rejecting
backgrounds~\cite{Cooper2011303}. A gamma ray from a background source may
deposit energy in one detector, exit the detector, and deposit energy in another
detector. These background events can be identified with the granularity tag.
Some radioactive decays, including the decay of \nuc{68}{Ge} to \nuc{68}{Ga},
may be identified not only by their multiple-site nature, but also by analysis
of correlations in time, energy, and location.
Ultra-clean fabrication efforts are producing components for the \textsc{Dem\-on\-strat\-or}.
Copper electroforming is underway at underground laboratories at a shallow site
at Pacific Northwest National Laboratory in Richland, WA, and at the 4850-foot
level of Sanford Underground Research Facility.
A prototype Parylene deposition system has produced low-background
Parylene-coated copper ribbon cable for readout of detector electronics.
Operating for five years, the \textsc{Dem\-on\-strat\-or}\ will achieve only a fraction of a tonne-year
of exposure, and less than one count is expected in the ROI. A thorough
understanding of the full \textsc{Dem\-on\-strat\-or}\ background energy spectrum will be required to
predict the background count rate in the ROI of a tonne-scale \nuc{76}{Ge}
experiment. By measuring the energy spectrum in a window up to 250 keV wide
surrounding the $0 \nu \beta \beta$\ Q-value, a sensitive measurement can be made with \textsc{Dem\-on\-strat\-or}\
and extrapolated to the tonne scale.
The Monte-Carlo physics simulation package \textsc{MaGe}~\cite{MaGe2011} is used to
simulate \textsc{Dem\-on\-strat\-or}\ detector response to backgrounds.
\textsc{MaGe}\ results are combined with information from the material assay campaign to
predict the \textsc{Dem\-on\-strat\-or}\ background energy spectrum.
\section{Conclusions}
\textsc{Majo\-ra\-na}\ is constructing the \textsc{Dem\-on\-strat\-or}, a 40-kg germanium detector array to search for
$0 \nu \beta \beta$\ of \nuc{76}{Ge}. Efforts in development of ultra-low-background
electronics and cabling, fabrication of radiopure components, material assay,
and modeling of backgrounds are underway. The \textsc{Dem\-on\-strat\-or}\ should confirm or refute a
previous claim of $0 \nu \beta \beta$\ observation and will determine whether a background
goal of one count per tonne-year is achievable for a tonne-scale experiment.
The \textsc{Dem\-on\-strat\-or}\ is expected to operate with enriched detectors in 2013.
\bibliographystyle{aipproc}
|
1,941,325,220,537 | arxiv | \section{Results}
\begin{figure}
\begin{center}
\includegraphics[width=1.0\columnwidth,clip=true]{Figure2.pdf}
\end{center}
\caption{
\textbf{SrO termination}
a) STM image, b) AFM image of the inset marked in (a) with the Sr/O atoms sketched in green/red, and point defects highlighted by the open green circles. c) STS spectra measured above the clean surface and above the point defects. d) Unit cell grid superimposed on the atomically-resolved AFM image.
}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=1.0\columnwidth,clip=true]{Figure3.pdf}
\end{center}
\caption{
\textbf{TiO$_2$ termination}
a) STM image, b) AFM image of the same area measured simultaneously. c) STS spectrum of the TiO$_2$ termination. The region close to the Fermi level (black line) was measured at a closer tip-sample distance than the rest of the spectrum (gray). d) Unit cell grid superimposed on the atomically-resolved AFM image.
}
\end{figure}
Our previous study on KTaO$_3$ \cite{SetvinScience2018} indicated that ferroelectricity induced in a perovskite can create a natural cleavage plane oriented perpendicular to the electric polarization. SrTiO$_3$ is an incipient ferroelectric material: Its ferroelectric Curie temperature lies below 0~K, but can be shifted even above room temperature \cite{ferro4} when the $c$-axis is sufficiently elongated to suppress the antiferroelectric lattice distortions \cite{Vanderbilt1995,ad1,ad3}. This can be achieved by application of compressive or tensile strain in a direction perpendicular or parallel to the intended polarization direction, respectively.
Based on these considerations we have designed a device for cleaving the SrTiO$_3$, sketched in Figure 1a. It is based on compressing the STO single crystal along the [110] direction from the side in two regions, while sparing a small fraction of the volume (shaded in Figure 1a) where the cleavage is intended to occur. The cleaving is performed by pushing the crystal ($\mathbf{F}_{\text{ext}}$) along the [110] axis, which induces additional strain, strongly confined in the shaded region (see Figure 1b). In contrast to unstrained crystals, samples properly pre-strained prior to the cleaving become brittle and readily cleave along the (001) plane. A successful cleavage is shown in Figure 1c. The surface is characterized by a train of terraces separated by macrospcopic steps oriented along the [110] direction (parallel to the external strain). A large-area STM image of a SrTiO$_3$ (001) surface cleaved by this method is displayed in Figure 1d. The surface consists of atomically flat terraces ranging from 100 nm up to several $\mu$m in size (see Figure S2 for more examples).
Using combined STM/AFM, we identified two distinct terminations. Figures 2 and 3 show areas terminated with top layers of SrO and TiO$_2$, respectively. Figure 2a shows a constant-height STM image, the area marked by the dashed square is imaged by AFM in Figure 2b. A clear, bulk-terminated (1x1) pattern is observed in AFM, while the STM image only shows electronic contrast. The lack of atomic resolution in STM is attributed to an absence of localized surface states associated with the atom positions. Superimposed on Figure 2b is a ball model of the (1$\times$1) SrO surface structure. This termination is not perfect as there are point defects in the form of missing atoms. We observe only a single type of vacancies, which, in principle, could be either missing O or Sr atoms. The defect type cannot be determined solely from AFM images, since the contrast achieved on ionic crystals is tip-dependent: Cation-terminated tips interact attractively with surface anions and repulsively with cations \cite{AFM1, AFM2}. An anion-terminated tip will show an inverted contrast (see Figure S3). The unit cell grid superimposed on the atomically-resolved AFM image of the SrO termination (Figure 2d) reveals that all the vacancies appear in equivalent lattice positions, with a concentration of $\eta$(Vac) = (14$\pm$2)\%. This concentration was constant and reproducible in all experiments.
Complementary to the vacancy-riddled SrO termination, the TiO$_2$ terminated surface is covered with point defects in the form of adatoms, seen in Figure 3b. These adatoms are imaged as depressions in the constant-height AFM image, reflecting the attractive chemical force with the tip. The underlying TiO$_2$ lattice is not resolved due to the dominant contribution of the geometrically protruding adatoms. The constant-height STM image (Figure 3a), obtained simultaneously with the atomically-resolved AFM image (Figure 3b), shows that the current signal stems almost exclusively from the adatoms. The brightest species in the STM image (Figure 3a) are probably linked to extrinsic impurities (like Nb dopants). The concentration of the adatoms $\eta$(Ad) = (14$\pm$2)\% equals the previously discussed vacancy concentration on the SrO-terminated surface areas, and the adatoms also sit in equivalent lattice positions (see Fig. 3d).
The electronic properties of the two surface terminations are dramatically different, see the scanning tunneling spectroscopy data in Figures 2c and 3c. The SrO termination appears as a wide-gap semiconductor, with the conduction band minimum located almost 2~eV above the Fermi level. On the other hand, the TiO$_2$ termination is metallic; the Fermi level crosses the bottom of the conduction band and a small density of states is detected throughought the whole band gap.
Based on their electronic fingerprints, the same concentration, and their specific lattice sites, we attribute the point defects present on both terminations to be strontium vacancies on the SrO termination and complementary strontium adatoms on the TiO$_2$ termination. Sr adatoms act as electron donors and are responsible for the metallic nature of the TiO$_2$ termination. The ionized Sr$^{2+}$ adatoms tend to be distributed across the surface to reduce electrostatic repulsion. Strontium vacancies act as negatively-charged acceptor impurities and are responsible for the upward shift of the SrO conduction band. The locally measured STS spectra above the Sr vacancies exhibit an upward shift of 0.2 V compared to the defect-free regions (the green curve in Fig. 2c). This defect-induced band bending is responsible for the empty-state STM image contrast in Figure 2a: The defects appear as dark depressions (marked by the green circles) because of the upwards band shift. Kelvin probe spectroscopy measurements above the Sr vacancies consistently show a band shift in the same direction, but weaker than in STS (see Figure S4).
The large-scale morphology of the as-cleaved surface, shown in Figures 4a--c provides valuable insight into the fracturing mechanism behind successful cleavage. The typical domain size of the SrO/TiO$_2$ terminations is much larger than the maximum scan range of our STM/AFM ($>$1~$\mu$m). Interestingly, steps do not switch the termination type and are mostly one unit cell high ($\approx$0.4~nm), or a multiple of that. Figure 4a shows a staircase of atomically flat terraces with TiO$_2$ termination, separated by unit-cell-high steps. Figure 4b shows an area with a more complex morphology containing multisteps; all the terraces here have the SrO termination. The surface termination switches exclusively at very high steps, on the order of 100 nm, which are oriented in the [110] direction (parallel to the applied strain). This is illustrated in Figure 4c, here a multistepstep $\approx$60~nm high separates the SrO termination (left) from the TiO$_2$ termination (right).
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\columnwidth]{Figure4.pdf}
\end{center}
\caption{
\textbf{Spatial distribution of surface terminations and proposed model for ferroelectricity-assisted cleavage}. a) TiO$_2$ termination with a series of unit-cell-high steps b) SrO termination with a steep 6 nm staircase. c) Change in termination associated with a $\approx$60~nm high step. The region highlighted with a broken line is displayed below after high-pass filtering. d) Line profiles measured along the blue, green, and red lines in panels a,b,c, respectively. e) Proposed scheme of the induced polarized domains and the fracturing mechanism. The red, dotted line shows the distribution of surface terminations with respect to the polarization orientation. d) Electrostatic potential in a ferroelectric, polar STO crystal with polarization compensation in the form of Sr adatoms and vacancies on opposing surfaces.}
\end{figure}
The observed distribution of the SrO and TiO$_2$ terminations indicates that large bulk domains might be involved in the cleavage process. We propose a mechanism based on strain-induced ferroelectricity, see Figure 4e. The main feature of the ferroelectric phase in perovskites is an either positive or negative displacement of the B-site transition metal atom along the [001] direction \cite{Vanderbilt1995}. The displacement direction is coherent within a ferroelectric domain and results in a spontaneous electric polarization \textbf{P} along the [001] direction (see the top row of Figure 4e). In a simple electrostatic picture, this distortion separates the formally electroneutral TiO$_2$ layers into two layers with formally equal and opposite charges, Ti$^{4+}$ and (O$_{2})^{4-}$. The distorted TiO$_6$ octahedra create inequivalent (001) planes suitable for cleaving (bottom scheme in Figure 4e), where the resulting surface termination (SrO/TiO$_2$) does not change within the whole ferroelectric domain. The surface termination can only switch at ferroelectric domain walls, which, at the surface, are associated with very large steps. The observed domain size is in the few $\mu$m range, and the orientation of the steps is consistent with observations made on various thin films grown on SrTiO$_3$, where the domain walls run along the main crystallographic directions and linear combinations thereof \cite{dw1,dw3,dw4}.
Our cleaving device is designed to induce the ferroelectric phase transition by a combination of compressive strain from the sample mount, and tensile strain induced by external cleaving force. SrTiO$_3$ requires biaxial compressive strain of 1\% to become ferroelectric at room temperature \cite{ferro4}, or even higher uniaxial compressive strain. In our setup used for $in-situ$ cleaving, the maximum achievable strain is estimated as $\approx$0.4\% (see Figure S5). This is clearly below the value necessary for the ferroelectric phase transition, but it already guarantees a good cleavage. The ferroelectric phase is therefore induced mainly by the high tensile strain associated with the crack propagation during the cleavage \cite{Griffith1921}. The compressive strain from the sample mount therefore serves to confine the tensile strain into a well-defined region, and it also allows for tuning the size of the ferroelectric domains. This is illustrated in Figure S6 on a different perovskite, KTaO$_3$, which forms smaller domains, suitable for STM imaging.
An interesting feature is the formation of Sr adatoms and vacancies in a well-defined concentration of $\eta$(Sr$_{\text{Ad}}$) = $\eta$(V$_{\text{Sr}}$) = (14$\pm$2)\% in all our successful cleaving experiments. This indicates that they are intrinsic to stabilizing the surface. The ideal STO bulk crystal consists of alternating, weakly polar SrO and TiO$_2$ layers, which would not cause a polar catastrophe when cleaved, since the bond breaking and an increase of bond covalency at the surface layer would serve as a sufficient polarity compensation mechanism \cite{pol1}. However, the cleavage occurs while the crystal is ferroelectric, and here the separation of Ti$^{4+}$ and (O$_2$)$^{4-}$ layers renders the resulting surfaces polar. The removal of Sr$^{2+}$ from the SrO and their addition to the TiO$_2$ termination therefore seems to serve as a mechanism for compenstaing the induced surface polarity \cite{pol2}.
A simplified model is sketched in Figure 4f. The ferroelectric, polar, STO crystal is represented as stacking of three inequivalent layers along the [001] direction: A negatively charged (O$_2$)$^{4-}$ layer, a positively charged Ti$^{4+}$ layer, and a formally electroneutral SrO layer, which we neglect in the electrostatic considerations. The Sr$^{2+}$ adatoms and V$_\mathrm{Sr}^{2-}$ vacancies are represented by a surface layer with a formal charge of $\pm2\times(0.14\pm0.2) \approx \pm0.3$ electrons per unit cell. The electrostatic potential resulting from the polarization will cancel out with the additional potential from the surface defects: \begin{math} |V|=\frac{4 \delta} {\epsilon}=\frac{0.3 c}{2\epsilon} \end{math}. Here $V$ is the drop of electrostatic potential on one unit cell distance, $\delta$ is the separation of the Ti and O$_2$ layers due to the applied strain, $\epsilon$ is the dielectric permittivity, and $c$ is the lattice constant along the [001] direction. This relation gives an estimate for the distortion $ \delta $ as 3.7\%~$c$, which seems slightly overestimated, but reasonable. The formation of charged surface defects therefore points to an interesting interplay between ferroelectricity and surface polarity. The cleavage occurs when the crystal is in its ferroelectric state, it is likely that the polarization remains in the near-surface region after the cleavage, stabilized by the charge of the surface defects.
In summary, we have presented a technique for preparing bulk-terminated (001) surfaces of SrTiO$_3$, providing single-domain regions of SrO and TiO$_2$ terminations several micrometers in size. Strain-induced ferroelectricity is the key phenomenon determining the crystal cleavage, and a precise control of the strain allows for controlling the domain size. Results obtained on the KTaO$_3$ crystals (Figure S6) indicate that the technique should be applicable to other incipient-ferroelectric perovskites as well.
{This work was supported by the Austrian Science Fund (FWF) Projects Wittgenstein Prize (Z 250), Doctoral College "Solids4Fun" (W 1234), and SFB "FOXSI" (F4505) }
\section{Materials and Methods}
The AFM/STM measurements were performed in an UHV chamber with a base pressure below 2 $\times$ 10$^{-11}$ mbar, equipped with a ScientaOmicron LTSTM/AFM at T=4.8 K, using the q-Plus setup with a separate wire \cite{Majzik2012} for the tunneling current and a differential cryogenic preamplifier \cite{GiessiblPreamp}. Etched tungsten tips were glued on the tuning fork and cleaned by self-sputtering in Ar atmosphere \cite{SetvinTips2012} prior to the experiment. The resonance frequency of the used qPlus cantilevers was ~77 kHz with the Q factor of $\approx$ 50000. SrTiO$_3$ samples (CrysTec) with 0.5\% wt Nb doping were used.
Samples polished along the (110) face were cut into rectangles of 3.5$\times$7 mm$^2$ and 1 mm thickness. The cleavage was performed along the (001) plane, $i.e.$, perpendicular to the polished surface. The profile in Fig. 1c was measured on a Bruker DektakXTL profiler.
|
1,941,325,220,538 | arxiv | \section{Introduction}
The accurate measurement of tip-surface force is central to the development of AFM toward a more quantitative form of microscopy. Determining tip-surface force from steady-state cantilever dynamics, as opposed to static bending, promises greater sensitivity, higher accuracy, and the ability to probe the viscous response of surfaces and other dissipative tip-surface interactions. We see the advancement of quantitative dynamic AFM as resting on the solution to three underlying problems: The first problem is the accurate measurement of cantilever motion, where calibration is the central issue. The second problem is the analysis of the measured motion to correctly determine the tip-surface force. This second problem involves nonlinear dynamics and a clear understanding of the cantilever as a transducer of tip-surface force. The third problem is the interpretation of the properly measured tip-surface force so as to provide useful, quantitative information about the material system studied. In this paper we discuss these three problems, with particular emphasis on the second.
We compare different modes of AFM force measurement in terms of their sensitivity and the ability to accurately calibrate the measurement. For this purpose it is useful to analyse the cantilever motion in the frequency domain. We do this by simulating the motion of an AFM cantilever using a point-mass, simple harmonic oscillator model, subject to a nonlinear tip-surface force. With modern numerical integrators we can accurately determine the motion of our model nonlinear dynamical system with the nonlinear tip-surface force is an input to the simulator. Various measurement scenarios and drive schemes can then be compared by examining the frequency content of the motion and its relation to the noise in typical AFM experiments. Such simulations are vital to understanding and improving AFM and they allow us to verify methods for reconstructing force from nonlinear motion. With simulated data we can work backward from the nonlinear oscillation to a {\em known force}. This is in contrast to experiment, where we do not know the actual nonlinear tip-surface force which contributed to the measured motion.
Before we begin our discussion of the different dynamic force measurement methods, we will make a quick review of quasi-static force measurement which is most common among AFM users. Quasi-static and dynamic force measurements both use the same basic AFM apparatus and it is possible to compare them within the context of one simple model. After considering the problem of reconstructing tip-surface force for both quasi-static measurement and the so-called fast-force-curve measurement, we will consider the case of driving the cantilever near a resonance. We describe a method called Intermodulation AFM that exploits frequency mixing by a nonlinear oscillator. The Intermodulation method allows one to reconstruct the tip-surface force by collecting response only in a narrow frequency band near resonance, where accuracy and sensitivity are greatest. It is explained how one can reconstruct both the conservative and dissipative forces between the tip and the surface from analysis of the intermodulation spectrum.
We have attempted to write this article in a pedagogical fashion with the hope that it will be useful to the growing community of scientists and engineers who both development and use methods of multi-frequency AFM. A mathematical notation for preforming matrix algebra on discrete spectra is developed, which is particularly well suited for the analysis of weakly nonlinear oscillations. While this paper deals explicitly with dynamic AFM, we would like to stress that the techniques developed herein are generally applicable to the many types of measurements that exploit the enhanced sensitivity of a transducer with a high quality resonance. We hope that our explanation of the intermodulation spectral measurement technique and the methods for analysis of the intermodulation spectra, will lead to more wide spread application of intermodulation spectroscopy in nanotechnology.
\section{The point mass, single eigenmode model}\label{sec:single_mode}
\begin{figure}[t]
\includegraphics[width=12cm]{coordinate_convention.png}
\caption{A simplified schematic diagram of a probe-scanning AFM which explains the coordinate convention used in the text.}
\label{coordinate_convention}
\end{figure}
Let us analyse the problem of tip motion in a coordinate system fixed to the inertial reference frame in which the sample is at rest, the so-called laboratory frame. In this coordinate system the vertical location of the tip is denoted $z$ and the vertical location of the tip when the cantilever is in its equilibrium position is denoted $h$ (see fig.~\ref{coordinate_convention}).
We will analyse the scaned-probe AFM, where the position $h(t)$ can be changed by moving the base of the AFM probe, either rapidly with a shaker piezo, or more slowly with the Z piezo in the AFM scanner. To model dynamic AFM, one typically reduces the spatially distributed dynamic bending of the cantilever to an equivalent point-mass model for the tip coordinate $z(t)$. Following this approach, we postulate that the cantilever exerts a vertical force on this point-mass, such that $\mathrm{f_{c}}=-k_c (z-h)$. We apply Newtons second law to this effective point-mass in the laboratory frame and write down an equation of motion for the tip including an inertial term, a damping term, a cantilever restoring force, and nonlinear tip-surface force $\mathrm{f}(z)$.
\begin{equation}
m \ddot{z} = - m \gamma_0 \dot{z} - k_c (z-h) + \mathrm{f}(z)
\label{equ_mot}
\end{equation}
Here the dots mean differentiation with respect to time and the damping parameter $\gamma_0$ is interpreted as the rate of momentum loss due to random collisions with molecules of air or fluid. In terms of the deflection $d(t)=z(t)-h(t)$ , the equation of motion becomes.
\begin{equation}
\frac{1}{\omega_0^2} \ddot{d} + \frac{1}{Q \omega_0} \dot{d} + d = \left[ - \frac{1}{\omega_0^2} \ddot{h} - \frac{1}{Q \omega_0} \dot{h} \right] + \frac{1}{k_{\rm c}}\mathrm{f}(h + d)
\label{equ_mot_2}
\end{equation}
where $\omega_0=\sqrt{k_c/m}$ is the resonant frequency and $Q=\omega_0/\gamma_0$ is the quality factor.
The equivalent point-mass model eq.~(\ref{equ_mot_2}) is only approximate, but it does capture the basic features of AFM cantilever dynamics under many realistic conditions of operation. In fact, under certain conditions of operation, eq.~(\ref{equ_mot_2}) is actually very accurate, as will be discussed below. The two terms in square brackets are often referred to as the "drive force". The first is an inertial force that arises because the cantilever has mass and we are shaking (accelerating) the base, and the second is a drag force due to the motion of the cantilever through the damping medium. In what follows, we will investigate the nonlinear dynamics of a cantilever driven by a moving base and interacting with a surface, by numerical integration of eq.~(\ref{equ_mot_2}).
A rigorous derivation of the equivalent point-mass model is a subtle and difficult problem that involves solving the Euler-Bernouli equation governing the beam dynamics in the non-intertial reference frame attached to the base \cite{Lee:NonLinDynamMicrolever:02}\cite{Rodriguez:PointMassHarmonics:02}\cite{Melcher:EquivPointMassModel:07} which is moving through a damping medium \cite{Sader:FreqResponseBeamViscous:98}. The solution to the resulting equation is a formidable task which we do not discuss here. We only note that the equation is separable in space and time, so that a particular solution for the dynamic deflection of the beam in the moving frame can be written as $u(x,t)=\Phi(x)q(t)$. The vibrational eigenmodes of the ideal, undamped beam, rigidly fixed at one end and free at the other end, constituent an orthogonal basis set of bending functions $\Phi_i(x)$ with which we could describe an arbitrary deflection of the free end of the beam. The time dependence of the ideal beam is described by a harmonic oscillator equation with a specific eigenfrequency for each bending function. These eigenmodes are coupled by the damping force, which is a function of $x$, and the non-linear tip-surface force, which acts at the free end of the beam $\mathrm{f}(z)=\mathrm{f}( h+\sum_i a_i\Phi_i(L_c)q_i(t))$. Thus, the general problem is that of a coupled, multi-dimensional nonlinear dynamical system, with two dimensions for each relevant eigenmode.
The problem of determining tip-surface force from cantilever motion in a realistic AFM experiment is therefore quite complex {\em if the motion involves many eigenmodes}. However, the problem is simplified enormously if all significant motion is contained in a narrow frequency band surrounding the fundamental eigenfrequency. Concentrating nonlinear motion to one resonance is possible with cantilevers because they do not have their eigenfrequencies at integer multiples of a fundamental. This is in contrast to other common resonators such as taught strings on a musical instrument or electromagnetic transmission-line-resonators which do have their eigenfrequencies at integer multiples of a fundamental. If the cantilever eigenmodes have high quality resonances, multiple modes will not be excited by the nonlinear tip-surface force which has frequency content at harmonics, or integer multiples of the drive frequency. When the quality factor is too low, the frequency content of the nonlinear force will overlap with and excite multiple modes of the cantilever, as observed for example in liquids \cite{Xu:ForceTappingLiquid:10}. Concentrating motion to the fundamental eigenmode is advantageous because the shape of this mode is particularity insensitive to the viscosity of the damping medium \cite{Sader:FreqResponseBeamViscous:98}, a fact which opens up the possibility of using thermal noise as a means of accurately calibrating all mode parameters and the deflection detector, with one simple noise measurement.
These considerations motivate a comparison of the different methods of dynamic force measurement in terms of the frequency content of the cantilever motion.
\section{Frequency domain analysis}\label{sec:frequency_domain}
In an AFM experiment the motion $d(t)$ is typically detected by optical means and converted to an electronic signal. This signal is digitized by sampling at equally spaced intervals $\Delta t$ for a finite time $N \Delta t$. We can represent the motion in this observation window in either the time domain or the frequency domain, where two are related by the Discrete Fourier transform (DFT) and its inverse.
\begin{eqnarray}
\hat{d}_k &\equiv &
\sum_{n=0}^{N-1} d(n \Delta t) e^{-\frac{2 \pi i}{N} k n} \label{DFT}\\
d_n & \equiv & \frac{1}{N} \sum_{k=0}^{N-1} \mathrm{\hat{d}}(k \Delta \omega) e^{+\frac{2 \pi i}{N} k n}
\end{eqnarray}
The time domain data $d_n$ consists of $N$ real data values, and the spectrum $\hat{d}_k=\hat{d}(k \Delta \omega)$ consists of $N/2$ complex numbers equally spaced in frequency by $\Delta \omega = 2 \pi / N \Delta t$ , where N is assumed to be even. In what follows, we will represent these discrete functions of frequency and time as vectors denoted by lower case bold letters, where a hat is used to denote functions of frequency
\begin{eqnarray}
\mathbf{\hat{d}} &=& \mathcal{F} \left[ \mathbf{d} \right] \\
\mathbf{d} &=& \mathcal{F}^{-1} \left[ \mathbf{\hat{d}} \right]
\end{eqnarray}
Let us first consider the frequency spectrum of the free motion within the context of the single eigenmode model. When the cantilever is well above the surface the nonlinear tip-surface force is zero. The DFT of equation (\ref{equ_mot_2}) with $\mathrm{f}=0$ can be written as a matrix equation for the discrete response spectrum of the free cantilever.
\begin{equation}
\mathbf{\hat{d}}^{\rm (free)} = \left( \mathbf{\hat{G}} - \mathbb{1} \right) \mathbf{\hat{h}}
\label{delta_free}
\end{equation}
The matrix $\mathbb{1}$ is the identity matrix and the diagonal matrix $\mathbf{\hat{G}}$ is the linear transfer function of the harmonic oscillator.
\begin{equation}
\hat{G}_{kl}=\begin{cases}
\begin{array}{c}
\left[ 1+i \left( \frac{k \Delta \omega}{\omega_0 Q} \right) - \left( \frac{k \Delta \omega}{\omega_0} \right)^2 \right]^{-1} \\ 0
\end{array} & \begin{array}{c}
k=l \\
k \neq l
\end{array}\end{cases}
\label{G}
\end{equation}
Multiplication by the complex numbers that are the diagonal elements of $(\mathbf{\hat{G}}-\mathbb{1})$, transfers the drive spectrum $\mathbf{\hat{h}}$, which is the motion of the base, to a response spectrum $\mathbf{\hat{d}}^{\rm (free)}$, which is the free deflection of the cantilever. The factor $\mathbb{1}$ is negligable when the quality factor is high and one is driving on or near resonance, where $\vert \mathbf{\hat{G}} \vert \simeq Q >> 1$. On the other hand, driving at frequencies well below a high $Q$ resonance, where $\vert \mathbf{\hat{G}} \vert \simeq 1$, generates no deflection of the free cantilever. Note that linearity of the response is expressed in the diagonal nature of the transfer matrix, which maps each frequency component of the drive vector $\mathbf{\hat{h}}$ to the same frequency in the response vector $\mathbf{\hat{d}}^{\rm (free)}$.
Consider next the single eigenmode dynamics while the cantilever is engaging a surface, where the nonlinear tip-surface force is not zero. Taking the DFT of the equation of motion (\ref{equ_mot_2}) leads to the following equation for the frequency spectrum of the engaged cantilever deflection $\mathbf{\hat{d}}$ .
\begin{equation}
\mathbf{\hat{d}} = \mathbf{\hat{d}}^{\rm (free)} + k_c^{-1}\mathbf{\hat{G}} \mathbf{\hat{f}}
\label{engaged_spectrum}
\end{equation}
Here we have assumed that the free spectrum and the engaged spectrum are both generated by the same drive $\mathbf{\hat{h}}$. In doing so we are neglecting the back-action of the tip-surface force on the actuator which is providing the drive. This is clearly allowed as the cantilever base, as well as the shaker and scanner piezo actuators, all have far greater mass than the effective mass of the cantilever eigenmode.
Due to the nonlinear nature of the tip-surface force, the engaged spectrum $\mathbf{\hat{d}}$ will have components in the response spectrum at frequencies where there is no drive. By measuring these frequency components, we can determine the nonlinear force, which can be seen by inverting eq.(\ref{engaged_spectrum}).
\begin{equation}
\mathbf{\hat{f}}=k_c \mathbf{\hat{G}^{-1}}(\mathbf{\hat{d}} - \mathbf{\hat{d}^{(free)}})
\label{inversion_1}
\end{equation}
Equation (\ref{inversion_1}) gives us the Fourier components of the tip-surface force, but we are actually interested in the tip-surface force as a function of the tip position $z$. We can find $\mathrm{f}(t)$ by applying the inverse Fourier transform to eq. (\ref{inversion_1}). If the time dependence of the tip-surface force is implicit, coming only from time dependence of the tip position, the force curve $\mathrm{f} \left( z(t) \right)$ is found by plotting $\mathrm{f}(t)$ versus $ z(t)=d(t)+h(t) $. Note that our assumption of implicit time dependence is only valid for a conservative tip-surface force. Non-conservative forces will be discussed in section \ref{sec:conservative_dissipative}.
\section{Sensitivity and Calibration}\label{sec:Sensitivity_Calibration}
Quantitative AFM must not only strive to fully understand the cantilever motion and its relation to all forces involved, but also the accurate calibration of the constants that are necessary for converting motion to force, namely $k$, $m$, and $\gamma$ for all relevant eigenmodes. Furthermore, quantitative AFM should strive to maximize the sensitivity of measurement. Naively, one might think that sensitivity is enhanced when using a softer cantilever (smaller $k$). It is true that the cantilever response (deflection of the free end) to a {\em static} tip-surface force increases with decreasing stiffness. However, when we take into account the cantilevers high quality factor resonance, we come to a different conclusion when considering the frequency dependence of the cantilevers response to a {\em dynamic} force.
The sensitivity of a measurement is expressed in terms of the signal-to-noise ratio (SNR). Within the context of the single eigenmode model, the frequency dependant SNR is given by,
\begin{equation}
\mathrm{SNR} = \frac{ k_c^{-1} \mathbf{\hat{G}} \mathbf{\hat{f}} }{
\sqrt{ k_c^{-2} \vert \mathbf{\hat{G}} \vert ^2 S_\mathrm{FF} B + \alpha ^2 S_{VV} B }}
\label{SNR}
\end{equation}
where $B$ [Hz] is the measurement bandwidth, $S_\mathrm{FF}$ [N$^2$/Hz] and $S_{VV}$ [V$^2/$Hz] are the power spectral densities of force and detector voltage fluctuations respectively, and $\alpha$ [nm/V] is the detector responsivity. The numerator (signal) is the frequency-dependant deflection caused by each frequency component of the tip-surface force. The denominator (noise) consists of two independent contributions, where the square amplitudes are added to get the total noise. The first noise term is the frequency-dependant cantilever response to the frequency-independent thermal noise force as given by the fluctuation dissipation theorem, $ S_\mathrm{FF}= 2 \mathrm{k_B} T m \gamma_0 $. The second noise term is the detector's voltage noise, which is taken to be frequency-independent in the band of interest, expressed as an equivalent cantilever deflection noise.
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{cantilever_noise-eps-converted-to.pdf}
\caption{A measurement of the noise in an AFM when the cantilever is far above a surface. In the narrow band of frequency around the fundamental eigenfrequency, the thermal noise force can be resolved above the detector noise due to large transfer gain $G(\omega)$ of the high $Q$ resonance. In this frequency band force measurement is most sensitive, limited only by the thermal noise force associated with the damping medium. From the noise measurement we can calibrate all parameters in the single eigenmode model, and the optical detector responsivity $\alpha$. In room temperature air, this cantilever had $k_c=25.8~\mathrm{N/m}$, $f_0=271~\mathrm{kHz}$ and $Q=500$. The thermal noise force was $\sqrt{S_\mathrm{FF}}=23~\mathrm{fN/\sqrt{Hz}}$ and the optical detector noise floor was $\alpha \sqrt{S_{VV}}=120~\mathrm{fm/\sqrt{Hz}}$ . }
\label{cantilever_noise}
\end{center}
\end{figure}
Figure ~\ref{cantilever_noise} shows a fit of the theory (denominator of eq.~(\ref{SNR})) to the measured noise for an undriven cantilever far above a surface, where there are no tip surface forces. Here we display actual experimental data (all other figures are simulations) taken at room temperature in air for an AFM cantilever having nominal $f_0=300$~kHz and $k_c=40$~N/m. The optical detection system was that of a Digital Instruments AFM and the cantilever was back side coated for enahnced reflectivity. The theory gives an excellent description of the measurement in the band of frequency near the cantilever resonance.
In the frequency band where the thermal noise force dominates over the noise of the detector, the SNR is independent of the cantilever stiffness $k_c$. In this band, no improvements in the detector will effect the sensitivity of force measurement. Improved force sensitivity can only be achieved by reducing the temperature, or by reducing the damping coefficient $m \gamma_0$, for example by using a smaller cantilever \cite{Walters:SmallCantilevers:96} or by working in vacuum. Signals collected in this frequency band therefore represent measurement at a fundamental limit of sensitivity for a particular cantilever and given experimental conditions. For the cantilever of fig.~\ref{cantilever_noise} this sensitivity limit in air at room temperature is 23~fN/$\sqrt{\mathrm{Hz}}$. We can compare this dynamic force sensitivity near resonance with the force sensitivity of quasi-static, or low frequency force measurement where $\vert \hat{G} \vert =1$, which is given by $k_c \alpha \sqrt{S_{VV}} = 3100$~fN/$\sqrt{\mathrm{Hz}}$. The SNR improves by a factor of 134 for signals measured near resonance.
Fitting the measured noise to eq.~(\ref{SNR}) provides an accurate way to determine the linear response function of the single eigenmode in question. For the fundamental eigenmode, not only the force transducers linear response function can be determined by thermal noise measurement, but also the optical lever responsivity $\alpha$ . Recently it was shown how calculations of fluid dynamic damping \cite{Sader:FreqResponseBeamViscous:98} can be combined with measurements of thermal noise \cite{Hutter:NoiseCalib:93} to determine both $\alpha$ and $k_c^{-1} \mathbf{\hat{G}}$ from one simple noise measurement \cite{Higins:NoninvasiveCalibration:06}. This so-called non-invasive method is by far the best calibration method for many reasons: It can be preformed without touching a surface, thus keeping the tip pristine; it directly applies fundamental theory in a proper regime of validity and does not require uncontrolled assumptions about tip-sliding and surface deformation; the calibration of all relevant constants, including $\alpha$, are traceable to one and the same measurement. In these respects, the non-invasive method is the most 'primary' of all AFM calibration methods. The method does however assume a single eigenmode model of the cantilevers Brownian motion, and is therefore only valid for frequencies near a resonance. Because the mode shape of higher bending modes is sensitive to the viscosity of the damping medium \cite{Sader:FreqResponseBeamViscous:98}, the application of this non-invasive calibration method to resonances other than the fundamental bending mode has proven difficult\cite{Lozano:CalibrationHigherModes:10}.
Thus, there is significant advantage to a method that can extract the tip-surface force by analyzing the motion only in a narrow band around the fundamental resonance of the cantilever. In this case we can accurately describe the motion in therms of the single eigenmode model, and we have a simple and direct means of calibrating the relevant constants in our equation of motion. Before discussing such a method in detail, we will review the more common methods of force measurement for the purpose of comparison.
\section{Quasi-static force measurement}\label{sec:quasi_static}
When the cantilever base is moved with a very slow linear ramp toward and away from the surface at a rate of 0.1-10 Hz, one can neglect all time derivatives of $h$ in eq.~(\ref{equ_mot_2}). If we also neglect the time derivatives of $d$, we are left with the equation of static force balance between the cantilever and the tip-surface force.
\begin{equation}
k_c d = \mathrm{f}(d+h)
\label{static_force_eq}
\end{equation}
Figure~\ref{static_force_curves} shows a simulation based on this equation. In the simulattion of fig.~\ref{static_force_curves} and throughout this article, we use the van der Waals - DMT model \cite{garcia:DynamicAFMReview:02} to simulate the tip surface force
\begin{equation}
\mathrm{f}(z)=\begin{cases}
\begin{array}{c}
-\frac{HR}{6z^2}\\
-\frac{HR}{6a_{0}^2}+\frac{4}{3}E^{*}\sqrt{R(a_0-z)^{3}}
\end{array} & \begin{array}{c}
z \geq a_0\\
z< a_0
\end{array}\end{cases}
\label{DMT}
\end{equation}
with the parameters: intermolecular distance $a_0=0.3$~nm, Hamaker constant $H=7.1 \times 10^{-20}$~J, effective modulus $E^*=1.0$~GPa, and tip radius $R=10$~nm. This model is derived for an ideal geometry with materials of uniform composition. The actual tip-surfaces forces in real experiments may differ significantly from this model, depending on the shape of the tip, the varying topography of the surface in question, the presence or absence of adsorbed molecules on the surface, and the possibility of other, non-van der Waals interactions that could attract the tip to the surface. Nevertheless, this simple model captures some basic features of realistic tip-surface forces, and it serves as a useful starting point for exploring the nonlinear motion of the AFM cantilever.
Under typical experimental conditions the static equilibrium becomes unstable very close to tip contact with the surface, where strong attractive forces cause a 'jump-to-contact' upon approaching the surface, and 'pull-off' event when retracting from the surface. This instability results in vertical jumps in a plot of the measured cantilever deflection $d$, versus the base position $h$ (left panel of fig.~\ref{static_force_curves}). If the spring constant $k_c$ is known, the measured quasi-static deflection gives us the tip-surface force by eq.~({\ref{static_force_eq}). If the base position $h$ is also known, we can plot this force versus the tip position $z=d+h$ as shown in the left panel of fig.~\ref{static_force_curves}. Here we can see that the jump-to-contact and pull-off events connect two stable solutions of eq.~(\ref{static_force_eq}) along a 'load line' with slope given by the cantilever stiffness, $k_c = 1$~N/m in this simulation. From fig.~\ref{static_force_curves} it is apparent that the softer the cantilever (smaller slope of the load line), the larger the bistable region in the measurement. The soft cantilever is unable to make a static force measurement on the unstable branch of the force curve, between the two jump events. When crossing the jump points, a rapid motion of the cantilever ensues, exciting high frequency components in the response spectrum $\mathbf{\hat{d}}$. This high frequency motion is typically filtered out electronically when measuring a static force curve.
\begin{figure}
\begin{center}
\includegraphics[width=16cm]{static_force_curves-eps-converted-to.pdf}
\caption{Simulation of a quasi-static force measurement. Left panel: The measured cantilever deflection $d$ plotted versus the position of the cantilever base $h$. Right panel: The tip-surface force plotted versus the tip position $z$. The tip-surface force model used in this simulation and all subsequent simulations is described by eq.~(\ref{DMT}) with parameters given in the text. The cantilever stiffness $k_c=1$~N/m .}
\label{static_force_curves}
\end{center}
\end{figure}
The accuracy of this quasi-static method requires that we calibrate the measurement of both $h$ and $d$. Measuring $h$ accurately requires a well-calibrated scanner. Measuring $d$ accurately requires a well calibrated detector. It is advantageous if both of these measurements are traceable back to one and the same calibration. One should be weary of systematic errors in the determination of $z=d+h$ if the measurement of $d$ is based on thermal noise calibration, and the measurement of $h$ relies on the scanner calibration. The conversion of the measured quasi-static deflection $d$ to force also requires a calibration of the static force constant, which is slightly different than the dynamic mode stiffness as determined by thermal noise calibration \cite{Walters:SmallCantilevers:96}. Finally, quasi-static force measurements suffer from the fact that the measured signals $d$ and $h$ are at very low frequencies where $1/f$ noise becomes a significant source of error.
\section{Fast force curves}\label{sec:fast_force_curves}
The slow nature of the quasi-static method makes it impractical for high-density measurements of the tip-surface force in the x-y plane, the so-called force-volume measurement.
There has therefore been considerable interest in increasing the speed of force measurement so that one can map the tip-surface force with high resolution. One approach to speeding up force measurement is to simply drive the base faster with a sinusoidal motion \cite{Krotil:pulse_force:99}. In the commercial modes that use this method, one typically drives the cantilever base with a frequency in the range 1-3 kHz, while measuring the cantilever deflection signal over a much broader frequency band.
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{fast_force_curve_70kHz-eps-converted-to.pdf}
\caption{Simulation of steady-state dynamics of a relatively soft cantilever, when driven by a sinusoidal base motion with frequency 1kHz. The tip-surface forces acting on this soft cantilever result in motion spread over a wide band where the cantilever transfer function is strongly frequency dependant. It is not possible to reconstruct the tip-surface force by filtering out the cantilever oscillations. Simulation parameters are: cantilever stiffness $k_c=1$N/m, resonance frequency $f_0=70$kHz, quality factor $Q=50$, tip surface force given in eq.~(\ref{DMT}) with parameters given below the equation. }
\label{fast_force_curve_70kHz}
\end{center}
\end{figure}
Figure~\ref{fast_force_curve_70kHz} shows a simulation of the cantilever dynamics for this mode of operation with the same tip-surface force and cantilever stiffness as for fig.~\ref{static_force_curves}. The cantilever resonance was assumed to have $f_0$=70~kHz and $Q$=50. Simulations were made by numerical integration of the nonlinear ordinary differential equation eq.~(\ref{equ_mot_2}) using CVODE, part of the Sundials suite of nonlinear solvers \cite{SUNDIALS}. Care was taken to properly treat the numerical integration in the neighbourhood of the contact point of the piece-wise-defined force model. To make the simulation more realistic we added noise to the simulated deflection signal. As explained in section \ref{sec:Sensitivity_Calibration}, the noise consisted of two contributions: a frequency-independent detector noise corresponding to an equivalent RMS deflection noise amplitude of 10~fm/$\sqrt{\text{Hz}}$; and a frequency-independent noise force with amplitude 23~fN/$\sqrt{\text{Hz}}$. The force noise is identical to that of fig.~\ref{cantilever_noise}, but the detector noise level is a factor of 15 lower. We choose this lower value of detector noise as it is achievable in the latest generation of commercial AFM's. The noise was added to the deflection signal after simulation of the noise-free nonlinear dynamics, so our simulation neglects the very small nonlinear response due to the noise force.
During one cycle of the base motion, the simulated deflection signal (fig.~\ref{fast_force_curve_70kHz} top panel) shows rapid oscillations at the resonance frequency of the cantilever. As the base approaches the surface there is a sudden downward deflection when the cantilever jumps to contact, followed by a broad hump where the cantilever is deflected in contact with the surface, and a sudden upward deflection at the pull-off point. The pull-off event excites the free oscillation of the cantilever at $f_0$, which rings down but does not completely extinguish before the next jump-to-contact event. The jump-to-contact also excites higher frequency oscillations characteristic of the tip being bound to the surface. In this simulation these oscillations decay with the same damping coefficient as the free oscillations, $\gamma_0$.
The middle panel of fig.~\ref{fast_force_curve_70kHz} shows the DFT of the deflection signal. The DFT was taken over exactly one period of the drive signal, so each point of the DFT corresponds to a harmonic of the drive frequency, $f_\mathrm{drive}=1$~kHz. In the spectrum one can see the free resonance frequency $f_0=70$~kHz, and a broader peak just below 175~kHz, containing jump-to-contact, pull-off, and oscillations in contact. Above 175~kHz the deflection signal is essentially detector noise.
The bottom panel of fig.~\ref{fast_force_curve_70kHz} shows the reconstruction of the tip-surface force from the spectrum using eq.~(\ref{inversion_1}). If we use the full response spectrum (blue line in the middle panel) the tip-surface force can be reconstructed as shown by the blue dots in the bottom panel of fig.~\ref{fast_force_curve_70kHz}. This inversion quite faithfully reproduces the actual tip-surface force used in the simulation (red curve) albeit with noise. We note that the reconstruction of the attractive part of the force curve, where the static force measurement method became unstable, only contains a few data points. It is not possible to better resolve this part of the force curve simply by faster sampling of the motion. One can see in the spectrum that we measure only noise at frequencies above 175~kHz. The measurement is not limited by the bandwidth of data acquisition, but rather by the mechanical bandwidth of the force transducer.
Here we should emphasize that in order to correctly reconstruct the tip-surface force, it is essential to capture the frequency components of the motion around resonance and above resonance, and it is necessary know the frequency dependence of the linear transfer function of the cantilever. Without the former we can not reconstruct the tip-surface force near contact which cause rapid acceleration. Without the later we can not separate the inertial, free damping and cantilever restoring forces from the tip-surface force, in their combined effect on cantilever motion. One should not simply filter out the cantilever oscillations as demonstrated by the green curves in fig.~\ref{fast_force_curve_70kHz}. Here we have applied a Hann filter with a cut-off frequency just below resonance. This filter nicely removes the 'ringing' in the deflection signal (green curve, top panel of fig.~\ref{fast_force_curve_70kHz}) while retaining a dip feature at jump-to-contact, and deeper dip before pull-off. A reconstruction of the tip-surface force from this filtered motion miss-represents the actual tip-surface force in the contact region completely (green curve, bottom panel fig.~\ref{fast_force_curve_70kHz}). The filtered force curve is some kind of approximately quasi-static force curve with rounded jump trajectories, and one should not interpret the different branches on approach and retract as a measurement of the tip-surface interaction.
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{fast_force_curve_300kHz-eps-converted-to.pdf}
\caption{Simulation of steady-state dynamics with the same tip-surface force as in fig.~\ref{fast_force_curve_70kHz}, but with a stiffer cantilever driven on resonance. For this case, all significant motion occurs well below resonance and the force measurement is essentially quasi-static. The stiffer cantilever gives poorer signal-to-noise ratio at these low frequencies. Simulation parameters are: cantilever stiffness $k_c=40$N/m, resonance frequency $f_0=300$kHz, quality factor $Q=468$, tip surface force given in eq.~(\ref{DMT}) with parameters below the equation. }
\label{fast_force_curve_300kHz}
\end{center}
\end{figure}
The only solution to the problem of measuring the strongly attractive tip-surface force is to use a stiffer cantilever. Increasing the stiffness for a fixed mass will increase the resonance frequency of the cantilever, thereby increasing its bandwidth as a force transducer. To demonstrate this we preformed a simulation identical to that of fig.~\ref{fast_force_curve_70kHz} in all respects, differing only in the cantilever which had stiffness $k_c=40$~N/m. The resonance was taken to have $f_0=300$~kHz, and $Q=468$, so that the noise force was the same $22$~fN/$\sqrt{\mathrm{Hz}}$. Figure~\ref{fast_force_curve_300kHz} shows the result of this simulation. The level of the deflection signal is smaller and the collision with the surface does not excite ringing above the noise level. The jump-to-contact and pull-off events give the same change in the deflection signal (apart from the noise). The high frequency part of the spectrum above $50$~kHz is only noise, so reconstruction of the tip-surface force from the full spectrum gives a rather noisy force curve (blue dots, bottom panel), this time however with more data in the contact region. If we apply the same filter with a cut-frequency of $60$~kHz, we see that the reconstructed force curve more faithfully reproduces the tip-surface force in the contact region. However the force curve suffers from rather much noise in the non-contact region and has difficulties reproducing the sharp kink at the contact point.
Increasing the stiffness caused the frequency content of the motion to occur well below resonance where we can assume $\mathbf{\hat{G}} \simeq \mathbb{1}$. In this case, the fast force curve is actually equivalent to rapidly taken quasi-static force curves. Comparison of the two simulations presented in this section demonstrates that in order to reconstruct tip-surface force from motion, it is necessary to know the transfer function $\mathbf{\hat{G}}$ over the entire frequency band where there is significant motion.
\section{Force measurement near resonance}\label{sec:force_meas_resonance}
When the frequency content of the motion is far from a resonance, one looses the advantage of the increased signal due to the large transfer gain of a high quality factor resonance ($\vert \hat{G}(\omega_0) \vert = Q$). Several methods have been devised which exploit this increased sensitivity. In one particularly popular method known as frequency modulation AFM, the cantilever is driven with one pure tone in a phase-locked loop (PLL). The PLL adjusts the drive frequency so that the phase difference between the drive and response is kept at a fixed value \cite{Albrecht:FMAFM:91}. A force-distance curve can be generated by analysing the frequency shift while slowly moving the base toward and away from the surface \cite{Giessibl:FMAFMreview:03}\cite{Sader:QuantitiveForceMeasurement:05}. While high density force-volume maps have been made using this method \cite{Albers:3DdataAcquisition:09}\cite{Gross:ChemStructureAFM:09}, they are quite time consuming to acquire.
Some measurement methods have been devised which excite the cantilever and measure at more than one resonance simultaneously \cite{martinez:2mode_afm:06}\cite{proksch:dual_acmode:06}\cite{Solares:TripleFreq:10}\cite{Martin:ProteinBimodal:11}. While interesting images can be made with these methods, monitoring response at two or three frequencies contains too little information to be able to fully reconstruct the force. Furthermore, the use of multiple eigenmodes complicates the dynamics considerably, making the force reconstruction problem and calibration much more difficult. Non-resonant, multi-frequency methods have also been devised which use one drive tone and measure the response at many harmonics (integer multiples) of the drive tone \cite{Legleiter:SPAMinFluids:06}\cite{sahin:torsional_cantilever:07}\cite{stark:InvertingDynForceMic_02}. When driving at resonance the response at these harmonics is effectively filtered out by the decaying transfer function ($G(\omega) \sim 1/\omega^2$ for $\omega > \omega_0$). In order to better read out response at higher harmonics additional eigenmodes of the cantilever have been used \cite{sahin:torsional_cantilever:07}\cite{Sarioglu:TappingModeForceCurves:12}. Here again, the use of multiple eigenmodes greatly complicates the force transducers dynamics and calibration. Finally, some techniques have considered two drive tones near a resonance with frequency modulation AFM \cite{Rodriguez:DART:07} and continuous band excitation near resonance \cite{Jesse:BandExcitation:07}, where the response was analysed within the context of linear dynamics.
\section{Intermodulation AFM}\label{sec:ImAFM}
The difficulties associated with broad band response caused by the nonlinear tip-surface force can be circumvented by exploiting the phenomena of frequency mixing, or intermodulation, which occurs in nonlinear systems that are driven by more than one pure tone. In this case, not only harmonics of the drive frequencies are generated by the nonlinearity, but also intermodulation products, which occur at integer linear combinations of the drive frequencies $\omega_1, \omega_2, \omega_3 \dots$,
\begin{equation}
\omega_{\rm IMP} = n_1 \omega_1 + n_2 \omega_2 + n_3 \omega_3 + \dots
\end{equation}
where $n_1, n_2, n_3 \dots$ are integers. The order of an intermodulation product is given by $\vert n_1 \vert + \vert n_2 \vert + \vert n_3 \vert + \dots$. With proper choice of the drive frequencies, we can induce a response where many intermodulation products of high order occur near resonance. This will ensure that information about strongly non-linear terms in the oscillator equation of motion can be measured with good SNR.
When measuring intermodulation products it is necessary that the drive frequencies be taken from a discrete set of tones which are the integer multiples of $\Delta \omega$. In this case, all intermodulation products will occur at frequencies that are contained in the DFT and one can therefore extract both the amplitude and phase of the response at intermodulation frequencies by calculating Fourier sums as in eq.~\ref{DFT}. This condition on the drive frequencies simply means that the applied drive waveform is periodic in the time window $T=2\pi / \Delta \omega $. The fundamental assumption we make in intermodulation AFM is that the weakly nonlinear motion will be periodic, with the same period as the drive waveform. We note that for strongly nonlinear motion, where bifurcations to sub-harmonic response occurs {\it en route} to chaos, our analysis based on the assumption of periodic response could break down.
Perhaps the most simple way to preform intermodulation AFM is to choose two drive tones, both integer multiples of a base tone, which are approximately centred around the cantilever resonance and separated in frequency by the linewidth of the resonance $\gamma_0$ \cite{Platz:ImAFM:08}. When the drive amplitudes are adjusted to give equal amplitude response at each frequency, the free oscillation of the cantilever will form a beating waveform in the time domain, as displayed in the top panel of fig.~\ref{intermodulation_force_curve}. Here the engaged response is simulated for the same van der Waals - DMT nonlinear force used for all previous simulations, with the stiffer cantilever, $k_c$=40 N/m, used for fig.~\ref{fast_force_curve_300kHz}. The response waveform of the cantilever engaging the surface (blue curve) has an envelope function slightly different from of the perfect beat, which can hardly be seen in the time-domain data. The spectrum however shows response not only at the two drive frequencies, but also several new peaks clustered around resonance, which are the intermodulation products of the two drive tones (inset of fig.~\ref{intermodulation_force_curve}, middle panel). Note that a cluster of intermodulation products can also be seen just above the noise floor, around the second harmonic of the center frequency, $2\bar{\omega}=(\omega_1+\omega_2)$. The amplitude of these peaks is however greatly diminished due to the fact that they occur off resonance.
The intermodulation spectrum near resonance represents essentially all the signal that is possible to measure above the detector noise floor. This data is only a {\em partial} spectrum of the actual motion. Detector noise and limited bandwidth of the force transducer inhibit our ability to measure the entire spectrum. If the entire spectrum could be measured in full detail, the reconstruction of force would be given by eq.~\ref{inversion_1}. The essential question then becomes: To what extent can we reconstruct the nonlinear force from analysis of the partial spectrum? The answer is that we can do an excellent job in spite of the narrow detection bandwidth because the motion spectrum contains many intermodulation products to rather high order. These intermodulation products contain much information about the nonlinearity that generated them.
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{intermodulation_force_curve.png}
\caption{Simulation of intermodulation AFM for the same cantilever and tip-surface force parameters as that of fig.~\ref{fast_force_curve_300kHz}, but now driven with two closely spaced tones centred on resonance. The free cantilever motion in the time domain is a beating waveform. The engaged motion has a slightly distorted beat envelope which appears in the frequency domain as intermodulation products, or mixing products of the two drive tones. The inset in the middle panel shows a detail of the engaged response spectrum near resonance. Only the amplitude is displayed, but the phase of each intermodulation product is also measured. Through analysis of this intermodulation spectrum we can reconstruct a polynomial approximation of the tip-surface force, or reconstruct the parameters of the force model used in the simulation.}
\label{intermodulation_force_curve}
\end{center}
\end{figure}
\section{Reconstruction from a partial spectrum}\label{sec:Reconstruction}
Suppose we have a partial spectrum $ \mathbf{\hat{d}}^{\mathrm{(part)}}$, which consists of $M$ components $\lbrace \mathrm{\hat{d}}_{k_1}, \mathrm{\hat{d}}_{k_2}, \dots, \mathrm{\hat{d}}_{k_M} \rbrace$ . We use this partial spectrum of engaged motion, the spectrum of free motion, and the calibrated transfer function to reconstruct the tip motion in the lab frame,
\begin{equation}
\mathbf{z} = \mathcal{F}^{-1} \left[ \mathbf{\hat{d}}^{\mathrm{(part)}} + ( \mathbf{\hat{G}} - \mathbb{\hat{1}} )^{-1} \mathbf{\hat{d}}^\mathrm{(free)} \right]
\label{partial_motion}
\end{equation}
In order that this be an accurate representation of the actual motion, it is important that our partial spectrum contains all significant peaks near resonance, and all peaks near higher harmonics of $\bar{\omega}$ are negligible. Here we note that because we are driving near resonance, we an easily measure both $\mathbf{\hat{d}}^\mathrm{(part)}$ and $\mathbf{\hat{d}}^{\rm (free)}$ from the same drive $\mathbf{\hat{h}}$, so it is possible to reconstruct the motion $z(t)$ from analysis of {\em only deflection signals}. There is no need for an independent measurement of the base motion $h(t)$. This means that the calibration of all measurements needed to generate $\mathrm{f}(z)$ are traceable to one thermal noise measurement which is easily performed before scanning, and can be frequently checked during a scan session. One does not rely in any way on the scanner calibration. For this reason, we will plot force directly in terms of the cantilever deflection in fig.~\ref{intermodulation_force_curve}.
One approach to reconstructing the nonlinear force is to assume a particular nonlinear force model described by a function that contains $P$ parameters $g_i$. We then define an error function which is the difference between the nonlinear force at the frequencies measured, and that calculated from the measured motion $\mathbf{z}$,
\begin{equation}
\mathbf{\hat{e}} = k_c \mathbf{\hat{G}}^{-1}
\left( \mathbf{\hat{d}}^{\mathrm{(part)}} -
\mathbf{\hat{d}}^{\mathrm{(free)}} \right) -
\mathcal{F} \left[ \mathrm{f}( g_1,g_1,\dots,g_P; \mathbf{z}) \right]
\end{equation}
Using numerical optimization we adjust the parameters of the model function so as to minimize the $M$ components of $\mathbf{\hat{e}}$ where we have data with good SNR \cite{Forchheimer:SpectFit:12}.
Figure~\ref{intermodulation_force_curve} shows the result of a least-square minimization of the sum $\sum_{k=1}^M(\mathrm{Re[\hat{e}}_k])^2+(\mathrm{Im[\hat{e}}_k])^2$ where the DMT model was assumed. The red curve is the actual force, and the green curve is the reconstructed force found by adjusting $P=4$ parameters of the DMT model to fit the noisy partial spectrum consisting of $M=28$ components around resonance. We do not adjust the tip radius because the DMT model does not depend on $R$ in an unambiguous way. We can see that in spite of the noise, the reconstruction is nearly perfect, missing the actual force only slightly near the contact point where the nonlinearity changes rapidly.
This reconstruction from simulated data with noise demonstrates that the partial spectrum does contain enough information with good enough SNR to fully reconstruct the force. It is however rather artificial in that we are starting with a model function which has the exact same form as the nonlinearity which actually caused the motion. In a real experiment we do not know what model best describes the true force. The spectral fitting method can therefore be dangerous to use without reference to the quality of the fit, because it will often converge to some parameters that describe a noiseless, ideal force curve having the shape of the model function. This type of analysis should therefore be complimented with an independent method, to be sure that the assumed model is physically meaningful. Nevertheless, if one does have a good model, this spectral fitting method represents a way to optimally extract the model parameters directly from the measured data.
The ultimate goal of quantitative dynamic AFM to devise a method to reconstruct force from motion, which does not depend on an assumed force model. This reconstruction is however an ill-posed problem, and certain assumptions must be made in order to determine tip-surface force from a partial representation of the motion. The art of finding a good reconstruction method is to make assumptions which are physically well motivated, but minimally constrain the possible nonlinear functions that can be achieved. For example, one may assume that the nonlinear tip-surface force can be well approximated by a polynomial in $z$ of degree $P$ on the finite interval of $z$ over which the cantilever oscillates \cite{Hutter:Reconstructing:2010}.
\begin{equation}
\mathrm{f}(z) \simeq k_{\rm c} \sum_{j=0}^{P-1} g_j z ^j
\textrm{ for }z \in \lbrace z_\mathrm{min} , z_\mathrm{max} \rbrace
\label{polynomial_force}
\end{equation}
Since this force is linear in the polynomial coefficients, its DFT can be written as a matrix equation,
\begin{equation}
\mathbf{\hat{f}} = k_\mathrm{c} \mathbf{\hat{H}} \mathbf{g}
\label{fhat_k}
\end{equation}
where the components of the vector $\mathbf{g}$ are the polynomial coefficients, and the matrix $\mathbf{\hat{H}} $ is a $M \times P$ matrix constructed from the DFT of $z^j$.
\begin{equation}
\hat{H}_{kj} = \mathcal{F}_k \left[ z ^j \right]
\label{H_matrix}
\end{equation}
The equation for the engaged response, eq.~(\ref{engaged_spectrum}), then becomes
\begin{equation}
\mathbf{\hat{d}} = \mathbf{\hat{d}}^{\rm (free)} + \mathbf{\hat{G}} \mathbf{\hat{H}} \mathbf{g}
\label{poly_d_hat}
\end{equation}
Taking only the measured intermodulation frequencies, we have a system of $M$ equations which can be inverted to solve for the polynomial coefficients,
\begin{equation}
\mathbf{g}=\mathbf{\hat{H}}^{+} \mathbf{\hat{G}}^{-1} (\mathbf{\hat{d}}^{\rm (part)} - \mathbf{\hat{d}}^{\rm free})
\label{poly_inversion}
\end{equation}
where the matrix $\mathbf{\hat{H}}^+$ is the pseudo inverse of $\mathbf{\hat{H}}$. The greater the number $M$ of Fourier coefficients in the partial spectrum $\mathbf{\hat{d}}^\text{(part)}$ which have good SNR, the larger the number $P$ of polynomial coefficients that can be determined. When $M>P$ the system of equations (\ref{poly_inversion}) becomes overdetermined and in this case the pseudo inverse finds the polynomial coefficients which best fit the measured data in a least-squared sense.
Applying this polynomial inversion to a partial spectrum such as that shown in middle panel inset of fig.~\ref{intermodulation_force_curve} will however not work directly. The drive scheme used to generate this spectrum produces only intermodulation products of odd order near resonance, where the dominant response comes from coefficients of $z^j$ with odd $j$ (first order response in perturbation theory). The partial spectrum of odd intermodulation products contains very little information about the even polynomial coefficients. We can however easily fix this problem by applying an additional, physically well-motivated constraint to our reconstruction algorithm. We know that the tip-surface interaction only occurs near one turning point of the cantilever oscillation. The constraint $\mathrm{f}=0$ for $d>0$ can then be used to determine the even $g$'s, given that the odd $g$'s are determined as described above.
The result of this reconstruction method is shown in the bottom panel of fig.~\ref{intermodulation_force_curve} with the blue line, where $P=19$ polynomial coefficients were determined from the intermodulation spectrum consisting of $M=28$ components. Over the entire interval of oscillation, the polynomial captures the shape of the DMT force curve quite well. The zoom of the contact region shows that the polynomial curve wiggles around the actual force curve. These wiggles are not due to the noise in the spectral data, they are rather the nature of a polynomial curve which is trying to approximate a function that is flat everywhere on the interval, except for one end where it changes rapidly and has a very sharp kink. Here we should point out that the piece-wise-defined force model used in the simulations is a very demanding nonlinearty to reconstruct. In real experiments we expect that actual forces will not have discontinuous force gradients as with this model function. Thus, the polynomial method can be quite good with experimental data. Using simulations such as these, we have determined that the polynomial generated by the inversion algorithm described above is very close to the optimal interpolation polynomial that approximates the nonlinear function. No significant improvement on this general method can be expected by assuming a different type of polynomial, for example a Chebychev polynomial \cite{Durig:InteractionSensing:00}.
The polynomial reconstruction has the distinct advantage that it captures the general shape of the force curve without assuming a particular force model. The equations are linear in the polynomial coefficients and the matrix inversion gives us a {\em unique} set of coefficients. The key step in the polynomial inversion is building the matrix $\mathbf{\hat{H}}$ and inverting it. The columns of $\mathbf{\hat{H}}$, eq.(\ref{H_matrix}), can be built up by recursive multiplication of the motion, eq.(\ref{partial_motion}), followed by a DFT. The DFT's can be done using the FFT alogrithm, and the matrix calculations are not very large, so the entire reconstruction algorithm is quite fast. The non-optimized Numpy code used to invert the data for fig.~\ref{intermodulation_force_curve} took less than 82 ms to run on a laptop. An AFM which is equipped to rapidly capture the intermodulation spectrum at each pixel allows the operator to get immediate feedback in the form of a calibrated force curve at any chosen image pixel \cite{Tholen:IMlockin:10}\cite{Intermodulation-Products}. Depending on the shape of the polynomial curve, the operator can switch inversion methods and quantitatively determine the physical parameters of an arbitrary force model which best fit the measured data.
\section{Conservative and dissipative forces}\label{sec:conservative_dissipative}
Our analysis of dynamic force measurements thus far has been based on simulations of the nonlinear dynamics of a cantilever with conservative tip-surface interactions. In real experiments we can expect that dissipative processes will be present in the tip-surface interaction, due to a non-elastic response of the material under deformation, the rearrangement of adsorbate molecules on the surface, or the making and breaking of chemical bonds when the tip makes and brakes contact with the surface. Many possible microscopic irreversible processes can give rise to dissipation, or energy loss of the oscillating cantilever. These dissipative interactions do not exist when the cantilever and surface forces are in static equilibrium. Dissipative interactions can only be probed with dynamic modes of AFM. Understanding the origin of and reconstructing dissipative interactions is one of the grand challenges of quantitiative AFM\cite{Sader:QuantitiveForceMeasurement:05}\cite{Durig:InteractionSensing:00}\cite{Raman:InvertingAmplitudePhase:08}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{intermodulation_disipative_force_curve.png}
\caption{Simulation of intermodulation AFM with the same parameters as in fig.~\ref{intermodulation_force_curve}, but this time with an additional Kelvin-Voigt surface damping force. The dissipative forces causes greater distortion of the free beating waveform. The intermodulation spectrum contains the information needed to reconstruct both the conservative force and the damping function. The reconstruction of the conservative force is completely uneffected by the dissipative interaction. Due to energy loss of the oscillator, the maximum tip-surface force has decreased in comparison with fig.~\ref{intermodulation_force_curve}. }
\label{intermodulation_disipative_force_curve}
\end{center}
\end{figure}
The intermodulation spectrum does contain information about dissipative tip-surface interactions, information which was not used in the reconstruction algorithms described previously for conservative forces. Within the context of the single eigenmode model which has only two degrees of freedom, $z$ and $\dot{z}$, dissipation can be described by a velocity-dependant tip-surface force. The determination of this force becomes the task of reconstructing an arbitrary nonlinear function $\mathrm{F}(z,\dot{z})$ from the measurable partial spectrum of intermodulation products. Let us assume a simple model, where the total tip-surface force can be described by a conservative force, plus a position-dependant viscous damping force.
\begin{equation}
\mathrm{F}(z,\dot{z}) = \mathrm{f}(z) + m\gamma(z)\dot{z}
\end{equation}
Approximating the damping function by a polynomial in $z$,
\begin{equation}
\gamma(z) \simeq \sum_{j=1}^{P} g^\mathrm{(dis)}z^j
\end{equation}
and using the same polynomial approximation for the conservative force with coefficients $\mathbf{g}^{\rm (con)}$, we arrive at an equation of motion similar to eq.(\ref{poly_d_hat}).
\begin{equation}
\mathbf{\hat{d}}=\mathbf{\hat{d}}^{{\rm (free)}}+\mathbf{\hat{G}\left\{ \mathbf{\hat{H}},\mathbf{\hat{I}}\right\} }\left\{ \begin{array}{c}
\mathbf{g^{\mathrm{(con)}}}\\
\mathbf{g^{\mathrm{(dis)}}}
\end{array}\right\}
\label{poly_d_hat2}
\end{equation}
where the braces denote the horizontal and vertical concatenation of the matrices and vectors respectively. The matrix $\mathbf{\hat{I}}$ is formed in a way analogous to the matrix $\mathbf{\hat{H}}$,
\begin{equation}
\hat{I}_{kj} =\omega_0^{-2} \mathcal{F}_k \left[ \dot{z} z ^j \right]
\end{equation}
Multiplication in the time domain is a convolution in the frequency domain, so the matrix $\mathbf{\hat{I}}$ can be calculated by convolution of the velocity vector $\hat{\dot{z}}_k=ik\Delta\omega \hat{z}_k$ with the columns of $\mathbf{\hat{H}}$.
The vector of polynomial coefficients describing the conservative force, $\mathbf{g^{\mathrm{(con)}}}$ and position dependant viscosity, $\mathbf{g^{\mathrm{(dis)}}}$ are then found by inverting eq.~(\ref{poly_d_hat2}),
\begin{equation}
\mathbf{\left\{ \begin{array}{c}
\mathbf{g^{\mathrm{(con)}}}\\
\mathbf{g^{\mathrm{(dis)}}}
\end{array}\right\} }=\mathbf{\left\{ \mathbf{\hat{H}},\mathbf{\hat{I}}\right\} ^{+}G^{-1}}\left(\mathbf{\hat{d}}-\mathbf{\hat{d}}^{{\rm (free)}}\right)
\end{equation}
Dissipative processes can also be incorporated in a model function describing the tip-surface force. If the dissipative process is parametrized, then the spectral fitting method can extract the parameter values which best fit the data. Because dissipation can be due to so many different factors, it is difficult to argue that one particular model is more valid than another. Nevertheless, one very attractive feature of the spectral fitting method is that it can accommodate essentially any interaction model, such as double-value functions and forces which turn on and off instantaneously depending on the motion history and the tip position and velocity. As long as the interaction can be programmed, numerical optimization can be preformed.
To demonstrate force reconstruction including dissipative forces, we simulate the nonlinear dynamics using the van der Waals - DMT - Kelvin-Voigt model for the tip-surface force \cite{melcher:061301},
\begin{equation}
\mathrm{f}(z)=\begin{cases}
\begin{array}{c}
-\frac{HR}{6z^2}\\
-\frac{HR}{6a_{0}^2}+\frac{4}{3}E^{*}\sqrt{R(a_0-z)^{3}} -\eta \dot{z}\sqrt{R(a_0-z)}
\end{array} & \begin{array}{c}
z \geq a_{0}\\
z<a_{0}
\end{array}\end{cases}
\label{DMT_KV}
\end{equation}
which is a simple modification to the previous model eq.~(\ref{DMT}) to include a viscous damping described by the parameter $\eta$, that increases as the square root of the penetration in the contact region.
Figure \ref{intermodulation_disipative_force_curve} shows the result of a simulation using this force. The simulation is identical to that done for fig.~\ref{intermodulation_force_curve}, the only difference being that we added the dissipative force with $\eta=1000$Pa-s. One can see a more significant change between the free oscillation and the engaged oscillation in the time domain (top panel). The frequency domain (middle panel) shows that nearly all motion is still contained in the narrow band around resonance, where several intermodulation products can be resolved. The reconstruction of the conservative force and damping function are shown in the bottom panels of fig.~\ref{intermodulation_disipative_force_curve} where we see that the spectral fitting method can accurately recover the nonlinear functions used to generate the simulated data. We note that the ability to recover the conservative force is completely unaffected by the addition of the dissipative force in the simulation. The polynomial damping function reconstructed from the intermodulation spectrum is able to capture the rapid turn-on of the dissipative interaction at the very end of the oscillation, but a polynomial can not fully approximate actual damping function used in the simulation. The inaccuracy in the reconstruction of the damping curve with both methods is greatest at the endpoint of the oscillation. This turning point, where the velocity goes to zero, is a singular point for any viscous damping model, so we can not expect to measure dissipation accurately in this region of the oscillation. Nevertheless, this simulation clearly demonstrates how intermodulation AFM enables one to determine the region of the oscillation cycle where the dissipation is occurring.
\section{Summary and conclusion}
In this paper we looked at the problem of determining tip-surface force from measurement and analysis of cantilever motion. In contrast to quasi-static measurement, dynamic measurement of the tip-surface force must properly account for the contributions of inertia and damping on the cantilever motion. We focused on the single eigenmode model as this is an excellent description of the cantilevers free linear dynamics in a frequency band surrounding a resonance. Our approach to understanding the engaged dynamics was to consider the full effect of the nonlinear tip-surface force, without approximating an effectively linear oscillator dynamics. Assuming that the steady-state response of the nonlinear oscillator will have the same periodicity as the drive signal, we considered a discrete frequency domain analysis of the motion. The effect of the nonlinearity on multiple drive tones contained in this discrete domain, was the generation intermodulation products that are also contained in the discrete domain.
A special case of Intermodulation AFM was simulated with a drive signal consisting of two pure tones close to resonance that generated a response with many odd-order intermodulation products near resonance. Using this scheme we can measure the amplitude and phase of intermodulation products to rather high order, in a frequency band where sensitivity is highest and where accurate calibration can be preformed. For these reasons, Intermodulation AFM stands out as being a very accurate and sensitive way to extract the tip-surface force. The information contained in the intermodulation spectrum allows one to reconstruct not only conservative, but also dissipative tip-surface forces. In this respect, dynamic force measurements offer a wealth of new possibilities for quantitative AFM that go far beyond what one can do with static force measurements.
In conclusion we would like to point out that the two methods of force reconstruction presented here, one based on numerical optimization with arbitrary force models and the other based on matrix inversion with a linear expansion of the force, are not the only possible methods of reconstructing nonlinearity from an intermodulation spectrum. One can look at intermodulation spectroscopy as technique for transposing information about the nonlinearity, to the narrow frequency band around resonance where it can be revealed above the detector noise floor. Various schemes for driving the nonlinear oscillator can be envisioned, which differently transpose information about the nonlinearity. Once transposed, the intermodulation spectrum can be analyzed in a variety of different ways to extract the desired information from the measurable nonlinear response. From this point of view the intermodulation measurement can be seen as a 'compression sensing technique' which efficiently extracts and stores relevant information from the detector signal, rejecting only noise. This compression aspect of intermodulation AFM gives it great advantage over competing force measurement methods in that it enables storage of the complete raw data set for offline analysis. Thus, different hypothesis can be tested on the same data set, and a much more detailed study of the tip-surface interaction at every image point can be made.
\section*{Acknowledgements}
We gratefully acknowledge financial support form the Swedish Research Council (VR) and the Swedish Governmental Agency for Innovation Systems (VINNOVA).
\section*{References}
\bibliographystyle{unsrt}
|
1,941,325,220,539 | arxiv |
\section{Introduction}
Tremendous strides have been made in estimating the 2D structure of in-the-wild scenes in terms of their constituent objects.
While recent work has also demonstrated impressive results in estimating 3D structures, particularly human bodies, the focus is often on bodies~\cite{hmr,kolotouros2019spin} and objects~\cite{choy20163d,girdhar16b} imaged in isolation or in controlled lab conditions~\cite{prox,PiGraph}.
To enable true 3D in-the-wild scene understanding, we argue that one must look at the {\em holistic} 3D scene, where objects and bodies can provide contextual cues for each other so as to correct local ambiguities. Consider the task of understanding the image in Figure \ref{fig:teaser}.
Independently estimated 3D poses of humans and objects are not necessarily consistent in the spatial arrangement of the 3D world of the scene (top row).
When processed holistically, one can produce far more plausible 3D arrangements by exploiting contextual cues, such as the fact that
humans tend to sit on park benches and ride bicycles rather than float mid-air.
In this paper, we present a method that can similarly recover the 3D spatial arrangement and shape of humans and objects in the scene from a single image.
We demonstrate our approach on challenging, in-the-wild images containing multiple and diverse human-object interactions. We propose an optimization framework that relies on automatically predicted 2D segmentation masks to recover the 3D pose, shape, and location of humans along with the 6-DoF pose and intrinsic scale of key objects in the scene.
Per-instance intrinsic scale allows one to convert each instance's local 3D coordinate system to coherent world coordinates, imbuing people and objects with a consistent notion of metric size.
There are three significant challenges to address. First is that the problem is inherently ill-posed as multiple 3D configurations can
result in the same 2D projection. It is attractive to make use of data-driven priors to resolve such ambiguities. But we immediately run into the second challenge: obtaining training data with 3D supervision is notoriously challenging, particularly for entire 3D scenes captured in-the-wild. Our key insight is that considering humans and objects jointly gives rise to 3D scene constraints that reduce ambiguity.
We make use of physical 3D constraints including a prior on the typical size of objects within a category. We also incorporate spatial constraints that encode typical modes of interactions with humans (e.g.~humans typically interact with a bicycle by grabbing its handlebars).
Our final challenge is that while there exists numerous mature technologies supporting 3D understanding of humans (including shape models and keypoint detectors), the same tools do not exist for the collective space of all objects.
In this paper, we take the first step toward building such tools by learning the natural size distributions of object categories without any supervision.
Our underlying thesis, bourne out by experiment, is that contextual cues arising from holistic processing of human-object arrangements can still provide enough information to understand objects in 3D.
We design an optimization-based framework, where we first reconstruct the humans and objects {\em locally} in each detected bounding box. For humans, we make use of state-of-the-art 3D human reconstruction output~\cite{joo2020exemplar}. For objects, we solve for the 6-DoF parameters of a category-specific 3D shape exemplar that fits the local 2D object instance segmentation mask \cite{kato2018nmr}. We then use a per-instance intrinsic scale to convert each local 3D prediction into a world coordinate frame by endowing metric size to each object and define a {\em global} objective function that scores different 3D object layouts, orientations, and shape exemplars. We operationalize constraints through loss terms in this objective.
We make use of gradient-based solvers to optimize for the globally consistent layout.
Although no ground truth is available for this task, we evaluate our approach qualitatively and quantitatively on the COCO-2017 dataset \cite{lin2014coco}, which contains challenging images of humans interacting with everyday objects obtained in uncontrolled settings.
We demonstrate the genericity of our approach by evaluating on objects from 8 categories of varying size and interaction types: baseball bats, bicycles, laptops, motorcycles, park benches, skateboards, surfboards, and tennis rackets.
\section{Related Work}
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{figures/overview.pdf}
\caption{\textbf{Overview of our method, PHOSA.} Given an image, we first detect instances of humans and objects~\cite{maskRCNN}. We predict the 3D pose and shape of each person~\cite{joo2020exemplar} and optimize for the 3D pose of each object by fitting to a segmentation mask~\cite{kato2018nmr}.
Then, we convert each 3D instance in its own local coordinate frame into world coordinates using an intrinsic scale. Using our Human-Object Spatial Arrangement optimization, we produce a globally consistent output, as shown here.
Our framework produces plausible reconstructions that capture realistic human-object interaction, preserve depth ordering, and obey physical constraints.
}
\label{fig:Overview}
\end{figure*}
\paragraph{\textbf{3D human pose and shape from a single image.}}
Recovering the 3D pose and shape of a person from a single image is a fundamentally
ambiguous task. As such, most methods employ statistical 3D body models with strong priors on shape learned from large-scale 3D scans and with known kinematic structure to model the articulation \cite{Anguelov:2005,zhou2010parametric,SMPL,SMPL-X:2019,joo2018}.
Seminal works in this area
\cite{grauman2003inferring,sigal2008combined,agarwal2006recovering} fit the parameters of a 3D body model to manually annotated silhouettes or keypoints \cite{sigal2008combined,guan2009estimating,zhou2010parametric}. Taking advantage of the progress in 2D pose estimation, \cite{SMPLify} propose a fully automatic approach where the parameters of the SMPL body model \cite{SMPL} are fit to automatically detected 2D joint locations in combination with shape and pose priors. More recent approaches employ expressive human models with faces and fingers \cite{xiang19monocular,pavlakos2019smlx}.
Another line of work develops a learning-based framework, using a feed-forward model to directly predict the parameters of the body model from a single image~\cite{Tung2017,pavlakos2018humanshape,hmr,omran2018nbf,varol2018bodynet,VNect_SIGGRAPH2017}.
\cite{guler2019holopose} combine human detection with 3D human pose and shape prediction. Much of the focus in such approaches is training on in-the-wild images of humans without paired 3D supervision. \cite{hmr} employ an adversarial prior on pose, \cite{pavlakos2018ordinal} explore using ordinal supervision, and \cite{pavlakos2019texturepose} use texture consistency.
To achieve state-of-the-art results, \cite{kolotouros2019spin,joo2020exemplar} have proposed hybrid approaches that combine feed-forward networks with optimization-based methods to improve 2D keypoint fit. In this work, we use the 3D regression network from \cite{joo2020exemplar} to recover the 3d pose and shape of humans.
Note that all of these approaches consider the human in isolation. More related to our work are methods that recover 3D pose and shape of multiple people \cite{zanfir18monocular,zanfir18deep,jiang2020coherent}. These approaches use collision constraints to avoid intersection, and use bottom-up grouping or ordinal-depth constraints to resolve ambiguities.
We take inspiration from these works for the collision loss and the depth ordering loss from \cite{jiang2020coherent}, but our focus is on humans and objects in this work.
\paragraph{\textbf{3D objects from a single image.}}
There has also been significant literature in single-view 3D object reconstruction. Earlier methods optimize a deformable shape model to image silhouettes \cite{lim2013parsing,kholgade20143d,aubry2014seeing,CSDM,izadinia2017im2cad}. Recent approaches train a deep network to predict the 3D shape from an image \cite{choy20163d,girdhar16b,psgn,groueix2018papier,park2019deepsdf,mescheder2019occupancy,chen2019learning}. Most of these approaches require 3D supervision or multi-view cues and are trained on synthetic datasets such as \cite{wu20153d,song2017semantic}. Many of these approaches reason about 3D object shape in isolation, while in this work we focus on their spatial arrangements. There are several works \cite{factored3dTulsiani17,kundu20183d,meshrcnn} that recover the 3D shape of multiple objects but still reason about them independently. More recently, \cite{kulkarni20193d} proposes a graph neural network to reason about the relationships between object to infer their layout, trained on a synthetic dataset with no humans. In this work we explore 3D spatial arrangements of humans and objects in the wild. As there is no 3D supervision for these images, we take the traditional category-based model-fitting approach to get the initial 6-DoF pose of the objects and refine their spatial arrangements in relation to humans and other objects in the scene.
\paragraph{\textbf{3D human-to-object interaction.}}
Related to our work are earlier approaches that infer about the 3D geometry and affordances from observing people interacting with scenes over time \cite{gupta20113d,delaitre2012scene,fouhey2014people,li2019estimating}.
These approaches are similar to our work in spirit in that they use the ability to perceive humans to understand the 3D properties of the scene.
The majority of the recent works rely on a pre-captured 3D scene to reason about 3D human-object interaction. \cite{PiGraph}
use RGB-D sensors to capture videos of people interacting with indoor scenes and use this data to learn a probabilistic model that reasons about how humans interact with its environment. Having access to 3D scenes provides scene constraints that improve 3D human pose perception \cite{yamamoto2000scene,rosenhahn2008markerless,kjellstrom2010tracking}.
The PROX system \cite{prox} demonstrates this idea through an optimization-based approach to improve 3D human pose estimation conditioned on a known 3D scene captured by RGB-D sensors. While we draw inspiration from their contact terms to model human-object interaction, we critically do not assume that 3D scenes are available.
\cite{Rosinol-RSS-20} is another recent approach that constructs a 3D scene graph of humans and objects from video and inertial data captured indoors.
We, on the other hand, experiment on single images captured in uncontrolled {\em in-the-wild} environments, often outdoors.
More related to our approach are those that operate on images. There are several hand-object papers that recover both 3D object and 3D hand configurations \cite{hasson2019learning}. In this work we focus on 3D spatial arrangements of humans and objects.
Imapper~\cite{monszpart2019imapper} uses priors built from RGB-D data \cite{PiGraph} to recover a plausible global 3D human motion and a global scene layout from an indoor video.
Most related to our work is \cite{chen2019holistic++}, who develop an approach that recovers a parse graph that represents the 3D human pose, 3D object, and scene layout from a single image. They similarly recover the spatial arrangements of humans and objects, but rely on synthetic 3D data to (a) learn priors over human-object interaction and (b) train 3D bounding box detectors that initialize the object locations. In this work we focus on recovering 3D spatial arrangements of humans and objects in the wild where no 3D supervision is available for 3D objects and humans and their layout. Due to the reliance on 3D scene capture and/or 3D synthetic data, many previous work on 3D human object interaction focus on indoor office scenes. By stepping out of this supervised realm, we are able to explore and analyze how 3D humans and objects interact in the wild.
\section{Method}
Our method takes a single RGB image as input and outputs humans and various categories of objects in a common 3D coordinate system. We begin by separately estimating 3D humans and 3D objects in each predicted bounding box provided by an object detector~\cite{maskRCNN}.
We use a state-of-the art 3D human pose estimator~\cite{joo2020exemplar} to obtain 3D humans in the form of a parametric 3D human model (SMPL \cite{SMPL}) (in \secref{sec:anthro_rcnn}), and use a differentiable renderer to obtain 3D object pose (6-DoF translation and orientation)
by fitting 3D mesh object models to predicted 2D segmentation masks~\cite{kirillov2019pointrend} (in \secref{sec:nmr_opt}).
The core idea of our method is to exploit the interaction between humans and objects to spatially arrange them in a common 3D coordinate system by optimizing for the per-instance \textit{intrinsic scale}, which specifies their metric size (in \secref{sec:human_object}). In particular, our method can also improve the performance of 3D pose estimation for objects by exploiting cues from the estimated 3D human pose. See Fig.~\ref{fig:Overview} for the overview of our method.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{figures/surfing_scale_figure.pdf}
\caption{\textbf{Ambiguity in scale.} Recovering a 3D scene from a 2D image is fundamentally ambiguous because multiple 3D interpretations can have the same 2D projection. Consider the photo of the surfer on the left. A large surfboard far away (c) and a small one closer (a) have the same 2D projection (second panel) as the correct interpretation (b). In this work, we aim to resolve this scale ambiguity by exploiting cues from human-object interaction, essentially using the human as a ``ruler."}
\label{fig:scale_fig}
\end{figure*}
\subsection{Estimating 3D Humans}
\label{sec:anthro_rcnn}
Given a bounding box for a human provided by a detection algorithm~\cite{maskRCNN}, we estimate the 3D shape and pose parameters of SMPL \cite{SMPL} using~\cite{joo2020exemplar}. The 3D human is parameterized by pose $\theta \in \mathbb{R}^{72}$ and shape $\beta \in \mathbb{R}^{10}$, as well as as a weak-perspective camera $\Pi = [\sigma, t_x, t_y] \in \mathbb{R}^3$ to project the mesh into image coordinates. To position the humans in the 3D space, we convert the weak-perspective camera to the perspective camera projection by assuming a fixed focal length $f$ for all images, where the distance of the person is determined by the reciprocal of the camera scale parameter $\sigma$. Thus, the 3D vertices of the SMPL model for the $i$-th human is represented as:
\begin{equation}
V^i_h = \mathcal{M}(\beta^i, \theta^i) + \begin{bmatrix}t^i_x & t^i_y & \nicefrac{f}{\sigma^i}\end{bmatrix},
\label{eq:SMPL_pers}
\end{equation}
where $\mathcal{M}$ is the differentiable SMPL mapping from pose and shape to a human mesh with 6890 vertices in meters. The SMPL shape parameter $\beta$ controls the height and size of the person. In practice, this is difficult to reliably estimate from an image since a tall, far-away person and a short, closeby person may project to similar image regions (\figref{fig:teaser} and \figref{fig:loss}).
To address this ambiguity, we fix the estimated SMPL pose and shape and introduce an additional per-human intrinsic scale parameter $s^j \in \mathbb{R}$
that changes the size and thus depth of the human in world coordinates: ${V^i_h}^* = s^j V^i_h$. While the shape parameter $\beta$ also captures size, we opt for this parameterization
as it can also be applied to objects (described below) and thus optimized to yield a globally consistent layout of the scene.
\subsection{Estimating 3D Objects}
\label{sec:nmr_opt}
We consider each object as a rigid body mesh model and estimate the 3D location $\mathbf{t} \in \mathbb{R}^3$, 3D orientation $\mathcal{R} \in SO(3)$, and an intrinsic scale $s\in\mathbb{R}$. The intrinsic scale converts the local coordinate frame of the template 3D mesh to the world frame. We consider single or multiple exemplar mesh models for each object category, pre-selected based on the shape variation within each category. For example, we use a single mesh for skateboards but four meshes for motorcycle. The mesh models are obtained from \cite{warehouse,free3d,kundu20183d}
and are pre-processed to have fewer faces (about 1000) to make optimization more efficient. See \figref{fig:part_labels} for some examples of mesh models and the supplementary for a full list. The 3D state of the $j$th object is represented as:
\begin{equation}
V^j_o = s^j \left(\mathcal{R}^j \mathcal{O} ( c^j, k^j ) + \mathbf{t}^j\right),
\end{equation}
where $\mathcal{O} ( c^j, k^j )$ specifies the $k^j$-th exemplar mesh for category $c^j$. Note that the object category $c^j$ is provided by the object detection algorithm~\cite{maskRCNN}, and $k^j$ is automatically determined in our optimization framework (by selecting the exemplar that minimizes reprojection error).
Our first goal is to estimate the 3D pose of each object independently. However, estimating 3D object pose in the wild is challenging because
(1) there are no existing parametric 3D models for target objects; (2) 2D keypoint annotations or 3D pose annotations for objects in the wild images are rare; and (3) occlusions are common in cluttered scenes, particularly those with humans.
We propose an optimization-based approach using a differentiable renderer~\cite{kato2018nmr} to fit the 3D object to instance masks from~\cite{kirillov2019pointrend} in a manner that is robust to partial occlusions. We began with an pixel-wise L2 loss over rendered silhouettes $S$ versus predicted masks $M$, but found that it ignored boundary details that were important for reliable pose estimation. We added a symmetric chamfer loss \cite{gavrila2000pedestrian} which focuses on boundary alignment, but found it computationally prohibitive since it required recomputing a distance transform of $S$ at each gradient iteration. We found good results with an L2 mask loss augmented with a {\em one-way} chamfer loss that computes the distance of each silhouette boundary pixel to the nearest mask boundary pixel, which requires computing a single distance transform once for the mask $M$. Given a
no-occlusion indicator $I$ (0 if pixel only corresponds to mask of different instance, 1 else), we write our loss as follows:
\begin{equation}
\loss{occ-sil} = \sum \left(I\circ S - M\right)^2 + \sum_{p\in E(I\circ S)} \min_{\hat{p}\in E(M)} \norm{p - \hat{p}_2}
\label{loss:occ_sil}
\end{equation}
where $E(M)$ computes the edge map of mask $M$. Note that this formulation can handle partial occlusions by object categories for which we do not have 3D models, as illustrated in \figref{fig:occ-sil-loss}. We also add an offscreen penalty to avoid degenerate solutions when minimizing the chamfer loss. To estimate the 3D object pose, we minimize the occlusion-aware silhouette loss:
\begin{equation}
\{ \mathcal{R}^j, \mathbf{t}^j \}^* = \operatorname*{argmin}_{\mathcal{R}, \mathbf{t} } \loss{occ-sil} \left( \Pi_{\text{sil}} (V^j_o), M^j \right),
\label{eq:sil}
\end{equation}
where $\Pi_{\text{sil}}$ is the silhouette rendering of a 3D mesh model via a perspective camera with a fixed focal length (same as $f$ in \eref{eq:SMPL_pers}) and $M^j$ is a 2D instance mask for the $j$-th object. We use PointRend \cite{kirillov2019pointrend} to compute the instance masks. See \figref{fig:occ-sil-loss} for a visualization and the supplementary for more implementation details. While this per-instance optimization provides a reasonable 3D pose estimate, the mask-based 3D object pose estimation is insufficient since
there remains a fundamental ambiguity in determining the global location relative to other objects or people, as shown in \figref{fig:scale_fig}. In other words, reasoning about instances in isolation cannot resolve ambiguity in the intrinsic scale of the object.
\begin{figure}[t]
\centering
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/occ_sil.pdf}
\caption{\textbf{Occlusion-Aware Silhouette Loss for optimizing object pose.} Given an image, a 3D mesh model, and instance masks, our occlusion-aware silhouette loss finds the 6-DoF pose that most closely matches the target mask (bottom right). To be more robust to partial occlusions, we use a no-occlusion indicator (top right) to ignore regions that correspond to other object instances, including those for which we do not have 3D mesh models (e.g.~dogs).}
\label{fig:occ-sil-loss}
\end{minipage}\hfill
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/meshes.pdf}
\caption{
\textbf{Part labels for fine-grained interaction.}
To model human-object interactions, we label each object mesh with interaction regions corresponding to parts of the human body. Each color-coded region on the person (top left) interacts with the matching colored region for each object.
The interaction loss pulls pairs of corresponding parts closer together. To better capture variation in shape, we can use multiple mesh instances for the same category (e.g.~the motorcycles shown on the right). See the supplementary in \secref{sec:supp} for all mesh models and interaction correspondences.
}
\label{fig:part_labels}
\end{minipage}
\end{figure}
\subsection{Modeling Human-Object Interaction for 3D Spatial Arrangement}
\label{sec:human_object}
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{figures/before_after.pdf}
\caption{\textbf{Recovering realistic human-object spatial arrangements by reasoning about depth ordering and interpenetration.} Given the image on the left as input, we first initialize the spatial arrangement by independently estimating the 3D human and object poses (Independent Composition).
By incorporating physical priors such as avoiding mesh inter-penetration as well as preserving the depth ordering inferred from the predicted segmentation mask, we can produce the much more plausible spatial arrangement shown on the right.}
\label{fig:loss}
\end{figure}
Reasoning about the 3D poses of humans and objects independently may produce inconsistent 3D scene arrangements.
In particular, objects suffer from a fundamental depth ambiguity: a large object further away can project to the same image coordinates as a small object closer to the camera (see \figref{fig:scale_fig}). As such, the absolute 3D depth cannot be estimated. The interactions between humans and objects can provide crucial cues to reason about the relative spatial arrangement among them. For example, knowing that two people are riding on the same bike suggests that they should have similar depths
in Fig~\ref{fig:Overview}. This pair-wise interaction cue can be propagated to determine the spatial arrangement of multiple humans and objects together.
Furthermore, given the fact that a wide range of 2D and 3D supervision exists for human pose,
we can leverage 3D human pose estimation to further adjust the orientation of the 3D object. For instance, knowing that a person is sitting down on the bench can provide a strong prior to determine the 3D orientation of the bench. Leveraging this requires two important steps: (1) identifying a human and an object that are interacting and (2) defining an objective function to correctly adjust their spatial arrangements.
\paragraph{\textbf{Identifying human-object interaction.}}
We hypothesize that an interacting person and object must be nearby in the world coordinates.
In our formulation, we solve for the 6-DoF object pose and the intrinsic scale parameter $s^j$ which places the object into world coordinates and imbues the objects with metric size.
We use 3D bounding box overlap between the person and object to determine whether the object is interacting with a person. The size of the per-category 3D bounding box in world coordinates is set larger for larger object categories. See the supplementary materials in \secref{sec:supp} for a full list of 3D box sizes.
Endowing a reasonable initial scale is important for identifying human-object interaction because
if the object is scaled to be too large or too small in size, it will not be nearby the person. We first initialize the scale using common sense reasoning, via an internet search to find the average size of objects (e.g.~baseball bats and bicycles are $\sim$0.9 meters and $\sim$2 meters long respectively). Through our proposed method, the per-instance intrinsic scales change during optimization. From the final distribution of scales obtained over the test set, we compute the empirical mean scale and repeat this process using this as the new initialization (\figref{fig:scales}).
\paragraph{\textbf{Objective function to optimize 3D spatial arrangements.}}
Our objective includes multiple terms to provide constraints for interacting humans and objects:
\begin{equation}
\label{eq:opt_function}
L = \lambda_1 \loss{occ-sil} + \lambda_2 \loss{interaction} + \lambda_3 \loss{scale} + \lambda_4 \loss{depth} + \lambda_5 \loss{collision} .
\end{equation}
We optimize \eqref{eq:opt_function} using a gradient-based optimizer~\cite{kingma2014adam} w.r.t.~intrinsic scale $s^i\in\mathbb{R}$ for the $i$-th human and intrinsic scale $s^j\in\mathbb{R}$, rotation $\mathcal{R}^j\in SO(3)$, and translation $\mathbf{t}^j\in \mathbb{R}^3$ for the $j$-th object instance jointly. The object poses are initialized from \secref{sec:nmr_opt}.
$\loss{occ-sil}$ is the same as \eqref{eq:sil} except without the chamfer loss which didn't help during joint optimization. We define the other terms below.
\textbf{Interaction loss}: We first introduce a coarse, instance-level interaction loss to pull the interacting object and person close together:
\begin{equation}
\loss{coarse inter} = \sum_{h\in \mathcal{H}, o\in \mathcal{O}} \mathbbm{1}(h, o)\norm{C(h) - C(o)}_2,
\end{equation}
where $\mathbbm{1}(h, o)$ identifies whether human $h$ and object $o$ are interacting according to the 3D bounding box overlap criteria described before.
Humans generally interact with objects in specific ways.
For example, humans hold tennis rackets by the handle. This can be used as a strong prior for human-object interaction and adjust their spatial arrangement. To do this, we annotate surface regions on the SMPL mesh and on our 3D object meshes where there is likely to be interaction, similar to PROX~\cite{prox}.
These include the hands, feet, and back of a person or the handlebars and seat of a bicycle, as shown in Fig.~\ref{fig:part_labels}.
To encode spatial priors about human-object interaction (e.g.~people grab bicycle handlebars by the hand and sit on the seat), we enumerate pairs of object and human part regions that interact (see supplementary for a full list).
We incorporate a fine-grained, parts-level interaction loss by using the part-labels (\figref{fig:part_labels}) to pull the interaction regions closer to achieve better alignment:
\begin{equation}
\loss{fine inter} = \sum_{h\in \mathcal{H}, o\in \mathcal{O}} \sum_{\substack{\mathcal{P}_h, \mathcal{P}_o \in \\ \mathcal{P}(h, o)}} \mathbbm{1}(\mathcal{P}_h, \mathcal{P}_o) \norm{C(\mathcal{P}_h) - C(\mathcal{P}_o)}_2,
\end{equation}
where $\mathcal{P}_h$ and $\mathcal{P}_o$ are the interaction regions on the person and object respectively. Note that we define the parts interaction indicator $\mathbbm{1}(\mathcal{P}_h, \mathcal{P}_o)$ using the same criteria as instances, i.e.~3D bounding box overlap. The interactions are recomputed at each iteration. Finally, $\loss{interaction}=\loss{coarse inter} + \loss{fine inter}$.
\textbf{Scale loss}:
We observe that there is a limit to the variation in size within a category. Thus, we incorporate a Gaussian prior on the intrinsic scales of instances in the same category using a category-specific mean scale:
\begin{equation}
\loss{scale} = \sum_c \sum_{j \in [|\mathcal{O}_c|]}\norm{s^j - \Bar{s}_c}_2.
\end{equation}
We initialize the intrinsic scale of all objects in category $c$ to $\bar{s}_c$.
The mean object scale $\bar{s}_c$ is initially set using common sense estimates of object size. In \figref{fig:scales}, we visualize the final distribution of object sizes learned for the COCO-2017 \cite{lin2014coco} test set after optimizing for human interaction. We then repeat the process with the empirical mean as a better initialization for $\bar{s}_c$.
We also incorporate the scale loss for the human scales $s^i$ with a mean of 1 (original size) and a small variance.
\begin{figure*}[t]
\centering
\includegraphics[width=0.9\textwidth]{figures/scale_histograms/intrinsic_scales_hist.pdf}
\caption{\textbf{Learned size distribution from human interaction:} Here, we visualize the distribution of object sizes across the COCO-2017~\cite{lin2014coco} test set at the end of optimization. The red caret denotes the size resulting from the hand-picked scale used for initialization. The blue line denotes the size produced by the empirical mean scale of all category instances at the end of optimization. We then use the empirical mean as the new initialization for intrinsic object scale.
}
\label{fig:scales}
\end{figure*}
\textbf{Ordinal Depth loss}:
The depth ordering inferred from the 3D placement should match that of the image.
While the correct depth ordering of people and objects would also minimize the occlusion-aware silhouette loss, we posit that the ordinal depth loss introduced in Jiang et al~\cite{jiang2020coherent} can help recover more accurate depth orderings from the modal masks. Using an ordinal depth can give smoother gradients to both the occluder and occluded object.
Formally, for each pair of instances, we compare the pixels at the intersections of the silhouettes with the segmentation mask. If at pixel $p$, instance $i$ is closer than instance $j$ but the segmentation masks at $p$ show $j$ and not $i$, then we apply a ranking loss on the depths of both instances at pixel $p$:
{\small
\begin{equation}
\loss{depth} =
{\sum_{o^i \in \mathcal{H} \cup \mathcal{O}} \sum_{o^j \in \mathcal{H} \cup \mathcal{O}} \sum_{\substack{p\in \text{Sil}(o^i) \\ \cap \text{Sil}(o^j)}} \mathbbm{1}(p, o^i, o^j)
\log\left(1+\exp(D_{o^j}(p) - D_{o^i}(p))\right)},
\end{equation}}
where $\text{Sil}(o)$ is the rendered silhouette of instance $o$, $D_o(p)$ is the depth of instance $o$ at pixel $p$,
and $\mathbbm{1}(p, o^i, o^j)$ is 1 if the segmentation label at pixel $p$ is $o^j$ but $D_{o^i}(p) < D_{o^j}(p)$.
See \cite{jiang2020coherent} for more details.
\textbf{Collision loss}: Promoting proximity between people and objects can exacerbate the problem of instances occupying the same 3D space. To address this, we penalize poses that would
human and/or object
interpenetration using the collision loss $\loss{collision}$ introduced in \cite{ballan2012motion,tzionas2016capturing}. We use a GPU implementation based on \cite{SMPL-X:2019} which detects colliding mesh triangles, computes a 3D distance field, and penalizes based on the depth of the penetration. See \cite{SMPL-X:2019} for more details.
\section{Evaluation}
\label{sec:quantitative}
In this section, we provide quantitative and qualitative analysis on the performance of our method on the COCO-2017 \cite{lin2014coco} dataset. We focus our evaluation on 8 categories: baseball bats, benches, bicycles, laptops, motorcycles, skateboards, surfboards, and tennis rackets.
These categories cover a significant variety in size, shape, and types of interaction with humans.
\subsection{Quantitative Analysis}
\begin{table*}[t]
\centering
\resizebox{\textwidth}{!}{
\footnotesize{
\begin{tabularx}{\textwidth}{m{25mm}<{\centering}@{}c*9{>{\centering\arraybackslash}X}@{}}
\toprule
Ours vs. & Bat & Bench & Bike & Laptop & Motor. & Skate.\ & Surf. & Tennis & Avg.\\\midrule
Indep. Comp.& 83 & 74 & 73 & 61 & 74 & 71 & 82 & 82 & 75.0\\
\cmidrule(lr){1-10}
No $\loss{occ-sil}$& 79 & 74 & 87 & 76 & 70 & 96 & 80 & 77 & 79.9\\
No $\loss{interaction}$& 82 & 59 & 57 & 46 & 71 & 71 & 76 & 68 & 66.3 \\
No $\loss{scale}$& 77 & 49 & 54 & 51 & 54 & 55 & 55 & 56 & 56.4\\
No $\loss{depth}$& 50 & 55 & 55 & 55 & 52 & 50 & 51 & 50 & 52.3 \\
No $\loss{collision}$& 52 & 40 & 51 & 51 & 50 & 52 & 50 & 50 & 49.5 \\
\bottomrule\\
\end{tabularx}
}
}
\caption{\textbf{Percentage of images for which our proposed method performs better on a subset of COCO 2017 test set.} In this table, we evaluate our approach against an independent composition and ablations of our method. The independent composition estimates human and object pose independently using \secref{sec:anthro_rcnn} and \secref{sec:nmr_opt} and sets the intrinsic scale to the empirical mean category scale learned by our method.
The ablations each drop one loss term from our proposed method.
In each row, we compute the average percentage of images for which our method performs better across a random sample of COCO-2017~\cite{lin2014coco} test set images. A number greater than 50 implies that our proposed method performs better than the independent composition or that the ablated loss term is beneficial for that category.
}
\label{tab:interaction}
\end{table*}
Since 3D ground truth annotations for both humans and objects do not exist for in-the-wild images, we used a forced-choice evaluation procedure on COCO-2017~\cite{lin2014coco} images.
To test the contribution of the holistic processing of human and object instances, we evaluate our approach against an ``independent composition," which uses our approach from \secref{sec:anthro_rcnn} and \secref{sec:nmr_opt} to independently estimate the human and object poses.
To make the independent composition competitive, we set the intrinsic scale to be the empirical mean per-category scale learned over the test set by our proposed method in \secref{sec:human_object}. This actually gives the independent composition global information
through human-object interaction. This is the best that can be done without considering all instances holistically.
We collected a random subset of images from the COCO 2017 test set in which at least one person and object overlap in 2D. For each of our object categories, we randomly sample 50 images with at least one instance of that category. For each image, annotators see the independent composition and the result of our proposed method in random order, marking whether our result looks better than, equal to, or worse than the independent composition (see supplementary for screenshots of our annotating tool). We compute the average percentage of images for which our method performs better (treating equal as 50) in \tabref{tab:interaction}. Overall, we find that our method safely outperforms the independent composition.
To evaluate the importance of the individual loss terms, we run an ablative study.
We run the same forced choice test for the full proposed method compared with dropping a single loss term in \tabref{tab:interaction}.
We find that omitting the occlusion-aware silhouette loss and the interaction loss has the most significant effect. Using the silhouette loss during global optimization ensures that the object poses continue to respect image evidence, and the interaction loss encodes the spatial arrangement of the object relative to the person. We did observe that the interaction loss occasionally pulls the object too aggressively toward the person for the laptop category.
The scale loss appears to have a positive effect for most categories. Note that because we initialized the scale to the empirical mean, the effects of the scale loss are not as pronounced as they would be if initialized to something else.
The depth ordering loss gave a moderate boost while the collision loss had a less pronounced effect. We attribute this to the collision loss operating only on the surface triangles and thus prone to getting stuck in local minima in which objects get embedded inside the person (especially for benches).
\subsection{Qualitative Analysis}
In \figref{fig:main_results}, we demonstrate the generality of our approach on the COCO 2017 dataset by reconstructing
the spatial arrangement of multiple people and objects engaged in a wide range of 3D human-object interactions.
We find that our method works on a variety of everyday objects with which people interact, ranging from handheld objects (baseball bats, tennis rackets, laptops) to full-sized objects (skateboards, bicycles, motorcycles) to large objects (surfboards, benches).
In the middle column of \figref{fig:main_results}, we visualize the spatial arrangement produced by the independent composition introduced in \secref{sec:quantitative}. We find that independently estimating human and object poses is often insufficient for resolving fundamental ambiguities in scale. Explicitly reasoning about human-object interaction produces more realistic spatial arrangements.
Please refer to the supplementary materials in \secref{sec:supp} for discussion of failure modes and significantly more qualitative results.
\section{Discussion}
In summary, we have found that 2D and 3D technologies for understanding objects and humans have advanced considerably. Armed with these advances, we believe the time is right to start tackling broader questions of holistic 3D scene-understanding---and moving such questions from the lab to uncontrolled in-the-wild imagery! Our qualitative analysis suggests that 3D human understanding has particularly matured, even for intricate interactions with objects in the wild. Detailed recovery of 3D object shape is still a challenge, as illustrated by our rather impoverished but surprisingly effective exemplar-based shape model. It would be transformative to learn statistical models (a ``SMPL-for-objects"), and we take the first step by learning the intrinsic scale distribution from data.
A number of conclusions somewhat surprised us. First, even though object shape understanding lacks some tools compared to its human counterpart (such as statistical 3D shape models and keypoint detectors), 2D object instance masks combined with a differentiable renderer and a 3D shape library proves to be a rather effective initialization for 3D object understanding. Perhaps even more remarkable is the scalability of such an approach. Adding new objects and defining their modes of interactions is relatively straightforward, because it is far easier to ``paint" annotations on 3D models than annotate individual image instances. Hence 3D shapes provide a convenient coordinate frame for {\em meta}-level supervision. This is dramatically different from the typical supervised pipeline, in which adding a new object category is typically quite involved.
While the ontology of objects will likely be diverse and continue to grow and evolve over time, humans will likely remain a consistent area of intense focus and targeted annotation. Because of this, we believe it will continue to be fruitful to pursue approaches that leverage contextual constraints from humans that act as ``rulers" to help reason about objects. In some sense, this philosophy harkens back to Protagoras's quote from Ancient Greece\,---``man is the measure of all things"!\\
\noindent \textbf{Acknowledgements:}
We thank Georgia Gkioxari and Shubham Tulsiani for insightful discussion and Victoria Dean and Gengshan Yang for useful feedback. We also thank Senthil Purushwalkam for deadline reminders.
This work was funded in part by the CMU Argo AI Center for Autonomous Vehicle Research.
\section{Supplemental Material}
\label{sec:supp}
In this section, we describe implementation details in \secref{sec:implementation_details} and the mesh processing pipeline in \secref{sec:mesh_processing}. We also include more qualitative results in \secref{sec:more_qualitative} and describe a few failure modes in \figref{fig:failure_modes}.
\subsection{Implementation details}
\label{sec:implementation_details}
We represent rotations for the object poses using the 6-DoF rotation representation introduced in \cite{zhou2019continuity}. We optimize the the occlusion aware silhouette loss using the ADAM optimizer~\cite{kingma2014adam} with learning rate 1e-3 for 100 iterations. We compute the edge maps $E(M)$ using $\text{MaxPool}(M) - M$ with a filter size of 7. Since the occlusion-aware silhouette loss is susceptible to getting stuck in local minima, we initialize with 10,000 randomly generated rotations and select the pose that produces the lowest loss value. For some categories (bicycle, bench, motorcycle), we found it beneficial to bias the sampling toward upright poses (elevation between -30 and 30 degrees, azimuth between 0 and 360 degrees).
We jointly optimize the 3D spatial arrangement loss \eqref{eq:opt_function} using ADAM with learning rate 1e-3 for 400 iterations. The trainable parameters are intrinsic scale $s^i\in\mathbb{R}$ for the $i$-th human and intrinsic scale $s^j\in\mathbb{R}$, rotation $\mathcal{R}^j\in SO(3)$, and translation $\mathbf{t}^j\in \mathbb{R}^3$ for the $j$-th object instance. The loss weights $\lambda_i$ are tuned qualitatively on the COCO-2017 val set. We initialized the optimization with the human poses estimated using \cite{joo2020exemplar} and the best object pose estimated in \secref{sec:nmr_opt} per object instance. To improve computational speed, we downsample the SMPL human meshes to 502 vertices and 1000 faces when computing losses.
A list of interaction parts pairs can be found in \tabref{tab:parts}, and an enumeration of the sizes of the 3D bounding boxes used to compute the interaction losses can be found in \tabref{tab:bbox_sizes}.
\begin{table}[h]
\centering
\footnotesize{
\begin{tabular}{ll}
\toprule
Category & Part Pairs (Object Part, Human Part)\\\midrule
Bat & (Handle, L Palm), (Handle, R Palm)\\
Bench & (Seat, Butt), (Seat Back, Back)\\
Bicycle & (Seat, Butt), (Handlebars, L Palm), (Handlebars, R Hand)\\
Laptop & (Laptop, L Palm), (Laptop, R Palm)\\
Motorcycle & (Seat, Butt), (Handlebars, L Palm), (Handlebars, R Palm)\\
Skateboard & (Skateboard, L Foot), (Skateboard, R Foot)\\
Surfboard & (Surfboard, L Foot), (Surfboard, R Foot), (Surfboard, L Palm)\\
& (Surfboard, R Palm)\\
Tennis Racket & (Handle, L Palm), (Handle, R Palm)\\
\bottomrule\\
\end{tabular}
}
\caption{\textbf{List of Parts Pairs used per category.} Each parts pair consists of a part of an object and a part of the human body. These parts pairs are used to assign human-object interactions.}
\label{tab:parts}
\end{table}
\begin{table}[h]
\centering
\footnotesize{
\begin{tabular}{lccc}
\toprule
Category & XY (Coarse) & XY (Fine) & Z Depth\\\midrule
Bat & 0.5 & 2.5 & 5\\
Bench & 0.3 & 0.5 & 10\\
Bicycle & 0 & 0.7 & 4\\
Laptop & 0.2 & 0 & 2.5\\
Motorcycle & 0 & 0.7 & 5\\
Skateboard & 0 & 0.5 & 10\\
Surfboard & 0.8 & 0.2 & 50\\
Tennis Racket & 0.4 & 2 & 5\\
\bottomrule\\
\end{tabular}
}
\caption{\textbf{Size of 3D Bounding Boxes.} To determine whether to apply the coarse interaction loss, we take the bounding box of the object and the bounding box of the person and expand each by the coarse expansion factor (Column 2). If the expanded bounding boxes overlap and the difference in the depths of the person and object is less than the depth threshold (Column 4), then we consider the person and object to be interacting. To determine whether to apply the fine interaction loss, we similarly take the bounding boxes corresponding to the object part and person part, expand the bounding boxes by the fine expansion factor (Column 3), and check for overlap. If the expanded bounding boxes overlap and the difference in depths of the parts is less than the depth threshold, then we consider the person part and object part to be interacting.}
\label{tab:bbox_sizes}
\end{table}
\subsection{Pre-processing Mesh Models}
\label{sec:mesh_processing}
We show all mesh instances that we built for each 3D category in \figref{fig:mesh_instances}. To better cover the shape variation or articulation within an object category, we use multiple mesh models for a few object categories (e.g., motorcycle, bench, and laptop). All the meshes are pre-processed to be watertight and are simplified with a low number of faces and uniform face size, to make the the optimization more efficient. For the pre-processing, we first fill in the holes of the raw mesh models (e.g. the holes in the wheels or tennis racket) to make the projection of the 3D models consistent with the silhouettes obtained by the instance segmentation algorithm~\cite{kirillov2019pointrend}. Then, we perform a TSDF fusion approach~\cite{Stutz2018ARXIV} that converts the raw meshes to be watertight and simplified. Finally, we reduce the number of mesh vertices using MeshLab~\cite{meshlab}.
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{figures/meshes_no_horses.jpg}
\caption{\textbf{Mesh models from various 3D object categories.} Here, we show all the mesh models that we used. \textbf{First row:} Motorcycle. \textbf{Third Row:} Laptop, Bench. \textbf{Fourth Row:} Bicycle, Skateboard, Tennis Racket, Baseball Bat, Surfboard.}
\label{fig:mesh_instances}
\end{figure}
\begin{figure}[t]
\centering
\fbox{\includegraphics[width=\textwidth]{figures/sup_evaluation_interface_blurred.jpg}}
\caption{\textbf{Screenshot from our comparison evaluation test interface.} Annotators were asked to evaluate which 3D arrangement looks more accurate, in this case, without and with the depth ordering loss for a picture of a person with a surfboard. Clockwise from top-left: original image, image with rendered projection, top-down view, frontal view. The arrangement highlighted in green is the one that is selected.}
\label{fig:evaluation_interface}
\end{figure}
\subsection{More Qualitative Results}
\label{sec:more_qualitative}
We show results on a large number of COCO images (test set) for each category evaluated in the main paper: baseball bats (\figref{fig:bat}), benches (\figref{fig:bench}), bicycles (\figref{fig:bicycle}), laptops (\figref{fig:laptop}), motorcycles (\figref{fig:motorcycle}), skateboards (\figref{fig:skateboard}), surfboards (\figref{fig:surfboard}), and tennis rackets (\figref{fig:tennis}).
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{supp_bat.pdf}
\caption{\textbf{Our output on COCO images with baseball bats.}}
\label{fig:bat}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{supp_bench.pdf}
\caption{\textbf{Our output on COCO images with benches.}}
\label{fig:bench}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{supp_bicycle.pdf}
\caption{\textbf{Our output on COCO images with bicycles.}}
\label{fig:bicycle}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{supp_laptop.pdf}
\caption{\textbf{Our output on COCO images with laptops.}}
\label{fig:laptop}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{supp_motorcycle.pdf}
\caption{\textbf{Our output on COCO images with motorcycles.}}
\label{fig:motorcycle}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{supp_skateboard.pdf}
\caption{\textbf{Our output on COCO images with skateboards.}}
\label{fig:skateboard}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{supp_surfboard.pdf}
\caption{\textbf{Our output on COCO images with surfboards.}}
\label{fig:surfboard}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{supp_tennis.pdf}
\caption{\textbf{Our output on COCO images with tennis rackets.}}
\label{fig:tennis}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}{\textwidth}
\includegraphics[width=\textwidth]{figures/failure_modes/human_pose_failure.jpg}
\caption{\textbf{Human pose failure.} Our human pose estimator sometimes incorrectly estimates the pose of the person, such as in this challenging snapshot of a tennis player performing a volley. In such cases, it can be difficult to reason properly about human-object interaction since the hand is misplaced and far from the tennis racket.\\
}
\label{fig:failure_human_pose}
\end{subfigure}
\begin{subfigure}{\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/failure_modes/object_pose_failure.jpg}
\caption{\textbf{Object pose failure.} The predicted masks are sometimes unreliable for estimating the pose of the object. In such cases, it difficult to recover a plausible scene reconstruction.\\
}
\label{fig:failure_object_pose}
\end{subfigure}
\begin{subfigure}{\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/failure_modes/scale_failure.jpg}
\caption{\textbf{Incorrect reasoning about interaction due to scale.} The interaction loss requires a reasonable scale initialization. Sometimes, objects in the real world can fall outside the expected scale distribution, such as in the case of this small bicycle.
}
\label{fig:failure_interaction}
\end{subfigure}
\caption{\textbf{Failure modes.} In this figure, we describe a few failure modes of our method.}
\label{fig:failure_modes}
\end{figure}
|
1,941,325,220,540 | arxiv | \section{Introduction}
The recent observation of gravitational waves (GWs) \cite{Ligo1, Ligo2} has established gravitational wave detection as an exciting new observational tool for cosmological phenomena. Terrestrial optical interferometers are sensitive to gravitational radiation at frequencies above roughly 10 Hz \cite{harry2010advanced}. In order to extend these techniques to lower frequencies where there are expected to be an abundance of signals, space-based optical interferometers such as the Laser Interferometer Space Antenna (LISA) have been proposed, and are currently under technological development \cite{LISA_proposal, LISApathfinder}.
More recently, space-based gravitational wave detectors based on optical transitions in cold atoms have been proposed as an alternative architecture \cite{2015arXiv150100996L, GravitationalRaman2009, yu2011gravitational, kolkowitz2016gravitational, vutha2015optical, hollberg2017optical, graham2013new, hogan2016atom, graham2016resonant}. These proposals rely on a combination of optical and atomic coherence to provide sensitivity to gravitational waves. While optical interferometers require three satellites to cancel laser phase noise, these proposals require only two.
We will focus here on two categories of the atom-based proposals: those described as ``atom interferometers" (AI) typified by \cite{graham2013new}, and those described as ``optical lattice clocks" \cite{kolkowitz2016gravitational}. These two types of proposal are illustrated in Fig.~\ref{fig:ExpDiag}.
AI-type proposals based on two-photon Raman transitions have also been proposed as a means of detecting gravitational waves, both in space-based and ground-based applications \cite{dimopoulos2009gravitational, canuel2014matter, PhysRevD.93.021101}. However, these proposals are not inherently insensitive to laser phase noise, and thus require noise cancellation techniques similar to a purely optical interferometer, such as the use of three interferometers. AI detectors based on two-photon Raman transitions may prove to be powerful tools for detecting gravitational waves, but here will focus exclusively on proposals that leverage long-lived optical atomic coherence on a single detection baseline with intrinsic insensitivity to laser phase noise.
In both clock and AI type proposals, two ensembles of atoms with a long-lived optically excited state are prepared in two satellites separated by a large distance. Laser pulses transmitted between the two satellites interact with the atoms in order to imprint the effects of a passing gravitational wave onto atomic observables. In both cases, the atoms are in a freely falling reference frame. In AI proposals, this is accomplished by simply preparing the atoms in a high vacuum environment. In clocks, the atoms are tightly trapped in an optical lattice formed by reflecting a laser off of a freely falling mirror that serves as an inertial reference, as in the proposed LISA interferometer \cite{LISApathfinder}. In AI proposals, the interaction between the atoms and the laser leads to recoil momentum kicks imparted to the atoms, while in clock proposals the photon momentum is absorbed by the much more massive inertial reference.
In the clock community, gravitational wave detection has been described as a frequency measurement of Doppler shifts that result from the stretching of space between the satellites \cite{kolkowitz2016gravitational, vutha2015optical}.
In the AI community, the stretching of space has been described as causing a phase shift between the laser phase and the atomic coherence \cite{graham2016resonant}.
A key element of AI proposals is enhanced sensitivity to gravity waves through the use of Large Momentum Transfer (LMT) pulses in which the atoms acquire many photon recoil momentum kicks. In contrast, the tight confinement of the atoms relative to the inertial reference mass in an optical lattice clock causes the photon recoil to be suppressed. This difference in particular, as well as the language used to describe the devices, would seem to indicate that the two sensors are somehow fundamentally different.
In sections II and III we provide a comparison of these two sensors to show that the fundamental mechanism for sensitivity is in fact the same for the clock and AI type detectors. In both types of sensors, laser pulses that couple a ground and long-lived optically excited state imprint their local phase onto an internal quantum superposition state of each atom. The atoms primarily act as a highly coherent phase memory that keeps track of these phase imprints and allows them to be read out through atomic observables.
By viewing clock-like detectors as phase memories, rather than simply as clocks whose sole capability is to measure frequency, we show in section IV that they support the implementation of LMT-like protocols with enhanced sensitivity even when negligible momentum is transferred to the atom. We further show in section V that the relevant phase information can actually be stored in stable ground states that evolve phases at acoustic rather than optical frequencies. This enables useful coherent evolution times beyond the lifetime of the optically excited state.
\begin{figure*}[!htb]
\includegraphics[width=6.75in]{Fig1_satellites.pdf}
\caption{Optical lattice clock and atom interferometer detectors for gravitational waves. Each type of proposal relies on two ensembles of atoms, one in each satellite. Lasers transmitted between the satellites encode phase shifts due to changing optical path lengths caused by a gravitational wave. The key difference between clock and atom interferometer proposals is that in clocks, atomic recoils due to the momentum of absorbed/emitted photons are supressed by tightly confining the atoms, while in atom interferometers the atoms are free to recoil. }
\label{fig:ExpDiag}
\end{figure*}
\section{Optical Path Length Changes due to Gravitational Waves}
\begin{figure*}[htb]
\includegraphics[width=6.75in]{Fig2_toy_clock_AI.pdf}
\caption{A toy model for gravitational wave detection. (a) Two atoms ($a$ and $b$) are addressed by a single laser resonant with a transition between the ground state $\ket{g}$ and a long-lived excited state $\ket{e}$. A phase modulator mimics the effect of a gravitational wave by modifying the optical path length between the two atoms. (b) For tightly confined atoms (clocks), a simple Ramsey sequence can be used to detect changes in optical path length. A first $\pi/2$ pulse imprints the laser phase on the two atoms when the path length is shifted by $L_1$, and a second $\pi/2$ pulse acts when the path length is shifted by $L_2$. Random variations in the laser phase ($\phi_1$ and $\phi_2$) are common to the two atoms, while phase shifts resulting from changes in path length are not. The signal from the phase modulator manifests as phase shift $\Phi_s = k (L_2-L_1)$ between the excited state probability oscillations of the two atoms.
(c) For unconfined atoms (atom interferometers), transitions between $\ket{g}$ and $\ket{e}$ are accompanied by momentum kicks, which necessitate an additional laser $\pi$ pulse in the middle of the sequence. The phase shift between excited state oscillations of the two atoms is now $\Phi_s = k (L_3 - 2 L_2 + L_1)$.}
\label{fig:ToyModel}
\end{figure*}
For both types of proposals, the important effect of a gravitational wave is its modification of the optical path length $L$ between the two satellites.
We will describe the effects of a gravitational wave at angular frequency $\omega_{g}$ by replacing the gravitational wave with a phase modulator that fills the space between the two satellites (e.g. an electro-optic modulator or EOM) sinusoidally driven at frequency $\omega_{g}$.
The drive applied to the modulator leads to a phase shift $\alpha(t)$ on laser light that is launched from one satellite and detected at the other satellite.
The light is launched from the first satellite at time $t$, but arrives at the second satellite where it is detected at time $t+T_d$, where $T_d$ is the nominal delay or transit time between satellites.
The total phase shift $\alpha(t_j)$ for a pulse launched at time $t_j$ is calculated by integrating along the optical path. We can define a net effective path length for the pulse $L+L_j= L+\alpha(t_j)/k$ where the small change in optical path length is
\begin{equation}
L_j = \frac{h c}{2 \omega_{g}}\left(\sin{(\omega_{g} (t_j+T_d))} - \sin{(\omega_{g} t_j)}\right) \, .
\end{equation}
\noindent We have made the approximation that the gravitational wave's strain is very small $h\ll1.$ In this expression, $c$ is the speed of light in the undriven modulator, $k=2 \pi/\lambda$ is the wavenumber of the laser light, and $\lambda$ is the laser wavelength.
\section{Detecting Changes in Optical Path Length}
The goal of this section is to understand how AI and clock-like sensors can be used to precisely estimate changes in the optical path length $L_j$. In our model, shown in Fig.~\ref{fig:ToyModel}, we consider two atoms labeled $a$ and $b$ that are separated by distance $L$. The atoms may either be cooled in free space (atom interferometers) or tightly confined (clocks). The atoms have a long-lived optical transition, such as those in alkaline-earth and similar atoms (for example, Sr and Yb atoms). A laser located near atom $a$ launches pulses of light that interact with both atoms with equal intensity.
The laser pulses interact with the atoms by applying so-called $\pi/2$ and $\pi$ pulses between ground $\ket{g}$ and excited $\ket{e}$ states. We will assume that the coupling or Rabi frequency is much larger than the atomic decay rate from $\ket{e}$ and any relevant Doppler shifts due to atomic motion. We will also assume for simplicity that the laser frequency is exactly equal to the atomic transition frequency. These assumptions rule out some of the capabilities of atom interferometers (such as addressing atoms that have experienced different recoils independently), but retains the basic mechanism for sensitivity. The effect of the laser pulses on the atoms written in a rotating frame at the atomic transition frequency can then be expressed using the operators:
\begin{equation}
\hat{R}_{\pi/2} (\phi) = \frac{1}{\sqrt{2}}\begin{bmatrix}
1 & -e^{-i (\phi + k x)} \\
e^{i (\phi + k x)} & 1 \\
\end{bmatrix}
\end{equation}
\begin{equation}\hat{R}_{\pi} (\phi) = \begin{bmatrix}
0 & -e^{-i (\phi + k x)} \\
e^{i (\phi + k x)} & 0 \\
\end{bmatrix}
\end{equation}
\noindent These operators act on the basis states :
\begin{equation}\ket{g}\otimes \Psi_g(x) = \begin{bmatrix}
1 \\
0 \\
\end{bmatrix} \otimes \Psi_g(x)
\end{equation}
\begin{equation}
\\ \ket{e}\otimes \Psi_e(x) = \begin{bmatrix}
0 \\
1 \\
\end{bmatrix} \otimes \Psi_e(x)
\end{equation}
\noindent that are a product of an internal state label $\ket{e}$ or $\ket{g}$, and an external state wavefunction $\Psi_{e,g}(x)$.
Here $x$ represents the distance from a fixed plane where the laser's phase is defined as $\phi$. The effect of these interactions is both to transfer amplitude between the internal states of the atoms and also to imprint the laser's local phase upon the transferred portion of the atom's wave function. The fact that $\Psi_g(x)$ may differ from $\Psi_e(x)$ indicates the possibility of entanglement between the internal and external degrees of freedom of the atom.
For simplicity, we account for laser frequency noise by allowing the laser phases to vary between pulses, but taking the actual laser frequency to be fixed at the atomic transition frequency such that $k$ is constant \cite{RelaxedAssumption}.
First, we consider the effect of these rotations for a clock-like sensor in which the atoms are confined to much less than the laser wavelength (known as the Lamb Dicke regime.) In this limit, we can think of the $e^{i(\phi + k x)}$ term as imprinting a spatially constant phase onto the atom, whose value is determined by the location of the atom along the laser's path. For example, a $\pi$ pulse applied to an atom tightly confined in a trap centered at position $x = A$ and in the ground state $\ket{\Psi_0}=\ket{g}\otimes \Psi_0(x)$ transfers the atom to the state $\ket{\Psi}\approx\ket{e}e^{- i(\phi + k A)}\otimes \Psi_0(x)$. The external state wave-function is to very good approximation unmodified by the pulse, but its internal wavefunction has acquired a net phase $\phi+ k A$.
In contrast, in AI sensors the atoms are not confined and a change in the internal state is accompanied by a change in the external state. In particular, one cannot neglect the variation of the optical phase factor over the spatial extent of the atomic wavefunction. For concreteness, a $\pi$ pulse applied to an unconfined atom centered at $x=A$ and in the ground state $\ket{\Psi_0}=\ket{g}\otimes \Psi_0(x)$ transfers the atom to the state $\ket{\Psi}=\ket{e}e^{- i\phi} \otimes e^{-i k x} \Psi_0(x)$. As written, the internal portion of the wavefunction appears identical to that of the confined case, but now the external wavefunction has a spatially varying phase corresponding to one photon's worth of momentum recoil.
A change in the optical path length $L_j$ between the atom and the laser producing the $\pi$-pulse manifests in the confined case as $\ket{\Psi}= e^{- i(\phi + k( L_j+A))}\ket{e}\otimes \Psi_0(x)$ and in the unconfined case as $\ket{\Psi}= e^{- i(\phi + k L_j)}\ket{e}\otimes \Psi_0(x)e^{- i k x}$. The sensitivity to changes in path lengths from gravitational waves is due to the imprinting of an additional phase $k L_j$, which is the same whether the atoms are confined or not. In the case of AI sensors one must add additional pulses to become insensitive to the terms associated with the photon recoil.
\subsection{Clock-like Detectors}
A clock-like gravitational wave detector with confined atoms could be used to detect the gravitational wave phase shift using a basic Ramsey sequence \cite{kolkowitz2016gravitational}, pictured in Fig.~\ref{fig:ToyModel}b. A change in the optical path length $L_j$ leads to a modification of the laser phase experienced by atom $b$ of $k L_j$. The phase experienced by atom $a$ is unmodified by the change in optical path length. In this sequence, the role of atom $a$ is then simply to record any variation of the phase of the laser itself due to technical sources of noise so that this laser phase noise can be subtracted out from the final measurement.
Stepping through the Ramsey measurement, atoms in both locations are initially prepared in $\ket{g}$. At time $t_1$, the laser drives the first $\pi/2$ pulse with phase $\phi=\phi_1$. we keep track of this phase only to demonstrate insensitivity to its value. The rotation applied to atom $a$ is
$R_a = R_{\pi/2} (\phi_1)$. When the same pulse arrives at atom $b$ it creates a rotation
$R_b = R_{\pi/2} (\phi_1+ k L_1)$.
At a later time $t_2$, a second $\pi/2$ pulse is applied to atom $a$ with a laser phase $\phi=\phi_2$: $R_a = R_{\pi/2} (\phi_2)$. The same pulse of light travels to atom $b$ to drive a $\pi/2$ pulse
$R_b = R_{\pi/2} (\phi_2 + k L_2)$.
The final signal that is detected is the difference in the probability for finding the atom in its excited and ground states. For atom $a$ this observable can be parameterized as $S_a= P_{ea} -P_{ga} = \cos{\Phi_{as}}$ and for atom $b$ as $S_b= P_{eb} -P_{gb} = \cos{\Phi_{bs}}$. For the above Ramsey sequence, the signal phases are given $\Phi_{as}= (\phi_2 - \phi_1)$ and $\Phi_{bs}= k L_2- k L_1 + (\phi_2 - \phi_1)$, The gravitational wave signal is given by extracting the difference of the phase of these two signals \cite{foster2002method}
\begin{equation}
\Phi_s= \Phi_{bs}-\Phi_{as}= k(L_2 - L_1)
\end{equation}
\noindent The key result is that the technical laser phase noise is canceled by having been recorded on both atoms and only the gravitational wave's signal remains.
\subsection{Atom Interferometer Detectors}
In an atom interferometer, the atoms are unconfined and a slightly more complicated sequence is needed. The initial $\pi/2$ pulse entangles internal and external degrees of freedom of the atom by imparting a momentum kick to the portion of the atomic wavefunction transferred to $\ket{e}$. If this momentum kick is not reversed, the portions of the atomic wavefunction will not be spatially overlapped to interfere at the time of the second $\pi/2$ pulse. The simplest solution is to add a $\pi$ pulse in the middle of the sequence (Fig.~\ref{fig:ToyModel}c). More complex sequences, with enhanced performance in different regimes are presented in \cite{graham2013new, graham2016resonant}, though the core mechanism for sensitivity is the same as this simple version.
Stepping through the simplest AI sequence, the first $\pi/2$ pulse, $\pi$ pulse and second $\pi/2$ pulse are launched from the laser at times $t_1$, $t_2$, and $t_3$, respectively. When the three pulses arrive at atom $b$ they will have experienced optical path length differences $L_1$, $L_2$, and $L_3$.
The signal that is extracted is the same as above, and depends only on the optical path lengths as
\begin{equation}
\Phi_s = k (L_3-2 L_2 + L_1)
\end{equation}
\noindent where the random fluctuations in laser phase $\phi_3$, $\phi_2$, and $\phi_1$ are again cancelled because they are common to both atoms. A similar analysis for the atom interferometer is presented in \cite{yu2011gravitational}.
\section{Enhanced sensitivity analogous to large-momentum-transfer}
We now propose a mechanism by which the signal recorded by the clock-like sensor can be enhanced. In atom interferometry in general, and AI based GW detectors in particular \cite{graham2013new}, large momentum transfer (LMT) is a crucial tool used to enhance the size of measured signals. Our enhancement mechanism is very similar to LMT, but is applied to confined atoms, so no momentum is actually transferred to the atoms. Instead, by allowing for multiple interactions between the lasers and atoms, a phase is repeatedly written in to the atomic coherence in a constructive manner to enhance the signal size $\Phi_s$.
\begin{figure}[!htb]
\includegraphics[width=3.375in]{TwoLasersEnhancedLotsofLabels.pdf}
\caption{LMT-like enhancement with confined atoms. (a) Atoms are addressed by counter-propagating lasers, both of which pass through the phase modulator that sits between the atoms. (b) Two blocks of $M$ pairs of $\pi$ pulses (blue boxes) are inserted between the $\pi/2$ pulses of the standard Ramsey sequence. Each pair of $\pi$ pulses contains a pulse originating from each direction. The duration of each block is assumed here to be short relative to the GW period, while a long evolution time (grey box) between the two blocks may be comparable to the GW period. (c) The additional pulses lead to a factor of $4M+1$ enhancement in the signal phase $\Phi_s$ compared to simple Ramsey sequence. }
\label{fig:LMT}
\end{figure}
The key to this enhancement sequence is to apply the laser pulses from alternating sides (as is proposed for LMT protocols for single-photon transitions with unconfined atoms \cite{graham2013new, hogan2016atom, graham2016resonant}) using one laser near atom $a$ and one laser near atom $b$, as illustrated in Fig.~\ref{fig:LMT}. Because the sign of the laser phase shift imprinted on the atoms is opposite in sign when the atoms are driven from $\ket{e}$ to $\ket{g}$ versus $\ket{g}$ to $\ket{e}$, if the same laser were used to try to imprint its phase multiple times, it would simply unwrite the phase that it had just written in. By interleaving $\pi$ pulses from a second laser, which does not encode the same gravitational wave phase shift as the first, we can ensure that the phase shift corresponding to the GW is always written in with the same sign. The phases of the laser pulses are always referenced to a plane near the laser from which they originate so that there is negligible GW contribution to this phase.
For simplicity, we treat the case where a set of pulses is applied in quick succession at the beginning of the measurement sequence, with launch times roughly equal to $t_1$ and with optical path length changes all equal to $L_1$. During a subsequent free evolution period $T_e$, much longer than the time over which the rotations were applied, the optical path length may change. At time $t_2=t_1+T_e$, a second set of rotations is quickly applied with launch times all roughly equal to $t_2$ such that all pulses experience an optical path length change equal to $L_2$ (a discussion of the effects of time delays is included in the appendix).
Consider the first group of pulses. From the perspective of one of the two atoms, the laser pulse that arrives from the distant laser encodes in its phase $k L_1$ the change in path length. This phase is then imprinted on the atomic superposition with a different sign between the portion that was transferred from $\ket{e}$ to $\ket{g}$ versus $\ket{g}$ to $\ket{e}$ such that the difference phase is $2 k L_1$. The next laser pulse that is launched from the locally situated laser resets the two portions of the atomic superposition to their original internal states. It imprints a phase shift of its own, but this phase does not encode information about the change in optical path length $L_1$, and the laser's phase noise cancels in the final differential signal phase $\Phi_s$. Because the two portions of the atomic superposition have been reset to their original internal states, a subsequent rotation from the distant laser will imprint a phase encoding the same path length difference $k L_1$ that adds constructively with the previously imprinted phase so that the total imprinted phase on the atom is now $2 \times 2 k L_1 $.
If we apply $M$ pairs of $\pi$ rotations after the first $\pi/2$ pulse and $M$ pairs of $\pi$ rotations before the last $\pi/2$ pulse (i.e. $4M$ total launched $\pi$ pulses), with each pair containing a rotation originating from both the left and right sides, the differential phase shift between the two output channels is enhanced by a factor of $(4M+1)$ relative to the simple Ramsey sequence presented above so that now
\begin{equation}
\Phi_s= (4 M+1) k(L_2-L_1)
\end{equation}
\noindent In the regime we consider here, even a single pair of LMT pulses (i.e. $4$ total $\pi$-pulses), improves the estimate of $(L_2-L_1)$ by a factor of 25 in variance. As a result, the same precision can be achieved with a reduction in required resources such as atom number or averaging time by a factor of 25. Alternatively, one could achieve the same sensitivity to gravitational waves with a 5 times shorter satellite separation or evolution time $T_e$ for reduced technical complexity or enhanced bandwidth for detecting gravitational waves, respectively.
This enhanced protocol is insensitive to laser phase noise for the same reason that the simple protocols presented in section III are: each laser pulse interacts with both interferometers, so any phase noise on the laser cancels in the differential signal.
Previous proposals for detectors using confined \cite{kolkowitz2016gravitational} and unconfined \cite{graham2016resonant} atoms include so-called dynamical decoupling (DD) sequences. A DD sequence would amount to applying a $\pi$ pulse from the same laser every time $t_j$ at which the magnitude of the optical path length change $\left|L_j\right|$ is maximal. Because the sign of $L_j$ alternates between pulses, the resulting imprinted phase shifts add constructively. This is conceptually similar to our LMT-like enhancement mechanism, except that we can switch the sign of the phase shift by alternating which laser applies the pulse instead of waiting for the sign of the GW to switch.
Dynamical decoupling is useful when the evolution time $T_e$ greatly exceeds the GW period $T_g$. In this regime, the enhancement in signal scales as $T_e/T_g \gg 1$. This reflects the fact that the phase shifts from $T_e/T_g$ cycles of the gravitational wave may be added constructively. There is no constraint on delay time $T_d$.
LMT-type sequences are useful when the period of the GW $T_g$ and total evolution time $T_e$ are long compared to the pulse transit time $T_d$. This is necessary in order to send multiple pulses back and forth between the two satellites before the path length changes caused by the gravitational wave switches sign. The full signal response versus gravitational wave frequency is considered in the appendix. The main result is that one can build a long baseline experiment and set $T_e=T_d$ using a straightforward Ramsey sequence, or one can use LMT-type sequences to dramatically shorten the baseline such that $T_e$ is the same but now $T_e\gg T_d$. In either case, the sensitivities are comparable. LMT-like enhancement allows one to address technical constraints that might be relaxed by operating with shorter baselines, at the expense of potentially introducing technical errors associated with the additional laser pulses.
Finally, as analyzed in \cite{graham2016resonant}, LMT can be combined with DD when both $T_e \gg T_g$ and $T_g \gg T_d$. Doing so amounts to varying from which satellite the $\pi$ pulses are sent.
\section{Evolution times longer than the excited state lifetime}
We now present a further enhancement that can be realized by treating the atoms as a phase memory, rather than a clock. In a typical clock, the atoms are considered to be a two-state system. Because external degrees of freedom ideally remain unchanged, clocks lack additional quantum labels to specify other states. Any pulse that interacts with one `arm' of the clock interferometer ($\ket{g}$) also interacts with the other ($\ket{e}$): the quantum mechanical amplitudes may be swapped between ground and excited states, but the portion of the wave-function in the ground state cannot be manipulated without also modifying the portion of the wave function in the excited state. In an AI, the external degrees of freedom of the atom provide additional labels that allow the two arms to be manipulated independently.
\begin{figure*}[!htb]
\includegraphics[width=6.5in]{Three_State.pdf}
\caption{Use of three-state system to allow atoms to stay in ground state for much of evolution time. (a) Our procedure utilizes two ground states, $\ket{g_1}$ and $\ket{g_2}$ and a single long-lived optically excited state $\ket{e}$. $\ket{g_1}$ and $\ket{g_2}$ can independently be coupled to $\ket{e}$ using laser pulses with different frequency and/or polarization (red and blue arrows). (b) After preparing an atomic superposition of $\ket{g_1}$ and $\ket{e}$ using a $\pi/2$ pulse from the left (atoms begin in $\ket{g_1}$), the portion of the atoms in $\ket{e}$ is transferred to $\ket{g_2}$ using a $\pi$ pulse from the right. The atoms are then in a superposition of $\ket{g_1}$ and $\ket{g_2}$ with phase dictated by the lasers and the path length $L_1$.
The sequence of pulses is reversed after a long evolution time, during which time the atoms are not susceptible to excited state decay. The phase shift in the excited state probabilities of the two atoms $\Phi_s = 2k(L_2-L_1)$ now encodes changes in the optical path length that occurred while the atoms were in the ground states. }
\label{fig:3State}
\end{figure*}
While in clock type proposals, a fraction of the atomic wave-function must always be in the excited state to achieve sensitivity, AI type detection sequences may include useful periods where both arms of the interferometer are in the ground state \cite{graham2013new}, allowing for evolution times greater than the excited state lifetime. This technique is particularly useful for enhancing the signal in the regime where the period of the GW exceeds the lifetime of the excited state.
In a recoil-free GW detector, additional internal states of the atoms can provide additional quantum labels that allow for the independent manipulation of only one arm of the interferometer. Here, we consider an atomic species like $^{171}$Yb with nuclear spin $I=1/2$. There are two ground states with nuclear spin projections $m_I=\pm 1/2$ that we we will label as $\ket{g_1}$ and $\ket{g_2}$. For our purposes, we care about a single excited state $\ket{e}$, as shown in Fig.~\ref{fig:3State}a. A small magnetic field is applied to define a quantization axis. Transitions may be driven between either $\ket{g_1}$ and $\ket{e}$ or $\ket{g_2}$ and $\ket{e}$ by applying laser light with different polarizations or frequencies.
An experimental sequence for such a protocol is shown in Fig.~\ref{fig:3State}b. A superposition is prepared by driving a $\pi/2$ pulse originating from the left between $\ket{g_1}$ and $\ket{e}$. A $\pi$ pulse originating from the right then transfers the population of $\ket{e}$ to $\ket{g_2}$. The atoms remain in this configuration while the path length is allowed to change during an evolution time $T_e$, which we take to be much larger than the delay time $T_d$. The pulse sequence is then reversed and the change in path length is read out via a population measurement. The resulting signal is $\Phi_s= 2k(L_2-L_1)$, again with no contribution from laser phase noise. Memory storage in ground states could be combined with LMT-type enhancement by inserting $\pi$ pulses on the $\ket{g_1}$ to $\ket{e}$ transition in between the $\pi$ pulses on the $\ket{g_2}$ to $\ket{e}$ transition and the initial and final $\pi/2$ pulses to realize further enhancement.
Because the atoms are in superpositions of ground states, there is no spontaneous emission during $T_e$. Instead, the relative phase that would normally exist between $\ket{g_1}$ and $\ket{e}$ is now stored between $\ket{g_1}$ and $\ket{g_2}$ during the evolution period $T_e$. In addition to extending the evolution time beyond the spontaneous-decay limited coherence time of the atoms, this technique of shelving the atoms in ground states provides insensitivity to collisional effects that may limit the atomic density or coherence time achievable in practice. Inelastic collisions between excited state atoms, which contribute to atom loss in dense systems, would be eliminated by storing the atoms in the ground states \cite{PhysRevLett.103.090801, bishof2011inelastic}. Because elastic collisional properties of alkaline-earth-like atoms are independent of nuclear spin state \cite{Zhang19092014, scazza2014observation}, mean-field shifts due to elastic collisions are common to the two ground states and will not lead to noise or dephasing. Other bias errors in clocks such as black-body, electrostatic, and lattice polarizability and hyperpolarizability shifts would also be suppressed. As in the AI case, this technique is applicable when the evolution time $T_e$ would otherwise be limited to much less than the gravitational wave period $T_g$.
The key result is that it is not necessary for the phase memory to exist in a frame accumulating phase at an optical frequency. It is only necessary that the interactions with the laser pulses happen between optical transitions, but these interactions can in principle represent a small fraction of the total evolution time.
This protocol may appear similar to those that use two-photon Raman transitions in that the atoms occupy two ground states during a long evolution time. However, Raman transitions require a laser pulse from each satellite to interact with an atomic ensemble at the same time in order to perform a rotation without populating the short-lived excited state.
As with single-photon transitions, each laser pulse must also interact with both interferometers in order to obtain a signal that is insensitive to laser phase noise. These two requirements lead to a conflict for detectors that use Raman transitions: to our knowledge, there is no protocol based on a two-satellite configuration in which each rotation is performed by simultaneously incident pulses from both satellites, and in which each pulse leads to the desired rotation in both interferometers.
Because the use of single-photon transitions as discussed here allows population to be stored in the excited state for a time of order $T_d$ or longer, the laser pulses from the two satellites need not be simultaneously incident on the atoms, enabling phase-noise insensitive protocols that do not appear to be possible with Raman transitions.
\section{Conclusion}
Gravitational waves create phase shifts on optical pulses, which we would like to detect as sensitively as possible. At the most basic level, one would like to store an optical pulse of light sent at time $t_1$ with phase shift $k L_1$ until a second pulse of light arrives with a different phase shift $k L_2$ at time $t_2$, then compare the phases of the two pulses. Note that this is a related mechanism to that employed by purely optical GW detectors, which compare the phase shifts experienced by light travelling on two different paths that experience opposite GW-induced phase shifts, rather than light that that travelled along the same path at two different times.
By providing a highly coherent phase memory with which the two pulses may interact, the atoms allow one to make time-separated phase comparisons of the two pulses in a manner that is insensitive to laser phase noise. This capability eliminates the need in atomic detectors for a third satellite that is required in purely optical interferometers.
Further, multiple pulses can be made to constructively imprint their phases onto the atoms by properly alternating their launch direction (LMT) or by synchronizing their launch times with the frequency of the gravitational wave one wishes to detect (DD). From a fundamental perspective, the ability to achieve LMT-type signal enhancement does not appear to require the transfer of momentum to the atoms or that the two lasers originate from different directions (although the latter is required for cancellation of technical sources of noise). In certain regimes, LMT allows the signal size to be increased such that the uncertainty in the estimate of the optical phase of interest $k(L_2-L_1)$ can be greatly reduced to well below the atomic standard quantum limit $1/\sqrt{N}$~rad on atomic phase resolution.
For gravitational wave detection, we do not need to measure the absolute frequency of a laser relative to an atomic transition frequency, as one would do in a clock. As we have shown, this allows for the construction of a ground state shelving protocol with reduced sensitivities to perturbations and evolution times $T_e$ greater than the optical transition lifetime.
Ultimately, the decision to use confined or unconfined atoms will depend on a myriad of technical considerations that we have no business weighing in on. From a fundamental perspective, both methods have the same mechanism for sensitivity: atom interferometers and clocks both sense changes in phase that result from changes in optical path length between the two satellites.
This interpretation should help distinguish to what degree differing expected sensitivities for future proposals are due to the choice of sensor architecture versus the specific parameters considered.
\section{ACKNOWLEDGMENTS}\label{ack}
We thank Shimon Kolkowitz, Jason Hogan and Peter Graham for important discussions and input.
All authors acknowledge financial support from DARPA QuASAR, ARO, NSF PFC, and NIST. J.R.K.C. acknowledges financial support from NSF GRFP. This work is supported by the National Science Foundation under Grant Number 1125844.
\section{Appendix: Accounting for Time Delays}
In the main text, we considered an LMT pulse sequence in which the time delay $T_d$ is much shorter than both the evolution time $T_e$ and gravitational wave period $T_g$. Rather than launching a series of LMT pulses at the beginning and at the end, one could in principle launch a continuous series of alternating $\pi$-pulses during the entire period of time between the $\pi/2$ pulses. For continuous LMT, we find that the signal size averaged over all phases of the gravitational wave is (to good approximation) given by:
\begin{equation}
\bar{\Phi}_s = 4 \sqrt{2} h \frac{\omega_l \left|\sin(\omega_g T_d /2) \right|}{ \omega_g^2 T_d} \sin^2( \omega_g T_e /4)
\end{equation}
\noindent where $\omega_l$ is the laser angular frequency. When $T_d\gg T_g$, the sensitivity falls of rapidly as $1/\omega_g^2$. However, in the limit $T_d\ll T_g $ the scaling changes to
\begin{equation}
\bar{\Phi}_s \approx 2 \sqrt{2} h \frac{\omega_l}{\omega_g} \sin^2( \omega_g T_e /4)
\end{equation}
\noindent For comparison, if the spacing of the satellites is increased until $T_d = T_e$, then one cannot use LMT and the Ramsey sequence yields an averaged signal:
\begin{equation}
\bar{\Phi}_s =\sqrt{2} h \frac{\omega_l}{\omega_g} \sin^2( \omega_g T_e /2)
\end{equation}
\noindent The oscillations in the signal versus $\omega_g$ differ by a factor of 2 and the envelope of the signal size is larger by a factor of 2 for LMT. Most importantly, approximately the same signal can be obtained using continuous LMT with of order $T_e/T_d$ $\pi$ pulses and an approximately $T_e/T_d$ times shorter satellite satellite spacing. It is likely that decreasing the spacing of the satellites will be advantageous for technical reasons, for instance, it allows for reduced requirements on laser power and pointing stability.
|
1,941,325,220,541 | arxiv | \section{Introduction}\label{sec:intro}
Random forests and bagging are some of the most widely used prediction methods~\citep{breiman1996, breiman2001}, and over the course of the past fifteen years, much progress has been made in analyzing their statistical performance~\citep{buhlmannyu,hallsamworth,biau2008,biau2012,scornetconsistency}. However, from a computational perspective, relatively little is understood about the \emph{algorithmic convergence} of these methods, and in practice, ad hoc criteria are generally used to assess this convergence.
To clarify the idea of algorithmic convergence, recall that when bagging and random forests are used for classification, a large collection of $t$ randomized classifiers is trained, and then new predictions are made by taking the plurality vote of the classifiers. If such a method is run several times on the same training data $\mathcal{D}$, the prediction error $\textsc{Err}_t$ of the ensemble will vary with each run, due to the randomized training algorithm. As the ensemble size increases $(t\to\infty)$ with $\mathcal{D}$ held fixed, the random variable $\textsc{Err}_t$ typically decreases and eventually stabilizes at a limiting value $\text{err}_{\infty}=\text{err}_{\infty}(\mathcal{D})$. In this way, an ensemble reaches algorithmic convergence when its prediction error nearly matches that of an infinite ensemble trained on the same data.
Meanwhile, with regard to computational cost, larger ensembles are more expensive to train, to store in memory, and to evaluate on unlabeled points. For this reason, it is desirable to have a quantitative guarantee that an ensemble of a given size will perform nearly as well as an infinite one.
This type of guarantee also prevents wasted computation, and assures the user that extra classifiers are unlikely to yield much improvement in accuracy.
\subsection{Contributions and related work}\label{sec:contrib}
To measure algorithmic convergence, we propose a new bootstrap method for approximating the distribution $\mathcal{L}(\sqrt{t}(\textsc{Err}_t-\text{err}_{\infty})|\mathcal{D})$ as $t\to\infty$. Such an approximation allows the user to decide when the algorithmic fluctuations of $\textsc{Err}_t$ around $\text{err}_{\infty}$ are negligible. If particular, if we refer to the \emph{algorithmic variance}
$$\sigma_t^2:=\operatorname{var}(\textsc{Err}_t|\mathcal{D})=\mathbb{E}[\textsc{Err}_t^2|\mathcal{D}]-(\mathbb{E}[\textsc{Err}_t|\mathcal{D}])^2,$$
as the variance of $\textsc{Err}_t$ due only the training algorithm, then the parameter $\sigma_t$ is a concrete measure of convergence that can be estimated via the bootstrap.
In addition, the computational cost of the method turns out to be quite modest, by virtue of an extrapolation technique, as described in Section~\ref{sec:practical}.
Although the bootstrap is an established approach to distributional approximation and variance estimation, our work applies the bootstrap in a relatively novel way. Namely, the method is based on ``bootstrapping an algorithm'', rather than ``bootstrapping data'' --- and in essence, we are applying an inferential method in order to serve a computational purpose. The opportunities for applying this perspective to other randomized algorithms can also be seen in the papers \citet{nocedal,lopes_matrix,lopes_LS}, which deal with stochastic gradient methods, as well as randomized versions of matrix multiplication and least-squares.
\paragraph{Bootstrap consistency} From a theoretical standpoint, our main result (Theorem~\ref{THM:CLT}) shows that the proposed method consistently approximates the distribution \smash{$\mathcal{L}(\sqrt{t}(\textsc{Err}_t-\text{err}_{\infty})|\mathcal{D})$} as $t\to\infty$ under a ``first-order model'' for randomized ensembles. The proof also offers a couple of theoretical contributions related to Hadamard differentiability and the functional delta method~\citep[Chapter 3.9]{vaartWellner}.
The first ingredient is a lifting operator $\boldsymbol L$, which transforms a univariate empirical c.d.f. \smash{$\mathbb{F}_t:[0,1]\to [0,1]$} into a multivariate analogue \smash{$\boldsymbol L(\mathbb{F}_t):\Delta\to\Delta$}, where $\Delta$ is a simplex.
In addition to having interesting properties in its own right, the lifting operator will allow us to represent $\textsc{Err}_t$ as a functional of an empirical process.
The second ingredient is the calculation of this functional's Hadamard derivative, which leads to
a surprising connection with the classical \emph{first variation formula} for smooth manifolds~\citep{simon,white}.\footnote{Further examples of problems where geometric analysis plays a role in understanding the performance of numerical algorithms may be found in the book~\cite{cucker}.}
To briefly comment on the role of this formula in our analysis, consider the following informal statement of it. Let $\mathcal{M}$ be a smooth $d$-dimensional manifold contained in $\mathbb{R}^d$, and let $\{f_{\delta}\}_{\delta \in(-1,1)}$ be a one-parameter family of diffeomorphisms \smash{$f_{\delta} : \mathcal{M}\to f_{\delta}(\mathcal{M})\subset \mathbb{R}^d$}, satisfying $f_{\delta}\to\text{id}_{\mathcal{M}}$ as $\delta\to 0$, where $\text{id}_{\mathcal{M}}$ denotes the identity map on $\mathcal{M}$. Then,
\begin{equation}\label{firstvarm}
\textstyle\frac{d}{d\delta} \operatorname{vol}(f_{\delta}(\mathcal{M}))\Big|_{\delta=0} = \displaystyle\int_{\mathcal{M}} \text{div}(Z)(\theta) d\theta,
\end{equation}
where $\operatorname{vol}(\cdot)$ is a volume measure, the symbol $\text{div}(Z)$ denotes the divergence of the vector field $Z(\theta):=\frac{\partial}{\partial \delta} f_{\delta}(\theta) \big|_{\delta=0}$, and the symbol $d\theta$ is a volume element on $\mathcal{M}$.
In our analysis, it is necessary to adapt this result to a situation where the maps $f_{\delta}$ are non-smooth, the manifold $\mathcal{M}$ is a non-smooth subset of Euclidean space, and the vector field $Z(\cdot)$ is a non-smooth Gaussian process. Furthermore, applying a version of Stokes' theorem to the right side of equation~\eqref{firstvarm} leads to a particular linear functional of $Z(\cdot)$, which turns out to be the Hadamard derivative relevant to understanding $\textsc{Err}_t$. A more detailed explanation of this connection is given below equation~\eqref{maindiff} in Appendix~\ref{sec:highlevel}.
\paragraph{Related work} In the setting of binary classification, a few papers analyze the bias $\mathbb{E}[\textsc{Err}_t-\text{err}_{\infty}|\mathcal{D}]$, and show that it converges at the fast rate of $1/t$ under various conditions~\citep{ngjordan,lopes2016,cannings2015}. A couple of other works study alternative measures of convergence. For instance, the paper~\citet{lam1997application} considers the probability that the majority vote commits an error at a fixed test point, and the paper~\citet{hernandez} provides an informal analysis of the probability that an ensemble of size $t$ disagrees with an infinite ensemble
at a random test point, but these approaches do not directly control $\textsc{Err}_t$. In addition, some empirical studies of algorithmic convergence may be found in the papers~\citet{oshiro,latinne}.
Among the references just mentioned, the ones that are most closely related to the current paper are~\citet{lopes2016} and~\citet{cannings2015}. These works derive theoretical upper bounds on $\operatorname{var}(\textsc{Err}_t|\mathcal{D})$ or $\operatorname{var}(\textsc{Err}_{t,l}|\mathcal{D})$, where $\textsc{Err}_{t,l}$ is the error rate on a particular class $l$ (cf. Section~\ref{sec:practical}). The paper~\citet{lopes2016} also proposes a method to estimate the unknown parameters in such bounds. In relation to these works, the current paper differs in two significant ways. First, we offer an approximation to the full distribution $\mathcal{L}(\textsc{Err}_t-\text{err}_{\infty}|\mathcal{D})$, and hence provide a \emph{direct estimate} of algorithmic variance, rather than a bound. Second, the method proposed here is relevant to a wider range of problems, since it can handle any number of classes, whereas the analyses in~\citet{lopes2016} and~\citet{cannings2015} are specialized to the binary setting. Moreover, the theoretical analysis of the bootstrap approach is entirely different from the previous techniques used in deriving variance bounds.
Outside of the setting of randomized ensemble classifiers, the papers \citet{sexton2009standard,arlot2014,wager2014confidence,mentch2016,scornet2016asymptotics} look at the algorithmic fluctuations of ensemble regression functions at a fixed test point.
\subsection{Background and setup}\label{sec:background}
We consider the general setting of a classification problem with $k\geq 2$ classes. The set of training data is denoted $\mathcal{D}:=\{(X_1,Y_1),\dots,(X_{n},Y_{n})\}$, which is contained in a sample space $\mathcal{X}\times \mathcal{Y}$. The feature space $\mathcal{X}$ is arbitrary, and the space of labels $\mathcal{Y}$ has cardinality $k$.
An ensemble of $t$ classifiers is denoted by $Q_i:\mathcal{X}\to\mathcal{Y}$, with $i=1,\dots,t$.
\noindent \paragraph{Randomized ensembles}
The key issue in studying the algorithmic convergence of bagging and random forests is randomization. In the method of bagging, randomization is introduced by generating random sets $\mathcal{D}_1^*, \dots,\mathcal{D}_t^*$, each of size $n$, via sampling with replacement from $\mathcal{D}$. For each $i=1,\dots,t$, a classifier $Q_i$ is trained on $\mathcal{D}_i^*$, with the same classification method being used each time. When each $Q_i$ is trained with a decision tree method (such as CART~\citep{CART}), the random forests procedure extends bagging by adding a randomized feature selection rule~\citep{breiman2001}.
It is helpful to note that the classifiers in bagging and random forests can be represented in a common way. Namely, there is a deterministic function, say $g$, such that for any fixed $x\in\mathcal{X}$, each classifier $Q_i$ can be written as
\begin{equation}\label{rfrep}
Q_i(x)=g(x, \mathcal{D}, \xi_i),
\end{equation}
where $\xi_1,\xi_2,\dots$, is an i.i.d. sequence of random objects, independent of $\mathcal{D}$, that specify the ``randomizing parameters'' of the classifiers (cf.\,\citet[Definition 1.1]{breiman2001}). For instance, in the case of bagging, the object $\xi_i$ specifies the randomly chosen points in $\mathcal{D}_i^*$.
Beyond bagging and random forests, our proposed method will be generally applicable to ensembles that can be represented in the form~\eqref{rfrep}, such as those in~\citet{tinkamho1998,dietterich2000,buhlmannyu}. This representation should be viewed abstractly, and it is not necessary for the function $g$ or the objects $\xi_1,\xi_2,\dots$ to be explicitly constructed in practice. Some further examples include a recent ensemble method based on random projections~\citep{cannings2015}, as well as the voting Gibbs classifier~\citep{ngjordan}, which is a Bayesian ensemble method based on posterior sampling.
More generally, if the functions $Q_1,Q_2,\dots$ are i.i.d.~conditionally on $\mathcal{D}$, then the ensemble can be represented in the form~\eqref{rfrep}, as long as the classifiers lie in a standard Borel space~\cite[Lemma 3.22]{Kallenberg}.
Lastly, it is important to note that the representation~\eqref{rfrep} generally does not hold for classifiers generated by boosting methods~\citep{schapire2012boosting}, for which the analysis of algorithmic convergence is quite different.
\noindent \paragraph{Plurality vote} For any $x\in\mathcal{X}$, we define the ensemble's \emph{plurality vote} as the label receiving the largest number of votes among $Q_1(x),\dots,Q_t(x)$. In the exceptional case of a tie, it will simplify technical matters to define the plurality vote as a symbol not contained in $\mathcal{Y}$, so that a tie always counts as an error. We also use the labeling scheme, $\mathcal{Y}:=\{\boldsymbol e_0,\dots,\boldsymbol e_{k-1}\}\subset\mathbb{R}^{k-1}$, where $\boldsymbol e_0:=\boldsymbol 0$, and $\boldsymbol e_l$ is the $l$th standard basis vector for $l\geq 1$. One benefit of this scheme is that the plurality vote is determined by the average of the labels, $\bar{Q}_t(x):=\textstyle\frac{1}{t}\sum_{i=1}^t Q_i(x)$. For this reason, we denote the plurality vote as ${\tt{V}}(\bar{Q}_t(x))$.
\noindent \paragraph{Error rate} Let $\nu=\mathcal{L}(X,Y)$ denote the distribution of a test point $(X,Y)$ in $\mathcal{X}\times \mathcal{Y}$, drawn independently of $\mathcal{D}$ and $Q_1,\dots,Q_t$.
Then, for a particular realization of the classifiers $Q_1,\dots,Q_t$, trained with the given set $\mathcal{D}$, the prediction error rate is defined as
\begin{equation}\label{errdef}
\textsc{Err}_{t}:=\int_{\mathcal{X}\times \mathcal{Y}} 1\{{\tt{V}}(\bar{Q}_t(x))\neq y\}d\nu(x,y) \ = \ \P\Big({\tt{V}}(\bar{Q}_t(X)) \neq Y \, \big| \, \mathcal{D}, \boldsymbol \xi_t\Big),
\end{equation}
where $\boldsymbol \xi_t:=(\xi_1,\dots,\xi_t)$. (Class-wise error rates $\textsc{Err}_{t,l}$, with $l=0,\dots,k-1$ will also be addressed in Section~\ref{sec:classwise}.)
Here, it is crucial to note that $\textsc{Err}_{t}$ is a random variable, since $\bar{Q}_t$ is a random function. Indeed, the integral above shows that $\textsc{Err}_t$ is a functional of $\bar{Q}_t$. Moreover, there are two sources of randomness to consider: the algorithmic randomness arising from $\boldsymbol \xi_t$, and the randomness arising from the training set $\mathcal{D}$. Going forward, we will focus on the algorithmic fluctuations of $\textsc{Err}_{t}$ due to $\boldsymbol \xi_t$, and our analysis will always be conditional on $\mathcal{D}$.
\noindent \paragraph{Algorithmic variance} At first sight, it might not be obvious how to interpret the algorithmic fluctuations of $\textsc{Err}_t$ when $\mathcal{D}$ is held fixed. These fluctuations are illustrated below in Figure~\ref{fig:intro}.
The left panel shows how $\textsc{Err}_{t}$ changes as decision trees are added incrementally during a single run of the random forests algorithm. For the purposes of illustration, if we run the algorithm repeatedly on $\mathcal{D}$ to train many ensembles, we obtain a large number of sample paths of $\textsc{Err}_{t}$ as a function of $t$, shown in the right panel.
Averaging the sample paths at each value of $t$ produces the red curve, representing $\mathbb{E}[\textsc{Err}_{t}\big| \mathcal{D}]$ with $\boldsymbol \xi_t$ averaged out. Furthermore, the
blue envelope curves for the sample paths are obtained by plotting \mbox{$\mathbb{E}[\textsc{Err}_{t}| \mathcal{D}]\pm3\sqrt{\operatorname{var}(\textsc{Err}_t|\mathcal{D})}$} as a function of $t$.
\begin{figure}[h!]
\centering
{\includegraphics[angle=0,
width=.44\linewidth, height=.44\linewidth]{./Figures/intro_plot_left_raw_24dec2017.pdf}} \ \ \ \
{\includegraphics[angle=0,
width=.44\linewidth,height=.44\linewidth]{./Figures/intro_plot_right_raw_24dec2017.pdf}}
%
\vspace{-0.1cm}
\caption{ Left panel: The fluctuations of $\textsc{Err}_{t}$ for a single run of random forests on the ``nursery data'' (cf. Section~\ref{sec:expt}). Right panel: The fluctuations of $\textsc{Err}_{t}$ for 1,000 runs of random forests on the same data (i.e. $1,\!000$ different realizations of the ensemble $Q_1,\dots,Q_t$). For each ensemble, the value $\textsc{Err}_t$ was approximated by testing on the set of points denoted $\mathcal{D}_{\textup{ground}}$, as described in Section~\ref{sec:expt}.
}
\label{fig:intro}
\end{figure}
\noindent \paragraph{Problem formulation}\label{probformulation}
Recall that the value $\text{err}_{\infty}=\text{err}_{\infty}(\mathcal{D})$ represents the ideal prediction error of an infinite ensemble trained on $\mathcal{D}$. Hence, a natural way of defining algorithmic convergence is to say that it occurs when $t$ is large enough so that the condition \smash{$|\textsc{Err}_t-\text{err}_{\infty}|\leq \epsilon$} holds with high probability, conditionally on $\mathcal{D}$, for some user-specified tolerance $\epsilon$. However, the immediate problem we face is that it is not obvious how to check such a condition in practice.
From the right panel of Figure~\ref{fig:intro}, we see that for most $t$, the inequality \smash{$|\textsc{Err}_t-\text{err}_{\infty}|\leq 3\sigma_t$} is highly likely to hold --- and this observation can be formalized using Theorem~\ref{THM:CLT} later on. For this reason, we propose to estimate $\sigma_t$ as a route to measuring algorithmic convergence. It is also important to note that estimating the quantiles of $\L(\textsc{Err}_t-\text{err}_{\infty}|\mathcal{D})$ would serve the same purpose, but for the sake of simplicity, we will focus on $\sigma_t$.
In particular, there are at least two ways that an estimate $\hat{\sigma}_t$ can be used in practice:
\label{probformulation}
\begin{enumerate}
\item \emph{Checking convergence for a given ensemble.} \label{probformulation}
If an ensemble of a given size $t_0$ has been trained, then convergence can be checked by asking whether or not the observable condition $3\hat{\sigma}_{t_0}\leq \epsilon$ holds. Additional comments on possible choices for $\epsilon$ will be given shortly.
\item \emph{Selecting $t$ dynamically.} In order to make the training process as computationally efficient as possible, it is desirable to select the smallest $t$ needed so that $|\textsc{Err}_t-\text{err}_{\infty}|\leq \epsilon$ is likely to hold. It turns out that this can be accomplished using an extrapolation technique, due to the fact that $\sigma_t$ tends to scale like $1/\sqrt{t}$ (cf. Theorem~\ref{THM:CLT}). More specifically, if the user trains a small initial ensemble of size $t_0$ and computes an estimate $\hat{\sigma}_{t_0}$, then ``future'' values of $\sigma_t$ for $t\gg t_0$ can be estimated at no additional cost with the re-scaled estimate $\sqrt{t_0/t}\,\hat{\sigma}_{t_0}$. In other words, it is possible to look ahead and predict how many additional classifiers are needed to achieve $3\sigma_t\leq \epsilon$. Additional details are given in Section~\ref{sec:extrap}.
\end{enumerate}
\paragraph{Sources of difficulty in estimating $\sigma_t$} Having described the basic formulation of the problem, it is important to identify what challenges are involved in estimating $\sigma_t$.
First, we must keep in mind that the parameter $\sigma_t$ describes how $\textsc{Err}_t$ fluctuates over \emph{repeated ensembles} generated from $\mathcal{D}$ --- and so it is not obvious that it is possible to estimate $\sigma_t$ from the output of a \emph{single ensemble}. Second,
the computational cost to estimate $\sigma_t$ should not outweigh the cost of training the ensemble, and consequently, the proposed method should be computationally efficient.
These two obstacles will be described in Sections~\ref{sec:consist} and~\ref{sec:extrap} respectively.
\paragraph{Remarks on the choice of error rate}
Instead of analyzing the random variable $\textsc{Err}_t$, a natural inclination would be to consider the ``unconditional'' error rate \smash{$\mathbb{E}[\textsc{Err}_t]=\P({\tt{V}}(\bar{Q}_t(X))\neq Y))$}, which is likely more familiar, and reflects averaging over both $\mathcal{D}$ and the randomized algorithm. Nevertheless, there are a few reasons why the ``conditional'' error $\textsc{Err}_t$ may be more suitable for analyzing algorithmic convergence. First, the notion of algorithmic convergence typically describes how much computation should be applied to a given input --- and in our context, the given input for the training algorithm is $\mathcal{D}$. Second, from an operational standpoint, once a user has trained an ensemble on a given dataset, their \emph{actual} probability of misclassifying a future test point is $\textsc{Err}_t$, rather than $\mathbb{E}[\textsc{Err}_t]$.
There is also a sense in which the convergence of $\mathbb{E}[\textsc{Err}_t]$ may be misleading. Existing theoretical results suggest that $\mathbb{E}[\textsc{Err}_t-\text{err}_{\infty}]$ converges at the fast rate of $1/t$, and in the binary case, \smash{$k=2$}, it can be shown that \smash{$\mathbb{E}[\textsc{Err}_t-\text{err}_{\infty}|\mathcal{D}]=\textstyle\frac{1}{t}c(\mathcal{D})+o(\textstyle\frac{1}{t})$,} for some number $c(\mathcal{D})$, under certain conditions~\citep{lopes2016,cannings2015}. However, in Theorem 1 of Section~\ref{sec:main}, we show that conditionally on $\mathcal{D}$, the difference $\textsc{Err}_t-\text{err}_{\infty}$ has fluctuations of order $1/\sqrt{t}$.
In this sense, if the choice of $t$ is guided only by $\mathbb{E}[\textsc{Err}_t]$ (rather than the fluctuations of $\textsc{Err}_t$), then the user may be misled into thinking that algorithmic convergence occurs much faster than it really does --- and this distinction is apparent from the red and blue curves in Figure~\ref{fig:intro}.
\paragraph{Outline}Our proposed bootstrap method is described in Section~\ref{sec:method}, and our main consistency result is given in Section~\ref{sec:main}. Practical considerations are discussed in Section~\ref{sec:practical}, numerical experiments are given in Section~\ref{sec:expt}, and conclusions are stated in Section~\ref{sec:conc}. The essential ideas of the proofs are explained in Appendices~\ref{app:mainthm} and~\ref{sec:highlevel}, while the technical arguments are given Appendices~\ref{app:variation}-\ref{lemmas}. Lastly, in Appendix~\ref{app:assumption}, we provide additional assessment of technical assumptions. All appendices are in the supplementary material.
\section{Method}\label{sec:method}
Based on the definition of $\textsc{Err}_t$ in equation~\eqref{errdef}, we may view $\textsc{Err}_t$ as a functional of $\bar{Q}_t$, denoted
\begin{equation*}
\textsc{Err}_t = \varphi(\bar{Q}_t).
\end{equation*}
From a statistical standpoint, the importance of this expression is that $\textsc{Err}_t$ is a functional of a sample mean, which makes it plausible that $\sigma_t$ is amenable to bootstrapping, provided that $\varphi$ is sufficiently smooth.
To describe the bootstrap method, let $(Q_1^*,\dots,Q_t^*)$ denote a random sample with replacement from the trained ensemble $(Q_1,\dots,Q_t)$, and put
$\bar{Q}_t^*(\cdot):=\textstyle\frac{1}{t}\textstyle\sum_{i=1}^t Q_i^*(\cdot).$
In turn, it would be natural to regard the quantity
\begin{equation}\label{errtstar}
\textsc{Err}_t^*:= \varphi(\bar{Q}_t^*)
\end{equation}
as a bootstrap sample of $\textsc{Err}_t$, but strictly speaking, this is an ``idealized'' bootstrap sample, because the functional $\varphi$ depends on the unknown test point distribution $\nu=\mathcal{L}(X,Y)$. Likewise, in Section~\ref{hoalg} below, we explain how each value $\varphi(\bar{Q}_t^*)$ can be estimated. So, in other words, if $\hat{\varphi}$ denotes an estimate of $\varphi$, then an estimate of $\textsc{Err}_t$ would be written as
\begin{equation*}
\hat{\textsc{Err}}_t:=\hat{\varphi}(\bar{Q}_t),
\end{equation*}
and the corresponding bootstrap sample is
\begin{equation*}
\hat{\textsc{Err}}_t^*:=\hat{\varphi}(\bar{Q}_t^*).
\end{equation*}
Altogether, a basic version of the proposed bootstrap algorithm is summarized as follows.
\paragraph{Algorithm 1 (Bootstrap estimation of $\sigma_t$)}
~\\
\vspace{-0.2cm}
\hrule
~\\[0.1cm]
\noindent \textbf{For} $b=1,\dots, B \, \textbf{:}$
\begin{itemize}
\item Sample $t$ classifiers $(Q_1^*,\dots,Q_t^*)$ with replacement from $(Q_1,\dots,Q_t)$.
\item Compute $z_b:= \hat{\varphi}(\bar{Q}_t^*)$.
\end{itemize}
\textbf{Return:} the sample standard deviation of $z_1,\dots,z_B$, denoted $\hat{\sigma}_t$.\\[-0.1cm]
\hrule
~\\
\vspace{-0.3cm}
\noindent
\paragraph{Remark} While the above algorithm is conceptually simple, it suppresses most of the implementation details, and these are explained below. Also note that in order to approximate quantiles of \smash{$\mathcal{L}(\textsc{Err}_t-\text{err}_{\infty}|\mathcal{D})$}, rather than $\sigma_t$, it is only necessary to modify the last step, by returning the desired quantile of the centered values $z_1-\bar{z},\dots,z_B-\bar{z}$, with $\bar{z}=\frac{1}{B}\sum_{b=1}^Bz_b$.
\subsection{Resampling algorithm with hold-out or ``out-of-bag'' points}\label{hoalg}
Here, we consider a version of Algorithm 1 where $\varphi$ is estimated implicitly with hold-out points, and then later on, we will explain how hold-out points can be avoided using ``out-of-bag'' (\textsc{oob}) points. To begin, suppose we have a hold-out set of size $m$, denoted \smash{$\mathcal{D}_{\text{hold}}:=\{(\tilde{X}_1,\tilde{Y}_1),\dots,(\tilde{X}_m,\tilde{Y}_m)\}$.} Next, consider an array $\tilde{A}$ of size $t\times m$, whose $i$th row $\tilde{\mathbf{a}}_i$ is given by the predicted labels of $Q_i$ on the hold-out points. That is,
\begin{equation}\label{atildedef}
\tilde{\mathbf{a}}_i:= {\Large{\boldsymbol [}} \, Q_i(\tilde{X}_1)\, , \dots, \, Q_i(\tilde{X}_m) \,{\Large{\boldsymbol ]} },
\end{equation}
and
\begin{eqnarray}
\tilde{A}:= \left[
\begin{array}{l}
- \tilde{\mathbf{a}}_1 - \\
\ \ \ \vdots\\
- \tilde{\mathbf{a}}_t -
\end{array}
\right].
\end{eqnarray}
The estimated error rate is easily computed as a function of this array, i.e.~$\hat{\textsc{Err}}_t = \hat{\textsc{Err}}_t(\tilde{A})$. To spell out the details, let $\tilde{A}_j=(Q_1(\tilde{X}_j),\dots,Q_t(\tilde{X}_j)$) denote the $j$th column of $\tilde{A}$, and with a slight abuse of our earlier notation, let ${\tt{V}}(\tilde{A}_j)$ denote the plurality vote of the labels in $\tilde{A}_j$. Then, the estimated error rate is obtained from simple column-wise operations on $\tilde{A}$,
\begin{equation}\label{errAdef}
\hat{\textsc{Err}}_t(\tilde{A}) := \frac{1}{m}\displaystyle\sum_{j=1}^m 1\{{\tt{V}}(\tilde{A}_j)\neq \tilde{Y}_j\}.
\end{equation}
In other words, $\hat{\textsc{Err}}_t(\tilde{A})$ is just the proportion of columns of $\tilde{A}$ for which plurality vote is incorrect. (Note that equation~\eqref{errAdef} is where $\varphi$ is implicitly estimated.) Finally, since there is a one-to-one correspondence between the rows $\tilde{\mathbf{a}}_i$ and the classifiers $Q_i$, the proposed method is equivalent to resampling the rows $\tilde{\mathbf{a}}_i$, as given below.
\paragraph{Algorithm 2 (Bootstrap estimation of $\sigma_t$ with hold-out points)}
~\\
\vspace{-0.2cm}
\hrule
~\\[0.1cm]
\noindent \textbf{For} $b=1,\dots, B \, \textbf{:}$
\begin{itemize}
\item Draw a $t\times m$ array $\tilde{A}^*$ whose rows $(\tilde{\mathbf{a}}_1^*,\dots,\tilde{\mathbf{a}}_t^*)$ are sampled with replacement from $(\tilde{\mathbf{a}}_1,\dots,\tilde{\mathbf{a}}_t)$.
\item Compute $z_b:= \hat{\textsc{Err}}(\tilde{A}^*)$.
\end{itemize}
\textbf{Return:} the sample standard deviation of $z_1,\dots,z_B$, denoted $\hat{\sigma}_t$.\\[0.0cm]
\hrule
~\\[-0.5cm]
\paragraph{Extension to OOB points} Since the use of a hold-out set is often undesirable in practice, we instead consider \textsc{oob} points --- which are a special feature of bagging and random forests. To briefly review this notion, recall that each classifier $Q_i$ is trained on a set of $n$ points $\mathcal{D}_i^*$ obtained by sampling with replacement from $\mathcal{D}$. Consequently, each set $\mathcal{D}_i^*$ excludes approximately $(1-\textstyle\frac{1}{n})^n\approx $ 37\% of the points in $\mathcal{D}$, and these excluded points may be used as test points for the particular classifier $Q_i$. If a point $X_j$ is excluded from $\mathcal{D}_i^*$, then we say ``the point $X_j$ is \textsc{oob} for the classifer $Q_i$'', and we write $i \in \textsc{\textsc{oob}}(X_j)$, where the set $\textsc{\textsc{oob}}(X_j)\subset\{1,\dots,t\}$ indexes the classifiers for which $X_j$ is \textsc{oob}.
In this notation, the error estimate $\hat{\textsc{Err}}_t(\tilde{A})$ in Algorithm 2 can be given an analogous definition in terms of \textsc{oob} points. Define a \emph{new} $t\times n$ array $A$ whose $i$th row is given by
\begin{equation*}
\mathbf{a}_i:=\big[ Q_i(X_1),\dots,Q_i(X_n)\big].
\end{equation*}
Next, letting $A_j$ be the $j$th column of $A$, define ${{\tt{V}}}_{\textsc{o}}(A_j)$ to be the plurality vote of the set of labels $\{Q_i(X_j) \, |\, i \in \textsc{\textsc{oob}}(X_j)\}$. (If this set of labels is empty, then we treat this case as a tie, but this is unimportant, since it occurs with probability $[1-(1-\textstyle\frac{1}{n})^{n}]^t \approx (0.63)^t$.)
So, by analogy with $\hat{\textsc{Err}}_{t}(\tilde{A})$, we define
\begin{equation}\label{errOOB}
\hat{\textsc{Err}}_{t,\textsc{o}}(A) :=\frac{1}{n}\sum_{j=1}^n 1\{{\tt{V}}_{\textsc{o}}(A_j)\neq Y_j\}.
\end{equation}
Hence, the \textsc{oob} version of Algorithm 2 may be implemented by simply interchanging $\tilde A$ and $A$, as well as $\hat{\textsc{Err}}_{t}(\tilde{A}^*)$ and $\hat{\textsc{Err}}_{t,\textsc{o}}(A^*)$. The essential point to notice is that the sum in equation~\eqref{errOOB} is now over the training points in $\mathcal{D}$, rather than over the hold-out set $\mathcal{D}_{\text{hold}}$, as in equation~\eqref{errAdef}.
\section{Main result}\label{sec:main}
Our main theoretical goal is to prove that the bootstrap yields a consistent approximation of $\mathcal{L}(\sqrt{t}(\textsc{Err}_t-\text{err}_{\infty})|\mathcal{D})$ as $t$ becomes large. Toward this goal, we will rely on two simplifications that are customary in analyses of bootstrap and ensemble methods. First, we will exclude the Monte-Carlo error arising from the finite number of $B$ bootstrap replicates, as well as the error arising from the estimation of $\textsc{Err}_t$. For this reason, our results do not formally require the training or hold-out points to be i.i.d. copies of the test point $(X,Y)$ --- but from a practical standpoint, it is natural to expect that this type of condition should hold in order for Algorithm 2 (or its \textsc{oob} version) to work well.
Second, we will analyze a simplified type of ensemble, which we will refer to as a \emph{first-order model}. This type of approach has been useful in gaining theoretical insights into the behavior of complex ensemble methods in a variety of previous works~\citet{biau2008,biau2012,arlot2014,linjeon,genuer,scornet2016asymptotics,scornetIEEE}. In our context, the value of this simplification is that it neatly packages the complexity of the base classifiers, and clarifies the relationship between $t$ and quality of the bootstrap approximation. Also, even with such simplifications, the theoretical problem of proving bootstrap consistency still leads to considerable technical challenges. Lastly, it is important to clarify that the first-order model is introduced only for theoretical analysis, and our proposed method does not rely on this model.
\subsection{A first-order model for randomized ensembles} Any randomized classifier $Q_1:\mathcal{X}\to \{\boldsymbol e_0,\dots,\boldsymbol e_{k-1}\}$ may be viewed as a stochastic process indexed by $\mathcal{X}$. From this viewpoint, we say that another randomized classifier \smash{$T_1:\mathcal{X}\to \{\boldsymbol e_0,\dots,\boldsymbol e_{k-1}\}$} is a \emph{first-order model} for $Q_1$ if it has the same marginal distributions as $Q_1$, conditionally on $\mathcal{D}$, which means
\begin{equation}\label{firsto}
\mathcal{L}(Q_1(x)|\mathcal{D}) = \ \mathcal{L}(T_1(x)|\mathcal{D}) \ \textup{ for all } x\in\mathcal{X}.
\end{equation}
Since $Q_1(x)$ takes values in the finite set of binary vectors $\{\boldsymbol e_0,\dots,\boldsymbol e_{k-1}\}$, the condition~\eqref{firsto} is equivalent to
\begin{equation}\label{firste}
\mathbb{E}[Q_1(x)|\mathcal{D}] \ = \ \mathbb{E}[T_1(x)|\mathcal{D}] \ \ \textup{ for all } x\in\mathcal{X},
\end{equation}
where the expectation is only over the algorithmic randomness in $Q_1$ and $T_1$.
A notable consequence of this matching condition is that the ensembles associated with $Q_1$ and $T_1$
have the \emph{same} error rates on average.
Indeed, if we let $\textsc{Err}_{t}'$ be the error rate associated with an ensemble of $t$ independent copies of $T_1$, then it turns out that
\begin{equation}\label{sameerr}
\begin{split}
\mathbb{E}[\textsc{Err}_{t}|\mathcal{D}]= \mathbb{E}[\textsc{Err}_{t}'|\mathcal{D}],
\end{split}
\end{equation}
for all $t\geq 1$, where $\textsc{Err}_t$ is the error rate for $Q_1,\dots,Q_t$, as before. (A short proof is given in Appendix~\ref{lemmas}.)
In this sense, a first-order model $T_1$ is a meaningful proxy for $Q_1$ with regard to statistical performance --- even though the internal mechanisms of $T_1$ may be simpler.
\subsubsection{Constructing a first-order model}
Having stated some basic properties that are satisfied by any first-order model, we now construct a particular version that is amenable to analysis. Interestingly, it is possible to start with an \emph{arbitrary} random classifier $Q_1:\mathcal{X}\to\{\boldsymbol e_0,\dots, \boldsymbol e_{k-1}\}$, and construct an associated $T_1$ in a relatively explicit way.
To do this, let $x\in\mathcal{X}$ be fixed, and consider the function
\begin{equation}\label{varthetadef}
\vartheta(x):=\mathbb{E}[Q_1(x)|\mathcal{D}],
\end{equation}
which takes values in the ``full-dimensional'' simplex $\Delta\subset\mathbb{R}^{k-1}$, defined by
\begin{equation*}
\Delta :=\Big\{\theta\in [0,1]^{k-1}\Big| \ \theta_1+\dots+\theta_{k-1}\leq 1 \Big\}.
\end{equation*}
For any fixed $\theta\in\Delta$, there is an associated partition of the unit interval into sub-intervals
\begin{equation*}
I_0(\theta)\cup \dots \cup I_{k-1}(\theta) = [0,1],
\end{equation*}
such that the width of interval $I_l(\theta)$ is equal to $\theta_l$ for $l\geq 1$. Namely, we put $I_1(\theta):=[0,\theta_1]$, and for $l=2,\dots,k-1$,
\begin{equation*}
I_l(\theta) := \Big((\theta_1+\cdots+\theta_{l-1})\, , \, (\theta_1+\cdots+\theta_{l})\Big].
\end{equation*}
Lastly, for $I_0$, we put
\begin{equation*}
I_0(\theta):=\big(\textstyle\sum_{l=1}^{k-1}\theta_l \, , \, 1\big].
\end{equation*}
Now, if we let $x\in\mathcal{X}$ be fixed, and let $U_1\sim $ Uniform$[0,1]$, then we define $T_1(x)\in \{\boldsymbol e_0,\dots,\boldsymbol e_{k-1}\}$ to have its $l$th coordinate equal to the following indicator variable
\begin{equation*}
[T_1(x)]_l:=1\big\{U_1\in I_l(\vartheta(x))\big\},
\end{equation*}
where $l=1,\dots,k-1$. It is simple to check that the first-order matching condition~\eqref{firste} holds, and so $T_1$ is indeed a first-order model of $Q_1$.
Furthermore, given that $T_1$ is defined in terms of a single random variable $U_1\sim$ Uniform$[0,1]$, we obtain a corresponding ``first-order ensemble'' $T_1,\dots,T_t$ via an i.i.d.~sample of uniform variables $U_1,\dots,U_t$, which are independent of $\mathcal{D}$. (The $l$th coordinate of the $i$th classifier $T_i$ is given by $[T_i(x)]_l=1\{U_i\in I_l(\vartheta(x))\}$.) Hence, with regard to the representation $Q_i(x)=g(x,\mathcal{D},\xi_i)$ in equation~\eqref{rfrep}, we may make the identification\label{thetaiui}
$$\xi_i=U_i,$$
when the first-order model holds with $Q_i=T_i$.
\paragraph{Remark} To mention a couple of clarifications, the variables \smash{$U_1,\dots,U_t$} are only used for the construction of a first-order model, and they play no role in the proposed method. Also, even though the ``randomizing parameters'' $U_1,\dots,U_t$ are independent of $\mathcal{D}$, the classifiers $T_1,\dots,T_t$ still depend on $\mathcal{D}$ through the function $\vartheta(x)=\mathbb{E}[Q_1(x)|\mathcal{D}]$.
\paragraph{Interpretation of first-order model}
To understand the statistical meaning of the first-order model, it is instructive to consider the simplest case of binary classification, $k=2$. In this case, $T_1(x)$ is a Bernoulli random variable, where $T_1(x)=1\{U_1\leq \vartheta(x)\}$. Since $\bar{Q}_t(x)\to \vartheta(x)$ almost surely as \smash{$t\to\infty$} (conditionally on $\mathcal{D}$), the majority vote of an infinite ensemble has a similar form, i.e. $1\{\textstyle\frac{1}{2}\leq \vartheta(x)\}$. Hence, the classifiers $\{T_i\}$ can be viewed as ``random perturbations'' of the asymptotic majority vote arising from $\{Q_i\}$. Furthermore, if we view the number $\vartheta(x)$ as score to be compared with a threshold, then the variable $U_i$ plays the role of a random threshold whose expected value is $\frac{1}{2}$. Lastly, even though the formula $T_1(x)=1\{U_1\leq \vartheta(x)\}$ might seem to yield a simplistic classifier, the complexity of $T_1$ is actually wrapped up in the function $\vartheta$.
%
Indeed, the matching condition~\eqref{firste} allows for the function $\vartheta$ to be arbitrary.
\subsection{Bootstrap consistency}\label{sec:consist}
We now state our main result, which asserts that the bootstrap ``works'' under the first-order model.
To give meaning to bootstrap consistency, we first review the notion of conditional weak convergence.
\paragraph{Conditional weak convergence}
Let $\lambda_0$ be a probability distribution on $\mathbb{R}$, and let $\{\lambda_{\boldsymbol \xi_t}\}_{t\geq 1}$ be a sequence of probability distributions on $\mathbb{R}$ that depend on the randomizing parameters \smash{$\boldsymbol \xi_t=(\xi_1,\dots,\xi_t)$}. Also, let $d_{\text{BL}}$ be the bounded Lipschitz metric for distributions on $\mathbb{R}$~\cite[Sec 1.12]{vaartWellner}, and let $\P_{\boldsymbol \xi}$ be the joint distribution of $(\xi_1,\xi_2,\dots)$. Then, as $t\to\infty$, we say that $\lambda_{\boldsymbol \xi_t}\xrightarrow{ \ w \ } \lambda_0$ in $\P_{\boldsymbol \xi}$-probability if the sequence $\{d_{\text{BL}}(\lambda_{\boldsymbol \xi_t},\lambda_0)\}_{t\geq 1}$ converges to 0 in $\P_{\boldsymbol \xi}$-probability.
\paragraph{Remark} If a test point $X$ is drawn from class $Y=\boldsymbol e_l$, then we denote the distribution of the random vector $\vartheta(X)$, conditionally on $\mathcal{D}$, as
\begin{equation*}
\mu_l:=\mathcal{L}(\vartheta(X)|\mathcal{D}, Y=\boldsymbol e_l),
\end{equation*}
which is a distribution on the simplex $\Delta\subset \mathbb{R}^{k-1}$. Since this distribution plays an important role in our analysis, it is worth noting that the properties of $\mu_l$ are \emph{not affected} by the assumption of a first-order model, since $\vartheta(x)=\mathbb{E}[T_1(x)|\mathcal{D}]=\mathbb{E}[Q_1(x)|\mathcal{D}]$ for all $x\in \mathcal{X}$. We will also assume that the measures $\mu_l$ satisfy the following extra regularity condition.
\begin{assumption}\label{ASSUMPTION}
For the given set $\mathcal{D}$, and each $l=0,\dots,k-1$, the distribution $\mu_l$ has a density $f_l:\Delta\to [0,\infty)$ with respect to Lebesgue measure on $\Delta$, and $f_l$ is continuous on $\Delta$. Also, if $\Delta^{\circ}$ denotes the interior of $\Delta$, then for each $l$, the density $f_l$ is $C^1$ on $\Delta^{\circ}$, and $\|\nabla f_l\|_2$ is bounded on $\Delta^{\circ}$.
\end{assumption}
To interpret this assumption, consider a situation where the class-wise test point distributions $\mathcal{L}(X|Y=\boldsymbol e_l)$ have smooth densities on $\mathcal{X}\subset\mathbb{R}^p$ with respect to Lebesgue measure. In this case, the density $f_l$ will exist as long as $\vartheta$ is sufficiently smooth (cf. Appendix~\ref{app:assumption}, Proposition~\ref{prop:assumption}). Still, Assumption~\ref{ASSUMPTION} might seem unrealistic in the context of random forests, because $\vartheta$ is obtained by averaging over all decision trees that can be generated from $\mathcal{D}$, and strictly speaking, this is a finite average of non-smooth functions.
However, due to the bagging mechanism in random forests, the space of trees that can be generated from $\mathcal{D}$ is very large, and consequently, the function $\vartheta$ represents a very fine-grained average. Indeed, the idea that bagging is actually a ``smoothing operation'' on non-smooth functions has received growing attention over the years~\citep{bujastuetzle2000,buhlmannyu,bujastuetzle2006,efron2014}, and the recent paper \citet{efron2014} states that bagging is ``also known as bootstrap smoothing''. In Appendix~\ref{app:assumption}, we provide additional assessment of Assumption~\ref{ASSUMPTION}, in terms of both theoretical and empirical examples.
%
%
\begin{thm}[Bootstrap consistency]\label{THM:CLT} \hspace{-.2cm}Suppose that the first-order model $Q_i=T_i$ holds for all $i\geq 1$, and that Assumption~\ref{ASSUMPTION} holds.
Then, for the given set $\mathcal{D}$, there are numbers $\textup{err}_{\infty}=\textup{err}_{\infty}(\mathcal{D})$ and $\sigma=\sigma(\mathcal{D})$ such that as $t\to\infty$,
\begin{equation}\label{basicclt}
\mathcal{L}\big(\sqrt{t}(\textup{Err}_t-\textup{err}_{\infty})\big|\mathcal{D}\big)
\xrightarrow{ \ \ w \ \ } N(0,\sigma^2),
\end{equation}
and furthermore,
\begin{equation*}
\mathcal{L}\big(\sqrt{t}(\textup{Err}^*_t-\textup{Err}_t)\big| \mathcal{D}, \boldsymbol \xi_t)
\xrightarrow{ \ \ w \ \ } N(0,\sigma^2) \text{ \ in \ $\P_{\boldsymbol \xi}$-probability.}
\end{equation*}
\end{thm}
\paragraph{Remarks}
In a nutshell, the proof of Theorem~\ref{THM:CLT} is composed of three pieces: showing that $\textsc{Err}_t$ can be represented as a functional of an empirical process (Appendix~\ref{sec:lift}), establishing the smoothness of this functional (Appendix~\ref{sec:hadamard}), and employing the functional delta method (Appendix~\ref{sec:delta}).
With regard to theoretical techniques, there are two novel aspects of the proof. The problem of deriving this functional is solved by introducing a certain lifting operator, while the problem of showing smoothness is based on a non-smooth instance of the first-variation formula, as well as some special properties of Bernstein polynomials. Lastly, it is worth mentioning that the core technical result of the paper is Theorem~\ref{THM:HADAMARD}.
To mention some consequences of Theorem~\ref{THM:CLT}, the fact that the limiting distribution of $\mathcal{L}(\textsc{Err}_t-\text{err}_{\infty} \big| \mathcal{D})$ has mean 0 shows that the fluctuations of $\textsc{Err}_t$ have more influence on algorithmic convergence than the bias $\mathbb{E}[\textsc{Err}_t-\text{err}_{\infty}\big| \mathcal{D}]$ (as illustrated in Figure~\ref{fig:intro}). Second, the limiting distribution motivates a convergence criterion of the form $3\sigma_t\leq \epsilon$, since asymptotic normality it indicates that the event \smash{$|\textsc{Err}_t-\text{err}_{\infty}|\leq 3\sigma_t$} should occur with high probability when $t$ is large, and again, this is apparent in Figure~\ref{fig:intro}. Lastly, the theorem implies that the quantiles of $\mathcal{L}(\textsc{Err}_t-\text{err}_{\infty}\big| \mathcal{D})$ agree asymptotically with their bootstrap counterparts. This is of interest, because quantiles allow the user to specify a bound on $\textsc{Err}_t-\text{err}_{\infty}$ that holds with a tunable probability.
Quantiles also provide an alternative route to estimating algorithmic variance, because if $r_t^*$ denotes the interquartile range of $\mathcal{L}(\textsc{Err}_t^*-\textsc{Err}_t\big| \mathcal{D},\boldsymbol \xi_t)$, then the theorem implies
$ \textstyle\frac{\sqrt{t}}{c}r_t^* \xrightarrow{ \ \ \ } \sigma \text{ in $\P_{\boldsymbol \xi}$-probability},$
where $c=\Phi^{-1}(3/4)-\Phi^{-1}(1/4)$.
\section{Practical considerations}~\label{sec:practical}
In this section, we discuss some considerations that arise when the proposed method is used in practice, such as the choice of error rate, the computational cost, and the choice of a stopping criterion for algorithmic convergence.
\subsection{Extension to class-wise error rates}\label{sec:classwise} In some applications, class-wise error rates may be of greater interest than the total error rate $\textsc{Err}_t$. For any $l=0,\dots,k-1$, let \smash{$\nu_l=\mathcal{L}(X|Y=\boldsymbol e_l)$} denote the distribution of the test point $X$ given that it is drawn from class $l$. Then, the error rate on class $l$ is defined as
\begin{equation}\label{classerrdef}
\textsc{Err}_{t,l}:=\int_{\mathcal{X}} 1\{{\tt{V}}(\bar{Q}_t(x))\neq \boldsymbol e_l\}d\nu_l(x) \ = \ \P\Big({\tt{V}}(\bar{Q}_t(X)) \neq \boldsymbol e_l \, \big| \, \mathcal{D}, \boldsymbol \xi_t, Y=\boldsymbol e_l\Big),
\end{equation}
and the corresponding algorithmic variance is
$$\sigma_{t,l}^2 :=\operatorname{var}(\textsc{Err}_{t,l}|\mathcal{D}).$$
In order to estimate $\sigma_{t,l}$, Algorithm~2 can be easily adapted using either hold-out or \textsc{oob} points from a particular class.
Our theoretical analysis also extends immediately to the estimation of $\sigma_{t,l}$ (cf. Section~\ref{sec:lift}).
\subsection{Computational cost and extrapolation}\label{sec:extrap}
A basic observation about Algorithm 2 is that it only relies on the array of predicted labels $\tilde{A}$ (or alternatively $A$). Consequently, the algorithm does not require any re-training of the classifiers. Also, with regard to computing the arrays $\tilde{A}$ or $A$, at least one of these is \emph{typically computed anyway} when evaluating an ensemble's performance with hold-out or \textsc{oob} points --- and so the cost of obtaining $\tilde{A}$ or $A$ will typically not be an added expense of Algorithm 2.
Thirdly, the algorithm can be parallelized, since the bootstrap replicates can be computed independently.
Lastly, the cost of the algorithm is \emph{dimension-free} with respect to the feature space $\mathcal{X}$, since all operations are on the arrays $\tilde A$ or $A$, whose sizes do not depend on the number of features.
To measure the cost of Algorithm 2 in terms of floating point operations, it is simple to check that at each iteration $b=1,\dots, B$, the cost of evaluating $\widehat{\textsc{Err}}_t(\tilde{A}^*)$ is of order $t\cdot m$, since $\tilde{A}$ has $m$ columns, and each evaluation of the plurality voting function has cost $\mathcal{O}(t)$. Hence, if the arrays $\tilde{A}$ or $A$ are viewed as given, and if $m=\mathcal{O}(n)$, then the cost of Algorithm 2 is $\mathcal{O}(B\cdot t \cdot n)$, for either the hold-out or \textsc{oob} versions. Below, we describe how this cost can be reduced using a basic form of extrapolation~\citep{bickelrichardson,sidi,brezinski}.
\paragraph{Saving on computation with extrapolation}
To explain the technique of extrapolation,
the first step produces an inexpensive estimate $\hat{\sigma}_{t_0}$ by applying Algorithm 2 to a small initial ensemble of size $t_0$. The second step then rescales $\hat{\sigma}_{t_0}$ so that it approximates $\sigma_t$ for $t\gg t_0$.
This rescaling relies on Theorem~\ref{THM:CLT}, which leads to the approximation,
$\sigma_t \approx \textstyle\frac{\sigma}{\sqrt{t}}$.
Consequently, we define the extrapolated estimate of $\sigma_t$ as
\begin{equation}\label{extrapcondition}
\hat{\sigma}_{t,\text{extrap}}:=\textstyle\frac{\sqrt{t_0}}{\sqrt{t}}\cdot \hat{\sigma}_{t_0}.
\end{equation}
In turn, if the user desires $3\sigma_t\leq \epsilon$ for some $\epsilon\in(0,1)$, then $t$ should be chosen so that
\begin{equation}\label{eq:epscriterion}
3\hat{\sigma}_{t,\text{extrap}}\leq \epsilon,
\end{equation}
which is equivalent to
$t\geq (\textstyle\frac{3\sqrt{t_0}}{\epsilon}\cdot\hat{\sigma}_{t_0})^2$.
In addition to applying Algorithm 2 to a smaller ensemble, a second computational benefit is that extrapolation allows the user to ``look ahead'' and dynamically determine how much extra computation is needed so that $\sigma_t$ is within a desired range. In Section~\ref{sec:expt}, some examples are given showing that $\sigma_{1,000}$ can be estimated well via extrapolation when $t_0=200$.
\paragraph{Comparison with the cost of training a random forest} Given that one of the main uses of Algorithm 2 is to control the size of a random forest, one would hope that the cost of Algorithm 2 is less than or similar to the cost of training a single ensemble.
In order to simplify this comparison, suppose that each tree in the ensemble is grown so that all nodes are split into exactly 2 child nodes (except for the leaves), and that all trees are grown to a common depth $d\geq 2$. Furthermore, suppose that $\mathcal{X}\subset \mathbb{R}^p$, and that random forests uses the default rule of randomly selecting from $\lceil \sqrt{p}\rceil$ features during each node split. Under these conditions, it is known that the cost of training a random forest with $t$ trees via CART is at least of order $t\cdot \sqrt{p}\cdot d\cdot n$~\cite[p.166]{CART}. Additional background on the details of random forests and decision trees may be found in the book~\citet{elements}.
Based on the reasoning just given, the cost of running Algorithm 2 does not exceed the cost of training $t$ trees, provided that
$$B =\mathcal{O}(\textstyle\frac{t}{t_0}\cdot\sqrt{p} \cdot d),$$
where the factor $\frac{t}{t_0}$ arises from the extrapolation speedup described earlier.
Moreover, with regard to the selection of $B$, our numerical examples in Section~\ref{sec:expt} show that the modest choice $B=50$ allows Algorithm 2 to perform well on a variety of datasets.
\paragraph{The choice of the threshold $\epsilon$} When using a criterion such as~\eqref{eq:epscriterion} in practice, the choice of the threshold $\epsilon$ will typically be unique to the user's goals. For instance, if the user desires that $\textsc{Err}_t$ is within 0.5\% of $\text{err}_{\infty}$, then the choice $\epsilon=.005$ would be appropriate. Another option is to choose $\epsilon$ from a relative standpoint, depending on the scale of the error. If the error is high (say $\mathbb{E}[\textsc{Err}_t|\mathcal{D}]$ is 40\%), then it may not be worth paying a large computational price to ensure that $3\sigma_t$ is less than 0.5\%. Conversely, if $\mathbb{E}[\textsc{Err}_{t}|\mathcal{D}]$ is 2\%, then it may be worthwhile to train a very large ensemble so that $\sigma_t$ is a fraction of 2\%. In either of these cases, the size of $\sigma_t$ could be controlled in a relative sense by selecting $t$ when $\hat{\sigma}_t\leq \eta \hat{m}_t$, where $\hat{m}_t$ is an estimate of $\mathbb{E}[\textsc{Err}_t|\mathcal{D}]$ obtained from a hold-out set, and $\eta\in (0,1)$ is a user-specified constant that measures the balance between computational cost and accuracy. But regardless of the user's preference for $\epsilon$, the more basic point to keep in mind is that the proposed method makes it possible for the user to have direct control over the relationship between $t$ and $\sigma_t$, and this type of control has not previously been available.
\section{Numerical Experiments}\label{sec:expt}
To illustrate our proposed method, we describe experiments in which the random forests method is applied to natural and synthetic datasets (6 in total). More specifically, we consider the task of estimating the parameter \mbox{3$\sigma_t=3\sqrt{\operatorname{var}(\textsc{Err}_t|\mathcal{D})}$}, as well as
$3\sigma_{t,l}=3\sqrt{\operatorname{var}(\textsc{Err}_{t,l}|\mathcal{D})}$.
Overall, the main purpose of the experiments is to show that the bootstrap can indeed produce accurate estimates of these parameters. A second purpose is to demonstrate the value of the extrapolation technique from Section~\ref{sec:extrap}.
\subsection{Design of experiments}\label{data:design}
Each of the 6 datasets were partitioned in the following way. First, each dataset was evenly split into a training set $\mathcal{D}$ and a ``ground truth'' set $\mathcal{D}_{\text{ground}}$, with nearly matching class proportions in $\mathcal{D}$ and $\mathcal{D}_{\text{ground}}$. (The reason that a substantial portion of data was set aside for $\mathcal{D}_{\text{ground}}$ was to ensure that ground truth values of $\sigma_t$ and $\sigma_{t,l}$ could be approximated using this set.) Next, a smaller set $\mathcal{D}_{\text{hold}}\subset\mathcal{D}_{\text{ground}}$ with cardinality satisfying $|\mathcal{D}_{\text{hold}}|/(|\mathcal{D}_{\text{hold}}|+|\mathcal{D}|)\approx 1/6$ was used as the hold-out set for implementing Algorithm 2. As before, the class proportions in $\mathcal{D}_{\text{hold}}$ and $\mathcal{D}$ were nearly matching. The smaller size of $\mathcal{D}_{\text{hold}}$ was chosen to illustrate the performance of the method when hold-out points are limited.
\paragraph{Ground truth values} After preparing $\mathcal{D}$, $\mathcal{D}_{\text{ground}}$, and $\mathcal{D}_{\text{hold}}$, a collection of 1,000 ensembles was trained on $\mathcal{D}$ by repeatedly running the random forests method. Each ensemble contained a total of 1,000 trees, trained under default settings from the package {\tt{randomForest}}~\citep{randomForestsCitation}. Also, we tested each ensemble on $\mathcal{D}_{\text{ground}}$ to approximate a corresponding sample path of $\textsc{Err}_t$ (like the ones shown in Figure~\ref{fig:intro}). Next, in order to obtain ``ground truth'' values for $\sigma_t$ with $t=1,\dots,1,\!000$, we used the sample standard deviation of the 1,000 sample paths at each $t$. (Ground truth values for each $\sigma_{t,l}$ were obtained analogously.)
\paragraph{Extrapolated estimates} With regard to our methodology, we applied the hold-out and \textsc{oob} versions of Algorithm 2 to each of the ensembles --- yielding 1,000 realizations of each type of estimate of $\sigma_t$. In each case, the number of bootstrap replicates was set to $B=50$, and we applied the extrapolation rule, starting from $t_0=200$. If we let $\hat{\sigma}_{200,\textsc{h}}$ and $\hat{\sigma}_{200,\textsc{o}}$ denote the initial hold-out and \textsc{oob} estimators, then the corresponding extrapolated estimators for $t\geq 200$ are given by
\begin{equation}\label{extrapests}
\hat{\sigma}_{t,\textsc{h},\text{extrap}}:=\textstyle\frac{\sqrt{200}}{\sqrt{t}} \hat{\sigma}_{200,\textsc{h}} \text{ \ \ \ and \ \ \ } \hat{\sigma}_{t,\textsc{o},\text{extrap}}:=\textstyle\frac{\sqrt{200}}{\sqrt{t}} \hat{\sigma}_{200,\textsc{o}}.
\end{equation}
Next, as a benchmark, we considered an enhanced version of the hold-out estimator, for which the entire ground truth set $\mathcal{D}_{\text{ground}}$ was
used in place of $\mathcal{D}_{\text{hold}}$. In other words, this benchmark reflects a situation where a much larger hold-out set is available, and it is referred to as the ``ground estimate'' in the plots. Its value based on $t_0=200$ is
denoted $\hat{\sigma}_{200,\textsc{g}}$, and for $t\geq 200$, we use
\begin{equation}\label{extrapground}
\hat{\sigma}_{t,\textsc{g},\text{extrap}}:=\textstyle\frac{\sqrt{200}}{\sqrt{t}} \hat{\sigma}_{200,\textsc{g}}
\end{equation}
to refer to its extrapolated version. Lastly, class-wise versions of all extrapolated estimators were computed in an analogous way.
\subsection{Description of datasets}\label{sec:dataprep}
The following datasets were each partitioned into $\mathcal{D}$, $\mathcal{D}_{\text{hold}}$ and $\mathcal{D}_{\text{ground}}$, as described above.
\paragraph{Census income data} A set of census records for 48,842 people were collected with 14 socioeconomic features (continuous and discrete)~\citep{uci}. Each record was labeled as 0 or 1, corresponding to low or high income. The proportions of the classes are approximately (.76,.24). As a pre-processing step, we excluded three features corresponding to work-class, occupation, and native country, due to a high proportion of missing values.
\paragraph{Connect-4 data} The observations represent 67,557 board positions in the two-person game ``connect-4''~\citep{uci}. For each position, a list of 42 categorical features are available, and each position is labeled as a draw $l=0$, loss $l=1$, or win $l=2$ for the first player, with the class proportions being approximately $(.10, .25,.65)$.
\paragraph{Nursery data}
This dataset was prepared from a set of 12,960 applications for admission to a nursery school~\citep{uci}. Each application was associated with a list of 8 (categorical) socioeconomic features. Originally, each application was labeled as one of five classes, but in order to achieve reasonable label balance, the last three categories were combined. This led to approximate class proportions $(1/3,1/3,1/3)$.
\paragraph{Online news data} A collection of 39,797 news articles from the website \smash{{\tt{mashable.com}}} were associated with 60 features (continuous and discrete). Each article was labeled based on the number of times it was shared: fewer than 1000 shares $(l=0)$, between 1,000 and 5,000 shares ($l=1$), and greater than 5,000 shares ($l=2$), with approximate class proportions $(.28,.59, .13)$.
\paragraph{Synthetic continuous data} Two classes of data points in $\mathbb{R}^{100}$, each of size 10,000, were obtained by drawing samples from the multivariate normal distributions $N(\boldsymbol \mu_0,\Sigma)$ and $N(\boldsymbol \mu_1,\Sigma)$ with a common covariance matrix $\Sigma$. The first mean vector was chosen to be $\boldsymbol \mu_0 = \boldsymbol 0\in\mathbb{R}^{100}$, and the second mean vector was constructed to be a sparse vector in the following way. Specifically, we sampled 10 numbers $(i_1,\dots,i_{10})$ without replacement from $\{1,\dots,100\}$, and the coordinates of $\boldsymbol \mu_1$ indexed by $(i_1,\dots,i_{10})$ were set to the value $.05$ (with all other coordinates were set to 0). Letting $U\Lambda U^{\top} $ denote the spectral decomposition of $\Sigma$, we selected the matrix of eigenvectors $U$ by sampling from the uniform (Haar) distribution on $100\times 100$ orthogonal matrices. The eigenvalues were chosen as $\Lambda=\text{diag}(\frac{1}{1^2},\frac{1}{2^2},\dots, \frac{1}{100^2})$.
\paragraph{Synthetic discrete data} Two classes of data points in $\mathbb{R}^{100}$, each of size 10,000, were obtained by drawing samples from the discrete distributions Multinomial$(N_0,\boldsymbol p_0)$ and Multinomial$(N_1,\boldsymbol p_1)$, where $N_l$ refers to the number of balls in 100 cells, and the cell probabilities are specified by $\boldsymbol p_l\in\mathbb{R}^{100}$. Specifically, we set $N_0=N_1=100$, and \smash{$\boldsymbol p_0=(\frac{1}{100},\dots,\frac{1}{100})$}. The vector $\boldsymbol p_1$ was obtained by perturbing $\boldsymbol p_0$ and then normalizing it. Namely, letting \smash{$\boldsymbol z\in\mathbb{R}^{100}$} be a vector of i.i.d. $N(0,1)$ variables, we defined the vector \smash{$\boldsymbol p_1= |\boldsymbol p_0+\textstyle\frac{1}{300} \boldsymbol z|/\|\boldsymbol p_0+\textstyle\frac{1}{300} \boldsymbol z\|_1$}, where $|\cdot|$ refers to coordinate-wise absolute value.
\subsection{Numerical results}\label{data:results}
\paragraph{Interpreting the plots}
For each dataset, we plot the ground truth value $3\sigma_t$ as a function of $t=1,\dots,1,\!000$, where the y-axis is expressed in units of \%, so that a value $3\sigma_t=.01$ is marked as 1\%.
Alongside each curve for $3\sigma_t$,
we plot the averages of $3\hat{\sigma}_{t,\textsc{o},\text{extrap}}$ (green) , $3\hat{\sigma}_{t,\textsc{h},\text{extrap}}$ (purple), and $3\hat{\sigma}_{t,\textsc{g},\text{extrap}}$ (orange) over their 1,000 realizations, with error bars indicating the spread between the 10th and 90th percentiles of the estimates. Here, the error bars are only given to illustrate the variance of the estimates, conditionally on $\mathcal{D}$, and they are not proposed as confidence intervals for $\sigma_t$. (Indeed, our main focus is on the fluctuations of $\textsc{Err}_t$, rather than the fluctuations of variance estimates.) Lastly, we plot results for the class-wise parameters $\sigma_{t,l}$ in the same manner, but in order to keep the number of plots manageable, we only display the class $l$ with the highest value of $3\sigma_{t,l}$ at $t=1,\!000$. This is reflected in the plots, since the values of $3\sigma_{t,l}$ for the chosen class $l$ are generally larger than $3\sigma_t$.
\paragraph{Walking through an example (Figure 2)} To explain the plots from the user's perspective, suppose the user trains an initial ensemble of $t_0=200$ classifiers with the `census income' data. (The following considerations will apply in the same way to the other datasets in Figures 3-7.) At this stage, the user may compute either of the estimators $\hat{\sigma}_{200,\textsc{o}}$ or $\hat{\sigma}_{200,\textsc{h}}$. In turn, the user may follow the definitions~\eqref{extrapests} to plot the extrapolated estimators for all $t\geq 200$ at no additional cost. These curves will look like the purple or green curves in the left panel of Figure 2, up to a small amount of variation indicated by the error bars.
If the user wants to select $t$ so that $3\sigma_t$ is at most, say 0.5\%, then the purple or green curves in the left panel of Figure~2 would tell the user that 200 classifiers are already sufficient, and no extra classifiers are needed (which is correct in this particular example). Alternatively, if the user happens to be interested in the class-wise error rate for $l=1$, and if the user wants $3\sigma_{t,1}$ to be at most 0.5\%, then the curve for the \textsc{oob} estimator accurately predicts that approximately 600 total (i.e.~400 extra) classifiers are needed. By contrast, the hold-out method is conservative, and indicates that approximately 1,000 total (i.e.~800 extra) classifiers should be trained. So, in other words, the hold-out estimator would still provide the user with the desired outcome, but at a higher computational cost.
\vspace{0.3cm}
\begin{figure*}[h!]
\centering
{\includegraphics[angle=0,
width=.45\linewidth,height=.45\linewidth]{./Figures/census_income_plot_6apr2018.pdf}} \ \ \
{\includegraphics[angle=0,
width=.47\linewidth,height=.47\linewidth]{./Figures/census_income_plot_class1_6apr2018.pdf}}
%
\caption{Results for census income data.}
\label{fig:abalone}
\end{figure*}
~\\
\vspace{.2cm}
\begin{figure*}[h!]
\centering
{\includegraphics[angle=0,
width=.45\linewidth,height=.45\linewidth]{./Figures/connect4_plot_6apr2018.pdf}} \ \ \
{\includegraphics[angle=0,
width=.47\linewidth,height=.47\linewidth]{./Figures/connect4_plot_class1_6apr2018.pdf}}
\caption{Results for connect-4 data.}
\label{fig:contraceptive}
\end{figure*}
\begin{figure*}[h!]
\centering
\ {\includegraphics[angle=0,
width=.48\linewidth,height=.48\linewidth]{./Figures/online_news_plot_6apr2018.pdf}} \hspace{-.2cm}
{\includegraphics[angle=0,
width=.49\linewidth,height=.49\linewidth]{./Figures/online_news_plot_class0_6apr2018.pdf}}
\vspace{-.3cm}
\caption{Results for online news data.}
\label{fig:contraceptive}
\end{figure*}
~\\
\begin{figure*}[h]
\centering
{\includegraphics[angle=0,
width=.45\linewidth,height=.45\linewidth]{./Figures/nursery_plot_6apr2018.pdf}} \ \ \
{\includegraphics[angle=0,
width=.47\linewidth,height=.47\linewidth]{./Figures/nursery_plot_class1_6apr2018.pdf}}
\caption{Results for nursery data.}
\label{fig:contraceptive}
\end{figure*}
\begin{figure*}[h!]
\centering
{\includegraphics[angle=0,
width=.45\linewidth,height=0.45\linewidth]{./Figures/synthetic_cont_plot_6apr2018.pdf}} \
{\includegraphics[angle=0,
width=.45\linewidth,height=.45\linewidth]{./Figures/synthetic_cont_plot_class1_6apr2018.pdf}}
\caption{Results for synthetic continuous data.}
\label{fig:contraceptive}
\end{figure*}
\begin{figure*}[h!]
\centering
{\includegraphics[angle=0,
width=.47\linewidth]{./Figures/synthetic_discrete_plot_6apr2018.pdf}}
{\includegraphics[angle=0,
width=.47\linewidth]{./Figures/synthetic_discrete_plot_class0_6apr2018.pdf}}
\caption{Results for synthetic discrete data.}
\label{fig:contraceptive}
\end{figure*}
\paragraph{Comments on numerical results} Considering all of the datasets collectively, the plots show that the extrapolated \textsc{oob} and ground estimators are generally quite accurate. Meanwhile, the hold-out estimator tends to be conservative, due to an upward bias. Consequently, the \textsc{oob} method should be viewed as preferable, since it is both more accurate, and does not require data to be held out. Nevertheless, when considering the hold-out estimator, it is worth noting that the effect of the bias actually diminishes with extrapolation, and even if the initial value $3\hat{\sigma}_{t_0,\textsc{h},\text{extrap}}$ has noticeable bias at $t_0=200$, it is possible for the extrapolated value $3\hat{\sigma}_{t,\textsc{h},\text{extrap}}$ to have relatively small bias at $t=1,\!000$.
To understand where the bias of the hold-out estimator comes from, imagine that two ensembles have been trained on the same data, and suppose their accuracy is compared on a small hold-out set. In this situation, it is possible for their observed error rates on the hold-out set to noticeably differ --- even if the true error rates are very close. For this reason, the small size of $\mathcal{D}_{\text{hold}}$ leads to greater variation among the estimated values $\widehat{\textsc{Err}}(\tilde{A}^*)$ generated in the hold-out version of Algorithm 2, which leads to an inflated estimate of $\sigma_t$. By contrast, the ground estimator suffers less from this bias because it relies on the much larger ground truth set $\mathcal{D}_{\text{ground}}$ in place of $\mathcal{D}_{\text{hold}}$. Similarly, the \textsc{oob} estimate of $\sigma_t$ is less susceptible to this bias, because it will typically use every point in the larger training set $\mathcal{D}$ as an ``effective test point'', preventing excess variation among the values $\widehat{\textsc{Err}}(A^*)$. (Even for small choices of $t_0$, all training points are likely to be an \textsc{oob} point for at least one classifier.)
One last point to mention is that many of the datasets have discrete features, which may violate the theoretical conditions in Assumption~\ref{ASSUMPTION}. Nevertheless, the presence or absence of discrete features does not seem to substantially affect on the performance of the estimators. So, to this extent, the bootstrap does not seem to depend too heavily on Assumption~\ref{ASSUMPTION}. (See Appendix~\ref{app:empirical} for further empirical assessment of that assumption.)
\section{Conclusion}\label{sec:conc}
We have studied the notion of algorithmic variance $\sigma_t^2=\operatorname{var}(\textsc{Err}_t|\mathcal{D})$ as a criterion for deciding when a randomized ensemble will perform nearly as well as an infinite one (trained on the same data). To estimate this parameter, we have developed a new bootstrap method, which allows the user to directly measure the convergence of randomized ensembles with a guarantee that has not previously been available.
With regard to practical considerations, we have shown that our bootstrap method can be enhanced in two ways. First, the use of a hold-out set can be avoided with the \textsc{oob} version of Algorithm 2, and our numerical results show that the \textsc{oob} version is preferable when hold-out points are scarce. Second, the extrapolation technique substantially reduces the cost of bootstrapping. Furthermore,
we have analyzed the cost of the method in terms of floating point operations to show that it compares favorably with the cost of training a single ensemble via random forests.
From a theoretical standpoint, we have analyzed the proposed method within the framework of a first-order model for randomized ensembles. In particular, for a generic ensemble whose classifiers are conditionally i.i.d.~given $\mathcal{D}$, there is a corresponding first-order model that matches the generic ensemble with respect to its average error rate $\mathbb{E}[\textsc{Err}_t|\mathcal{D}]$. Under this setup, our main result shows that the proposed method consistently approximates $\mathcal{L}(\sqrt{t}(\textsc{Err}_t-\text{err}_{\infty})|\mathcal{D})$ as $t\to\infty$. Some extensions of this result could include generalizations to other models of randomized ensembles (e.g., along the lines of~\citet{biau2008,biau2012,scornetconsistency,scornetIEEE}), as well as corresponding results in the context of regression ensembles, which we hope to pursue in future work. More generally, due to the versatility of bootstrap methods, our approach may also be relevant to measuring the convergence of other randomized algorithms, as in~\cite{lopes_matrix,lopes_LS}.
\section*{Acknowledgements} The author thanks Peter Bickel, Philip Kegelmeyer, and Debashis Paul for helpful discussions. In addition, the author thanks the editors and referees for their valuable feedback, which significantly improved the paper.
\bibliographystyle{imsart-nameyear}
|
1,941,325,220,542 | arxiv | \section{Introduction}
Reverse engineering is important in cyber security for purposes such as analyzing malicious code and finding vulnerabilities in software. The process of reverse engineering involves first gaining an overview of the functions in a given binary, which often requires context of what they do and prior experience \cite{observationsRE}. Although there are many mature tools in a reverse engineer's tool belt to disassemble, decompile and analyze binaries\cite{ghidra,kvroustek2017retdec,capa}, there is a gap in open source tooling that performs one-to-many comparisons in binary code against a known code base to quickly gain awareness of what each function might do.
\subsection{The Reverse Engineering Process}
The process of (software) reverse engineering is performed to determine the original purpose of a piece of software, usually in binary form without source code available. There are various tools and techniques used by reverse engineers, such as disassemblers, decompilers and debuggers to both statically and dynamically inspect the software. An observational study of different processes reverse engineers take\cite{observationsRE} revealed that they approach unfamiliar programs from a non-linear, fact-finding perspective. They make hypotheses about program functionality using existing ``beacons" (clues and patterns), then focus on proving or disproving them.
During the aforementioned study, when asked to articulate the process they take when approaching a new program to reverse, all of the participants (N=16) responded that they start by seeking a high-level program view, and that the results from this step guides which part of the program to prioritize for more complex investigation. Having established initial questions and hypotheses based on the overview step, they then perform sub-component scanning to only review small parts of the program (e.g. functions) at a time to produce more refined hypotheses and questions, before finally moving onto in-depth static and dynamic analysis to prove their theories and answer their questions.
The primary aim of BCD is to add tooling into the first stage of the reverse engineering process: to provide an overview of unknown functions in the binary in context of known functions in the database, to better guide the reverse engineer's efforts in the second phase (sub-component scanning).
\subsection{Binary Code Similarity}
Compiled binary code is ISA (Instruction Set Architecture) specific, with popular architectures being Intel x86/64, ARM, MIPS etc. There has been many approaches in literature for binary code similarity, some able to detect cross-architecture similarities. An in-depth survey of this field was done by Haq and Caballero\cite{haq2021survey}. Being able to detect cross-architecture similarities has a big advantage, given that malware compiled for devices using non-mainstream architectures for Internet of Things and Operational Technology are on the rise.
Approaches taken by prior works to perform cross-architecture binary code similarity can be summarised roughly by the following list (and often used in combination):
\begin{enumerate}
\item Comparing CFGs (Control Flow Graphs) or ASTs (Abstract Syntax Trees) using graph edit distance or subgraph isomorphism. This approach is very popular among works targeting cross-architecture vulnerability detection\cite{feng2016scalableGENIUS,eschweiler2016discovre}.
\item Training machine learning models based on embedding program features such as instructions, Input/Output pairs or CFGs into vectors\cite{xu2017neuralGEMINI}, and/or using a neural network (such as LTSM and Siamese networks)\cite{zuo2019neuralInnereye,xu2017neuralGEMINI};
\item Using Locality Sensitive Hashing approaches to compare syntactic or semantic instruction features \cite{feng2016scalableGENIUS,pewny2015crossMultiMH}
\item Overcome architecture-independence by first lifting the assembly code to an Intermediate Representation \cite{pewny2015crossMultiMH,ding2016kam1n0,chandramohan2016bingo}
\end{enumerate}
There has been an increase in cross-architecture binary similarity approaches focused around identifying the same vulnerabilities across different architectures\cite{feng2016scalableGENIUS,xu2017neuralGEMINI,zuo2019neuralInnereye}, such as firmware images, due to the rise of Internet of Things. However, there hasn't been much work that utilizes binary code similarity specifically for aiding the process of understanding binary code in a reverse engineering context; the closest is Rendevouz \cite{khoo2013rendezvous}, which specifically aimed to build a binary code search engine. Rendevouz utilized DynInst, an open source binary instrumentation library to perform code analysis, and its approach is not cross-architecture. Unfortunately, the implementation was never released to public, which are the two issues this work aims to address.
\subsection{Intermediate Representations}
Intermediate Representations (IR) are used in both the compilation and decompilation process\cite{cifuentes1994reverse}, to serve as a layer between low level architecture dependent assembly code and high-level source code to make control flow analysis and data flow analysis easier. Examples of IRs appear everywhere in compilers, static and dynamic analysis tools, such as LLVM IR, Ghidra's PCODE, Valgrind's VEX etc.
IR is very useful for eliminating architecture dependence for code similarity comparisons, as it can both be compiled from high level source code and lifted from low level assembly code to provide a common ground for comparison.
\subsection{Locality Sensitive Hashing}
Cryptographic hashing algorithms (such as MD5 and SHA1) are used widely to compare equivalence in two pieces of data. However, they do not capture similarities in data. There are two common ``fuzzy" hashing approaches that detect similarities in data: Locality Sensitive Hashing (LSH) and Context-Triggered Piecewise Hashing (CTPH).
Locality Sensitive Hashing hashes data in same sized pieces; as in, a piece of data (such as \texttt{Hello World}) is divided into a number of tokens (\texttt{Hel, llo, loW, Wor..}) and then each token is hashed, producing a set of hashes. Then different techniques are used to reduce the dimensions of the set to a smaller set or a single number, by using pseudo-random selection (e.g. MinHash\cite{broder1997resemblanceMinHash} randomly chooses the $p$ numbers in the set using a consistent seed).
CTPH is similar, but instead of hashing individual tokens over a rolling window, it is a rolling hash algorithm that produces a pseudo-random output based only on the current context of an input \cite{2006ssdeep}.
Fuzzy hashing approaches have been utilized before in detecting binary similarity, such as in Multi-MH\cite{pewny2015crossMultiMH}, which used MinHash to perform similarity searching across function input/output pairs.
\section{Approach}
\label{approach}
The general approach of BCD to construct a cross-architecture binary code similarity database is as follows:
\begin{enumerate}
\item Take the input binary, preferably with symbols (not stripped)
\item Lift it to LLVM IR using RetDec\cite{kvroustek2017retdec}, an open source decompiler
\item Extract individual functions out of the LLVM IR code, and convert it into a set of instruction tokens through a series of normalization transforms
\item Run the transformed IR tokens through a fuzzy hashing algorithm
\item Store the hashes and functions in a database, and compare functions against each other using the hashes
\item For faster indexed lookups, index functions using hashes as the index into a hash table and perform hash lookups to find similar functions
\end{enumerate}
A comparison of hashing functions are performed at the end of step 5 using function names as ground truth (hence why binaries with symbols are preferred). The dataset used to compare the hashing functions are samples of Mirai botnet across 9 different architectures (arm4vl, arm5vl, arm6vl, i586, i686, x86, mips, mipsel, powerpc), totaling to 893116 compared function pairs across all architectures, with 3473 having the same name.
Four different fuzzy hashing algorithms are compared: MinHash\cite{broder1997resemblanceMinHash}, SimHash\cite{charikar2002similaritySimHash}, SSDEEP (CTPH)\cite{2006ssdeep} and TLSH\cite{oliver2013tlsh}.
\subsection{IR Normalization}
\label{irnormal}
Normalization of the output LLVM IR instructions needed to be done to aid cross-architecture similarity searching, because the LLVM IR lifting process can introduce ISA-specific differences such as different integer word sizes across 16,32 and 64 bit architectures, offset order based stack variable and global variable names (since RetDec uses Single Static Assignment\cite{van2007staticSSA} when lifting assembly to LLVM IR, assigning new variable names on each use).
Different ISAs may have different number of stack variables due to different number of registers and stack sizes being used, leading to change in the IR code when lifted from the assembly code.
The goal of the IR Normalization phase is to capture only the important semantic information that maximizes chances that a function would be matched across a different ISA. The transformations applied to the lifted LLVM IR code are:
\begin{itemize}
\item Single Static Assignment references and register names (e.g. \texttt{\%2}, \texttt{\%eax}) are replaced with a generic reference token \texttt{\%r}
\item Generated variable names with offset numbers are replaced with only the variable type (e.g. \texttt{\%stack\_var\_-4} replaced with \texttt{\%stack\_var})
\item Integer types and sizes dropped, since data type recovery is imperfect and word sizes differ between ISAs (e.g. \texttt{i16}, \texttt{i32} are dropped)
\item Instruction addresses (for debugging), comments and other extraneous artifacts generated by the tool are dropped
\end{itemize}
For the sake of demonstration, this snippet of \href{https://gist.github.com/rverton/a44fc8ca67ab9ec32089#file-rc4-c-L54}{RC4 encryption function}\cite{rc4gist} (which calls two other stages of RC4) from Github is compiled into three different architectures (ARM, MIPS and x86) with GCC then lifted by RetDec and fed into the normalization process.
The C code of the function is:
\begin{lstlisting}[language=C]
int RC4(char *key, char *plaintext, unsigned char *ciphertext) {
unsigned char S[N];
KSA(key, S);
PRGA(S, plaintext, ciphertext);
return 0;
}
\end{lstlisting}
The resulting LLVM IR function snippets in respective ISAs are shown below.
\newpage
\textbf{ARM}
\begin{lstlisting}[language=llvm]
define i32 @RC4(i32
dec_label_pc_107e8:
!insn.addr !59
i32
ret i32 0, !insn.addr !61
}
\end{lstlisting}
\textbf{MIPS}
\begin{lstlisting}[language=llvm]
define i32 @RC4(i32
dec_label_pc_400b44:
, !insn.addr !59
i32
ret i32 0, !insn.addr !61
}
\end{lstlisting}
\textbf{x86}
\begin{lstlisting}[language=llvm]
define i32 @RC4(i32
dec_label_pc_1425:
!insn.addr !98
i32
store i32 0, i32*
br i1
!insn.addr !101
dec_label_pc_14a7: ; preds =
store i32
br label
dec_label_pc_14ac: ; preds =
ret i32
}
\end{lstlisting}
It is obvious from above that code inserted by the compiler for individual ISA such as stack canary checking for x86 (\texttt{\_\_stack\_chk\_fail\_local}) are also captured in the lifting process, which made the lifted LLVM IR from x86 a bit more different than ARM and MIPS.
Then IR tokens after the transformations from each ISA are shown below:
\textbf{ARM}
\begin{lstlisting}[language=LLVM]
ret 0,
\end{lstlisting}
\textbf{MIPS}
\begin{lstlisting}[language=LLVM]
ret 0,
\end{lstlisting}
\textbf{x86}
\begin{lstlisting}[language=LLVM]
store 0,
br i1
store
br label
ret
\end{lstlisting}
As shown, after the transformation process, the LLVM IR tokens from ARM and MIPS are exactly the same. This is due to their original similarity in the first place, since they are both RISC (Reduced Instruction Set Computer) architectures. However the difference from x86 is quite large, as x86 is better utilized by the compiler (GCC) with stack protection mechanisms and is a CISC (Complex Instruction Set Computer) architecture. The extra compiler artefacts could be removed to improve cross-architecture similarity between x86 and the other RISC architectures, but is not done here to preserve inserted code following the principle of least surprise.
\subsection{Comparing Hashing Algorithms}
There are fundamental differences between how the four hashing algorithms are used to compare similarity of input data, therefore different ways of applying them to compare similarity of the transformed IR tokens.
MinHash\cite{broder1997resemblanceMinHash} turns data into $p$ hashes by turning data into shingles, and pseudo-randomly choosing $p$ shingles to hash by splitting the data using a fixed sized rolling window, where $p$ is the number of permutations (with a consistent seed so that the same input always produces the same selection). The distance between two inputs are calculated using the Jaccard distance, as in, the size of intersection of the two sets of $p$ hashes. Two inputs can only be compared for similarity if the MinHash sets were generated with the same $p$. For example, if $p = 64$, $64$ numbers are generated for each input. If two inputs have $18$ numbers in common with their MinHash sets, then the similarity is $18 / 64 = 0.28125$. This value is between 0 and 1, and an arbitrary threshold can be set to decide if two inputs are similar or not.
SimHash\cite{charikar2002similaritySimHash} uses rounding algorithms to output a sketch for estimating the cosine similarity measure between inputs. Given two inputs, SimHash produces a $t$-bit vector (e.g. 64 bit number), where the cosine distance is 0 to $t$, from 0 being the two inputs are nearly the same to $t$ being extremely different. For this experiment, 64 bit SimHash values are used, which means distance is between 0 to 64 and the smaller the distance, the more similar two inputs are.
SSDEEP/CTPH\cite{2006ssdeep} is widely used in the industry for matching similar malware. Adopted from the spamsum\cite{tridgell2002spamsum} algorithm, has two components: the rolling hash and the traditional hash (FNV). The rolling hash updates on each byte of the input, and at certain trigger points generates a traditional hash of the block of data, and appends the base64 encoded six Least Significant Bytes of the traditional hash to the digest. The trigger points are hit at each blocksize, which is calculated initially based on a minimum blocksize and the length of the input. SSDEEP processes the input twice; once using the length of the input to generate the initial blocksize, and once using half of it, generating two digests (one for ``chunk" and one for ``doublechunk"). Two digests are then compared using string edit distance, producing a re-scaled score of 0-100. Two inputs are considered similar if the matching score is bigger than 0. Note that two inputs can only be compared using SSDEEP if their blocksize is the same. This means inputs with large size differences will not match.
TLSH\cite{oliver2013tlsh} aims to trump the industry standard of SSDEEP by addressing some of its limitations: 1) the fact that it is very easy to ``trick" SSDEEP into generating completely different digests given targeted byte swaps in the input (while remaining largely similar) 2) it has a much lower detection ratio due to the use of a scaled similarity ratio that shows a match when score is bigger than 0. TLSH processes the input data using a sliding window of size 5 to populate buckets of triplets (e.g. for 5 bytes A,B,C,D,E, form buckets ABC, ABD, ABE, ACD ..), then select 6 out of 10 possible triplets to apply the Pearson hash\cite{pearson1990fast} on. The distance between two TLSH hashes are calculated using modular difference between the quartile ratio of similar items in buckets. The algorithm is not fully described here; the sake of brevity please refer to the TLSH paper for more details. The notable difference is that TLSH, much like SimHash outputs a distance metric instead of a similarity score, and the distance score is unbounded. A distance of 0 means that they two inputs are very similar, and anything more than that means they are more different (at which point a variable threshold can be applied, much like SimHash). The paper reports that false positives significantly increase past the distance of 100. The limitations of TLSH is that only data more than 50 bytes are eligible for hashing.
To compare the effectiveness of the four hashing algorithms, which all can have variable thresholds, ROC (Receiver Operating Characteristic) curves are used with a selected range of thresholds specific to each hashing algorithm (for unbounded distance scores such as TLSH, up to twice the range of the advised threshold is used, which is $100 * 2$).
\begin{table}[t]
\caption{\textbf{Comparison of hashing algorithms with score/distance thresholds (n=893116,Positives=3473)}}
\label{hashtable}
\hspace{-2cm}
\resizebox{18cm}{!}{
\begin{tabular}{@{}lll|lll|lll|lll@{}}
\toprule
MinHash & TPR & FPR & SimHash & TPR & FPR & SSDEEP & TPR & FPR & TLSH & TPR & FPR \\ \midrule
5 & 0.998 & 0.919 & 39 & 0.992 & 0.984 & 5 & 0.625 & 0.05 & 200 & 0.944 & 0.671 \\
10 & 0.993 & 0.826 & 37 & 0.971 & 0.946 & 10 & 0.625 & 0.05 & 190 & 0.926 & 0.609 \\
15 & 0.976 & 0.728 & 35 & 0.926 & 0.866 & 15 & 0.62 & 0.05 & 180 & 0.908 & 0.545 \\
20 & 0.962 & 0.624 & 33 & 0.872 & 0.737 & 20 & 0.605 & 0.049 & 170 & 0.879 & 0.478 \\
25 & 0.95 & 0.519 & 31 & 0.808 & 0.574 & 25 & 0.594 & 0.047 & 160 & 0.843 & 0.412 \\
30 & 0.926 & 0.379 & 29 & 0.725 & 0.403 & 30 & 0.584 & 0.043 & 150 & 0.804 & 0.347 \\
35 & 0.907 & 0.278 & 27 & 0.65 & 0.272 & 35 & 0.574 & 0.037 & 140 & 0.768 & 0.286 \\
40 & 0.886 & 0.188 & 25 & 0.615 & 0.195 & 40 & 0.56 & 0.031 & 130 & 0.72 & 0.231 \\
45 & 0.854 & 0.116 & 23 & 0.593 & 0.158 & 45 & 0.543 & 0.026 & 120 & 0.674 & 0.18 \\
50 & 0.791 & 0.066 & 21 & 0.579 & 0.14 & 50 & 0.508 & 0.02 & 110 & 0.621 & 0.137 \\
55 & 0.681 & 0.028 & 19 & 0.565 & 0.124 & 55 & 0.468 & 0.017 & 100 & 0.572 & 0.101 \\
60 & 0.604 & 0.017 & 17 & 0.553 & 0.102 & 60 & 0.434 & 0.014 & 90 & 0.523 & 0.071 \\
65 & 0.533 & 0.011 & 15 & 0.533 & 0.075 & 65 & 0.371 & 0.011 & 80 & 0.479 & 0.047 \\
70 & 0.463 & 0.008 & 13 & 0.516 & 0.049 & 70 & 0.319 & 0.008 & 70 & 0.423 & 0.029 \\
75 & 0.396 & 0.007 & 11 & 0.488 & 0.032 & 75 & 0.266 & 0.005 & 60 & 0.36 & 0.016 \\
80 & 0.321 & 0.004 & 9 & 0.462 & 0.023 & 80 & 0.236 & 0.003 & 50 & 0.322 & 0.008 \\
85 & 0.271 & 0.003 & 7 & 0.43 & 0.019 & 85 & 0.204 & 0.002 & 40 & 0.272 & 0.003 \\
90 & 0.23 & 0.002 & 5 & 0.356 & 0.013 & 90 & 0.186 & 0.002 & 30 & 0.224 & 0.001 \\
95 & 0.2 & 0.001 & 3 & 0.238 & 0.005 & 95 & 0.165 & 0.001 & 20 & 0.191 & 0.001 \\
100 & 0.169 & 0.001 & 1 & 0.174 & 0.001 & 100 & 0 & 0 & 10 & 0.161 & 0 \\ \bottomrule
\end{tabular}}
\end{table}
Table \ref{hashtable} shows the True Positive Rate and False Positives Rate of the different algorithms. Note that for SimHash and TLSH, the thresholds are distance based (smaller means more similar), while for MinHash and SSDEEP they are normalized scores from 0-100 percent (bigger means more similar).
Figure \ref{roc} plots the TPR (True Positive Rate) against the FPR (False Positive Rate) under varying thresholds using a ROC (Receiver Operating Characteristic) curve, which describes how well each algorithm performs under different thresholds (the bigger the Area Under the Curve, the better). From the graph it's evident that MinHash performs that best in this scenario, with TLSH in second and SSDEEP doing the worst (although SSDEEP had a very low False Positive Rate generally, its match rate was too low).
\begin{center}
\begin{figure}[ht]
\caption{ROC Curve}
\hspace{+2.5cm}
\includegraphics[width=8cm]{roc.png}
\label{roc}
\end{figure}
\end{center}
\subsection{Indexing and Searching}
As MinHash\cite{broder1997resemblanceMinHash} was the algorithm with the best ROC Curve and AUC (Area Under the Curve), it is chosen to implement a faster, more user friendly binary code search database. MinHash can be indexed to produce a hash table and enable O(1) lookups of each hash, since it produces a set of $p$ hashes from an input. After hashing the normalized LLVM IR tokens from a function and running them through MinHash, a set of $p$ integer hashes are produced. Each integer can be used as an index to store the function in a hash table, where the key is the integer and the value is a list of functions which contain the integer in its MinHash output hash.
So, given the MinHash output from a function's LLVM IR tokens, lookup of a function can be performed in best case $O(1)$ speed as $p$ is a constant hyperparameter defined for MinHash. For a given binary with $n$ functions, lookup of similar functions with any matching hash in the entire database can be performed in $O(n)$, and if a threshold has to be applied to the number of hashes matched to determine Jaccard distance, we can store the count of each function matched in a hash table using the function name as the index, and perform at most $F$ hash table lookups to determine if the count of hashes matched is over a certain threshold for $F$ being the total number of functions in the database, making the worst case for getting similar functions in the database for a given binary $O(n * F * p)$.
The algorithm for indexing and lookup are described in Python pseudo-code below.
\begin{lstlisting}[language=python]
def index(func_llvmir):
func_tokens = tokenize(func_llvmir) #the IR normalization process
func_minhash = minhash(func_tokens) # produce minhash set
for h in func_minhash:
if db.get(h) == None:
db[h] = []
db[h].append(file_func)
def lookup(func_llvmir):
func_tokens = tokenize(func_llvmir)
func_minhash = minhash(func_tokens)
matches = {}
for h in func_minhash:
if db.get(h) != None:
matched = db.get(h)
for func in matched:
matches[func] += 1
\end{lstlisting}
The hyperparameter $p$, number of hashes in the output produced by MinHash can be tuned for a balance of accuracy and speed trade-offs (doubling $p$ means doubling the amount of hash lookups).
To find the best threshold, a small range of $p$ are used to test accuracy (in this case, due to the severe class imbalance in the dataset of having a lot more functions with different than same names, Balanced Accuracy (BA) is used to measure success: $\frac{(TPR + TNR)}{2}$
The $TPR$ (True Positive Rate) and $TNR$ (True Negative Rate) are calculated using a threshold of $0.5$. As shown in Table \ref{permu}, $p = 256$ is the optimal value.
\begin{table}[h]
\caption{Permutations}
\label{permu}
\begin{center}
\begin{tabular}{@{}llll@{}}
\toprule
\textbf{$p$} & \textbf{TPR} & \textbf{TNR} & \textbf{Balanced Accuracy} \\ \midrule
64 & 0.791 & 0.934 & 0.863 \\
128 & 0.817 & 0.929 & 0.873 \\
\textbf{256} & 0.822 & 0.928 & \textbf{0.875} \\
512 & 0.806 & 0.944 & 0.875 \\ \bottomrule
\end{tabular}
\end{center}
\end{table}
\subsection{Cross-Architecture Containment Detection}
From the experiments above it is clear that the LLVM IR + MinHash approach performs well to detect similar functions across architectures. However, can it detect code in smaller functions being embedded bigger ones? (Detecting binary code containment instead of just similarity/resemblance.) This is important and useful in aiding the reverse engineering process for several reasons:
\begin{enumerate}
\item Code from a function could be copied and adapted, in part or in whole, into another bigger function
\item Malware authors often insert dead code or no-operation instructions into start / end / middle of functions to avoid detection
\item To patch a vulnerability, developers often add or remove code from an existing function for additional security, sometimes changing its size significantly.
\end{enumerate}
In theory, this should be detectable via MinHash, since the algorithm selects pseudo-random fixed sized pieces (shingles) from the input to hash and include in the resulting set \cite{broder1997resemblanceMinHash} in order to detect both resemblance and containment. The limitation to MinHash is that since random pieces are selected, there will always be data loss if the number of shingles are bigger than the hyperparameter $p$. This also means the bigger $p$ is, the better containment detection should be, and that smaller functions should have higher accuracy than larger ones.
To validate if this approach works for both cross-architecture similarity and containment, an experiment is done by modifying the RC4 code used in section \ref{irnormal} \cite{rc4gist} and combining the PRGA, KSA and RC4 wrapper function into one. The resulting combined function is listed below and compiled into Intel x86, ARM and MIPS architecture for comparison.
\footnote{See the original code for the separate KSA and PRGA functions \cite{rc4gist}.}
\begin{lstlisting}[language=C]
int RC4(char *key, char *plaintext, unsigned char *ciphertext) {
unsigned char S[N];
int len = strlen(key);
int j = 0;
for(int i = 0; i < N; i++)
S[i] = i;
for(int i = 0; i < N; i++) {
j = (j + S[i] + key[i
swap(&S[i], &S[j]);
}
int i = 0;
j = 0;
for(size_t n = 0, len = strlen(plaintext); n < len; n++) {
i = (i + 1)
j = (j + S[i])
swap(&S[i], &S[j]);
int rnd = S[(S[i] + S[j])
ciphertext[n] = rnd ^ plaintext[n];
}
// original wrapper calls commented out
// KSA(key, S);
// PRGA(S, plaintext, ciphertext);
return 0;
}
\end{lstlisting}
To compare both cross-architecture resemblance and containment, the same process described in section \ref{approach} is applied to the original PRGA and KSA function, as well as the new combined RC4 function, and the MinHash outputs are compared across the different architectures they have been compiled in to produce a Jaccard similarity score between the contained and combined function in each architecture. To demonstrate the effects of changing the hyperparam $p$, the comparison has been done using multiple $p$ values.
Function names could not be used as ground truth in x86 as the RetDec lifting process had a critical flaw that lead to the correct function body being split up from the correct name to one unnamed function immediately below it (in memory address). This threat to validity is discussed further in section \ref{threatsvalid}. Since this is a small dataset, the correct x86 functions were manually identified and compared to ensure results are sound.
\begin{table}[]
\caption{Cross-Architecture Containment and Resemblance, ($p$=64)}
\label{contain64}
\begin{tabular}{l|rrr}
\multicolumn{1}{c|}{} & \multicolumn{1}{c}{\textbf{Combined ARM}} & \multicolumn{1}{c}{\textbf{Combined MIPS}} & \multicolumn{1}{c}{\textbf{Combined x86}} \\ \hline
KSA ARM & \textbf{0.65625} & 0.53125 & 0.421875 \\
KSA MIPS & 0.53125 & \textbf{0.734375} & 0.484375 \\
KSA x86 & 0.25 & 0.328125 & \textbf{0.796875} \\
PRGA ARM & \textbf{0.625} & 0.5 & 0.390625 \\
PRGA MIPS & 0.546875 & \textbf{0.78125} & 0.40625 \\
PRGA x86 & 0.46875 & 0.515625 & \textbf{0.578125}
\end{tabular}
\end{table}
\begin{table}[]
\caption{Cross-Architecture Containment and Resemblance, ($p$=128)}
\label{contain128}
\begin{tabular}{l|rrr}
\textbf{} & \multicolumn{1}{c}{\textbf{Combined ARM}} & \multicolumn{1}{c}{\textbf{Combined MIPS}} & \multicolumn{1}{c}{\textbf{Combined x86}} \\ \hline
KSA ARM & \textbf{0.703125} & 0.59375 & 0.4609375 \\
KSA MIPS & 0.59375 & \textbf{0.7578125} & 0.5 \\
KSA x86 & 0.3125 & 0.3359375 & \textbf{0.7890625} \\
PRGA ARM & \textbf{0.6875} & 0.5703125 & 0.4296875 \\
PRGA MIPS & 0.5703125 & \textbf{0.8359375} & 0.3984375 \\
PRGA x86 & 0.5234375 & 0.53125 & \textbf{0.59375}
\end{tabular}
\end{table}
\begin{table}[]
\caption{Cross-Architecture Containment and Resemblance, ($p$=256)}
\begin{tabular}{l|rrr}
\textbf{} & \multicolumn{1}{c}{\textbf{Combined ARM}} & \multicolumn{1}{c}{\textbf{Combined MIPS}} & \multicolumn{1}{c}{\textbf{Combined x86}} \\ \hline
KSA ARM & \textbf{0.69921875} & 0.57421875 & 0.47265625 \\
KSA MIPS & 0.58984375 & \textbf{0.765625} & 0.48828125 \\
KSA x86 & 0.3359375 & 0.34375 & \textbf{0.77734375} \\
PRGA ARM & \textbf{0.7109375} & 0.60546875 & 0.41796875 \\
PRGA MIPS & 0.59375 & \textbf{0.84375} & 0.3984375 \\
PRGA x86 & 0.5546875 & 0.5703125 & \textbf{0.5703125}
\end{tabular}
\end{table}
\begin{table}[]
\caption{Cross-Architecture Containment and Resemblance, ($p$=512)}
\begin{tabular}{l|rrr}
\textbf{} & \multicolumn{1}{c}{\textbf{Combined ARM}} & \multicolumn{1}{c}{\textbf{Combined MIPS}} & \multicolumn{1}{c}{\textbf{Combined x86}} \\ \hline
KSA ARM & \textbf{0.716796875} & 0.599609375 & 0.4453125 \\
KSA MIPS & 0.59765625 & \textbf{0.78515625} & 0.453125 \\
KSA x86 & 0.3046875 & 0.326171875 & \textbf{0.771484375} \\
PRGA ARM & \textbf{0.748046875} & 0.634765625 & 0.40625 \\
PRGA MIPS & 0.6171875 & \textbf{0.85546875} & 0.388671875 \\
PRGA x86 & 0.533203125 & 0.55859375 & \textbf{0.5625}
\end{tabular}
\end{table}
\begin{table}[]
\caption{Cross-Architecture Containment and Resemblance, ($p$=1024)}
\label{contain1024}
\begin{tabular}{l|rrr}
\textbf{} & \multicolumn{1}{c}{\textbf{Combined ARM}} & \multicolumn{1}{c}{\textbf{Combined MIPS}} & \multicolumn{1}{c}{\textbf{Combined x86}} \\ \hline
KSA ARM & \textbf{0.712890625} & 0.576171875 & 0.435546875 \\
KSA MIPS & 0.5849609375 & \textbf{0.7919921875} & 0.4248046875 \\
KSA x86 & 0.314453125 & 0.322265625 & \textbf{0.7734375} \\
PRGA ARM & \textbf{0.751953125} & 0.619140625 & 0.388671875 \\
PRGA MIPS & 0.611328125 & \textbf{0.8603515625} & 0.3740234375 \\
PRGA x86 & 0.529296875 & 0.541015625 & \textbf{0.556640625}
\end{tabular}
\end{table}
The same comparisons were ran across $p$ values 64,128,256,512 and 1024, shown from tables \ref{contain64} to \ref{contain1024}. The containment comparison scores across the same architecture are in bold, and those values show that same-architecture containment detection works. Cross-architecture containment detection was successful between MIPS and ARM (50 to 60 percent match), which are both RISC (Reduced Instruction Set Computer) architectures; but was only moderately successful (above 40 percent) when those two architectures are compared against x86, a CISC (Complex Instruction Set Computer) architecture, evidently shown by the average scores between the three architectures across increasing $p$ values on table \ref{avgcontain}.
The \href{https://gist.github.com/rverton/a44fc8ca67ab9ec32089#file-rc4-c-L18}{KSA function} (line 18 in \cite{rc4gist}) is significantly smaller in size when compiled to ARM and MIPS versus in x86. This is because the Key Scheduling Algorithm is mainly an I/O bound function with many memory movement instructions in loops, which produces more instructions in a RISC architecture that does not have looped memory movement mnemonics unlike in x86. Since the x86 compiled assembly is smaller, and the instructions are different, the cross-architecture detection rates are very low for the KSA function as seen on tables \ref{contain64} to \ref{contain1024}. On the other hand, the \href{https://gist.github.com/rverton/a44fc8ca67ab9ec32089#file-rc4-c-L35}{PRGA function} mostly consists of mathematical operations (like modulus and XOR), which produces more similar tokens when lifted to LLVM IR, and thus have a higher similarity between RISC and CISC architectures. These findings show that BCD's approach captures underlying design differences between instruction set architectures (ISAs), which can affect cross-architecture comparison results between classes of ISAs.
It is also worth noting that increasing the $p$ value does not mean the detection rate will always increase; since MinHash selects $p$ random shingles to include, having more shingles provide more accurate similarity scores, but not always consistently due to the random selection (on table \ref{avgcontain}, the average score increased significantly after $p > 64$, but then fluctuates.
\begin{table}[bt]
\caption{Average containment detection scores}
\label{avgcontain}
\begin{center}
\begin{tabular}{r|rrr}
\multicolumn{1}{l|}{\textbf{p}} & \multicolumn{1}{l}{\textbf{arm vs mips}} & \multicolumn{1}{l}{\textbf{arm vs x86}} & \multicolumn{1}{l}{\textbf{mips vs x86}} \\ \hline
64 & 0.52734375 & 0.3828125 & 0.43359375 \\
128 & 0.58203125 & 0.431640625 & 0.44140625 \\
256 & 0.5908203125 & 0.4453125 & 0.4501953125 \\
512 & 0.6123046875 & 0.4223632813 & 0.431640625 \\
1024 & 0.5979003906 & 0.4169921875 & 0.4155273438
\end{tabular}
\end{center}
\end{table}
\subsection{Usability}
As the goal of BCD is to aid reverse engineers in their initial stages of the RE process, the availability and the usability of code is key. To the loss of the reverse engineering field, many prior works in Binary Code Similarity did not open source their code (only 16 out of 70 evaluated approaches were open sourced according to the 2021 survey\cite{haq2021survey}).
To make BCD as usable as possible, two different interfaces to use BCD are made available: a command line program that takes arguments (as many reverse engineers like using the terminal, and command line programs can be integrated into automation tools such as plugins and CI/CD pipelines), as well as a web interface so that BCD can be ran over the network as a shared service for analysts and less advanced users. The command line program is \texttt{bcd.py}, and the Flask web application is \texttt{server.py}, and they are both open sourced on GitHub \href{https://github.com/h4sh5/bcddb}{here}.
To index functions in a binary using the command line program, simply use the index mode and point the program either at the path of the binary or the lifted LLVM IR file (if the binary is passed, it will automatically be lifted with RetDec\cite{retdecGH}:
\texttt{./bcd.py -i /path/to/binary}
To search all functions in a target binary, simply do the same but using \texttt{-s} (by default, the program uses search mode):
\texttt{./bcd.py -s /path/to/binary}
The web interface allows users to upload a binary and perform index or searching. For each function in the target binary, the functions that matched it are ranked based on the similarity score.
\subsection{Usage Recommendations}
BCD performs many-to-many function comparisons under the hood; as in, for each function in an input binary, it will compare the functions against all known functions in the database using MinHash. This means that the more functions are in a database, the longer comparisons will take. Therefore we recommend building different databases containing different types of functions to compare against, where each function has its original symbol name or is labelled. For example:
\begin{itemize}
\item A database for known Linux library calls (index libc with symbols)
\item A database for known cryptographic functions commonly used in malware
\item A database with known malware samples
\item A database with known functions vulnerable to a certain type of bug
\end{itemize}
Building separate databases for each use-case can not only reduce search time, but also make BCD act as a function tagging / multi-class classifier system.
Another good use case is to compile an open-source project (with different compilers, optimization levels and architectures), then to search unknown functions against it to detect open-source functions in the binary; that way, function matches with high scores can effectively be ``decompiled". It can also be useful for code-clone detection purposes, such as detecting violation against open source licensing.
\section{Evaluation}
\subsection{Efficiency}
Efficiency testing was done on the hash table implementation of BCD (\texttt{bcd.py}) instead of the experiment script (which used a very slow one pair at a time comparison approach to compare functions).
The dataset used is coreutils 8.32\cite{coreutils8.32}, compiled across four different architectures (x64, aarch64, mips and powerpc), totally to 441 binaries and 81668 lifted functions. Out of those 441 binaries, 436 was successfully lifted and the rest had errors.
Time taken to lift each executable to LLVM IR varies as BCD uses on the efficiency of RetDec, but is generally quite fast ($<$ 1 minute), for example, lifting \texttt{/bin/ls} took 10 seconds and \texttt{/bin/sh} 7 seconds in x64 architecture.
Indexing the entire dataset already lifted to LLVM IR and constructing the hash table took 212.68 seconds. That means on average, indexing each binary's LLVM IR code into the database takes 0.48 seconds and indexing each function takes 0.0026 seconds (2.6 milliseconds). The data set produced 1363199 keys (each one a number in the MinHash set) in total (using $p=256$).
Searching functions in the database was tested by taking binaries to search against the database. The database hash table object is saved as a Python pickle file for speed of saving and loading. The database in total was 132MB in size, and took 15 seconds to load into memory. An example binary used was \texttt{/usr/bin/test} on a x64 Linux system, which was 55KB in size, had 227 lifted LLVM IR functions, and took 128 seconds to perform many-to-many comparisons (227 functions in the test binary against 81668 functions in the database). The same test done on \texttt{/usr/bin/gcc}, with 2616 lifted functions, took 1634.37 seconds.
If we divide time by the number of total function pairs compared (which would be $227\times81668=18538636$ and $2616\times81668 = 213643488$ respectively), each function pair comparison only takes 0.006-0.007 milliseconds. This shows that BCD is scalable when it comes to one-to-many binary comparisons and many-to-many binary function searches.
\subsection{Accuracy}
Accuracy testing was done using the same dataset, by extracting 10000 random functions that had symbols (not unnamed functions, which would be labeled \texttt{@function\_xxxx} by RetDec. Each function selected would be compared against the same number of same name functions and the different name functions to ensure classes are balanced. At threshold $t=0.5$ and permutations $p=256$, BCD achieved 0.808 accuracy, 0.910 precision and 0.685 recall ($F_1=0.782$, $F_2=0.721$).
\section{Threats to Validity}
\label{threatsvalid}
One of the advantages of doing binary code similarity over source code similarity is that ``What You See Is Not What You eXecute" (WYSINWYX)\cite{wysinwyxWhatYouSee} - the compilation process is complicated and may add many additional code into the output binary. By lifting the compiled code into an IR that captures extra compiler outputs, these compilation ``side-effects" can be detected. However, due to time constraints, BCD has not been evaluated finding similar code against cross-optimization levels or cross-compiler differences.
The tool used to lift binary code to LLVM IR, RetDec, has flaws that can affect the results of BCD significantly. For example, the GCC compiler inserts \texttt{endbr} or \texttt{endbr64} instructions in Intel 32/64 bit architecture as a control flow integrity feature; but at times a bug in RetDec causes this to cut a decompiled function short, causing the function body in the decompiled code to be incomplete. Given that BCD uses function names as ground truth for accuracy measures, this can skew the results. However, as all prior work in this field, BCD relies on external tools for disassembling and lifting binary code, therefore propagating any mistakes the tools make into the results. It is worth noting that the general framework of BCD, since it uses string manipulation and fuzzy hashing of lifted IR code, can work with any other intermediate representation, source code as well as assembly code to perform comparisons.
\section{Conclusion and Future Work}
In this paper, a framework for detecting binary code similarity across different CPU architecture was designed, implemented and released as usable code. We compared multiple fuzzy hashing algorithms for accuracy to be used in this framework, and built an efficient approach to search similar (including embedded/contained) binary code in a large database of cross-architecture binary functions. As BCD aims to aid reverse engineers, usability considerations in the reverse engineering process are discussed along with use case suggestions.
In the future, cross-optimization and cross-compiler evaluations can be performed using BCD, and improving the framework to increase effectiveness of results especially by further fine tuning the IR Normalization process. Usability and design considerations around specific use cases for the binary code similarity matching should be further researched, as many prior works in this field did not publish their code as usable tooling nor consider the use cases in a reverse engineer centered manner.
\bibliographystyle{ieeetr}
|
1,941,325,220,543 | arxiv | \section{Details of experiments} \label{app:empirical}
\subsection{Datasets}
For $k$-nearest neighbhours, $k$-means and hierarchical clustering,
we empirically evaluate the performance of MAB-based algorithms on two real
world high-dimensional datasets: tiny-imagenet \citep{imagenet}, and $10x$
single cell RNA-Seq dataset \citep{10xdata}.
Tiny-imagenet consists of $100$k images of size $64\times64$ with
$3$ channels thus living in
$12288$ dimensional space.
We downsize these images to evaluate our method on smaller dimensions.
We chose imagenet because it is one of the most popular benchmark
dataset in the field of computer vision -- where many applications use nearest
neighbhours, clustering in their pipelines.
To further exhibit
the robustness of our approach across applications,
our second dataset is a single
cell RNA-Seq dataset of \citet{10xdata} from the field of computational
biology. The scale of this dataset is also large both in sample size and
dimension and additionally it is a very sparse ($7\%$). For our empirical
evaluations we randomly sub-sample $100$k points from this dataset. We
note that each
row lies in $28$k dimensional space.
For Maximum Mutual information feature selection we evaluated the performance
of MAB-based algorithm on smaller sized Greenhouse gas sensor dataset of
\citet{ghgdata}. This was the largest
dataset we could find with continuous features and continuous response.
We restricted ourselves to such datasets, because we wanted to
exhibit the applicability of our approach with sophisticated
estimators (like the Kozachenko-Leonenko based estimator
for mutual information).
This dataset consist of $4,578$ rows of $2,950$
dimensions i.e, it has readings from $2950$ sensors for at $4578$ time points.
For a fixed sensor, say $s_1$, our goal is to find the sensor (among the other
$2949$ sensors) with maximum mutual information.
\subsection{Implementation and evaluation}
We implemented the code in C++. The results and figures of the paper can be
reproduced from our
\href{https://github.com/govinda-kamath/combinatorial_MAB}{github
repository}.
A schematic showing the
structure of implementation is shown in Figure \ref{fig:implementation}.
\begin{figure}[H]
\centering
\includegraphics[width=.7\linewidth]{Implementation.png}
\caption{The implementation consists of three modules. The first module
loads the data, cleans it and puts it in a form that it can be processed.
The second module uses the data and designs the arms
corresponding to the problem at hand and then uses a multi-armed bandit
algorithm from the third module.}\label{fig:implementation}
\end{figure}
\texttt{UCB} has only one tuning parameter, For all the experiments we set $\delta=0.01$ in the UCB algorithm. We evaluate the accuracy of the algorithms and control it
as follows:
\begin{enumerate}
\item \textbf{$k$-nearest neighbors: } For $n$ points,
let the true $k$ nearest neighbors of point $i$ be the set $\texttt{NN}_i^{*}$
(evaluated using the naive algorithm)
and let the answer given by bandit-based algorithm be
the set $\texttt{NN}_i^{\text{MAB}}$. We define the accuracy by
$\frac{1}{n}\sum_{i=0}^n\mathbb{E}\Big[ \mathbf{1}\big\{\texttt{NN}_i^{*}=\texttt{NN}_i^{\text{MAB}} \big\} \Big]$.
For all of the $k$-nearest neighbour experiments, we
obtain an accuracy of $> 99\%$ for the bandits based algorithm
using $\delta=0.01$.
\item \textbf{$k$ means}: For $n$ points and $k$ cluster centres,
let for point $i$ let the nearest cluster be $c_i^{*}$ and the answer returned
by bandit-based method be $c_i^{\text{MAB}}$. We define the accuracy by
$\frac{1}{n}\sum_{i=0}^n\mathbb{E}\Big[\mathbf{1}\big\{ c_i^{*}=c_i^{\text{MAB}} \big\} \Big]$.
For all of the $k$-means experiments, we
obtain an accuracy of $> 99\%$ for the bandits based algorithm
using $\delta=0.01$.
\item \textbf{Maximal Mutual Information}: For dataset with $d$ features, let the feature with maximal mutual information with the label be $f^{*}$ and let the answer given by MAB-based algorithm be $f^{\text{MAB}}$.
We define the accuracy be $\mathbb{E}\Big[ \mathbf{1}\big\{ f^{*}=f^{\text{MAB}} \big\} \Big]$. For all of the Maximal Mutual Information experiments,
we
obtain an accuracy of $> 99\%$ for the bandits based algorithm
using $\delta=0.01$.
\item \textbf{Hierarchical clustering}:
For $n$ points let the $T^{*}$ be the tree graph given by the the brute hierarchical method, let $T^{\text{MAB}}$ be the tree graph given by MAB-based algorithm and let $T^{\text{random}}$ be a random tree graph with $n$ leaves. We measure the accuracy of $T^{\text{mab}}$ as
$$ 1-\frac{\sum_{i=0}^n\sum_{j=0}^{i-1}\mathbb{E}\Big[\big|\texttt{dist}^*(i,j)-\texttt{dist}^{\text{MAB}}(i,j)\big|\Big]}{\sum_{i=0}^n\sum_{j=0}^{i-1}\mathbb{E}\Big[\big|\texttt{dist}^*(i,j)-\texttt{dist}^{\text{random}}(i,j)\big|\Big]},$$ where $\texttt{dist}^*$, $\texttt{dist}^{\text{MAB}}$, $\texttt{dist}^{\text{random}}$ is the distance between point $i$ and $j$ in tree graph $T^*$, $T^{\text{MAB}}$ and $T^{\text{random}}$ respectively. For all hierarchical clustering experiments, our accuracy $>90\%$ and adjusted random score is $>0.7$.
\end{enumerate}
In the above expressions, the expectation is taken over the randomness in the sampling
to obtain the estimators.
We evaluate the algorithms based on the effective number of dimensions
or points used by estimators in the algorithm. For $k$-nearest neighbors,
$k$-means, hierarchical clustering this is the \textit{effective number of distance
evaluations}.
The effective number of distance computations between two points
(which are non-sparse)
is the fraction of dimensions used by the algorithm to
estimate distance between two points. The effective number
of distance computations in an algorithm
is the sum of the effective number of distance
computations between all pair points in the algorithm.
This is a reasonable metric as the amount of
work done between
distance computations is at most $O(\log n)$.
For the sparse case we define the
effective number of distance computations between two points
is the fraction of the non-zero co-ordinates used by the algorithm to
estimate distance between two points.
The effective number
of distance computations in an algorithm
is the sum of the effective number of distance
computations between all pair points in the algorithm.
For the Maximum Mutual information estimation problem we use
the \textit{effective number of samples } as
the metric to measure performance. This sum of the number of
samples used in the all the estimators.
\subsection{Exploration vs Exploitation}
The bandit based algorithm for $k$ nearest neighbors
algorithm evaluates true distance only for a handful
of contenders.
For the rest, it only
computes an approximation of
their distances. For point
$x_1$ (in $d=12$k) from imagenet ,
Figure \ref{fig:knn_working} shows that the UCB
achieves a 'good' trade-off between exploration
and exploitation for $k$ nearest neighbour problem -
it evaluates distance along all the co-ordinates (measures
the exact distance ) to the close points whereas for far off
points it calculates distances only along very few
co-ordinates (to obtain a coarse
estimate of the distance).
These observations generalise to the other three applications.
\begin{figure}[H]
\centering
\includegraphics[width=.9\linewidth]{distance_comp.pdf}
\caption{The number of distance
evaluations for $k$-nearest neighbor of a point
with othe points in its neighborhood.}\label{fig:knn_working}
\end{figure}
\subsection{Practice}
\paragraph{Speed-accuracy tradeoff:}
By setting $\delta$ to large values (say $0.1$),
UCB will maintain small confidence intervals
around the mean estimates of each arm and this will reduce the total arm pulls by UCB. Therefore, one
can trade off accuracy for faster running time.
Further
a deterministic running time
can be acheived if one
fixes the number of arm pulls for UCB.
This might be desirable in certain applications.
\paragraph{Confidence intervals:}
UCB heavily relies on the accuracy of its confidence intervals, whose estimation depends on the value of $\sigma$. Overestimating the value of $\sigma$ (likewise confidence intervals) increases running time whereas underestimating $\sigma$ decreases the accuracy.
Hence having a good estimate of $\sigma$ is crucial
for the bandit based algorithm to perform well.
We do it by maintaining a (running) estimate of
the mean and the second moment for every arm.
For instance which are application/dataset specific,
if one can obtain tighter confidence intervals, it will vastly improve the running time.
Estimating confidence intervals using other techniques could potentially improve the performance.
\paragraph{Others miscellaneous observations}
\begin{enumerate}
\item All the arms are maintained in a priority queue sorted by the lower confidence bounds of the arms. This gives us $\log n$ overhead in the running time over the number of arm pulls.
\item In our current implementation we sample the arms without replacement for the sake of simplicity. However,
we note that sampling without replacement would improve both the accuracy and the running time of UCB.
\item We have not taken cache management into consideration in our implementation. Thus the gains in complexity are not directly reflected in the gain in running times.
\end{enumerate}
\section{Details of Hierarchical Clustering}\label{app:heirarchical}
In the main text, we have explained hierarchical
clustering briefly. Here we elaborate on it.
Hierarchical clustering is an algorithm that is
widely used in data analysis
\multicitep{[Chapter 7]{LesRajUll} , [Chapter 14]{HasTibFri}}.
This is especially popular
if the number of clusters is not known apriori.
In general one assumes a set of $n$ points and
a distance/similarity measure to be the input to an
algorithm doing hierarchical clustering. The output of
the algorithm is usually a dendrogram (or a binary tree)
with $n$ leaves.
One approach to do this are \textit{agglomerative
algorithms} which combines points in a bottom up manner.
Such algorithms start of with the $n$ points
each being a cluster of one point. At every
step, the algorithm merges the two ``closest''
clusters to form one cluster. It keeps
doing this till there is only one cluster left.
This has a natural binary tree representation
where the leaves are the the $n$ points, the
root is the cluster of all points, and all
intermediate vertices are clusters formed at
intermediate steps. The two children of any vertex in
the tree are the clusters that were merged to form
the that cluster.
For two clusters which consist of only one
point, the notion of
closeness is unambiguously defined by the distance
metric. For clusters with more than one point, the
distance between clusters can be defined in a
variety of ways. Three popular ways \citep{MurCon,MosWan} are
\begin{enumerate}
\item Average linkage: which is the average
distance between points in the two clusters.
\item Single linkage: where the distance between
two clusters is the minimum of the pairwise distances
between points in the two clusters.
\item Complete linkage: where the distance between
two clusters is defined as the pairwise distances
between points in the two clusters.
\end{enumerate}
We mention that there is another class of algorithms
to do hierarchical clustering called divisive algorithms
which take a top down approach of starting from one cluster
and dividing it up to get better clusters.
In the sequel we will be considering an average linkage
based agglomerate algorithm to do hierarchical clustering.
At every step, instead of computing pairwise distances
between every pair of clusters (present in that step)
and picking the pair with the smallest pairwise distance,
we pose it in the framework of
adaptive Monte Carlo optimization problem.
In this formulation,
pairs of clusters are
considered as arms, and an arm pull being one randomly
picked co-ordinate-wise
evaluation of distances between two randomly picked points
one in each cluster.
We now introduce some notation.
Let $x_1, x_2, \cdots, x_n$ be $n$
points in $\mathbb{R}^d$ that we want to cluster.
We define a {\em cluster} to be a set of points
that have been merged by the algorithm. At the start
of the algorithm, each cluster is a singleton point
and is
indexed by a singleton set corresponding to the point
in the cluster.
Let $\mathcal{C}_t$ be the set of
clusters in the system at time $t$.
At the $t$-th step of the algorithm,
we have done $t-1$ merges and thus have
$n-t+1$ clusters in the system.
Hence $|\mathcal{C}_t|= n-t+1$.
Further, note that at any step $t$ in the algorithm, the
union of the clusters $\mathcal{C}_t$ gives us $[n]$
because each of the points is contained in exactly
one of the cluster.
We denote each arm in the system by an unordered
pair of clusters. We denote $A(C, C')$
to be the arm of clusters
$C$ and $C'$.
Let $\mathcal{A}_{t}$ be the set of arms in
the system at time $t$. Thus,
$\mathcal{A}_t = \{ A(C,C') \big| C, C' \in \mathcal{C}_t, C \ne C'\}$.
Clearly we have that $|\mathcal{A}_t| =
\binom{|\mathcal{C}_t|}{2}$. The algorithm used
is shown in \ref{alg:UCB-heirarchical}
(We overload notation in \texttt{UCB} to
allow MAX\_PULLS to be different for
each arm and is equal to the
dimension multiplied with the product
of the cluster size of each arm).
\begin{algorithm}
\caption{\texttt{Dynamic-UCB} \label{alg:UCB-heirarchical}}
\begin{algorithmic}[1]
\State Points $x_1,\cdot, x_n$ are input to the algorithm.
Each point is a cluster.
\State Create arms for every pair of points. Let the arms in
the system now be $\mathcal{A}_1$
\State Initialise estimates for points
\For{ $t$ in $1:n-2$}
\State Run UCB($\mathcal{A}_t$,$d_t$,$1$). Pick the winning Arm $A^*(C_t, C_t')$.
\Comment{$d_t$ is the total set of co-ordinates.}
\State Delete arms contain at least one of
$C_t$ and $C_t'$ from the the system.
\State Merge the $C_t, C_t'$ to form a new cluster.
\State Add arms corresponding to the newly formed $C_t \cup C_t'$ and
the other clusters in the system.
\State Initialise these
new arms.
\State $\mathcal{A}_{t+1}$ are the new arms in the system.
\EndFor
\end{algorithmic}
\end{algorithm}
In this notation, we have that at the beginning of
the algorithm,
\begin{align*}
\mathcal{C}_1 &= \{ \{1\}, \{2\}, \cdots, \{n\} \},\\
\mathcal{A}_1 &= \{ A(\{i\}, \{j\}) \big| 1 \le i < j \le n \}.
\end{align*}
Let $A^*(C_t, C_t')$ be the arm that wins the MAB
contest at time $t$.
Thus the clusters that are removed from the system
at time $t$
are $C_t$ and $C_t'$ and the new cluster added is
$\Tilde{C}_{t+1}= C_t \cup C_t'$. In other words,
\begin{align*}
\mathcal{C}_{t+1} &= (\mathcal{C}_{t}- \{ C_t, C_t'\} ) \cup \{\Tilde{C}_{t+1}\}.
\end{align*}
Thus at time $t+1$ the system contains arms for all unordered pairs
of clusters in $\mathcal{C}_{t+1}$. However we note that
$\mathcal{C}_{t+1}$ contains all but two clusters
in $\mathcal{C}_t$ and contains only one cluster
not in $\mathcal{C}_t$. Thus most of
the arms in the system at time $t$ are also in
the system at time $t+1$ and hence a lot of
computation is carried over from time $t$.
The arms deleted from the system
after time $t$ are of the form $A(C_t, C)$ and $A(C_t, C)$
for $C\in \mathcal{C}_{t}- \{ C_t, C_t'\}$.
Therefore,
\begin{align*}
\mathcal{D}_t &= \{A(C_t, C_t')\} \cup \big\{ A(C, C') \big| C\in\{ C_t, C_t'\} , C'\in \mathcal{C}_{t}- \{ C_t, C_t'\} \big\}, \\
\implies |\mathcal{D}_t| &= 1 + 2(n-t-1) = 2(n-t)-1.
\end{align*}
The arms added to the system before time $t+1$
are of the form $A(\tilde{C}_{t+1},C)$ for
$C\in \mathcal{C}_{t}- \{ C_t, C_t'\}$.
Thus we have that the arms at time $t+1$
is given by,
\begin{align*}
\mathcal{A}_{t+1} &= (\mathcal{A}_t - \mathcal{D}_t) \cup \mathcal{N}_{t+1},
\end{align*}
where $\mathcal{D}_t$ is the set of arms deleted after time $t$
and $\mathcal{N}_{t+1}$ is the set of arms that are added before
time $t+1$.
Therefore
\begin{align*}
\mathcal{N}_t &= \{A(\tilde{C}_t, C) \big| C \in \mathcal{C}_{t}- \{ C_t, C_t'\} \}, \\
\implies |\mathcal{N}_t| &= n-t-1.
\end{align*}
Thus the total number of unique arms in the system at all times
is $M$
\begin{align*}
M &= \binom{n}{2} + \sum_{i=1}^{n-2} n-i-1,\\
&= \binom{n}{2} + \binom{n-1}{2},\\
&< n^2.
\end{align*}
To reiterate, an arm $A^*(C_t, C_t')$ is killed at
step $t$ of the algorithm if either $A^*(C_t, C_t')$
is the winner of the MAB at step $t$ or
the winning arm is of the form $A^*(C_t, D)$ or
$A^*(C_t',D)$ for some other
cluster $D$ which is in the system at time $t$.
Let the winning mean at step $t$ be $\mu^*_t$.
\begin{lem} \label{lem:increasing_mu}
$\mu^*_1 \le \mu^*_2 \le \cdots \le \mu^*_{n-3} \le \mu^*_{n-2}$
\end{lem}
\begin{proof}
We first claim that the true means each of the arms added before
step $t$ is at least as large as one of the arms
deleted after step $t-1$. Once we show this,
the proof is immediate as all arms in the system at time
$t-1$ had their means to be at least $\mu^*_{t-1}$,
and the new arms added have their true means to be
at least $\mu^*_{t-1}$. So all the arms in the
system at time $t$ have their true means to be
at least $\mu^*_{t-1}$ giving us that the mean of the
winning arm at time $t$, $\mu^*_{t}$ is
at least $\mu^*_{t-1}$, that is $\mu^*_{t} \ge \mu^*_{t-1}$
proving the result.
To see the claim, let $A^*(C_t, C_t')$ be the arm that won
the MAB at time $t-1$. For any cluster $D$ (which is not
$C$ and $C'$), we delete arms $A^*(C_t,D)$ and $A(C'_t,D)$
and add arm $A(C_t\cup C_t',D)$. We note that as we are
in the average linkage paradigm, the true mean
of $A(C_t\cup C_t',D)$, $\mu_{A(C_t\cup C_t',D)}$ is a convex
combination of the true means of $A(C_t,D)$ and $A(C_t',D)$.
More concretely,
\begin{align*}
\mu_{A(C\cup C',D)} &= \frac{|C|}{|C|+|C'|} \mu_{A(C,D)} + \frac{|C'|}{|C|+|C'|} \mu_{A(C',D)},\\
&\ge \min (\mu_{A(C,D)}, \mu_{A(C',D)}),
\end{align*}
where we have used the fact that as $C$ and
$C'$ are two clusters in the system at the same time
they are disjoint. This proves the claim
and thus gives us the lemma.
\end{proof}
Let $\tau(A(C,C'))$ denote the time at which arm
$A(C,C')$ goes out of the system. This can happen
either because the arm wins that round of MAB
or some arm containing one of cluster $C$ or $C'$
wins.
We note that if arm $A(C,C')$ is introduced
before step $t_1$, then following our previous analysis,
the number of distance evaluations
at step $t_1$ is
\begin{align*}
\min \left( \frac{ 24\sigma^2 \log (nd)}{(\mu_{A(C,C')}- \mu^*_{t_1})^2}
, d |C| |C'|\right)
\end{align*}
The number of additional co-ordinate wise distance
evaluations at time $t_1+1$ (if $A(C, C')$ is
not delete after time $t_1$) is then
\begin{align*}
\left[\min \left( \frac{24\sigma^2 \log nd}{(\mu_{A(C,C')}- \mu^*_{t_1+1})^2}
, d |C| |C'|\right) - \min \left( \frac{24\sigma^2 \log n}{(\mu_{A(C,C')}- \mu^*_{t_1})^2}
, d |C| |C'|\right)\right]\\
\end{align*}
where Lemma \ref{lem:increasing_mu} gives us that the
quantity above is non-negative.
This gives us that the total number of distance
evaluations at time $\tau(A(C,C'))$, continuing in this manner is
thus
\begin{align*}
n(A(C,C')) &= \min \left( \frac{24\sigma^2 \log nd}{(\mu_{A(C,C')}- \mu^*_{t_1})^2}
, d |C| |C'|\right) \\
& + \sum_{i=t_1+1}^{\tau(A(C,C'))} \left[\min \left( \frac{24\sigma^2 \log nd}{(\mu_{A(C,C')}- \mu^*_{t_1+1})^2}
, d |C| |C'|\right) - \min \left( \frac{24\sigma^2 \log nd}{(\mu_{A(C,C')}- \mu^*_{t_1})^2}
, d |C| |C'|\right)\right],\\
&= \min \left( \frac{24\sigma^2 \log nd}{(\mu_{A(C,C')}- \mu^*_{\tau(A(C,C'))})^2}
, d |C| |C'|\right),
\end{align*}
We note that following all our previous analysis
we have that if the number of
distance evaluations on $A(C,C')$
is at most
\begin{align*}
\min \left( \frac{24\sigma^2 \log nd}{(\mu_{A(C,C')}- \mu^*_{\tau(A(C,C')})^2}
, d |C| |C'|\right),
\end{align*}
If $\mu_{A(C, C')}- \mu^*_{\tau(A(C,C')}$
can be thought of as constants independent of $n$,
then this gives us that the algorithm needs $O(n^2 \log n)$
co-ordinate wise distance computations.
More generally at step $t$, if $A(C_t, C_t')$
is the winning arm, then the number of evaluations
in arms discarded at step $t$ apart from the winners
is given by
\begin{align*}
\sum_{A(C,C') \in \mathcal{D}_t - A(C_t, C_t')}
\min \left( \frac{24\sigma^2 \log nd}{(\mu_{A(C,C')}- \mu^*_{t})^2}
, d |C| |C'|\right)
\end{align*}
If we assume that $\mu_{A(C,C')} \sim \mathcal{N}(\mu, 1)$
then this is $O(n\log n)$ at each step.
This gives us that the total number of distance evaluations
on non-winning arms is $O(n^2 \log nd)$.
We further note that the total number of distance evaluations
is also at most the twice the number distance evaluations
spent on the non-winning arms because the cost of the winning
and the second best arms are the same.
This gives us that we have $O(n^2 \log nd)$ co-ordinate
wise distance evaluations.
While the above model is not a very compelling model
of our problem, it does give us some intuition
as to why the algorithm is observed to run with
$O(n^2 \log nd)$ co-ordinate wise distance evaluations.
\begin{note}
We note that if we reuse the distances evaluated in
the deleted arms which are not winners, all distances
evaluated end up being used in distinguishing one
of the winners or a second best arm. This intuitively
tells us that we are not wasting distance evaluations.
\end{note}
\section{Comparison with projection based dimensionality reduction} \label{app:jl}
A natural algorithm for improving
the scaling of
$k$-nearest neighbors
is to project the data into lower
dimensional space using Johnson-Lindenstrauss
and computing the $k$-nearest neighbors.
If one projects a set of $n$ points onto
$\frac{\log n}{\epsilon^2}$ dimensional
space (obtained by a random linear transformation),
then one preserves the distances
between the any to points to a multiplicative factor
of $(1\pm \epsilon)$. Thus using this method,
we obtain a $(1 + \epsilon)$ approximation for
$k$-nearest neighbors. This
algorithm would need $O(nd\frac{\log n}{\epsilon^2})$
computations for projecting data
down, and then would require $O(n^2 \frac{\log n}{\epsilon^2} )$
to obtain the $k$-nearest neighbors.
On the other hand the adaptive sampling based algorithm
would take
$O(n^2\frac{\log n}{\epsilon})$ (assuming $n > d $)
to obtain
an $1 +\epsilon$ approximation, giving it a better scaling.
However we note that this comes at the expense of assuming
mild
regularity conditions (which we have shown that
the data satisfies) that the Johnson-Lindenstrauss
based algorithm (which only
assumes boundedness).
\section{Details on $k$-means}\label{app:kmeans}
Here we have $k$ centroids. We consider the number of
co-ordinate wise distance evaluated to compute the nearest
of the $k$ centroids to the first of $n$ points of a data-set.
Let $i^*$ be the centroid closet to point $1$
and for $i\in \{1,2,\cdots,k\}$, let $\Delta_i$ be
the difference between the distance between
centroid $i$ and the centroid $i^*$.
However, we use $\delta = 1 - \frac{1}{n^3d}$ here.
Theorem \ref{thm:knn1} directly gives us that:
\begin{cor}
With probability $1- \Theta( \frac{1}{n^2})$, this
algorithm
find the closests of $k$ centroids of point $1$ is
with at most
\begin{align*}
M \le \sum_{i=1; i\neq i^*}^{k} \left( \left( \frac{48 \sigma^2}{\Delta_i^2} \log (nd) \right) \wedge 2d \right)
\end{align*}
co-ordinate wise distance evaluations. The running time is
\begin{align*}
\log(k)\sum_{i=1;i\neq i^*}^{k}\left( \left( \frac{48 \sigma^2}{\Delta_i^2} \log (nd) \right) \wedge 2d \right)
\end{align*}
\end{cor}
If we again assume that $\Delta_i \sim \mathcal{N}(\gamma,1)$,
then we have from a direct consequence of
Theorem \ref{thm:knnfull}
\begin{cor}
If we assume that $\Delta_i \sim \mathcal{N}(\gamma, 1)$,
for some $\gamma$, $d = O(n)$ and $k$ is a constant ,
then we have that the expected number of co-ordinate wise distance evaluations (over randomness in $\Delta_i$) over all points,
\begin{align*}
\mathbb{E}[M] \le O(nk \log (nd)).
\end{align*}
with high probability over randomness
in the algorithm.
\end{cor}
The proof is trivial since $\delta_i$'s are $O(1)$ quantities.
We now consider the $(1+ \epsilon)$ approximation
algorithm as before. The definition of the approximation
is the same of before, with $\mu^*$ is the distance
between point $1$ and it's nearest centroid.
\begin{cor}
The number of co-ordinate wise distance evaluations
to find the nearest centroid of point $1$ upto
a $(1+\epsilon)$-approximation is
at most
\begin{align*}
M \le \sum_{i=1}^{k} \left( \frac{24 \sigma^2}{\left(\max(\Delta_i,\epsilon \mu^{*})\right)^2} \log (nd) \right) = \sum_{i=1}^{k} \left( \left(\frac{\sigma}{\mu^{*}}\right)^2 \frac{24 }{\max\left(\left(\frac{\Delta_i}{\mu^{*}}\right)^2, \epsilon^2\right)} \log
(nd) \right)
\end{align*}
\end{cor}
If we again assume that $\Delta_i \sim \mathcal{N}(\gamma,1)$,
then we have from a direct consequence of
Theorem \ref{thm:knnapproxfull}
\begin{cor}
If we assume that $\Delta_i \sim \mathcal{N}(\gamma,1)$,
then the expected number of distance evaluations
(over randomness in $\Delta_i$) is
\begin{align*}
\mathbb{E}[M] \le O\left( \frac{k \log kd}{\epsilon}\right)
\end{align*}
with probability $1-\Theta(\frac{1}{k^2})$ over
randomness in the algorithm.
\end{cor}
\section{Details on $k$-nearest neighbors}\label{app:knn}
To recap the setting we note that for posing $k$-nearest
neighbour graph construction in our framework, we run a
multi-armed bandit with each arm being one of the remaining
$n-1$ points.
We begin by noting that
we measure the performance of algorithms in
\textit{effective number of
distance evaluations}. We define effective number of distance
evaluations between two points as the fraction of co-ordinates
involved in estimating the distance between these two points. The
effective number of distance evaluations of an algorithm is the
sum of effective number of distance evaluations between
every pair of points in the algorithm.
We define the number of \textit{coordinate-wise
distance evaluations} of an
algorithm
to be $d$ times the
the effective number of
distance evaluations
of the algorithm.
Without loss of generality, we assume that we are finding
the $k$ nearest neighbors of point $1$.
In this case we have $n-1$ arms, one corresponding
to each of points $2, \cdots, n$.
Further the mean of each arm can be evaluated
exactly in a maximum of
$d$ pulls.
Let $f(i)$ be used to denote the distance between point
$1$ and point $i$.
The expectation of each pull of arm $i$ is
the distance between point $i$ and point $1$.
The estimator after $\ell$ pulls of
arm $i$
is given by
\begin{align}\label{eq:knn}
\Hat{f}_{\ell} (i) &= \frac{1}{\ell} \sum_{j=1}^{\ell} (x_{1,t_j}-x_{i, t_j})^2,
\end{align}
where $t_j$ are sampled from $[d]$ with replacement
for $j \in [\ell]$. The update of the
estimator at step $\ell+1$ is
\begin{align}\label{eq:knn_update}
\Hat{f}_{\ell+1} (i) = \frac{1}{\ell + 1} \left( \ell \Hat{f}_{\ell} (i)
+ (x_{1,t_{\ell+1}}-x_{i, t_{\ell+1}})^2 \right),
\end{align}
We assume that the estimators are $\sigma$-subgaussian, for
some $\sigma$ independent of $n$. In practice, we estimate $\sigma$ from few initial samples and update it after every pull. We construct
the $1-\delta$ confidence intervals of $\hat{f}_{\ell}(i)$ as,
\begin{align} \label{eq:CI}
C(\ell) &= \begin{cases}
\sqrt{\frac{2 \sigma^2 \log \frac{2}{\delta}}{\ell}} & \text{ if } \ell \le d\\
0 &\text{ if } \ell > d
\end{cases},\\
f(i) &\in \left[ \hat{f}_{\ell}(i) - C(\ell), \hat{f}_{\ell}(i) + C(\ell) \right],
\text{ w.p. } 1 - \delta.
\end{align}
We assume that the estimate of $\sigma$ is the true value. Further
we set $\delta = \frac{2}{n^3d}$. Then we have that,
\begin{lem}\label{lem:knn}
With probability $1- \frac{1}{n^2}$, each one of our
true values lies in the their confidence intervals during the
run of the algorithm.
\end{lem}
Let $i_k^*$ be the $k$-th nearest neighbors
of point $1$, and define
\begin{align*}
\Delta_i = \max(0,f(i)-f(i_k^*))
\end{align*}
Using this lemma, we now state the following theorem,
\begin{thm}[Restating Theorem \ref{thm:knn1}]
With probability $1-\frac{1}{n^2}$,
Algorithm \ref{alg:UCB-knn} returns the
$k$-nearest neighbors of point $1$
with at most
\begin{align} \label{eq:knn_proof_statement}
M \le \sum_{i=2}^{n} \left( \left( \frac{24 \sigma^2}{\Delta_i^2} \log (nd) \right) \wedge 2d \right)
\end{align}
distance evaluations.
This takes
\begin{align} \label{eq:knn_proof_time}
O\left( \sum_{i=2}^{n}\log (n) \left( \left( \frac{\sigma^2 \log (nd)}{\Delta_i^2} \right) \wedge d \right)
\right)
\end{align}
time.
\end{thm}
\begin{proof}
Let $\ell_i(t)$, be the number of times, arm $i$ is pulled
till iteration $t$.
Let $i_1^*, i_2^*, \cdots, i_k^*$ be
the $k$ one
nearest neighbors of point $1$ in order.
For simplicity of notation, let us define
\begin{align*}
\hat{\mu}_{i}(t) &= \hat{f}_{\ell_i(t)}(i),\\
C_i(t) &= C(\ell_i(t)),\\
\Delta_i^{(w)} &= \max(0,f(i)-f(i_w^*))
\end{align*}
In words, $\hat{\mu}_{i}(t)$ is simply the estimate of the
mean of arm $i$ at the $t$-th iteration of the
algorithm, and $2C_i(t)$ is the width of the
$(1-\delta)$ confidence interval of arm $i$ at iteration $t$
of the algorithm. For any point $i$ which is not
the $k$ nearest neighbors,
$\Delta_i^{(w)}$ is the difference in distance of point $i$
to point $1$ and the $w$-th nearest neighbour.
Also clearly from this definition,
$ \Delta_i^{(1)} \ge \Delta_i^{(2)} \ge \cdots \ge \Delta_i^{(k)}$.
Further $\Delta_i^{(k)}= \Delta_i$ defined above.
Let us first analyse the running of the algorithm till
it found the first nearest neighbour of point $1$ to
start with.
We observe
that if we choose to update
arm $i\neq i^*_1$ at time $t$, then we have
\begin{align*}
\hat{\mu}_i(t) - C_i(t) \leq \hat{\mu}_{i^*_1}(t) - C_{i^*_1}(t).
\end{align*}
For this to occur, at least one of the following three events must occur:
\begin{align*}
& \mathcal{E}_1 = \left\{ \hat{\mu}_{i^*_1}(t) \geq \mu_{i^*_1}(t)+C_{i^*_1}(t) \right\},\\
& \mathcal{E}_2 = \left\{ \hat{\mu}_i(t) \leq \mu_i(t)-C_i(t) \right\}, \\
& \mathcal{E}_3 = \left\{ \Delta_i^{(1)} = \mu_i - \mu_{i^*_1} \leq 2 C_i(t) \right\}.
\end{align*}
To see this, note that if none of
$\mathcal{E}_1, \mathcal{E}_2, \mathcal{E}_3$
occur,
we have
\begin{align*}
\hat{\mu}_i(t) - C_i(t) \overset{(a)}{>} \mu_i - 2 C_i(t)\overset{(b)}{>}\mu_{i^*_1} \overset{(c)}{>} \hat{\mu}_{i^*_1} - C_{i^*_1}(t),
\end{align*}
where $(a)$, $(b)$, and $(c)$ follow because $\mathcal{E}_2$,
$\mathcal{E}_3$, and $\mathcal{E}_1$ do not hold
respectively.
We note that as we compute
$\left(1- \frac{2}{n^3d}\right)-$confidence intervals
at most $d$ times for each point. Thus we have
at most $n^2d$ computations of
$\left(1- \frac{2}{n^3d}\right)-$confidence intervals
in total.
Thus $\mathcal{E}_1$ and $\mathcal{E}_2$
do not occur during any iteration
with probability $1-\frac{2}{n^2}$,
because
\begin{align} \label{eq:sg_bound}
\text{w.p.}~\left(1-\frac{2}{n^2}\right): \vert f(i) - \hat{\mu}_i(t) \vert \leq C_i(t),~\forall~i\in[n],~\forall~t.
\end{align}
This also implies that
with
probability $1- \Theta\left(\frac{1}{n^2}\right)$
the
algorithm does not stop unless
the event $\mathcal{E}_3$,
a deterministic condition, stops occurring.
Let $\zeta_i^{(1)}$ be the
iteration of the algorithm when it
evaluates a distance to point $i$ for
the last time before declaring the $1$-nearest
neighbour.
From the previous discussion, we
have that the algorithm stops evaluating
distances to
points $i$ when the following holds.
\begin{align*}
& C_i(\zeta_i^{(1)}) \le \frac{\Delta_i^{(1)}}{2}
\implies \frac{\Delta_i^{(1)}}{2} \ge \sqrt{\frac{2 \sigma^2 \log n^3d}{T_i(\zeta_i^{(1)})}} \text{ or } C_i(\zeta_i^{(1)}) = 0, \\
&\implies T_i(\zeta_i^{(1)}) \ge \frac{8 \sigma^2}{(\Delta_i^{(1)})^2} \log
(n^3d) \text{ or } T_i(\zeta_i^{(1)}) \ge 2d.
\end{align*}
We note that the above is true for $i^*_1$ as well
as we have at most $2d$ distance evaluations there.
As the bandit algorithm progresses, let us similarly define,
$\zeta_i^{(w)}$ be the
iteration of the algorithm when it
evaluates a distance to point $i$ for
the last time before declaring the $w$-th-nearest
neighbour.
By the same computation as above, we have that,
\begin{align*}
T_i(\zeta_i^{(w)}) \ge \frac{8 \sigma^2}{(\Delta_i^{(w)})^2} \log
(n^3d) \text{ or } T_i(\zeta_i^{(w)}) \ge 2d.
\end{align*}
We further note that the total number of distance computations
is
\begin{align*}
M &= \sum_{i=2}^{n} T_i(\zeta_i^{(k)}),\\
&\le \sum_{i=2}^{n}\left( \left( \frac{8 \sigma^2}{(\Delta_i^{(k)})^2} \log (n^3d) \right) \wedge 2d \right),\\
&\overset{(d)}{=} \sum_{i=2}^{n}\left( \left( \frac{8 \sigma^2}{\Delta_i^2} \log (n^3d) \right) \wedge 2d \right),\\
&\le \sum_{i=2}^{n}\left( \left( \frac{24 \sigma^2}{\Delta_i^2} \log (nd) \right) \wedge 2d \right),
\end{align*}
where $(d)$ follows from the definition of $\Delta_i$.
Thus with probability $(1- \Theta\left(\frac{1}{n^2}\right))$, the algorithm
returns $i^*$ as the medoid with at most $M$ distance
evaluations, where
\begin{align*}
M \le \sum_{i = 2}^{n} T_i(\zeta_i)
\le \sum_{i = 2}^{n} \left( \left( \frac{24 \sigma^2}{\Delta_i^2} \log (nd) \right) \wedge 2d \right).
\end{align*}
giving us the claim in
Eq. \eqref{eq:knn_proof_statement}.
We note that each step of the multi-armed bandit
can be implemented in $O(\log n)$ time
using a priority queue giving us the result of
\eqref{eq:knn_proof_time}
\end{proof}
\begin{thm}\label{thm:knnfull}
If we further assume that $\Delta_i \sim \mathcal{N}(\gamma, 1)$,
for some $\gamma$, and $d = c n$, for $c \in [0,1]$,
then we have that the expected number of co-ordinate wise distance evaluations (over randomness in $\Delta_i$),
\begin{align*}
\mathbb{E}[M] \le O(n \log (nd) + kd).
\end{align*}
with probability $1-\Theta(\frac{1}{n^2})$ over randomness
in the algorithm.
This gives us that the entire $k$-nearest neighbors
takes $O(n^2\log (nd) + nkd )$ in expectation
over randomness in the $\Delta_i$
co-ordinate wise distance evaluations
with probability $1-\Theta(\frac{1}{n})$ over
randomness in the algorithm.
\end{thm}
\begin{proof}
This follows directly from the Appendix A of \citet*{BagKamNtrZhaTse}
\end{proof}
If instead one is just interested in approximating the $k$
nearest neighbors of a point, then one can use a slightly modified
version of the UCB algorithm of Algorithm \ref{alg:genUCB}. By
approximating $k$ nearest neighbors, we mean picking points
that are within a distance $\epsilon$ additively
of the $k$-nearest neighbors
of point $i$. We can do this by modifying line $10$
of Algorithm \ref{alg:genUCB} to stop
if the width of the confidence
interval of the best arm is of width less than $\epsilon$
times its UCB.
As before let $i_1^*, \cdots, i_k^*$ be the $k$ nearest
neighbors of point $1$,and let their distances be
$f(i^*_w) = \mu_{i^*_w}$ for $w \in \{1, \cdots, k\}$. Also
let $\Delta_i$ be as before.
In this case with the assumptions as above, we have the following result
\begin{thm}[Restatement of Theorem \ref{thm:knnapprox}]
With probability
$1-\frac{1}{n^2}$
finds the $k$-nearest neighbor of point $1$ upto
a $(1+\epsilon)$-approximation with at most
\begin{align} \label{eq:knn_proof_approx}
M \le \sum_{i=2}^{n} \left( \frac{24 \sigma^2}{\left(\max(\Delta_i,\epsilon \mu_{i^*_k})\right)^2} \log (nd) \right) = \sum_{i=2}^{n} \left( \left(\frac{\sigma}{\mu_{i^*_k}}\right)^2 \frac{24 }{\max\left(\left(\frac{\Delta_i}{\mu_{i^*_k}}\right)^2, \epsilon^2\right)} \log
(nd) \right)
\end{align}
co-ordinate wise distance computations with
a running time of
\begin{align}\label{eq:knn_proof_approx_time}
O\left( \sum_{i=2}^{n} \log(n)\left( \frac{\sigma^2 \log (nd) }{\left(\max(\Delta_i,\epsilon \mu_{k}^*)\right)^2} \right) \right)
\end{align}
\end{thm}
\begin{proof}
Let $\ell_i(t)$, be the number of times, arm $i$ is pulled
till iteration $i$.
Let $i_1^*, i_2^*, \cdots, i_k^*$ be
the $k$ one
nearest neighbors of point $1$ in order.
For simplicity of notation, let us define
\begin{align*}
\hat{\mu}_{i}(t) &= \hat{f}_{\ell_i(t)}(i),\\
C_i(t) &= C(\ell_i(t)),\\
\Delta_i^{(w)} &= \max(0,f(i)-f(i_w^*))
\end{align*}
In words, $\hat{\mu}_{i}(t)$ is simply the estimate of the
mean of arm $i$ at the $t$-th iteration of the
algorithm, and $2C_i(t)$ is the width of the
$(1-\delta)$ confidence interval of arm $i$ at iteration $t$
of the algorithm. For any point $i$ which is not
the $k$ nearest neighbors,
$\Delta_i^{(w)}$ is the difference in distance of point $i$
to point $1$ and the $w$-th nearest neighbour.
Also clearly from this definition,
$ \Delta_i^{(1)} \ge \Delta_i^{(2)} \ge \cdots \ge \Delta_i^{(k)}$.
Further $\Delta_i^{(k)}= \Delta_i$ defined above.
Let us first analyse the running of the algorithm till
it found the first nearest neighbour of point $1$ to
start with.
Define
\begin{align*}
J_{\epsilon}^{(1)} &= \{ i \in \{2,\cdots, n\} \bigg| \Delta_i^{(1)} \le f(i^*_1) \epsilon\},\\
K_{\epsilon}^{(1)} &= \{2,\cdots, n\} - J_{\epsilon}^{(1)},
\end{align*}
In words, we have that $J_{\epsilon}^{(1)}$ are the points whose
distance from point $1$ is less than $(1+\epsilon)$ times
the distance of the nearest neighbour. Thus returning
any one of these points satisfies our approximation requirements.
$K_{\epsilon}^{(1)}$ is other points.
We observe
that if we choose to update
arm $i \in K_{\epsilon}^{(1)}$ at time $t$, then we have
\begin{align*}
\hat{\mu}_i(t) - C_i(t) \leq \hat{\mu}_{i^*_1}(t) - C_{i^*_1}(t).
\end{align*}
For this to occur, at least one of the following three events must occur:
\begin{align*}
& \mathcal{E}_1 = \left\{ \hat{\mu}_{i^*_1}(t) \geq \mu_{i^*_1}(t)+C_{i^*_1}(t) \right\},\\
& \mathcal{E}_2 = \left\{ \hat{\mu}_i(t) \leq \mu_i(t)-C_i(t) \right\}, \\
& \mathcal{E}_3 = \left\{ \Delta_i^{(1)} = \mu_i - \mu_{i^*_1} \leq 2 C_i(t) \right\}.
\end{align*}
To see this, note that if none of
$\mathcal{E}_1, \mathcal{E}_2, \mathcal{E}_3$
occur,
we have
\begin{align*}
\hat{\mu}_i(t) - C_i(t) \overset{(a)}{>} \mu_i - 2 C_i(t)\overset{(b)}{>}\mu_{i^*_1} \overset{(c)}{>} \hat{\mu}_{i^*_1} - C_{i^*_1}(t),
\end{align*}
where $(a)$, $(b)$, and $(c)$ follow because $\mathcal{E}_2$,
$\mathcal{E}_3$, and $\mathcal{E}_1$ do not hold
respectively.
We note that as we compute
$\left(1- \frac{2}{n^3d}\right)-$confidence intervals
at most $d$ times for each point. Thus we have
at most $n^2d$ computations of
$\left(1- \frac{2}{n^3d}\right)-$confidence intervals
in total.
Thus $\mathcal{E}_1$ and $\mathcal{E}_2$
do not occur during any iteration
with probability $1-\frac{2}{n^2}$,
because
\begin{align} \label{eq:sg_bound}
\text{w.p.}~\left(1-\frac{2}{n^2}\right): \vert f(i) - \hat{\mu}_i(t) \vert \leq C_i(t),~\forall~i\in[n],~\forall~t.
\end{align}
This also implies that
with
probability $1- \Theta\left(\frac{1}{n^2}\right)$
the
algorithm does not stop unless
the event $\mathcal{E}_3$,
a deterministic condition stops occurring.
Let $\zeta_i^{(1)}$ be the
iteration of the algorithm when it
evaluates a distance to point $i$ for
the last time before declaring the $1$-nearest
neighbour.
From the previous discussion, for arms in $J_{\epsilon}^{(1)}$ we
have that the algorithm stops evaluating
distances to
points $i$ when the following holds.
\begin{align*}
& C_i(\zeta_i^{(1)}) \le \frac{\Delta_i^{(1)}}{2} \\
\implies &\frac{\Delta_i^{(1)}}{2} \ge \sqrt{\frac{2 \sigma^2 \log n^3}{T_i(\zeta_i^{(1)})}} \\
\implies& T_i(\zeta_i^{(1)}) \ge \frac{8 \sigma^2}{(\Delta_i^{(1)})^2} \log
(n^3d),
\end{align*}
where we use the fact that $\Delta_i^{(1)} \ge \epsilon f(i^*_1)$.
Further note that if none of $\mathcal{E}_1$, $\mathcal{E}_2$
$\mathcal{E}_3$, occur than no arm in $K_{\epsilon}^{(1)}$ would
be pulled when the width of their confidence interval is less than
$\epsilon f(i^*_1)$. To see this, note that,
\begin{align*}
\epsilon(\mu_i + C_i(t)) > \epsilon f(i^*_1) \overset{(d)}{>} \Delta_i^{(1)}
\overset{(e)}{>} 2C_i(t) = \text{Width of confidence interval},
\end{align*}
where $(d)$ follows from the fact that $i \in K_{\epsilon}^{(1)}$
and $(e)$ follows from the fact that $\mathcal{E}_3$ does not occur.
Thus we exit only when one of the arms in $K_{\epsilon}^{(1)}$ is pulled
and each of them is pulled. Thus for each of the arms in $K_{\epsilon}^{(1)}$
the following holds:
\begin{align*}
& C_i(\zeta_i^{(1)}) \le \frac{\epsilon \mu_{i^*_1}}{2} \\
\implies &\frac{\epsilon \mu_{i^*_1}}{2} \ge \sqrt{\frac{2 \sigma^2 \log n^3}{T_i(\zeta_i^{(1)})}} \\
\implies& T_i(\zeta_i^{(1)}) \ge \frac{8 \sigma^2}{(\epsilon \mu_{i^*_1})^2} \log
(n^3d).
\end{align*}
Thus we can using the definition of $K_{\epsilon}^{(1)}$ and $K_{\epsilon}^{(1)}$, we can write
this more succinctly as
\begin{align*}
T_i(\zeta_i^{(1)}) \ge \frac{8 \sigma^2}{\left(\max(\Delta_i^{(1)},\epsilon \mu_{i^*_1})\right)^2} \log
(n^3d).
\end{align*}
As the bandit algorithm progresses, let us similarly define,
$\zeta_i^{(w)}$ be the
iteration of the algorithm when it
evaluates a distance to point $i$ for
the last time before declaring the $w$-th-nearest
neighbour.
By the same computation as above, we have that,
\begin{align*}
T_i(\zeta_i^{(w)}) \ge \frac{8 \sigma^2}{\left(\max(\Delta_i^{(w)},\epsilon \mu_{i^*_w})\right)^2} \log
(n^3d).
\end{align*}
We further note using the fact that $T_i(\zeta_i^{(w)})$
are increasing as function of $w$,
that the total number of distance computations
is
\begin{align*}
M &= \sum_{i=2}^{n} T_i(\zeta_i^{(k)}),\\
&\le \sum_{i=2}^{n}\left( \frac{8 \sigma^2}{\left(\max(\Delta_i^{(k)},\epsilon \mu_{i^*_k})\right)^2}\log (n^3d) \right) ,\\
&\overset{(d)}{=} \sum_{i=2}^{n} \left( \frac{8 \sigma^2}{\left(\max(\Delta_i,\epsilon \mu_{i^*_k})\right)^2} \log (n^3d) \right) ,\\
&\le \sum_{i=2}^{n} \left( \frac{24 \sigma^2}{\left(\max(\Delta_i,\epsilon \mu_{i^*_k})\right)^2} \log (nd) \right),
\end{align*}
where $(d)$ follows from the definition of $\Delta_i$.
We note that each step of the multi-armed bandit
can be implemented in $O(\log n)$ time
using a priority queue giving us the result of
\eqref{eq:knn_proof_approx_time}.
\end{proof}
\begin{thm}\label{thm:knnapproxfull}
If we assume that $\Delta_i \sim \mathcal{N}(\gamma,1)$,
then the expected number of distance evaluations
(over randomness in $\Delta_i$) is
\begin{align*}
\mathbb{E}[M] \le O\left( \frac{n \log nd}{\epsilon}\right)
\end{align*}
with probability $1-\Theta(\frac{1}{n^2})$ over
randomness in the algorithm.
This gives us that the entire $k$-nearest neighbors
takes $O(\frac{n^2\log (nd)}{\epsilon})$
in expectation
over randomness in the $\Delta_i$
co-ordinate wise distance evaluations
with probability $1-\Theta(\frac{1}{n})$ over
randomness in the algorithm.
\end{thm}
\begin{proof}[Sketch of proof]
For Gaussian $\Delta_i$, the
\begin{align*}
\min_i \gamma - \Delta_i \rightarrow \sqrt{2\log n}.
\end{align*}
Thus the fraction of points in within $(1+\epsilon)$
of the minimum is at most $e^{-1-\epsilon}$. Thus the
work done for these points is $O(\frac{1}{\epsilon})$.
Similarly, one can handle the rest of the points as well.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:knn}]
Let $\ell_i(t)$, be the number of times, arm $i$ is pulled
till iteration $i$.
We first show that
$[\hat{f}_i(\ell_i(t))-C(\ell_i(t)), \hat{f}_i(\ell_i(t))+C(\ell_i(t))]$
are true
$\left(1-\frac{2}{n^3d}\right)$-confidence
intervals of $f(i)$ at every iteration $t$.
We observe that if point $i$ is picked
by less than $d$ times at
time t, then $\ell_i(t)$ is equal to the number
of times the point is picked. Further
$C_i(t)$ is the true $(1-\delta)$-confidence interval from
the assumption of $\sigma$-sub-gaussainity on
the estimators.
However, if point $i$ is picked for the
$n$-th time at iteration $t$ (
line $8$ of Algorithm \ref{alg:genUCB})
then the empirical mean is computed
by evaluating the distance exactly
with
$d$ co-ordinate wise distance computations.
As we know the
mean distance of point $i$
exactly, $C_i(t) = 0$ is
still the true confidence interval.
As throughout the algorithm we can compute at most
$n^2d$ confidence intervals of length $1-\frac{1}{n^3d}$,
with probability $1-\frac{1}{n^2}$, the true value
is
always in the computed confidence intervals.
\end{proof}
\section{Details on Maximum Mutual Information based feature selection }\label{app:mmi}
\begin{algorithm}
\caption{\texttt{MMI UCB} \label{alg:UCB-mmi}}
\begin{algorithmic}[1]
\State $d$ features with n samples:
$x_1,\cdot, x_d$ are and a response $y$ input to the algorithm.
\State Consider arms $\mathcal{A}_{\ell}, 1 \le \ell \ne i \le n$
with estimators defined as in Eq. \eqref{eq:mmi}.
\State The most relevant feature to the response is given by
\texttt{UCB}$(\{\mathcal{A}_{\ell}, 1 \le \ell \ne i \le n\},n,1)$, using maximum instead of minimum.
\Comment{MAX\_PULLS= d}
\end{algorithmic}
\end{algorithm}
Let $i^*$ be the feature whose mutual information
with the target is the largest. Define
\begin{align*}
\Delta_i = f(i^*)-f(i),
\end{align*}
where $f(i)$ is the empirical estimate
of the mutual information of the feature $i$
with the target.
\begin{thm}
With probability $1-\frac{1}{d^2}$,
Algorithm \ref{alg:UCB-mmi} returns the
feature of maximum mutual information
of point $1$
with at most
\begin{align}
M \le \sum_{i=2}^{n} \left( \left( \frac{24 \sigma^2}{\Delta_i^2} \log (nd) \right) \wedge 2n \right)
\end{align}
effective number of computations (which is the total number of points used in estimating the mutual information).
This takes
\begin{align} \label{eq:knn_proof_time}
O\left( \sum_{i=1}^{d}\log (d) \left( \left( \frac{\sigma^2 \log (nd)}{\Delta_i^2} \right) \wedge n \right)
\right)
\end{align}
time.
\end{thm}
\begin{thm}
If we further assume that $\Delta_i \sim \mathcal{N}(\gamma, 1)$,
for some $\gamma$, and $d = c n$, for $c \in [0,1]$,
then we have that the expected number of effective number of computations (over randomness in $\Delta_i$),
\begin{align*}
\mathbb{E}[M] \le O(d \log (nd) + n).
\end{align*}
with probability $1-\Theta(\frac{1}{d^2})$ over randomness
in the algorithm of returning the
correct answer. This takes
$O(d \log (nd) \log(d) + n\log(d))$
\end{thm}
\section{Details about the sparse estimator}\label{app:sparsity}
Let the set non-zero entries of $x_0$ and $x_1$ be $S_0 \subseteq [d]$ and $S_1 \subseteq [d]$, with $|S_0| = n_0$ and $|S_1| = n_1$. We show that the sparse estimator, $\tilde{d}(x_0, x_1, k)$, is an unbiased estimator for the squared eucleadian distance between $x_0$ and $x_1$ i.e,
$$\mathbb{E}\left[\tilde{d}(x_0, x_1, k)\right] = \frac{1}{d}\sum_{j\in S_0\cup S_1}\left(x_{0,j}-x_{1,j}\right)^2,$$
For the sake of the reader's convenience we re-defining:
\begin{align} \label{eq:sparse_estimator_re}
\tilde{d}(x_0, x_1, k) &= \frac{|S_0| }{2d}(x_{0,t_k^{(0)}}-x_{1,t_k^{(0)}})^2 (1+ \mathbb{I}(x_{1,t_k^{(0)}}=0)) +
\frac{|S_1|}{2d}(x_{0,t_k^{(1)}}-x_{1,t_k^{(1)}})^2
(1+ \mathbb{I}(x_{0,t_k^{(1)}}=0)),
\end{align}
where $t_k^{(0)}$ is uniformly sampled from $S_0$, and $t_k^{(1)}$ is uniformly sampled from $S_1$.
Taking the expectation of the first term in the RHS of equation \eqref{eq:sparse_estimator_re} we obtain
\begin{align}
\mathbb{E}\Big[\frac{|S_0|}{2d}(x_{0,t_k^{(0)}}-x_{1,t_k^{(0)}})^2 &(1+ \mathbb{I}(x_{1,t_k^{(0)}}=0)) \Big] \nonumber \\
&= \mathbb{E}\left[\frac{|S_0|}{2d}(x_{0,t_k^{(0)}}-x_{1,t_k^{(0)}})^2\right] +\mathbb{E}\left[\frac{|S_0|}{2d}(x_{0,t_k^{(0)}}-x_{1,t_k^{(0)}})^2\mathbb{I}(x_{1,t_k^{(0)}}=0))\right]\nonumber \\
&\overset{(a)}{=} \frac{1}{2d}\sum_{j\in S_0}\left(x_{0,j}-x_{1,j}\right)^2+\frac{1}{2d}\sum_{j\in S_0 -
S_1}\left(x_{0,j}-x_{1,j}\right)^2,\label{eq:spare_estimator_expectation_1}
\end{align}
where (a) follows because $t_k$ is uniformly sampled from set $S_0$. Similarly for the second term in equation \eqref{eq:sparse_estimator_re} we have
\begin{align}\label{eq:spare_estimator_expectation_2}
\mathbb{E}\left[\frac{|S_1|}{2d}(x_{0,t_k^{(1)}}-x_{1,t_k^{(1)}})^2 (1+ \mathbb{I}(x_{0,t_k^{(1)}}=0)) \right] &= \frac{1}{2d}\sum_{j\in S_1}\left(x_{0,j}-x_{1,j}\right)^2+\frac{1}{2d}\sum_{j\in S_0-S_1}\left(x_{0,j}-x_{1,j}\right)^2,
\end{align}
Using equations \eqref{eq:spare_estimator_expectation_1} and \eqref{eq:spare_estimator_expectation_2} in equation \eqref{eq:sparse_estimator_re}, we obtain
\begin{align*}
\mathbb{E}\left[\tilde{d}(x_0, x_1, k)\right] =& \frac{1}{2d}\sum_{j\in S_0}\left(x_{0,j}-x_{1,j}\right)^2+\frac{1}{2d}\sum_{j\in S_0-
S_1}\left(x_{0,j}-x_{1,j}\right)^2\\
&+ \frac{1}{2d}\sum_{j\in S_1}\left(x_{0,j}-x_{1,j}\right)^2+\frac{1}{2d}\sum_{j\in S_0-
S_1}\left(x_{0,j}-x_{1,j}\right)^2\\
=& \frac{1}{d}\sum_{j\in S_0\cup S_1}\left(x_{0,j}-x_{1,j}\right)^2
\end{align*}
\subsection{The Update step}
We begin by noting
that sparse vectors are usually stored in
a data-structure with two structures:
a vector $v_{\text{data}}$ storing the non-zero entries
and
a dictionary (or unordered map)
with maps the co-ordinate in the original
vector to the co-ordinate in the $v_{\text{data}}$.
An entry with $0$ is not in the dictionary.
We have that we can check membership of key in a
dictionary as well find the value of the key
if it is present in the dictionary
in $O(1)$ amortised time. Further we can
generate a random key in the dictionary
also in $O(1)$ time.
\section{The modified UCB Algorithm} \label{app:UCB}
Let $\mathcal{A}_{\ell}, \ell \in [n]$ is a set of arms
defined by the routine (say $k$-nearest neighbors) that calls the function \texttt{UCB}.
The routine calling
function \texttt{UCB} also defines the methods to pull arms, update
the estimates of their mean, and construct confidence
intervals around these estimates. Further,
it also provides
a method to evaluate the means
of the arm exactly.
Let $\mu_i$ be
the true value of the mean of arm $i$. Let $\hat{\mu}_i(t)$
be the estimate of the mean of arm $i$ at the $t$-th iteration
of the algorithm. Let
$\left[\hat{\mu}_i(t) - C_i(t),\hat{\mu}_i(t) + C_i(t)\right]$
be the $(1-\delta)$-confidence interval for $\mu_i$
maintained
at the $t$-th iteration of the algorithm.
The \texttt{UCB} algorithm defined below is essentially
the UCB of \citet{LaiRob} with an extra condition
of evaluating the mean of the arm exactly
and setting the widths of the confidence interval
to $0$ if it
is pulled more than MAX\_PULLS times. $k$ is the
number of best arms returned by the algorithm.
\begin{algorithm}
\caption{\texttt{UCB} ($\{\mathcal{A}_{\ell}: \ell \in [n]\}, \text{MAX\_PULLS}, k$) \label{alg:genUCB}}
\begin{algorithmic}[1]
\State For each arm $\{\mathcal{A}_{\ell}: \ell \in [n]\}$,
compute a $(1-\delta)$ confidence intervals
by $\log (n)$ steps of the estimator to obtain:
$[\hat{\mu}_j(1)- C_j(1),\hat{\mu}_j(1)- C_j(1) ]$
\State $\mathcal{B} = \{ \}$ \Comment{Set of $k$ best arms}
\State $\mathcal{S} = \{\mathcal{A}_{\ell}: \ell \in [n]\}$
\Comment{Set of arms under consideration}
\While {TRUE}
\State At iteration $t$, pick arm $\mathcal{A}_t$ that
minimises $\hat{\mu}_j(t-1)-C_j(t-1)$ among arms in $\mathcal{S}$.
\If {arm $\mathcal{A}_t$ is
evaluated less than $\text{MAX\_PULLS}$ times}
\State improve the confidence interval
and mean estimate of the arm $\mathcal{A}_t$ by updating the the estimator
by one more step.
\Else
\State Set
$\hat{\mu}_{\mathcal{A}_t}(t)$ to be brute force evaluation
of the mean of the arm and set
$C_{\mathcal{A}_t}(t)=0$.
\EndIf
\If {Arm $\mathcal{A}_t$ is
such that $\forall \mathcal{A}_i \neq \mathcal{A}_t$,
$\hat{\mu}_{\mathcal{A}_t}(t) + C_{\mathcal{A}_t}(t) < \hat{\mu}_{\mathcal{A}_i}(t) - C_{\mathcal{A}_i}(t)$}
\State Add $\mathcal{A}_t$ to $\mathcal{B}$
\State Remove $\mathcal{A}_t$ from the set of
arms under consideration, that is
$\mathcal{S} = \mathcal{S} - \{\mathcal{A}_t \}$.
\If{$|\mathcal{B}| = k$}
\Return $\mathcal{B}$
\EndIf
\EndIf
\EndWhile
\end{algorithmic}
\end{algorithm}
\input{app_empirical}
\section{More examples in adaptive Monte Carlo optimization framework}\label{app:examples}
\paragraph{The medoid finding problem of
\citet{BagKamNtrZhaTse} : } The problem
they considered was finding medoids.
Given a set of $n$ points in $\mathbb{R}^d$,
$x_1, x_2, \cdots, x_n$ the medoid is defined
as the point with the smallest average
distance from all other points in the ensemble
(with respect to some distance).
For
simplicity of exposition we only present
this in the context of $\ell_1$ distance.
\footnote{We discuss how other distances fall in our
framework in Appendix \ref{app:ext}.}
\begin{align*}
\min_{i \in [n]} f(i) &= \min_{i \in [n]} \frac{1}{n} \sum_{j=1}^n \sum_{k=1}^d |x_{i,k}-x_{j,k}|
\end{align*}
\citet{BagKamNtrZhaTse} assume that for any point $i$,
the distances to all other points is $\sigma$
sub-gaussian (where $\sigma$ is an absolute constant)
which allows them to get the estimator,
\begin{align*}
\hat{f}_{\ell}(i) &= \frac{1}{\ell} \sum_{j=1}^{\ell}
\sum_{k=1}^d |x_{i,k}-x_{t_j,k}|,
\end{align*}
where $t_j$ is uniformly sampled from $[n]-\{i\}$
for $j \in [\ell]$.
The sub-Gaussianity assumption gives that this
estimators $\hat{f}_{\ell}(i) \sim \mathcal{N}(f(i), \frac{\sigma^2}{\ell})$. They pose the problem as
a multi-armed bandit problem with each point $i$
being an arm, and pulling an arm corresponding
to updating the estimator, which can be done in
constant time. They then run the UCB
algorithm of Appendix \ref{app:UCB} and report that
this gives considerable computational savings.
\paragraph{Monte-carlo tree search of \citet{KocSze}: } Here
the authors aim to to
carry out planning to decide the best possible policy at
at state with respect to a finite horizon. The
classical monte-carlo planning algorithms sample actions
in each state and run the simulations for some fixed time
and pick the action with best rewards.
The authors propose treating each action as an arm and
running a bandit at each state to decide on the
best action.
\paragraph{Hyper-parameter tuning of
\citet{JamTal} and \citet{LiJamEtAl}
: }We note that the hyperparameter tuning
example of \citet{JamTal} and \citet{LiJamEtAl} can also be
put into this framework with a twist. As discussed
before, the authors are trying to find the hyperparameters
with minimum validation error for a deep neural
network. Each configuration of hyperparameters
is an arm, and pulling an
arm corresponds to running training for an epoch
and measuring change in validation error.
However, unlike our other examples, we it is not
easy to come up with confidence intervals around
the estimate, which makes the UCB algorithm
untenable. However a successive elimination based
algorithm like that of \citet{EveManMan} can
provide computational savings here, and
\citet{LiJamEtAl} present an algorithm similar
in flavour.
\paragraph{Non-convex optimzation : }We
note that non-convex optimization
algorithms, especially in alternating
minimization based algorithms
where one starts with
different starting points and picks the one with
the best objective can in general be put in such
a framework, and one can use successive elimination
to obtain computational savings. This falls into
the framework of \citep{JamTal,LiJamEtAl}.
\paragraph{Others: } Quickhull is a divide and conquer based method to compute convex hull of $n$ points. The divide step of this algorithm involves first finds farthest points on each side of a hyperplane. One could improve the running time of this step by posing it in the framework of adaptive Monte Carlo optimization. Graph centrality measures such as betweenness centrality, closeness centrality can also be modeled in the adaptive Monte Carlo framework.
\section{An extension: When we have a nice function
of a good estimator}\label{app:ext}
Consider a smooth
function $g : \mathbb{R} \rightarrow \mathbb{R}$. We can generalise the adaptive Monte Carlo optimization
framework of Eq. \eqref{eq:prob_formulation_intro}
to solve
\begin{align*}
\min_{i \in \mathcal{I}} g(f(i))
\end{align*}
Let $\hat{f}_l(i)$ be an estimator of $f(i)$.
We use the delta method
to approximate
\begin{align}\label{eq:delta_prob_formulation}
\sqrt{n}\left(g\left(f(i)\right) - g(\hat{f}_l(i))\right) &\approx
\sqrt{n} g'(\hat{f}_l(i)) (\hat{f}_l(i) - f(i))
\end{align}
which gives us that,
\begin{align} \label{eq:delta_method}
\sqrt{n}\left(g(f(i)) - g(\hat{f}_l(i))\right) &\rightarrow
\mathcal{N}(0, n g'(\hat{f}_l(i))^2 \text{Var}(\hat{f}_l(i)))
\end{align}
This allows us to construct confidence intervals
around $g$ using estimators of $f$. This
allows us to extend the
objective to additive functions
composed with smooth functions.
If $g$ is assumed to be uniformly continuous
in the closure of the range of $f$, we have
that $|g'| \le M$ for some known constant $M$
(independent of $n$).
We further assume that
$|g''| \le \kappa$ for
some known constant $\kappa$
(independent of $n$).
For $g(i)$, $i\in\mathcal{I},$ we define the sequence of estimators $\{\hat{g}_l(i)|l\in \mathbb{N}\}$ where $\hat{g}_l(i) := g(\hat{f}_l(i))$ and whose confidence
intervals can be obtained using equation \eqref{eq:delta_method}. We then use this in the
adaptive Monte Carlo optimization framework
to solve \eqref{eq:delta_prob_formulation}.
We note that the estimators $\hat{g}_l(i)$
are are not necessarily unbiased and can have a bias of
approximately $\kappa \text{Var}(\hat{f}_n(x))$.
An example here would be to compute medoids
on some distance like the $\ell_2$ distance.
\input{app_knn}
\input{app_kmeans}
\input{app_hierarchical}
\input{app_mmi}
\input{app_jl}
\section{Discussions}\label{sec:disc}
We show in our empirical and theoretical results that adaptive Monte-Carlo optimization is a powerful and broad framework that can provide significant computational savings in many basic machine learning problems.
Some generalizations of the framework
are discussed in Appendix \ref{app:ext}.
Modern datasets are often large in both sample size and dimensions.
Our algorithm provides gains in one of the two.
In applications such as $k$-nearest neighbors
as KGraph \citep{KGraph},
Lanczos bisection based methods \citep{fast-knn},
Locality sensitive hashing
based methods \citep{AndInd, AndIndEtAl}
which provides savings in the sample size, whereas
our algorithm provides savings in the dimensions. Integrating both these approaches to obtain savings in both sample size and dimension is an open question.
\section{Maximum Mutual Information Feature Selection} \label{sec:entropy}
In all applications
we considered so far,
$f(i)$ was a simple function and
constructing confidence intervals for estimates
of $f(i)$ was easy.
Here we consider the case where $f(i)$
is itself a sophisticated estimator of
an underlying statistical quantity: mutual information,
in the context of maximum mutual information feature
selection.
A reasonably popular
feature selection
procedure is Maximum Mutual Information Criterion \citep{YanMoo,PenLonDin}.
In this scheme, given a set of features
and a target, one picks feature that
has the highest estimated mutual information
with the target as the most relevant feature.
Approaches to pick multiple features
include Max-Relevance Min-Redundancy \citep{PenLonDin},
Joint Mutual Information \citep{YanMoo} or Chow-Liu trees
\citep{ChowLiu} etc. Here, we will focus on applying the adaptive Monte Carlo optimization framework to the problem of picking a single most informative feature. The framework can also be applied accelrate the approaches which pick multiple features on a sequential basis, such as building Chow-Liu trees.
As before, we have an $(n \times d)$ data matrix $X$
corresponding to $n$ points each with $d$ features. Let us
also assume we have an $n$-vector $Y$ corresponding
to the target.
Here we focus on picking the single most relevant
feature to the target.
\footnote{If both the features and the target are discrete,
then one can use the plug-in estimators
and put the problem into our framework as well.}
If both the features and the target are continuous,
a clever non-parametric function of \citet{KozLeo}
is often used used to empirically estimate
mutual information.
On putting it into the framework (with min replaced by max), every
feature is an arm. $f(i)$ is an
empirical mutual information of the
feature $i$ with $Y$.
We next describe the $f(i)$ used in this application.
We then rely on the estimator
of differential entropy by \citet{KozLeo}
to obtain $f(i)$.
Let $Z_1, \cdots, Z_k$ be $k$ iid samples of a
random variable. Define
$R_i = \min_{j \in [k] - \{ i \}} \|Z_j-Z_i\|_2.$
Then the Kozachenko-Leonenko estimator
is given by
$\hat{h}_k (\{Z_1,\cdots, Z_k\}) = \frac{1}{k} \sum_{i=1}^{k} \log R_i + c_d,$
for an absolute constant $c_d$ depending
only on dimension of the random variable.
Given an iid sequence of $k$ pairs of random variables
$\{(W_1,Z_1), \cdots, (W_k, Z_k)\}$, we
estimate the mutual information between
the two as
\begin{align*}
\hat{I}_k(\{W_i\}_{1}^k; \{Z_i\}_{1}^k)
&= \hat{h}_k(\{W_i\}_{1}^k)
+ \hat{h}_k( \{Z_i\}_{1}^k)
-\hat{h}_k(\{(W_i,Z_i)\}_{1}^k)
\end{align*}
$f(i)$ and it's estimators $\hat{f}_{\ell}(i)$ are given by:
\vspace{-0.09in}
\begin{align*}
f(i) &= \Hat{I}_n(\{X_{t,i}\}_{t=1}^{n}, \{Y_{t}\}_{t=1}^{n}),\\
\hat{f}_{\ell}(i) &= \Hat{I}_{\ell}(\{X_{t_k,i}\}_{k=1}^{\ell}, \{Y_{t_k}\}_{k=1}^{\ell})
\end{align*}
We obtain
the confidence intervals using a central limit theorem
derived for this estimator \citep{DelFou,BicBre}.
While updating the estimators cheaply is not
as trivial as in previous applications, it
can be done cheaply as explained in Appendix \ref{app:mmi}.
We consider the
Greenhouse gas sensor dataset of \citet{ghgdata}
obtained from the UCI repository \citep{UCIrep}.
This consists of greenhouse gas (GHG) concentrations
measured on $2921$ sensors. We consider one of the
sensors as the target and empirically find that we get
a saving of around $20$\texttt{x} in the effective
number of
computations (which is the total number of points
used in estimating the mutual information) to
compute maximum mutual information as shown in Figure
\ref{fig:knn-intro}$(c)$. \vspace{-3mm}
\section{Agglomerative Algorithms for Hierarchical clustering}\label{sec:hierarchical}
We next illustrate
incorporating
the adaptive monte carlo optimization
framework into sequential algorithms to
obtain computational savings. We use hierarchical
clustering as an example.
Hierarchical clustering is an algorithm that is
widely used in data analysis
\multicitep{[Chapter 7]{LesRajUll}}.
This is especially popular
if the number of clusters is not known apriori, or the
data has hierarchical structure.
Given a set of $n$ points and
a distance measure
hierarchical clustering finds a
binary tree
with $n$ leaves
to represent the data with
each leaf corresponding to a point,
and
the distance
between two points
along the tree
showing the dis-similarity
between them.
A class of algorithms for hierachical
clustering are \textit{agglomerative
algorithms} which combine
points in a bottom up manner.
Such algorithms start of with the $n$ clusters,
each corresponding to a point. At every
step, the algorithm merges the two ``closest''
clusters to form a single cluster
until there is only one cluster remaining.
This naturally sets up a binary tree representation
where the each leaf is a cluster with $1$ point, the
root is the cluster of all points, and all
intermediate vertices are clusters formed at
intermediate steps. The two children of any vertex in
the tree are the clusters that were merged to form
the that cluster.
For two clusters which consist of only one
point, the notion of
closeness is unambiguously defined by the distance
metric. For clusters with more than one point,
one way to define the
distance between clusters is the \textit{average linkage}
which is the average distance between points in the cluster.
Each step of this algorithm computes
the pair of clusters with smallest distance
between them. Each step can be posed as a bandit problem
with the arms being each pair of clusters in the system
at that step.
If we consider squared euclidean
distances as our distance metric
(other distances can be handled
using the trick
discussed in Appendix \ref{app:ext}),
then
pulling an arm corresponds to evaluating the
square of
difference of two co-ordinates in two points,
randomly chosen each from one of the two clusters corresponding
to the arm. More detailed analysis is presented in
Appendix \ref{app:heirarchical}.
On running this on the tiny-imagenet validation
dataset of $10k$
$64\times 64$ RGB images, we obtain a gain of
more than $30$\texttt{x} over the naive algorithm
in terms of the effective number of distance computations as
shown in Figure \ref{fig:knn-intro}.
\begin{note}
The algorithm reduces the running time of the
algorithm from $O(n^2(d + \log (n)) )$ to
$O(n^2 \log^2 (nd))$ under some
regularity assumptions discussed
in Appendix \ref{app:heirarchical}.
\end{note}
\vspace{-3mm}
\section{Introduction}
The use of random sampling to efficiently estimate a deterministic quantity dates back to the Buffon's needle experiment of Mario Lazzarini in 1901, and was later developed into the celebrated {\em Monte-Carlo method} by Stanislav Ulam. In computer science, this idea has been applied to develop techniques such as
locality-sensitive hashing
\citep{GioIndMot} and sketching \citep{AloMatSze}.
More recently it has been used for fast
approximations of matrix operations like
matrix multiplication, low rank approximation
and matrix decomposition \citep{DriKanMah1,DriKanMah2,DriKanMah3},
to approximate hessians for convex
optimization
\citep{PilWai16, PilWai17}.
These techniques find wide applications in machine learning.
In this paper, we consider the use of random sampling to solve discrete optimization problems of the form:
\begin{align}\label{eq:prob_formulation_intro}
\min_{i \in \mathcal{I}} f(i)
\end{align}
where $\mathcal{I}$ is a large but finite set and $f$ is an expensive-to-compute function. A direct application of the Monte-Carlo method to this problem is to generate enough samples to estimate each value $f(i)$ accurately and then compute the minimum of the estimates. However, this is computationally inefficient since the values $f(i)$'s which are much larger than the minimum need not be estimated as accurately as the values $f(i)$'s which are closer to the minimum. Instead, a more efficient procedure is to first estimate all the $f(i)$'s crudely using few samples and then {\em adaptively} focus the sampling on the more promising candidates. We term this general approach {\em adaptive Monte-Carlo optimization}.
Instances of this general approach have already been appeared in machine learning:
\begin{itemize}[leftmargin=*]
\item The Monte-Carlo tree search method \citep{OR_folks,KocSze} to solve large-scale Markov decision problems. Here, the set $\mathcal{I}$ is the set of all possible actions at a state $s$ and $f(i) = Q(i,s)$, the expected value of the total reward by taking action $i$ at state $s$ and then following the optimal policy. The problem of interest is estimating the action that maximizes reward, i.e. solving (\ref{eq:prob_formulation_intro}) (but with minimization replaced by maximization) at some state. The Monte-Carlo tree search method is an improvement over the non-adaptive Monte-Carlo planning method of \citet{KeaManNg}, and is a central component of modern reinforcement learning systems such as AlphaZero \citep{alpha0}.
\item Adaptive hyper-parameter tuning for deep neural networks \citep{JamTal,LiJamEtAl}. Here, the set $\mathcal{I}$ consists of the possible hyper-parameter configurations and $f(i)$ is the validation performance of the neural network under the $i$th hyper-parameter configuration. The problem is to find the best hyper-parameter configuration using the validation data.
\item Computing the medoid (exemplar) of a large number $n$ of points in high-dimensional space \citep{BagKamNtrZhaTse}. Here, the set $\mathcal{I}$ consists of all the points in the dataset and $f(i)$ is the average of the distances $d_{ij}$'s of point $i$ to all the other points, i.e.
\vspace{-2mm}
$$f(i) = \frac{1}{n-1}\sum_j d_{ij}.$$
The medoid is the point in the dataset that minimizes the average distance to the other points.
\end{itemize}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{intro.pdf}
\caption{Performance gain of adaptive Monte Carlo optimization algorithms over
the brute force algorithm
in terms of number of computations in four applications: (a) $k$-nearest
neighbors graph construction, (b) $k$-means, (c) hierarchical clustering
(d) minimum mutual information feature selection as a function of number
of points and the dimensions. We see a gain roughly linear in
dimensions in the first three and linear in
sample size in the last, consistent with the theoretical results in Table \ref{tab:comp}.
For $k$-nearest neighbors, $k$-means and
hierarchical clustering we use the tiny-imagenet
dataset.
For Maximum Mutual information we use the UCI Green-house Gas dataset.
More details are in Appendix \ref{app:empirical}.
}\label{fig:knn-intro}
\vspace{-3 mm}
\end{figure}
\begin{table}
\centering
\caption{Scaling of naive algorithms and
those designed using the adaptive Monte Carlo
framework with
regularity assumptions on the data.
$n$: number of points, $d$: dimensions/feature size.
In each application we obtain a gain in one of $n$ or $d$
as indicated in the fourth column.
This is also reflected empirically in
Fig \ref{fig:knn-intro}. }\label{tab:comp}
\begin{tabular}{llll}\hline
Application & Brute time & Our running time & Appx. Gain\\ \hline
$k$-nearest neighbors & $O(n^2d)$ & $O\left(n^2\log^2(nd)+knd \log(n) \right)$ & $ O(d)$ \\
$k$-means (an assignment step) & $O(nkd)$ & $O(nk \log (dn)\log(k))$ & $ O(d)$ \\
Hierarchical clustering& $O(n^2d)$ & $O\left(n^2\log^2(nd)\right)$ & $ O(d)$ \\
Max mutual information & $O(nd)$ & $O\left(d\log(nd\right)+n)$ & $ O(n)$ \\\hline
\end{tabular}
\vspace{-3 mm}
\end{table}
Each of these three problems is solved by a reformulation into a {\em multi-armed bandit} problem, where each arm corresponds to each element in the set $\mathcal{I}$, each arm's reward corresponds to $f(i)$, and each pull of arm $i$ corresponds to generating a sample to update the estimate of $f(i)$. For example, in the medoid problem, each pull of the arm $i$ corresponds to the random sampling of another point $j$ and obtaining a distance $d_{ij}$ from the point $i$,
and then using it to update the estimate of the average distance of point $i$ to the other points. Well-known algorithms developed for multi-arm bandits like the upper confidence bound (UCB) algorithm \citep{LaiRob} can be used to derive good adaptive sampling strategies. Computation savings of one or more orders-of-magnitude can sometimes be obtained by using such an algorithm. For example, under some mild conditions on the the distances, the computation cost of the medoid of $n$ points can be reduced from a naive exact computation cost of $O(n^2)$ to a cost of $O(n \log n)$ using the UCB algorithm with a high probability correctness guarantee \citep{BagKamNtrZhaTse}. To achieve within an accuracy of $1+\epsilon$ times the minimum, such an algorithm only needs $O(1/\epsilon)$ distance computations in contrast to $O(1/\epsilon^2)$ computations needed by a non-adaptive sampling method \citep{EppWan}.
Curiously, although the solution approaches for these three problems are similar, their individual development is very tied to their specific context and is done in isolation of each other. The goal of the present paper is to advocate a broader and more unified view of adaptive Monte-Carlo optimization as a general framework which can be applied to a wide range of other problems in machine learning and beyond. Such opportunities are enhanced particularly by the fact that modern datasets often consist of not only large number of samples but also a large number of features, or dimensions. Thus, both computation across all features for a sample as well as a computation across all samples for a feature can be expensive, and their costs can be significantly ameliorated by adaptive Monte-Carlo optimization.
We support our thesis by applying the framework to develop new methods for several problems and evaluate their performance empirically
on large datasets (Figure \ref{fig:knn-intro}) as well as theoretically (Table \ref{tab:comp}).
\begin{itemize}[leftmargin=*]
\item \textit{Nearest neighbor} :
The basic problem of computing the nearest
neighbor of a point among $n$ points in $d$ dimensional space is expensive if both $n$ and $d$ are large. For many common distance measures which are additive, i.e. the distance between two points $x$ and $y$ can be written as:
$$ \rho(x,y) = \sum_{j=1}^d \rho_j(x_j,y_j),$$
where $x_j,y_j$ is the $j$th component of the point $x$ and $y$ respective, the adaptive
Monte-Carlo optimization framework can be applied. The set $\mathcal{I}$ is the set of all contending points, $f(i)$ is the distance of point $i$ to the point of interest, and each sample for $f(i)$ is an evaluation of the distance from point $i$ to the point of interest on a randomly selected coordinate. We apply this accelerated nearest neighbor subroutine to improve $k$-nearest neighbors graph construction as well as the assignment step of Lloyd iteration in the $k$-means algorithm.
\item \textit{Hierarchical
clustering} :
Hierarchical clustering algorithms takes $n$ points in $d$-dimensional space
and then deriving a binary tree
representing the closeness among points.
Agglomerative algorithms start off with $n$ singleton
clusters each corresponding to point and
in every step the two closest
clusters are merged to form a new cluster, until there is one cluster left.
Finding the two closest clusters to merge at each step can be
put into the adaptive Monte-Carlo optimization
framework.
At step $t$ of the algorithm,
$\mathcal{I}$ is the set of all pairs of
remaining clusters at that step, $f(i)$ is the
average distance between points in the $i$th pair of clusters. Between two clusters $C_1$ and $C_2$:
$$ f(i) = \frac{1}{|C_1| |C_2|}\sum_{x \in C_1, y\in C_2} \rho(x,y) = \frac{1}{|C_1| |C_2|}\sum_{x\in C_1, y\in C_2} \sum_j \rho(x_j, y_j),$$
assuming that the distance measure is additive. Each sample for $f(i)$ can be obtained by randomly sampling along {\em both} points and co-ordinates.
This application is an example of a dynamic algorithm in which each step is
an adaptive Monte-Carlo optimization but the estimates
are carried forward from one step to the next.
\item \textit{Maximum mutual information
feature selection} :
Consider a supervised learning problem with $n$ training samples and $d$ features plus one
target variable. The features and the target variables are both continuous-valued.
The goal is to select the
feature with the maximum mutual information
to the target.
We pose this as an adaptive Monte-Carlo optimization, with
$\mathcal{I}$ as the set of $d$ features.
Here $f(i)$ is an empirical estimate of the mutual information between the feature $i$
and the target, calculated from all the sample points. Because the features are continuous-valued, this estimate is not in an additive form across the sample points. Nevertheless we can randomly sample and efficiently update estimates of $f(i)$.
This example shows that even complicated non-additive $f(i)$'s
can be put into the adaptive Monte-Carlo optimization framework. Moreover, unlike the previous two examples where the set
$\mathcal{I}$ represents the sample points and the computation of $f(i)$ is along the features (dimensions), here $\mathcal{I}$ represents the features and the computation of $f(i)$ is along the sample points. It shows the broad applicability of the framework.
\end{itemize}
In Sections \ref{sec:knn}-\ref{sec:entropy},
we pose the applications discussed above in the adaptive
Monte Carlo optimization framework and derive algorithms to
solve them. In Section \ref{sec:sparsity} we discuss
leveraging sparsity of the dataset in the adaptive
Monte Carlo optimization framework. We then
conclude with some discussion in Section \ref{sec:disc}.
\section{$k$-nearest neighbors graph construction and $k$-means} \label{sec:knn}
\subsection{$k$-nearest neighbor graph construction}
\textit{$k$-nearest neighbor graph construction}
is the problem of constructing
$k$-nearest neighbor graphs from $n$
unlabelled data points with respect to some distance
metric \citep[Chapter 13]{HasTibFri}.
This is used as a subroutine in many unsupervised
learning algorithms such as
Iso-map \citep{TenSilLan}, Local linear
Embedding \citep{RowLaw},
Hessian Eigenmaps \citep{DonGri},
some algorithms in spectral clustering
\citep{LinMisEtAl}
among others.
In recent years there has been a lot of work in
construction of
$k$-nearest neighbor graphs
on high dimensional data such as that on
the YFCC100M dataset
in \citet{JohDouHer}.
While dimensionality
reduction is often helpful in applications like
embedding graphs \citep{wang2016structural}
and
videos \citep{ramanathan2015learning},
the dimensions of the space embedded in is also
in a few thousand, which makes constructing
$k$ nearest neighbor graphs
a problem of interest in high dimensions.
For low dimensions there are some heuristics
that are known to perform well such as
$k$-$d$ trees \citep{BentleyKDTrees} and
ball-trees \citep{OmoBallTree}.
One popular algorithm
there is the KGraph \citep{KGraph} which is a
heuristic algorithm
improving over \citet{DonChaLi}.
In
this section we consider the $k$-nearest
neighbor graph construction problem, pose
it in the adaptive Monte Carlo optimization framework,
and then
evaluate performance of the algorithm
we obtain.
Let us assume that we
have an $(n \times d)$ data matrix $X$ corresponding
to $n$ points: $x_1, \cdots, x_n$ $\in \ \mathbb{R}^d$.
Our goal is to find the nearest neighbor
graph in $\ell_p$
distance. As a subroutine, let us consider finding
the $k$ nearest neighbors of point $1$. Without
loss of generality let us consider $\ell_2$ distances
We consider
each of the points $\{2, \cdots, n\}$ as arms.
As the $k$-nearest neighbors
with $\ell_2$ distances is the same as finding
$k$-nearest neighbors with squared euclidean
distances, we consider $f(i)$ as
\vspace{-0.08in}
\begin{align*}
f(i) &= \frac{1}{d} \sum_{t=1}^{d} (x_{1,t}- x_{i,t})^2,
\end{align*}
and the objective is to find $k$
points with the
smallest $\ell_p$ distances to point $1$.
Here and in applications to
follow, we
design a sequence of estimators for $f(i)$ with
increasing accuracy:
$\{ \hat{f}_{\ell}(i) \}$,
such that: 1) we can use evaluations of
$\hat{f}_{\ell}(i)$ to construct confidence
intervals on $f(i)$
(which often involves making some regularity
assumptions about the data), and 2) updating
$f_{\ell}(i)$ to $f_{\ell+1}(i)$ is computationally
cheap.
An estimator after $\ell$ pulls of
arm $i$ (for squared euclidean distances) would be
\vspace{-0.08in}
\begin{align}\label{eq:knn1}
\hat{f}_{\ell} (i) &= \frac{1}{\ell} \sum_{k=1}^{\ell}
(x_{1,t_k}- x_{i,t_k})^2,
\end{align}
\vspace{-0.04in}
where $t_1, \cdots, t_{\ell}$ are uniformly
sampled from $[d]$ with replacement.
The update of the
estimator at step $\ell+1$ is
\begin{align}\label{eq:knn_update}
\Hat{f}_{\ell+1} (i) = \frac{\ell}{\ell + 1} \Hat{f}_{\ell} (i)
+ \frac{1}{\ell + 1} (x_{1,t_{\ell+1}}-x_{i, t_{\ell+1}})^2,
\end{align}
which takes $O(1)$ time, where $j_{\ell+1}$ is sampled
from $[d]$, uniformly at random.
We further assume that the the estimator
of Eq. \eqref{eq:knn1} has $\sigma$-
sub-Gaussian tails \footnote{$\sigma$ being independent
of $n$} which allows us to construct confidence intervals.
Figure \ref{fig:potpourri} (b)
shows that such an assumption is reasonable \footnote{In practice we
estimate sigma from some random samples at the beginning of the run and update it as the algorithm progresses. Details in
Appendix \ref{app:empirical}.}.
Thus we have reformulated the $k$-nearest neighbor
as a multi-armed bandit problem using the adaptive
Monte Carlo optimization framework.
Throughout this manuscript,
we restrict ourselves
to using a slightly modified version of
the standard
Upper-Confidence Bound algorithm of \citet{LaiRob}
to solve the bandit problem.
We note that in reformulations obtained
from the adaptive Monte Carlo optimization
framework, there exists
an integer
MAX\_PULLS such that
the
total computation required to
compute $\hat{f}_{\text{MAX\_PULLS}}(i)$
(over all updates) is
equal to the computation required to
compute $f(i)$ exactly.
We thus
evaluate $f(i)$ exactly if
arm $i$ is pulled MAX\_PULLS times and
set the confidence intervals to be of zero
width giving us
the only modification we make to the
algorithm of \citet{LaiRob}.
Given a set of $n$ arms $\mathcal{A}$,
the MAX-PULLS, the number of
arms to be returned $k$
and a procedure to update them to
construct confidence intervals
we present this Algorithm \ref{alg:genUCB} of
Appendix \ref{app:UCB} and call it \texttt{UCB}.
We find $k$-nearest neighbors
using the algorithm
defined in Algorithm \ref{alg:UCB-knn}.
This leverages the \texttt{UCB}
sub-routine of Appendix \ref{app:UCB}.
\begin{algorithm}
\caption{\texttt{k-nearest-neighbors UCB} \label{alg:UCB-knn}}
\begin{algorithmic}[1]
\State $n$ points in $\mathbb{R}^d$:
$x_1,\cdot, x_n$ are input to the algorithm.
\For{$i \in [n]$ }
\State Consider arms $\mathcal{A}_{\ell}, 1 \le \ell \ne i \le n$
with estimators defined as in Eq. \eqref{eq:knn1}
and confidence intervals constructed using $\sigma$-
sub-gaussianity.
\State neighbors of $x_i$ are given by \texttt{UCB}$(\{\mathcal{A}_{\ell}, 1 \le \ell \ne i \le n\},d,k)$ \Comment{MAX\_PULLS $= d$}
\EndFor
\end{algorithmic}
\end{algorithm}
Let $\mu^*_k$ be the distance
of the $k$-th nearest neighbor to point $1$.
Let $\mu_i$ be the distance of point $i$ from point $1$.
Define,
$\Delta_i = \max(0, \mu_i - \mu^*_k)$. $\Delta_i$
is $0$ for the $k$ nearest neigbours of point $1$
and for the other points it measures how much
larger than $\mu^*_k$ their distance to point $1$ is.
In the
\begin{thm} \label{thm:knn1}
With probability $1-\frac{1}{n^2}$,
Algorithm \ref{alg:UCB-knn} returns the
$k$-nearest neighbors of point $1$ in
$O\left( \sum_{i=2}^{n}\log (n) \left( \left( \frac{\sigma^2 \log (nd)}{\Delta_i^2} \right) \wedge d \right)
\right)$
time.
\end{thm}
\begin{thm}\label{thm:knnapprox}
We can modify Algorithm \ref{alg:UCB-knn}
to obtain an algorithm, which with probability
$1-\frac{1}{n^2}$
finds the $k$-nearest neighbor of point $1$ upto
a $(1+\epsilon)$-approximation is
$O\left( \sum_{i=2}^{n} \log(n)\left( \frac{\sigma^2 \log (nd) }{\left(\max(\Delta_i,\epsilon \mu_{k}^*)\right)^2} \right) \right)$
\end{thm}
\begin{note}
The running time of the above
algorithm
for finding $k$-nearest
neighbors of all $n$ points
is $O(nk \log (dn)\log(k))$
under some natural assumptions on
distribution of $\Delta_i$
which is again better than
the running time of the naive algorithm
which is
$O(n^2 d )$. Detailed analysis is
in Appendix \ref{app:knn}.
\end{note}
\begin{figure}
\centering
\includegraphics[width=0.99\linewidth]{figure2.pdf}
\caption{(a) This shows the gain of our algorithm \ref{alg:UCB-knn} and that
obtained by KGraph on the tiny-imagenet dataset, (b) shows
the gain obtained by our algorithm \ref{alg:UCB-knn}
on sparse gene dataset of $10$\texttt{x}-genomics. (c) Histogram of co-ordinate wise distances for randomly chosen pair of points in the
dense dataset (imagenet) and sparse dataset (10x genomics).}\label{fig:potpourri}
\vspace{-0.5cm}
\end{figure}
\vspace{-0.5cm}
\paragraph{Empirical Performance :}
Since the bandit based algorithms do not
compute exact distances between any two points,
we measure the performance of algorithms in terms of
\textit{effective number of distance evaluations}.
We define effective number of distance evaluations
between two points
as
the fraction of co-ordinates involved
in estimating the distance between
these two points. The
effective number of distance evaluations
of an algorithm is the sum of
effective number of distance evaluations between every
pair of points in the algorithm.
For example, note that the effective
number of distance evaluations to
find the distance between two vectors exactly is $1$
as all the co-ordinates are used.
In Figure \ref{fig:knn-intro} $(a)$, we show the
gain of the bandit based algorithm
over the brute-force algorithm in
terms of the effective number of distance evaluations
on the tiny-imagenet training
data. We vary the number of images
from $10k$ to $100k$ and their dimensions
from $700$ to $12k$.
We obtain a gain of more than $100$\texttt{x} over
the naive algorithm.
We also notice that the gain over the naive algorithm
increases almost linearly
with dimensions, giving us that this algorithm behaves
in an almost dimension free manner. We hasten to add
that the gain in absolute running time in our proof-of-concept implementation (which does not have systems level optimization)
was around $2$\texttt{x}, but there is scope of
significant improvement
with more careful cache management and parallelization.
A state of the art algorithm for computing the $k$-nearest
neighbor graph is KGraph \citep{KGraph}.
We compare our
performance with KGraph on the tiny-imagenet dataset in
Figure \ref{fig:potpourri} $(a)$.
KGraph and the bandit based algorithm
operate in two fundamentally different
ways.
While KGraph improves its effective number of distance
evaluations by computing the distances between fewer
pairs of
points using the fact that neighborhoods of
neighboring points have large
intersections
(giving it good scaling with
the $n$), our algorithm
improves its effective number of distance evaluations
by computing distances between all pairs of points
approximately
(giving it good scaling with $d$).
Combining the two approaches to improve scaling
with both $n$ and $d$ is an avenue for further research.
\begin{note}
Another natural algorithm for computing $k$-nearest
neighbors would be to first project the data into
lower dimensional space using Johnson-Lindenstrauss
and then computing $k$-nearest neighbors in the
lower dimensional space. We discuss connections
to this in Appendix \ref{app:jl}
\end{note}
\subsection{$k$-Means}\label{sec:kmeans}
The canonical algorithm for computing $k$-means:
Lloyd's algorithm \citep{Llo82}, starts with $k$ initial
centroids and then iteratively alternates between two steps:
\textit{the assignment step }: where each of the $n$ points
is assigned to the centroid closest to it, and
\textit{the update step }:
where the centroids are updated.
For a standard implementation
every iteration of $k$-means,
the assignment step takes $O(kd)$
time per point and the update step is much
cheaper and
takes $O(d)$ time per point, where
$d$ is the ambient dimension
of the $n$ points.
We note that one can view the assignment step of $k$ means
as finding the nearest neighbor of each of
the $n$ points
among the $k$ centroids. For each point,
this can thus be posed as a nearest
neighbor problem with $k$ arms.
Under mild assumptions, this algorithm has a running
time of $O((k \log (dn)) \log(k))$ per point.
We analyse this formally in Appendix \ref{app:kmeans}.
As shown in Figure \ref{fig:knn-intro} $(b)$, we note that
we get an improvement of 30-50\texttt{x}
in terms of effective
number of distance evaluations compared
to the naive algorithm
on
the tiny-imagenet dataset, with $k=100$.
\vspace{-3mm}
\section{Improved Estimators for Sparse Datasets} \label{sec:sparsity}
In this section we illustrate that
the performance of algorithms from
our framework can be improved significantly
by designing better estimators that exploit
structure in the data using sparsity
as an example.
Algorithms based
on the adaptive Monte Carlo
optimization framework
adaptively sample
a fraction of data to obtain estimates
of
$f(i)$ to solve the minimization problem.
Often we have to deal with sparse datasets.
Ideally we would
want to design estimators $\hat{f}_{\ell}(i)$ to
leverage the sparsity and
only sample the non-zero entries of the data.
For example in the single cell
gene dataset of \citet{10xdata}, only
$7\%$ of the entries are non-zero,
even though the points live in $28k$
dimensions.
Consider estimating
the squared euclidean distance between sparse points
$x_0$ and $x_1$ lying in $d$ dimensional space with
$n_0$ and $n_1$
non-zero co-ordinates respectively.
Rather than sampling over all co-ordinates,
we would like to sample co-ordinates which
are non-zero in either one of the two points.
However computing the the set union
of the non-zero co-ordinates of $x_0$ and $x_1$,
which would take $O(n_0 + n_1)$ time neutralizing
any gain obtained using our framework.
We
circumvent
this
expensive computation by using the following estimator
which does not use the set union.
\vspace{-0.08 in}
\begin{align*}
\hat{d}_{\ell}(x_0,x_1) &= \frac{1}{\ell} \sum_{k=1}^{\ell} \tilde{d}(x_0, x_1,k),
\end{align*}
\vspace{-0.04 in}
where the $\tilde{d}(x_0, x_1, k)$ is obtained by first
sampling
$t_k^{(0)}$, $t_k^{(1)}$ from the non-zero
co-ordinates of $x_0$ and $x_1$ and
\vspace{-0.08 in}
\begin{align*}
\tilde{d}(x_0, x_1, k) &= \frac{n_0}{2d}(x_{0,t_k^{(0)}}-x_{1,t_k^{(0)}})^2 (1+ \mathbb{I}(x_{1,t_k^{(0)}}=0)) +
\frac{n_1}{2d}(x_{0,t_k^{(1)}}-x_{1,t_k^{(1)}})^2
(1+ \mathbb{I}(x_{1,t_k^{(1)}}=0))
\end{align*}
\vspace{-0.04 in}
A more elaborate way of thinking about this estimator,
is that at each step we follow the following procedure:
\begin{enumerate}
\item Sample a non-zero co-ordinate of $x_0$ and $x_1$. Let
the co ordinates obtained be $t_k^{(0)}$ and $t_k^{(1)}$ respectively.
\item Check if $t_k^{(0)}$ is a non-zero co-ordinate of $x_1$.
Based on this compute
\begin{align*}
\tilde{d}^{(0)}(x_0, x_1, k) &= \begin{cases}
\frac{n_0}{2d}(x_{0,t_k^{(0)}}-x_{1,t_k^{(0)}})^2 & \text{ if }x_{1,t_k^{(0)}}\ne 0,\\
\frac{n_0}{d}x_{0,t_k^{(0)}}^2 & \text{ if }x_{1,t_k^{(0)}}= 0.
\end{cases}
\end{align*}
$\tilde{d}^{(1)}(x_0, x_1, k)$ is similarly computed and
depends upon if $t_k^{(1)}$ is a non-zero co-ordinate of $x_0$.
\item We thus have that,
\begin{align*}
\tilde{d}(x_0, x_1, k) &= \tilde{d}^{(0)}(x_0, x_1, k) + \tilde{d}^{(1)}(x_0, x_1, k).
\end{align*}
\end{enumerate}
We next claim that the the estimator is unbiased.
Let the set non-zero co-ordinates
of $x_0$ and $x_1$ be $S_0 \subseteq [d]$
and $S_1 \subseteq [d]$, with $|S_0| = n_0$ and $|S_1| = n_1$.
To see this note that, as $t_k^{(0)}$ is sampled
uniformly from the $S_0$,
every element of $S_0$ is picked with
probability $\frac{1}{n_0}$.
Thus, we have that
\begin{align*}
\mathbb{E}[\tilde{d}^{(0)}(x_0, x_1, k)] &=
\frac{1}{d} \sum_{j\in S_0 - S_1} (x_{0,j})^2 + \frac{1}{2d} \sum_{j\in S_0 \cap S_1} (x_{0,j} - x_{1,j})^2.
\end{align*}
With an analogous evaluation of $x_1$, we have that,
\begin{align*}
\mathbb{E}[\tilde{d}(x_0, x_1, k)] &= \frac{1}{d} \sum_{j\in S_0 - S_1} (x_{0,j})^2 + \frac{1}{d} \sum_{j\in S_0 \cap S_1} (x_{0,j} - x_{1,j})^2 +
\frac{1}{d} \sum_{j\in S_1 - S_0} (x_{1,j})^2,\\
&\overset{(a)}{=} \frac{1}{d} \sum_{j\in S_0 \cup S_1} (x_{0,j} - x_{1,j})^2,\\
&= \frac{1}{d} \sum_{j\in [d]} (x_{0,j} - x_{1,j})^2,
\end{align*}
which is the distance between the two points.
We note that $(a)$ follows from the definition of $S_0$
and $S_1$.
We note that this estimator thus needs only two
non-trivial operations: $1)$ sampling
from the non-zero co-ordinates of a sparse vector, and $2)$
checking if a co-ordinate is non-zero in a sparse vector.
To see that these can be done in $O(1)$ time, we elaborate
on how sparse vectors are usually stored in
standard implementations.
Sparse vectors consist of a data structure
a data-structure with two object:
a vector $v_{\text{data}}$ storing the non-zero entries
and
a dictionary (or unordered map)
with maps the co-ordinate in the original
vector to the co-ordinate in the $v_{\text{data}}$.
We have that we can check membership of key in a
dictionary as well find the value of the key
if it is present in the dictionary
in $O(1)$ amortised time. Further we can
generate a random key in the dictionary
also in $O(1)$ time. This gives us that
$\tilde{d}$ can be computed in
$O(1)$ time.
As shown in Figure
\ref{fig:potpourri}$(c), (d)$, we
obtain $3$\texttt{x} gain
over brute force
in the effective
number of distance evaluations for $k$-nearest
neighbours and $k$-means on
the dataset of \citet{10xdata}.
The baseline considered here takes
sparsity into account.
We also observe if we had used the estimator of Section
\ref{sec:knn}
we would not improve over
the brute force algorithm. \vspace{-3mm}
|
1,941,325,220,544 | arxiv | \section{Introduction}
The Standard Model (SM) provides a successful description of physics up to the weak scale. However, it provides some 18 parameters which are input by hand to fit experiment data. Most of these input parameters are associated with flavor physics and are included to parameterize the fermion mass hierarchy, Cabibbo-Kobayashi-Maskawa (CKM) angles, and neutrino oscillations. Many theories, either supersymmetric (SUSY) or non-supersymmetric, are constructed to address the flavor problem and, hopefully, make predictions on new physics. Among these theories beyond the Standard Model (SM), supersymmetric grand unification provides an elegant framework that explains not only the gauge quantum numbers of fermions transforming under the SM gauge group $SU(3)_C\times SU(2)_L\times U(1)_Y$, but also the prediction of $\alpha_s(M_Z)$. This remarkable success of the prediction of $\alpha_s(M_Z)$ motivates further exploration of SUSY grand unification \cite{SUSYunification}.
Among the ideas of grand unification, gauge groups such as $SU(5)$, $E_6$, and $SO(10)$ are frequently used in GUT model construction \cite{SUSYGUT,GUTbook}. However, there are reasons that make $SO(10)$ theories more attractive than others. First, $SO(10)$ is the smallest group in which all matter fields in one family can fit into one irreducible representation. Second, the two light Higgs doublets needed in any SUSY theory fit into one {\bf 10} of SO(10). This allows the Yukawa couplings of up-type and down-type quarks to be determined by Clebsch-Gordan coefficients, thus making $SO(10)$ theories more predictive.
There is a problem with this approach, however. Typical SUSY $SO(10)$ models need to use Higgs fields in higher representations, the {\bf 126} or {\bf 45}, to achieve successful GUT relations for Yukawa matrices. These representations are complex in their own right, and theories which contain tensor fields of rank higher than two cannot be constructed from the simplest string-derived GUT theories \cite{StringyGutReview,GGfromString}. This motivates the use of extended GUT gauge groups such as $G\times G$ or $G\times G\times G$, where $G$ denotes the usual GUT group, in SUSY GUT model construction \cite{GG,GGG,Chou555}.
Supersymmetric GUT models based on the gauge groups $SO(10) \times SO(10)$ and $SO(10)\times SO(10)\times SO(10)$ have been discussed in the literature \cite{GG,GGG}. In these models, the breaking of the GUT gauge group was done when fundamental Higgs fields in the $(10, 10)$ representation, acquire their vacuum expectation values (VEVs) along the embedded diagonal subgroup directions of $SO(10)\times SO(10)$ and $SO(10)\times SO(10)\times SO(10)$, while the spinorial Higgs fields $\Psi_i, \bar \Psi_i$ acquire VEVs along $SU(5)$-preserving directions. Four sets of the $(10, 10)$ fields carrying charges of different discrete symmetries were introduced; the large number of fields is needed not only to achieve the desirable Higgs doublet-triplet splitting, but also give the desirable asymmetry between the up and down quark mass matrices. As a result, typical predictions of SUSY GUT $SO(10)$ models, such as the top-bottom Yukawa unification $\lambda_t = \lambda_b$, and Clebsch-Gordan relations in Yukawa matrices are not valid in their models.
In this paper, we follow the idea of using $SO(10)\times SO(10)$ and $SO(10)\times SO(10)\times SO(10)$ as the SUSY GUT gauge groups. However, we show that the traditional merits of the SUSY GUT $SO(10)$ models can be preserved in our $SO(10)\times SO(10)$ and $SO(10)\times SO(10)\times SO(10)$ model construction. Although it is motivated from the string constructions, our model construction is self-contained and does not make explicit reference to string theories. In our models, all Higgs fields are in the fundamental representations of the gauge groups and no rank two tensors of any $SO(10)$ gauge group are required.
In section \ref{sec:EffAd}, we show that the extended GUT gauge group breaking can be implemented when Higgs fields acquire VEVs along diagonal $SO(10)_D$ directions, diagonal $SU(5)_D\times U(1)_D$ directions, or other diagonal directions. Most importantly, we argue that the effective adjoint fields for each $SO(10)_i$ group can be formed by combining two VEV-acquiring Higgs fields. In section \ref{sec:SUSY1010}, we construct an explicit model based on $SO(10)\times SO(10)$. We show that the Higgs doublet-triplet problem is naturally solved through the Dimopoulos-Wilczek mechanism \cite{DWmechanism} without destabilizing the gauge hierarchy. The doublet-triplet splitting mechanism also guarantees strong suppression of proton decay, since the contributions from heavy Higgsino triplet exchange diagrams are absent or highly suppressed. We also show that this model gives Yukawa matrices of the type similar to Georgi-Jarlskog ansatz. An explicit model which was analyzed by Anderson {\it et al.} \cite{OperatorSO10} is constructed by using effective adjoint operators. In section \ref{sec:SUSY101010}, we present an $SO(10)\times SO(10)\times SO(10)$ model with each family of matter multiplets transforming under different $SO(10)$ groups. In section \ref{sec:Conclusion10}, we make our conclusion.
\section{Effective adjoint operators for $SO(10)$}
\label{sec:EffAd}
As pointed out in the literature \cite{GG,GGG}, the breaking of extended GUT gauge groups $G\times G$ and $G \times G \times G$ can be achieved by a set of Higgs fields in the fundamental representation. For example, an $SO(10)_1 \times SO(10)_2$ model breaks down to its diagonal subgroups when fields in the fundamental representation $(10, 10)$ develop VEVs. We will denote $(10, 10)$ fields in this paper as $S$ or $Z$ depending on the VEV patterns. We denote fields with the following three canonical patterns of VEVs $<S_X>$, $<S_{B-L}>$, $<S_{T_{3R}}>$, corresponding to
\begin{eqnarray}
<S_X>={v_D \over \sqrt{10}} \cdot {\scriptsize ( \begin{array}{cc}
1&0\\
0&1
\end{array} )}
\otimes \mbox{diag}(1,1,1,1,1) \\
<S_{B-L}>={v_G \over \sqrt{10}} \cdot {\scriptsize ( \begin{array}{cc}
1&0\\
0&1
\end{array} )}
\otimes \mbox{diag}(a,a,a,0,0) \\
<S_{T_{3R}}>={v_G \over \sqrt{10}} \cdot{\scriptsize ( \begin{array}{cc}
1&0\\
0&1
\end{array} )}
\otimes \mbox{diag}(0,0,0,b,b).
\label{SVEVs}
\end{eqnarray}
\noindent The VEVs of $S_X$, $S_{B-L}$, and $S_{T_{3R}}$ break $SO(10)_1\times SO(10)_2$ down to its embedded diagonal subgroups $SO(10)_D$, $SO(6)_D \times SO(4)_1 \times SO(4)_2$, and $SO(6)_1\times SO(6)_2\times SO(4)_D$ respectively. Usually, a tree level superpotential has many SUSY vacua which include the VEVs in Eq. (\ref{SVEVs}); a typical form includes
\begin{equation}
W \supset {\lambda M \over 2}\mbox{Tr}(SS^T)+{A \over 4M}(\mbox{Tr}(SS^T))^2+{B \over 4M}\mbox{Tr}(SS^TSS^T).
\label{SVEVW}
\end{equation}
However, there are other SUSY vacua which lie along the direction of the embedded diagonal $SU(5)_D\times U(1)_D$, or other directions such as $SU(3)_D\times U(1)_D\times SO(4)_1\times SO(4)_2$ and $SO(6)_1\times SO(6)_2\times SU(2)_D\times U(1)_D$. We denote the associated $(10, 10)$ fields as $Z$ and again refer to the VEV patterns using subscripts:
\begin{eqnarray}
<Z_X>&=&{v_{10} \over \sqrt{10}} \cdot {\scriptsize ( \begin{array}{ccc}
0&1\\
-1&0
\end{array} )}
\otimes \mbox{diag}(2,2,2,2,2) \nonumber \\
<Z_{B-L}>&=&{v_5 \over \sqrt{10}} \cdot {\scriptsize ( \begin{array}{ccc}
0&1\\
-1&0
\end{array} )}
\otimes \mbox{diag}(2a/3,2a/3,2a/3,0,0) \nonumber \\
<Z_{T_{3R}}>&=&{v_5 \over \sqrt{10}} \cdot{\scriptsize ( \begin{array}{ccc}
0&1\\
-1&0
\end{array} )}
\otimes \mbox{diag}(0,0,0,b/2,b/2).
\label{ZVEVs}
\end{eqnarray}
As an alternative to generating scales or VEVs by minimizing a tree level superpotential, it has been shown that the scale $<S_X>$ could also be dynamically generated through a strongly coupled supersymmetric dynamics \cite{DynamicalScale}. Following the same line of thinking, we can introduce two supersymmetric gauge groups $SU(N_c)$ and $Sp(n_c)$ with fields in their fundamental representations $q(N_c,1,10,1)$, ${\bar q}({\bar N_c},1,1,10)$, and $Q(1,2n_c,1,10)$, where the numbers in brackets denote the dimensionality of each field under the two strong groups and the GUT gauge group $SO(10)_1 \times SO(10)_2$. With the imposition of some discrete symmetry, say $Z_N \times Z_K$, that keeps the field $S_X$ from coupling directly to $Z_X$, the lowest order of tree level superpotential is given by
\begin{eqnarray}
W_{tree} = {A\over{2NM^{2N-3}}}S_X^{2N}+{B\over{2KM^{2K-3}}}Z_X^{2K}+\lambda_1 S_X q{\bar q}+{\lambda_2 \over M} (S_X Z_X)^{ab}Q^aQ^b \label{ZStree}
\end{eqnarray}
\noindent where $A$ and $B$ are coefficients, $M$ is the superheavy scale or Planck scale, and $\lambda_i$ denote the dimensionless coupling constants. To this should be added the dynamical superpotential resulting from the strong dynamics,
\begin{eqnarray}
W_{dyn}={C\over{N_c-10}}\left[ {\Lambda_1^{3N_c-10} \over \mbox{det}q{\bar q} } \right]^{1\over {N_c-10}}+{D \over {n_c+1-5}}\left[ {\Lambda_2^{3n_c -2} \over \mbox{Pf}(QQ) } \right]^{1 \over {n_c -4}}.
\label{StrongPotential}
\end{eqnarray}
\noindent By stabilizing the superpotential in Eqs (\ref{ZStree}) and (\ref{StrongPotential}) along the $<S_X>$ and $<Z_X>$ directions, we obtain the following equations for the VEVs:
\begin{eqnarray}
{A\over M^{2N-3}}s^{2N-1}+{5C \over (n_c+1) s}\left[{\lambda_2 sz\over M\Lambda_2}\right]^{5\over{n_c+1}}\Lambda_2^3+{10D \over N_c s}\left[{\lambda_1 s\over \Lambda_1}\right]^{10\over N_c}\Lambda_1^3=0 \label{StrongVEV1}\\
{B\over M^{2K-3}}z^{2K-1}+{5C \over (n_c+1) z}\left[{\lambda_2 sz\over M\Lambda_2}\right]^{5\over{n_c+1}}\Lambda_2^3=0,
\label{StrongVEV2}
\end{eqnarray}
\noindent where $s=v_D/\sqrt{10}$ and $z=2v_{10}/\sqrt{10}$. It is easily seen that solving Eq.s (\ref{StrongVEV1}) and (\ref{StrongVEV2}) could lead to nonzero $v_{D}$ and $v_{10}$ when either one of the following conditions is satisfied:
\begin{eqnarray}
{2(n_c+1) \over N_c} &>& 1 + {10 \over N_c}, \hspace{0.5cm} \mbox{or} \\
{2(n_c+1) \over N_c} &\leq& 1.
\label{NonzeroVEV}
\end{eqnarray}
Given VEVs of the $S$ and $Z$ fields, we can form effective rank two tensors which carry quantum numbers of the gauge group $SO(10)_2$ by combining any two of the $S$ and $Z$ fields. In this way, we can form effective adjoint operators of $SO(10)_2$, which we call $\Sigma$ and $\Sigma '$, given by
\begin{eqnarray}
\Sigma^{bc}_X &\equiv& {1\over M}\mbox{Tr}_1(Z^T_XS_X)={1\over M}Z^{ab}_XS_X^{ac}, \nonumber\\
\Sigma^{bc}_{B-L}&\equiv& {1\over M}\mbox{Tr}_1(Z^T_{B-L}S_X)={1\over M}Z^{ab}_{B-L}S_X^{ac}, \nonumber \\
\Sigma^{bc}_{T_{3R}}&\equiv& {1\over M}\mbox{Tr}_1(Z^T_{T_{3R}}S_X)={1\over M}Z^{ab}_{T_{3R}}S_X^{ac}, \nonumber\\
{\Sigma'}^{bc}_{B-L} &\equiv& {1\over M}\mbox{Tr}_1(S^T_{B-L}Z_X)={1\over M}S^{ab}_{B-L}Z_X^{ac}\ \nonumber \\
{\Sigma'}^{bc}_{T_{3R}} &\equiv& {1\over M}\mbox{Tr}_1(S^T_{T_{3R}}Z_X)={1\over M}S^{ab}_{T_{3R}}Z_X^{ac}
\label{operator45}
\end{eqnarray}
\noindent We can also form effective identity operators of $SO(10)_2$, such as $I={1\over M}\mbox{Tr}_1(S_X^T S_X)$ or $I'={1\over M}\mbox{Tr}_1(Z_X^T Z_X)$. Reciprocally, we can form effective adjoint and identity operators of $SO(10)_1$. All of these effective tensors can arise physically from integrating out heavy states which transform under one of the $SO(10)$'s. For example, we can generate the structure $\mbox{Tr}_1(Z^TZ')$ by integrating out the heavy states $10_1$ and $10_1 '$ from the following superpotential:
\begin{eqnarray}
M_110_110_1'+10_1Z10_2+10_1'Z'10_2' \longrightarrow {1\over M_1}10_2\mbox{Tr}_1(Z^TZ')10_2'
\label{OP45}
\end{eqnarray}
\noindent Once we are equipped with these effective rank two tensors, it is possible to construct supersymmetric GUT models with realistic fermion masses and CKM angles. A systematic analysis of the construction of $SO(10)$ GUT models has been done by Anderson {\it et al.} \cite{OperatorSO10}. Our treatment, with the $\Sigma$, $\Sigma'$, $I$ and $I'$ effective fields, now maps directly onto that analysis.
\section{A SUSY $SO(10)\times SO(10)$ GUT model}
\label{sec:SUSY1010}
In this section, we present an example based on the $S_X$ and $Z$ VEVs which demonstrates that typical SUSY $SO(10)$ GUT predictions can actually be preserved in $SO(10)_1 \times SO(10)_2$ gauge theories with experimentally acceptable Yukawa matrices. We assume four fundamental Higgs fields $S_X$, $Z_X$, $Z_{B-L}$, and $Z_{T_{3R}}$ of representation dimensionality $(10, 10)$ in our $SO(10)\times SO(10)$ GUT model. We construct the superpotential so that each of the $(10, 10)$ Higgs fields acquires a VEV along the indicated direction as described in Section \ref{sec:EffAd}.
\subsection{Higgs doublet-triplet splitting}
The Higgs structure is constructed by the requirement of Higgs doublet-triplet splitting. Higgs triplets, if they are not heavy enough, could contribute to the evolution of the gauge couplings, and thus spoil the unification of the gauge couplings. In addition, Higgsino triplets may also mediate fast proton decay. So we might begin by analyzing the constraints imposed by the splitting mechanism.
In conventional $SO(10)$ models, Higgs triplet fields may acquire heavy masses by coupling to the adjoint fields which have their VEVs along the $B-L$ direction
\begin{eqnarray}
W(H_1, H_2)&=&H_1 A H_2, \hspace{1cm} \mbox{with} \nonumber \\
<A>&=&V\cdot {\scriptsize ( \begin{array}{ccc}
0&1\\
-1&0
\end{array} )} \cdot \mbox{diag} (1,1,1,0,0),
\label{DWform}
\end{eqnarray}
\noindent where $H_1$ and $H_2$ are the fundamental Higgs fields, and $A$ denotes the adjoint Higgs field which acquires its VEV of the Dimopolous-Wilczek forms. As seen from Eq. (\ref{DWform}), the triplet fields in $H_1$ and $H_2$ get heavy masses $V$ and splitted from their doublet partners.
In our $SO(10)\times SO(10)$ model, among the four fundamental Higgs, $Z_{B-L}$ and $Z_{T_{3R}}$ acquire their VEVs of the Dimopoulos-Wilczek (DW) forms through the stabilization of a tree level superpotential as in Eq. (\ref{SVEVW}). However, the DW forms of VEVs may be seriously destabilized when some cross coupling terms, such as $\mbox{Tr}(Z_{B-L}^T Z_{T_{3R}}) \equiv Z_{B-L}^{ab} Z_{T_{3R}}^{ab}$, $ \mbox{Tr}(Z_X^T Z_{B-L})$, $\mbox{Tr}(Z_X^T Z_{T_{3R}})$, and $\mbox{Tr}(Z_X^T Z_{B-L})$ are present in the superpotential. For instance, the presence of the term $\mbox{Tr}(Z_X^T Z_{T_{3R}})$ would destabilizes the gauge hierarchy in $Z_{T_{3R}}$ since the F-flatness condition $\mbox{F}_{Z_{T_{3R}}}=0$ would give a term proportional to $Z_X$. As a result, these cross coupling terms must be excluded to implement the DW mechanism for the Higgs doublet-triplet splitting problem. Although SUSY allows unwanted superpotential terms to be dropped by hand, it is less arbitrary to forbid them by a discrete symmetry.
Barr \cite{Stability55} has suggested that a discrete symmetry may do the job of forbidding the above cross coupling terms. In our model, there is a possible choice $K=Z_2^{T_{3R}} \times Z_2^{T_{3R}'} \times Z_2^{B-L}\times Z_5^1 \times Z_5^2$, and under which the various $Z$ fields transform as
\begin{eqnarray}
Z_2^{T_{3R}}&:& \hspace{0.6cm} Z_{T_{3R}} \rightarrow -Z_{T_{3R}} \nonumber \\
Z_2^{T_{3R}'}&:& \hspace{0.6cm} Z_{T_{3R}} \rightarrow -Z_{T_{3R}} \nonumber \\
Z_2^{B-L}&:& \hspace{0.6cm} Z_{B-L} \rightarrow -Z_{B-L} \nonumber \\
Z_5^1&:& \hspace{0.6cm} S_X \rightarrow e^{6 \pi i/5} S_X \nonumber \\
Z_5^2&:& \hspace{0.6cm} (S_X, Z_X) \rightarrow e^{2\pi i/5}(S_X, Z_X).
\label{Zsymmetry}
\end{eqnarray}
\noindent The $Z_2^{B-L}$ and $Z_2^{T_{3R}}$ symmetries in Eq. (\ref{Zsymmetry}) are designed to forbid the dangerous cross coupling superpotential terms noted above but still allow the coupling terms at the quartic level
\begin{eqnarray}
\mbox{Tr}(Z_{B-L}Z_{B-L}^T)\mbox{Tr}(Z_{T_{3R}} Z_{T_{3R}}^T)&,& \hspace{0.5cm} [\mbox{Tr}(Z_{B-L} Z_{T_{3R}}^T) ]^2 \nonumber \\
\mbox{Tr}(Z_{B-L}Z_{B-L}^T Z_{T_{3R}} Z_{T_{3R}}^T)&,& \hspace{0.5cm} \mbox{Tr}(Z_{B-L}Z_{T_{3R}}^T Z_{B-L}Z_{T_{3R}}^T).
\label{QuarticTerms}
\end{eqnarray}
\noindent The terms in Eq. (\ref{QuarticTerms}) might change the values of the scales appearing in $<Z_{B-L}>$ and $<Z_{T_{3R}}>$, but they do not destabilize the DW forms of VEVs. The last two terms of Eq. (\ref{QuarticTerms}) would have zero contribution to the F-flatness conditions. However, they remove the would-be Goldstone modes that are not eaten by the gauge bosons in the fields $Z_{B-L}$ and $Z_{T_{3R}}$. The $Z_2^{T_{3R}'}$ discrete symmetry prevents the effective identity operator $\mbox{Tr}_1(Z_X^T Z_{T_{3R}})$ from coupling to the spinorial superheavy states $\Psi_1$ and $\bar \Psi_7$ in our model. However, this $Z_2^{T_{3R}'}$ symmetry is basically construction-dependent and may not be necessarily introduced into our $SO(10)\times SO(10)$ model. We will come to this point again when discussing fermion spectrum in the next subsection. In general, the symmetry $K$ would keep fields $S_X$ and $Z_X$ from coupling to $Z_{B-L}$ and $Z_{T_{3R}}$ up to a very high order, {\it e.g.} $\mbox{Tr}(Z_XZ_{B-L})\mbox{Tr}(Z_XZ_{B-L}S_X^8)$ as implied by Table \ref{tab:1010model}. Thus the DW forms of VEVs are protected up to corrections of the order of the weak scale when the GUT gauge group breaking parameters $v_D/M$, $v_{10}/M$ and $v_5/M$ are sufficiently small.
An explicit superpotential giving Higgs doublet-triplet splitting by the above mechanism is:
\begin{equation}
W_{DT}=10_H Z_{B-L} 10_{H'}+{1\over M}10_{H'}S_XZ_{T_{3R}}10_{H''}+X10_{H''}10_{H''}
\label{DTpotential}
\end{equation}
\noindent where $10_H$, $10_{H'}$ and $10_{H''}$ denote Higgs fields in (1,10), (10,1), (10,1) representations respectively. $X$ is a gauge singlet\footnote%
{The singlet $X$ may or may not be the effective rank two tensor fields $I$ or $I'$ depending on how the $K$ symmetry is chosen in our model.}
that acquires a GUT scale VEV and this makes $10_{H''}$ superheavy. The introduction of the singlet $X$ is required by the fact that if $10_{H''}10_{H''}$ is a singlet and present in superpotential, so is the non-renormalizable term $S_XS_X10_{H'}10_{H'}$. This term $S_XS_X10_{H'}10_{H'}$, if exists, will give superheavy mass to the triplet states living in $10_{H'}$ and generates heavy Higgsino triplets exchange diagrams that mediate proton decay and spoil the strong suppression of proton decay. As in generating the effective rank two tensors, the non-normalizable term in Eq. (\ref{DTpotential}) may rise from integrating out heavy states in the (1, 10) representation. The insertion of the field $S_X$ in this term is designed to protect $10_{H}$ from coupling $10_{H''}$ to a high order level. In order to achieve DW mechanism, these Higgs fields must transform non-trivially under the discrete symmetry $K$. In general, there are many possible ways of assigning $K$ charges to all fields in our model. One assignment for the $K$ charges is given and can be found in Table \ref{tab:1010model}
Here it is clear that the discrete symmetry $Z_5^1 \times Z_5^2$ would play the role of forbiding unwanted terms in superpotential. According to Table \ref{tab:1010model}, the Higgs mass terms $M_{HH}10_H10_H$ and $M_{H'H'}10_{H'}10_{H'}$ are forbidden by this discrete symmetry up to $(<X S_X^8>/M^8+<X^3S_X^2Z_X^2/M^6>)10_H10_H$ and $<S_X^2X^4>/M^{5}10_{H'}10_{H'}$ respectively, and $M_{HH''}10_H10_{H''}$ are very highly suppressed by the discrete symmetry $K$. Therefore, up to the order of weak scale, the mass matrix $M_{H_T}$ and $M_{H_D}$ for Higgs triplets and doublets are given as
\begin{eqnarray}
M_{H_T}&=&\left({\small \begin{array}{ccc}
0&<Z_{B-L}>&0\\
<Z_{B-L}>&{<S_X^2X^4> \over M^5}&0\\
0&0&<X>
\end{array}} \right), \nonumber \\
M_{H_D}&=&\left( {\small \begin{array}{ccc}
0&0&0\\
0&{<S_X^2X^4> \over M^5}&<{S_X Z_{T_{3R}}\over M}>\\
0&<{S_X Z_{T_{3R}}\over M}>&<X>
\end{array} }\right).
\label{DTmatrices}
\end{eqnarray}
\noindent Therefore, as from Eq. (\ref{DTmatrices}), only one pair of the doublets in $10_{H}$ would remain light after the breaking of the GUT gauge group. However,
as seen from Eq. (\ref{DTmatrices}), one pair of the heavy Higgs triplets may receive a GUT scale $M_G\sim v_5$ mass, while the corresponding Higgs doublets fields receive a mass $v_D^2 v_5^2/<X>$ which is less than the scale $v_5$. This may affect the gauge unification in our model depending upon the scale hierarchy between the two masses. Actually, this mass discrepancy results from forbidding dangerous high order nonrenormalizable operators which also contribute to the mass matrices $M_{H_T}$ and $M_{H_D}$. If we assume that only renormalizable terms in superpotential are allowed at the superheavy scale $M$ and all high order terms are generated from the Heavy Fermion Exchange mechanism (HFE) \cite{HFE}, than the scale ratio $<S_X>/M$ can be of order $O(1)$ thus all heavy Higgs states, doublets and triplets, would have GUT scale masses $m\sim v_5$ and we have the gauge unification as that of the minimal supersymmetric standard model (MSSM). In this paper, we simply assume negligible effects on gauge unification caused by the mass discrepancy among the heavy Higgs multiplets.
Finally, we discuss the implications for proton decay. Eq. (\ref{DTmatrices}) also implies a strong suppression of proton decay. Since the high order operator $S_X^2X^410_{H'}10_{H'}/M^5 $ is present, then the dimension 5 operators (dimension 4 in superpotential) that mediate proton decay are formed by exchanging heavy Higgsino triplets
\begin{equation}
{\lambda \over M^*}QQQL, \label{PdecayStrength},
\end{equation}
\noindent with the effective mass $M^*$
\begin{equation}
M^*\approx {M^5 <Z_{B-L}>^2 \over <S_X^2X^4>} \sim 10^{31} \mbox{ GeV} >> M_{pl}.
\label{PdecaySterngth}
\end{equation}
\noindent Here we use $Q$ and $L$ to represent the associated quarks and leptons in proton decay processes. The estimated value for $M^*$ in Eq. (\ref{PdecayStrength}) is obtained by assuming $M\sim M_{pl}$, $<Z_{B-L}>/<S_X> \sim 10^{-2}$, and $<X>/M\sim 10^{-4}$. The coupling strength parameter $\lambda \sim 10^{-7}$ comes from multiplying the associating Yukawa coupling constants in the color-Higgs exchange Feynman diagrams. To saturate current experiment limits on proton decay \cite{ParticleBook},
, the coupling strength $\lambda/M^*$ for the dimension 5 operators should be no large than about $10^{-24} \mbox{ GeV}^{-1}$. Obviously, the estimated strength in Eq. (\ref{PdecayStrength}) is far more less than the limit, therefore proton decay is highly suppressed in our model.
\subsection {Fermion masses}
Anderson {\it et al.} \cite{OperatorSO10} showed that, with adjoint operators $\Sigma$ in a SUSY GUT $SO(10)$ gauge theory, experimentally acceptable fermion mass spectrum as well as CKM angles can be obtained when these fields acquire their VEVs and break the GUT $SO(10)$ gauge group down to Standard Model gauge group. We can generate the same Yukawa matrices by using effective higher dimension operators. These can be obtained by integrating out heavy fields. Then, following the choices made by Anderson {\it et al.}, we show that viable fermion mass matrices, such as those incorporating Georgi-Jarlskog ansatz \cite{OperatorSO10,Georgi}, can be constructed in the $SO(10)_1\times SO(10)_2$ model.
We need to introduce additional heavy fields in the {\bf 16} and ${\bf {\overline {16}}}$ of $SO(10)_2$. We assume that all other matter multiplets also transform under the gauge group $SO(10)_2$. From Table \ref{tab:1010model} , it is easy to see that non-renormalizable terms at the quartic level, for instance the $\Psi_1 \mbox{Tr}_1(Z_X^T S_X) \bar\Psi_1$, are allowed to occur in our $SO(10)_1\times SO(10)_2$ model. This term may come from integrating out a pair of superheavy spinorial fields $\Psi_1'(16,1)$ and $\bar\Psi_1'(\overline {16},1)$ from the renormalizable superpotential
\begin{equation}
W \supset M_1'\Psi_1' {\bar \Psi_1'}+\Psi_1S_X{\bar \Psi_1'}+\Psi_1' Z_X {\bar \Psi_1},
\label{3to4level}
\end{equation}
\noindent where $M_1'$ denotes the super-heavy mass of $\Psi_1'$ and $\bar\Psi_1'$. At the renormalizable level with the generated quartic terms, the most general tree level superpotential consistent with the discrete symmetry $K$ in Table \ref{tab:1010model} and responsible for giving masses to quarks and leptons has the form
\begin{eqnarray}
W_{mass} &\supset& 16_316_310_H+16_3\Sigma_{B-L}{\bar\Psi_2}+16_2{\Psi_1}10_H+16_2\Sigma_X{\bar\Psi_8}+16_1\Sigma_X{\bar\Psi_3}\nonumber \\
&+&\Psi_1\Sigma_X{\bar\Psi_1}+\Psi_2\Sigma_X{\bar\Psi_2}+\Psi_2\Sigma_{B-L}{\bar\Psi_1}+\Psi_3\Sigma_X{\bar\Psi_4}+\Psi_4\Sigma_X{\bar\Psi_5}\nonumber \\
&+&\Psi_5 \Psi_6 10_H+{\bar\Psi_6}\Sigma_X\Psi_7+{\bar\Psi_7}\Sigma_X\Psi_8+X_S16_2{\bar \Psi_2} \nonumber \\
&+&\sum^8_{i=3}\Psi_i \cdot I \cdot {\bar \Psi_i},
\label{WforMass}
\end{eqnarray}
\noindent where the gauge singlet field $X_S$ is introduced to give mass to $16_2$ and $\bar\Psi_2$ when acquiring a superheavy VEV.
From Eq. (\ref{WforMass}), only the third family matter multiplet $16_3$ could get a mass of weak scale due to the discrete symmetry $K$. When the effective adjoint operators $\Sigma_X$ and $\Sigma_{B-L}$ acquire their VEVs, the spinorial fields $\Psi_i, \bar\Psi_i$ become heavy and can be integrated out in the low energy effective theory. The higher dimension operators $O_{ij}$ that give masses to matter quarks and leptons are thus generated after diagonalizing the mass matrices of these superheavy spinorial fields \cite{OperatorSO10}.
\begin{eqnarray}
O_{23}&=&16_2 10_H {\Sigma_{B-L}^2 \over \Sigma_X^2}16_3 \nonumber \\
O_{22}&=&16_2 10_H {X_S \Sigma_{B-L}\over \Sigma_X^2} 16_2 \nonumber \\
O_{12}&=&16_1 ({\Sigma_X \over I})^3 10_H ({\Sigma_X \over I})^3 16_2.
\label{SO10OPs}
\end{eqnarray}
\noindent The generation for the $O_{ij}$ operators is much easier to be seen from the diagrams in Fig.(\ref{fig:OijDiagram}). As seen from Eq. (\ref{SO10OPs}), fermion mass hierarchy is explained due to the hierarchy of the GUT breaking scales $M>v_{D}>v_{10}>v_{5}$. The effective adjoint operators $\Sigma_X$ and $\Sigma_{B-L}$ act on fermion states and give different quantum numbers to the states as described in Table \ref{tab:quantum45}. As a result, Eq. (\ref{SO10OPs}) leads to the following typical Georgi-Jarlskog ansatz for the Yukawa matrices at GUT scale
\begin{eqnarray}
M_u&=&<H>\left[ \begin{array} {ccc}
0&{1\over 27}C&0\\
{1\over 27}C&0&B\\
0&B&A
\end{array} \right], \hspace{1cm}
M_d=<\bar H>\left[ \begin{array} {ccc}
0&-C&0\\
-C&E&B\\
0&{1\over 9}B&A
\end{array} \right], \nonumber \\
M_e&=&<\bar H>\left[ \begin{array} {ccc}
0&-C&0\\
-C&3E&B\\
0&9B&A
\end{array} \right],
\label{GJMatrices}
\end{eqnarray}
\noindent with $A\approx O(1)$, $B\approx v_5^2/v_{10}^2$, $C\approx 27(v_{10}^6/v_D^6)$, and $E\approx v_5M<X_S>/(v_D v_{10}^2)$. As seen from Eq. (\ref{GJMatrices}), we have the following successful GUT relations
\begin{eqnarray}
\lambda_t=\lambda_b&=&\lambda_{\tau} \label{taubottom}\\
\lambda_{22}^u:\lambda_{22}^d:\lambda_{22}^e&=&0:1:3 \label{lambda22}\\
m_{\tau}=m_{b}, \hspace{0.5cm}, m_{\mu}&\approx& 3m_s, \hspace{0.5cm} m_d\approx 3m_e, \label{GJrelations}
\end{eqnarray}
\noindent where $\lambda$'s denote the effective Yukawa coupling constants for corresponding mass operators.
Conclusively, it is suggested that the breaking of our GUT model is arranged as
\begin{eqnarray}
SO(10)_1\times SO(10)_2 &{\stackrel {v_D}{\longrightarrow}}& SO(10)_D \nonumber \\
&{\stackrel {v_{10}}{\longrightarrow}}& SU(5)_D\times U(1)_D \nonumber \\
&{\stackrel {v_5}{\longrightarrow} }& SU(3)_C\times SU(2)_L\times U(1)_Y,
\label{breaking1}
\end{eqnarray}
\noindent with approximate ratios $v_D/M \sim 1/30$, $v_{10}/v_D\sim O(10^{-1})$, $v_5/v_{10}\sim O(10^{-1})$, and $<X_S>=<X>\approx v_Dv_5/M$. Detailed analysis for the mass operators $O_{ij}$ can be found in \cite{OperatorSO10}, and will not be discussed in this paper.
As in more familiar GUT $SO(10)$ models, we can also analyze the neutrino masses in our $SO(10)_1\times SO(10)_2$ model. First we observe that the matrix $M_{\nu^c \nu}$ for Dirac masses of neutrinos has a nonzero 22 entity also coming from $O_{22}$, and is far from identical to the up-quark mass matrix.
\begin{equation}
M_{\nu^c \nu}=<H>\cdot \left[
\begin{array}{ccc}
0&-125C&0 \\
-125C&-{6 \over 25} E&B\\
0&{9\over 25}B&A
\end{array} \right]
\label{Mneu}
\end{equation}
\noindent Since $125C$ is almost the same order of magnitude as $A$, the Dirac mass matrix for neutrino is no longer as hierarchical as quark and charged lepton mass matrices. To form Majorana mass for the right handed neutrinos, we introduce a set of spinorial Higgs fields $\Psi_{V_i}(1,16)$, $\bar \Psi_{V_i}(1, \overline {16})$ which VEVs preserve the $SU(5)_2$ subgroup of $SO(10)_2$. In general, the following neutrino mass operators can also be formed from heavy fermion exchanges
\begin{equation}
{1\over M}\sum_{i,j}(\bar \Psi_{V_i} \Gamma_a^{\over {(126)}} \bar \Psi_{V_j})(16_i \Gamma_a^{(126)} 16_j),
\label{MR}
\end{equation}
\noindent where $i, j$ are flavor indices. For simplicity, we would assume the Majorana mass matrix $M_R$ for right handed neutrinos to be a diagonal matrix with eigenmasses $m_{R_1}$, $m_{R_2}$ and $m_{R_3}$. Thus, from Eqs (\ref{Mneu}) and (\ref{MR}), the effective left handed Majorana mass matrix is
\begin{equation}
M_{\nu \nu}\approx M_{\nu^c \nu}^{+} M_R^{-1} M_{\nu^c \nu}.
\label{ML}
\end{equation}
\noindent Taking $C/E \approx 0.22 \approx 6/25$ as implied by the Cabibbo angle, it thus lead to the following three Majorana eigenmasses for left handed neutrinos
\begin{equation}
m_{\nu_{\tau}}\approx {m_t^2 \over m_{R_3}}, \hspace{0.6cm} m_{\nu_{\mu}}\approx {(125 m_d tan\beta)^2 \over m_{R_2}}, \hspace{0.6cm} m_{\nu_{e}}\approx {(125 m_d tan\beta)^2 \over m_{R_1}}.
\label{MLeigenvals}
\end{equation}
\noindent Although all the VEVs of $\Psi_{V_i}$ need not to be the same, we might take all $<\Psi_{V_i}>$ to be equal to $v_{10}$ for illustration. This gives $m_{R_i}=m_R\approx 2\times 10^{14} \mbox{ GeV}$ and leads to $m_{\nu_{\tau}} \sim 1/20 \mbox{ eV}$, $m_{\nu_u} \approx m_{\nu_e} \sim O(10^{-3} \mbox{ eV})$ for $\mbox{tan}\beta=45$. Visible ${\nu_{\tau}}-{\nu_{\mu}}$ and ${\nu_{\tau}}-{\nu_{e}}$ oscillations with very small neutrino oscillation angles $\mbox{sin}^22\theta_{\tau \mu}^{osc}$ and $\mbox{sin}^22\theta_{\tau e}^{osc}$ are favored when taking such assumption. However, other mass spectra for left handed neutrinos as well as large neutrino mixing angles may be obtained \cite {SO10neutrino,U1xFlavor}. This is because the right handed neutrino mass matrix $M_R$ may itself be nontrivial and have a hierarchical
structure in our $SO(10)\times SO(10)$ model.
\section{An SO(10) $\times$ SO(10) $\times$ SO(10) model}
\label{sec:SUSY101010}
It is straightforward to extend the GUT gauge group to $SO(10)_1\times SO(10)_2\times SO(10)_3$ and have all matter multiplets transform under one of the $SO(10)$ gauge groups. However, this extension is basically a replication of the SUSY GUT $SO(10)_1\times SO(10)_2$ model described in the previous sections. Different from the above direct generalization, in this section, we assign each matter multiplet $16_i$ to transform under different gauge group $SO(10)_i$. We also assume the existence of the three Higgs fields $10_H(1,1,10)$, $10_{H'}(1,10,1)$, and $10_{H''}(1,10,1)$, and a set of fundamental Higgs fields $S_X(1,10,10)$, $Z_X(1,10,10)$, $Z_{B-L}(1,10,10)$, $Z_{T_{3R}}(1,10,10)$, $Z_{B-L}'(10,10,1)$, and $Z_{T_{3R}}'(10,10,1)$ for implementing the DW mechanism. The complete set of assignment is shown in Table \ref{tab:103fields}.
The fundamental Higgs fields acquire their VEVs along some GUT breaking directions as described in the previous sections. To protect the DW forms of the VEVS, some discrete symmetries above the GUT scales must typically be expected to restrict possible tree level superpotential terms. Without giving the discrete symmetries explicitly, we note that the superpotential responsible for giving heavy masses to Higgs triplet fields must be restricted to the following form:
\begin{eqnarray}
10_H Z_{B-L} 10_{H'}+{1\over M}10_{H'}S_X Z_{T_{3R}}10_{H''}+X10_{H''}10_{H''},
\label{DWGGG}
\end{eqnarray}
\noindent where $M$ denotes the superheavy scale and $X$ is again a gauge singlet with a GUT scale VEV. By the HFE mechanism mentioned in Section \ref{sec:SUSY1010}, the second term in Eq. (\ref{DWGGG}) may also come from integrating out some superheavy states. In the worst case, if all allowed nonrenormalizable operators are present in superpotential, the gauge hierarchy as well as the DW forms of VEVs could still be protected up to a very high order by some discrete symmetries. As a result, only the pair of the Higgs doublets states in $10_H$ remain light down to the weak scale and proton decay could be suppressed strongly.
In the following, we will briefly discuss the construction of realistic fermion mass matrices without going into details of how the fields transform under the needed discrete symmetries.
As usual, only the third family of matter multiplet $16_3$ gets a weak scale mass through the tree level dimension four operator $O_{33}=16_316_310_H$. Other $O_{ij}$ operators are generically nonrenormalizable because $16_1$ and $16_2$ both carry no $SO(10)_3$ gauge quantum numbers. However, it is impossible to form $O_{ij}$ operators for the off-diagonal entries of fermion mass matrices by simply using matter multiplets and the Higgs fields in fundamental representations. A set of additional heavy fields in the ${\bf 16+{\overline {16}}}$, $\Psi_{V_i}$ and $\bar \Psi_{V_i}$, which transform under the GUT gauge group as $(16,1,1)$, $({\overline {16}},1,1)$, $(1, 16, 1)$, $(1, {\overline {16}}, 1)$, $(1,1,16)$ and $(1,1,{\overline {16}})$, must be introduced into the model and acquire VEVs along the $SU(5)_i$ singlet directions. The VEV's can be stabilized by the superpotential of the following form \cite{linearW}
\begin{equation}
Y(\Psi_V {\bar \Psi_V})^2/M_V^2+ f(Y),
\label{VEV16}
\end{equation}
\noindent where Y is a singlet field and $f(Y)$ is a polynomial function that contains a linear term. Notice that the would-be Goldstone modes in the spinors $\Psi_{V_i}$ and $\bar \Psi_{V_i}$ can be removed by adding more terms to the superpotential \cite{linearW}.
Although there are many possible nonrenormalizable operators which may or may not survive from imposing the discrete symmetries, the following high dimensional operators could also arise from the HFE mechanism, and are interesting because they may help to realize the Gerogi-Jarlskog type of Yukawa matrices in the model.
\begin{eqnarray}
O_{23}&=&(\Psi_{V_2} \cdot S_X \cdot S_X \cdot 16_2) \cdot (\Psi_3 \cdot 10_H \cdot {1 \over \Sigma_X^2}16_3)\\
O_{22}&=&(16_2 \cdot S_X \cdot {\Sigma_{B-L} \over \Sigma_X^2}16_2) \cdot 10_H \\
O_{12}^{(1)}&=&(\Psi_{V_1} \cdot Z_{B-L}' \cdot 16_1) \cdot (\Psi_{V_2} \cdot S_X \cdot 16_2) \cdot 10_H \\
O_{12}^{(2)}&=&(\Psi_{V_1}' \cdot Z_{T_{3R}}' \cdot 16_1) \cdot (\Psi_{V_2} \cdot S_X \cdot 16_2) \cdot 10_H \\
O_{12}^{(3)}&=&(\Psi_{V_1} \cdot Z_{B-L}'\cdot Z_{B-L}' \cdot 16_1) \cdot (\Psi_{V_2}' \cdot S_X \cdot {\Sigma_X^3 \over I^3} 16_2) \cdot 10_H \\
O_{12}^{(4)}&=&(\Psi_{V_1} \cdot Z_{T_{3R}}' \cdot Z_{T_{3R}}' \cdot 16_1) \cdot (\Psi_{V_2}' \cdot S_X \cdot {\Sigma_X^3 \over I^3}16_2) \cdot 10_H \\
O_{12}^{(5)}&=&(\Psi_{V_1}' \cdot Z_{B-L}'\cdot Z_{T_{3R}}' \cdot 16_1) \cdot (\Psi_{V_2}' \cdot S_X \cdot {\Sigma_X^3 \over I^3} 16_2) \cdot 10_H
\label{OPGGG}
\end{eqnarray}
\noindent Again, the effective adjoint operator $\Sigma_X$ of the gauge group $SO(10)_2$ gives different quantum numbers to the fermion states in the matter multiplets $16_i$. Since the Higgs fields $Z'_{B-L}$ and $Z'_{T_{3R}}$ must at least carry different charges of some global $Z_2$ symmetry to avoid the breaking of gauge hierarchy, we thus need two additional VEV-acquiring spinors ${\Psi'}_{V_{i}}$ and $\overline {\Psi'}_{V_{i}}$ , where $i=1,2$, to make the operators $O_{12}$ respect the $Z_2$ symmetry.
Let us parametrize the contributions of the operators $16_3 16_3 10_H$ and the $O_{ij}$ as $A, B, E,....$. In the case that only $16_3 16_3 10_H$ and the $O_{ij}$ operators give dominant contributions to fermion masses, the fermion mass matrices become
\begin{eqnarray}
M_u&=&<H>\left[ \begin{array} {ccc}
0&C^{(3)}&0\\
C^{(5)}&0&B\\
0&B&A
\end{array} \right], \nonumber \\
M_d&=&<\bar H>\left[ \begin{array} {ccc}
0&C^{(1)}&0\\
27C^{(5)}&E&0\\
0&{1\over 9}B&A
\end{array} \right], \nonumber \\
M_e&=&<\bar H>\left[ \begin{array} {ccc}
0&27C^{(4)}&0\\
C^{(2)}&3E&{1\over 9}B\\
0&0&A
\end{array} \right], \nonumber \\
M_{\nu^c \nu}&=&<H>\cdot \left[
\begin{array}{ccc}
0&27(C^{(3)}+C^{(4)}+C^{(5)})&0 \\
C^{(2)}&-{6 \over 25} E&{1\over 9}B\\
0&0&A
\end{array} \right],
\label{MassGGG}
\end{eqnarray}
\noindent where $A$, $B$, $E$, and $C^{(i)}$ come from the contribution of the operators $O_{33}$, $O_{23}$, $O_{22}$, and $O_{12}^{(i)}$ respectively. Again, the numbers shown in Eq. (\ref{MassGGG}) are Clebsch-Gordan coefficients. From the mass ratio $m_u/m_d$, we may estimate that the ratio $C^{(3)}/C^{(1)} \approx 1/27$. Therefore, to realize an experimentally acceptable fermion mass matrix, as implied from Eq. (\ref{MassGGG}), the breakdown of the GUT gauge group $SO(10)_1 \times SO(10)_2 \times SO(10)_3$ may take the following steps
\begin{eqnarray}
[SO(10)]^3 &\longrightarrow& SU(5)_1 \times SU(5)_2 \times SU(5)_3 \hspace{0.6cm} \mbox{at} <\Psi_{V_i}> \sim M \nonumber \\
&\longrightarrow& SU(3)_C \times SU(2)_L \times U(1)_Y \hspace{0.6cm} \mbox{at} \hspace{0.3cm} v_5 \approx {1\over 30}M,
\end{eqnarray}
\noindent where $M\approx 6\times 10^{17} \mbox{ GeV}$, $v_D \approx v_{10}\approx v_5$, and $<X_S>/v_5 \sim <X_S'>/v_5 \sim 10^{-1}$ are assumed. In this GUT group breaking scenario, the $SO(10)\times SO(10)\times SO(10)$ GUT gauge group would first be broken down to $SU(5)\times SU(5)\times SU(5)$ by the spinorial Higgs fields $\Psi_{V_i}$ and $\bar\Psi_{V_i}$, and then breaks into the embedded diagonal subgroup $SU(3)_C \times SU(2)_L\times U(1)_Y$ at the GUT scale $M_G \approx v_5$.
Neutrinos may acquire masses by the same mechanism described in the previous section. A set of spinorial Higgs fields with non-vanishing VEVs along the $SU(5)_i$-preserving directions are necessary for giving Majorana masses to right handed neutrinos. However, none of the spinorial Higgs fields used in constructing the operators $O_{ij}$ can be used in giving a Majorana mass to right handed $\tau$ neutrino $\nu_{\tau}$ since, otherwise, we would get the Majorana mass for left handed $\tau$ neutrino $m_{\nu_{\tau}} \approx m_t^2/M \approx 1/6 \times 10^{-4} \mbox{eV}\gg m_{\nu_{\mu}}, m_{\tau_{e}}$, which is disfavored by recent SuperKamiokande data \cite{SuperK}. Thus a new pair of spinorial Higgs fields $\Psi_{V_3}'$ and $\Psi_{V_3}'$ would be needed to give an acceptable mass to $\nu_{\tau}^c \nu_{\tau}^c$
\begin{eqnarray}
{1\over M} (\bar \Psi_{V_3}' \Gamma_a^{\over {(126)}} \bar \Psi_{V_3}')(16_3 \Gamma_a^{(126)} 16_3).
\label{neutauR}
\end{eqnarray}
\noindent with $<\Psi_{V_3}'>=<\bar \Psi_{V_3}'>\approx v_5$.
As before, a non-trivial Majorana mass matrix $M^R_{\nu}$ for right handed neutrinos may be present in the model, and heavily influence the Majorana mass spectrum as well as neutrino mixing angles of left handed neutrinos. We will not discuss this problem in detail in this paper.
\section{Conclusion}
\label{sec:Conclusion10}
Typical SUSY $SO(10)$ GUT models require a variety of rank two tensor fields, such as the fields in ${\bf 45}$ and ${\bf 54}$ representations, to be phenomenologically successful. These rank two tensors, when they acquire their VEVs and break the gauge $SO(10)$ group, also play important roles in implementing the Dimopoulos-Wilczek mechanism and in deriving experimentally acceptable Yukawa matrices. However, these representations are complicated, and it is usually difficult for all the needed rank two tensor fields to be generated by a simple string construction.
In this paper, we have shown that it is possible not only to implement the DW mechanism but also to provide experimentally acceptable Yukawa matrices. In our $SO(10)\times SO(10)$ and $SO(10)\times SO(10)\times SO(10)$ GUT models, without introducing any rank two tensor fields, the Higgs doublet-triplet splitting problem is naturally solved with strong suppression of proton decay when some Higgs fields of fundamental representations acquiring their VEVs in Dimopoulos-Wilczek forms. Also, unlike other $SO(10)\times SO(10)$ and $SO(10)\times SO(10)\times SO(10)$ models in the literatures \cite{GG,GGG}, effective adjoint operators of at least one of the $SO(10)$ gauge group can be formed when combining the $S$ and one of the $Z$ fields in our model. This allows us to construct realistic fermion mass matrices with successful GUT relations such as top-bottom-tau unification $\lambda_t = \lambda_b =\lambda_{\tau}$, $m_{\mu}=3m_s$, and $m_d=3m_e$.
On the neutrino mass problem, as in conventional $SO(10)$ theories, some spinorial Higgs fields in the {\bf 16} representation of the corresponding $SO(10)_i$ gauge group are necessary for making effective $\nu^c \nu^c$ mass operators. When acquiring VEVs that preserve subgroups $SU(5)_i$ for each corresponding $SO(10)_i$, these spinorial Higgs fields give superheavy Majorana masses to right handed neutrinos. Small Majorana masses for left handed neutrinos are thus generated from see-saw mechanism. However, further understandings on the neutrino sector will be needed in our models for constructing the mass matrix for right handed neutrinos, and also for understanding the mass hierarchy/splitting as well as the mixing angles among left handed neutrinos.
\subsection*{Acknowledgments}
The author thanks Michael E. Peskin for very helpful comments. This work was supported by Department of Energy under contract DE-AC03-76SF00515.
|
1,941,325,220,545 | arxiv |
\section{Introduction}
\label{sec:intro}
Statistical methods for novelty detection are becoming increasingly popular in recent literature. Similar to standard supervised classifiers, these models are trained on a fully labeled dataset and subsequently employed to group unlabeled data. In addition, novelty detectors allow for the discovery and consequent classification of samples showcasing patterns not previously observed in the learning phase.
For example, novelty detection models have been successfully employed in uncovering unknown adulterants in food authenticity studies \citep{Cappozzo2019e,Fop2022}, collective anomalies in high energy particle physics \citep{Vatanen}, and novel communities in social network analyses \citep{Bouveyron}.\\
Formally, we define a \emph{novelty detector} as a classifier trained on a set (the training set) characterized by a specific number of classes that is used to predict the labels of a second set (the test set). For a detailed account of the topic, the interested reader is referred to the reviews of \cite{Markou2003b} and \cite{Markou2003}, where the former is devoted explicitly to statistical methods for novelty detection.
Within the statistical approach, different methodologies have been proposed to construct novelty detectors. Recently, \cite{denti2021two} introduced a Bayesian Robust Adaptive Novelty Detector (Brand). Brand is a semiparametric classifier divided into two stages, focused on training and test datasets, respectively.
In the first stage, a robust estimator is applied to each class of the labeled training set. By doing so, Brand recovers the representative traits of each known class. The second step consists in fitting a semiparametric nested mixture to the test set: the hierarchical structure of the model specifies a convex combination between terms already observed in the training set and a novelty term, where the latter is decomposed into a potentially unbounded number of novel classes. At this point, Brand uses the information extracted from the first phase to elicit reliable priors for the known components.
In their article, \cite{denti2021two} devised an MCMC algorithm for posterior estimation. The algorithm is based on a variation of the slice sampler \citep{Kalli2011} for nonparametric mixtures, which avoids the ex-ante specification of the number of previously unseen components, reflecting the expected ignorance about the structure of the novelty term. As a result, the algorithm obtains good performance as it targets the true posterior without resorting to any truncation. Nonetheless, as is often the case when full MCMC inference is performed, it becomes excessively slow when applied to large multidimensional datasets.\\
In this paper, we aim to scale up the applicability of Brand by adopting a variational inference approach, vastly improving its computational efficiency. Variational inference is an estimating technique that approximates a complex probability distribution by resorting to optimization \citep{Jordan1999, Ormerod2010}, which has received significant attention within the statistical literature in the past decade \citep{Blei2017}. In the Bayesian set-up, the distribution of interest is, of course, the posterior distribution. Variational inference applied to Bayesian problems (also known as variational Bayes, VB from now on) induces a paradigm shift in the approximation of a complicated posterior: we switch from a simulation problem (MCMC) to an optimization one. Broadly speaking, the VB approach starts with the specification of a class of simple distributions called the variational family. Then, within this specified family, one looks for the member that minimizes a suitable distributional divergence (e.g., the Kullbak-Lielber divergence) from the actual posterior. By dealing with a minimization problem instead of a simulation one, we can considerably scale up the applicability of Brand, obtaining results for datasets with thousands of observations measured in high-dimension in a fraction of the time needed by MCMC techniques.
Variational techniques have showcased the potential to enhance the relevance of the Bayesian framework even to large datasets, sparking interest in its theoretical properties \citep[e.g.,][]{Wang2012a,Nieman2022,Ray2022}, and its applicability to, for instance, logistic regression \citep{Jaakkola2000,Rigon}, network analysis \citep{Aliverti2022}, and, more in general, non-conjugate \citep{Wang2012} and advanced models \citep{Zhang2019}.
The present paper is structured as follows. In Section \ref{sec:background}, we review Brand and provide a summary of the variational Bayes approach. In Section \ref{sec:algo}, we discuss the algorithm and the hyperparameters needed for the VB version of the considered model. Section \ref{sec:simstud} reports extensive simulation studies that examine the efficiency and robustness of our method. Then, in Section \ref{sec:appl}, we present an application to the \texttt{Statlog} dataset, openly available from the UCI dataset repository. In this context, novelty detection is used to discover novel types of soil starting from satellite images. We remark that this analysis would have been prohibitive with a classical MCMC approach. Lastly, Section \ref{sec:disc} concludes the manuscript.
\section{Background: Bayesian novelty detection and variational Bayes}
\label{sec:background}
This section introduces the two stages in which Brand articulates and briefly presents the core concepts of the variational Bayes approach.
\subsection{The Brand model}
In what follows, we review the two-stage procedure proposed in \cite{denti2021two}. The first stage centers around a fully labeled learning set from which we extract robust information to set up a Bayesian semiparametric model in the second stage.
More specifically, consider a labeled training dataset with $n$ observations grouped into $J$ classes. This paper will focus on classes distributed as multivariate Gaussians, but one can readily extend the model using different distributions. To this regard, we will write $\mathcal{N}\left(\cdot\mid\boldsymbol{\Theta}\right)$ to indicate a Multivariate Gaussian density with generic location and scale parameters $\boldsymbol{\Theta}=\left(\boldsymbol{\mu},\Sigma\right)$. Within each of the $J$ training classes, we separately apply the Minimum Regularized Covariance Determinant (MRCD) estimator \citep{Boudt2020} to retrieve robust estimates for the associated mean vector and covariance matrix. In detail, the MRCD searches for a subset of observations with fixed size whose regularized sample covariance has the lowest possible determinant. MRCD estimates are then defined as the multivariate location and regularized scatter based on the so-identified subset. Compared to the popular Minimum Covariance Determinant \citep{Driessen1999, Hubert2018}, MRCD takes advantage of regularization with a pre-specified positive definite target matrix to ensure the existence of the solution even when the data dimension exceeds the sample size. This feature implies that the robustness properties of the original procedure are preserved whilst being applicable also to ``$p>n$'' problems. Such a characteristic is paramount in the context considered in the present paper, where we specifically aim to scale up the applicability of Brand to high-dimensional scenarios. We will denote the estimates obtained for class $j$, $j=1,\ldots,J,$ with $\bar{\Theta}^{MRCD}_j = \left(\hat {\boldsymbol{\mu}}^{MRCD}_j,\hat{\boldsymbol{\Sigma}}^{MRCD}_j\right)$. These quantities will be employed to elicit informative priors for the $J$ known components in the second stage. Specifically, in so doing, outliers and label noise that might be present in the training set will not hamper the Bayesian mixture model specification hereafter reported.
In the second stage, we estimate a semiparametric Bayesian classifier on a test set with $M$ unlabeled observations. We want to build a novelty detector, i.e., a model that can discern between ``known'' units - which follow a pattern already observed in the training set - or ``novelties''. At this point, the likelihood for each observation is a simple two-group mixture between a generic novelty component $f_{nov}$ and a density reflecting the behavior found in the training set $f_{obs}$. Given stage 1, it is immediate to specify $f_{obs}$ as a mixture of $J$ multivariate Gaussians, whose hyperpriors elicitation will be guided by each of the robust estimates $\bar{\Theta}^{MRCD}_j$. Therefore, we can now write the likelihood for the generic observation $\boldsymbol{y}_m$, where $m=1,\ldots, M$, as a mixture of $J+1$ distributions:
\begin{equation}
\mathcal{L}\left(\boldsymbol{y}_m \mid \boldsymbol{\Theta}^{obs}, \boldsymbol{\pi}\right) = \pi_0 f_{nov} + \sum_{j=1}^{J} \pi_j\mathcal{N}\left(\boldsymbol{y}_m\mid \boldsymbol{\Theta}^{obs}_{j}\right).
\label{eq:genlik}
\end{equation}
Estimating this model allows us to either allocate each of the $M$ observations into one of the previously observed $J$ Gaussian classes or flag it as a novelty generated from an unknown distribution $f_{nov}$.
The specification of $f_{nov}$ is more delicate. To specify a flexible distribution that would reflect our ignorance regarding the novelty term, we employ a Dirichlet process mixture model with Gaussian kernels \citep{Escobar1995}. It is a mixture model where the mixing distribution is sampled from a Dirichlet process \citep{Ferguson1973}, characterized by concentration parameter $\gamma$ and base measure $H$. In other words, we have that $f_{nov}(\boldsymbol{y}_m) = \sum_{k=1}^\infty \omega_k \mathcal{N}\left(\boldsymbol{y}_m \mid \boldsymbol{\Theta}^{nov}_{k}\right)$, where the atoms are sampled from the base measure, i.e., $\boldsymbol{\Theta}^{nov}\sim H$, and the mixture weights follow the so-called stick-breaking construction \citep{Sethuraman1994a}, where $w_k = v_k\prod_{l<k}(1-v_l)$ and $v_l\sim Beta(1,\gamma)$. To indicate this process, we will write $\boldsymbol{\omega}\sim SB(\gamma)$. Thus, the likelihood in \eqref{eq:genlik} becomes, for $m=1,\ldots,M$,
\begin{equation}
\mathcal{L}\left(\boldsymbol{y}_m \mid \boldsymbol{\Theta}, \boldsymbol{\pi}\right) = \pi_0 \left[\sum_{k=1}^\infty \omega_k \mathcal{N}\left(\boldsymbol{y}_m \mid \boldsymbol{\Theta}^{nov}_{k}\right)\right] + \sum_{j=1}^{J} \pi_j\mathcal{N}\left(\boldsymbol{y}_m \mid\boldsymbol{\Theta}_{j}\right).
\nonumber
\end{equation}
This nested-mixture expression of the likelihood highlights a two-fold advantage. First, it is highly flexible, effectively capturing departures from the known patterns and flagging them as novelties. Second, the mixture nature of $f_{nov}$ allows to automatically cluster the novelties, capturing potential patterns that may arise. Furthermore, clusters in the novelty terms characterized by very small sizes can be interpreted as simple outliers.
Notice that the previous nested-mixture likelihood can be conveniently re-expressed as:
\begin{align}
\mathcal{L}\left(\boldsymbol{y}_m \mid \boldsymbol{\Theta}^{obs}, \boldsymbol{\pi}\right) &= \sum_{k=1}^{\infty} \tilde{\pi}_k \mathcal{N}\left(\boldsymbol{y}_m\mid \tilde{\boldsymbol{\Theta}}_{k}\right),
\label{eq:marg_lik}
\end{align}
where, for $k \in \mathbb{N}$, we define $\tilde\pi_k = {\pi_k}^{\mathbbm{1}{\{0 < k \leq J\}}}(\pi_0\omega_{k-J})^{\mathbbm{1}{\{k \geq J\}}}$ and
$\tilde{\boldsymbol{\Theta}}_k = (\boldsymbol{\Theta}_k^{obs})^{\mathbbm{1}{\{0 < k \leq J\}}}(\boldsymbol{\Theta}_{k}^{nov})^{\mathbbm{1}{\{k \geq J\}}}$. Note that, without loss of generality, we regard the first $J$ mixture components as the known components and all the remaining ones as novelties.
Finally, as customary in mixture models, to ease the computation, we augment the model by introducing the auxiliary variables $\xi_m\in \mathbb{N}$, for $m=1,
\ldots,M$, where $\xi_m = l$ means that the $m$-th observation has been assigned to the $l$-th component. Therefore, the model in \eqref{eq:marg_lik} simplifies to
\begin{align}
\boldsymbol{y}_m \mid \xi_m,\tilde{\boldsymbol{\Theta}}, \boldsymbol{\pi}\sim & \mathcal{N}\left(\boldsymbol{y}_m\mid \tilde{\boldsymbol{\Theta}}_{\xi_m}\right),\quad \quad
{\xi_m}\mid{\tilde{\boldsymbol{\pi}} \overset{iid}{\sim} \sum_{k=1}^\infty{\tilde\pi_k\delta_k(\cdot)} }.
\label{eq:aux_spec}
\end{align}
We complete our Bayesian model with the following prior specifications for the weights and the atoms:
\begin{equation}
\begin{aligned}
\boldsymbol{\pi} &\sim {Dirichlet}(\alpha_0, \alpha_1, ..., \alpha_J),\quad \quad
\boldsymbol{\omega} \sim {SB}(\gamma),\\
\boldsymbol{\Theta}_k^{obs} &\sim {\mathcal{NIW}}(\boldsymbol{\mu}^{{obs}}_k, \nu^{{obs}}_k, \lambda^{{obs}}_k, \boldsymbol{\Psi}^{{obs}}_k), \quad \quad k = 1,\ldots,J,\\
\boldsymbol{\Theta}_k^{nov} &\sim {\mathcal{NIW}}({\boldsymbol{\mu}_0^ {nov}}, \nu_0^{ {nov}}, \lambda_0^{ {nov}}, \boldsymbol{\Psi}_0^{nov}), \quad \quad k = J+1,\ldots, \infty,
\label{eq:prior_NIW}
\end{aligned}
\end{equation}
where $\mathcal{NIW}$ indicates a Normal Inverse-Wishart distribution. To ease the notation, let $\boldsymbol{\Theta} = \{\boldsymbol{\Theta}_k^{obs}\}_{k=1}^J \cup \{\boldsymbol{\Theta}_k^{nov}\}_{k=J+1}^\infty$, $\boldsymbol{\varrho}_k = (\boldsymbol{\mu}^{{obs}}_k, \nu^{{obs}}_k, \lambda^{{obs}}_k, \boldsymbol{\Psi}^{{obs}}_k)$ and $\boldsymbol{\varrho}_0 = (\boldsymbol{\mu}^{nov}_0, \nu^{nov}_0, \lambda^{nov}_0, \boldsymbol{\Psi}^{nov}_0)$. We remark that the values for the hyperparameters $\{\boldsymbol{\varrho}_k^{obs}\}_{k = 1}^J$ are defined according to the robust estimates obtained in the first stage. See \cite{denti2021two} for more details about the hyperparameters specification.
Given the model, it is easy to derive the joint law of the data and the parameters, which is proportional to the posterior distribution of interest $p\left({\boldsymbol{\Theta}}, \boldsymbol{\pi},\boldsymbol{\xi} \mid \boldsymbol{y} \right)$.
Therefore, the posterior we target for inference is proportional to:
\begin{align}
p\left({\boldsymbol{\Theta}}, \boldsymbol{\pi},\boldsymbol{\xi} \mid \boldsymbol{y} \right) \propto &
\prod_{m=1}^M\left[ \prod_{k=1}^J\mathcal{N}(\boldsymbol{y}_m\mid\boldsymbol\Theta_k^{obs})^{\mathbbm{1}_{\xi_m=k}}\prod_{k=J+1}^{\infty}\mathcal{N}(\boldsymbol{y}_m\mid\boldsymbol\Theta_k^{nov})^{\mathbbm{1}_{\xi_m=k}}\right] \times
\nonumber \\ & \prod_{k=1}^J\mathcal{\mathcal{NIW}}(\boldsymbol\Theta_k^{obs}\mid \boldsymbol{\varrho}_k) \prod_{k=J+1}^{\infty}{\mathcal{NIW}}(\boldsymbol\Theta_k^{nov}\mid\boldsymbol{\varrho}_0)\times \nonumber \\ & \prod_{m=1}^M \left[ \prod_{k=1}^J(\pi_k)^{\mathbbm{1}_{\xi_m=k}}\prod_{k=J+1}^{\infty}\pi_0\left( v_{k-J}\prod_{h=1}^{k-J-1}(1-v_h)\right)^{\mathbbm{1}_{\xi_m=k}}\right] \times \nonumber
\\
& \prod_{k=0}^J(\pi_k)^{\alpha_k-1}\prod_{k=1}^{\infty}(1-v_k)^{\gamma-1}.
\label{eq:post}
\end{align}
The left panel of Figure \ref{fig:brand_vb} contains a diagram that summarizes how Brand works.
In the following, we will devise a VB approach to approximate \eqref{eq:post} in a timely and efficient manner. The following subsection briefly outlines the general strategy underlying a mean-field variational approach, while the thorough derivation of the algorithm used to estimate Brand is deferred to Section \ref{sec:algo}.
\begin{figure}[t!]
\centering
\includegraphics[width = \linewidth]{figures_main/FIG1.pdf}
\caption{Diagrams depicting how Brand (left panel) and variational inference (right panel) work, respectively.}
\label{fig:brand_vb}
\end{figure}
\subsection{A short summary of mean-field variational Bayes}
Working in a Bayesian setting, we ultimately seek to estimate the posterior distribution to draw inference. Unfortunately, the expression in \eqref{eq:post} is not available in closed form, and therefore we need to rely on approximations.
MCMC algorithms, which simulate draws from \eqref{eq:post}, could be prohibitively slow when applied to large datasets. To this extent, we resort to variational inference to recasts the approximation of \eqref{eq:post} into an optimization problem. For notational simplicity, while we present the basic ideas of VB, let us denote a generic target distribution with $p(\boldsymbol{\theta}\mid \boldsymbol{y})\propto p(\boldsymbol{\theta}, \boldsymbol{y})$.
As in \cite{Blei2017}, we focus on a mean-field variational family $\mathcal{Q}$, a set containing distributions for the parameters of interest $\boldsymbol{\theta}$ that are all mutually independent: $\mathcal{Q} = \{q_{\boldsymbol{\zeta}}(\boldsymbol{\theta}): q_{\boldsymbol{\zeta}}\boldsymbol{\theta} = \prod_jq_{\zeta_j} ({\theta}_j)\}$. The postulated independence dramatically simplifies the problem at hand. Notice that each member of $\mathcal{Q}$ depends on a set of variational parameters denoted by $\boldsymbol{\zeta}$.
We seek, among the members of this family, the candidate that provides the best approximation of our posterior distribution $p(\boldsymbol{\theta}\mid \boldsymbol{y})$. Herein, the goodness of the approximation is quantified by the Kullback-Leibler (KL) divergence. Thus, we aim to find the member of the variational family $\mathcal{Q}$ that minimizes the KL divergence between the actual posterior distribution and the variational approximation.
The KL divergence ${D}_{KL} (\cdot\mid\mid\cdot)$ can be written as ${D}_{KL} (p(\boldsymbol{\theta} \mid \boldsymbol{y})\mid\mid q_{\boldsymbol{\zeta}}(\boldsymbol{\theta}))) = \mathbb{E}_q[\log{q_{\boldsymbol{\zeta}}(\boldsymbol{\theta})}] - \mathbb{E}_q[\log{p(\boldsymbol{\theta},\boldsymbol{y})}]+\log p(\boldsymbol{y}).$ Unfortunately, we cannot easily compute the \emph{evidence} $p(\boldsymbol{y})$. However, the evidence does not depend on any variational parameter and can be treated as fixed w.r.t. $\boldsymbol{\theta}$ during the optimization process. We then re-formulate the problem into an equivalent one, the maximization of the \textit{Evidence Lower Bound} (ELBO), which is fully computable:
\begin{equation}
ELBO(q) = \mathbb{E}_q[\log{p(\boldsymbol{z},\boldsymbol{y})}] - \mathbb{E}_q[\log{q_{\boldsymbol{\zeta}}(\boldsymbol{z})}].
\label{eq:elbo}
\end{equation}
Maximizing \eqref{eq:elbo} is equivalent to minimize the aforementioned KL divergence.\\
To detect the optimal member $q^\star\in \mathcal{Q}$, we employ a widely used algorithm called \emph{Coordinate Ascent Variational Inference} (CAVI). It consists of a one-variable-at-a-time optimization procedure. Indeed, exploiting the independence postulated by the mean-field property, one can show that the optimal variational distribution for the parameter $\theta_j$ is given by:
\begin{equation}
q^\star_{\zeta_j}(\theta_j)\propto
\exp\{\mathbb{E}_{-j}\left[\log p(\theta_j, \boldsymbol{\theta}_{-j},\boldsymbol{y})\right]\},
\label{eq:step}
\end{equation}
where $\boldsymbol{\theta}_{-j} = \{\theta_l\}_{l\neq j}$ and the expected value is taken w.r.t. the densities of $\boldsymbol{\theta}_{-j}$. The CAVI algorithm iteratively computes \eqref{eq:step} for every $j$ until the ELBO does not register any significant improvement. The basic idea behind the CAVI algorithm is depicted in the right-half of Figure \ref{fig:brand_vb}.\\
In the next subsections, we will derive the CAVI updates for the variational approximation of our model, along with the corresponding expression of the ELBO.
We call the resulting algorithm the variational Brand, or \texttt{VBrand}.
\section{Variational Bayes for the Brand model}
\label{sec:algo}
In this section, we tailor the generic variational inference algorithm to our specific case. First, we write our variational approximation, highlighting the dependence of each distribution on its specific variational parameters, collected in $\boldsymbol{\zeta} =\left( \boldsymbol{\eta},\boldsymbol{\varphi},\boldsymbol{a},\boldsymbol{b},\boldsymbol{\rho}^{obs},\boldsymbol{\rho}^{nov}
\right)$. The factorized form we adopt is:
\begin{equation}\begin{split}
q_{\boldsymbol{\zeta}}( \boldsymbol{\xi}, \boldsymbol{\pi}, \boldsymbol{v}, \boldsymbol{\Theta}^{obs}, \boldsymbol{\Theta}^{nov}) =& q_{\boldsymbol{\eta}}(\boldsymbol{\pi}) \prod_{m=1}^M q_{\boldsymbol{\varphi}^{(m)}}(\xi^{(m)}) \prod_{k=1}^{T-1} q_{a_k,b_k}(v_k)\times \\
&\prod_{k=1}^J q_{\boldsymbol{\rho}_k^{obs}} (\boldsymbol{\Theta}^{obs}_k)
\prod_{k=J+1}^{J+T} q_{\boldsymbol{\rho}_k^{nov}} (\boldsymbol{\Theta}^{nov}_k).
\end{split}
\label{eq:Vfam}
\end{equation}
In Equation \eqref{eq:Vfam}, we truncated the stick-breaking representation of the Dirichlet Process on the novelty term at a pre-specified threshold $T$, as suggested in \cite{Blei2006}. This implies that $q(v_T = 1) = 1$ and that all the variational mixture weights indexed by $t > T $ are equal to zero.
Then, we can exploit a key property of VB. Note that all the full conditionals of the Brand model have closed-form expressions (c.f.r. Section S1 of the Supplementary Material) and belong to the exponential family. This feature greatly simplifies the search for the variational solution. Indeed, it ensures that the corresponding optimal variational distributions belong to the same family of the corresponding full conditional, with properly updated variational parameters. Therefore, we can already state that $q_{\boldsymbol{\eta}}(\boldsymbol{\pi})$ is the density function of a Dirichlet distribution, $q_{\boldsymbol{\varphi}^{(m)}}(\xi^{(m)})$ follows a Categorical distribution,
each $q_{a_k,b_k}(v_k)$ follows a Beta distributions, and
$q_{\boldsymbol{\rho}_k^{obs}} (\boldsymbol{\Theta}^{obs}_k)$ and $q_{\boldsymbol{\rho}_k^{nov}} (\boldsymbol{\Theta}^{nov}_k)$ are both distributed according to a Normal Inverse-Wishart.\\
\subsection{The CAVI parameters update}
Once the parametric expressions for the members of the variational family are obtained, we can derive the explicit formulas to optimize the parameters via the CAVI algorithm. In this subsection, we state the updating rules that have to be iteratively computed to fit \texttt{VBrand}.
As we will observe, the set of responsibilities $\{\varphi_k^{(m)}\}_{k = 1}^{J+T}$, i.e., the variational probabilities $\varphi^{(m)}_k=q(\xi^{(m)}=k)$, will play a central role in all the steps. In detail, the CAVI steps are as follows:
\begin{enumerate}
\item $q^\star_{\boldsymbol{\eta}}(\boldsymbol{\pi})$ is the density of a $Dirichlet(\eta_0,\eta_1,\ldots,\eta_J)$, where
\begin{equation}
\eta_0 =
\alpha_0 + \sum_{m=1}^M\sum_{l=J+1}^{J+T}\varphi_l^{(m)},
\quad \;
\eta_j = \alpha_j + \sum_{m=1}^M\varphi_j^{(m)}, \; \text{ for }j=1,\ldots,J.\\
\label{eq:upd_eta}
\end{equation}
Here, each hyperparameter linked to a known component is updated with the sum of the responsibilities of the data belonging to the same specific known component. Likewise, the variational novelty probability hyperparameter $\eta_0$ contains the sum of the responsibilities of all data belonging to all the novelty terms.\\
\item Each $q^\star_{a_k,b_k}(v_k)$, for $k=1,\ldots,T-1$, is the density of a $Beta(a_k,b_k)$. The update for these variational parameters is given by:
\begin{equation}
a_k = 1 + \sum_{m=1}^M\varphi_{J+k}^{(m)}, \quad b_k = \gamma + \sum_{l=k+1}^T\sum_{m=1}^M\varphi_{J+l}^{(m)}.
\label{eq:upd_ab_beta}
\end{equation}
Here, as expected from the stick-breaking construction, the first parameter of the variational $Beta$ distribution is updated with the sum of the probabilities of each point belonging to a specific novelty cluster $k$. At the same time, the second parameter is updated with the sum of variational probabilities of belonging to one of the next novelty components.\\
\item Both $q^\star_{\boldsymbol{\rho}_k^{nov}} (\boldsymbol{\Theta}^{nov}_k)$ and
$q^\star_{\boldsymbol{\rho}_k^{obs}} (\boldsymbol{\Theta}^{obs}_k)$ are Nomal Inverse-Wishart densities. Let us start with the updating rules of the known components.
Note that each variational parameter $\boldsymbol{\rho}_k^{obs}$ is a shorthand for $\left(\boldsymbol{m}^{obs}_k,\ell^{obs}_k, u^{obs}_k, \boldsymbol{S}^{obs}_k \right)$. These parameters have the same interpretation as the parameters of \eqref{eq:prior_NIW}, contained in $\boldsymbol{\varrho}_k$. So we have
\begin{equation}
\begin{aligned}
\boldsymbol{m}^{obs}_k &= \frac{1}{\lambda^{obs}_k+\sum_{m=1}^M\varphi_{k}^{(m)}}\left(\lambda^{obs}_k\boldsymbol{\mu}^{obs}_k + \sum_{m=1}^M\boldsymbol{y}_m\varphi_{k}^{(m)}\right), \\
\ell_k &= \lambda_k^{obs} + \sum_{m=1}^M\varphi_{k}^{(m)},\quad
u^{obs}_k = \nu_K^{obs} + \sum_{m=1}^M\varphi_{k}^{(m)},\\
\boldsymbol{S}^{obs}_k &= \boldsymbol{\Psi}^{obs}_k + \sum_{m=1}^M \hat{\boldsymbol{\Sigma}}_{k}^{(m)} + \frac{\lambda^{obs}_k\sum_{m=1}^M\varphi_{k}^{(m)}}{\lambda^{obs}_k+\sum_{m=1}^M\varphi_{k}^{(m)}}(\overline{\boldsymbol{y}}_k-\boldsymbol{\mu}^{obs}_k)^T(\overline{\boldsymbol{y}}_k-\boldsymbol{\mu}^{obs}_k),
\label{eq:upd_NIW}
\end{aligned}
\end{equation}
where we defined $\hat{\boldsymbol{\Sigma}}_{k}^{(m)} = (\boldsymbol{y}_m-\overline{\boldsymbol{y}}_k)(\boldsymbol{y}_m-\overline{\boldsymbol{y}}_k)^T\varphi_{k}^{(m)}$ and
$\overline{\boldsymbol{y}}_k = \sum_{m=1}^M\boldsymbol{y}_m\varphi_{k}^{(m)}/\sum_{m=1}^M\varphi_{k}^{(m)}.$ The update for the parameters in $\boldsymbol{\rho}_k^{nov}$ follows the same structure, with the hyperprior parameters in $\boldsymbol{\varrho}^{nov}$ carefully substituted to $\boldsymbol{\varrho}^{obs}$.\\
\item Updating the responsibilities $\{\varphi_k^{(m)}\}_{k = 1}^{J+T}$ for $m=1,\ldots,M,$ is the most challenging step of the algorithm, given the nested nature of the mixture in \eqref{eq:marg_lik}. We recall that, for a given $m$, the distribution $q_{\boldsymbol{\varphi}^{(m)}}(\xi^{(m)})$ is categorical with $J+T$ levels. Thus, we need to compute the values for the $J+T$ corresponding probabilities. For the known classes $k=1,\ldots,J$, we have
\begin{equation}
\log \varphi_k^{(m)} \propto
\mathbb{E}[\log{\pi_k}] + \mathbb{E}[\log{\mathcal{N}(\boldsymbol{y}_m\mid\boldsymbol{\Theta}_k^{obs})}],
\label{eq:upd_obs}
\end{equation}
while for the novelty terms $k=J+1,\ldots,J+T$, we have
\begin{equation}
\begin{aligned}
\log\varphi_k^{(m)} \propto \:\: & \mathbb{E}[\log{\pi_0}] + \mathbb{E}[\log{v_{k-J}}] + \sum_{l=1}^{k-J-1}\mathbb{E}[\log\left(1-v_{l}\right)] + \\ & \mathbb{E}[\log{\mathcal{N}(\boldsymbol{y}_m\mid\boldsymbol{\Theta}_{k}^{nov})}].
\label{eq:upd_nov}
\end{aligned}
\end{equation}
For the sake of conciseness, we report the explicit expression for all the terms of \eqref{eq:upd_obs} and \eqref{eq:upd_nov} in Section S2 of the Supplementary Material.
This means that the probability of datum $\boldsymbol{y}_m$ to belong to cluster $k$ depends on the likelihood of $\boldsymbol{y}_m$ under that same cluster and on the overall relevance of $k^{th}$ cluster. Such relevance is determined as the expected value of the relative component of the ${Dirichlet}$ distributed $\boldsymbol\pi$ and, for novelties, the stick-breaking weight, here unrolled in its ${Beta}$-distributed components.\\
\end{enumerate}
\subsection{The expression of the ELBO for \texttt{VBrand}}
In this subsection, we report the terms that need to be derived to obtain the ELBO in Equation \eqref{eq:elbo} for \texttt{VBrand} model. We start by computing the first term of Equation \eqref{eq:elbo}, which takes the following form:
\begin{equation*}\begin{split}
\mathbb{E}[\log{p}] = & \sum_{m=1}^M\left(\sum_{k=1}^J f_{1}^{(m,k)} + \sum_{k=J+1}^{J+T} f_{2}^{(k)} \right) + \sum_{k=1}^J f_{3}^{(k)} + \sum_{k=J+1}^{J+T} f_{4}^{(k)} +\\& \sum_{m=1}^M\left(\sum_{k=1}^J f_{5}^{(m,k)} + \sum_{k=J+1}^{J+T} f_{6}^{(m,k)} \right) + \sum_{k=0}^{J} f_{7}^{(k)} + \sum_{l=1}^{T} f_{8}^{(l)} + const,
\end{split}\end{equation*}
where the quantities $\{f_k\}_{k=1}^8$ have the following expressions (note we have suppressed the superscripts to ease the notation, and that $\boldsymbol{\psi}(\cdot)$ indicates the digamma function):
\begin{equation*}
\begin{aligned}
f_{1} &= \varphi_{k}^{(m)}\mathbb{E}[\log{{N}(\boldsymbol{y}_m\mid\boldsymbol{\Theta}_k^{obs})}],&\quad
f_{2} &= \varphi_{k}^{(m)}\mathbb{E}[\log{{N}(\boldsymbol{y}_m\mid\boldsymbol{\Theta}_k^{nov})}], \\
f_{3} &= \mathbb{E}[\log{{\mathcal{NIW}}(\boldsymbol{\Theta}_k^{obs}\mid \boldsymbol{\varrho}_k)}], & \quad f_{4} &= \mathbb{E}[\log{{\mathcal{NIW}}(\boldsymbol{\Theta}_k^{nov}\mid \boldsymbol{\varrho}_0})].
\end{aligned}
\end{equation*}
Moreover, we have $f_{5} = \varphi_{k}^{(m)}\left(\boldsymbol{\psi}\left(\eta_k\right)-\boldsymbol{\psi}\left(\sum_{j=0}^J\eta_j\right)\right)$,
\begin{equation*}
\begin{aligned}
f_{6} = \varphi_{k}^{(m)}\left[\boldsymbol{\psi}(\eta_0)-\boldsymbol{\psi}(\sum_{j=0}^J\eta_j) + \boldsymbol{\psi}(a_{k-J}+b_{k-J}) + \sum_{h=1}^{k-J-1}\boldsymbol{\psi}(b_{h})-\boldsymbol{\psi}(a_{h}+b_{h})\right],
\end{aligned}
\end{equation*}
and, lastly, $f_{7} = (\alpha_k-1)(\boldsymbol{\psi}(\eta_k)-\boldsymbol{\psi}(\sum_{j=0}^J\eta_j)),$ and $f_{8} = (\gamma-1)(\boldsymbol{\psi}(b_l)-\boldsymbol{\psi}(a_l+b_l))$. \\
The second term of Equation \eqref{eq:elbo} can be written as
\begin{equation*}
\mathbb{E}[\log{q}] = \sum_{m=1}^{M}\left(h_1^{(m)} + h_2^{(m)} + \sum_{k=1}^{T}h_3^{(m,k)} + \sum_{k=1}^{J}h_4^{(m,k)} + \sum_{k=1}^{T}h_5^{(m,k)}\right),
\end{equation*}
with $h_{1} = \sum_{k=1}^{J+T}\varphi_{k}^{(m)}\ln{\varphi_{k}^{(m)}} - \ln{\sum_{k=0}^{J+T}\varphi_{k}^{(m)}}$,
$h_{4} = \mathbb{E}[\log{\mathcal{NIW}(\boldsymbol{\Theta}_k^{obs}\mid \boldsymbol{\rho}_k^{obs})}]$, \\
$h_{5}^{(m,k)} = \mathbb{E}[\log{\mathcal{NIW}(\boldsymbol{\Theta}_k^{nov}\mid \boldsymbol{\rho}_k^{nov})}]$, and, finally,
\begin{equation*}
\begin{aligned}
h_{2} =& \sum_{j=0}^{J}(\eta_j-1)\left(\boldsymbol{\psi}(\eta_j)-\boldsymbol{\psi}(\sum_{j=0}^J\eta_j)\right) +
\ln{\Gamma\left(\sum_{j=0}^J{\eta_j}\right)} - \sum_{j=0}^{J}\ln{\Gamma(\eta_j)},\\
h_{3} =& (a_k-1)(\boldsymbol{\psi}(a_k)-\boldsymbol{\psi}(a_k+b_k)) +\\ & (b_k-1)(\boldsymbol{\psi}(b_k)-\boldsymbol{\psi}(a_k+b_k)) - \ln{\left(\frac{\Gamma(a_k)\Gamma(b_k)}{\Gamma(a_k+b_k)}\right)}.
\end{aligned}
\end{equation*}
Additional details are deferred to Section S2 of the Supplementary Material. \\
At each step of the algorithm, one needs to update the parameters according to the rules \eqref{eq:upd_eta}-\eqref{eq:upd_nov} and to evaluate all the terms in $\{f_k\}_{k=1}^8$ and $\{h_k\}_{k=1}^5$. Albeit faster than MCMC, this articulates over numerous steps. To further reduce the overall computing time, an \texttt{R} package relying on an efficient \texttt{C++} implementation has been implemented. The package is openly available at the GitHub repository \url{JacopoGhirri/VarBRAND}. In the same repository, the interested reader can find all the \texttt{R} scripts written to run the simulation studies and real data application that we will discuss in the following sections.
\section{Simulation studies}
\label{sec:simstud}
In this section, we report the performance of our variational algorithm on a range of simulated datasets. In particular, our simulation study is articulated into three different experiments. Each experiment investigates a different aspect of the model while altogether providing a multi-faceted description of its performance.
The first experiment focuses on the scaling capabilities of our proposal; the second compares the results and efficiency of \texttt{VBrand} with the original slice sampler, while the last one assesses the sensitivity of the recovered partition to the hyperprior specification. For the MCMC algorithm, we follow the default hyperprior setting suggested in \cite{denti2021two}. Moreover, we run the slice sampler for 20,000 iterations, discarding the first 10,000 as burn-in. As for \texttt{VBrand}, we use the same hyperprior specifications used for the MCMC (unless otherwise stated), while we use $k$-means estimation to initialize the novelty terms' means. We set a threshold $\varepsilon = 10^{-9}$ as stopping rule.
\begin{table}[t!]
\centering
\begin{tabular}{cccccccc}
\toprule
SS1 & $C_1$ & $C_2$ & $C_3$ & $C_4^*$ & $C_5^*$ & $C_6^*$ & $C_7^*$ \\
\midrule
$\boldsymbol{\mu}_k$ & (-5,-5) & (-4,-4) & (4,4) & (0,0) & (5,-10) & (5,-10) & (-10,-10)\\
$\sigma^2$ & 1 & 2 & 2 & 1 & 1 & 1 & 0.1\\
$\rho$ & 0.9 & 0 & 0 & -0.75 & 0.9 & 0.9 & 0\\
$n_{Tr}$ & 300 & 300 & 300 & -- &--& --&--\\
$n_{Te}$ & 200 & 200& 250& 90& 100& 100& 60\\
\midrule
SS2 & $C_1$ & $C_2$ & $C_3^*$ & $C_4^*$ & $C_5^*$ & --& -- \\
\midrule
$\boldsymbol{\mu}_k$ & (2,2) & (-2,-2) & (2,-2) & (-2,2) & (0,0) & -- & --\\
$n_{Tr}$ & 500 & 500 & -- & -- &-- & -- & --\\
$n_{Te}$ & 200 & 200& 200 & 200 & 200 & -- & --\\
\midrule
SS3 & $C_1$ & $C_2$ & $C_3$ & $C_4^*$ & $C_5^*$ & $C_6^*$ & -- \\
\midrule
$\boldsymbol{\mu}_k$ & (-5,5) & (5,5) & (5,-5) & (0,0) & (0,0) & (-5,5) & --\\
$\sigma^2$ $^\dagger$ & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 1.5 & -- \\
$\rho$ & 0 & 0 & 0 & 0.8 & -0.8 & 0 & --\\
$n_{Tr}$ & 50 & 50 & 50 & -- &-- & -- & --\\
$n_{Te}$ & 1950 & 1950 & 1950 & 2000 & 2000 & 100 & --\\
\bottomrule
\end{tabular}
\caption{Characteristics of the synthetic datasets used in the three simulation studies (SS1$-$SS3). The components flagged with $^*$ are novelties. $^\dagger$ the variances reported in this table refer to the low overlap scenario. For the high overlap scenario, we consider $\sigma^{2\star} = 6\sigma^2$.}
\label{tab:simuall}
\end{table}
\subsection{Classification performance}
\label{simu::scaling}
We test the classification performance of our proposal by applying \texttt{VBrand} to a sequence of increasingly complex variations of a synthetic dataset. We monitor the computation time and different clustering metrics to provide a complete picture of the overall performance. In particular, we compute the Adjusted Rand Index (ARI, \cite{Hubert1985}), the Adjusted Mutual Information (AMI, \cite{Vinh2009}), and the Fowlkes-Mallows Index (FMI, \cite{Fowlkes1983}). While the first two metrics correct the effect of agreement solely due to chance, the latter performs well also if noise is added to an existing partition. Thereupon, the joint inspection of these three quantities aim to provide a complete picture of the results.
The considered data generating process (DGP) for this experiment is based on a mixture of 7 bivariate Normals. In detail, the first three components represent the known classes, appearing in both training and test sets, while the remaining $4$ are deemed to be novelties, present only in the test set. The components are characterized by different mean vectors $\boldsymbol{\mu}_k$, correlation matrices $(\sigma_k^2,\rho_k)$, and cardinalities in the training ($n_{Tr}$) and test ($n_{Te}$). The main attributes of this DGP are summarised in the first block, named SS1, of Table \ref{tab:simuall}.
\begin{comment}
\begin{table}[th!]
\centering
\begin{tabular}{cccccccc}
\toprule
Component & $C_1$ & $C_2$ & $C_3$ & $C_4^*$ & $C_5^*$ & $C_6^*$ & $C_7^*$ \\
\midrule
$\boldsymbol{\mu}_k$ & (-5,-5) & (-4,-4) & (4,4) & (0,0) & (5,-10) & (5,-10) & (-10,-10)\\
$\sigma^2$ & 1 & 2 & 2 & 1 & 1 & 1 & 0.1\\
$\rho$ & 0.9 & 0 & 0 & -0.75 & 0.9 & 0.9 & 0\\
$n_{Tr}$ & 300 & 300 & 300 & -- &--& --&--\\
$n_{Te}$ & 200 & 200& 250& 90& 100& 100& 60\\
\bottomrule
\end{tabular}
\caption{Characteristics of the base dataset used for the scalability simulation study. The components flagged with $^*$ are novelties.}
\label{tab:simu1}
\end{table}
\end{comment}
Starting from this basic mechanism, we subsequently increment the difficulty of the classification tasks. Specifically,
we increase both the data dimension and the sample size as follows:
\begin{itemize}
\item \emph{Sample size}: while keeping unaltered the mixture proportions, we consider the sample sizes $\tilde{n}_k=q\cdot n_k$, with multiplicative factor $q\in \{0.5, 1, 2.5, 5, 10\}$ in both the training and test sets;
\item \emph{Data dimensionality}: we augment the dimensionality $p$ of the problem by considering $p\in \{2, 3, 5, 7, 10\}$. Each added dimension (above the second) comprises independent realization from a standard Gaussian. Note that the resulting datasets define a particularly challenging discrimination task: all the information needed to distinguish the different components is contained in the first two dimensions. In contrast, the remaining ones only display overlapping noise.
\end{itemize}
For each combination of sample size and dataset dimension, we generate 50 replications of each simulated dataset and summarize the results by computing the means and the standard errors of the chosen metrics.
\begin{figure}[t]
\centering
\includegraphics[width = \linewidth]{figures_main/summary_full_VB.pdf}
\caption{Performance metrics and elapsed time obtained by the \texttt{VBrand} algorithm stratified by number of variables $p$ and size scaling $n$. The dots represent the averages obtained over 50 replicates of the simulated experiment, while the vertical bars display the associated standard errors.}
\label{fig:scaling_VB}
\end{figure}
Results for this experiment are reported in Figure \ref{fig:scaling_VB}.
We immediately notice that the clustering performances deteriorate as the dimensionality of the problem increases. This trend is expected, especially given the induced overlap in the added dimensions. However, the classification abilities of our method remain consistent and satisfactory across all metrics. Indeed, as we can see from the three panels at the top, ARI, AMI, and FMI are all strictly above 70\% across all scenarios. This outcome indicates that not only are the known classes correctly identified and clustered as such, but the flexible nonparametric component effectively captures the novelty term.
The computation time (bottom panel) grows exponentially as a function of the test set cardinality. Interestingly, the increment of data dimensionality does not significantly impact the computational costs, suggesting an effective scalability of our proposal to high dimensional problems. Indeed, even when the test size is in the order of tens of thousands, the devised CAVI algorithm always reaches convergence in less than half a minute. Lastly, we remark that the time needed for convergence is sensitive to the different initialization provided to the algorithm, explaining the high variance that characterized the computational costs.
\subsection{Comparison with MCMC} \label{sec:comp_with_MCMC}
We now compare the variational and the MCMC approaches for approximating the Brand posterior. The MCMC algorithm we consider is the modified slice-sampler introduced in \cite{denti2021two}. We compare the estimating approaches leveraging on the same DGP highlighted in the previous section and slightly modified accordingly.
In detail, we consider five spherical bivariate Gaussian to generate the classes, out of which three are deemed as novelties: the basic details of the resulting DGP can be found in the second block of Table \ref{tab:simuall} (SS2).
We consider 50 different scenarios resulting from the interactions of the levels of following three attributes:
\begin{itemize}
\item \emph{Simple vs. complex scenarios}: we set the variance of all the mixture components to either $\sigma_S^2=0.2$ or $\sigma_C^2=0.75$ (the only exception being $\sigma_{C_5^*}^2=0.375$). The former value implies clear separation among the elements in the simple scenario. In contrast, the latter variance defines the complex case, where we induce some overlap that may hinder the classification. A descriptive plot is displayed in Section S3 of the Supplementary Material;
\item \emph{Sample size}: we modify the default sample size $n = 1000$ by considering different multiplicative factors in $q\in \{0.5, 1, 2.5,5,10\}$, thus obtaining datasets ranging from $500$ to $10000$ observations.
\item \emph{Data dimensionality}: we augment the dimensionality $p$ of the problem by considering $p\in \{2, 3, 5, 7, 10\}$. The dimensionality augmentation is carried out as described in Section \ref{simu::scaling}.
\end{itemize}
We assess the classification performances with the same metrics previously introduced. For each of the 50 simulated scenarios, we perform 50 Monte Carlo replicates to assess the variation in the performances. A summary of the classification results under the complex scenario are reported in Figure \ref{fig:vb_vs_mcmc_ind}. The panels show that the slice sampler always outperforms the VB implementation in the two-dimensional case. However, as the dimensionality of the dataset increases the MCMC performance rapidly drops, while \texttt{VBrand} always obtains good clustering recovery, irrespective of the data dimensionality and the sample size. Similar results are obtained under the simple scenario, for which a summarizing plot can be found in Section S3 of the Supplementary Material.
\begin{figure}[t!]
\centering
\includegraphics[width = \linewidth]{figures_main/inds_VBvsMCMC_mox_easy0.pdf}
\caption{Performance metrics (MCMC in red, variational Bayes - VB - in blue) stratified by number of variables $p$ and sample size scaling factor $q$ under the complex scenario. The dots represent the averages obtained over 50 replicates of the simulated experiment, while the vertical bars display the associated standard errors.}
\label{fig:vb_vs_mcmc_ind}
\end{figure}
Figure \ref{fig:vb_vs_mcmc_time} compares the algorithms in terms of computation time. As expected, \texttt{VBrand} provides results in just a fraction of the time required by the MCMC approach, being approximately two orders of magnitude faster.
\begin{figure}[ht]
\centering
\includegraphics[width = \linewidth]{figures_main/time_VBvsMCMC.pdf}
\caption{Computational time in seconds (MCMC in red, variational Bayes - VB - in blue) grouped by number of variables $p$, sample size scaling factor $q$, and type of scenario. The dots represent the averages obtained over 50 replicates of the simulated experiment, while the vertical bars display the associated standard errors.}
\label{fig:vb_vs_mcmc_time}
\end{figure}
These results cast light not only on the apparent gain in computational speed when using the variational approach, which is expected by any means, but also on the superior recovering of the underlying partition in the test set in more complex scenarios.
\subsection{Sensitivity Analysis}
Finally, we investigate the sensitivity of the model classification to different hyperprior specifications. Two pairs of crucial hyperparameters may considerably affect the clustering results. The first two are $\boldsymbol{\alpha}$ and $\gamma$, which drive the parametric and nonparametric mixture weights, respectively. The second pair is given by the $\mathcal{NIW}$ precision $\lambda^{nov}_0$ and degrees of freedom $\nu^{nov}_0$ of the novelty components.
To assess their impact, we devise a sensitivity analysis considering each possible combination of the following hyperparameters: $\boldsymbol{\alpha} \in \{0.1,0.55,1\}$, $\gamma\in\{1,5.5,10\}$, $\lambda^{nov}_0\in\{1,5,10\}$, and $\nu^{nov}_0\in \{4,52,100\}$, thus defining 81 scenarios.
We fit \texttt{VBrand} to a dataset composed of five Normals in a bivariate domain, considering a fixed sample size for both training and test sets. We chose a small sample size for the training set to limit the informativeness of the robust estimation procedure.
Moreover, for each combination of the hyperparameters, we consider both low and high values for the variances of the mixing components, obtaining scenarios with low overlap (LOV) and high overlap (HOV), respectively. Additional details about the data-generating process can be found in the third block of Table \ref{tab:simuall} (SS3). For this experiment, we compare the retrieved partitions in terms of ARI, as done in Sections \ref{simu::scaling} and \ref{sec:comp_with_MCMC}, and by monitoring the $F1$ score, i.e., the harmonic mean of precision and recall. The results are reported in Figure \ref{fig:SA}. We immediately notice by inspecting the panels that for the LOV case the method performs well regardless of the combination of hyper-parameters chosen for the prior specification. In the HOV scenario the recovery of the underlying true data partition is less effective, as it is nevertheless expected. In particular, it seems that setting a high value for the degrees of freedom $\nu^{nov}_0$ produces a slight drop in the ARI metric. This behavior is due to the extra flexibility allowed to the novelty component, by which some of the units belonging to the known groups are incorrectly captured by the novelty term.
\begin{figure}[h!]
\centering
\includegraphics[width=\linewidth]{figures_main/inds_SA_ALL.pdf}
\caption{Classification metrics obtained over 50 replicates for 81 combinations of the hyperparameters $\boldsymbol{\alpha}$, $\gamma$, $\lambda^{nov}_0$, and $\nu^{nov}_0$ under the low overlap (LOV) and high overlap (HOV) cases. The dots represent the averages obtained over 50 replicates of the simulated experiment, while the vertical bars display the associated standard errors.}
\label{fig:SA}
\end{figure}
In summary, our proposal increases the scalability of the approach introduced in \cite{denti2021two}, allowing for the original novelty detector to be successfully applied to complex scenarios in a more timely manner, without being it affected by the hyper-parameters specification.
\section{Application to novel soil type detection}
\label{sec:appl}
For our application, we consider the \texttt{Statlog (Landsat Satellite)} Data Set, publicly obtained from the UCI machine-learning repository\footnote{https://archive.ics.uci.edu/ml/datasets/Statlog+\%28Landsat+Satellite\%29}. It consists of a collection of observations from satellite images of different soils. Each image contains four spectral measurements over a 3x3 grid, recorded to classify the soil type captured in the picture. There are a total of six different soil types recorded: Red Soil (RS), Grey Soil (GS), Damp Grey Soil (DGS), Very Damp Grey Soil (VDGS), Cotton Crop (CC), and Soil with Vegetation Stubble (SVS).
We frame the original classification problem in a novelty detection task by removing the images of CC and SVS from the training set, leaving these groups in the test set to be detected as novelties.
Even when performing a simple classification, a method that can account for the possible presence of previously unseen scenarios can be of paramount utility in many fields. For example, new plants \citep{Christenhusz2016}, animals \citep{Camilo2011}, and viruses \citep{Woolhouse2012} are progressively discovered every year. Likewise, related to our application, landscapes present novelties at increasing rates \citep{Finsinger2017}.
Moreover, a scalable model that can discern and separate outlying observations is necessary when dealing with real-world data, allowing the results to be robust to outliers or otherwise irregular observations. Once these observations are flagged, they can be the objective of future investigations. Thus, our novelty detection application to the \texttt{Statlog} dataset is a nontrivial example that may inspire future applications of our method.
The original data are already split into training and test sets. After removing the CC and SVS classes from the training set, we obtain a training set of 3486 observations. The test set instead contains 2000 instances. Each observation includes the four spectral values recorded over the 9-pixel grids. Therefore, we will model these data with a semiparametric mixture of 36-dimensional multivariate Normals.
Given the large dimensionality of the dataset, the application of the MCMC estimation approach is problematic in terms of both required memory and computational time. Indeeed, estimating the model via slice sampler becomes unfeasible for most laptop computers. Moreover, we recall that the MCMC approach showed some numerical instabilities in our simulation studies when applied to large dimensional datasets.
We apply \texttt{VBrand} adopting a mixture with full covariance matrices to capture the potential dependence across the different pixels. Being primarily interested in clustering, we first rescale the data to avoid numerical problems. In detail, we divided all the values in both training and test by 4.5 to reduce the dispersion. Indeed, before the correction, the variabilities within groups ranged from 25.40 to 228.04. After, the within-group variability ranges from 1.25 to 11.26, significantly improving the stability of the algorithm.
Since the variational techniques are likely to find a locally optimal solution, we run the CAVI algorithm 200 times adopting different initializations. For each run, we obtain different random starting point as follows:
\begin{itemize}
\item we set the centers for the novelty NIWs equal to the centers returned by a $k$-means algorithm performed over the whole test set, with $k$ being equal to the chosen truncation;
\item the Dirichlet parameters, $\boldsymbol{\nu}^{nov}$ and $\boldsymbol{\lambda}^{nov}$ are randomly selected. In particular, we sample the Dirichlet parameters from $\left(0.1, 1\right)$ and the $\mathcal{NIW}$ parameters from $\left(1,10\right)$ through Latin Hypercube sampling;
\end{itemize}
The other variational hyperparameters are assumed fixed, equal to the corresponding prior hyperparameters. In Section S4 of the Supplementary Material, we report a detailed list of the hyperparameter specifications that we adopted for this analysis.
As a final result, we select the run with the highest ELBO value, being the one whose variational distribution has the lowest KL divergence from the true posterior. The top-left panel of Figure \ref{fig:t-results} shows the ELBO trends for all the run we performed.
\begin{figure}[t!]
\centering
\includegraphics[width= \linewidth]{figures_main/results.pdf}
\caption{Top-left panel: collection of ELBO trajectories obtained via CAVI updates starting from 200 different random intializations; the trajectory providing the highest ELBO is highlighted in blue (the $y$ axis is truncated for improved visualization). Top-right panel: projection of the test dataset onto a two-dimensional space via the \texttt{tSNE} algorithm. Bottom panel: heatmap of the resulting confusion matrix.}
\label{fig:t-results}
\end{figure}
The bottom panel of Figure \ref{fig:t-results} reports the resulting confusion matrix: we observe that the algorithm successfully detected both novelties, achieving a satisfactory classification performance of the previously observed soil types. However, the model struggles with classifying the DGS instances, often mistaken for GS or VDGS. Such difficulty is explained by the overlap between these groups, as shown by the visualization of the test set obtained via the \texttt{tSNE} projection \citep{Hinton2008}, reported in the top-right panel of Figure \ref{fig:t-results}: from the plot, we see that it is not straightforward to establish clear boundaries between GS, DGS, and VDGS soil types.
Overall, \texttt{VBrand} captures the main traits of the data and flags some observations as outliers (e.g., Novelty clusters 4 and 5), which may warrant further investigation.
All in all, our variational approach provides a good clustering solution in a few seconds and it is fast enough to allow for a brute-force search for a better, albeit only locally optimal, solution employing multiple initializations.
\section{Discussion and conclusions}
\label{sec:disc}
In this paper we introduced \texttt{VBrand}, a variational Bayes algorithm for novelty detection, to classify instances of a test set that may conceal classes not observed in a training set. We showed how \texttt{VBrand} outperforms the previously proposed slice sampler implementation in terms of both computational time and robustness of the estimates. The application to soil data provides an example of the versatility of our method in a context where the MCMC algorithm fails because of the large dimensionality of the problem.\\
Our results pave the way for many possible extensions. First, given recent developments in Bayesian nonparametric literature, the variational algorithm can be enriched by adding a hyperprior distribution for the concentration parameter of the novelty DP \citep{Ascolani2022}. While in practice \texttt{VBrand} already obtains very good classification performance, this addition would lead to a consistent model for the number of true clusters. Second, we can consider different likelihood specifications and develop variational inference novelty detectors for, but not limited to, functional or graphical data. Third, at the expense of efficiency, we can explore more complex specifications for the variational distributions, as in structured variational inference. Albeit potentially slower, this choice would lead to an algorithm that could better capture the complex structure of the posterior distribution we are targeting. Finally, we can resort to stochastic variational inference algorithms \citep{Hoffman2003}, to scale up \texttt{VBrand}'s applicability to massive datasets that could benefit from novelty detection techniques.
\section{Acknowledgments}
The authors are grateful to Fabio Lavezzo for his help in the early-stage research that led to the development of this article.
\section{Full conditionals for Brand with multivariate Gaussian components}
\label{suppsec::gibbs}
Let $X\mid \cdots$ denote the full conditional distribution for the random variable $X$. Then, the full conditionals for the parameters of the model introduced in \cite{denti2021two} are as follows:
\begin{itemize}
\item The memberships labels are updated according to
\begin{equation*}
\mathbb{P}\left[\xi_m = k \mid \cdots\right] = \begin{cases}
\pi_k \cdot \mathcal{N}(\boldsymbol{y}_m\mid\boldsymbol\Theta_k^{obs})\cdot\mathbbm{1}_{\xi_m = k} &\quad\; k=1,\;...\;,J,\\
\pi_0\cdot v_k \left(\prod_{l=1}^{k-J-1}(1-v_l)\right)
\cdot \mathcal{N}(\boldsymbol{y}_m\mid\boldsymbol\Theta_k^{nov})\cdot\mathbbm{1}_{\xi_m = k}, &\quad k\geq J+1.\\
\end{cases}
\end{equation*}
\item The weights of the parametric mixtures and the stick-breaking variables are sampled from
\begin{equation*}
\boldsymbol\pi\mid \cdots \overset{}{\sim} Dirichlet\left(\{\alpha_k+n_k\}_{k=0}^J\right),
\quad \quad
v_k\mid \cdots \overset{}{\sim} Beta\left( n_{J+k}+1,\gamma+\sum_{l=k+1}^{\infty} n_{J+l}\right),
\end{equation*}
\item The mixture locations are updated according to
$
\boldsymbol{\Theta}_k^{q}\mid\cdots \overset{}{\sim} \mathcal\mathcal{NIW}(\boldsymbol{\mu}_n,\lambda_n,\nu_n,\boldsymbol{\Psi}_n)$, with $q\in\{obs, nov\}$, where
\begin{equation*}\begin{aligned}
& n_k = \begin{cases}
\#[\xi_m=k] & \text{if $k$ $\geq$ 1 },\\
\#[\xi_m>J] & \text{if $k$ = 0 },\\
\end{cases} \quad \quad
\boldsymbol\mu_n = \frac{\lambda_0\cdot\boldsymbol{\mu}_0 + \sum_{m=1}^{M}\boldsymbol{y}_m\cdot\mathbbm{1}_{\xi_m=k}}{\lambda_0+n_k},\\
& \lambda_n=\lambda_0+n_k,\quad \quad \nu_n=\nu_0+n_k,\\
& \boldsymbol\Psi_n=\boldsymbol\Psi_0+ \boldsymbol{S} + \frac{\lambda_0 \cdot n_k}{\lambda_0+n_k}\left(\frac{\sum_{m=1}^{M} \boldsymbol{y}_m\cdot\mathbbm{1}_{\xi_m=k}}{n_k} - \boldsymbol{\mu}_0\right)\left(\frac{\sum_{m=1}^{M} \boldsymbol{y}_m\cdot\mathbbm{1}_{\xi_m=k}}{n_k} - \boldsymbol{\mu}_0\right)^T,\\
& \boldsymbol{S}=\sum_{m=1}^{M}(\boldsymbol{y}_m-\overline{\boldsymbol{y}}^k)\cdot(\boldsymbol{y}_m-\overline{\boldsymbol{y}}^k)^T\cdot\mathbbm{1}_{\xi_m=k},
\quad \quad \overline{\boldsymbol{y}}_k = \frac{\sum_{m=1}^M\boldsymbol{y}_m\cdot\mathbbm{1}_{\xi_m=k}}{\sum_{m=1}^M\mathbbm{1}_{\xi_m=k}}.\\
\end{aligned}\end{equation*}
Here, ($\boldsymbol{\mu}_0$, $\lambda_0$, $\nu_0$, $\boldsymbol\Psi_0$) indicate the hyperparameters of the relative ${NIW}$ distribution, according to which atom $\boldsymbol{\Theta}_k^{q}$ is being considered.\\
\end{itemize}
\section{Additional details regarding the VBrand algorithm}
\label{suppsec::cavi_elbo}
\subsection{Quantities useful to devise the CAVI updating rules}
Here, we report some additional formulas to clarify the derivation of the CAVI algorithm for VBrand. In our implementation, we used the following well-known expected values:
\begin{itemize}
\item $
\mathbb{E}[\log{\pi_k }] = \boldsymbol{\psi}(\eta_k)-\boldsymbol{\psi}(\overline{\eta}),$ with $\overline{\eta} = \sum_k \eta_k$
\item $
\mathbb{E}[\log{v_{k-J}}] = \boldsymbol{\psi}(a_{k-J})-\boldsymbol{\psi}(a_{k-J}+b_{k-J})$ and $ \mathbb{E}[\log{(1-v_{l})}] = \boldsymbol{\psi}(b_{l})-\boldsymbol{\psi}(a_{l}+b_{l})$,
\item If $\Sigma^{-1} \sim \textit{Wishart}$, we have that:
\begin{equation*}
\mathbb{E}[\log{\mid\Sigma^{-1}\mid}] = -p\log{2} - \log{\mid\boldsymbol{\Psi}_k^{-1}\mid} - \sum_{l=1}^p\boldsymbol{\psi}\left(\frac{\nu_k-l+1}{2}\right).
\end{equation*}
\end{itemize}
Given the previous results and the updating rules derived in Section \ref{suppsec::gibbs}, we can easily derive the value for the negative entropy of a multivariate Normal distribution for $q\in\{obs,nov\}$:
\begin{equation*}\begin{split}
\mathbb{E}[\log{\mathcal{N}(\boldsymbol{y}_m\mid\boldsymbol{\Theta}_k^{q})}] &= \frac{1}{2}\left(-p\log{2\pi} + \sum_{l=1}^p\boldsymbol{\psi}\left(\frac{\nu_k-l+1}{2}\right) + p\log{2} + \log{\mid\boldsymbol{\Psi}_k^{-1}\mid} \right) + \\& -\frac{1}{2} \left( \frac{p}{\lambda_k} +
\nu_k(\boldsymbol{y}_m-\boldsymbol{\mu}_k)^T\boldsymbol{\Psi}_k^{-1}(\boldsymbol{y}_m-\boldsymbol{\mu}_k)\right).
\end{split}\end{equation*}
\subsection{Quantities useful to derive the Elbo components}
In the following formulas, $Tr(\cdot)$ denotes the trace of a matrix. Moreover, the expression of the function $B(\cdot,\cdot)$ can be found in \cite{Bishop2006}, Appendix B, p.693.
\begin{equation*}\begin{split}
\mathbb{E}[\log( \mathcal{NIW}(\boldsymbol{\Theta}_k^{obs}\mid\boldsymbol{\varrho}_k
))] =& \frac{1}{2}\big(p\log{\frac{\lambda_k^{obs}}{2\pi}} + \sum_{i=1}^p\psi(\frac{u_k^{obs}-i+1}{2}) + \log{\mid\boldsymbol{S}_k^{obs^{-1}}\mid} +\\
&- p\frac{\lambda_k^{obs}}{l_k^{obs}} - \lambda_k^{obs}u_k^{obs}(\boldsymbol{m}_k^{obs}-\boldsymbol{\mu}_k^{obs})^T\boldsymbol{S}_k^{obs^{-1}}(\boldsymbol{m}_k^{obs}-\boldsymbol{\mu}_k^{obs})\big) +\\
&+\log{B(\boldsymbol{\Psi}_k^{obs^{-1}},\nu_k^{obs})} - \frac{1}{2}u_k^{obs} Tr(\boldsymbol{\Psi}_k^{obs}\boldsymbol{S}_k^{obs^{-1}}) + const,
\end{split}\end{equation*}
\begin{equation*}\begin{split}
\mathbb{E}[\log( \mathcal{NIW}(\boldsymbol{\Theta}_k^{nov}\mid\boldsymbol{\varrho}_0
))] =& \frac{1}{2}\big(p\log{\frac{\lambda_0^{nov}}{2\pi}} + \sum_{i=1}^p\psi(\frac{u_k^{nov}-i+1}{2}) + \log{\mid\boldsymbol{S}_k^{nov^{-1}}\mid} +\\
&- p\frac{\lambda_0^{nov}}{l_k^{nov}} - \lambda_0^{nov}u_k^{nov}(\boldsymbol{m}_k^{nov}-\boldsymbol{\mu}_0^{nov})^T\boldsymbol{S}_k^{nov^{-1}}(\boldsymbol{m}_k^{nov}-\boldsymbol{\mu}_0^{nov})\big) +\\
&+\log{B(\boldsymbol{\Psi}_0^{nov^{-1}},\nu_0^{nov})} - \frac{1}{2}u_0^{nov} Tr(\boldsymbol{\Psi}_0^{nov}\boldsymbol{S}_k^{nov^{-1}}) + const,
\end{split}\end{equation*}
\begin{equation*}\begin{aligned}
\mathbb{E}[\log( \mathcal{NIW}(\boldsymbol{\Theta}_k^{obs}\mid\boldsymbol{\rho}_k^{obs}))]\ = \frac{u_k^{obs}}{2}\log{\mid\boldsymbol{S}_k^{obs}\mid} - \frac{u_k^{obs} p}{2}\log{2} - \log\Gamma_p\left(\frac{u_k^{obs}}{2}\right) + \frac{p}{2}\log{l_k^{obs}} + \\
+ \frac{u_k^{obs} + p + 2}{2} \left(\log{\mid\boldsymbol{S}_k^{obs^{-1}}\mid} + \sum_{i=1}^p\boldsymbol{\psi}\left(\frac{u_k^{obs}-i+1}{2}\right)\right) - \frac{u_k^{obs}}{2} p,
\end{aligned}\end{equation*}
\begin{equation*}\begin{aligned}
\mathbb{E}[\log( \mathcal{NIW}(\boldsymbol{\Theta}_k^{nov}\mid\boldsymbol{\rho}_k^{nov}))]\ = \frac{u_k^{nov}}{2}\log{\mid\boldsymbol{S}_k^{nov}\mid} - \frac{u_k^{nov} p}{2}\log{2} - \log\Gamma_p\left(\frac{u_k^{nov}}{2}\right) + \frac{p}{2}\log{l_k^{nov}} + \\
+ \frac{u_k^{nov} + p + 2}{2} \left(\log{\mid\boldsymbol{S}_k^{nov^{-1}}\mid} + \sum_{i=1}^p\boldsymbol{\psi}\left(\frac{u_k^{nov}-i+1}{2}\right)\right) - \frac{u_k^{nov}}{2} p.
\end{aligned}\end{equation*}
\newpage
\section{Additional figures}
\label{suppsec::addfig}
\begin{figure}[!ht]
\centering
\includegraphics[width =.8\linewidth]{figures_suppmat/comp_vs_simp.pdf}
\caption{Example of the first two dimensions of the generated data under the simulation study described in Section 4.2 of the main paper.}
\label{fig:supp_simple_vs_complex}
\end{figure}
\FloatBarrier
\begin{figure}[!ht]
\centering
\includegraphics[width = .8\linewidth]{figures_suppmat/inds_VBvsMCMC_mox_easy1.pdf}
\caption{Results of the simulation study described in Section 4.2 of the main paper under the simple scenario.}
\label{fig:supp_simple_res}
\end{figure}
\FloatBarrier
\clearpage
\section{Hyperparameter initializations for the novel soil detection analysis}
\label{suppsec::soil}
We report a detailed list of the hyperparameters initialization of the VBrand model used for the novelty detection of soil type in Section 5 of the main paper.
\begin{enumerate}
\item The truncation is set at $T=10$. \item The concentration parameter of the stick-breaking process is set to $\gamma=5$.
\item The hyperparameters of the Dirichlet distribution are all set to $\alpha_j = 0.1\;\;\forall j$, inducing the same probability of being in each known component and of being a novelty.
\item The $\mathcal{NIW}$ prior for the novelty locations is parameterized as follows: $\boldsymbol{\mu}_0^{nov}$ is set equal to the grand mean of the training set, to allow an hyperprior specification that does not rely on the test set; the precision is given as
${\lambda}_0^{nov} = 0.1$
the degrees of freedom ${\nu}_0^{nov}= p+2$, where $p$ is the dimensionality of the dataset. This is the minimum integer value that is allowed to have so that the $\mathcal{IW}$ distribution has a finite mean. Finally, the variance is set to be $(p+1)$ times the training sample covariance matrix. This choice implies that the expected variance sampled from the $\mathcal{NIW}$ is an inflated version of the overall sample covariance matrix of the training set.
\item For the known components, the mean vector, and the variance-covariance matrices are estimated via the MRCD procedure. To put high mass on these estimates, for all the known groups, we set $\lambda_k^{obs}= 200$ and $\nu^{obs}_k = p+1+200$.
\end{enumerate}
\section*{Supplementary Material}
\input{core_suppmat}
\end{document} |
1,941,325,220,546 | arxiv | \section{Introduction}
Given a complete filtered probability space $(\Omega,\F,\FF,P)$ satisfying the usual hypotheses, let $R$ be an optional process of class $(D)$, and consider the optimal stopping problem
\begin{equation}\label{os}\tag{OS}
\maximize\quad ER_\tau\quad\ovr\quad \tau\in\T,
\end{equation}
where $\T$ is the set of stopping times with values in $[0,T]\cup\{T+\}$ and $R$ is defined to be zero on $T+$. We allow $T$ to be $\infty$ in which case $[0,T]$ is interpreted as the one-point compactification of the positive reals.
Without further conditions, optimal stopping times need not exist (take any deterministic process $R$ whose supremum is not attained).
Theorem~II.2 of Bismut and Skalli~\cite{bs77} establishes the existence for bounded reward processes $R$ such that $R\ge\cev R$ and $\vec R\le\pp R$. Here,
\[
\vec R_t:=\limsup_{s\upto t} R_s\quad\text{and}\quad\cev R_t:=\limsup_{s\downto t} R_s,
\]
the {\em left-} and {\em right-upper semicontinuous regularizations} of $R$, respectively. Bismut and Skalli mention on page 301 that, instead of boundedness, it would suffice to assume that $R$ is of class $(D)$.
In order to extend the above, we study the ``optimal quasi-stopping problem''
\begin{equation}\label{oqs}\tag{OQS}
\maximize\quad E[R_\tau+\vec R_{\tilde \tau}]\quad\ovr\quad (\tau,\tilde\tau)\in\hat\T,
\end{equation}
where $\hat\T$ is the set of {\em quasi-stopping times} (``split stopping time'' in Dellacherie and Meyer~\cite{dm82}) defined by
\[
\hat\T:=\{(\tau,\tilde\tau)\in\T\times\T_p \mid \tilde\tau>0,\ \tau\vee\tilde\tau =T+ \},
\]
where $\T_p$ is the set of predictable times. When $R$ is cadlag, $\vec R=R_-$, and our formulation of the quasi-optimal stopping coincides with that of Bismut~\cite{bis79}. Our main result gives the existence of optimal quasi-stopping times when $R\ge\cev R$. When $R \ge \cev R$ and $\vec R\le \pp R$, we obtain the existence for \eqref{os} thus extending the existence result of \cite[Theorem~II.2]{bs77} to possibly unbounded processes $R$ as suggested already on page 301 of \cite{bs77}.
Our existence proofs are based on functional analytical arguments that avoid the use of Snell envelopes which are used in most analyses of optimal stopping. Our strategy is to first look at a convex relaxation of the problem. This turns out be a linear optimization problem over a compact convex set of random measures whose extremal points can be identified with (quasi-)stopping times. As soon as the objective is upper semicontinuous on this set, Krein-Milman theorem gives the existence of (quasi-)stopping times. Sufficient conditions for upper semicontinuity are obtained as a simple application of the main result of Perkki\"o and Trevino~\cite{pt18a}. The overall approach was suggested already on page 287 of Bismut~\cite{bis79b} in the case of optimal stopping. We extended the strategy (and provide explicit derivations) to quasi-optimal stopping for a merely right-upper semicontinuous reward process.
The last section of the paper develops a dual problem and optimality conditions for optimal (quasi-)stopping problems. The dual variables turn out to be martingales that dominate $R$. As a simple consequence, we obtain the duality result of Davis and Karatzas~\cite{dk94} in a more general setting where the reward process $R$ is merely of class $(D)$.
\section{Regular processes}\label{sec:regular}
In this section, the reward process $R$ is assumed to be {\em regular}, i.e.\ of class $(D)$ such that the left-continuous version $R_-$ and the predictable projection $\pp R$ of $R$ are indistinguishable; see e.g.\ \cite{bis78} or \cite[Remark~50.d]{dm82}. Our analysis will be based on the fact that the space of regular processes is a Banach space whose dual can be identified with optional measures of essentially bounded variation; see Theorem~\ref{thm:reg1} below.
The space $M$ of Radon measures may be identified with the space $X_0$ of left-continuous functions of bounded variation on $\reals_+$ which are constant on $(T,\infty]$ and $x_0=0$. Indeed, for every $x\in X_0$, there exists a unique $Dx\in M$ such that $x_t=Dx([0,t))$ for all $t\in\reals$. Thus $x\mapsto Dx$ defines a linear isomorphism between $X_0$ and $M$. The value of $x$ for $t>T$ will be denoted by $x_{T+}$. Similarly, the space $\M^\infty$ of optional random measures with essentially bounded total variation may be identified with the space $\N_0^\infty$ of adapted processes $x$ with $x\in X_0$ almost surely and $Dx\in\M^\infty$.
Let $C$ the space of continuous functions on $[0,T]$ equipped with the supremum norm and let $L^1(C)$ be the space of (not necessarily adapted) continuous processes $y$ with $E\|y\|<\infty$. The norm $E\|y\|$ makes $L^1(C)$ into a Banach space whose dual can be identified with the space $L^\infty(M)$ of random measures whose pathwise total variation is essentially bounded. The following result is essentially from \cite{bis78}; see \cite[Theorem~8]{pp18c} or \cite[Corollary~16]{pp18b}. It provides the functional analytic setting for analyzing optimal stopping with regular processes.
\begin{theorem}\label{thm:reg1}
The space $\R^1$ of regular processes equipped with the norm
\[
\|y\|_{\R^1}:=\sup_{\tau\in\T}E|y_\tau|
\]
is Banach and its dual can be identified with $\M^\infty$ through the bilinear form
\[
\langle y,u\rangle =E\int ydu.
\]
The optional projection is a continuous surjection of $L^1(C)$ to $\R^1$ and its adjoint is the embedding of $\M^\infty$ to $L^\infty(M)$. The norm of $\R^1$ is equivalent to
\[
p(y):= \inf_{z\in L^1(C)} \{E\|z\| \mid \op z = y\}
\]
which has the dual representation
\[
p(y)=\sup\{\langle y,u\rangle\,|\,\esssup(\|u\|)\le 1\}.
\]
\end{theorem}
We first write the optimal stopping problem as
\[
\maximize\quad \langle R, Dx\rangle\quad\ovr\quad x\in\C_e,
\]
where
\[
\C_e:=\{x\in\N_0^\infty\,|\, Dx\in\M^\infty_+,\ x_t\in\{0,1\}\}.
\]
The equation $\tau(\omega) = \inf\{t\in\reals\mid x_t(\omega)\ge 1\}$ gives a one-to-one correspondence between the elements of $\T$ and $\C_e$. Consider also the convex relaxation
\begin{equation}\label{Rcr}\tag{ROS}
\maximize\quad \langle R,Dx\rangle\quad\ovr\quad x\in\C,
\end{equation}
where
\[
\C:=\{x\in\N_0^\infty\,|\, Dx\in\M^\infty_+,\ x_{T+}\le 1\}.
\]
Clearly, $\C_e\subset\C$ so the optimum value of optimal stopping is dominated by the optimum value of the relaxation. The elements of $\C$ are {\em randomized stopping times} in the sense of Baxter and Chacon~\cite[Section~2]{bc77}.
Recall that $x\in\C$ is an {\em extreme point} of $\C$ if it cannot be expressed as a convex combination of two points of $\C$ different from $x$.
\begin{lemma}\label{lem:km}
The set $\C$ is convex, $\sigma(\N_0^\infty,\R^1)$-compact and $\C_e$ is the set of its extreme points.
\end{lemma}
\begin{proof}
The set $\C$ is a closed convex set of the unit ball that $\N_0^\infty$ has as the dual of the Banach space $\R^1$. The compactness thus follows from Banach-Alaoglu. It is easily shown that the elements of $\C_e$ are extreme points of $\C$. On the other hand, if $x\notin\C_e$ there exists an $\bar s\in(0,1)$ such that the processes
\[
x^1_t:=\frac{1}{\bar s}[x_t\wedge\bar s]\quad\text{and}\quad x^2_t:=\frac{1}{1-\bar s}[(x_t-\bar s)\vee 0]
\]
are different elements of $\C$. Since $x=\bar s x^1+(1-\bar s)x^2$, it is not an extreme point of $\C$.
\end{proof}
Since the function $x\mapsto\langle R,Dx\rangle$ is continuous, the compactness of $\C$ in Lemma~\ref{lem:km} implies that the maximum in \eqref{Rcr} is attained. The fact that the maximum is attained at a genuine stopping time follows from the characterization of the extreme points in Lemma~\ref{lem:km} and the following variant of the Krein-Millman theorem; see e.g.~\cite[Theorem~25.9]{cho69}.
\begin{theorem}[Bauer's maximum principle]\label{thm:bauer}
In a locally convex Hausdorff topological vector space, an upper semicontinuous (usc) convex function on a compact convex set $K$ attains its maximum at an extremal point of $K$.
\end{theorem}
Combining Lemma~\ref{lem:km} and Theorem~\ref{thm:bauer} gives the following.
\begin{theorem}\label{thm:os}
Optimal stopping time in \eqref{os} exists for every $R\in\R^1$.
\end{theorem}
The above seems to have been first proved in Bismut and Skalli~\cite[Theorem~I.3]{bs77}, which says that a stopping time defined in terms of the Snell envelope of the regular process $R$ is optimal. Their proof assumes bounded reward $R$ but they note on page~301 that it actually suffices that $R$ be of class $(D)$. The proof of Bismut and Skalli builds on the (nontrivial) existence of a Snell envelope and further limiting arguments involving sequences of stopping times. In contrast, our proof is based on elementary functional analytic arguments in the Banach space setting of Theorem~\ref{thm:reg1}, which is of independent interest.
Note that $x$ solves the relaxed optimal stopping problem if and only if $R$ is {\em normal} to $\C$ at $x$, i.e.\ if $R\in\partial\delta_\C(x)$ or equivalently $x\in\partial\sigma_\C(R)$, where
\[
\sigma_\C(R) = \sup_{x\in\C}\langle R,Dx\rangle.
\]
Here, $\partial$ denotes the {\em subdifferential} of a function; see e.g.\ \cite{roc74}. If $R$ is nonnegative, we have $\sigma_\C(R)=\|R\|_{\R^1}$ (by Krein--Milman) and the optimal solutions of the relaxed stopping problem are simply the subgradients of the $\R^1$-norm at $R$.
\section{Cadlag processes}\label{sec:cadlag}
This section extends the previous section to optimal quasi-stopping problems when the reward process $R$ is merely {\em cadlag and of class $(D)$}. In this case, optimal stopping times need not exist (see the discussion on page 1) but we will prove the existence of a quasi-stopping time by functional analytic arguments analogous to those in Section~\ref{sec:regular}.
The Banach space of cadlag functions equipped with the supremum norm will be denoted by $D$. The space of purely discontinuous Borel measures will be denoted by $\tilde M$. The dual of $D$ can be identified with $M\times\tilde M$ through the bilinear form
\[
\langle y,(u,\tilde u)\rangle := \int ydu + \int y_-d\tilde u
\]
and the dual norm is given by
\[
\sup_{y\in D}\left\{\left.\int ydu + \int y_-d\tilde u\,\right|\,\|y\|\le 1\right\}=\|u\|+\|\tilde u\|,
\]
where $\|u\|$ denotes the total variation norm on $M$. This can be deduced from \cite[Theorem~1]{pes95} or seen as the deterministic special case of \cite[Theorem~VII.65]{dm82} combined with \cite[Remark~VII.4(a)]{dm82}.
The following result from \cite{pp18b} provides the functional analytic setting for analyzing quasi-stopping problems with cadlag processes of class $(D)$.
\begin{theorem}\label{thm:L1}
The space $\D^1$ of optional cadlag processes of class $(D)$ equipped with the norm
\[
\|y\|_{\D^1}:=\sup_{\tau\in\T}E|y_\tau|
\]
is Banach and its dual can be identified with
\[
\hat\M^\infty:=\{(u,\tilde u)\in L^\infty(M\times\tilde M)\mid u\text{ is optional},\, \tilde u\text{ is predictable}\}
\]
through the bilinear form
\[
\langle y,(u,\tilde u)\rangle =E\left[\int ydu+\int y_-d\tilde u\right].
\]
The optional projection is a continuous surjection of $L^1(D)$ to $\D^1$ and its adjoint is the embedding of $\hat\M^\infty$ to $L^\infty(M\times \tilde M)$. The norm of $\D^1$ is equivalent to
\[
p(y):= \inf_{z\in L^1(D)} \{E\|z\| \mid \op z = y\},
\]
which has the dual representation
\[
p(y)=\sup\{\langle y,(u,\tilde u)\rangle\,|\,\esssup(\|u\|+\|\tilde u\|)\le 1\}.
\]
\end{theorem}
The space $M\times\tilde M$ may be identified with the space $\hat X_0$ of (not necessarily left-continuous) functions $x:\reals_+\to\reals$ of bounded variation which are constant on $(T,\infty]$ and have $x_0=0$. Indeed, every $x\in\hat X_0$ can be written uniquely as
\[
x_t = Dx([0,t)) + \tilde Dx([0,t]),
\]
where $\tilde Dx\in\tilde M$ and $Dx\in M$ are the measures associated with the functions $\tilde x_t :=\sum_{s\le t} (x_s-x_{s-})$ and $x-\tilde x$, respectively. The linear mapping $x\mapsto(Dx,\tilde Dx)$ defines an isomorphism between $\hat X_0$ and $M\times\tilde M$. The value of $x$ for $t>T$ will be denoted by $x_{T+}$. Similarly, the space $\hat \M^\infty$ may be identified with the space $\hat\N_0^\infty$ of predictable processes $x$ with $x\in \hat X_0$ almost surely and $(Dx,\tilde Dx)\in\hat\M^\infty$.
Problem \eqref{oqs} can be written as
\[
\maximize\quad \langle R, (Dx,\tilde Dx)\rangle\quad\ovr\quad x\in\hat\C_e,
\]
where
\[
\hat\C_e:=\{x\in\hat\N_0^\infty\,|\, (Dx,\tilde Dx)\in\hat\M^\infty_+,\ x_t\in\{0,1\}\}.
\]
Indeed, the equations $\tau(\omega) = \inf\{t\in\reals\mid x_t(\omega)\ge 1\}$ and $\tilde\tau(\omega) = \inf\{t\in\reals\mid x_t-x_{t-}(\omega)\ge 1\}$ give a one-to-one correspondence between the elements of $\hat\T$ and $\hat\C_e$.
Consider also the convex relaxation
\begin{equation}\label{Dcr}\tag{ROQS}
\maximize\quad \langle R,(Dx,\tilde Dx)\rangle\quad\ovr\quad x\in\hat\C,
\end{equation}
where
\[
\hat\C:=\{x\in\hat\N_0^\infty\,|\, (Dx,\tilde Dx)\in\hat\M^\infty_+,\ x_{T+}\le 1\}.
\]
\begin{lemma}\label{lem:cadkm}
The set $\hat\C$ is convex, $\sigma(\hat \M^\infty,\D^1)$-compact and the set of quasi-stopping times $\hat \C_e$ is its extreme points. Moreover, the set of stopping times is $\sigma(\hat \M^\infty,\D^1)$-dense in $\hat \C_e$ and, thus, $\C$ is $\sigma(\hat \M^\infty,\D^1)$-dense in $\hat\C$.
\end{lemma}
\begin{proof}
The set $\hat\C$ is a closed convex set of the unit ball that $\hat\N_0^\infty$ has as the dual of the Banach space $\D^1$. The compactness thus follows from Banach-Alaoglu. It is easily shown that the elements of $\hat\C_e$ are extreme points of $\hat\C$.
If $x\notin\hat\C_e$, there exist $\bar s\in(0,1)$ such that
\begin{align*}
x^1_t &:=\frac{1}{\bar s }[x_t\wedge\bar s],\quad\quad x^2_t:=\frac{1}{1-\bar s}[(x_t-\bar s)\vee 0]
\end{align*}
are distinguishable processes that belong to $\hat\C$. Since $x=\bar s x^1+(1-\bar s)x^2$, $x$ is not an extremal in $\hat C$.
To prove the last claim, let $(\tau,\tilde\tau)$ be a quasi-stopping time and $(\tau^\nu)$ an announcing sequence for $\tilde\tau$. We then have
\[
\langle(\delta_{\tau\wedge\tau^\nu},0),y\rangle\to\langle(\delta_\tau,\delta_{\tilde\tau}),y\rangle
\]
for every $y\in\D^1$.
\end{proof}
Just like in Section~\ref{sec:regular}, a combination of Lemma~\ref{lem:cadkm} and Theorem~\ref{thm:bauer} gives the following existence result which was established in Bismut~\cite{bis79} using more elaborate techniques based on the existence of Snell envelopes.
\begin{theorem}\label{thm:os}
If $R\in\D^1$, then optimal quasi-stopping time in \eqref{oqs} exists and the optimal values of \eqref{oqs}, \eqref{os} and \eqref{Dcr} are all equal.
\end{theorem}
As another implication of Lemma~\ref{lem:cadkm} and Theorem~\ref{thm:L1}, we recover the following result of Bismut which says that the seminorms in Theorem~\ref{thm:L1} are not just equivalent but equal
\begin{theorem}[{\cite[Theorem~4]{bis78}}]\label{thm:equiv1}
For every $y\in\D^1$,
\[
\|y\|_{\D^1}=\inf_{z\in L^1(D)}\{E\|z\|_D\mid \op z = y\}.
\]
\end{theorem}
\begin{proof}
The expression on the right is the seminorm $p$ in Theorem~\ref{thm:L1} with the dual representation
\[
p(y)=p(|y|)=\sup_{x\in\hat\C}\langle |y|,(Dx,\tilde Dx)\rangle
\]
which, by Theorem~\ref{thm:os}, equals the left side.
\end{proof}
Combining the above with Theorem~\ref{thm:reg1} gives a simple proof of the following.
\begin{theorem}[{\cite[Theorem~3]{bis78}}]\label{thm:equivreg}
For every $y\in\R^1$,
\[
\|y\|_{\R^1}=\inf_{z\in L^1(C)}\{E\|z\|_D\mid \op z = y\}.
\]
\end{theorem}
\begin{proof}
By Jensen's inequality, the left side is less than the right which is the seminorm $p$ in Theorem~\ref{thm:reg1} with the dual representation
\begin{align*}
p(y) &= \sup\{\langle y,u\rangle\,|\,\esssup(\|u\|)\le 1\}\\
&\le \sup\{\langle y,(u,\tilde u)\rangle\,|\,\esssup(\|u\|+\|\tilde u\|)\le 1\}\\
&=\sup_{x\in\hat\C}\langle |y|,(Dx,\tilde Dx)\rangle,
\end{align*}
which, again by Theorem~\ref{thm:os}, equals the left side.
\end{proof}
\section{Non-cadlag processes}\label{sec:usc}
This section gives a further extension to cases where the reward process is not necessarily cadlag but merely {\em right-upper semicontinuous} (right-usc) in the sense that $R\ge\cev R$. In this case, the objective of the relaxed quasi-optimal stopping problem \eqref{Dcr} need not be continuous. The following lemma says that it is, nevertheless, upper semicontinuous, so Bauer's maximum principle still applies.
\begin{lemma}\label{lem:usc}
If $R$ is right-usc and of class $(D)$, then the functional
\begin{align*}
\hat\J(u,\tilde u)=\begin{cases}
E\left[\int Rdu + \int \vec R d\tilde u\right]\quad &\text{if }(u,\tilde u)\in\hat\M^\infty_+\\
-\infty\quad&\text{otherwise}
\end{cases}
\end{align*}
is $\sigma(\hat\M^\infty,\D^1)$-usc.
\end{lemma}
\begin{proof}
Recalling that every optional process of class $(D)$ has a majorant in $\D^1$ (see \cite[Remark 25, Appendix I]{dm82}), the first example in \cite[Section~8]{pt18a} shows, with obvious changes of signs, that $\hat\J$ is usc.
\end{proof}
Combining Lemma~\ref{lem:usc} with Theorem~\ref{thm:bauer} gives the existence of a relaxed quasi-stopping time at an extreme point of $\C$ which, by Lemma~\ref{lem:cadkm}, is a quasi-stopping time. We thus obtain the following.
\begin{theorem}\label{thm:usc}
If $R$ is right-usc and of class $(D)$, then \eqref{oqs} has a solution.
\end{theorem}
We have not been able find the above result in the literature but it can be derived from Theorem~2.39 of El Karoui~\cite{elk81} on ``divided stopping times'' (temps d'arret divis\'es). A recent analysis of divided stopping times can be found in Bank and Besslich \cite{bb18a}. These works extend Bismut's approach on optimal quasi-stopping by dropping the assumption of right-continuity and augmenting quasi-stopping times with a third component that acts on the right limit of the reward process. Much like Bismut's approach, \cite{elk81,bb18a} build on the existence of a Snell envelope.
Theorem~\ref{thm:usc} yields the existence of an optimal stopping time when the reward process $R$ is {\em subregular} in the sense that it is right-usc, of class $(D)$ and $\vec R\le\pp R$.
\begin{theorem}\label{thm:uscreg}
If $R$ is subregular, then \eqref{os} has a solution and its optimum value equals that of \eqref{oqs}.
\end{theorem}
\begin{proof}
Clearly, the optimum value of \eqref{oqs} is at least that of \eqref{os} while for subregular $R$,
\[
E[R_\tau+\vec R_{\tilde\tau}] \le E[R_\tau+\pp R_{\tilde\tau}] = E[R_\tau+R_{\tilde\tau}] = ER_{\tau\wedge\tilde\tau},
\]
where the first equality holds by the definition of predictable projection. The claim now follows from Theorem~\ref{thm:usc}.
\end{proof}
The above seems to have been first established in Bismut and Skalli~\cite[Section~II]{bs77} for bounded $R$ (again, they mention on page 301 that, instead of boundedness, it would suffice to assume that $R$ is of class $(D)$).
Regularity properties are preserved under compositions with convex functions much like martingale properties. Indeed, if $R$ is regular and $g$ is a real-valued convex function on $\reals$ then $g(R)$ is subregular as soon as it is of class $(D)$. Indeed, for any $\tau\in\T_p$, conditional Jensen's inequality gives
\[
E[g(\vec R_\tau)\one_{\tau<+\infty}]= E[g(\pp R_\tau)\one_{\tau<+\infty}] \le E[g(R_\tau)\one_{\tau<+\infty}].
\]
Similarly, if $R$ is subregular and $g$ is a real-valued increasing convex function, then $g(R)$ is subregular as soon as the composition is of class $(D)$.
\section{Duality}
We end this paper by giving optimality conditions and a dual problem for the optimal stopping problems. The derivations are based on the conjugate duality framework of~\cite{roc74} which addresses convex optimization in general locally convex vector spaces.
The results below establish the existence of dual solutions without assuming the existence of optimal (quasi-)stopping times. They hold without any path properties as long as the reward process $R$ is of class $(D)$.
We denote the space of martingales of class $(D)$ by $\R^1_m$.
\begin{theorem}\label{thm:osdual}
Let $R$ be of class $(D)$. Then the optimum values of \eqref{oqs} and \eqref{os} coincide and equal that of
\begin{equation}\label{d}\tag{DOS}
\inf\{EM_0 \mid M\in\R^1_m,\ R\le M\},
\end{equation}
where the infimum is attained.
Moreover, $x\in\hat\C$ is optimal in the convex relaxation of \eqref{oqs} if and only if there exists $M\in\R^1_m$ with $R\le M$ and
\begin{align}
\int (M-R)dx+\int(M_--\vec R)d\tilde x &= 0,\label{eq:oc1}\\
x_{T+}=1\quad\text{or}\quad M_T&=0\label{eq:oc2}
\end{align}
almost surely. Thus, $(\tau,\tilde\tau)\in\hat\T$ is optimal in \eqref{oqs} if and only if there exists $M\in\R^1_m$ with $R\le M$, $M_\tau=R_\tau$, $M_{\tilde \tau_-} =\vec R_{\tilde\tau}$ and almost surely either $\tau+\tilde\tau<\infty+$ or $M_T=0$.
In particular, $x\in\C$ is optimal in the convex relaxation of \eqref{os} if and only if there exists $M\in\R^1_m$ with $R\le M$ and
\begin{align*}
\int (M-R)dx&= 0,\\
x_{T+}=1\quad\text{or}\quad M_T&=0
\end{align*}
almost surely. Thus, $\tau\in\T$ is optimal in \eqref{os} if and only if there exists $M\in\R^1_m$ with $R\le M$, $M_\tau=R_\tau$ and almost surely either $\tau<\infty+$ or $M_T=0$.
\end{theorem}
\begin{proof}
By \cite[Remark~25, Appendix~I]{dm82}, there are measurable processes $z$ and $\tilde z$ such that $R =\op z$, $\vec R=\op{\tilde z}$ and $E[\sup_tz_t+\sup_t\tilde z_t]<\infty$. The optimum value and optimal solutions of \eqref{oqs} coincide with those of
\begin{align}\label{eq:ros}
&\maximize_{x\in\hat\N^\infty}\quad E\left[\hat \J(Dx,\tilde Dx) - \rho(x_{T+}-1)^+\right],
\end{align}
where $\rho:=\sup_tz_t+\sup_t\tilde z_t+1$ and $\hat\J$ is defined as in Lemma~\ref{lem:usc}. Indeed, if $x$ is feasible in \eqref{eq:ros} then $\bar x:=x\wedge 1$ is feasible in \eqref{oqs} and since $x-\bar x$ is an increasing process with $(x-\bar x)_{T+}=(x_{T+}-1)^+$, we get
\begin{align*}
\hat\J(D\bar x,\tilde D\bar x) &= \hat\J(Dx,\tilde Dx) - \hat\J(D(x-\bar x),\tilde D(x-\bar x))\\
&\ge \hat\J(Dx,\tilde Dx) - E\rho(x_{T+}-1)^+.
\end{align*}
Problem~\eqref{eq:ros} fits the general conjugate duality framework of \cite{roc74} with $U=L^\infty$, $Y=L^1$ and
\[
F(x,w) = -\hat\J(Dx,\tilde Dx) + E\rho(x_{T+}+w-1)^+.
\]
By \cite[Theorem~22]{roc74}, $w\to F(0,w)$ is continuous on $L^\infty$ in the Mackey topology that it has as the dual of $L^1$. Thus, by \cite[Theorem~17]{roc74}, the optimum value of \eqref{eq:ros} coincides with the infimum of the dual objective
\[
g(y) :=-\inf_{x\in\hat\N^\infty} L(x,y),
\]
where $L(x,y) :=\inf_{w\in L^\infty}\{F(x,w)-Ewy\}$, and moreover, the infimum of $g$ is attained. By the interchange rule \cite[Theorem~14.60]{rw98},
\begin{align*}
L(x,y)&=
\begin{cases}
+\infty & \text{if $x\notin\hat\N^\infty_+$},\\
-\hat\J(Dx,\tilde Dx) + E\left[\inf_{u\in\reals}\{\rho(x_{T+}+u-1)^+ - uy\}\right]&\text{otherwise}\\
\end{cases}\\
&=
\begin{cases}
+\infty & \text{if $x\notin\hat\N^\infty_+$},\\
-\hat\J(Dx,\tilde Dx)+ E\left[x_{T+}y-y - \delta_{[0,\rho]}(y)\right]&\text{otherwise}.
\end{cases}
\end{align*}
We have
\[
E[x_{T+}y] = E[\int(y\one)dx+\int(y\one)d\tilde x] = \langle M,(Dx,\tilde Dx)\rangle,
\]
where $M=\op(y\one)\in\R^1_m$. Thus,
\[
L(x,y) =
\begin{cases}
+\infty & \text{if $x\notin\hat\N^\infty_+$},\\
-\hat\J(Dx,\tilde Dx) + \langle M,(Dx,\tilde Dx)\rangle - EM_T & \text{if $x\in\hat\N^\infty_+$ and $0\le M_T\le\rho$,}\\
-\infty & \text{otherwise}.
\end{cases}
\]
The dual objective can be written as
\begin{align*}
g(y) &=
\begin{cases}
EM_0 & \text{if $0\le M_T\le\rho$, $M\ge R$ and $M_-\ge\vec R$},\\
+\infty & \text{otherwise}.
\end{cases}
\end{align*}
Since $M$ is cadlag, $M_-\ge\vec R$ holds automatically when $M\ge R$. In summary, the optimum value of \eqref{oqs} equals that of \eqref{d}.
The dual problem of \eqref{os} is obtained similarly by defining
\[
F(x,w) = -\J(Dx) + E\rho(x_{T+}+w-1)^+.
\]
The function $w\to F(0,w)$ is again Mackey-continuous on $L^\infty$ and one finds that the dual is again \eqref{d}. Thus, the optimum value of \eqref{os} equals that of \eqref{d}.
As to the optimality conditions, \cite[Theorem~15]{roc74} says that $x$ is optimal in \eqref{eq:ros} and $y$ is optimal in the dual if and only i
\[
0\in\partial_x L(x,y),\quad 0\in\partial_y[-L](x,y).
\]
The former means that $x\in\hat\N^\infty_+$, $M\ge R$ and
\[
\int (M-R)dx=0, \quad \int (M_--\vec R)d\tilde x=0\quad P\text{-a.s.}
\]
By the interchange rule for subdifferentials (\cite[Theorem~21c]{roc74}), the latter is equivalent to \eqref{eq:oc2}.
\end{proof}
Note that for any martingale $M\in\R^1_m$,
\[
\sup_{\tau\in\T}ER_\tau = \sup_{\tau\in\T}E(R_\tau+M_T-M_{\tau})\le E\sup_{t\in[0,T]}(R_t+M_T-M_t),
\]
where the last expression is dominated by $EM_0$ if $R\le M$. Thus,
\begin{align*}
\sup_{\tau\in\T}ER_\tau &\le \inf_{M\in\R^1_m}E\sup_{t\in[0,T]}(R_t+M_T-M_t)\\
&\le \inf_{M\in\R^1_m}\{E\sup_{t\in[0,T]}(R_t+M_T-M_t)\,|\,R\le M\}\\
&\le \inf_{M\in\R^1_m}\{EM_0\,|\,R\le M\},
\end{align*}
where, by Theorem~\ref{thm:os}, the last expression equals the first one as soon as $R$ is of class $(D)$. The optimum value of the stopping problem then equals
\[
\inf_{M\in\R^1_m}E\sup_{t\in[0,T]}(R_t+M_T-M_t).
\]
This is the dual problem derived in Davis and Karatzas~\cite{dk94} and Rogers~\cite{rog2}. Note also that if $Y$ is the Snell envelope of $R$ (the smallest supermartingale that dominates $R$), then the martingale part $M$ in the Doob--Meyer decomposition $Y=M-A$ is dual optimal. These facts were obtained in \cite{dk94} and \cite{rog2} under the assumptions that $\sup_tR_t$ is integrable.
\bibliographystyle{plain}
|
1,941,325,220,547 | arxiv | \section{Introduction}
\label{sect:intro}
Persistence is one of the key features a database management system (DBMS) must fulfil.
Such features are found in the well-known acronym ACID,
where in particular D stands for Durability (i.e., persistence of data along different user sessions) \cite{silberschatz2010database}.
This way, updates in the database must be persistent in a non-volatile memory, as secondary storage (typically, the file system that the host operating system provides).
Whereas persistence in relational DBMS's are given for granted, deductive databases have been traditionally implemented as in-memory database systems (as, e.g., DLV~\cite{dlv-2006}, XSB~\cite{DBLP:journals/tplp/SwiftW12}, bddbddb~\cite{bddbddb-2005}, Smodels~\cite{DBLP:journals/corr/cs-AI-0003033}, DES~\cite{DBLP:conf/aplas/Saenz-PerezCG11}, \ldots)
Some logic programming systems also allow persistent predicates, as Ciao Prolog does \cite{DBLP:conf/padl/CorreasGCCH04} (but only for the extensional part of the database).
In this work, we present an approach for adding predicate persistence to the deductive system DES (\url{des.sourceforge.net}) \cite{DBLP:conf/aplas/Saenz-PerezCG11} relying on external SQL DBMS's via ODBC bridges.
Enabling persistence leads to several advantages:
1) Persistent predicates with transparent handling, also allowing updates.
Both the extensional (EDB, i.e., facts) and intensional (IDB, i.e., rules) databases can be persistent.
2) Interactively declare and undeclare predicates as persistent.
Applications for this include database migration (cf. Section \ref{sect:migration}).
3) Mix both deductive solving and external SQL solving.
On the one hand, the system takes advantage of the external database performance (in particular, table indexing is not yet provided by DES) and scalability.
On the other hand, queries that are not supported in an external database (as hypothetical queries or recursive queries in some systems) can be solved by the deductive engine.
So, one can use DES as a front-end to an external database and try extended SQL queries that add expressiveness to the external SQL language (cf. Sections \ref{sect:extending-expressivity} and \ref{sect:business-intelligence}).
4) Database interoperability.
As several ODBC connections are allowed at a time, different predicates can be made persistent in different DBMS's, which allows interoperability among external relational engines and the local deductive engine, therefore enabling business intelligence applications (cf. Section \ref{sect:business-intelligence}).
5) Face applications with large amounts of data which do not fit in memory.
Predicates are no longer limited by available memory (consider, for instance, a 32bit OS with scarce memory); instead, persistent predicates are using as much secondary storage as needed and provided by the underlying external database.
Predicate size limit is therefore moved to the external database.
Nonetheless, a few deductive systems also integrated persistence or database connections, as DLV$^{DB}$ \cite{Terracina:2008:ERQ:1348834.1348835},
MyYapDB \cite{DBLP:conf/jelia/FerreiraR04}, and LDL++~\cite{ldl-2003}.
One point that makes DES different from others is the ability to declare on-the-fly a given predicate as persistent and to drop such a declaration.
This is accomplished by means of assertions, which together with a wide bunch of commands, make this system amenable for rapid experimenting and prototyping.
In addition, since predicates can be understood as relations, and DES enjoys SQL, relational algebra (RA) and Datalog as query languages (SQL and RA are translated into Datalog), a persistent predicate can be used in any language and a given query can mix persistent predicates located at different databases.
Those systems neither support full-fledged duplicates (including rules as duplicate providers), nor null-related operations, nor top-N queries, nor ordering metapredicates, nor several query languages accessing the same database (including Datalog, SQL, and extended relational algebra) as DES does \cite{DBLP:conf/aplas/Saenz-PerezCG11}.
Such features are required for supporting the already available expressiveness of current relational database systems.
In addition, no system support hypothetical queries and views for decision support applications \cite{sae13c-ictai13}.
Organization of this paper proceeds as follows.
Section \ref{sect:persistence} describes our approach to persistence, including in Subsection \ref{sect:intermixing} a description of intermixing query solving as available as the result of embodying external DBMS access into the deductive engine, as well as some optimizations.
Section \ref{sect:applications} lists some applications for which persistence in a deductive system is well-suited.
Next, Section \ref{sect:performance} compares performance of this system w.r.t. DBMS's, and the extra work needed to handle persistent data.
Finally, Section \ref{sect:conclusions} summarizes some conclusions and points out future work.
\section{Enabling Persistence}
\label{sect:persistence}
For a given predicate to be made persistent in an external SQL database, type information must be provided because SQL is strong-typed.
As DES allows optional types for predicates (which are compatible with those of SQL)
the system can take advantage of known type information for persistence.
Note that, although the predicate to be made persistent has no type information, it may depends on others that do.
This means that the declared or inferred type information for such a predicate must be consistent with other's types.
To this end, a type consistency check is performed whenever a predicate is to be made persistent.
\subsection{Declaring Persistence}
\label{sect:declaring-persistence}
We propose an assertion as a basic declaration for a persistent predicate, similar to \cite{DBLP:conf/padl/CorreasGCCH04}.
The general form of a persistence assertion is as follows:
\medskip
\noindent
{\fontsize{10.5}{11}
{\tt
:- persistent({\em PredSpec},{\em Connection})
}
}
\medskip
\noindent where {\tt {\em PredSpec}} is a predicate schema specification and the optional argument {\tt {\em Connection}} is an ODBC connection identifier.
{\tt {\em PredSpec}} can be either the pattern {\tt {\em PredName}}/{\tt {\em Arity}} or {\tt {\em PredName}}({\tt {\em Schema}}), where {\tt {\em Schema}} is the predicate schema, specified as: {\tt {\em ArgName$_1$}}:{\tt {\em Type$_1$}}, \ldots, {\tt {\em ArgName$_n$}}:{\tt {\em Type$_n$}}, where {\tt {\em ArgName$_i$}} are the argument names and {\tt {\em Type$_i$}} are their (optional) types for an $n$-ary predicate ($n>0$).
If a connection name is not provided, the name of the current open database is used, which must be an ODBC connection.
An ODBC connection is identified by a name defined at the OS level, and opening a connection in DES means to make it the current database and that any relation (either a view or a table) defined in a DBMS is allowed as any other relation (predicate) in the deductive local database \verb+$des+.
A predicate can be made persistent only in one external database.
Any rule belonging to the definition of a predicate \verb+p+ which is being made persistent is expected, in general, to involve calls to other predicates (either directly or indirectly).
Each callee (such other called predicate) can be:
\begin{itemize}
\item An existing relation in the external database.
\item A persistent predicate loaded in the local database.
\item A persistent predicate not yet loaded in the local database.
\item A non-persistent predicate.
\end{itemize}
For the first two cases, besides making \verb+p+ persistent, nothing else is performed when processing its persistence assertion.
For the third case, a persistent predicate is automatically restored in the local database, i.e., it is made available to the deductive engine.
For the fourth case, each non-persistent predicate is automatically made persistent if types match; otherwise, an error is raised.
This is needed for the external database to be aware of a predicate only known by the deductive engine so far, as this database will be eventually involved in computing the meaning of \verb+p+.
\subsection{Implementing Persistence}
In general, a predicate is defined by extensional rules (i.e., facts) and intensional rules (including both head and body).
DES stores facts in a table and defines a view for the intensional rules.
For a predicate \verb+p+, a view with the same name as the predicate is created as the union of a table \verb+p_des_table+ (storing its extensional rules) and the equivalent SQL query for the remaining intensional rules.
This table is created resorting to the type information associated to \verb+p+.
So, given that a predicate \verb+p+ is composed of its extensional part {\tt {\em P}$_{ex}$} and its intensional part {\tt {\em P}$_{in}$},
each extensional rule in {\tt {\em P}$_{in}$} is mapped to a tuple in the table \verb+p_des_table+.
Let $\eval{\texttt{p}}{SQL}$ be the meaning of the view \verb+p+ in an SQL system,
and $\eval{\texttt{p}}{DL}$ be the meaning of the predicate \verb+p+ in the DES system, then:
\medskip
$\eval{\texttt{p}}{DL} = \eval{\texttt{p}}{SQL}$
\medskip
\noindent where the view \verb+p+ is defined by the following SQL query:
\medskip
{\fontsize{10.5}{11}
\noindent {\tt
SELECT * FROM p\_des\_table UNION ALL \em{DL\_to\_SQL}(P$_{in}$)
}
}
\medskip
\noindent and {\tt {\em DL\_to\_SQL(P$_{in}$)}} is the function that translates a set of rules {\tt {\em P}$_{in}$} into an SQL query.
To this end, we have resorted to Draxler's Prolog to SQL compiler \cite{draxler1992powerful} (PL2SQL from now on), which is able to translate a Prolog goal into an SQL query.
Interfacing to this compiler is performed by the predicate {\tt translate(+{\em ProjectionTerm},+{\em PrologGoal},-{\em SQLQuery})}, where its arguments are, respectively, for: specifying the attributes that are to be retrieved from the database, defining the selection restrictions and join conditions, and representing the SQL query as a term.
So, a rule composed of a head {\tt {\em H}} and a body {\tt {\em B}} can be translated into an SQL query {\tt {\em S}} with the call {\tt translate({\em H},{\em B},{\em S})}.
Writing this as the function $dx\_translate(\texttt{\em R}_i)$, which is applied to a rule $\texttt{\em R}_i \equiv \texttt{\em H}_i :- \texttt{\em B}_i$ and returns its translated SQL query, and being {\tt {\em P$_{in}$} = \{$\texttt{\em R}_1, \ldots, \texttt{\em R}_n$\}}, then:
\medskip
{\fontsize{10.5}{11}
\noindent
{\tt
{\em DL\_to\_SQL}({\em P}$_{in}$) =
{\tt $dx\_translate(\texttt{\em R}_1)$ UNION ALL \ldots UNION ALL $dx\_translate(\texttt{\em R}_n)$}
}
}
\medskip
PL2SQL is able to translate goals with conjunctions, disjunctions, negated goals, shared variables, arithmetic expressions in the built-in \verb+is+, and comparison operations, among others.
We have extended this compiler (PL2SQL$^+$ from now on) in order to deal with: Different, specific-DBMS-vendor code (including identifier delimiters and from-less SQL statements), the translation of facts, the mapping of some missing comparison operators, the inclusion of arithmetic functions to build expressions, and to reject both unsafe \cite{Ullman88} and recursive rules.
For instance, Access uses brackets as delimiters whereas MySQL uses back quotes.
Also, Oracle does not support from-less SQL statements and requires a reference to the table \verb+dual+, in contrast to other systems as PostgreSQL, which do not require it to deliver a one-tuple result (usually for evaluating expressions).
The predicate \verb+translate+ does not deal with true goals as they would involve a from-less SQL statement.
True goals are needed for translating facts, and so, we added support for this.
We have included arithmetic functions for the compilation of arithmetic expressions, including trigonometric functions (\verb+sin+, \verb+cos+, \ldots) and others (\verb+abs+, \ldots).
However, the support of such functions depends on whether the concrete SQL system supports them as well.
PL2SQL requires safe rules but it does not provide a check, so that we have included such a check to reject unsafe rules.
Recursive rules are not translated because not all DBMS's support recursive SQL statements (further DES releases might deal with specific code for recursive rules for particular DBMS's supporting recursion, as DB2 and SQL Server).
Figure \ref{fig:datalog-compiler} summarizes the syntax of valid inputs to PL2SQL$^+$ which are eventually represented as SQL statements.
Note that propositional predicates are not supported because relational databases require relations with arity greater than 0.
\begin{figure}
\begin{center}
\begin{tabular}{lll}
$fact$ & ::= & $p(c_1,\ldots,c_n)$ \\
$rule$ & ::= & $l ~ \texttt{:-} ~ l_1, \ldots, l_n$ \\
$l$ & ::= & $p(a_1,\ldots,a_n)$ \\
$l_i$ & ::= & $l ~|~ \texttt{not} ~ l ~|~ a_1 \lozenge a_2 ~|~ v ~ \texttt{is} ~~ e_1$ \\
$\lozenge$ & ::= & $\texttt{=} ~|~ \verb+\=+ ~|~ \texttt{<} ~|~ \texttt{=<} ~|~ \texttt{>} ~|~ \texttt{>=}$ \\
$e_i$ & ::= & $a ~|~ e_1 \blacklozenge e_2 ~|~ f(e_1,\ldots,e_n)$ \\
$\blacklozenge$ & ::= & $\texttt{+} ~|~ \texttt{-} ~|~ \texttt{*} ~|~ \texttt{/}$ \\
$f$ & ::= & $\texttt{sin} ~|~ \texttt{cos} ~|~ \texttt{abs} ~|~ \ldots$\\
\end{tabular}
\vspace*{2mm}
\begin{tabular}{cc}
$p$ is a predicate symbol. &
$c_i$ are constants, $i\geq 1$.\\
$l_i$ are literals, $i\geq 1$.&
$l$ is a term with depth 1.\\
$v$ is a variable. &
$a_i$ are either variables or constants, $i\geq 1$.\\
$e_i$ are arithmetic expressions. &
$rule$ is required to be safe and non recursive.\\
\end{tabular}
\vspace*{2mm}
\small{\em{True type symbols and pipes denote terminals and alternatives, respectively.}}
\end{center}
\caption{Valid Inputs to PL2SQL$^+$}
\label{fig:datalog-compiler}
\end{figure}
DES preprocesses Datalog rules before they can be eventually executed.
Preprocessing includes source-to-source transformations for translating several built-ins, including disjunction, outer joins, relational algebra division, top-N queries and others.
Rules sent to PL2SQL$^+$ are the result of these transformations, so that several built-ins that are not supported by PL2SQL can be processed by DES, as outer joins (left, right and full).
As well, there are other built-ins that PL2SQL$^+$ can deal with but which are not passed by DES up to now (as aggregates and grouping).
Non-valid rules for PL2SQL$^+$ but otherwise valid for DES are kept in the local database for their execution.
In such a case, the deductive engine couples its own processing with the processing of the external database in the following way.
Let a predicate \verb+p+ be defined by a set of rules {\tt {\em S}} that can be externally processed and other set of rules {\tt {\em D}} that cannot.
Then, the meaning of \verb+p+ is computed as the union of the meanings of both sets of rules:
\medskip
$\eval{\texttt{p}}{} = \eval{\texttt{\em S}}{SQL} \cup \eval{\texttt{\em D}}{DL}$
\medskip
Rules in $\texttt{\em D}$ are therefore not included in {\tt {\em P}$_{in}$} in the call to {\tt {\em DL\_to\_SQL}} as described above, and they are otherwise stored as regular in-memory Datalog rules and processed by the deductive engine.
Thus, all the deductive computing power is preserved when either the external DBMS lacks some features as, e.g., recursion (e.g., MySQL and MS Access), or a predicate contains some non-valid rules for PL2SQL$^+$.
\subsection{An Example}
\label{sect:example}
As an example, let's consider the Datalog predicates \verb+ancestor+, \verb+mother+, and \verb+parent+, the DBMS MySQL, and a table \verb+father+ already created and populated in this external database.
MySQL:
{\fontsize{10.5}{11}
\begin{verbatim}
CREATE TABLE father(father VARCHAR(20),child VARCHAR(20));
INSERT INTO father VALUES('tom','amy');
...
\end{verbatim}
}
DES:
{\fontsize{10.5}{11}
\begin{verbatim}
:-type(mother(mother:string,child:string)).
mother(grace,amy).
...
:-type(parent(parent:string,child:string)).
parent(X,Y) :- father(X,Y) ; mother(X,Y).
:-type(ancestor(ancestor:string,descendant:string)).
ancestor(X,Y) :- parent(X,Y).
ancestor(X,Y) :- parent(X,Z), ancestor(Z,Y).
\end{verbatim}
}
Then, if we submit the assertion \verb+:-persistent(ancestor/2)+ when the current opened database is MySQL, we get the following excerpt of the DES verbose output:
{\fontsize{10.5}{11}
\begin{verbatim}
Warning: Recursive rule cannot be transferred to external database
(kept in local database for its processing):
ancestor(X,Y) :- parent(X,Z), ancestor(Z,Y).
Info: Predicate mother/2 made persistent.
Info: Predicate parent/2 made persistent.
Info: Predicate ancestor/2 made persistent.
\end{verbatim}
}
Recalling Section \ref{sect:declaring-persistence}, declaring the persistence of \verb+ancestor/2+ involves to make persistent both \verb+mother/2+ and \verb+parent/2+ because, in particular, the first rule of \verb+ancestor/2+ includes a call to \verb+parent/2+, and the second call of \verb+parent/2+ is to \verb+mother/2+.
Even when \verb+parent/2+ includes a call to \verb+father/2+, the latter predicate is not made persistent because there exist the table \verb+father/2+ in the external database already.
The resulting views\footnote{They can be displayed, for instance, with the command {\tt /dbschema \$des}.} after processing the assertion are:
{\fontsize{10.5}{11}
\begin{verbatim}
CREATE VIEW mother(mother,child) AS
SELECT * FROM mother_des_table;
CREATE VIEW parent(parent,child) AS
(SELECT * FROM parent_des_table) UNION ALL
(SELECT rel1.mother,rel1.child FROM mother AS rel1) UNION ALL
(SELECT rel1.father,rel1.child FROM father AS rel1);
CREATE VIEW ancestor(ancestor,descendant) AS
(SELECT * FROM ancestor_des_table) UNION ALL
(SELECT rel1.parent,rel1.child FROM parent AS rel1);
\end{verbatim}
}
Note that, on the one hand, and as a difference with other systems as $DLV^{DB}$, these views are not materialized.
On the other hand, DES allows to project such intensional rules to the external database by contrast to Ciao, which only project extensional rules.
Processing a top-level call either to \verb+father/2+, or \verb+mother/2+ or \verb+parent/2+ is computed by the external database.
However, a call to \verb+ancestor/2+ is processed both by the external database because of its first rule involving a call to \verb+parent+, and by the local deductive engine due to the local rule (the recursive one which cannot be processed by MySQL), as it will be explained in Section \ref{sect:intermixing}.
All intensional rules (both valid and non-valid inputs to PL2SQL$^+$) of a persistent predicate \verb+p+ are externally stored as metadata information in a table named \verb+p_des_metadata+ to allow to recover original rules when removing a persistence assertion (cf. Section \ref{subsect:restoring_persistence}).
For instance, the contents of this table for \verb+parent+ are \footnote{Note that as a result of DES preprocessing, the rule with the disjunction has been translated into two rules.}:
{\fontsize{10.5}{11}
\begin{verbatim}
parent_des_metadata('parent(X,Y):-father(X,Y).').
parent_des_metadata('parent(X,Y):-mother(X,Y).').
\end{verbatim}
}
Whilst the contents of \verb+mother_des_table+ are its extensional rules (the facts \verb+mother(grace,amy)+, \ldots), the contents of \verb+parent_des_table+ and \verb+ancestor_des_table+ are empty (unless a fact is asserted in any of the corresponding predicates).
Note that, as \verb+father+ is a table in the external database, if we {\em assert} a new tuple {\em t} for it, it will be only loaded in the local database, instead of externally stored if it was a persistent predicate\footnote{Of course, {\em inserting} a tuple in the external table will store it in the DBMS.}.
In both cases, the top-level query \verb+father(X,Y)+ would return the same tuples (either for the table or for the persistent predicate), but upon restoring persistence of \verb+ancestor/2+, the tuple {\em t} would not be restored for the table \verb+father+.
\subsection{Updating Persistent Predicates}
Updating a persistent predicate {\tt p} is possible with the commands \verb+/assert+ and \verb+/retract+, which allow to insert and delete a rule, respectively,
and their counterpart SQL statements \verb+INSERT+ and \verb+DELETE+, which allow to insert and delete, respectively, a batch of tuples (either extensionally or intensionally).
Implementing the update of the IDB part of a persistent predicate amounts to retrieve the current external view corresponding to the persistent predicate, drop it, and create a new one with the update.
The update of the EDB part (insert or delete a tuple) is simply performed to the external table with an appropriate SQL statement (\verb+INSERT INTO ...+ or \verb+DELETE FROM ...+).
Each update is tuple-by-tuple, even when batch updates via select statements are processed.
For each update, if constraint checking is enabled, any strong constraint defined at the deductive level is checked.
Note that the view update problem \cite{silberschatz2010database} is not an issue because our approach to insertions and deletions of tuples in a persistent predicate \verb+p+ amounts to modify the extensional part of \verb+p+, which is stored in the table \verb+p_des_table+.
This is a different approach to DBMS's where a relation defined by a view only consists of an intensional definition, so that trying to update a view involves updating the relations (other views and tables) it depends on, and this can be done is some situations but not in general.
\subsection{Restoring and Removing a Persistent Predicate}
\label{subsect:restoring_persistence}
Once a predicate \verb+p+ has been made persistent in a given session, the state of \verb+p+ can be restored in a next session (i.e., after the updates --assertions or retractions-- on \verb+p+)\footnote{Cf. transaction logic \cite{Bonner94anoverview} to model states in logic programming.}.
It is simply done by submitting again the same assertion as used to make \verb+p+ persistent for the first time.
Note, however, that if there exists any rule for \verb+p+ in the in-memory database already, it will not be removed but stored as persistent in the external database.
Also, a given predicate can be made non-persistent by dropping its assertion, as, e.g.:
{\fontsize{10.5}{11}
\begin{verbatim}
DES> /drop_assertion :-persistent(p(a:int),mysql)
\end{verbatim}
}
This retrieves all the facts stored in the external database, stores them back in the in-memory database, removes them from the external database, and the original rules, as they were asserted (in its compiled Datalog form) are recovered from the table \verb+p_des_metadata+.
The view and tables for predicate \verb+p+ are dropped.
\subsection{Intermixing Query Solving}
\label{sect:intermixing}
As already introduced, persistence enables to couple external DBMS processing with deductive engine processing.
DES implements a top-down-driven, bottom-up fixpoint computation with tabling \cite{DBLP:conf/aplas/Saenz-PerezCG11}, which follows the ideas found in \cite{SD91,Dietrich87,TS86}.
This mechanism is implemented as described in \cite{Sae13a,sae13c-ictai13}.
In particular, the predicate {\tt solve\_goal} solves a goal (built-ins and user-defined predicates).
The following clause of this predicate is responsible of using program rules to solve a goal corresponding to a user predicate (where arguments which are not relevant for illustration purposes have been removed):
{\fontsize{10.5}{11}
\begin{verbatim}
solve_goal(G) :- datalog((G:-B),_Source), solve(B).
\end{verbatim}
}
This predicate selects a program rule matching the goal via backtracking and solves the rule body as a call to the the predicate {\tt solve}.
Such program rules are loaded in the dynamic predicate {\tt datalog}.
In order to allow external relations to be used as user predicates, this dynamic predicate is overloaded with the following clause, which in turn calls {\tt datalog\_rdb}:
{\fontsize{10.5}{11}
\begin{verbatim}
datalog(Rule,rdb(Connection)) :-
datalog_rdb(Rule,rdb(Connection)).
datalog_rdb(R,Source) :-
datalog_rdb_single_np(R,Source) ;
datalog_rdb_all_np(R,Source) ;
datalog_rdb_single_p(R,Source) ;
datalog_rdb_all_p(R,Source).
\end{verbatim}
}
The predicate {\tt datalog\_rdb} identifies two possible sources: non-persistent and persistent predicates.
Also, it identifies whether a particular predicate is called or otherwise all predicates are requested.
In the last case, all external relations must be retrieved, and predicates {\tt datalog\_rdb\_all\_np} and {\tt datalog\_rdb\_all\_p} implement this via backtracking.
The (simplified) implementation of the predicate {\tt datalog\_rdb\_single\_p} (a single, concrete, persistent predicate) for an external ODBC connection {\tt Conn} is as follows:
{\fontsize{10.5}{11}
\begin{verbatim}
datalog_rdb_single_p(R,RuleId,rdb(Conn)) :-
my_persistent(Connection,TypedSchema),
functor(TypedSchema,TableName,Arity),
R =.. [Name|Columns],
length(Columns,Arity),
schema_to_colnames(TypedSchema,ColNames),
sql_rdb_datasource(Conn,Name,ColNames,Columns,SQLstr),
my_odbc_dql_query_fetch_row(Conn,SQLstr,Row),
Row=..[_AnswerRel|Columns].
\end{verbatim}
}
The predicate {\tt sql\_rdb\_datasource} builds an SQL statement which returns rows for a relation under a connection matching the input column values ({\tt Columns} is the list of variables and/or constants for the query).
As an example, the query {\tt ancestor(A,amy)} for the example in Section \ref{sect:example} generates the following SQL statement (notice that the identifier delimiters in this DBMS do not follow standard SQL):
{\fontsize{10.5}{11}
\begin{verbatim}
SELECT * FROM `ancestor` WHERE `descendant`='amy'
\end{verbatim}
}
The predicate {\tt my\_odbc\_dql\_query\_fetch\_row} returns rows, one-by-one, via backtracking for this SQL statement.
Note that, for this simple example, row filtering is performed by the external engine.
Recall that this persistent predicate consists of two program rules:
{\fontsize{10.5}{11}
\begin{verbatim}
ancestor(X,Y) :- parent(X,Y).
ancestor(X,Y) :- parent(X,Z), ancestor(Z,Y).
\end{verbatim}
}
The first one was loaded in the external database as the view:
{\fontsize{10.5}{11}
\begin{verbatim}
CREATE VIEW ancestor(ancestor,descendant) AS
(SELECT * FROM ancestor_des_table) UNION ALL
(SELECT rel1.parent,rel1.child FROM parent AS rel1);
\end{verbatim}
}
\noindent and the second one was loaded in the local deductive database, as the dynamic clause:
{\fontsize{10.5}{11}
\begin{verbatim}
datalog((ancestor(X,Y) :- parent(X,Z), ancestor(Z,Y)),source)
\end{verbatim}
}
So, the fixpoint mechanism uses in the call to \verb+datalog+ both the non-recursive rule from the external database via {\tt datalog\_rdb\_single\_p}, and the recursive rule via the dynamic clause.
Concluding, the predicate {\tt datalog} provides to the deductive query solving not only the rules which are local, but also the rules which are externally stored and processed, retrieved via the predicate {\tt datalog\_rdb}, therefore enabling intermixed query solving.
\subsection{Fixpoint Optimizations}
\label{subsect:optimization}
We list some already implemented optimizations which are key to avoid retrieving the same tuple from the external database several times due to fixpoint iterations.
They can be independently enabled and disabled with commands to test their impact on performance.
\begin{itemize}
\item {\bf Complete Computations}.
Each call during the computation of a stratum is remembered in addition to its outcome (in the answer table).
Even when the calls are removed in each fixpoint iteration, most general ones do persist as a collateral data structure to be used for saving computations should any of them is called again during either computing a higher stratum or a subsequent query solving.
If a call is marked as a completed computation, it is not even tried if called again.
This means the following two points:
1) During the computation of the memo function, calls already computed are not tried to be solved again, and only the entries in the memo table are returned.
2) Moreover, computing the memo function is completely avoided if a subsuming already-computed call can be found.
In the first case, that saves solving goals in computing the memo function.
In the second case, that completely saves fixpoint computation.
\item {\bf Extensional Predicates}.
There is no need to iteratively compute extensional predicates and, therefore, no fixpoint computation is needed for them.
They are known from the predicate dependency graph simply because they occur in the graph without incoming arcs.
For them, a linear fetching is enough to derive their meanings.
\item {\bf Non-Recursive Predicates}.
Memoization comes at the cost of maintaining a cache which can be wasteful in some cases.
A top-level goal involving non-recursive predicates are computed by only caching the top-level goal, avoiding memorizing dependent calls.
This allows a fast solving by looking for all the answers of the goal, and finally storing the results in the memo table.
\end{itemize}
\section{Applications}
\label{sect:applications}
This section lists several applications derived from supporting persistence in DES as it includes some features which are not available in external DBMS's, such as hypothetical queries, extended recursion, and intermixed query solving.
\subsection{Database Interoperability}
Persistence allows for database interoperability as each persistent predicate is mapped to an ODBC connection and several connections can be opened simultaneously.
First scenario is for a persistent predicate \verb+p+ in a given connection and opening another connection from another database.
Then, both the predicate \verb+p+ and the relations defined in the latter connection are visible for the deductive database.
This is in contrast to other systems (as, e.g., $DLV^{DB}$) that need to explicitly state what relations from the external database are visible.
Here, no extra effort is needed.
Second scenario is for several persistent predicates which are mapped to different connections.
As they are visible for the deductive engine, all of them can be part of a query solved by the deductive engine.
Recall that any external view will be still processed by the external DBMS.
\subsection{Extending DBMS Expressivity}
\label{sect:extending-expressivity}
The more expressive SQL and Datalog languages as provided by DES can improve the expressiveness of the external database when acting as a front-end.
For instance, let's consider MySQL, which does not support recursive queries up to its current version 5.6.
The following predicate can be made persistent in this DBMS even when it is recursive:
{\fontsize{10.5}{11}
\begin{verbatim}
DES> :-persistent(path(a:int,b:int),mysql)
DES> /assert path(1,2)
DES> /assert path(2,3)
DES> /assert path(X,Y):-path(X,Z),path(Z,Y)
Warning: Recursive rule cannot be transferred to external database (kept
in local database for its processing):
path(X,Y) :- path(X,Z), path(Z,Y).
DES> path(X,Y)
{ path(1,2), path(1,3), path(2,3) }
\end{verbatim}
}
Here, non-recursive rules are stored in the external database whereas the recursive one is kept in the local database.
External rules are processed by MySQL and local rules by the deductive engine.
Though the recursive rule is not externally processed, it is externally stored as metadata, therefore ensuring its persistence between sessions.
In addition to Datalog, DES includes support for SQL for its local deductive database.
To this end, on the one hand, SQL data definition statements are executed and metadata (as the name and type of table fields) is stored as assertions.
On the other hand, SQL queries are translated into Datalog and executed by the deductive engine.
The supported SQL dialect includes features which are not found in current DBMS's, as non-linear recursive queries, hypothetical views and queries, and the relational algebra division operator.
Therefore, DES is able to compute more queries than a DBMS:
For instance, neither MS SQL Server nor IBM DB2 allow cycles in a path without compromising termination.
Also, recursive and stratifiable SQL queries do not fully allow \verb+EXCEPT+ such as in MS SQL Server and IBM DB2.
Another limitation is linear recursion: The rules in the last example above cannot be expressed in any DBMS as there are several recursive calls.
To name another, \verb+UNION ALL+ is enforced in those SQL's, so that just \verb+UNION+ (discarding duplicates) is not allowed.
For instance, the following recursive query is rejected in any current commercial DBMS, but accepted by DES:
{\fontsize{10.5}{11}
\begin{verbatim}
DES> CREATE TABLE edge(a int, b int);
DES> INSERT INTO edge VALUES (1,2),(2,3),(1,3);
DES> :-persistent(edge/2,mysql).
DES> :-persistent(path(a:int,b:int),mysql).
DES> WITH RECURSIVE path(a, b) AS
SELECT * FROM edge
UNION --Discard duplicates (ALL not required)
SELECT p1.a,p2.b FROM path p1, path p2 WHERE p1.b=p2.a
SELECT * FROM path;
Warning: Recursive rule cannot be transferred to external database
(kept in local database for its processing):
path_2_1(A,B) :- path(A,C), path(C,B).
answer(path.a:number(integer), path.b:number(integer)) ->
{ answer(1,2), answer(1,3), answer(2,3) }
\end{verbatim}
}
In this example, \verb+edge+ becomes a Datalog typed (and populated) relation because it is defined with the DES SQL dialect in the local deductive database, and it has been made persistent, as well as \verb+path+ (which is also typed because of the persistence assertion, but not populated).
The \verb+WITH+ statement allows to declare temporary relations.
In this case, the result of the compilation of the SQL query definition of \verb+path+ are temporary Datalog rules which are added to the persistent predicate \verb+path+ (note that the recursive part is not transferred to the external database):
{\fontsize{10.5}{11}
\begin{verbatim}
path(A,B) :- distinct(path_2_1(A,B)).
path_2_1(A,B) :- edge(A,B).
path_2_1(A,B) :- path(A,C), path(C,B).
\end{verbatim}
}
\noindent and the SQL query \verb+SELECT * FROM path+ is compiled to:
{\fontsize{10.5}{11}
\begin{verbatim}
answer(A,B) :- path(A,B).
\end{verbatim}
}
After executing the goal \verb+answer(A,B)+ for solving the SQL query, the temporary Datalog rules are removed.
Adding {\tt ALL} to {\tt UNION} to the same query for keeping duplicates makes to include the tuple {\tt answer(1,3)} twice in the result.
\subsection{Business Intelligence}
\label{sect:business-intelligence}
Business intelligence refers to systems which provide decision support \cite{Watson:2007:CSB:1300761.1301970} by using data integration, data warehousing, analytic processing and other techniques.
In particular, one of these techniques refer to ``what-if" applications.
DES also supports a novel SQL feature: Hypothetical SQL queries.
Such queries are useful, for instance, in decision support systems as they allow to submit a query by assuming some knowledge which is not in the database.
Such knowledge can be either new data assumed for relations (both tables and views) and also new production rules.
For example, and following the above system session, the tuple \verb+(3,1)+ is assumed to be in the relation \verb+path+, and then this relation is queried:
{\fontsize{10.5}{11}
\begin{verbatim}
DES> ASSUME SELECT 3,1 IN path(a,b) SELECT * FROM path;
answer(path.a:number(integer),path.b:number(integer)) ->
{ answer(1,1), answer(1,2), answer(1,3), answer(2,1), answer(2,2),
answer(2,3), answer(3,1), answer(3,2), answer(3,3) }
\end{verbatim}
}
As an example of adding a production rule, let's suppose a relation {\tt flight} and a view {\tt connect} for locations connected by direct flights:
{\fontsize{10.5}{11}
\begin{verbatim}
DES> CREATE TABLE flight(ori STRING, dest STRING, duration INT);
DES> INSERT INTO flight VALUES ('Madrid','Paris',90),
('Paris','Oslo',100), ('Madrid','London',110);
DES> CREATE VIEW connect(ori,dest) AS SELECT ori,dest FROM flight;
DES> :-persistent(connect/2,access) -- This also makes 'flight' persistent
DES> SELECT * FROM connect;
answer(connect.ori:string(real),connect.dest:string(real)) ->
{ answer('Madrid','London'), answer('Paris','Oslo'),
answer('Madrid','Paris') }
\end{verbatim}
}
Then, if we assume that connections are allowed with transits, we can submit the following hypothetical query (where the assumed SQL statement is recursive):
{\fontsize{10.5}{11}
\begin{verbatim}
DES> ASSUME
(SELECT flight.ori,connect.dest
FROM flight,connect
WHERE flight.dest = connect.ori)
IN
connect(ori,dest)
SELECT * FROM connect;
answer(connect.ori:string(real),connect.dest:string(real)) ->
{ answer('Madrid','London'),answer('Madrid','Oslo'),
answer('Madrid','Paris'), answer('Paris','Oslo')}
\end{verbatim}
}
Also, several assumptions for different relations can be defined in the same query.
\subsection{Migrating Data}
\label{sect:migration}
Once a predicate has been made persistent in a given connection, dropping its persistent assertion retrieves all data and schema from the external database into the in-memory Prolog database.
A successive persistent assertion for the same predicate in a different connection dumps it to the new external database.
These two steps, therefore, implement the migration from one database to another, which can be of different vendors.
For instance, let's consider the following session, which dumps data from MS Access to MySQL:
{\fontsize{10.5}{11}
\begin{verbatim}
DES> :-persistent(p(a:int),access)
DES> /drop_assertion :-persistent(p(a:int),access)
DES> :-persistent(p(a:int),mysql)
\end{verbatim}
}
\section{Performance}
\label{sect:performance}
In this section we analyze how queries involving persistent predicates perform w.r.t. native SQL queries, and the overhead caused by persistence w.r.t. the in-memory (Prolog-implemented) database.
As relational database systems, three widely-used systems have been chosen with a default configuration: The non-active desktop database MS Access (version 2003 with ODBC driver 4.00.6305.00), the mid-range, open-source Oracle MySQL (version 5.5.9 with ODBC driver 5.01.08.00), and the full-edged, commercial IBM DB2 (version 10.1.0 with ODBC driver 10.01.00.872).
All times are given in milliseconds and have been run on an Intel Core2 Quad CPU at 2.4GHz and 3GB RAM, running Windows XP 32bit SP3.
Each test has been run 10 times, the maximum and the minimum numbers have been discarded, and then the average has been computed.
Also, as Access quickly fragments the single file it uses for persistence, and this heavily impacts performance, each running of the benchmarks in this system is preceded by a defragmentation (though, the time for performing this has not been included in the numbers).
All optimizations, as listed in Section \ref{sect:intermixing}, are enabled.
Some results are collected in Table \ref{table:performance1}.
The tests consist of, first, inserting 1,000 tuples in a relation with a numeric field (columns with heading {\em Insert$_n$} and {\em Insert$_p$}, for native queries and persistent predicates, respectively).
The Datalog commands are {\tt /assert t({\em i})} and the SQL update queries are {\tt INSERT INTO t VALUES({\em i})} (1 $\leq$ {\tt {\em i}} $\leq$ 1,000).
\begin{table}[b!]
\begin{center}
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c|r@{}@{}rr@{}@{}rr@{}@{}r|r@{}@{}rr@{}@{}rr@{}@{}r|}
\cline{1-7}
{\em System} & \multicolumn{2}{|c}{\em Insert$_n$} & \multicolumn{2}{c}{\em Select$_n$} & \multicolumn{2}{c|}{\em Join$_n$} &\\
\hline
DES 3.2 & 359 & & 773 & & 3,627 & & \multicolumn{2}{c}{\em Insert$_p$} & \multicolumn{2}{c}{\em Select$_p$} & \multicolumn{2}{c|}{\em Join$_p$}\\
\cline{8-13}
Access & 439 & (1.22) & 1,014 & (1.31) & 7,303 & (2.01) & 1,102 & (3.07$\diamond$2.51) & 2,138 & (2.77$\diamond$2.11) & 17,270 & (4.76$\diamond$2.36)\\
MySQL & 9,950 & (27.72) & 1,160 & (1.50) & 13,183 & (3.63) & 10,279 & (28.63$\diamond$1.03) & 2,364 & (3.06$\diamond$2.04) & 22,305 & (6.15$\diamond$1.69)\\
DB2 & 1,264 & (3.52) & 1,018 & (1.32) & 9,057 & (2.50) & 1,869 & (5.21$\diamond$1.48) & 2,260 & (2.92$\diamond$2.22) & 18,637 & (5.14$\diamond$2.06)\\
\hline
\end{tabular}
}
\end{center}
\caption{Results for in-memory DES, DBMS's and Persistent Predicates}
\label{table:performance1}
\end{table}
Then, 1,000 select queries are issued (columns {\em Select$_n$} and {\em Select$_p$}).
The $i$-th select query asks for the $i$-th value stored in the table, so that all values are requested by independent queries.
The Datalog queries are {\tt t({\em i})} and the SQL select queries are {\tt SELECT a FROM t WHERE a={\em i}} (1 $\leq$ {\tt {\em i}} $\leq$ 1,000).
Next, a single query which computes an autojoin is submitted (columns {\em Join$_n$} and {\em Join$_p$}) which yields one million tuples in the result set.
The Datalog queries are {\tt t(X),t(Y)} and the SQL select queries are {\tt SELECT * FROM t AS t1,t AS t2}.
First line below headings of this table collects the results of the in-memory deductive database DES (Datalog commands and queries), with no persistence.
The next three lines in the block with subscripts $n$ in the headings (referred to as 'block $n$' from now on) show the results for the native queries in each DBMS (SQL {\tt INSERT} and {\tt SELECT} queries).
The three lines in the block with subscripts $p$ in the headings (referred to as 'block $p$' from now on) show the results for the Datalog commands (\verb+/assert+ {\tt t({\em i})}) and queries ({\tt t({\em i})} and \verb+t(X),t(Y)+) when the relation {\tt t} has been made persistent in each external DBMS.
Then, this table allows, first, to compare the in-memory, state-less system DES w.r.t. the relational, durable DBMS's (ratio values enclosed between parentheses in block $n$ as the time for each DBMS divided by the time for DES).
Second, to examine the overhead of persistence by confronting the results in the line DES and the results in the block $p$ for each DBMS (first ratio value enclosed between parentheses in the table as the time for DES divided by the time for each DBMS).
And, third, to compare the results of DES as a persistent database w.r.t. each DBMS for dealing with the same actions (inserting and retrieving data), by confronting the results in block $p$ and block $n$ for each DBMS (second ratio value enclosed between parentheses in the table as the time for the time in block $p$ divided by the corresponding time in block $n$).
For the {\tt SELECT} queries, we focus on retrieving to the main memory the results but without actually displaying it in order to elide the display time.
For the deductive database, this means that each tuple in the result is computed and stored in the answer table but it is not displayed.
For the relational databases, this means that a single ODBC cursor connection is used for a single query and each tuple in its result is retrieved to main memory, but not displayed.
With respect to the native queries (focusing at block $n$) a first observation is that insertions (column {\em Insert$_n$}) in the in-memory deductive database are, as expected, faster than for DBMS's.
However, Access is very fast as it is more oriented towards a file system (it is not an active database) and its time is comparable to that of DES (Access is only 22\% slower).
Another observation is that MySQL takes much more time for updates than DB2 (9,950/1,264 $\approx$ 8 times slower) and Access (9,950/439 $\approx$ 22.6 times slower), but it performs close to them for the batch of 1,000 select queries.
(This behaviour can also be observed in queries to persistent data.)
A third observation is that computations for select operations (columns {\em Select}$_n$ and {\em Join}$_n$) in the in-memory deductive database are faster than in DBMS's.
While for 1,000 queries in Datalog (column {\em Select}$_n$) there is a speed-up of up to 1.50, in the single query (column {\em Join}$_n$) this grows up to 3.63 (both for MySQL).
Queries to persistent data (focusing at block $p$) show two factors: 1) The performance of queries involving persistence w.r.t. their counterpart native SQL queries, and 2) The overhead caused by persistence in the deductive system for the different DBMS's.
With respect to factor 1, by comparing native queries to queries to persistent data, we observe that the cost for inserting tuples by using a persistent predicate w.r.t. a native SQL {\tt INSERT} statement ranges from a negligible ratio of 1.03 (MySQL) to 2.51 (Access).
Also, the overhead for computing 1,000 queries with a Datalog query on a persistent predicate w.r.t. its counterpart native SQL select statement, is around 2 times for all DBMS's.
And for the autojoin, the ratio ranges from 1.69 for MySQL to 2.36 for Access.
With respect to factor 2, insertions require a ratio ranging from 3.07 to 5.21 for Access and DB2, respectively, whereas for DB2 a huge ratio of 28.63 is found.
Managing individual insert statements via cursor connections is hard in this case.
However, the overhead comes from the connection itself as the code to access the different external databases is the same.
The select queries perform quite homogeneously with ratios from 2.76 to 3.06, in accordance to factor 1.
Last, for the autojoin, the ratio ranges from 4.76 to 6.15.
Finally, Table \ref{table:performance2} shows the cost for creating and removing persistence for each DBMS.
The column {\em Create} shows the time for creating a persistent predicate where its 1,000 tuples are in the in-memory database.
This amounts to store each in-memory tuple in the external database, so that numbers are similar to that of the column {\em Insert}$_n$.
Dropping the persistent assertion, as shown in column {\em Drop}, takes a small time.
Recall that this operation also retrieve the 1,000 tuples to the in-memory database.
The difference between the cost of creating and dropping the assertion lies in that the former submits 1,000 SQL queries while the latter submits a single SQL query.
Thus, the cost of opening and closing cursor connections is therefore noticeable.
\begin{table}[hb]
\begin{center}
\begin{tabular}{|c|rr|}
\hline
{\em DBMS} & {\em Create} & {\em Drop}\\
\hline
Access & 1,256 & 31\\
MySQL & 10,523 & 74\\
DB2 & 1,926 & 172\\
\hline
\end{tabular}
\end{center}
\caption{Creating and Removing Persistence}
\label{table:performance2}
\end{table}
\section{Conclusions}
\label{sect:conclusions}
This paper has shown how persistence is supported by a tabled-based deductive system.
This work includes extended language features that might be amenable to try even projected to such external databases.
Although this system was targeted at teaching and not to performance, some numbers have been taken to assess its applicability.
When comparing the times taken by the queries relating persistent predicates w.r.t. their counterpart native SQL queries, ratios from 1.03 up to 2.51 are got, which overall shows the overhead of using the deductive persistent system w.r.t. the SQL systems.
When comparing the times taken by the queries relating persistent predicates w.r.t. their counterpart in-memory queries, higher ratios have been found, from 2.76 up to 6.15, and an extreme case of 28.63 due to the costly insertions through the ODBC bridge.
These results suggest that the cost of persistence might be worthwhile depending on the DBMS and the application.
Differences between this system and others can be highlighted, besides those which were already noted in the introduction and along the paper.
For instance, predicates in DLV$^{DB}$ are translated into materialized relations, i.e., a predicate is mapped to a table and the predicate extension
is inserted in this table, which opens up the view maintenance problem.
Ciao Prolog is only able to make the extensional part of a predicate to persist, disabling the possibility of surrogating the solving of views for intensional rules.
MyYapDB (for *unixes) is not understood as implementing persistence, instead, it allows to connect to the external MySQL DBMS, making external relations available to YAP as if they were usual predicates.
This is similar to what DES does simply by opening an ODBC connection, which automatically makes visible all the external relations (not only in MySQL but for any other DBMS and OS).
LDL++ was retired in favor of DeAL, and currently there is no information about its connection to external databases, though in \cite{ldl-2003} such a connection was very briefly described for the former.
As for future work, built-ins supported by the compiler \cite{draxler1992powerful} but not passed by DES can be included in forthcoming releases.
Also, query clustering can be useful (cf. \cite{DBLP:conf/padl/CorreasGCCH04}), i.e., identifying those complex subgoals that can be mapped to a single SQL query, therefore improving the results for queries as the autojoin, by reducing the number of cursors.
Rules with linear recursive queries supported by the external DBMS can be allowed to be projected.
Since the deductive engine is not as efficient as others \cite{DBLP:journals/tplp/SwiftW12}, it can be improved or replaced with an existing one but upgraded to deal with extra features (as nulls and duplicates).
Finally, the current implementation has been tested for several DBMS's, including Access, SQL Server, MySQL, and DB2.
Although the connection to such external databases is via the ODBC bridge which presents a common interface to SQL, some tweaks depending on the particular SQL dialect should be made in order to cope with other DBMS's.
\section*{Acknowledgements}
This work has been partially supported by the Spanish projects CAVI-ART (TIN2013-44742-C4-3-R), STAMP (TIN2008-06622-C03-01), Prometidos-CM (S2009TIC-1465), GPD (UCM-BSCH-GR35/10-A-910502), and the Department Ingenier\'{\i}a del Software e Inteligencia Artificial at University Complutense of Madrid.
Also thanks to the anonymous referees who helped in improving this paper and the system DES.
|
1,941,325,220,548 | arxiv | \section{Introduction}\label{intro}
A wealth of experimental data generated at the Relativistic Heavy Ion Collider have shown that high-$p_T$ hadrons in
central A+A collisions are significantly suppressed in comparison with those in binary-scaled p+p collisions
\cite{Adcox:2001jp,Adler:2002xw}. Moreover, the disappearance of back-to-back azimuthal correlations has been observed
in central A+A collisions, in contrast to the strong back-to-back correlations in p+p and peripheral A+A collisions
\cite{Adler:2002tq}. Those results have been attributed to the strong interaction between hard partonic jets produced
in the early stage of the collisions and the hot and dense nuclear medium created in the collisions, and they are
commonly referred to as jet quenching \cite{Gyulassy:1993hr}.
A lot of effort has gone into quantitatively calculating the quenching of jets taking place inside the soft nuclear
matter. Experimentally, jet quenching can be quantitatively described by measurements of various quantities. One of
those is the nuclear modification factor $R_{AA}$ of single-hadron spectra in A+A collisions with respect to that in
p+p interactions scaled by the number of binary collisions. In Ref. \cite{Qin:2007zz} we presented a systematic
calculation of $R_{AA}$ for pions in central and non-central Au+Au collisions at $\sqrt{s} = 200$~AGeV by applying the
Arnold, Moore and Yaffe (AMY) \cite{Arnold:2001ms,Arnold:2001ba,Arnold:2002ja} formalism, where the bulk properties of
the medium created in those collision are described by (3+1)-dimensional relativistic ideal hydrodynamics
\cite{Nonaka:2006yn}, which has been shown to give a good description of bulk properties at RHIC. Here, we present our
calculation of both radiative and collisional energy loss in the same framework and compare the relative contribution
to the nuclear suppression at RHIC \cite{Qin:2007rn,Qin:2008ea}.
Additional information on jet quenching may be obtained by performing correlation measurements, i.e., studying the
production of high-$p_T$ hadrons associated with trigger high-$p_T$ particles. One of the most promising suggested triggers is a
high $p_T$ photon \cite{Wang:1996yh,Wang:1996pe}. Hard scatterings in the early stage of the collisions are an
important source of photons. If we trigger on such photons, the transverse energy of the away-side
associated jet will be determined (note, however, that at next-to-leading order, the kinematics will get contributions from
additional $2\to 3$ processes; we postpone this discussion to a later study.).
It is noted that high-$p_T$ photons may be produced from other sources in A+A
collisions, such as those involving jet--plasma interaction during jet propagation in the medium and
fragmentation of surviving jets after their passing through the medium. In this work, we present a comprehensive study of
jet--photon correlations by taking into account all relevant sources for high $p_T$ photon production.
\section{Jet energy loss}\label{sec:1}
In our approach, jets (quarks and gluons) evolve in the nuclear medium according to a set of Fokker--Planck
type rate equations for their momentum distributions $P(E,t) = {dN(E,t)}/{dE}$. Here we write
down the generic
form of the rate equations \cite{Jeon:2003gi,Turbide:2005fk},
\begin{eqnarray}\label{FP-eq}
\frac{dP_j(E,t)}{dt} \!&=&\! \! \sum_{ab} \! \int \! d\omega \left[P_a(E+\omega,t) \frac{d\Gamma_{a\to
j}(E+\omega,\omega)}{d\omega dt} \right. \nonumber\\ && \left. - P_j(E,t)\frac{d\Gamma_{j\to b}(E,\omega)}{d\omega
dt}\right], \ \ \ \ \ \
\end{eqnarray}
where ${d\Gamma^{j\to a}(E,\omega)}/{d\omega dt}$ is the transition rate for the partonic process $j\to a$, with $E$
the initial jet energy and $\omega$ the energy lost in the process. The $\omega<0$ part of the integration
incorporates the energy-gain channels. The radiative and collisional parts of the transition rates have been
extensively discussed in \cite{Qin:2007zz,Qin:2007rn}.
The initial jet momentum profiles may be computed from pQCD in the factorized formalism. To obtain the momentum
spectra of the produced high-$p_T$ hadrons in Au+Au collisions at RHIC, the energy loss of partonic jets in the nuclear
medium must be taken into account. This is performed by calculating the medium-modified fragmentation function
$\tilde{D}_{h/j}(z,\vec{r}_\bot, \phi)$ for a single jet,
\begin{eqnarray}
\label{medium_frag_fun} \tilde{D}_{h/j}(z,\vec{r}_\bot, \phi) \!&=&\!\! \sum_{j'} \!\int\! dp_{j'} \frac{z'}{z}
D_{h/j'}(z') P(p_{j'}|p_j,\vec{r}_\bot, \phi), \ \ \
\end{eqnarray}
where $D_{h/j}(z)$ is the vacuum fragmentation function, and $z = p_h / p_{j}$ and $z' = p_h / p_{j'}$ are two momentum
fractions with $p_h$ the hadron momentum and $p_{j}$($p_{j'}$) the initial (final) jet momentum.
In the above equation, $P(p_{j'}|p_j,\vec{r}_\bot,
\phi)$ is obtained by solving Eq.~(\ref{FP-eq}), representing the probability of obtaining a jet $j'$ with momentum
$p_{j'}$ from a given jet $j$ with momentum $p_j$. It depends on the path taken by the parton and the medium
profile along that path, which in turn depends on the location of the jet origin $r_\bot$ and its propagation angle
$\phi$ with respect to the reaction plane. Therefore, one must convolve the above expression over all transverse
positions and directions to obtain the final hadron spectra. The initial hard parton densities are determined from the
overlap geometry between two nuclei in the transverse plane of the collision zone.
\begin{figure}[htb]
\begin{center}
\resizebox{0.78\linewidth}{!}{%
\includegraphics{raa_radiative_vs_collisional.eps}
}\end{center} \caption{The nuclear modification factor $R_{AA}$ for neutral pions in central and mid-central collisions
(from Ref. \cite{Qin:2007rn}). The dashed curves account for only radiative energy loss, the dash-dotted curves for only
collisional energy loss and the solid curves incorporate both radiative and collisional energy loss mechanisms.
} \label{raa}
\end{figure}
The calculation of $R_{AA}$ for pions measured at mid-rapidity is shown in Fig. \ref{raa} for two different impact
parameters, 2.4 fm and 7.5 fm. The experimental data are taken from PHENIX \cite{Adler:2002xw} for the most central
(0--5\%) and mid-central (20--30\%) collisions. The strong coupling constant $\alpha_s$ is the only
free parameter in our model and it is adjusted in such a way that $R_{AA}$ in most central
collisions (Fig. \ref{raa}, upper panel) is described. The same value, $\alpha_s=0.27$, is used throughout this work for the calculation of photon
production and jet-photon correlations. In this figure, we calculate and compare the relative contribution of induced
gluon radiations and elastic collisions to the final pion $R_{AA}$. One finds that the overall magnitude of $R_{AA}$ is
sensitive to both radiative and collisional energy loss while the shape does not show a strong sensitivity.
\section{Photon production} \label{sec:2}
In nucleus--nucleus collisions, high-$p_T$ photons may be produced from a variety of sources, namely direct
photons, fragmentation photons and jet--plasma photons. Direct photons are produced from hard scatterings in the early
stage of the nuclear collisions, and may be computed from pQCD in the factorized formalism. Fragmentation photons are
produced from the surviving jets after their passing through the soft medium and may be obtained in a
way analogous to the calculation of high-$p_T$ hadrons by replacing the medium-modified hadron fragmentation functions
by appropriated photon fragmentation function.
Jet--plasma photons are those produced from the processes involving the interaction between jets and
the surrounding medium when jets are traversing the plasma. They include induced photon radiation
(bremsstrahlung photons) and the conversion from high energy jets by $2\to 2$ scatterings (conversion photons).
Jet--plasma photons may be
calculated by incorporating an additional evolution equations for photons in Eq.~(\ref{FP-eq}),
\begin{eqnarray}
\label{photon_evolve} \frac{dP^{\rm jet}_\gamma(E,t)}{dt} \!&=&\!\! \int \! d\omega P_{q\bar{q}}(E{+}\omega,t)
\frac{d\Gamma^{\rm jet}_{q\to \gamma}(E{+}\omega,\omega)}{d\omega dt}, \ \ \ \ \ \
\end{eqnarray}
where $d\Gamma^{\rm jet}_{q\to \gamma}(E,\omega)/d\omega dt$ are the transition rates for jet--plasma channels,
including photon bremsstrahlung processes and jet--photon conversion processes, $\Gamma^{\rm jet}_{q\to \gamma} =
\Gamma^{\rm brem}_{q\to \gamma} + \Gamma^{\rm conv}_{q\to \gamma}$. The transition rates for induced photon radiation
have been discussed in Ref. \cite{Arnold:2001ms,Arnold:2001ba,Arnold:2002ja}. The transition rates for jet--photon
conversion processes may be inferred from the photon emission rates in those processes,
\begin{eqnarray}
\frac{d\Gamma^{\rm conv}_{q\to \gamma}(E,\omega)}{d\omega dt} \!&=&\!\! \sum_f \left(\frac{e_f}{e}\right)^2 \frac{2\pi
\alpha_e \alpha_s T^2}{3E} \nonumber\\ && \times \left(\frac{1}{2} \ln \frac{ET}{m_q^2} + C_{2\to 2} \right)
\delta(\omega) . \ \ \
\end{eqnarray}
where $m_q^2 = g_s^2 T^2 / 6$ is the thermal quark mass. In the limit of $E \gg T$, $C_{2\to 2} \approx
-0.36149$ is a constant \cite{Arnold:2002ja}.
The $\delta$ function generates the constraint that the incoming quarks (or anti-quarks)
experience no energy loss in the conversion processes.
\begin{figure}\begin{center}
\resizebox{0.78\linewidth}{!}{%
\includegraphics{photon_yield_3d_0_10.eps}
}\end{center} \caption{ The contributions from different channels to the photon production in Au+Au collisions at
RHIC for $b=2.4$~fm compared with most 0--10\% PHENIX data. }
\label{photon_yield}
\end{figure}
In Fig. \ref{photon_yield}, the high-$p_T$ photon production from different channels is shown for most central 0--10\%
Au+Au collisions ($b=2.4$~fm) at RHIC, where the number of binary collisions is taken to be $\langle N_{\rm coll}
\rangle = 955$ \cite{Adler:2003qi}. Our calculations are in good agreement with the data taken from PHENIX
\cite{Isobe:2007ku}. While the direct photons dominate at the very high $p_T$ regime ($p_T > 6$~GeV), the presence of the
jet--plasma interaction is important for describing the total photon yield in Au+Au collisions at RHIC, especially
in the intermediate $p_T$ ($p_T \approx$ 3--6~GeV). The thermal photons are expected to be dominant at very low $p_T$
\cite{Turbide:2005fk,Turbide:2007mi}.
\section{Photon-tagged jets} \label{sec:3}
In correlation studies, one of the most interested observables is the per-trigger yield $P(p_T^{\rm asso}|p_T^{\rm
trig})$,
which represents the conditional probability of producing an associated particle with momentum $p_T^{\rm asso}$ given
a trigger particle with momentum $p_T^{\rm trig}$,
\begin{eqnarray}
P(p_T^{\rm asso}|p_T^{\rm trig}) = \frac{P(p_T^{\rm trig},p_T^{\rm asso})}{P(p_T^{\rm trig})},
\end{eqnarray}
where $P(p_T^{\rm asso})$ and $P(p_T^{\rm trig},p_T^{\rm asso})$ are the single-particle and two-particle joint
probability distributions. As for high-$p_T$ photon--hadron correlations, the associated hadrons are produced from the
fragmentation of survival jets after their passing through the nuclear medium, while the triggered photons may come
from various sources, as has been discussed in the previous section.
Therefore, the photon--hadron correlations depend on jet-photon correlations and the energy loss of the photon-tagged
jets.
To take into account all photon sources, we may write the photon yield as the sum of various parts, each associated
with a specific source,
\begin{eqnarray}
P(p_T^\gamma|p_T^j) = \sum_{\rm src} P(p_T^\gamma, {\rm src}|p_T^j),
\end{eqnarray}
where the sources include direct photons, fragmentation photons and jet--plasma photons. As for direct photons, the
probability distribution is $P(p_T^\gamma, {\rm dir}|p_T^j) = \delta(p_T^\gamma-p_T^j)$, since
the direct photons have the same momenta as the away-side jets at the production time. The probability function for
fragmentation photons is related to the medium-modified photon fragmentation function by $P(p_T^\gamma, {\rm
frag}|p_T^j)=\tilde{D}_{\gamma/j}(z) /p_T^j$, where $z=p_T^j/p_T^\gamma$ is the momentum fraction. As for jet--plasma
photons, the probability function $P(p_T^\gamma, {\rm jet}|p_T^j)$ may be obtained by solving Eq.~(\ref{photon_evolve})
directly.
\begin{figure}[htb]
\begin{center}
\resizebox{0.78\linewidth}{!}{%
\includegraphics{initial_jet_distribution_15_sum.eps}
}\end{center} \caption{The contributions from fragmentation photon and jet--plasma photon parts to the initial jet
momentum distribution at the production time when we trigger on a photon with momentum $p_T^\gamma=15~{\rm GeV}$ in most
central Au+Au collisions at RHIC.} \label{initial_jet_15}
\end{figure}
It would be useful to first take a look at the initial jet momentum profiles at production time, given a trigger photon
with certain transverse momentum. To compare the relative contributions from different photon sources, we may break the
per-trigger yield of the initial jets into different parts,
\begin{eqnarray}
P(p_T^j|p_T^\gamma) = \sum_{\rm src} P(p_T^j, {\rm src}|p_T^\gamma).
\end{eqnarray}
The above probability function $P(p_T^j, {\rm src}|p_T^\gamma)$ is plotted in Fig. \ref{initial_jet_15} for various
photon sources, where the trigger photons have fixed momenta, $p_T^\gamma=15$~GeV. It is obvious that the
probability distribution of jets (quarks plus gluons) tagged by direct photons is a $\delta$ function with some
normalization weight determined by the fraction of direct photons to total photons at $15$~GeV. The momentum profiles
of jets tagged by fragmentation photons and jet--plasma photons are shown by solid and dashed curves.
We find that the initial jets may have higher or lower momenta than the trigger photons. This is attributed to the
energy loss or gain experienced by the jets traversing the thermal medium before producing photons by fragmentation and direct
jet--plasma interaction. It is found that jets tagged by fragmentation photons dominate at high momentum regime, while
jets tagged by jet--plasma photons is prevalent at the relatively lower momentum regime except at $15$~GeV.
\begin{figure}[htb]
\begin{center}
\resizebox{0.78\linewidth}{!}{%
\includegraphics{contribution_to_yield_per_trigger_15.eps}
}\end{center} \caption{Various contributions to per-trigger yield of the away-side hadrons when we trigger a photon
with momentum $p_T^\gamma=15~{\rm GeV}$ in most central Au+Au collisions at RHIC. }
\label{contribution_15}
\end{figure}
By applying the medium-modified fragmentation function in Eq.~(\ref{medium_frag_fun}) to the above jet momentum
profiles, we may obtain the momentum profiles of the associated hadrons. Also, the per-trigger yield of associated
hadrons may be written as the sum of different parts associated with different photon sources,
\begin{eqnarray}
P(p_T^h|p_T^\gamma) = \sum_{\rm src} P(p_T^h, {\rm src}|p_T^\gamma).
\end{eqnarray}
In Fig. \ref{contribution_15}, the above probability function $P(p_T^h, {\rm src}|p_T^\gamma)$ for different sources
of high-$p_T$ photons are plotted as a function of hadron transverse momenta. We observe that the away-side hadrons at
relatively lower $p_T$ regime are mostly produced from those jets that are tagged by direct photons. However, a large amount of
away-side hadrons at the higher $p_T$ regime come from those jets tagged by jet--plasma photons and fragmentation photons.
Especially, close to the trigger photon momentum, the away side hadron production is dominated by those jets tagged by
fragmentation photons. It is noted that the away-side hadrons could have higher momenta than the trigger photon (not
shown).
In the above manuscript, we only present some results for photon-tagged jets in Au+Au collisions at RHIC energies. The
nuclear modification of the photon-triggered hadron production may be obtained by comparing with similar calculations
for p+p collisions. Details of such calculations as well as the comparison to the experimental data will be presented
in an upcoming publication \cite{Qin:2008}.
\section{Conclusions and discussions}
We first calculate the radiative and collisional energy loss of hard partons traversing the
quark--gluon plasma and compare the respective size of these contributions to the nuclear suppression in Au+Au
collisions at RHIC. The evolution of the thermal medium created in those collisions is modeled by utilizing
(3+1)-dimensional relativistic hydrodynamics. It is found that the magnitude of $R_{AA}$ is sensitive to the inclusion
of both radiative and collisional energy loss.
Then we apply the same formalism to study the energy loss of photon-tagged jets in Au+Au collisions at RHIC. For high-$p_T$ photon production, we take into account direct photons as well as fragmentation photons and jet--plasma photons.
Our results illustrate that the interaction between hard partons and the soft medium are significant for the
description of photon data at RHIC. As for the energy loss of photon-tagged jets, we observe the effects
due to jet--plasma interaction and fragmentation. Especially, these sources show a significant contribution to the
photon-hadron correlations at the high-$p_T$ regime for the associated hadrons. Therefore, it is important to include
all photon sources for a full understanding of high-$p_T$ photon--hadron correlations at RHIC. We
conclude that both high transverse momentum jets and photons as well as their correlations are very helpful in
probing the interaction between hard jets and the surrounding hot and dense medium created in
relativistic heavy-ion collisions.
\section{Acknowledgements}
We thank C. Nonaka and S. A. Bass for providing their hydrodynamical evolution calculation \cite{Nonaka:2006yn}.
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada
and by the Fonds Nature et Technologies of Quebec.
|
1,941,325,220,549 | arxiv | \section{ Introduction}
If there are compact extra dimensions it is then possible to associate
4-dimensional scalars with the {\it extra} components of gauge bosons;
these scalars can then be
responsible for the breaking of electroweak (EW)
symmetry~\cite{Hosotani:1988bm}, playing
the same role as the Standard Model\ Higgs doublet. In this so-called
Gauge-Higgs (GH) unification
one first proposes a non-Abelian gauge symmetry in the full
D-dimensional space time (where ${\rm{D}}=4+n$),
and assumes $n$ of these are compactified, usually over an orbifold
\cite{Hebecker:2001jb}
(for a review see {\it e.g.} \cite{Quiros:2003gg}).
The compact space together with boundary conditions are then chosen
to insure the presence of light 4-dimensional vector modes
corresponding to a low-energy
$\su3 \times \su2 \times U(1)$ gauge group, together with a 4-dimensional
scalar sector that leads to the right symmetry breaking pattern
to $\su3 \times U(1)_{EM}$. Models constructed within this
scenario involve two high energy scales:
the compactification radius $L$ and
an ultraviolet (UV) cutoff $\Lambda$,
beyond which the model ceases to be perturbative. This UV cutoff appears
because the models are non-renormalizable;
it is not hard to insure $ \Lambda L \gg 1 $ as required by
consistency~\cite{Chacko:1999hg,Grzadkowski:2007xm}.
Among the virtues of such a scheme, in addition to the absence of fundamental
scalars, is the possible solution to the hierarchy
problem~\cite{Hatanaka:1998wr,Sakamoto:2006wf}. Another feature is the fact that the low-energy
effective scalar potential, which is responsible for the EW symmetry
breaking, is determined by the group structure, fermion
content, and boundary conditions of the theory.
Several realizations~\cite{Hall:2001zb,Burdman:2002se,
Haba:2002vc,Gogoladze:2003bb,Scrucca:2003ut,Haba:2004qf,
Haba:2004jd,Chiang:2010hy,lim:2007xh} of this mechanism have been achieved with various
degrees of success.
5-dimensional models generally predict a very light Higgs boson. This is due to
the absence of a tree-level quartic term in the scalar
potential~\cite{Haba:2004bh,Gogoladze:2007ey}, which can be ameliorated by
considering models with 6 (or more) dimensions.
These models also typically predict a tree-level value of
$ \sin^2 \theta_{\rm w} $ ($\theta_w$ denotes the weak mixing angle)
higher than the experimental observation of $ \sim1/4 $;
in fact, for 5-dimensional gauge theories, it has been
shown~\cite{Grzadkowski:2006tp} that any model that assumes a low-energy Standard Model (SM)
gauge boson spectrum only, is inconsistent with the requirements $\sin^2\theta_W
\simeq 1/4$ and $\rho \equiv m_W^2/(m_Z^2\cos^2\theta_W)=1$
at tree level (equivalently, a vanishing tree-level
oblique T-parameter). This problem has been addressed
by introducing additional $U(1)$
factors~\cite{Antoniadis:2001cv,Scrucca:2003ra,Cacciapaglia:2005da} and/or brane
kinetic terms~\cite{Burdman:2002se,Agashe:2004rs,Aranda:2005ze}.
In this paper, following the analysis presented in~\cite{Grzadkowski:2006tp}, we
determine to what extent GH models naturally satisfy the following
set of low-energy constraints:
\bit
\item The only gauge bosons with masses at the EW mass scale are those
present in the Standard Model.
\item $\rho = 1$ (at tree-level), {\it i.e.} we look for models that contain only
isodoublet
or isosinglet 4-dimensional scalars~\footnote{While isodoublets are not the only
``$\rho$-safe'' representations, they {\em are} the only ones with this property
that occur in the models being considered.},
\item Contain representations that can accommodate all Standard Model\ particles;
in particular at least one representation should contain
isodoublet states of hypercharge $1/6 $, corresponding to the
left-handed quark fields.
\item Lead to acceptable values of $s_{\rm w}^2$.
\eit
For all simple groups we determine whether or not these
requirements can be fulfilled by the group structure itself without allowing
additions to the models such as fundamental scalars or brane couplings. We
present the analysis and results for an arbitrary number of extra dimensions and
all compactification schemes. Our approach is group theoretical supplemented by
the above four experimentally-motivated conditions.
We find that it is relatively easy to satisfy the first two requirements,
while the third one is obeyed in only a relatively small set of models
(listed in eq. \ref{eq:models}). It proved impossible, in addition, to
satisfy $ s_{\rm w}^2\sim1/4 $:
in the absence of brane couplings all GH models must either contain
additional light gauge bosons (with masses $\sim m_{\rm W,Z}$),
or must have a tree-level weak-mixing angle
substantially different form the observed one, or cannot accommodate quarks.
The masses of the additional
light gauge bosons are of the same order as those of the $W$ and $Z$ and
are subject to similar radiative corrections, so we do not expect
them to be split form the electroweak scale by loop effects;
hence models that exhibit
such particles appear phenomenologically excluded. One of the main results of
this paper is to provide a simple way of determining whether these
undesirable states are present without a reference to the compactification
scheme.
In contrast, radiative corrections to weak-mixing angle can be substantial and
can easily lead to a 50\%\ (or larger)
effect~\cite{Dienes:1998vh,Dienes:1998vg}. For this
reason we will first determine gauge groups that comply with the first three
conditions above and then discuss the possibility of obtaining a
phenomenologically viable weak-mixing angle at the electroweak scale.
In this paper we concentrate on the gauge-boson structure of the theory; fermions
will be discussed only in connection with their possible $\su2\timesU(1) $ quantum
numbers. The possibility of creating specific models that accommodate the right
light fermion with the correct $\su3\times\su2\timesU(1)$ quantum numbers, and -
more challenging - the observed Yukawa couplings, will not be studied here. It is
known that, even if absent at tree-level, radiative contributions generate
4-dimensional couplings on the orbifold fixed points~\cite{Georgi:2000ks}. Given a
specific realization of the GH scenario one must take this into account, and
determine the extent of the associated effects on the low-energy theory --
especially if it is assumed that the Standard Model\ fermions are localized in one of these
fixed points . This investigation requires a specific choice of model (including
gauge field, fermion representations, orbifold compactification and periodicity
conditions on the fields) and lies beyond the scope of the model-independent
restrictions presented here.
The paper is organized as follows: after a brief description of the conventions
used in our analysis, we present the general gauge and space time setup in Section~\ref{s2}.
The gauge transformations and Kaluza-Klein decomposition of the fields, as well as
the analysis regarding light vector bosons and
the necessary conditions for their absence are presented in Section~\ref{s3}.
In section~\ref{s5} we present the
vector boson mass matrix and the requirements needed for matter fields. Finally
we present our results in section~\ref{s6}. We have also included a couple of
appendixes at the end with details of the calculations.
\section{ Lagrangians and symmetries} \label{s2}
The material presented in this section is not new, it is included
for convenience and in order to introduce the notation that will
be used throughout the paper.
\subsection{Conventions}
In describing the group structure of the models being considered we
find it convenient to use a canonical basis where we
denote the Cartan generators by $ C_i $
(or, when appropriate as a vector {\bf C}), and the root generators by
$ E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} $. These satisfy
\beq \label{eq:generatoralgebra}
[ {\bf C}, E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}] = {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}\,; \quad
[E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}},E_{\-{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}} ] = {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} \cdot {\bf C} \,;\quad
[E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}, E_{\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}} ] = N_{{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}, {\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}}} E_{{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} + {\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}}} ~~
({\bm\beta}} % \def\betbf{{\pmb{$\beta$}} + {\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}} \not= {\bf0})
\eeq
where we will not need the explicit form of $N_{{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}, {\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}}}$
\cite{Gilmore:1974}.
Note that in this basis the structure constants are not
completely antisymmetric.
We will denote the simple roots by
$ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i $ and the corresponding fundamental weights
by $ {\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_j $ (see \cite{Gilmore:1974,Georgi:2000ks} for details), with
\beq
{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i \cdot {\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_k = \frac12} %\def\half{{1\over2} \left| {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i \right|^2 \delta_{ik}
{\quad\Rightarrow\quad} {\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_k = \sum_j \left(a^{-1}\right)_{j k} {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^j
\eeq
(no sum over $i$ in the first expression) where
\beq
a^{ij} = 2 \frac{{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i\cdot{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^j }{{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i\cdot{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i }
\label{eq:cart.mat}
\eeq
(no sum over $i$)
is called the Cartan matrix (which is not symmetric in general).
We also find convenient
to introduce the rescaled weights
\beq
\tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_j = \frac2{|{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^j|^2} {\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_j
{\quad\Rightarrow\quad}
{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i \cdot \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_j = \delta_{ij}
\label{eq:def.of.mu}
\eeq
(no sum over $j$ in the first expression) .
For the adjoint representation we assume that the
corresponding matrices are normalized according to
(using the same symbol for the generator as for its
representation)
\beq
{\rm tr} C_i C_j = \delta_{i,j} \,, \quad
{\rm tr} E_{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} = \delta_{{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}+{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}, 0} \,, \quad
{\rm tr} E_{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} C_i =0 \,,
\label{eq:norm}
\eeq
which ensures that $ (-1/4) {\rm tr}
|\partial_\mu A_\nu - \partial_\nu A_\mu|^2,~
A_\mu = \sum_i A_\mu^i C_i + \sum_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} A_\mu^{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} $,
is properly normalized.
When no confusion will arise, we will
refer to a gauge field and it associated generator interchangeably;
for example we will often refer to a
generator as ``corresponding'' to a light vector boson, by which
we imply that the gauge field associated with that generator
will have a light mode.
\subsection{Compactifications and the space group $ \Gamma$}
We consider theories defined on a $4+n$ dimensional
space time of the form $ {\mathbbm M}}%\def\mBB{ \hbox{{\mysmallii I}}\!\hbox{{\mysmallii M}} \times ({\mathbbm R}}%\def\rBB{ \hbox{{\mysmallii I}}\!\hbox{{\mysmallii R}} ^n/\Gamma) $ where
${\mathbbm M}}%\def\mBB{ \hbox{{\mysmallii I}}\!\hbox{{\mysmallii M}} $ denotes the usual 4-dimensional Minkowski space
and $ \Gamma $ denotes a discrete group~\cite{Hosotani:1988bm}
(for a review see \cite{Quiros:2003gg}) whose elements
$ \gamma \in \Gamma $ act on $ {\mathbbm R}}%\def\rBB{ \hbox{{\mysmallii I}}\!\hbox{{\mysmallii R}} ^n$ as follows~\footnote{The
notation is borrowed from solid state literature, for an accessible
introduction see \cite{Koster:1956}.}:
\beq
\gamma = \{ \mathfrak{r}|{\bf l}\} {\quad\Rightarrow\quad} \{ \mathfrak{r}|{\bf l}\} {\bf y} = \mathfrak{r} {\bf y} + {\bf l} \ ,
\eeq
where the $\mathfrak{r}$ are $ n \times n $ orthogonal matrices,
{\bf y}\ denote the coordinates of $ {\mathbbm R}}%\def\rBB{ \hbox{{\mysmallii I}}\!\hbox{{\mysmallii R}} ^n$, and the $ {\bf l}$
are $n$-dimensional translation vectors
(the notation is the same as
the one used in solid state for crystal groups).
We assume that $ \Gamma $ acts trivially on $ {\mathbbm M}}%\def\mBB{ \hbox{{\mysmallii I}}\!\hbox{{\mysmallii M}} $:
$ \gamma x^\mu = x^\mu $ where $ x^\mu$ denote the coordinates of $ {\mathbbm M}}%\def\mBB{ \hbox{{\mysmallii I}}\!\hbox{{\mysmallii M}} $.
The multiplication rule for the elements in $ \Gamma $
can be easily derived from their action on {\bf y},
for example
\beq
\{ \mathfrak{r}'|{\bf l}'\} \{ \mathfrak{r}|{\bf l}\} = \{\mathfrak{r}'\mathfrak{r} | \mathfrak{r}'{\bf l} + {\bf l}'\} \,;
\quad \{ \mathfrak{r}|{\bf l}\}^{-1} = \{ \mathfrak{r}^{-1} | - \mathfrak{r}^{-1} {\bf l}\} \ .
\eeq
The group $ \Gamma $ is assumed to have an
(Abelian) translation subgroup
$ \Theta $ composed of all elements of the form
$ \{ {\mathbbm1} | {\bf t} \} $, where the translations vectors {\bf t}\
are linear combinations of a
set of basis vectors $ \{ {\bf t}_ i \} $ with integer coefficients,
\beq
\Theta = \left\{ \{ {\mathbbm1} |\sum k_i {\bf t}_i\}, ~k_i={\rm integer}
\right\} \,;
\eeq
note that, in general, the vector
{\bf l}\ in $ \{\mathfrak{r}|{\bf l}\}$ need not be a translation when
$ \mathfrak{r}\not={\mathbbm1}$. Using the multiplication
rule we find that
\beq
\{ \mathfrak{r}|{\bf l}\}^{-1} \{ {\mathbbm1} |{\bf t}\} \{ \mathfrak{r}|{\bf l}\}= \{ {\mathbbm1} |\mathfrak{r}^{-1} {\bf t}\}
{\quad\Rightarrow\quad} \mathfrak{r}^{-1} {\bf t} \in \Theta \ .
\eeq
It follows that $ \Theta $ is an invariant subgroup, and that
for all rotations $\mathfrak{r}$ and translations {\bf t}\ the
vector $ \mathfrak{r} {\bf t}$ is also a translation.
\subsection{Gauge-field Lagrangian and automorphisms}
We denote gauge vector fields by $A^M_a = (A^\mu_a,A^m_a)$ with the Greek
indices associated with non-compact directions and Latin lower case
indices with compact directions; we often write $ {\bf A}_a =
(A^{m=1}_a, \ldots, A^n_a)$.
We assume the following action of $ \Gamma $ on the $A^M_a$:
\bea
A^\mu_a(x,{\bf y}') &=& {\mathbbm V}(\gamma)_{a b} A^\mu_b(x,{\bf y}) \,; \qquad
{\bf y}' = \gamma{\bf y} = \{ \mathfrak{r}|{\bf l}\} {\bf y} = \mathfrak{r} {\bf y} + {\bf l} \cr
{\bf A}_a(x,{\bf y}') &=& {\mathbbm V}(\gamma)_{ab} \mathfrak{R}(\gamma) {\bf A}_b(x,{\bf y})
\label{eq:qaction}
\eea
for all $ \gamma \in \Gamma $, where the matrices $\mathfrak{R}$ act on the indices
associated with the compact directions:
$ (\mathfrak{R}{\bf A}_b)^m = \mathfrak{R}^m{}_l A_b^l $.
Defining the curvature tensor by~\footnote{The structure constants obey
$
f_{a b c} = - f_{b a c}
$
but, in general $ f_{a b c} + f_{a c b } \not =0 $
(though there are bases where this does hold)}
\beq
F^{MN}_a = \partial^M A^N_a - \partial^N A^M_a + g f_{b c a} A^M_b A^N_c \ ,
\eeq
we find that the $F^2$ term in the Lagrangian will
be invariant under (\ref{eq:qaction}) provided
\bea
\mathfrak{r} = \mathfrak{R}(\gamma); ~ \gamma= \{ \mathfrak{r} | {\bf l} \} && \mathfrak{R}(\gamma){}^T \mathfrak{R}(\gamma) = {\mathbbm1} \ , \cr
f_{a'b'c'} = f_{abc} {\mathbbm V}(\gamma)_{a a '} {\mathbbm V}(\gamma)_{b b'}
{\mathbbm V}^\dagger(\gamma)_{c' c} &&
{\mathbbm V}(\gamma){}^\dagger {\mathbbm V}(\gamma) = {\mathbbm1} \ ,
\label{eq:autom}
\eea
so that $ {\mathbbm V}$ must be an automorphism of the
gauge group $ {\cal G} $~\cite{Hebecker:2001jb}.
Note also that
the equality $ \mathfrak{R}(\gamma) = \mathfrak{r} $ implies $\mathfrak{R}(\gamma)$ is
independent of the vector $ {\bf l} $. In particular
\beq
\mathfrak{R}(\{ {\mathbbm1} | {\bf t} \} ) = {\mathbbm1} \ .
\label{eq:R.under.T}
\eeq
\section{KK expansions} \label{s3}
\subsection{Consequences of covariance under translations}
The matrices $ {\mathbbm V}(\{{\mathbbm1}|{\bf t}\})$ carry a representation of $\Theta
$. Since $ \Theta $ is an Abelian group it has only one-dimensional
irreducible representation s, so we can choose a basis where $ {\mathbbm V}(\{{\mathbbm1}|{\bf t}\})_{a b} =
v_a({\bf t}) \delta_{a b} $ for all {\bf t}. Since the vectors {\bf t}\ are of the
form $ \sum_i k_i {\bf t}_i$ for some integers $k_i $,
and since $ \{ {\mathbbm1} | {\bf t}_i \} \in \Gamma$, then
\beq
v_a({\bf t}) = \prod_i
\left[ v_a({\bf t}_i) \right]^{k_i} \ ,
\eeq
while the unitarity of the representation requires
\beq
v_a({\bf t}_i)= e^{ i c_{a i}}, \quad c_{a i} \rightarrow \hbox{real},
~~ |c_{a i} | < \pi \ .
\eeq
In this basis, using (\ref{eq:qaction}) and (\ref{eq:R.under.T}),
\beq
A_a^N(x, {\bf y}+ {\bf t}) = e^{i \sum_i k_i c_{a i}} A_a^N(x,{\bf y}) \ .
\eeq
Let $ \{ {\bm\kappa}} % \def\kapbf{{\pmb{$\kappa$}}_j\} $ be a set of linearly independent vectors
dual to the $ {\bf t}_i$, that is, $ {\bm\kappa}} % \def\kapbf{{\pmb{$\kappa$}}_j \cdot {\bf t}_i = \delta_{ij} $.
Then we can expand
\bea
A^N_a(x,{\bf y}) &=& e^{ i {\bf Q}_a \cdot {\bf y}}
\sum_{\bf n} e^{2 \pi i {\bf y} \cdot \sum_j n_j {\bm\kappa}} % \def\kapbf{{\pmb{$\kappa$}}_j} \tilde A^N_a(x,{\bf n}); \quad
{\bf Q}_a = \sum_j c_{a j} {\bm\kappa}} % \def\kapbf{{\pmb{$\kappa$}}_j \ ,
\label{eq:KK.expansion}
\eea
which corresponds to the usual expansion in Kaluza-Klein
(KK) modes~\cite{Hosotani:1988bm,Quiros:2003gg}.
It follows that $ \tilde A^N_a(x,{\bf n}) $ will have a
the tree-level mass $ \sim | {\bf Q}_a + 2 \pi \sum n_j
{\bm\kappa}} % \def\kapbf{{\pmb{$\kappa$}}_j | $; we will assume that the scales associated with the $
{\bm\kappa}} % \def\kapbf{{\pmb{$\kappa$}}_i $ are large compared to the electroweak
scale. In this case
all light modes $ \tilde A^N_a $ must correspond to $ {\bf n} =0 $
and $ {\bf Q}_a =0 $. But $ {\bf Q}_a =0 $ requires $ c_{a i } = 0 $,
which implies that gauge fields that have light modes are
translationally invariant.
Since $ \Theta $ is an invariant subgroup we find that
for all $ \gamma$,
$ {\mathbbm V}(\gamma )_{ab} =0 $ {\em unless } ${\bf Q}_a = {\bf Q}_b $.
In particular the $ {\mathbbm V} $ do not mix the light
and heavy KK modes. Using this and substituting
(\ref{eq:KK.expansion}) in (\ref{eq:qaction})
we find that the light modes must obey
\beq
{\mathbbm V}(\gamma)_{a b} \tilde A^\mu_b(x,0)
=
\tilde A^\mu_a(x,0) \qquad
{\mathbbm V}(\gamma)_{a b} \mathfrak{R}(\gamma) \tilde{\bf A}_b(x,0)
=
\tilde{\bf A}_a(x,0) \ ,
\label{eq:mzero}
\eeq
that is, light 4-dimensional vectors are associated with the ``trivial''
subspace where $ {\mathbbm V}= {\mathbbm1} $, while light 4-dimensional scalars are
associated with the subspace where $ {\mathbbm V} \otimes \mathfrak{R} = {\mathbbm1} $.
One of the challenges in constructing a realistic theory
of this type is to find a ``compactifying'' group $ \Gamma $,
a gauge group $ {\cal G} $ and representations $ {\mathbbm V}$
and $ \mathfrak{R} $ such that the solutions to (\ref{eq:mzero})
will correspond to the bosonic sector of the Standard Model\ (possibly
with an extended scalar sector). While it is always
easy to ensure that a given field has a light mode,
the relations (\ref{eq:autom}) often imply the presence
of {\em additional} light vectors which, as we will see,
have masses of the same order as the Standard Model\ gauge bosons.
Requiring the absence of such undesirable light particles
severely restricts the choice of gauge groups $ {\cal G} $.
The gauge transformations are
\beq
A^N \rightarrow U^\dagger \left( i \partial^N + A^N \right) U; \qquad
A^N = g A^N_a T_a\,, ~~ U \in {\cal G}
\eeq
equivalently,
\beq
A^N_a \rightarrow {\cal U}^N_a + {\mathbbm U}_{ab} A^N_b;
\qquad
T_a {\cal U}^N_a = i U^\dagger \partial^N U,
\quad
T_a {\mathbbm U}_{ab} = U^\dagger T_b U \ .
\label{eq:gauge.transf}
\eeq
Consider now those $ {\cal U},~{\mathbbm U}$ that depend only on $x$. Using
(\ref{eq:KK.expansion}) in (\ref{eq:gauge.transf}) implies
that $ {\cal U}^\mu_a \not=0 $ if $ {\bf Q}_a =0 $; and
$ {\mathbbm U}_{ab}(x) \not=0 $ requires $
{\bf Q}_a - {\bf Q}_b = 2 \pi \bar n_j\up{a b} {\bm\kappa}} % \def\kapbf{{\pmb{$\kappa$}}_j$
with $ \bar n_i\up{a b} $ integers;
in particular if $ {\bf Q}_a =0 $, then either $ {\bf Q}_b =0 $
or else $ {\mathbbm U}_{ab}(x) =0 $.
Then, for the light fields obeying (\ref{eq:mzero}),
\bea
\tilde A^\mu_a(x,{\bf0}) &\rightarrow& {\cal U}^\mu_a(x)+ \sum_b
{\mathbbm U}_{ab}(x) \tilde A^\mu_b(x,{\bf0}) \cr
\tilde{\bf A}_a(x,{\bf0})
&\rightarrow& \sum_b {\mathbbm U}_{ab}(x)
\tilde{\bf A}_b(x,{\bf0})
\eea
showing that when $ {\cal U}^\mu_a \not =0 $ the
$ \tilde A^\mu_a(x,{\bf n}=0) $ transform as gauge fields
and, because of (\ref{eq:mzero}) these also
transform trivially under $ \Gamma $.
We denote these as the light gauge fields, and the set of
{\bf y}-independent transformations as the light gauge group $G$.
The light gauge group $G$ is a subgroup of ${\cal G}$, so that all
fields can be classified according to their $G$ representation.
For a realistic theory~\footnote{Models with additional $U(1)$
factors appear frequently, but in the absence of brane couplings
or bulk-propagating scalar fields,
the masses of the additional vector bosons are unacceptably low.}
$ G = \su3 \times\su2 \times U(1) $.
As in the 5-dimensional case~\cite{Grzadkowski:2006tp} the $\su2$ generators
$ J^{0,\pm} $ are specified
by choosing a root $ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}$, while the hypercharge
generator $Y$ can be taken
as a linear combination of Cartan generators:
\beq
J^0 = \inv{|{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}|} \hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}\cdot{\bf C} \,, \quad
J^\pm = \frac{\sqrt{2}}{|{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}|} E_{\pm{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}} \,;
\qquad Y = {\hbox{\egothfamily y}} \cdot {\bf C} \,.
\eeq
Though $ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} $ can be any root, there is always a freedom to relabel
axes and axis directions, so that many choices are
equivalent. A straightforward inspection of
the various classical groups
shows that, in fact, $ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} $ can be taken to be one of
the simple roots.
Using these definitions we can find the isospin $s$ and
$z$-component isospin $s_z$ of any root $ {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}$:
\beq
s = \frac{n_+ + n_- }2 \,, \qquad
s_z = \frac{n_- - n_+ }2 \ ,
\label{eq:def.of.s.sz}
\eeq
where $n_\pm $ are non-negative integers such that
${\bm\beta}} % \def\betbf{{\pmb{$\beta$}} + k {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}$ and $-n_n \le k \le n_+ $ are roots but
${\bm\beta}} % \def\betbf{{\pmb{$\beta$}} \pm (n_\pm +1) {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} $ are not.
If $ {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} $ is also a simple root then $n_- =0 $
so that $ s = - s_z = n_+/2 $.
The vector {\hbox{\egothfamily y}}\ is unspecified, except for the requirements that
the model should contain isodoublets of hypercharge $1/2 $ that can acquire
a vacuum expectation value, and that $ J^{0,\pm}$ commute with $Y$:
\beq
\left[ J^\pm , Y \right] =0 {\quad\Rightarrow\quad} {\hbox{\egothfamily y}}\cdot{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} =0
\eeq
Concerning the light 4-dimensional scalars we require that they
give rise to $ \rho = 1$ at tree level. Since these scalars are
associated with the adjoint generators their isospin can be read-off
form the Cartan matrix of the gauge group $ {\cal G} $; for the
simple groups a direct examination shows that only isospin $ \le 3/2 $
will occur (see appendix~\ref{sec:class.groups}), and so this constraint by
itself does not rule out any of them.
The scalars are then taken to be associated with specific root
generators $ E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} $ and a
realistic model must ensure that only the isospin $1/2 $
modes can acquire a vacuum expectation value.
Requiring consistency between (\ref{eq:qaction}) and (\ref{eq:gauge.transf})
implies
\bea
&& {\cal U}^\mu_a(x,{\bf y}') = {\mathbbm V}_{ab}(\gamma) {\cal U}^\mu_b(x,{\bf y}) \qquad
{\cal U}^m_a(x,{\bf y}') = \mathfrak{R}^m{}_l {\mathbbm V}_{ab}(\gamma) {\cal U}^l_b(x,{\bf y}) \cr
&& {\mathbbm U}(x,{\bf y}') = {\mathbbm V}(\gamma) {\mathbbm U}(x,{\bf y}) {\mathbbm V}^{-1}(\gamma) \ ,
\eea
where $ {\bf y}' = \mathfrak{r} {\bf y} + {\bf l},~~ \gamma=\{\mathfrak{r}|{\bf l}\} $. In
particular for $ {\mathbbm U} \in G$, $ {\mathbbm U} $ is independent of $x$
and
\beq
\left[ {\mathbbm U}(x) , {\mathbbm V}(\gamma)\right] = 0 \ .
\eeq
This relation implies (by Schur's lemma) that such $ {\mathbbm U}$ do
not mix $ \Gamma $ irreducible representation s carried by the $ {\mathbbm V} $, and the
$ {\mathbbm V} $ do not mix $G$ irreducible representation s carried by the $ {\mathbbm U} $. In
particular, the $ {\mathbbm V} $ will not mix generators that
have different $G$ quantum numbers.
Consider now the set of generators that are $G$ singlets, denoted
by $ T\up{\mathfrak{s}}_S$, and a set of generators $\tilde T_r $ that have
fixed $G$ quantum numbers (in our case, fixed hypercharge,
isospin and $z$-isospin component). Then it follows that
$ {\mathbbm V}_{S a} =0 $ unless $ a = R $, corresponding to some $ T\up{\mathfrak{s}}_R$,
and $ {\mathbbm V}_{r a} = 0 $ unless $ a = s $, corresponding to some
$ \tilde T_s $. The automorphism condition then implies
(using a basis where
$ f_{abc}$ is antisymmetric in $a$ and $b$ but not in all 3 indices)
\beq
f_{T r s} = {\mathbbm V}_{T' T} {\mathbbm V}_{r' r}
{\mathbbm V}^{-1}_{s s'} f_{T' r' s'}
{\quad\Rightarrow\quad} \sum_r f_{T r r} = \sum_{T'} {\mathbbm V}_{T' T} \sum_{r'}
f_{T' r' r'} \ .
\eeq
Then
\beq
F_S = F_R {\mathbbm V}_{R S} \,, \quad
\hbox{\rm where} \quad
F_S = \sum_r f_{S r r} \ .
\eeq
Now define
\beq
\tilde T\up{\mathfrak{s}} = \sum_S F_S T\up{\mathfrak{s}}_S
= \sum_{S\,r} f_{S r r} T\up{\mathfrak{s}}_S \ ,
\eeq
which is an $G$ singlet, while under $ \Gamma $
\beq
\tildeT\up{\mathfrak{s}} \rightarrow \sum {\mathbbm V}_{ R S} T\up{\mathfrak{s}}_S F_R =
\tildeT\up{\mathfrak{s}} \ ,
\eeq
so that this generator is also a $ \Gamma $ singlet.
This generator depends on the choice of $ \tilde T_r $ so that
there will be a $ \tildeT\up{\mathfrak{s}}$ for each set of $G$ quantum numbers.
Note that any linear combination of the $ \tildeT\up{\mathfrak{s}} $
will also be a $G$ and $ \Gamma $ singlet.
Now, the expression for $ \tildeT\up{\mathfrak{s}} $ involves $ f_{S r r} $,
which is not zero only if the commutator $ [T\up{\mathfrak{s}}_S , \tilde T_r ] $
has a term proportional to $ \tilde T_r $ itself. Looking now
at the commutators in terms of roots and Cartan generators it is
clear that this can happen only if $ T\up{\mathfrak{s}}_S $ is a linear combination
of Cartan generators and $ \tilde T_r $ is a root generator:
\beq
\left[ {\bf v} \cdot {\bf C}, E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} \right] = {\bf v} \cdot {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}
{\quad\Rightarrow\quad} f_{{\bf v}\cdot{\bf C} \, E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} \, E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}} = - i {\bf v} \cdot {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} \ ,
\eeq
so we identify $ T\up{\mathfrak{s}}_S $ with $ {\bf v} \cdot {\bf C} $, which
will be an $ \su2$ singlet provided
\beq
{\bf v}\cdot{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} =0 \ .
\eeq
It is also a $U(1)$ singlet since the
hypercharge generator is of the form $ {\hbox{\egothfamily y}} \cdot {\bf C} $.
Consider now a series of vectors $ \hat {\bf v}_S $ perpendicular to
$ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} $ and satisfying
\beq
\sum_S \hat{\bf v}_S \otimes \hat{\bf v}_S = {\mathbbm1} - \hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} \otimes \hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} \ .
\eeq
Then we can take $ T\up{\mathfrak{s}}_S = \hat{\bf v}_S \cdot{\bf C}$ and
the $G$ and $ \Gamma $ singlet generators are of the form
\beq
\sumprime_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}
\sum_S\left(\hat{\bf v}_S \cdot {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}\right) \hat{\bf v}_S \cdot {\bf C}
= \left(\sumprime_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}_\perp \right)\cdot {\bf C} \ ,
\label{eq:sum.1}
\eeq
where $ {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}_\perp = {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} - (\hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} \cdot {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}) \hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}$, and
the prime indicates that the sum is over all roots with
a specific set of $G$ quantum numbers. In the following we write
\beq
\sumprime_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}_\perp = {\hbox{\egothfamily y}}'_q,~ ~
q = \{ h ,s,s_z\}
\label{eq:sum.2}
\eeq
Since any two roots $ {\bm\beta}} % \def\betbf{{\pmb{$\beta$}},~{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}' $
in the same $ \su2$ multiplet satisfy $ {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} = {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}'
+ n {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} $, it follows that $ {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}_\perp = {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}'_\perp $.
If $ {\bm\beta}} % \def\betbf{{\pmb{$\beta$}},~ {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}'$ carry $G$ quantum numbers
$q = \{h, s, s_z \},~
q' = \{h, s, s_z' \} $ respectively, then
$ {\hbox{\egothfamily y}}'_q = {\hbox{\egothfamily y}}'_{q'} $. Note also that
\beq
{\hbox{\egothfamily y}}'_q = - {\hbox{\egothfamily y}}'_{\bar q} \,, \quad q = \{h , s , s_z \}, ~~
\bar q = \{ -h, s, -s_z \} \,,
\eeq
so that we can restrict ourselves to vectors $ {\hbox{\egothfamily y}}' $ associated
with positive hypercharge, $ h> 0 $, roots.
The generators $ {\hbox{\egothfamily y}}'_q \cdot {\bf C} $
are {\em necessarily} $G$ and $ \Gamma $ singlets~\footnote{
It is possible to follow the same argument
when $ G = \su2 $, the hope being that one
of the ${\hbox{\egothfamily y}}'_{s,s_z}$ can be used as {\hbox{\egothfamily y}}. This case,
however, is trivial since all the $ {\hbox{\egothfamily y}}'_{s,s_z}$
necessarily vanish (see~\ref{sec:ap2}),
which is related to the fact that
$\su2 $ has no complex representations.}: given a choice of $ {\cal G} $ and an
embedding of $G$ it is impossible to
have a light gauge group of rank 4, unless
all the vectors $ {\hbox{\egothfamily y}}_q' $ are proportional to $ {\hbox{\egothfamily y}} $.
The fact that a given choice of the Standard Model\ group as a
subgroup of $ {\cal G} $ will in general imply the presence
of additional light vector bosons, independently of the
compactification details of the model, is one of the
central results of this paper.
The vectors ${\hbox{\egothfamily y}}'_q$ need not be linearly independent,
nor do they have to be independent of {\hbox{\egothfamily y}}. Still,
by taking appropriate linear combinations we can find
and orthonormal subset
$ \hat{\hbox{\egothfamily y}}_r, ~ r=0,1, \ldots , R $ with $ \hat{\hbox{\egothfamily y}}_0 = \hat{\hbox{\egothfamily y}} $ and
$\hat{\hbox{\egothfamily y}}_r \cdot \hat {\hbox{\egothfamily y}}_s = \delta_{rs} $. The generators
$ \hat{\hbox{\egothfamily y}}_r \cdot {\bf C} $ also are $G$ and $ \Gamma $ singlets,
each generating a $U(1) $ subgroup that, by (\ref{eq:mzero}),
corresponds to a light neutral vector boson. This
result is {\em independent} of the choice
of $ \Gamma $ and the representations $ {\mathbbm V}$ and $ \mathfrak{R} $;
number $R$ of additional light vector bosons
is determined solely by the gauge structure of the
theory~\footnote{
This refers to the smallest number of light vector bosons;
specific models, of course, may have additional ones;
for example, if $ {\mathbbm V}(\gamma ) = {\mathbbm1} $
for all $ \gamma $, {\em all} $A^\mu_a$ will have light
modes.}.
We will show below that these $R$ vector
bosons may acquire masses thorough spontaneous symmetry breaking, but these are
of the order of the $W$ and $Z$ vector boson masses; in
particular models with $R \ge 1 $ are excluded phenomenologically.
One must therefore choose $G$ (that is,
hypercharge generator $Y = {\hbox{\egothfamily y}}\cdot{\bf C}$
and $ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} $) and $ {\cal G} $ such that $R=0 $, which proves to be
a stringent constraint on $ {\cal G} $ and the embedding of $G$.
\subsection{Necessary conditions for the absence of undesirable singlets}
\label{sec:nec.con}
In this section we list several requirements that gauge-Higgs unified theories
must meet in order to be phenomenologically viable
(at tree level). These
conditions are derived in appendix \ref{sec:group.theo}; here we
only list the results that are relevant to the rest of the paper.
We begin by noting that we can always assume that all the simple
roots have non-negative hypercharge and that
the root $ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} $, which defines
the Standard Model\ $\su2$ subgroup, is one of the simple roots
(see appendix~\ref{sec:ap1}),
\beq
{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} = {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^\isa \ .
\eeq
We can now divide the simple roots into 3 categories
{\it(i)} Those with positive hypercharge, $ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^{i_k}: ~~
{\hbox{\egothfamily y}}\cdot {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^{i_k} = h^k > 0 $. {\it(ii)}
Those, like $ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^\isa$, with zero hypercharge and non-zero isospin,
$ {\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}}^r: ~~
{\hbox{\egothfamily y}} \cdot{\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}}^r =0,~ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}\cdot{\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}}^r \not= 0$
(for $ {\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}}^r \not= {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^\isa $).
{\it(iii)} Those that are $G$-singlets: ${\bm\zeta}} % \def\zetbf{{\pmb{$\zeta$}}^l: ~~
{\hbox{\egothfamily y}} \cdot{\bm\zeta}} % \def\zetbf{{\pmb{$\zeta$}}^l = {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}\cdot{\bm\zeta}} % \def\zetbf{{\pmb{$\zeta$}}^l =0$.
It then follows that
\beq
{\hbox{\egothfamily y}} = \sum h^k \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_{i_k} \ .
\eeq
We have argued above that in order not to have additional
light vector bosons we must have $ {\hbox{\egothfamily y}}'_q \propto {\hbox{\egothfamily y}} $ for all $q$.
A straightforward application of Lie algebra theory (see appendix~\ref{sec:ap3})
shows that necessary conditions for this to occur are:
\bit
\item All simple roots must be either
isodoublets~\footnote{Strictly speaking the condition is
for all non-isosinglets roots to transform according to
the same $\su2$ representation; but we noted earlier that
these representations must have isospin $1/2,~1$ or
$3/2 $, of which only the first produce $ \rho= 1$ at
tree-level.} or isosinglets. This can be satisfied for any simple group.
\item All the isosinglets must have zero hypercharge.
\item All the isodoublets must have the same non-zero hypercharge:
\beq
{\hbox{\egothfamily y}} = h \sum_{\rm isodoublets} \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_{i_k} \ .
\label{eq:hyp.vec}
\eeq
In particular, there should be no roots of the type $ {\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}}^r$
except $ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} $ itself.
\eit
These conditions are useful in that they eliminate a large
number of groups; still, even when met, the low-energy spectrum must
be derived explicitly in order to insure the absence of light $Z'$
vectors.
It is worth pointing out that a choice of the $\su2$ subgroup
determines the set of simple roots that transform as isodoublets,
which in turn determines the specific embedding of the Standard Model\
$U(1)$ in $ {\cal G} $ given by (\ref{eq:hyp.vec}). Hence the ${\hbox{\egothfamily y}}'$
must be obtained for each choice of $ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} $ in order to determine
the viability of a given model.
\section{The weak mixing angle and hypercharge}\label{s5}
\subsection{Vector-boson mass matrix}
The (canonically normalized) light vector bosons
correspond to the zero modes of the gauge fields associated
with the generators $\hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} \cdot{\bf C} $, $E_{\pm{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}} $
and $ \hat{\hbox{\egothfamily y}}_r \cdot {\bf C} $, where $ \{ \hat{\hbox{\egothfamily y}}_0
, \ldots , \hat{\hbox{\egothfamily y}}_R \} $
denote the orthonormal basis for the subspace generated by $ {\hbox{\egothfamily y}}$
introduced in eq.(~\ref{eq:sum.2} (when there are no additional
$Z'$ bosons $R=0$ and $ \hat{\hbox{\egothfamily y}} = \hat{\hbox{\egothfamily y}}_0$);
we denote the corresponding zero modes
by $ W^0,~W^\pm $ and $B\up r$ respectively.
Following~\cite{Grzadkowski:2006tp}, but
allowing for the presence of more than one $\su2$-singlet
gauge boson, we expand
\bea
A_\mu &=& W_\mu^+ E_{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} + W_\mu^- E_{-{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}} + W_\mu^0 \hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}\cdot{\bf C}
+ \sum_{r=0}^R B\up r_\mu \hat{\hbox{\egothfamily y}}_r \cdot{\bf C} + \cdots \cr
A_n &=& \sum_{{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} > 0} \left( {\phi_{n,{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}} E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} + {\phi_{n,{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}}^* E_{- {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}} \right)
+ \cdots
\eea
where the sum over $ {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}$ is over those fields such that $
\vevof{\phi_{n,{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}} \not =0 $, and the ellipsis denote fields with
masses of the order $ | {\bm\kappa}} % \def\kapbf{{\pmb{$\kappa$}} | $ as discussed in section
\ref{s3}.
Then, using (\ref{eq:norm}),
\bea
- \hbox{tr} \left[ A_\mu , A_n \right]^2 &=& \sum_{{{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} > 0};
~{\rm isodoublets}} \left| {\phi_{n,{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}} \right|^2
\left\{ \frac12} %\def\half{{1\over2} {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^2 W^+ \cdot W^- + \left(W_\mu^0 \hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} \cdot
{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} + \sum_{r=0}^R B\up r_\mu \hat{\hbox{\egothfamily y}}_r \cdot {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} \right)^2 \right\} \cr
&& \quad + \sum_{{{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} > 0};
~G-{\rm singlets}} \left| {\phi_{n,{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}} \right|^2
\left(\sum_{r=0}^R B\up r_\mu \hat{\hbox{\egothfamily y}}_r \cdot {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} \right)^2 \ .
\label{eq:vbm}
\eea
It is shown in appendix \ref{sec:ap2} that
if $E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}$ is a singlet under $G$, then
$ \hat{\hbox{\egothfamily y}}_r \cdot{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} =0 $, so that the $G$-singlet contribution
to the mass term vanishes, and the mass term is simply
\beq
\sum_{{{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} > 0};
~{\rm isodoublets}} \left| {\phi_{n,{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}} \right|^2
\left\{ \frac12} %\def\half{{1\over2} {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^2 W^+ \cdot W^- + \left(W_\mu^0 \hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} \cdot
{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} + \sum_{r=0}^R B\up r_\mu \hat{\hbox{\egothfamily y}}_r \cdot {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} \right)^2 \right\} \ .
\eeq
This shows that when $R\ge1 $ there will be additional vector bosons
with mass of the same order as that of the $W$ and $Z$. In particular,
such models contain no mechanism through which the
tree-level mass of the additional
vectors can be pushed above the experimental limits, {\em irrespective
of the compactification scheme or of the fermion content of the
theory}~\footnote{At least as long as the model is weakly coupled at low energies.}.
This could be corrected by introducing scalars (either in the usual way
or using antisymmetric tensor fields), but in this case the motivation
and the attraction of this type of theories largely disappears.
For the case where $ R=0 $ we assume that the effective potential
for the 4-dimensional scalars insures these get a vacuum expectation value\ that
preserves the charge generator $ Q = J^0 + Y $, that is
\beq
\vevof{\phi_{n,{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}} \not=0 {\quad\Rightarrow\quad} {\hbox{\egothfamily y}}\cdot{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} =\frac12} %\def\half{{1\over2} , \quad
s = \frac{{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}\cdot{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}{|{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}|^2} = - \frac12} %\def\half{{1\over2} \ ,
\label{eq:sz.and.h}
\eeq
which fixes the normalization of {\hbox{\egothfamily y}}.
Writing $ B\up0 = B,~{\hbox{\egothfamily y}}_0 = {\hbox{\egothfamily y}} $,
and using $ \hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} \cdot{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} = - |{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}|/2 $,
we find
\beq
\sum_{{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} > 0} \left| {\phi_{n,{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}} \right|^2
\left\{ \frac12} %\def\half{{1\over2} {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^2 W^+ \cdot W^- + \inv4 {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^2
\left(W_\mu^0 - \inv{|{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}| |{\hbox{\egothfamily y}}|} B_\mu \right)^2 \right\} \ .
\eeq
From this we read off the tangent of the weak-mixing angle, $t_{\rm w}$:
\beq
\tan(\theta_{\rm w}) = t_{\rm w} = \inv{|{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}| |{\hbox{\egothfamily y}}|} \ .
\label{eq:tw}
\eeq
The properly-normalized
$Z$ boson field is then $ Z_\mu = \cos \theta_{\rm w}\, W_\mu^0
- \sin \theta_{\rm w} \, B_\mu$, and the vector-boson masses are given by
\beq
M_{\rm W}^2 = \frac{{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^2}2 \sum_{{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} > 0} \left| {\phi_{n,{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}} \right|^2
\,, \qquad
M_{\rm Z}^2 = \frac{ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^2}{2 \cos^2 \theta_{\rm w}} \sum_{{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} > 0} \left| {\phi_{n,{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}} \right|^2
\eeq
so that indeed $ \rho = M_{\rm Z} \cos \theta_{\rm w}/M_{\rm W} = 1 $,
as expected.
\subsection{ Matter fields}
Up to this point we have not discussed the possible
effects derived from the introduction of matter fields,
and which will further restrict the number of allowed
theories.
In viable models it should be possible
to choose the fermion content such that it includes
states with the Standard Model\ quantum numbers of the observed
quarks and leptons. In particular the choices of ${\cal G} $
and $G$ must be such that there are fermion representations
containing states with hypercharge $1/6 $.
The highest weight of a representation can be written as
\beq
{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_{\rm max} = \sum m_i {\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_i =
\sum \left( a^{-1} m \right)_j {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^j \ ,
\eeq
where the $ m_i $ are integers. A generic weight in this irreducible representation\
is obtained by applying lowering operators $ E_{-{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i} $
associated with the simple roots. The general weight then has the form
\beq
{\bm\mu}} % \def\mubf{{\pmb{$\mu$}} = \sum m_j {\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_j - \sum k_i {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i \ .
\label{eq:gen.mu}
\eeq
Now, though all the entries in the Cartan matrix
(\ref{eq:cart.mat}) $a$ are integers, this is not true
for $ a^{-1} $; still there is an integer $N$ such that
\beq
N \left(a^{-1} \right)_{ij} = {\rm integer} \ ,
\eeq
with $N$ given in table \ref{tab:t1}
(see also appendix \ref{sec:class.groups}); also, all entries
in $ a^{-1}$ are positive.
\begin{table}[h]
$$
\begin{array}{|c|c|}
\hline
{\rm Group} & N \cr \hline
\su n & n \cr
G_2,~F_4, ~ E_8 & 1 \cr
Sp(2n), ~SO(2n+1), ~ SO(4n), E_7 & 2 \cr
E_6 & 3 \cr
SO(4n+2) & 4 \cr \hline
\end{array}
$$
\caption{Values of $N$ (see text) for
several simple Lie algebras.}
\label{tab:t1}
\end{table}
It follows that the weight of any state is of the form
$ {\bm\mu}} % \def\mubf{{\pmb{$\mu$}} = \sum (n_i/N) {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i $ where the $n_i $ are integers; then
\beq
Y \ket{\bm\mu}} % \def\mubf{{\pmb{$\mu$}} = ({\hbox{\egothfamily y}}\cdot{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}) \ket{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}
= \left( \inv N \sum_i n_i {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i \cdot {\hbox{\egothfamily y}} \right) \ket{\bm\mu}} % \def\mubf{{\pmb{$\mu$}} \ .
\eeq
In order to accommodate quarks a {\em necessary} condition is
to have
\beq
{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i \cdot {\hbox{\egothfamily y}} = \frac N{ 6 \times {\rm integer} }
\label{eq:q.hyp}
\eeq
for at least one simple root;
which proves useful in
eliminating many groups.
The simplest way to find
sufficient conditions is to obtain $ {\bm\mu}} % \def\mubf{{\pmb{$\mu$}} \cdot {\hbox{\egothfamily y}} $
using (\ref{eq:hyp.vec}) and (\ref{eq:gen.mu}),
\beq
{\bm\mu}} % \def\mubf{{\pmb{$\mu$}} \cdot {\hbox{\egothfamily y}} = h \sum_{\rm isodoublets}
\left( a^{-1} m - k \right)_{i_l},
\label{eq:exact.hyp}
\eeq
from which it can be determined by inspection whether
$ {\hbox{\egothfamily y}}\cdot{\bm\mu}} % \def\mubf{{\pmb{$\mu$}} = 1/6 $ occurs in any given irreducible representation.
\section{Results and prospects}\label{s6}
From separately considering the various possibilities
(see appendix \ref{sec:class.groups}) for the unitary,
orthogonal and symplectic groups, as well as for the
exceptional groups $G_2,~F_4,~E_6,~E_7$, and $E_8$,
we find that the
models that can accommodate quarks, and do not
necessarily contain undesirable light $Z' $ bosons,
are those given in the following table:
\beq
\begin{array}{|c||c|c|c|}
\hline
{\rm group} & s_{\rm w}^2 & {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} & {\hbox{\egothfamily y}} \cr \hline
\su{3l} & 3l/(6l-2) & {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^1 & \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_2/2 \cr \hline
SO(2n+1) & 3/4 & {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^1 & \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_2/6 \cr\hline
G_2 & 3/4 & {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^1 & \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_2/6 \cr\hline
F_4 & 3/4 & {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^1 & \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_2/6 \cr\hline
E_6 & 3/8 & {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^{1,5} & \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_{2,3}/2 \cr\hline
E_7 & 3/4,~3/5 & {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^{1,7} & \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_{2,3}/6 \cr\hline
E_8 & 9/16,~3/8 &{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^{1,8} & \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_{2,3}/6 \cr\hline
\end{array}
\label{eq:models}
\eeq
where the conventions for the simple roots and weights
are given in appendix \ref{sec:class.groups}.
Phenomenologically these models face additional obstacles.
First is the large discrepancy between the tree-level
value of $ s_{\rm w}^2 $ in (\ref{eq:models})
and its low-energy experimental result of $ \sim1/4 $.
The renormalization group (RG) running between the electroweak and
the compactification scales cannot account for this difference,
except, possibly for the $E_6$ and one $E_8$ model, in which case
$ 1/L $ would be of the same order as the GUT scale. There
is, however, also a contribution from the RG evolution
between the compactification scale and the UV cutoff $ \Lambda $
which may account for this~\cite{Dienes:1998vh,Dienes:1998vg}.
Discussing this in detail falls outside
the scope of this paper, but will be investigated in a future
publication.
In addition there is the remaining question of whether there
are specific compactifications for which only the desired
modes satisfy (\ref{eq:mzero}); that is, whether there is
a specific compactification choice for which $G$ is the
full light gauge group. Again, we will not discuss this
in general but indicate the manner in which this can be realized
for one of the models considered, based on $ {\cal G} = \su6 $. For this case
we need to find a group $ \Gamma $ and a representation $ {\mathbbm V} $
such that (\ref{eq:mzero}) has solutions only for the
zero modes associated with
the ``light'' generators $ E_{\pm {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^1},
~ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^1 \cdot {\bf C} $ and $ {\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_2 \cdot {\bf C} $.
To construct $ \Gamma $ note that $\su6$ has two $\su2$
subgroups generated by
$ E_{\pm {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i}, ~{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i \cdot {\bf C}, ~ i=3,5$
that commute with each other and with $G$. The group
elements
$ \exp[ (i \pi/2) ( E_{{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i} + E_{-{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i}) ]$,
$ \exp[ (\pi/2) ( E_{{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i} - E_{-{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i}) ]$, and
$\exp[ (i \pi/2) {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i \cdot {\bf C} ]$,
correspond to rotations by an angle $ \pi $ and
generate a $ {\mathbbm Z}_4 \times {\mathbbm Z}_4 $ discrete subgroup under
which only the generators of $G$ are invariant, and which
we choose as $ \Gamma $, while for $ {\mathbbm V}$ we choose the
adjoint representation of these rotations~\footnote{This
argument can be extended to all models based
on $ {\cal G} = \su{3l} $ for $l$ even.}. Constructing the
representation $ \mathfrak{R} $ then requires first the
identification of the modes that should get a vacuum expectation value\
in order to have the right pattern of spontaneous symmetry breaking\ at low
energies, and then choosing the dimensionality of the
compact space that can accommodate these matrices.
Again, this detailed calculation falls outside the scope of the present paper.
In the above considerations we have avoided discussing the difficulties
associated with Yukawa couplings, mainly because our emphasis was in obtaining
the right vector-boson and scalar sector spectrum. If all fields are bulk-propagating
these couplings are determined in terms of the gauge coupling constants and
the choice of $ \Gamma $ the representations $ {\mathbbm V} $ and $\mathfrak{R} $.
Whether a model can be found that can accommodate the complicated
fermion-mass structure without the introduction of large brane
couplings~\footnote{Loop corrections in general induce brane couplings
\cite{Georgi:2000ks}, some of which may have power-dependence on
the UV cutoff $ \Lambda$, but whether these affect the light
scalar modes depends on the details of the model.}
will be investigated in a future publication.
\acknowledgements
JW is grateful to B. Grzadkowski for various interesting comments and insights.
This work was supported by in part UC-MEXUS under Grant No. CN-08-205,
by CONACYT and by the U. S. Department of Energy under
Grant No. DEFG03- 94ER40837.
\section{ Introduction}
If there are compact extra dimensions it is then possible to associate
4-dimensional scalars with the {\it extra} components of gauge bosons;
these scalars can then be
responsible for the breaking of electroweak (EW)
symmetry~\cite{Hosotani:1988bm}, playing
the same role as the Standard Model\ Higgs doublet. In this so-called
Gauge-Higgs (GH) unification
one first proposes a non-Abelian gauge symmetry in the full
D-dimensional space time (where ${\rm{D}}=4+n$),
and assumes $n$ of these are compactified, usually over an orbifold
\cite{Hebecker:2001jb}
(for a review see {\it e.g.} \cite{Quiros:2003gg}).
The compact space together with boundary conditions are then chosen
to insure the presence of light 4-dimensional vector modes
corresponding to a low-energy
$\su3 \times \su2 \times U(1)$ gauge group, together with a 4-dimensional
scalar sector that leads to the right symmetry breaking pattern
to $\su3 \times U(1)_{EM}$. Models constructed within this
scenario involve two high energy scales:
the compactification radius $L$ and
an ultraviolet (UV) cutoff $\Lambda$,
beyond which the model ceases to be perturbative. This UV cutoff appears
because the models are non-renormalizable;
it is not hard to insure $ \Lambda L \gg 1 $ as required by
consistency~\cite{Chacko:1999hg,Grzadkowski:2007xm}.
Among the virtues of such a scheme, in addition to the absence of fundamental
scalars, is the possible solution to the hierarchy
problem~\cite{Sakamoto:2006wf}. Another feature is the fact that the low-energy
effective scalar potential, which is responsible for the EW symmetry
breaking, is determined by the group structure, fermion
content, and boundary conditions of the theory.
Several realizations~\cite{Hall:2001zb,Burdman:2002se,
Haba:2002vc,Gogoladze:2003bb,Scrucca:2003ut,Haba:2004qf,
Haba:2004jd,Chiang:2010hy} of this mechanism have been achieved with various
degrees of success.
5-dimensional models generally predict a very light Higgs boson. This is due to
the absence of a tree-level quartic term in the scalar
potential~\cite{Haba:2004bh,Gogoladze:2007ey}, which can be ameliorated by
considering models with 6 (or more) dimensions.
These models also typically predict a tree-level value of
$ \sin^2 \theta_{\rm w} $ ($\theta_w$ denotes the weak mixing angle)
higher than the experimental observation of $ \sim1/4 $;
in fact, for 5-dimensional gauge theories, it has been
shown~\cite{Grzadkowski:2006tp} that any model that assumes a low-energy Standard Model (SM)
gauge boson spectrum only, is inconsistent with the requirements $\sin^2\theta_W
\simeq 1/4$ and $\rho \equiv m_W^2/(m_Z^2\cos^2\theta_W)=1$
at tree level (equivalently, a vanishing tree-level
oblique T-parameter). This problem has been addressed
by introducing additional $U(1)$
factors~\cite{Antoniadis:2001cv,Scrucca:2003ra,Cacciapaglia:2005da} and/or brane
kinetic terms~\cite{Burdman:2002se,Agashe:2004rs,Aranda:2005ze}.
In this paper, following the analysis presented in~\cite{Grzadkowski:2006tp}, we
determine to what extent GH models naturally satisfy the following
set of low-energy constraints:
\bit
\item The only gauge bosons with masses at the EW mass scale are those
present in the Standard Model.
\item $\rho = 1$ (at tree-level), {\it i.e.} we look for models that contain only
isodoublet
or isosinglet 4-dimensional scalars~\footnote{While isodoublets are not the only
``$\rho$-safe'' representations, they {\em are} the only ones with this property
that occur in the models being considered.},
\item Contain representations that can accommodate all Standard Model\ particles;
in particular at least one representation should contain
isodoublet states of hypercharge $1/6 $, corresponding to the
left-handed quark fields.
\item Lead to acceptable values of $s_{\rm w}^2$.
\eit
For all simple groups we determine whether or not these
requirements can be fulfilled by the group structure itself without allowing
additions to the models such as fundamental scalars or brane couplings. We
present the analysis and results for an arbitrary number of extra dimensions and
all compactification schemes. Our approach is group theoretical supplemented by
the above four experimentally-motivated conditions.
We find that it is relatively easy to satisfy the first two requirements,
while the third one is obeyed in only a relatively small set of models
(listed in eq. \ref{eq:models}). It proved impossible, in addition, to
satisfy $ s_{\rm w}^2\sim1/4 $:
in the absence of brane couplings all GH models must either contain
additional light gauge bosons (with masses $\sim m_{\rm W,Z}$),
or must have a tree-level weak-mixing angle
substantially different form the observed one, or cannot accommodate quarks.
The masses of the additional
light gauge bosons are of the same order as those of the $W$ and $Z$ and
are subject to similar radiative corrections, so we do not expect
them to be split form the electroweak scale by loop effects;
hence models that exhibit
such particles appear phenomenologically excluded. One of the main results of
this paper is to provide a simple way of determining whether these
undesirable states are present without a reference to the compactification
scheme.
In contrast, radiative corrections to weak-mixing angle can be substantial and
can easily lead to a 50\%\ (or larger)
effect~\cite{Dienes:1998vh,Dienes:1998vg}. For this
reason we will first determine gauge groups that comply with the first three
conditions above and then discuss the possibility of obtaining a
phenomenologically viable weak-mixing angle at the electroweak scale.
In this paper we concentrate on the gauge-boson structure of the theory; fermions
will be discussed only in connection with their possible $\su2\timesU(1) $ quantum
numbers. The possibility of creating specific models that accommodate the right
light fermion with the correct $\su3\times\su2\timesU(1)$ quantum numbers, and -
more challenging - the observed Yukawa couplings, will not be studied here. It is
known that, even if absent at tree-level, radiative contributions generate
4-dimensional couplings on the orbifold fixed points~\cite{Georgi:2000ks}. Given a
specific realization of the GH scenario one must take this into account, and
determine the extent of the associated effects on the low-energy theory --
especially if it is assumed that the Standard Model\ fermions are localized in one of these
fixed points . This investigation requires a specific choice of model (including
gauge field, fermion representations, orbifold compactification and periodicity
conditions on the fields) and lies beyond the scope of the model-independent
restrictions presented here.
The paper is organized as follows: after a brief description of the conventions
used in our analysis, we present the general gauge and space time setup in
section~\ref{s2}. Section~\ref{s3} discusses the Kaluza-Klein decomposition of
the fields as well as their gauge transformations. The analysis regarding light
vector bosons is discussed in detail in Section~\ref{s4} where we also present
the necessary conditions for their absence. In section~\ref{s5} we present the
vector boson mass matrix and the requirements needed for matter fields. Finally
we present our results in section~\ref{s6}. We have also included a couple of
appendixes at the end with details of the calculations.
\section{ Lagrangians and symmetries} \label{s2}
The material presented in this section is not new, it is included
for convenience and in order to introduce the notation that will
be used throughout the paper.
\subsection{Conventions}
In describing the group structure of the models being considered we
find it convenient to use a canonical basis where we
denote the Cartan generators by $ C_i $
(or, when appropriate as a vector {\bf C}), and the root generators by
$ E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} $. These satisfy
\beq \label{eq:generatoralgebra}
[ {\bf C}, E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}] = {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}\,; \quad
[E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}},E_{\-{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}} ] = {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} \cdot {\bf C} \,;\quad
[E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}, E_{\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}} ] = N_{{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}, {\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}}} E_{{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} + {\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}}} ~~
({\bm\beta}} % \def\betbf{{\pmb{$\beta$}} + {\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}} \not= {\bf0})
\eeq
where we will not need the explicit form of $N_{{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}, {\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}}}$
\cite{Gilmore:1974}.
Note that in this basis the structure constants are not
completely antisymmetric.
We will denote the simple roots by
$ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i $ and the corresponding fundamental weights
by $ {\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_j $ (see \cite{Gilmore:1974,Georgi:2000ks} for details), with
\beq
{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i \cdot {\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_k = \frac12} %\def\half{{1\over2} \left| {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i \right|^2 \delta_{ik}
{\quad\Rightarrow\quad} {\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_k = \sum_j \left(a^{-1}\right)_{j k} {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^j
\eeq
(no sum over $i$ in the first expression) where
\beq
a^{ij} = 2 \frac{{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i\cdot{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^j }{{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i\cdot{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i }
\label{eq:cart.mat}
\eeq
(no sum over $i$)
is called the Cartan matrix (which is not symmetric in general).
We also find convenient
to introduce the rescaled weights
\beq
\tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_j = \frac2{|{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^j|^2} {\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_j
{\quad\Rightarrow\quad}
{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i \cdot \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_j = \delta_{ij}
\label{eq:def.of.mu}
\eeq
(no sum over $j$ in the first expression) .
For the adjoint representation we assume that the
corresponding matrices are normalized according to
(using the same symbol for the generator as for its
representation)
\beq
{\rm tr} C_i C_j = \delta_{i,j} \,, \quad
{\rm tr} E_{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} = \delta_{{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}+{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}, 0} \,, \quad
{\rm tr} E_{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} C_i =0 \,,
\label{eq:norm}
\eeq
which ensures that $ (-1/4) {\rm tr}
|\partial_\mu A_\nu - \partial_\nu A_\mu|^2,~
A_\mu = \sum_i A_\mu^i C_i + \sum_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} A_\mu^{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} $,
is properly normalized.
When no confusion will arise, we will
refer to a gauge field and it associated generator interchangeably;
for example we will often refer to a
generator as ``corresponding'' to a light vector boson, by which
we imply that the gauge field associated with that generator
will have a light mode.
\subsection{Compactifications and the space group $ \Gamma$}
We consider theories defined on a $4+n$ dimensional
space time of the form $ {\mathbbm M}}%\def\mBB{ \hbox{{\mysmallii I}}\!\hbox{{\mysmallii M}} \times ({\mathbbm R}}%\def\rBB{ \hbox{{\mysmallii I}}\!\hbox{{\mysmallii R}} ^n/\Gamma) $ where
${\mathbbm M}}%\def\mBB{ \hbox{{\mysmallii I}}\!\hbox{{\mysmallii M}} $ denotes the usual 4-dimensional Minkowski space
and $ \Gamma $ denotes a discrete group~\cite{Hosotani:1988bm}
(for a review see \cite{Quiros:2003gg}) whose elements
$ \gamma \in \Gamma $ act on $ {\mathbbm R}}%\def\rBB{ \hbox{{\mysmallii I}}\!\hbox{{\mysmallii R}} ^n$ as follows~\footnote{The
notation is borrowed from solid state literature, for an accessible
introduction see \cite{Koster:1956}.}:
\beq
\gamma = \{ \mathfrak{r}|{\bf l}\} {\quad\Rightarrow\quad} \{ \mathfrak{r}|{\bf l}\} {\bf y} = \mathfrak{r} {\bf y} + {\bf l} \ ,
\eeq
where the $\mathfrak{r}$ are $ n \times n $ orthogonal matrices,
{\bf y}\ denote the coordinates of $ {\mathbbm R}}%\def\rBB{ \hbox{{\mysmallii I}}\!\hbox{{\mysmallii R}} ^n$, and the $ {\bf l}$
are $n$-dimensional translation vectors
(the notation is the same as
the one used in solid state for crystal groups).
We assume that $ \Gamma $ acts trivially on $ {\mathbbm M}}%\def\mBB{ \hbox{{\mysmallii I}}\!\hbox{{\mysmallii M}} $:
$ \gamma x^\mu = x^\mu $ where $ x^\mu$ denote the coordinates of $ {\mathbbm M}}%\def\mBB{ \hbox{{\mysmallii I}}\!\hbox{{\mysmallii M}} $.
The multiplication rule for the elements in $ \Gamma $
can be easily derived from their action on {\bf y},
for example
\beq
\{ \mathfrak{r}'|{\bf l}'\} \{ \mathfrak{r}|{\bf l}\} = \{\mathfrak{r}'\mathfrak{r} | \mathfrak{r}'{\bf l} + {\bf l}'\} \,;
\quad \{ \mathfrak{r}|{\bf l}\}^{-1} = \{ \mathfrak{r}^{-1} | - \mathfrak{r}^{-1} {\bf l}\} \ .
\eeq
The group $ \Gamma $ is assumed to have an
(Abelian) translation subgroup
$ \Theta $ composed of all elements of the form
$ \{ {\mathbbm1} | {\bf t} \} $, where the translations vectors {\bf t}\
are linear combinations of a
set of basis vectors $ \{ {\bf t}_ i \} $ with integer coefficients,
\beq
\Theta = \left\{ \{ {\mathbbm1} |\sum k_i {\bf t}_i\}, ~k_i={\rm integer}
\right\} \,;
\eeq
note that, in general, the vector
{\bf l}\ in $ \{\mathfrak{r}|{\bf l}\}$ need not be a translation when
$ \mathfrak{r}\not={\mathbbm1}$. Using the multiplication
rule we find that
\beq
\{ \mathfrak{r}|{\bf l}\}^{-1} \{ {\mathbbm1} |{\bf t}\} \{ \mathfrak{r}|{\bf l}\}= \{ {\mathbbm1} |\mathfrak{r}^{-1} {\bf t}\}
{\quad\Rightarrow\quad} \mathfrak{r}^{-1} {\bf t} \in \Theta \ .
\eeq
It follows that $ \Theta $ is an invariant subgroup, and that
for all rotations $\mathfrak{r}$ and translations {\bf t}\ the
vector $ \mathfrak{r} {\bf t}$ is also a translation.
\subsection{Gauge-field Lagrangian and automorphisms}
We denote gauge vector fields by $A^M_a = (A^\mu_a,A^m_a)$ with the Greek
indices associated with non-compact directions and Latin lower case
indices with compact directions; we often write $ {\bf A}_a =
(A^{m=1}_a, \ldots, A^n_a)$.
We assume the following action of $ \Gamma $ on the $A^M_a$:
\bea
A^\mu_a(x,{\bf y}') &=& {\mathbbm V}(\gamma)_{a b} A^\mu_b(x,{\bf y}) \,; \qquad
{\bf y}' = \gamma{\bf y} = \{ \mathfrak{r}|{\bf l}\} {\bf y} = \mathfrak{r} {\bf y} + {\bf l} \cr
{\bf A}_a(x,{\bf y}') &=& {\mathbbm V}(\gamma)_{ab} \mathfrak{R}(\gamma) {\bf A}_b(x,{\bf y})
\label{eq:qaction}
\eea
for all $ \gamma \in \Gamma $, where the matrices $\mathfrak{R}$ act on the indices
associated with the compact directions:
$ (\mathfrak{R}{\bf A}_b)^m = \mathfrak{R}^m{}_l A_b^l $.
Defining the curvature tensor by~\footnote{The structure constants obey
$
f_{a b c} = - f_{b a c}
$
but, in general $ f_{a b c} + f_{a c b } \not =0 $
(though there are bases where this does hold)}
\beq
F^{MN}_a = \partial^M A^N_a - \partial^N A^M_a + g f_{b c a} A^M_b A^N_c \ ,
\eeq
we find that the $F^2$ term in the Lagrangian will
be invariant under (\ref{eq:qaction}) provided
\bea
\mathfrak{r} = \mathfrak{R}(\gamma); ~ \gamma= \{ \mathfrak{r} | {\bf l} \} && \mathfrak{R}(\gamma){}^T \mathfrak{R}(\gamma) = {\mathbbm1} \ , \cr
f_{a'b'c'} = f_{abc} {\mathbbm V}(\gamma)_{a a '} {\mathbbm V}(\gamma)_{b b'}
{\mathbbm V}^\dagger(\gamma)_{c' c} &&
{\mathbbm V}(\gamma){}^\dagger {\mathbbm V}(\gamma) = {\mathbbm1} \ ,
\label{eq:autom}
\eea
so that $ {\mathbbm V}$ must be an automorphism of the
gauge group $ {\cal G} $~\cite{Hebecker:2001jb}.
Note also that
the equality $ \mathfrak{R}(\gamma) = \mathfrak{r} $ implies $\mathfrak{R}(\gamma)$ is
independent of the vector $ {\bf l} $. In particular
\beq
\mathfrak{R}(\{ {\mathbbm1} | {\bf t} \} ) = {\mathbbm1} \ .
\label{eq:R.under.T}
\eeq
\section{KK expansions} \label{s3}
\subsection{Consequences of covariance under translations}
The matrices $ {\mathbbm V}(\{{\mathbbm1}|{\bf t}\})$ carry a representation of $\Theta
$. Since $ \Theta $ is an Abelian group it has only one-dimensional
irreducible representation s, so we can choose a basis where $ {\mathbbm V}(\{{\mathbbm1}|{\bf t}\})_{a b} =
v_a({\bf t}) \delta_{a b} $ for all {\bf t}. Since the vectors {\bf t}\ are of the
form $ \sum_i k_i {\bf t}_i$ for some integers $k_i $,
and since $ \{ {\mathbbm1} | {\bf t}_i \} \in \Gamma$, then
\beq
v_a({\bf t}) = \prod_i
\left[ v_a({\bf t}_i) \right]^{k_i} \ ,
\eeq
while the unitarity of the representation requires
\beq
v_a({\bf t}_i)= e^{ i c_{a i}}, \quad c_{a i} \rightarrow \hbox{real},
~~ |c_{a i} | < \pi \ .
\eeq
In this basis, using (\ref{eq:qaction}) and (\ref{eq:R.under.T}),
\beq
A_a^N(x, {\bf y}+ {\bf t}) = e^{i \sum_i k_i c_{a i}} A_a^N(x,{\bf y}) \ .
\eeq
Let $ \{ {\bm\kappa}} % \def\kapbf{{\pmb{$\kappa$}}_j\} $ be a set of linearly independent vectors
dual to the $ {\bf t}_i$, that is, $ {\bm\kappa}} % \def\kapbf{{\pmb{$\kappa$}}_j \cdot {\bf t}_i = \delta_{ij} $.
Then we can expand
\bea
A^N_a(x,{\bf y}) &=& e^{ i {\bf Q}_a \cdot {\bf y}}
\sum_{\bf n} e^{2 \pi i {\bf y} \cdot \sum_j n_j {\bm\kappa}} % \def\kapbf{{\pmb{$\kappa$}}_j} \tilde A^N_a(x,{\bf n}); \quad
{\bf Q}_a = \sum_j c_{a j} {\bm\kappa}} % \def\kapbf{{\pmb{$\kappa$}}_j \ ,
\label{eq:KK.expansion}
\eea
which corresponds to the usual expansion in Kaluza-Klein
(KK) modes~\cite{Hosotani:1988bm,Quiros:2003gg}.
It follows that $ \tilde A^N_a(x,{\bf n}) $ will have a
the tree-level mass $ \sim | {\bf Q}_a + 2 \pi \sum n_j
{\bm\kappa}} % \def\kapbf{{\pmb{$\kappa$}}_j | $; we will assume that the scales associated with the $
{\bm\kappa}} % \def\kapbf{{\pmb{$\kappa$}}_i $ are large compared to the electroweak
scale. In this case
all light modes $ \tilde A^N_a $ must correspond to $ {\bf n} =0 $
and $ {\bf Q}_a =0 $. But $ {\bf Q}_a =0 $ requires $ c_{a i } = 0 $,
which implies that gauge fields that have light modes are
translationally invariant.
Since $ \Theta $ is an invariant subgroup we find that
for all $ \gamma$,
$ {\mathbbm V}(\gamma )_{ab} =0 $ {\em unless } ${\bf Q}_a = {\bf Q}_b $.
In particular the $ {\mathbbm V} $ do not mix the light
and heavy KK modes. Using this and substituting
(\ref{eq:KK.expansion}) in (\ref{eq:qaction})
we find that the light modes must obey
\beq
{\mathbbm V}(\gamma)_{a b} \tilde A^\mu_b(x,0)
=
\tilde A^\mu_a(x,0) \qquad
{\mathbbm V}(\gamma)_{a b} \mathfrak{R}(\gamma) \tilde{\bf A}_b(x,0)
=
\tilde{\bf A}_a(x,0) \ ,
\label{eq:mzero}
\eeq
that is, light 4-dimensional vectors are associated with the ``trivial''
subspace where $ {\mathbbm V}= {\mathbbm1} $, while light 4-dimensional scalars are
associated with the subspace where $ {\mathbbm V} \otimes \mathfrak{R} = {\mathbbm1} $.
One of the challenges in constructing a realistic theory
of this type is to find a ``compactifying'' group $ \Gamma $,
a gauge group $ {\cal G} $ and representations $ {\mathbbm V}$
and $ \mathfrak{R} $ such that the solutions to (\ref{eq:mzero})
will correspond to the bosonic sector of the Standard Model\ (possibly
with an extended scalar sector). While it is always
easy to ensure that a given field has a light mode,
the relations (\ref{eq:autom}) often imply the presence
of {\em additional} light vectors which, as we will see,
have masses of the same order as the Standard Model\ gauge bosons.
Requiring the absence of such undesirable light particles
severely restricts the choice of gauge groups $ {\cal G} $.
\subsection{Gauge transformations and KK expansions}
The gauge transformations are
\beq
A^N \rightarrow U^\dagger \left( i \partial^N + A^N \right) U; \qquad
A^N = g A^N_a T_a\,, ~~ U \in {\cal G}
\eeq
equivalently,
\beq
A^N_a \rightarrow {\cal U}^N_a + {\mathbbm U}_{ab} A^N_b;
\qquad
T_a {\cal U}^N_a = i U^\dagger \partial^N U,
\quad
T_a {\mathbbm U}_{ab} = U^\dagger T_b U \ .
\label{eq:gauge.transf}
\eeq
Consider now those $ {\cal U},~{\mathbbm U}$ that depend only on $x$. Using
(\ref{eq:KK.expansion}) in (\ref{eq:gauge.transf}) implies
that $ {\cal U}^\mu_a \not=0 $ if $ {\bf Q}_a =0 $; and
$ {\mathbbm U}_{ab}(x) \not=0 $ requires $
{\bf Q}_a - {\bf Q}_b = 2 \pi \bar n_j\up{a b} {\bm\kappa}} % \def\kapbf{{\pmb{$\kappa$}}_j$
with $ \bar n_i\up{a b} $ integers;
in particular if $ {\bf Q}_a =0 $, then either $ {\bf Q}_b =0 $
or else $ {\mathbbm U}_{ab}(x) =0 $.
Then, for the light fields obeying (\ref{eq:mzero}),
\bea
\tilde A^\mu_a(x,{\bf0}) &\rightarrow& {\cal U}^\mu_a(x)+ \sum_b
{\mathbbm U}_{ab}(x) \tilde A^\mu_b(x,{\bf0}) \cr
\tilde{\bf A}_a(x,{\bf0})
&\rightarrow& \sum_b {\mathbbm U}_{ab}(x)
\tilde{\bf A}_b(x,{\bf0})
\eea
showing that when $ {\cal U}^\mu_a \not =0 $ the
$ \tilde A^\mu_a(x,{\bf n}=0) $ transform as gauge fields
and, because of (\ref{eq:mzero}) these also
transform trivially under $ \Gamma $.
We denote these as the light gauge fields, and the set of
{\bf y}-independent transformations as the light gauge group $G$.
The light gauge group $G$ is a subgroup of ${\cal G}$, so that all
fields can be classified according to their $G$ representation.
For a realistic theory~\footnote{Models with additional $U(1)$
factors appear frequently, but in the absence of brane couplings
or bulk-propagating scalar fields,
the masses of the additional vector bosons are unacceptably low.}
$ G = \su3 \times\su2 \times U(1) $.
\subsection{Choice of $\su2 \times U(1)$}
As in the 5-dimensional case~\cite{Grzadkowski:2006tp} the $\su2$ generators
$ J^{0,\pm} $ are specified
by choosing a root $ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}$, while the hypercharge
generator $Y$ can be taken
as a linear combination of Cartan generators:
\beq
J^0 = \inv{|{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}|} \hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}\cdot{\bf C} \,, \quad
J^\pm = \frac{\sqrt{2}}{|{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}|} E_{\pm{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}} \,;
\qquad Y = {\hbox{\egothfamily y}} \cdot {\bf C} \,.
\eeq
Though $ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} $ can be any root, there is always a freedom to relabel
axes and axis directions, so that many choices are
equivalent. A straightforward inspection of
the various classical groups
shows that, in fact, $ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} $ can be taken to be one of
the simple roots.
Using these definitions we can find the isospin $s$ and
$z$-component isospin $s_z$ of any root $ {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}$:
\beq
s = \frac{n_+ + n_- }2 \,, \qquad
s_z = \frac{n_- - n_+ }2 \ ,
\label{eq:def.of.s.sz}
\eeq
where $n_\pm $ are non-negative integers such that
${\bm\beta}} % \def\betbf{{\pmb{$\beta$}} + k {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}$ and $-n_n \le k \le n_+ $ are roots but
${\bm\beta}} % \def\betbf{{\pmb{$\beta$}} \pm (n_\pm +1) {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} $ are not.
If $ {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} $ is also a simple root then $n_- =0 $
so that $ s = - s_z = n_+/2 $.
The vector {\hbox{\egothfamily y}}\ is unspecified, except for the requirements that
the model should contain isodoublets of hypercharge $1/2 $ that can acquire
a vacuum expectation value, and that $ J^{0,\pm}$ commute with $Y$:
\beq
\left[ J^\pm , Y \right] =0 {\quad\Rightarrow\quad} {\hbox{\egothfamily y}}\cdot{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} =0
\eeq
Concerning the light 4-dimensional scalars we require that they
give rise to $ \rho = 1$ at tree level. Since these scalars are
associated with the adjoint generators their isospin can be read-off
form the Cartan matrix of the gauge group $ {\cal G} $; for the
simple groups a direct examination shows that only isospin $ \le 3/2 $
will occur. The scalars are then taken to be associated with specific root
generators $ E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} $ and a
realistic model must ensure that only the isospin $1/2 $
modes can acquire a vacuum expectation value.
\section{ Interplay between $G$ and $ \Gamma $}\label{s4}
\subsection{The frequent occurrence of additional light neutral gauge bosons}
\label{sec:light}
Requiring consistency between (\ref{eq:qaction}) and (\ref{eq:gauge.transf})
implies
\bea
&& {\cal U}^\mu_a(x,{\bf y}') = {\mathbbm V}_{ab}(\gamma) {\cal U}^\mu_b(x,{\bf y}) \qquad
{\cal U}^m_a(x,{\bf y}') = \mathfrak{R}^m{}_l {\mathbbm V}_{ab}(\gamma) {\cal U}^l_b(x,{\bf y}) \cr
&& {\mathbbm U}(x,{\bf y}') = {\mathbbm V}(\gamma) {\mathbbm U}(x,{\bf y}) {\mathbbm V}^{-1}(\gamma) \ ,
\eea
where $ {\bf y}' = \mathfrak{r} {\bf y} + {\bf l},~~ \gamma=\{\mathfrak{r}|{\bf l}\} $. In
particular for $ {\mathbbm U} \in G$, $ {\mathbbm U} $ is independent of $x$
and
\beq
\left[ {\mathbbm U}(x) , {\mathbbm V}(\gamma)\right] = 0 \ .
\eeq
This relation implies (by Schur's lemma) that such $ {\mathbbm U}$ do
not mix $ \Gamma $ irreducible representation s carried by the $ {\mathbbm V} $, and the
$ {\mathbbm V} $ do not mix $G$ irreducible representation s carried by the $ {\mathbbm U} $. In
particular, the $ {\mathbbm V} $ will not mix generators that
have different $G$ quantum numbers.
Consider now the set of generators that are $G$ singlets, denoted
by $ T\up{\mathfrak{s}}_S$, and a set of generators $\tilde T_r $ that have
fixed $G$ quantum numbers (in our case, fixed hypercharge,
isospin and $z$-isospin component). Then it follows that
$ {\mathbbm V}_{S a} =0 $ unless $ a = R $, corresponding to some $ T\up{\mathfrak{s}}_R$,
and $ {\mathbbm V}_{r a} = 0 $ unless $ a = s $, corresponding to some
$ \tilde T_s $. The automorphism condition then implies
(using a basis where
$ f_{abc}$ is antisymmetric in $a$ and $b$ but not in all 3 indices)
\beq
f_{T r s} = {\mathbbm V}_{T' T} {\mathbbm V}_{r' r}
{\mathbbm V}^{-1}_{s s'} f_{T' r' s'}
{\quad\Rightarrow\quad} \sum_r f_{T r r} = \sum_{T'} {\mathbbm V}_{T' T} \sum_{r'}
f_{T' r' r'} \ .
\eeq
Then
\beq
F_S = F_R {\mathbbm V}_{R S} \,, \quad
\hbox{\rm where} \quad
F_S = \sum_r f_{S r r} \ .
\eeq
Now define
\beq
\tilde T\up{\mathfrak{s}} = \sum_S F_S T\up{\mathfrak{s}}_S
= \sum_{S\,r} f_{S r r} T\up{\mathfrak{s}}_S \ ,
\eeq
which is an $G$ singlet, while under $ \Gamma $
\beq
\tildeT\up{\mathfrak{s}} \rightarrow \sum {\mathbbm V}_{ R S} T\up{\mathfrak{s}}_S F_R =
\tildeT\up{\mathfrak{s}} \ ,
\eeq
so that this generator is also a $ \Gamma $ singlet.
This generator depends on the choice of $ \tilde T_r $ so that
there will be a $ \tildeT\up{\mathfrak{s}}$ for each set of $G$ quantum numbers.
Note that any linear combination of the $ \tildeT\up{\mathfrak{s}} $
will also be a $G$ and $ \Gamma $ singlet.
Now, the expression for $ \tildeT\up{\mathfrak{s}} $ involves $ f_{S r r} $,
which is not zero only if the commutator $ [T\up{\mathfrak{s}}_S , \tilde T_r ] $
has a term proportional to $ \tilde T_r $ itself. Looking now
at the commutators in terms of roots and Cartan generators it is
clear that this can happen only if $ T\up{\mathfrak{s}}_S $ is a linear combination
of Cartan generators and $ \tilde T_r $ is a root generator:
\beq
\left[ {\bf v} \cdot {\bf C}, E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} \right] = {\bf v} \cdot {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}
{\quad\Rightarrow\quad} f_{{\bf v}\cdot{\bf C} \, E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} \, E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}} = - i {\bf v} \cdot {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} \ ,
\eeq
so we identify $ T\up{\mathfrak{s}}_S $ with $ {\bf v} \cdot {\bf C} $, which
will be an $ \su2$ singlet provided
\beq
{\bf v}\cdot{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} =0 \ .
\eeq
It is also a $U(1)$ singlet since the
hypercharge generator is of the form $ {\hbox{\egothfamily y}} \cdot {\bf C} $.
Consider now a series of vectors $ \hat {\bf v}_S $ perpendicular to
$ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} $ and satisfying
\beq
\sum_S \hat{\bf v}_S \otimes \hat{\bf v}_S = {\mathbbm1} - \hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} \otimes \hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} \ .
\eeq
Then we can take $ T\up{\mathfrak{s}}_S = \hat{\bf v}_S \cdot{\bf C}$ and
the $G$ and $ \Gamma $ singlet generators are of the form
\beq
\sumprime_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}
\sum_S\left(\hat{\bf v}_S \cdot {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}\right) \hat{\bf v}_S \cdot {\bf C}
= \left(\sumprime_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}_\perp \right)\cdot {\bf C} \ ,
\label{eq:sum.1}
\eeq
where $ {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}_\perp = {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} - (\hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} \cdot {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}) \hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}$, and
the prime indicates that the sum is over all roots with
a specific set of $G$ quantum numbers. In the following we write
\beq
\sumprime_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}_\perp = {\hbox{\egothfamily y}}'_q,~ ~
q = \{ h ,s,s_z\}
\label{eq:sum.2}
\eeq
Since any two roots $ {\bm\beta}} % \def\betbf{{\pmb{$\beta$}},~{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}' $
in the same $ \su2$ multiplet satisfy $ {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} = {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}'
+ n {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} $, it follows that $ {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}_\perp = {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}'_\perp $.
If $ {\bm\beta}} % \def\betbf{{\pmb{$\beta$}},~ {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}'$ carry $G$ quantum numbers
$q = \{h, s, s_z \},~
q' = \{h, s, s_z' \} $ respectively, then
$ {\hbox{\egothfamily y}}'_q = {\hbox{\egothfamily y}}'_{q'} $. Note also that
\beq
{\hbox{\egothfamily y}}'_q = - {\hbox{\egothfamily y}}'_{\bar q} \,, \quad q = \{h , s , s_z \}, ~~
\bar q = \{ -h, s, -s_z \} \,,
\eeq
so that we can restrict ourselves to vectors $ {\hbox{\egothfamily y}}' $ associated
with positive hypercharge, $ h> 0 $, roots.
The generators $ {\hbox{\egothfamily y}}'_q \cdot {\bf C} $
are {\em necessarily} $G$ and $ \Gamma $ singlets~\footnote{
It is possible to follow the same argument
when $ G = \su2 $, the hope being that one
of the ${\hbox{\egothfamily y}}'_{s,s_z}$ can be used as {\hbox{\egothfamily y}}. This case,
however, is trivial since all the $ {\hbox{\egothfamily y}}'_{s,s_z}$
necessarily vanish (see appendix~\ref{sec:ap2}),
which is related to the fact that
$\su2 $ has no complex representations.}: given a choice of $ {\cal G} $ and an
embedding of $G$ it is impossible to
have a light gauge group of rank 4, unless
all the vectors $ {\hbox{\egothfamily y}}_q' $ are proportional to $ {\hbox{\egothfamily y}} $.
The fact that a given choice of the Standard Model\ group as a
subgroup of $ {\cal G} $ will in general imply the presence
of additional light vector bosons, independently of the
compactification details of the model, is one of the
central results of this paper.
The vectors ${\hbox{\egothfamily y}}'_q$ need not be linearly independent,
nor do they have to be independent of {\hbox{\egothfamily y}}. Still,
by taking appropriate linear combinations we can find
and orthonormal subset
$ \hat{\hbox{\egothfamily y}}_r, ~ r=0,1, \ldots , R $ with $ \hat{\hbox{\egothfamily y}}_0 = \hat{\hbox{\egothfamily y}} $ and
$\hat{\hbox{\egothfamily y}}_r \cdot \hat {\hbox{\egothfamily y}}_s = \delta_{rs} $. The generators
$ \hat{\hbox{\egothfamily y}}_r \cdot {\bf C} $ also are $G$ and $ \Gamma $ singlets,
each generating a $U(1) $ subgroup that, by (\ref{eq:mzero}),
corresponds to a light neutral vector boson. This
result is {\em independent} of the choice
of $ \Gamma $ and the representations $ {\mathbbm V}$ and $ \mathfrak{R} $;
number $R$ of additional light vector bosons
is determined solely by the gauge structure of the
theory~\footnote{
This refers to the smallest number of light vector bosons;
specific models, of course, may have additional ones;
for example, if $ {\mathbbm V}(\gamma ) = {\mathbbm1} $
for all $ \gamma $, {\em all} $A^\mu_a$ will have light
modes.}.
We will show below that these $R$ vector
bosons may acquire masses thorough spontaneous symmetry breaking, but these are
of the order of the $W$ and $Z$ vector boson masses; in
particular models with $R \ge 1 $ are excluded phenomenologically.
One must therefore choose $G$ (that is,
hypercharge generator $Y = {\hbox{\egothfamily y}}\cdot{\bf C}$
and $ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} $) and $ {\cal G} $ such that $R=0 $, which proves to be
a stringent constraint on $ {\cal G} $ and the embedding of $G$.
\subsection{Necessary conditions for the absence of undesirable singlets}
\label{sec:nec.con}
In this section we list several requirements that gauge-Higgs unified theories
must meet in order to be phenomenologically viable
(at tree level). These
conditions are derived in appendix \ref{sec:group.theo}; here we
only list the results that are relevant to the rest of the paper.
We begin by noting that we can always assume that all the simple
roots have non-negative hypercharge and that
the root $ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} $, which defines
the Standard Model\ $\su2$ subgroup, is one of the simple roots
(see appendix~\ref{sec:ap1}),
\beq
{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} = {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^\isa \ .
\eeq
We can now divide the simple roots into 3 categories
{\it(i)} Those with positive hypercharge, $ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^{i_k}: ~~
{\hbox{\egothfamily y}}\cdot {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^{i_k} = h^k > 0 $. {\it(ii)}
Those, like $ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^\isa$, with zero hypercharge and non-zero isospin,
$ {\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}}^r: ~~
{\hbox{\egothfamily y}} \cdot{\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}}^r =0,~ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}\cdot{\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}}^r \not= 0$
(for $ {\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}}^r \not= {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^\isa $).
{\it(iii)} Those that are $G$-singlets: ${\bm\zeta}} % \def\zetbf{{\pmb{$\zeta$}}^l: ~~
{\hbox{\egothfamily y}} \cdot{\bm\zeta}} % \def\zetbf{{\pmb{$\zeta$}}^l = {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}\cdot{\bm\zeta}} % \def\zetbf{{\pmb{$\zeta$}}^l =0$.
It then follows that
\beq
{\hbox{\egothfamily y}} = \sum h^k \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_{i_k} \ .
\eeq
We have argued above that in order not to have additional
light vector bosons we must have $ {\hbox{\egothfamily y}}'_q \propto {\hbox{\egothfamily y}} $ for all $q$.
A straightforward application of Lie algebra theory (see appendix~\ref{sec:ap3})
shows that necessary conditions for this to occur are:
\bit
\item All simple roots must be either
isodoublets~\footnote{Strictly speaking the condition is
for all non-isosinglets roots to transform according to
the same $\su2$ representation; but we noted earlier that
these representations must have isospin $1/2,~1$ or
$3/2 $, of which only the first produce $ \rho= 1$ at
tree-level.} or isosinglets.
\item All the isosinglets must have zero hypercharge.
\item All the isodoublets must have the same non-zero hypercharge:
\beq
{\hbox{\egothfamily y}} = h \sum_{\rm isodoublets} \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_{i_k} \ .
\label{eq:hyp.vec}
\eeq
In particular, there should be no roots of the type $ {\bm\gamma}} % \def\gambf{{\pmb{$\gamma$}}^r$
except $ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} $ itself.
\eit
These conditions are useful in that they eliminate a large
number of groups; still, even when met, the low-energy spectrum must
be derived explicitly in order to insure the absence of light $Z'$
vectors.
It is worth pointing out that a choice of the $\su2$ subgroup
determines the set of simple roots that transform as isodoublets,
which in turn determines the specific embedding of the Standard Model\
$U(1)$ in $ {\cal G} $ given by (\ref{eq:hyp.vec}). Hence the ${\hbox{\egothfamily y}}'$
must be obtained for each choice of $ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} $ in order to determine
the viability of a given model.
\section{The weak mixing angle and hypercharge}\label{s5}
\subsection{Vector-boson mass matrix}
The (canonically normalized) light vector bosons
correspond to the zero modes of the gauge fields associated
with the generators $\hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} \cdot{\bf C} $, $E_{\pm{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}} $
and $ \hat{\hbox{\egothfamily y}}_r \cdot {\bf C} $, where $ \{ \hat{\hbox{\egothfamily y}}_0
, \ldots , \hat{\hbox{\egothfamily y}}_R \} $
denote the orthonormal basis for the subspace generated by $ {\hbox{\egothfamily y}}$
introduced at the end of section \ref{sec:light} (when there are no additional
$Z'$ bosons $R=0$ and $ \hat{\hbox{\egothfamily y}} = \hat{\hbox{\egothfamily y}}_0$);
we denote the corresponding zero modes
by $ W^0,~W^\pm $ and $B\up r$ respectively.
Following~\cite{Grzadkowski:2006tp}, but
allowing for the presence of more than one $\su2$-singlet
gauge boson, we expand
\bea
A_\mu &=& W_\mu^+ E_{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} + W_\mu^- E_{-{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}} + W_\mu^0 \hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}\cdot{\bf C}
+ \sum_{r=0}^R B\up r_\mu \hat{\hbox{\egothfamily y}}_r \cdot{\bf C} + \cdots \cr
A_n &=& \sum_{{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} > 0} \left( {\phi_{n,{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}} E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} + {\phi_{n,{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}}^* E_{- {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}} \right)
+ \cdots
\eea
where the sum over $ {\bm\beta}} % \def\betbf{{\pmb{$\beta$}}$ is over those fields such that $
\vevof{\phi_{n,{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}} \not =0 $, and the ellipsis denote fields with
masses of the order $ | {\bm\kappa}} % \def\kapbf{{\pmb{$\kappa$}} | $ as discussed in section
\ref{s3}.
Then, using (\ref{eq:norm}),
\bea
- \hbox{tr} \left[ A_\mu , A_n \right]^2 &=& \sum_{{{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} > 0};
~{\rm isodoublets}} \left| {\phi_{n,{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}} \right|^2
\left\{ \frac12} %\def\half{{1\over2} {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^2 W^+ \cdot W^- + \left(W_\mu^0 \hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} \cdot
{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} + \sum_{r=0}^R B\up r_\mu \hat{\hbox{\egothfamily y}}_r \cdot {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} \right)^2 \right\} \cr
&& \quad + \sum_{{{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} > 0};
~G-{\rm singlets}} \left| {\phi_{n,{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}} \right|^2
\left(\sum_{r=0}^R B\up r_\mu \hat{\hbox{\egothfamily y}}_r \cdot {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} \right)^2 \ .
\label{eq:vbm}
\eea
It is shown in appendix \ref{sec:ap2} that
if $E_{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}$ is a singlet under $G$, then
$ \hat{\hbox{\egothfamily y}}_r \cdot{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} =0 $, so that the $G$-singlet contribution
to the mass term vanishes, and the mass term is simply
\beq
\sum_{{{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} > 0};
~{\rm isodoublets}} \left| {\phi_{n,{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}} \right|^2
\left\{ \frac12} %\def\half{{1\over2} {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^2 W^+ \cdot W^- + \left(W_\mu^0 \hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} \cdot
{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} + \sum_{r=0}^R B\up r_\mu \hat{\hbox{\egothfamily y}}_r \cdot {\bm\beta}} % \def\betbf{{\pmb{$\beta$}} \right)^2 \right\} \ .
\eeq
This shows that when $R\ge1 $ there will be additional vector bosons
with mass of the same order as that of the $W$ and $Z$. In particular,
such models contain no mechanism through which the
tree-level mass of the additional
vectors can be pushed above the experimental limits, {\em irrespective
of the compactification scheme or of the fermion content of the
theory}~\footnote{At least as long as the model is weakly coupled at low energies.}.
This could be corrected by introducing scalars (either in the usual way
or using antisymmetric tensor fields), but in this case the motivation
and the attraction of this type of theories largely disappears.
For the case where $ R=0 $ we assume that the effective potential
for the 4-dimensional scalars insures these get a vacuum expectation value\ that
preserves the charge generator $ Q = J^0 + Y $, that is
\beq
\vevof{\phi_{n,{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}} \not=0 {\quad\Rightarrow\quad} {\hbox{\egothfamily y}}\cdot{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} =\frac12} %\def\half{{1\over2} , \quad
s = \frac{{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}\cdot{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}{|{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}|^2} = - \frac12} %\def\half{{1\over2} \ ,
\label{eq:sz.and.h}
\eeq
which fixes the normalization of {\hbox{\egothfamily y}}.
Writing $ B\up0 = B,~{\hbox{\egothfamily y}}_0 = {\hbox{\egothfamily y}} $,
and using $ \hat{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} \cdot{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} = - |{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}|/2 $,
we find
\beq
\sum_{{\bm\beta}} % \def\betbf{{\pmb{$\beta$}} > 0} \left| {\phi_{n,{\bm\beta}} % \def\betbf{{\pmb{$\beta$}}}} \right|^2
\left\{ \frac12} %\def\half{{1\over2} {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^2 W^+ \cdot W^- + \inv4 {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^2
\left(W_\mu^0 - \inv{|{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}| |{\hbox{\egothfamily y}}|} B_\mu \right)^2 \right\} \ .
\eeq
From this we read off the tangent of the weak-mixing angle, $t_{\rm w}$:
\beq
t_{\rm w} = \inv{|{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}| |{\hbox{\egothfamily y}}|} \ .
\label{eq:tw}
\eeq
\subsection{ Matter fields}
Up to this point we have not discussed the possible
effects derived from the introduction of matter fields,
and which will further restrict the number of allowed
theories.
In viable models it should be possible
to choose the fermion content such that it includes
states with the Standard Model\ quantum numbers of the observed
quarks and leptons. In particular the choices of ${\cal G} $
and $G$ must be such that there are fermion representations
containing states with hypercharge $1/6 $.
The highest weight of a representation can be written as
\beq
{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_{\rm max} = \sum m_i {\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_i =
\sum \left( a^{-1} m \right)_j {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^j \ ,
\eeq
where the $ m_i $ are integers. A generic weight in this irreducible representation\
is obtained by applying lowering operators $ E_{-{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i} $
associated with the simple roots. The general weight then has the form
\beq
{\bm\mu}} % \def\mubf{{\pmb{$\mu$}} = \sum m_j {\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_j - \sum k_i {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i \ .
\label{eq:gen.mu}
\eeq
Now, though all the entries in the Cartan matrix
(\ref{eq:cart.mat}) $a$ are integers, this is not true
for $ a^{-1} $; still there is an integer $N$ such that
\beq
N \left(a^{-1} \right)_{ij} = {\rm integer} \ ,
\eeq
with $N$ given in table \ref{tab:t1}
(see also appendix \ref{sec:class.groups}); also, all entries
in $ a^{-1}$ are positive.
\begin{table}[h]
$$
\begin{array}{|c|c|}
\hline
{\rm Group} & N \cr \hline
\su n & n \cr
G_2,~F_4, ~ E_8 & 1 \cr
Sp(2n), ~SO(2n+1), ~ SO(4n), E_7 & 2 \cr
E_6 & 3 \cr
SO(4n+2) & 4 \cr \hline
\end{array}
$$
\caption{Values of $N$ (see text) for
several simple Lie algebras.}
\label{tab:t1}
\end{table}
It follows that the weight of any state is of the form
$ {\bm\mu}} % \def\mubf{{\pmb{$\mu$}} = \sum (n_i/N) {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i $ where the $n_i $ are integers; then
\beq
Y \ket{\bm\mu}} % \def\mubf{{\pmb{$\mu$}} = ({\hbox{\egothfamily y}}\cdot{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}) \ket{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}
= \left( \inv N \sum_i n_i {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i \cdot {\hbox{\egothfamily y}} \right) \ket{\bm\mu}} % \def\mubf{{\pmb{$\mu$}} \ .
\eeq
In order to accommodate quarks a {\em necessary} condition is
to have
\beq
{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i \cdot {\hbox{\egothfamily y}} = \frac N{ 6 \times {\rm integer} }
\label{eq:q.hyp}
\eeq
for at least one simple root;
which proves useful in
eliminating many groups.
The simplest way to find
sufficient conditions is to obtain $ {\bm\mu}} % \def\mubf{{\pmb{$\mu$}} \cdot {\hbox{\egothfamily y}} $
using (\ref{eq:hyp.vec}) and (\ref{eq:gen.mu}),
\beq
{\bm\mu}} % \def\mubf{{\pmb{$\mu$}} \cdot {\hbox{\egothfamily y}} = h \sum_{\rm isodoublets}
\left( a^{-1} m - k \right)_{i_l},
\label{eq:exact.hyp}
\eeq
from which it can be determined by inspection whether
$ {\hbox{\egothfamily y}}\cdot{\bm\mu}} % \def\mubf{{\pmb{$\mu$}} = 1/6 $ occurs in any given irreducible representation.
\section{Results and prospects}\label{s6}
From separately considering the various possibilities
(see appendix \ref{sec:class.groups}) for the unitary,
orthogonal and symplectic groups, as well as for the
exceptional groups $G_2,~F_4,~E_6,~E_7$, and $E_8$,
we find that the
models that can accommodate quarks, and do not
necessarily contain undesirable light $Z' $ bosons,
are those given in the following table:
\beq
\begin{array}{|c||c|c|c|}
\hline
{\rm group} & s_{\rm w}^2 & {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}} & {\hbox{\egothfamily y}} \cr \hline
\su{3l} & 3l/(6l-2) & {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^1 & \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_2/2 \cr \hline
SO(2n+1) & 3/4 & {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^1 & \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_2/6 \cr\hline
G_2 & 3/4 & {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^1 & \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_2/6 \cr\hline
F_4 & 3/4 & {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^1 & \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_2/6 \cr\hline
E_6 & 3/8 & {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^{1,5} & \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_{2,3}/2 \cr\hline
E_7 & 3/4,~3/5 & {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^{1,7} & \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_{2,3}/6 \cr\hline
E_8 & 9/16,~3/8 &{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^{1,8} & \tilde{\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_{2,3}/6 \cr\hline
\end{array}
\label{eq:models}
\eeq
where the conventions for the simple roots and weights
are given in appendix \ref{sec:class.groups}.
Phenomenologically these models face additional obstacles.
First is the large discrepancy between the tree-level
value of $ s_{\rm w}^2 $ in (\ref{eq:models})
and its low-energy experimental result of $ \sim1/4 $.
The renormalization group (RG) running between the electroweak and
the compactification scales cannot account for this difference,
except, possibly for the $E_6$ and one $E_8$ model, in which case
$ 1/L $ would be of the same order as the GUT scale. There
is, however, also a contribution from the RG evolution
between the compactification scale and the UV cutoff $ \Lambda $
which may account for this~\cite{Dienes:1998vh,Dienes:1998vg}.
Discussing this in detail falls outside
the scope of this paper, but will be investigated in a future
publication.
In addition there is the remaining question of whether there
are specific compactifications for which only the desired
modes satisfy (\ref{eq:mzero}); that is, whether there is
a specific compactification choice for which $G$ is the
full light gauge group. Again, we will not discuss this
in general but indicate the manner in which this can be realized
for one of the models considered, based on $ {\cal G} = \su6 $. For this case
we need to find a group $ \Gamma $ and a representation $ {\mathbbm V} $
such that (\ref{eq:mzero}) has solutions only for the
zero modes associated with
the ``light'' generators $ E_{\pm {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^1},
~ {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^1 \cdot {\bf C} $ and $ {\bm\mu}} % \def\mubf{{\pmb{$\mu$}}_2 \cdot {\bf C} $.
To construct $ \Gamma $ note that $\su6$ has two $\su2$
subgroups generated by
$ E_{\pm {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i}, ~{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i \cdot {\bf C}, ~ i=3,5$
that commute with each other and with $G$. The group
elements
$ \exp[ (i \pi/2) ( E_{{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i} + E_{-{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i}) ]$,
$ \exp[ (\pi/2) ( E_{{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i} - E_{-{\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i}) ]$, and
$\exp[ (i \pi/2) {\bm\alpha}} % \def\alpbf{{\pmb{$\alpha$}}^i \cdot {\bf C} ]$,
correspond to rotations by an angle $ \pi $ and
generate a $ {\mathbbm Z}_4 \times {\mathbbm Z}_4 $ discrete subgroup under
which only the generators of $G$ are invariant, and which
we choose as $ \Gamma $, while for $ {\mathbbm V}$ we choose the
adjoint representation of these rotations~\footnote{This
argument can be extended to all models based
on $ {\cal G} = \su{3l} $ for $l$ even.}. Constructing the
representation $ \mathfrak{R} $ then requires first the
identification of the modes that should get a vacuum expectation value\
in order to have the right pattern of spontaneous symmetry breaking\ at low
energies, and then choosing the dimensionality of the
compact space that can accommodate these matrices.
Again, this detailed calculation falls outside the scope of the present paper.
In the above considerations we have avoided discussing the difficulties
associated with Yukawa couplings, mainly because our emphasis was in obtaining
the right vector-boson and scalar sector spectrum. If all fields are bulk-propagating
these couplings are determined in terms of the gauge coupling constants and
the choice of $ \Gamma $ the representations $ {\mathbbm V} $ and $\mathfrak{R} $.
Whether a model can be found that can accommodate the complicated
fermion-mass structure without the introduction of large brane
couplings~\footnote{Loop corrections in general induce brane couplings
\cite{Georgi:2000ks}, some of which may have power-dependence on
the UV cutoff $ \Lambda$, but whether these affect the light
scalar modes depends on the details of the model.}
will be investigated in a future publication.
\acknowledgements
This work was supported by in part UC-MEXUS under Grant No. CN-08-205,
by CONACYT and by the U. S. Department of Energy under
Grant No. DEFG03- 94ER40837.
|
1,941,325,220,550 | arxiv | \section{Introduction} Let us consider a singular point of a germ of analytic vector field $\bm{X} $ on $({\mathbb C},0)$. If the singular point is simple, then the germ of vector field is analytically linearizable. If the singular point is multiple, also called \emph{parabolic}, then the vector field is analytically conjugate to any one of the following normal forms:
\begin{align}
\bm{X}(x) &=\left( x^{k+1}-\mu x^{2k+1} \right)\pp{x},\label{nf0_1}\\
\bm{X}(x) &= \frac{x^{k+1}}{1+\mu x^k}\pp{x},\label{nf0_2}
\end{align}
where $\mu=\mathrm{Res}_{x=0}\bm{X}^{-1}$ is the residue of the dual form.
The first normal form is more frequent in the older works of the Russian school. The second one is easier to manipulate;
for instance, the rectifying coordinate (time coordinate) $\int\bm{X}^{-1}$ is simple to calculate.
The next natural question is to consider normal forms for unfoldings of germs of analytic vector fields at a singular point.
When the singular point is simple the normal form of an unfolding is linear, and hence unique. When the singular point is parabolic, Kostov proved that the following standard deformation of \eqref{nf0_1} is versal \cite{Kostov}:
\begin{equation}
\bm{X}_1(x;y) =\left( x^{k+1}+y_{k-1} x^{k-1} + \ldots +y_1x+y_0 - (\mu + y_{2k+1}) x^{2k+1}\right)\pp{x}.\label{nf_1}\end{equation}
The proof uses that \eqref{nf_1} is an infinitesimal deformation of \eqref{nf0_1}, and then calls for the machinery of Martinet's Reduction Lemma (see for instance \cite{Arnold0}).
The philosophy behind this normal form is two-fold. First, a parabolic point of codimension $k$ is the merging of $k+1$ simple singular points, each having its own eigenvalue, which is an analytic invariant. Hence it is natural that a full unfolding would involve $k+1$ parameters. Second, the geometry of an unfolding of a parabolic point is simple, hence the convergence to the normal form.
Kostov's normal form is very important for many bifurcations problems. For instance, when one studies the unfolding of a parabolic point of a germ of $1$-diffeomorphism of $({\mathbb C},0)$ (i.e. a multiple fixed point), then a formal normal form is given by the time one map of a vector field of the form \eqref{nf_1}. The change of coordinate to this normal form diverges and the obstruction to the convergence is the classifying object of the unfoldings (see for instance \cite{MRR}, \cite{Rousseau}, and \cite{Ribon-f, Ribon-a}). The same normal form is used to classify germs of unfoldings of 2-dimensional vector fields in $({\mathbb C}^2,0)$ with either a saddle-node or a resonant saddle point: indeed the vector fields are orbitally analytically equivalent if and only if the holonomy map of the separatrices (the strong separatrices in the case of a saddle-node) are conjugate (see \cite{RC} and \cite{Rousseau-Teyssier}).
Very soon, other normal forms equivalent to \eqref{nf_1} appeared in the literature without proof:
\begin{align}
\bm{X}_2(x;y) &= \left(x^{k+1}+y_{k-1} x^{k-1} + \ldots +y_1x+y_0\right)\!\left(1 - (\mu + y_{2k+1}) x^k\right)\pp{x},\label{nf_2}\\
\bm{X}_3(x;y) &= \frac{x^{k+1}+y_{k-1} x^{k-1} + \ldots +y_1x+y_0}{1+(\mu+y_{2k+1}) x^k}\pp{x},\label{nf_3}
\end{align}
and they are all called \emph{Kostov's theorem}. In practice, most authors use the normal form \eqref{nf_3}, which is much more suitable for computations.
The paper \cite{Rousseau-Teyssier} indirectly suggests that the normal form \eqref{nf_3} is universal by showing that the normal form \eqref{nf_3} associated to a generic $k$-parameter unfolding of a parabolic point of codimension $k$ is unique up to the action of the group ${\mathbb Z}/k{\mathbb Z}$ of rotations of order $k$. This uniqueness property is extremely important in all classification problems of unfoldings under conjugacy or analytic equivalence: it shows that the parameters of the normal forms are essentially unique and hence analytic invariants of the unfoldings.
Hence, to show that two unfoldings are analytically equivalent, the first step is to change to the canonical parameters and it then suffices to study the equivalence problem for fixed values of the parameters.
In this paper, we provide self-contained proofs that the three normal forms \eqref{nf_1}, \eqref{nf_2} and \eqref{nf_3} are unique up to the action of the group ${\mathbb Z}/k{\mathbb Z}$, and universal.
These self-contained proofs are useful for further generalizations, for instance when the vector field has some symmetry or reversibility property, and also for the formal case, the $\mathcal C^\infty$-case, and mixed cases where the variable is analytic and the dependence on the parameters is only real-analytic: this mixed case occurs when one considers bifurcations of antiholomorphic parabolic fixed points (i.e. $f(x) = \bar x \pm \bar x^{k+1} + o(\bar x^{k+1})$).
As a second part of the paper, we briefly address the real analytic, formal, and smooth cases.
In the first two cases, each of the corresponding unfoldings is universal.
In the smooth case, we give an explicit example showing that the unfolding is only versal and cannot be universal, namely the two vector fields
$\bm{X}(x;\lambda)=(x^2+\lambda^2)\tpp{x}$, and
$\bm{X}'(x;\lambda)=(x^2+(\lambda+\omega(\lambda))^2)\tpp{x}$ are $\mathcal C^\infty$-conjugate when $\omega(\lambda)$ is infinitely flat at $\lambda=0$.
Let us explain one difference with the analytic case. In the latter case the eigenvalues at the singular points are complex $\mathcal C^1$-invariants and
for a given set of $k+1$ eigenvalues at the singular points, there are only a finite number of solutions for the $y_j$ in any of the normal forms \eqref{nf_1}, \eqref{nf_2} and \eqref{nf_3} with the prescribed eigenvalues.
In the smooth case, only the eigenvalues at the real singular points are $\mathcal C^1$-invariants and we can smoothly glue anything at the complex singular points. The two systems of our counterexample have purely imaginary singular points. An open question is to know if we have universal unfoldings in the smooth case when all the singular points are real.
The original articles \cite{Kostov, Kostov-real} of Kostov cover a much more general case of deformations of differential forms of real power $\alpha$.
However, our goal is not to redo what has been done well by Kostov, but to provide an elementary and self-contained proof in the case of vector fields, that is power $\alpha=-1$, which is why we do not discuss the other cases. Nevertheless, we believe that our proof of the uniqueness in the formal/analytic case, which is missing in Kostov's article, could be well adapted to general $\alpha$.
\section{The analytic theory}
The following definitions are classical: see for instance \cite{Arnold1}.
\medskip
\begin{definition}~
\begin{enumerate}[wide=0pt, leftmargin=\parindent]
\item Two germs of analytic (resp. real analytic, formal, $\mathcal C^\infty$) parametric families of vector fields $\bm{X}(x;\lambda)$, $\bm{X}'(x';\lambda)$ depending on a same parameter $\lambda$
are \emph{conjugate} if there exists an analytic (resp. real analytic, formal, $\mathcal C^\infty$) invertible change of coordinate
\begin{equation}\label{eq:transformation}
x'=\phi(x;\lambda),
\end{equation}
changing one family to the other. We write $\bm{X}=\phi^*\bm{X}'$ as a \emph{pullback} of $\bm{X}'$.
\item Let $\lambda\mapsto \lambda'=\psi(\lambda)$ be a germ of analytic (resp. real analytic, formal, $\mathcal C^\infty$) map (not necessarily invertible), then $\bm{X}(x,\lambda)=\bm{X}'(x,\psi(\lambda))$ is a family \emph{induced} from $\bm{X}'$.
\item A parametric family of vector fields $\bm{X}(x;\lambda)$ is a \emph{deformation} of $\bm{X}(x;0)$. Two deformations $\bm{X}(x;\lambda)$, $\bm{X}'(x,\lambda)$ of the same initial vector field $\bm{X}(x;0)=\bm{X}'(x;0)$ with the same parameter $\lambda$ are \emph{equivalent} (as deformations) if the two families are conjugate by means of an invertible transformation \eqref{eq:transformation} with $\phi(x;0)\equiv x$.
\item A deformation $\bm{X}'(x,\lambda')$ of $\bm{X}'(x,0)$ is \emph{versal} if any other deformation $\bm{X}(x,\lambda)$ of $\bm{X}'(x,0)= \bm{X}(x,0)$ is equivalent to one induced from it.
It is \emph{universal} if the inducing map $\lambda'=\psi(\lambda)$ is unique.
\end{enumerate}
\end{definition}
In this section we provide a self-contained proof of the following theorem:
\begin{theorem}\label{thm1}
In the analytic case, for $k\geq 1$, the deformation \eqref{nf_3} of \eqref{nf0_2} is universal.
\end{theorem}
\begin{corollary}\label{cor:Kostov}
In the analytic case, for $k\geq 1$, the deformations \eqref{nf_1} and \eqref{nf_2} of \eqref{nf0_1} are universal.
\end{corollary}
As explained in the introduction, the proof of the versality is due to Kostov (for \eqref{nf_1} see \cite{Kostov},
while for \eqref{nf_3} it has been often stated in literature without explicit proof, see e.g. \cite[p.116]{Arnold}),
and the uniqueness comes from \cite{Rousseau-Teyssier}.
Theorem~\ref{thm1} can be rephrased in more precise terms as the following theorem of which it is a direct consequence.
\begin{theorem}\label{theorem:Kostov}~
\begin{enumerate}[label=(\roman*), leftmargin=\parindent, itemindent=0em, itemsep=4pt, topsep=4pt, partopsep=0pt, parsep=0pt]
\item \textnormal{(Kostov \cite{Kostov})} Any analytic germ of a family of vector fields $\tilde \bm{X}(x,\lambda)$ depending on a multi-parameter $\lambda$ unfolding $\tilde \bm{X}(x;0)=x^{k+1}\frac{1}{\omega(x)}\pp{x}$, $\omega(0)\neq 0$, $k\geq 0$, is analytically conjugate
to a family of the form
\begin{align} \label{Kostov_form0}
\bm{X}(x;\lambda)&=c(\lambda)x\pp{x}, & k&=0,\\
\bm{X}(x;\lambda)&=\frac{x^{k+1}+y_{k-1}(\lambda)x^{k-1}+\ldots+y_0(\lambda)}{1+\mu(\lambda)x^k}\pp{x},& k&\geq 1,
\label{Kostov_form}
\end{align}
with $y_0(0)=\ldots =y_{k-1}(0)=0$, where
\[
\mu(\lambda)=-\mathrm{Res}_{x=\infty}\bm{X}(x;\lambda)^{-1}
\]
is the sum of the residues of $\tilde\bm{X}(x;\lambda)^{-1}$ over its local polar locus around the origin.
\item \textnormal{(Rousseau, Teyssier \cite[Theorem 3.5]{Rousseau-Teyssier}, \cite[Theorem 7.2]{Klimes-Rousseau})}
The normal form \eqref{Kostov_form0} for $k=0$ and \eqref{Kostov_form} for $k=1$ are unique, while the normal form
\eqref{Kostov_form} for $k > 1$ is unique up to the action of $x\mapsto e^{2\pi i\frac{l}{k}}x$, $l\in{\mathbb Z}_k$,
$$y_{j}(\lambda)\mapsto e^{-2\pi i\frac{(j-1)l}{k}}y_{j}(\lambda),\quad j=0,\ldots,k-1.$$
More precisely, if $x\mapsto\phi(x,\lambda)$ is a transformation between two vector fields \eqref{Kostov_form}, fixing $\lambda$,
then
\[
\phi(x,\lambda)=\begin{cases} e^{t(\lambda)}x, & \text{if }\ k=0,\\
e^{2\pi i\frac{l}{k}} \exp(t(\lambda)\bm{X})(x;\lambda), & \text{if }\ k\geq 1,\end{cases}
\]
for some $l\in{\mathbb Z}_k$ and some analytic germ $t(\lambda)$.
\end{enumerate}
\end{theorem}
The first step in proving Theorem~\ref{theorem:Kostov} is the following ``prenormal form'', which can be also found for example in \cite[Proposition~5.13]{Ribon-f}.
\begin{proposition}[Prenormal form] \label{prop:prenormal}
Any germ of a family of vector fields $\tilde \bm{X}(x,\lambda)$ depending on a multi-parameter $\lambda$ unfolding $\tilde \bm{X}(x;0)=(cx^{k+1}+\ldots)\pp{x}$, $k\geq 1$, is analytically conjugate to a family of the form
\begin{equation}
\bm{X}(x;\lambda)=\frac{x^{k+1}+y_{k-1}(\lambda)x^{k-1}+\ldots+y_0(\lambda)}{1+u_0(\lambda)+\ldots+u_{k-1}(\lambda)x^{k-1}+\mu(\lambda)x^k}\pp{x},
\label{pol_form}
\end{equation}
where $y_j(0)=0=u_j(0)$, $j=0,\ldots,k-1$, and
\[
\mu(\lambda)=-\mathrm{Res}_{x=\infty}\bm{X}(x;\lambda)^{-1}.
\]
\end{proposition}
\begin{proof}
First, let us transform $\tilde \bm{X}(x;0)=x^{k+1}\frac{1}{\omega(x)}\pp{x}$ to a form $\bm{X}(x;0)=\frac{x^{k+1}}{1+\mu(0)x^k}\pp{x}$.
Up to a linear change $x\mapsto ax$, $a\in{\mathbb C}\smallsetminus\{0\}$, we can assume $\omega(0)=1$.
Write $\omega(x)=1+\omega_1x+\ldots+\omega_kx^k+x^{k+1}r(x)$,
let $\mu(0):=\omega_k$, and let
$\alpha(x):=\int\tilde \bm{X}(x;0)^{-1}-\bm{X}(x;0)^{-1}=-\tfrac{\omega_1}{k-1}x^{1-k}-\ldots-\frac{\omega_{k-1}}{1}x^{-1}+\int_0^xr(x)dx$.
Then $\alpha$ is a meromorphic germ with pole of order at most $k-1$ at the origin, and the desired transformation is provided by Lemma~\ref{lemma} below.
By Weierstrass preparation and division theorem, any family $\tilde \bm{X}(x,\lambda)$ can be written in the form
$\tilde \bm{X}(x,\lambda)=\frac{P(x;\lambda)}{Q(x;\lambda)+P(x,\lambda)R(x,\lambda)}\pp{x}$, for some Weierstrass polynomials $P(x;\lambda)=x^{k+1}+y_{k-1}(\lambda)x^{k-1}+\ldots+y_0(\lambda)$,
$Q(x;\lambda)=1+u_0(\lambda)+\ldots+u_{k-1}(\lambda)x^{k-1}+\mu(\lambda)x^k$, and some analytic germ $R(x;\lambda)$. Let $\alpha(x,\lambda)=\int R(x;\lambda)dx$, then
\[\tilde \bm{X}(x,\lambda)=\frac{\bm{X}(x,\lambda)}{1+\bm{X}(x,\lambda).\alpha(x,\lambda)}\]
for $\bm{X}(x;\lambda)=\frac{P(x;\lambda)}{Q(x;\lambda)}\pp{x}$ of the form \eqref{pol_form}.
The result follows from Lemma~\ref{lemma}.
\end{proof}
The following lemma is classical (see for example \cite[Proposition 2.2]{Teyssier} to which it is essentially equivalent).
\begin{lemma}\label{lemma}
Let $\bm{X}_0,\ \bm{X}_1$ be two germs of analytic families of vector fields vanishing at the origin, and assume there exists an analytic germ $\alpha(x,\lambda)$ such that $\bm{X}_1=\frac{\bm{X}_0}{1+\bm{X}_0.\alpha}$.
Then the flow map of the vector field $\bm{Y}(x,t;\lambda)=\pp{t}-\frac{\alpha \bm{X}_0}{1+t\bm{X}_0.\alpha}$
\[ \phi_1(x,\lambda)=x\circ\exp(\bm{Y})\Big|_{t=0}, \]
conjugates $\bm{X}_1$ with $\bm{X}_0=\phi_1^*\bm{X}_1$.
The statement is also true if $\alpha$ is meromorphic such that $\bm{X}_0.\alpha$ and $\alpha\bm{X}_0$ are analytic and vanish for $(x,\lambda)=0$ (so that the flow of $\bm{Y}$ is defined for all $t\in[0,1]$).
\end{lemma}
\begin{proof}
On the one hand, if $\bm{X}_t:=\frac{\bm{X}_0}{1+t\bm{X}_0.\alpha}$ and $\bm{Y}=\pp{t}-\frac{\alpha \bm{X}_0}{1+t\bm{X}_0.\alpha}$
are vector fields in $x,t,\lambda$, then $[\bm{Y},\bm{X}_t]=0$, which means that the flow $\exp(s\bm{Y}):(x,t)\mapsto(\Phi_s(x,t),t+s)$ of $\bm{Y}$ preserves $\bm{X}_t=\Phi_s^*\bm{X}_{t+s}$. In particular $\phi_s(x):=\Phi_s(x,0)$ is such that $\phi_s^*\bm{X}_s=\bm{X}_0$.
\end{proof}
\begin{proposition}\label{prop:normalisation}
Consider two families of vector fields $\bm{X}_0$ and $\bm{X}_1$ of the form
\begin{equation}
\bm{X}_t=\frac{x^{k+1}+y_{k-1}x^{k-1}+\ldots+y_0}{1+t(u_0+\ldots+u_{k-1}x^{k-1})+\mu x^k}\pp{x},
\label{pol_form2}
\end{equation}
depending on parameters $(y,u,\mu)$.
Then there exists an analytic transformation $(x,y)\mapsto\big(\phi(x,y;u,\mu),\psi(y;u,\mu)\big)$ tangent at identity at $(x,y)=0$ that conjugates
$\bm{X}_1$ to $\bm{X}_0$.
\end{proposition}
\begin{proof}
Let $\bm{X}_t=\frac{P(x,y)}{Q(x,t;u,\mu)}\pp{x}$ be as above \eqref{pol_form2}.
We want to construct a family of transformations depending analytically on $t\in[0,1]$ between $\bm{X}_0$ and $\bm{X}_t$, defined by a flow of a vector field $\bm{Y}$ of the form
\[
\bm{Y}=\pp{t}+\sum_{j=0}^{k-1}\xi_j(t,y;u,\mu)\pp{y_j}+\frac{H(x,t,y;u,\mu)}{Q(x,t;u,\mu)}\pp{x},
\]
for some $\xi_j$ and $H$, such that $[\bm{Y},\bm{X}_t]=0$, that is
\[
-\frac{UP}{Q^2}+\frac{\Xi}{Q}+\frac{H}{Q}\pp{x}\left(\frac{P}{Q}\right)-\frac{P}{Q}\pp{x}\left(\frac{H}{Q}\right)=0,
\]
where $U(x;u)=u_0+\ldots+u_{k-1}x^{k-1}$ and $\Xi(x;\xi)=\xi_0+\ldots+\xi_{k-1}x^{k-1}$, which is equivalent to
\begin{equation}
H\tpp{x}P-P\tpp{x}H+Q\Xi=UP.
\label{homological_eq}
\end{equation}
We see that we can choose $H$ as a polynomial in $x$:
\[
H=h_0(t,y;u,\mu)+\ldots+h_k(t,y;u,\mu)x^k.
\]
Write $UP=b_0(y;u)+\ldots+b_{2k}(y;u)x^{2k}$,
then the equation \eqref{homological_eq} takes the form of a non-homogeneous linear system for the coefficients $(\xi,h)$:
\[
A(t,y;u,\mu)\begin{pmatrix}\xi\\ h\end{pmatrix}=b(y;u).
\]
For $y=u=0$ the equation \eqref{homological_eq} is
\[
(k+1)x^kH-x^{k+1}\tpp{x}H+(1+\mu x^k)\Xi=0,
\]
hence
\[
A(t,0;0,\mu)=
\begin{pmatrix} 1&& &0&& &0\\ &\ddots& &&\ddots& &\\ &&1 &&&0 &0\\ \mu&& &k+1&& &0\\ &\ddots& &&\ddots& &\\ &&\mu &&&2 &0\\ 0&\ldots&0 &0&\ldots&0 &1 \end{pmatrix}.
\]
This means that $A(t,y;u,\mu)$ is invertible for $(t,\mu)$ from any compact in ${\mathbb C}\times{\mathbb C}$ if $|y|,\ |u|$ are small enough.
Since $b(0;0)=0$, the constructed vector field $\bm{Y}(x,t,y;u,\mu)$ is such that $\bm{Y}(0,t,0;0,\mu)=\pp{t}$ and its flow is well-defined for all $|t|\leq 1$ as long as $|y|,\ |u|$ are small enough.
\end{proof}
\medskip
\begin{proof}[Proof of Theorem~\ref{theorem:Kostov}]~
\begin{enumerate}[label=(\roman*), leftmargin=\parindent, itemindent=0em, itemsep=4pt, topsep=4pt, partopsep=0pt, parsep=0pt]
\item The existence of an analytic normalizing transformation to \eqref{Kostov_form} when $k\geq1$ follows directly from Propositions~\ref{prop:prenormal} and~\ref{prop:normalisation}. For $k=0$ it follows from Lemma~\ref{lemma}.
\item Let us prove the uniqueness.
For $k=0$ it is obvious.
For $k>0$, let $\phi(x;\lambda)$ be a transformation between
$\bm{X}=\frac{P(x;\lambda)}{1+\mu(\lambda) x^k}\pp{x}$ and $\bm{X}'=\frac{P'(x;\lambda)}{1+\mu'(\lambda) x^k}\pp{x}$, preserving the parameter $\lambda$, and such that $\phi^*\bm{X}=\bm{X}'$.
By the invariance of the residue, $\mu(\lambda)=\mu'(\lambda)$.
Let $\phi(x;0)=cx+\ldots$ for some $c\neq 0$; necessarily $c=e^{2\pi i\frac{l}{k}}$ for some $l\in{\mathbb Z}_k$. Up to precomposition with a map $x\mapsto cx$,
we can assume that $c=1$ and that $\phi(x;0)=x+\ldots$ is tangent to identity.
Let
$$G(x,t, \lambda)=\exp(-t \bm{X})\circ \phi(x,\lambda),$$
and
$$K(t,\lambda)=\frac{\partial^{k+1}G}{\partial x^{k+1}}\big|_{x=0}.$$
The map $K$ is analytic and $\frac{\partial K}{\partial t}(t,0)= -(k+1)!\neq0$.
For $\lambda=0$, there exists $t_0$ such that $K(t_0,0)=0$ (in fact $t_0=\frac{1}{(k+1)!}\frac{\partial^{k+1}\phi}{\partial x^{k+1}}(0;0)$ since $\exp(t\bm{X}(x;0))=x+tx^{k+1}+\ldots$).
By the implicit function theorem, there exists a unique function $t(\lambda)$ such that $K(t(\lambda), \lambda)\equiv 0$. Then considering the new transformation $\psi=\exp(-t(\lambda)X)\circ\phi$, it suffices to proves that $\psi\equiv id$. This is done by the infinite descent.
Let $\psi(x,\lambda)=x+f(x;\lambda)$, where $\frac{\partial ^{k+1} f}{\partial x^{k+1}}\equiv 0$.
Denote $\mathcal I_\lambda$ the ideal of analytic functions of $(x;\lambda)$ that vanish when $\lambda=0$. To show that $\psi(x,0)\equiv x$ it suffices to show that $f\in\mathcal I_\lambda^n$ for all $n$.
For $\lambda=0$ both vector fields $\bm{X}(x;0)$ and $\bm{X}'(x;0)$ are equal to $\frac{x^{k+1}}{1+\mu(0)x^k}\pp{x}$,
and it is easy to verify that $\psi(x,0)\equiv x$ (for instance using power series), which gives us the induction hypothesis $f\in \mathcal I_\lambda$.
Suppose now that $f\in \mathcal I_\lambda^n$.
Developing the right side of the transformation equation
\begin{equation*}
\tfrac{P}{1+\mu x^k}\tpp{x}\psi=\tfrac{P'\circ\psi}{1+\mu\psi^k},
\end{equation*}
we have
\[\tfrac{P}{1+\mu x^k}\cdot\tpp{x}(x+f)=\tfrac{P'}{1+\mu x^k}+ f\cdot\tpp{x}\tfrac{P'}{1+\mu x^k}\mod \mathcal I_\lambda^{n+1},\]
from which
\[P-P'=-x^{k+1}\tpp{x}f +f\cdot\big((k+1)x^k-\tfrac{k\mu(0)x^{2k}}{1+\mu(0)x^k}\big) \mod \mathcal I_\lambda^{n+1}.\]
The left side being a polynomial of order $\leq k-1$ in $x$, this means that both sides vanish modulo $\mathcal I_\lambda^{n+1}$.
Therefore on the left side $P=P'\mod \mathcal I_\lambda^{n+1}$, while the right side can be rewritten as
\[-(1+\mu(0)x^k)x\tpp{x}f +f\cdot ((k+1)+\mu(0)x^{k})\equiv 0 \mod \mathcal I_\lambda^{n+1},\]
Putting $f=\sum_{j=0}^\infty f_j x^j$, yields
\[\sum_{j=0}^{k-1} (k+1-j)f_j x^{j} +\sum^\infty_{j=k} (k+1-j)(f_j+\mu(0)f_{j-k})x^{j} \equiv 0 \mod \mathcal I_\lambda^{n+1},\]
from which we get that all $f_j\in\mathcal I_\lambda^{n+1}$ since $f_{k+1}\equiv0$. \qedhere
\end{enumerate}
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:Kostov}]
Consider the two families \eqref{nf_1} and \eqref{nf_3}
\begin{align*}
\bm{X}_1(x;y) &= x^{k+1}+y_{k-1} x^{k-1} + \ldots +y_1x+y_0 - (\mu + y_{2k+1}) x^{2k+1}\pp{x},\\
\bm{X}_3(x';y') &= \frac{x'^{k+1}+y'_{k-1} x'^{k-1} + \ldots +y'_1x'+y'_0}{1+(\mu+y'_{2k+1}) x'^k}\pp{x'}.
\end{align*}
By Theorem~\ref{theorem:Kostov} we know that there exists a map
\[x'=\phi(x,y),\quad y_j'=\psi_j(y),\ j=0,\ldots,k-1,\ 2k+1, \]
such that $\bm{X}_1(x,y)=\phi^*\bm{X}_3(x,\psi(y))$, that is, such that
\[\begin{split}
&\frac{\phi^{k+1}+\psi_{k-1} \phi^{k-1} + \ldots +\psi_1\phi+\psi_0}{1+(\mu+\psi_{2k+1}) \phi^k}= \\
&\qquad\qquad=\left(x^{k+1}+y_{k-1}x^{k-1} + \ldots +y_1 x+y_0 - (\mu + y_{2k+1})x^{2k+1}\right)\pp{x}\phi.
\end{split}\]
We want to show that $\psi$ is invertible.
For $y=0$ we have
\[\frac{\phi(x,0)^{k+1}}{1+\mu \phi(x,0)^k}=\left(x^{k+1}-\mu x^{2k+1}\right)\pp{x}\phi(x,0),\]
and (up to a pre-composition with a flow map of $\bm{X}_1$ killing the term in $x^{k+1}$) we can assume that
\[\phi(x,0)=x+\mu^2\tfrac{1}{k}x^{2k+1}+O(x^{2k+2}).\]
Write $\phi(x,y)=\phi(x,0)+f(x,y)$, with $f(x,y)=\sum_{l=0}^\infty f_l(y)x^l$, and denote $\mathcal I_y$ the ideal of functions that vanish when $y=0$.
Then calculating modulo $\mathcal I_y^2$:
\[\begin{split}
&\frac{\phi(x,0)^{k+1}+\psi_{k-1} \phi(x,0)^{k-1} + \ldots +\psi_1\phi(x,0)+\psi_0}{1+(\mu+\psi_{2k+1}) \phi(x,0)^k}\\
&\qquad +
\frac{(k+1)\phi(x,0)^k+\mu\,\phi(x,0)\color{black}\xspace^{2k}}{(1+\mu\,\phi(x,0)^k)^2}f(x,y)=\\
&=\left(x^{k+1}+y_{k-1}x^{k-1} + \ldots +y_1 x+y_0 - (\mu + y_{2k+1})x^{2k+1}\right)\pp{x}\phi(x,0)\\
&\qquad +\left(x^{k+1}-\mu x^{2k+1}\right)\pp{x}f(x,y) \mod\mathcal I_y^2.
\end{split}\]
Comparing the coefficients of $x^j$, $j=0,\ldots,k-1$, on both sides we have
\[\psi_j=y_j\mod\mathcal I_y^2,\quad j=0,\ldots,k-1,\]
for $j=k+1$
\[-\psi_1\mu +(k+1)f_1(y)=f_1(y) \mod\mathcal I_y^2,\]
and for $j=2k+1$
\[\begin{split}
&-\mu-\psi_{2k+1}+\mu^2\tfrac{1}{k}\psi_1-(2k+1)\mu f_1+(k+1)f_{k+1}=\\
&\qquad =-\mu - y_{2k+1}+\mu^2\tfrac{2k+1}{k}y_1+(k+1)f_{k+1}-\mu f_1\mod\mathcal I_y^2,
\end{split}\]
from which
\[\psi_{2k+1}=y_{2k+1}-4\mu^2y_1 \mod\mathcal I_y^2.\]
This means that the transformation $(x,y)\mapsto(\phi(x,y),\psi(y))$ is invertible for small $x,y$.
Similarly for the families \eqref{nf_2} and \eqref{nf_3}
\end{proof}
\section{Real analytic, formal and smooth theory}
\begin{theorem}[Real analytic theory]\label{theorem:Kostov-real}~
The statement of Theorem~\ref{theorem:Kostov} is also true in the real analytic setting,
with the exception that \eqref{Kostov_form} needs to be replaced by
\begin{equation} \label{Kostov_form-real}
\bm{X}_{\text{real}}(x;\lambda)=\frac{x^{k+1}+y_{k-1}(\lambda)x^{k-1}+\ldots+y_0(\lambda)}{(\pm 1)^{k+1}+\mu(\lambda)x^k}\pp{x}.
\end{equation}
Consequently, the real analytic parametric family
\begin{equation}
\bm{X}_{3,\text{real}}(x;y) = \frac{x^{k+1}+y_{k-1} x^{k-1} + \ldots +y_0}{(\pm 1)^{k+1}+(\mu+y_{2k+1}) x^k}\pp{x},\label{nf_4}
\end{equation}
is a universal real analytic deformation for $\bm{X}_{3,\text{real}}(x;0)$.
\end{theorem}
\begin{proof}
Assuming the initial vector field $\tilde\bm{X}$ is real analytic then so are all the transformations of Propositions~\ref{prop:prenormal} and \ref{prop:normalisation} with the only exception: the leading coefficient of $\tilde\bm{X}(x;0)=\big(cx^{k+1}+\ldots\big)\pp{x}$ can be brought to either $c=\pm 1$ if $k$ is even, and $c=1$ if $k$ is odd.
\end{proof}
\begin{corollary}
If we consider deformations which are symmetric (resp. antisymmetric (also called reversible)) with respect to the real axis, then their associated universal deformations
\begin{align*}
\bm{X}_1'(x;y) &=c\left(x^{k+1}+y_{k-1} x^{k-1} + \ldots +y_1x+y_0 - c\,(\mu+y_{2k+1}) x^{2k+1}\right)\pp{x},\\
\bm{X}_2'(x;y) &= c\left(x^{k+1}+y_{k-1} x^{k-1} + \ldots +y_1x+y_0\right)\!\left(1 - c\,(\mu+y_{2k+1}) x^k\right)\pp{x},\\
\bm{X}_3'(x;y) &= \frac{x^{k+1}+y_{k-1} x^{k-1} + \ldots +y_1x+y_0}{c+(\mu+y_{2k+1}) x^k}\pp{x},
\end{align*}
$y_0,\ldots y_{k-1},cy_{2k+1}\in {\mathbb R}$, $c\mu\in{\mathbb R}$, with $c=(\pm 1)^{k+1}$ (resp. $c=i\,(\pm 1)^{k+1}$), have the same property, and the conjugacy commutes with the symmetry.
\end{corollary}
\begin{theorem}[Formal theory]\label{theorem:Kostov-formal}~
\begin{enumerate}[leftmargin=\parindent, itemindent=0em, itemsep=4pt, topsep=4pt, partopsep=0pt, parsep=0pt]
\item The statement of Theorem~\ref{theorem:Kostov} and therefore of Theorem~\ref{thm1} is also true in the formal setting, of formal parametric germs of vector fields and formal transformations \eqref{eq:transformation}, where by formal we mean a formal power series in $(x,\lambda)$.
In the formal real case (i.e. the series have real coefficients), then the normal form is given by \eqref{Kostov_form-real}.
\item \textnormal{(Ribon \cite[Proposition 6.1]{Ribon-f})}
Two \emph{analytic} germs of vector fields $\bm{X},\bm{X}'$ are formally conjugate if and only if they are analytically conjugate.
Moreover, denoting $\hat{\mathcal I}_\lambda$ the ideal of formal series that vanish when $\lambda=0$, if $\hat\phi(x,\lambda)$ is a formal conjugating transformation, then for any $n>0$ there exists an analytic conjugacy $\phi_n(x,\lambda)$, $\phi_n^*\bm{X}'=\bm{X}$, such that $\phi_n=\hat\phi\mod\hat{\mathcal I}_\lambda^n$.
\end{enumerate}
\end{theorem}
The second statement is an analogue of the Artin approximation theorem.
\begin{proof
\begin{enumerate}[leftmargin=\parindent, itemindent=0em, itemsep=4pt, topsep=4pt, partopsep=0pt, parsep=0pt]
\item The proof follows exactly the same lines. The key fact is that a formal flow map of a formal vector field
\[
\hat\bm{Y}=\pp{t}+\sum_{j=0}^{k-1}\hat\xi_j(t,y;u,\mu)\pp{y_j}+\hat F(x,t,y;u,\mu)\pp{x},
\]
which is analytic in $t$ and the parameters $(u,\mu)$ is well defined: see Lemma~\ref{lemma:formalflow} below.
\item This is a consequence of the uniqueness of the normal form \eqref{Kostov_form}: each analytic germ of parametric vector field is analytically conjugate to a normal form \eqref{Kostov_form}, and two such normal forms are formally conjugate if and only if they are conjugate by a rotation $x\mapsto e^{2\pi i\frac{l}{k}}x$, $l\in{\mathbb Z}_k$, which is analytic.
Moreover, the formal conjugacy is a composition of the rotation and of a formal time $\hat t(\lambda)$-flow map of the vector field. Replacing $\hat t(\lambda)$ with an analytic $t_n(\lambda)=\hat t(\lambda)\mod\hat{\mathcal I}_\lambda^n$ does the trick. \qedhere
\end{enumerate}
\end{proof}
\begin{lemma}\label{lemma:formalflow}
Let $\hat\bm{Y}=\tpp{t}+\hat\bm{Z}(z,t)$, where $\hat\bm{Z}(z,t)$ be a formal vector field in $z\in{\mathbb C}^p$ with coefficients entire in $t\in{\mathbb C}$, that vanishes at $z=0$: $\hat\bm{Z}(0,t)=0$.
Then $\hat\bm{Y}$ has a well-defined formal flow $z\circ\hat\exp(s\hat\bm{Y})=\sum_{n=0}^{+\infty}\frac{s^n}{n!}\hat\bm{Y}^n.z$ for all $(s,t)\in{\mathbb C}^2$.
\end{lemma}
\begin{proof}
For any $n\in{\mathbb Z}_{\geq 0}$, the $n$-jet with respect to the variable $z$ of $\hat\bm{Y}$ is an entire vector field $j_z^n\hat\bm{Y}(z,t)$ in ${\mathbb C}^p\times{\mathbb C}$ with well defined flow $z\circ\exp(s\, j_z^n\hat\bm{Y})$ fixing the origin in $z$.
For any $m\leq n$, the $m$-jet of this flow agrees with the $m$-jet of the one for $m$:
$j_z^m\big(z\circ\exp(s\,j_z^n\hat\bm{Y})\big)=j_z^m\big(z\circ\exp(s\,j_z^m\hat\bm{Y})\big)$,
meaning that they converge in the Krull topology as $n\to+\infty$ to a well-defined formal flow map $z\circ\hat\exp(s\hat\bm{Y})$.
See also \cite[Theorem~3.9]{IlYa}.
\end{proof}
\begin{theorem}[$\mathcal C^\infty$-smooth theory, Kostov \cite{Kostov-real}]\label{theorem:Kostov-real}~\\
In real $\mathcal C^\infty$-smooth setting, the deformation $\bm{X}_{3,\text{real}}(x;y)$ \eqref{nf_4} is a versal deformation of the normal form vector field $\bm{X}_{3,\text{real}}(x;0)$.
\end{theorem}
\begin{proof} The only purely analytic tools used in the proof of the existence of a normalizing transformation were the Weierstrass preparation and division theorems (used in the proof of Proposition~\ref{prop:normalisation}), which have their counterpart in the $\mathcal C^\infty$-setting in the Malgrange preparation and division theorems \cite{Malgrange, GG}.
\end{proof}
The deformation \eqref{nf_4} is not universal in the $\mathcal C^\infty$-setting in general.
The issue is the non-uniqueness in the Malgrange division and the lack of control over the potential non-real singularities in the family.
\begin{example}
The deformations
$\bm{X}(x;\lambda)=(x^2+\lambda^2)\tpp{x}$, and
$\bm{X}'(x;\lambda)=\big(x^2+(\lambda+\omega(\lambda))^2\big)\tpp{x}$, where $\omega(\lambda)$ is infinitely flat at $\lambda=0$
(i.e. $\big(\tpp{\lambda}\big)^n\omega\big|_{\lambda=0}=0$, for all $n\in{\mathbb Z}_{\geq 0}$),
are $\mathcal C^\infty$-equivalent by means of a conjugacy $\phi(x;\lambda)$ with $\Delta(x;\lambda):=\phi(x;\lambda)-x$ infinitely flat along $\lambda=0$.
Indeed we first change $x\mapsto \frac{\lambda+\omega(\lambda)}{\lambda}x =C(\lambda)x$ into $\bm{X}'$, thus transforming it into
$\bm{X}''(x;\lambda)=\frac1{C(\lambda)}\bm{X}(x;\lambda)$. Note that $C(\lambda) = 1+\frac{\omega(\lambda)}{\lambda}= 1+\mu(\lambda)$, with $\mu(\lambda)$ infinitely flat.
We then apply Lemma~\ref{lemma}. We look for a germ $\alpha(x;\lambda)$ such that
$1+\bm{X}.\alpha= C(\lambda)$, which is equivalent to $(x^2+\lambda^2)\tpp{x}\alpha= \mu(\lambda)$. This equation has the odd solution
$$\begin{cases}
\alpha(x;\lambda) = \frac{\mu(\lambda)}{\lambda} \arctan \frac{x}{\lambda},&\lambda\neq0,\\
0, &\lambda=0.\end{cases}$$
The function $\alpha$ is obviously $\mathcal C^\infty$ since $\arctan\frac{x}{\lambda}$ is bounded, and each derivative of $\arctan\frac{x}{\lambda}$ grows no faster than $(x^2+\lambda^2)^{-n}<\lambda^{-2n}$ for some $n$ depending on the derivative.
The flow of the vector field
\[\bm{Y}=\pp{t}-\frac{\alpha\bm{X}}{1+t\bm{X}.\alpha}=
\pp{t}-\frac{\alpha(x,\lambda)(x^2+\lambda^2)}{1+t\mu(\lambda)}\pp{x}\]
is well defined and $\mathcal C^\infty$-smooth for $t\in[0,1]$ as long as $|\mu(\lambda)|<1$.
\end{example}
\begin{remark} The deformations
$\bm{X}(x;\lambda)=(x^2-\lambda^2)\tpp{x}$, and
$\bm{X}'(x;\lambda)=\big(x^2-(\lambda+\omega(\lambda))^2\big)\tpp{x}$, where $\omega(\lambda)$ is infinitely flat at $\lambda=0$, are not $\mathcal C^\infty$-conjugate. Indeed, the eigenvalues at the singular points are $\mathcal C^1$ invariants.
\end{remark}
\begin{problem}
Can we expect uniqueness of the induced coefficients in \eqref{Kostov_form-real} in the special case when the deformation is such that it has $k+1$ merging real singularities when counted with multiplicity?
\end{problem}
\bibliographystyle{alpha}
\footnotesize
|
1,941,325,220,551 | arxiv | \section{Introduction}
The purpose of the article is to discuss the existence and multiplicity of weak solutions to the following singular problem:
\begin{equation*}
(P_\la)\;
\left\{\begin{array}{rllll}
(-\De)^s u
&=u^{-q}+ \la \left(\ds \int_{\Om}\frac{|u|^{2^*_\mu}(y)}{|x-y|^{\mu}}dy\right) |u|^{2^*_\mu-2}u, \quad u
>0 \; \text{in}\;
\Om,\\
u&=0 \; \text{ in } \R \setminus \Om,
\end{array}
\right.
\end{equation*}
for all $q>0, N\ge 2s, s \in (0, 1), 2_{\mu}^{*}=\frac{2N-\mu}{N-2s}$ and $\Om$ is a bounded domain in $\mathbb R^N$ with smooth boundary. Here the operator $(-\De)^s$ is the fractional Laplacian defined as
\begin{align*}
(-\De)^s u(x)= - \text{P.V. } \int_{ \R} \frac{u(x)-u(y)}{|x-y|^{N+2s}} dy
\end{align*}
where P.V denotes the Cauchy principal value.\\
The problems involving singular nonlinearity have a
very long history. In the pioneering work \cite{crandal}, Crandall, Rabinowitz and Tartar \cite{crandal} proved the existence of a solution of classical elliptic PDE with singular nonlinearity using the approximation arguments. Later many researchers studied the problems involving singular nonlinearity. Haitao \cite{hai} studied the following problem
\begin{align}\label{si96}
-\De u = au^{-q}+b u^{\frac{N+2}{N-2}}, u>0\text{ in } \Om, \quad u =0 \text{ on } \pa \Om
\end{align}
where $\Om\subset \R(N\geq 3)$ is a smooth bounded domain. If $a=\la$ and $b=1$, and $q\in (0,1)$, authors proved a global multiplicity result. While in \cite{adi1,dhanya1}, researchers improvised the results of \cite{hai} and proved the global multiplicity result for $q \in (0,3)$.
In \cite{hirano1}, Hirano, Saccon, and Shioji studied the problem \eqref{si96} with $a=\la$ and $b=1$, and $q\in (0,1)$. Using the well known splitting Nehari manifold method, authors proved the multiplicity of solutions for small $\la$. While in \cite{hirano}, authors studied the problem for all $q>0$, $a=1~b=\la$, and established a global multiplicity result using the nonsmooth analysis. For more details on singular type problems, we refer to \cite{coc,ghergu,GST2,GST, hai,hern} and references therein.
The study of nonlinear elliptic problems with critical terms motivated by Hardy-Littlewood-Sobolev inequality started long back and attracted lot of researchers due to its wide applications. Indeed, it was originated in the framework of various physical models. One of the first applications was found in H. Fr\"ohlich and S. Pekar model of the polaron, where free electrons in an ionic lattice interact with photons associated to the deformations of the lattice or with the polarization that it creates on the medium \cite{fro1,fro2}. In the modeling of one component plasma, Ph. Choquard gave the model which involves Choquard equation \cite{ehleib}. Later on such nonlinear problems are called Choquard equations and many researchers studied these type of problems to understand the existence, uniqueness, radial symmetry and regularity of the solutions \cite{moroz1,moroz2,moroz5} and references therein.
Pertaining the Choquard type critical exponent problems on bounded domains, Gao and Yang \cite{yang} studied the Brezis-Nirenberg type existence and nonexistence results with Choquard critical nonlinearities. In \cite{cassani}, \cite{seok} and\cite{tang} are studied a Brezis-Nirenberg type problem with uppercritical growth, concentration profiles of ground states and existence of semiclassical states respectively.
Nonlocal problems involving fractional Laplacian challenged a lot of researchers due to the large spectrum of applications. Consider the following problem
\begin{align}\label{si66}
(-\De)^s u = f(x,u) \text{ in } \Om ,\quad u=0 \text{ in } \R\setminus \Om
\end{align}
where $f$ is a Carath\'eodory function. The questions of existence, multiplicity and regularity of solutions to problem \eqref{si66} have been extensively studied in \cite{servadei, valdi} and references therein.
Concerning the existence and multiplicity of solutions to doubly nonlocal problems, a lot of works have been done. For a detailed state of art, one can refer \cite{chen,avenia,ts} and references therein.
On the other hand, Barrios et al. \cite{barrios} started the work on nonlocal equations with singular nonlinearity. Precisely, \cite{barrios} deal with the existence of solutions to the following problem
\begin{align}\label{si67}
(-\De)^s u = \la \left(\frac{a(x)}{u^{r}}+ f(x,u) \right) \text{ in } \Om,\quad u=0 \text{ in } \R\setminus \Om
\end{align}
where $\Om$ is a bounded domain with smooth boundary, $N>2s,\; 0<s<1,\; r,\la>0,\; f(x,u)\sim u^p,\; 1<p<2^*_s -1$. In the spirit of \cite{crandal}, here authors first prove the existence of solutions $u_n$ to the equation with singular term $1/u^r$ replaced by $1/(u+1/n)^{r}$ and use the uniform estimates on the sequence $\{u_n\}$ to finally prove the existence of a solution to \eqref{si67}.
Furthermore, authors prove some Sobolev regularity, in particular for $r>1$ that $u^{\frac{r+1}{2}} \in X_0$.
In case of $s=1$, optimal Sobolev regularity was established in \cite{bocardo} and \cite{gia} for semilinear and quasilinear elliptic type problems respectively. But in case of $0<s<1$, the question of optimal Sobolev regularity still remained an open question. The regularity issue is of independent interest. In the recent times, Adimurthi, Giacomoni and Santra \cite{adi} studied the problem \eqref{si67} with $f=0$ and complement the results of \cite{barrios}. In particular they obtained the boundary behaviour and H\"older regularity of the classical solution. Then exploiting this asymptotic behavior, authors obtained multiplicity of classical solutions by global bifurcation method in the framework of weighted spaces for \eqref{si67} with subcritical $f$.
Regarding the critical case, Giacomoni, Mukherjee and Sreenadh \cite{GMS} studied the problem \eqref{si67} with $a=1/\la, \; r>0,$ and $f(x,u)\sim u^{2^{*}_{s}-1}$. Here authors extended the techniques of \cite{hirano} in fractional framework and proved the existence and multiplicity of solutions in $C^\al_{\text{loc}}(\Om) \cap L^\infty(\Om)$ for some $\al>0$. Recently, authors \cite{gts1} proved the global multiplicity result for \eqref{si67} with $a=1/\la, p=2^*_s-1 $ and $r(2s-1)<(1+2s)$ for energy solutions.
Concerning the doubly nonlocal problem with singular operators, in \cite{newpaper}, we studied the regularity results for the problems of th type $(P_\la)$ with $0<q<1$.
But the questions of existence, multiplicity of solutions to the problem $(P_\la)$ was a completely open problem even when $s=1$. Also the question of (H\"older, Sobolev) regularity of solutions for $q\geq 1$ still remained as an open problem.
Motivated by the above discussion, in this article we answer the open problems stated above with an unified approach. More precisely, we consider a more general definition of weak solutions as compared to definition of \eqref{si67} in \cite{adi}. By establishing a new comparison principle (see Lemma \ref{lemsi17}) we prove that any very weak solution is actually a classical solution. This is a significant extension of the regularity results obtained in \cite{adi}. The question of optimal Sobolev regularity is also answered in our article (see Lemma \ref{lemsi16} and Theorem \ref{thmsi5}). To prove Lemma \ref{lemsi16}, we exploit suitably the boundary behavior of the weak solution of problem \eqref{si5} and the Hardy's inequality.
The crucial comparison principle in Lemma \ref{lemsi17} is obtained with a careful use of suitable testing functions to tackle the $H^s_{loc}$ phenomena. In case of $s=1$, this result was established in \cite{canino} with a slightly different approach. We first prove the $L^\infty(\Om)$ estimate for solutions of $(P_\la)$ by establishing the relation between the solutions of $(P_\la)$ and $(\widetilde{P_{\la}})$ (See Section 2).
The techniques used here can be applied in a more general context and are of independent interest. Next, using the results of \cite{adi} and Lemma \ref{lemsi17} we prove the asymptotic behavior and optimal H\"older and Sobolev regularity of weak solutions.
In this paper we have given a consolidated approach to prove the global multiplicity result for the problem $(P_\la)$ exploiting convex properties of the singular nonlinearity and the geometry of the energy functional.
To the best of our knowledge there is no previous contribution which deals the Choquard problem with singular nonlinearity. Further, the results proved in this article are new and novel even in case of $s=1$ where the approach can be closely adapted.
For simplicity of illustration, we set some notations. We denote $\|u\|_{L^p(\Om)} $ by $|u|_p$, $\|u\|_{X_0}$ by $\|u\|$, $[u]_{H^s(A)}= \int_{A}\int_{A}\frac{(u(x)-u(y))^2}{|x-y|^{N+2s}}~dxdy$. The positive constants $C,c_1,c_2\cdots $ values change case by case.
Turning to the paper organization: In Section 2, we define the function spaces, give some preliminaries of nonsmooth analysis and further state the main results of the article. In Section 3, we establish a very weak comparison principle. In Section 4, we established the regularity of solutions to $(P_\la)$. In sections 5 and 6 , we prove the existence of first and second solution to $(P_\la)$.
\section{Preliminary results and statement of main results}
We recall the Hardy-Littlewood-Sobolev Inequality which is foundational in study of Choquard problems of the type $(P_\la)$
\begin{Proposition} \cite{leib}
Let $t,r>1$ and $0<\mu <N$ with $1/t+\mu/N+1/r=2$, $f\in L^t(\mathbb{R}^N)$ and $h\in L^r(\mathbb{R}^N)$. There exists a sharp constant $C(t,r,\mu,N)$ independent of $f,h$, such that
\begin{equation*}\label{co9}
\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{f(x)h(y)}{|x-y|^{\mu}}~dydx \leq C(t,r,\mu,N) |f|_{t}|h|_{r}.
\end{equation*}
\end{Proposition}
Consider the space
\begin{align*}\label{si68}
X_0:= \{u \in H^{s}(\R): u=0 \text{ a.e in } \R \setminus \Om \},
\end{align*}
equipped with the inner product
\begin{align*}
\ld u,v \rd = \iint_{Q} \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+2s}}~dxdy.
\end{align*}
where $Q= \mathbb{R}^{2N} \setminus (\Om^c\times \Om^c)$. From the embedding results (\cite{servadei}), the space $X_0$ is continuously embedded into $L^r(\R)$ with $ r\in [1,2^*_s]$ where $2^*_s= \frac{2N}{N-2s}$. The embedding is compact if and only if $r < 2^{*}_{s}$.
The best constant $S$ of the classical Sobolev embedding is defined
\begin{align*}
S= \inf_{u \in X_0\setminus \{ 0\}} \frac{\int_{\R}\int_{\R} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}~dxdy }{\left( \int_{ \Om} |u|^{2^*_s}\right)^{2/2^*_s}}.
\end{align*}
Consequently, we define
\begin{align*}
S_{H,L}= \inf_{u \in X_0\setminus \{ 0\}} \frac{\int_{\R}\int_{\R} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}~dxdy }{\left( \int_{\R}\int_{\R} \frac{|u|^{2^*_\mu}(x)|u|^{2^*_\mu}(y)}{|x-y|^\mu} ~dxdy \right)^{1/2^*_\mu}}.
\end{align*}
\begin{Lemma}\label{lulem13}
\cite{ts}
The constant $S_{H,L}$ is achieved if and only if
\begin{align*}
u=C\left(\frac{b}{b^2+|x-a|^2}\right)^{\frac{N-2s}{2}}
\end{align*}
where $C>0$ is a fixed constant , $a\in \mathbb{R}^N$ and $b\in (0,\infty)$ are parameters. Moreover,
\begin{align*}
S= S_{H,L} \left(C(N,\mu)\right)^{\frac{N-2s}{2N -\mu}}.
\end{align*}
\end{Lemma}
\begin{Definition}
For $\phi \in C_0(\overline{\Om})$ with $\phi >0$ in $\Om$, the set $C_\phi(\Om)$ is defined as
\begin{align*}
C_\phi(\Om)= \{ u \in C_0(\overline{\Om})\; :\; \text{there exists } c \geq 0 \text{ such that } |u(x)|\leq c\phi(x), \text{ for all } x \in \Om \},
\end{align*}
endowed with the natural norm $\bigg\|\ds \frac{u}{\phi}\bigg\| _{L^{\infty}(\Om)}$.
\end{Definition}
\begin{Definition}
The positive cone of $C_\phi(\Om)$ is the open convex subset of $C_\phi(\Om)$ defined as
\begin{align*}
C_\phi^+(\Om)= \left\{ u \in C_\phi(\Om)\; :\; \inf_{x \in \Om} \frac{u(x)}{\phi(x)}>0 \right\}.
\end{align*}
\end{Definition}
The barrier function $\phi_q$ is defined as follows:
\begin{equation*}\label{si3}
\begin{aligned}
\phi_q= \left\{
\begin{array}{ll}
\phi_1 & \text{ if }0<q <1, \\
\phi_1\left(\log\left( \frac{A}{\phi_1}\right) \right)^{\frac{1}{2}} & \text{ if } q=1, \\
\phi_1^{\frac{2}{q+1}} & \text{ if } q>1,\\
\end{array}
\right.
\end{aligned}
\end{equation*}
where $\phi_1$ is the normalized ($\|\phi_1\|_{L^\infty(\Om)}=1$) eigenfunction corresponding to the smallest eigenvalue of $(-\De)^{s}$ on $X_0$ and $A> \text{diam}(\Om)$. We recall that $\phi_1 \in C^{s}(\R)$ and $ \phi_1
\in C_{d^{s}}^+(\Om)$ (See Proposition 1.1 and Theorem 1.2 of \cite{RS}).
Before giving the definition of weak solution to $(P_\la)$ we discuss the solution of
the following purely singular problem
\begin{equation}\label{si5}
(-\De)^s u
=u^{-q},\; u>0 \; \text{in}\;
\Om,
u=0 \; \text{ in } \R \setminus \Om.
\end{equation}
From \cite{adi} we know the following result.
\begin{Proposition}\label{propsi1}
Let $q>0$. Then there exists $ \overline{u} \in L^\infty(\Om) \cap C^+_{\phi_q} $ classical solution to \eqref{si5}. Moreover, $\overline{u}$ has the following properties:
\begin{enumerate}
\item [(i)] $\overline{u} \in X_0$ if and only if $q(2s-1) < (2s+1)$ and in this case we have unique classical solution to \eqref{si5}.
\item [(ii)] $\overline{u} \in C^\ga(\R) $ where
\begin{equation}\label{si4}
\begin{aligned}
\ga= \left\{
\begin{array}{ll}
s & \text{ if }q<1, \\
s-\e & \text{ if } q=1, \text{ for all } \e>0 \text{ small enough}, \\
\frac{1}{q+1} & \text{ if } q>1.\\
\end{array}
\right.
\end{aligned}
\end{equation}
\end{enumerate}
\end{Proposition}
\begin{Remark}\label{remsi1}
We remark that since $\overline{u} \in L^\infty(\Om) \cap C^+_{\phi_q} \cap C^\ga(\R)$. So we can achieve the interior $C^\infty$ regularity. That is for any compact set $\Om^\prime \subset \Om$ we have $\overline{u} \in C^\infty(\Om^\prime)$. From this one can easily prove the fact that $\overline{u} \in H^s_{\text{loc}}(\Om)$.
\end{Remark}
\begin{Lemma}\label{lemsi16}
(a) If $q(2s-1) \geq (2s+1)$ then $\overline{u}^\ga \in X_0$ if and only if $\ga>\frac{(2s-1)(q+1)}{4s}$. Moreover the lower bound on $\ga$ is optimal in the sense that $u^\ga \not \in X_0$ if $\ga\leq \frac{(2s-1)(q+1)}{4s}$. \\
(b) $(\overline{u}-\e)^+ \in X_0$ for all $\e>0$.
\end{Lemma}
\begin{proof}
(a) Let $\xi(x)= x^\ga,\; \ga >1$. Observe that $\xi$ is convex and differentiable function on $\mathbb R^+$. Hence using this and the fact that $\phi_ 1 \in C_{d^{s}}^+(\Om)$ and $\overline{u} \in C^+_{\phi_q}$, we deduce that
\begin{align*}
\|\xi(\overline{u})\|^2= \ld (-\De)^s \xi(\overline{u}), \xi(\overline{u})\rd & \leq \int_{\Om} \xi(\overline{u})\xi^\prime(\overline{u}) (-\De)^s \overline{u}~dx\\
& = \int_{\Om} \ga \overline{u}^{2\ga-q-1}~dx\\
& \leq C \int_{\Om} d^{\frac{2s(2\ga-q-1)}{q+1}}~dx.
\end{align*}
We know that $d^{\frac{2s(2\ga-q-1)}{q+1}} \in L^1(\Om)$ if and only if $\ga > \frac{(2s-1)(q+1)}{4s}$. This settles first part of the proof. For the second part, let $\ga\leq \frac{(2s-1)(q+1)}{4s}$ and if possible let $\overline{u}^\ga \in X_0$.
Consider
\begin{align*}
\int_{\Om} \frac{\overline{u}^{2\ga}}{d^{2s}}~dx \leq C\int_{\Om} d^{\frac{4s\ga}{q+1}-2s}~dx = \infty
\end{align*}
It contradicts the fact that $\overline{u}^\ga \in X_0$ and then satisfies the Hardy inequality.\\
(b) Let $A= \{x: \overline{u}(x)>\e \}$ then
\begin{align*}
\|(\overline{u}-\e)^+\|^2& \leq \int_{A}\int_{A}\frac{|\overline{u}(x)-\overline{u}(y)|^2}{|x-y|^{N+2s}}~dxdy +2 \int_{\R\setminus A} \int_{A}\frac{|\overline{u}(x)-\overline{u}(y)|^2}{|x-y|^{N+2s}}~dxdy\\
& \leq \kappa_1 \int_{A}\int_{A}\frac{|\overline{u}^\ga(x)-\overline{u}^\ga(y)|^2}{|x-y|^{N+2s}}~dxdy +2\kappa_2 \int_{\R\setminus A} \int_{A}\frac{|\overline{u}^\ga(x)-\overline{u}^\ga(y)|^2}{|x-y|^{N+2s}}~dxdy \\
& \leq (\kappa_1+\kappa_2) \int_{Q}\frac{|\overline{u}^\ga(x)-\overline{u}^\ga(y)|^2}{|x-y|^{N+2s}}~dxdy < \infty.
\end{align*}
By the mean value theorem and convexity arguments, one can easily prove the existence of $\kappa_1, \kappa_2$ such that $\kappa_1, \kappa_2 \geq \e^{\ga -1}$. Hence the proof is complete. \QED
\end{proof}
The energy functional associated to the probelm $(P_\la)$ is
\begin{align*}
I(u)= \frac12 \int_{Q} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}~dxdy - \frac{1}{1-q} \int_{\Om}|u|^{1-q}~dx - \frac{\la}{22^*_\mu} \iint_{\Om\times \Om}\frac{|u|^{2^*_\mu}|u|^{2^*_\mu}}{|x-y|^{\mu}}~dxdy.
\end{align*}
Though the functional $I$ is continuous on $X_0$ when $0 < q < 1$ but if $q \geq 1$, the functional $I$ is not even finite at all points of $X_0$.
Also $I$ it can be shown that $I$ is not G\^ateaux differentiable at all points of $X_0$.
The doubly nonlocal nature of the problem $(P_\la)$ and the lack of regularity of $I$ force to use to introduce a quite general definition of weak solution. The Lemma \ref{lemsi16} motivates the following definition of weak solution to the problem $(P_\la)$.
\begin{Definition}\label{defisi1}
A function $u \in H^s_{\text{loc}}(\Om)\cap L^{2^*_s}(\Om)$ is said to be a weak solution of $(P_\la)$ if the following hold:
\begin{enumerate}
\item [(i)] there exists $m_K>0$ such that $ u >m_K$ for any compact set $ K \subset \Om$.
\item [(ii)] For any $ \phi \in C_c^\infty(\Om)$,
\begin{align*}
\langle u, \phi \rangle= \int_{\Om} u^{-q}\phi ~dx+ \la\iint_{\Om\times \Om} \frac{u^{2^*_\mu}(x)u^{2^*_\mu-1}(y)\phi(y)}{|x-y|^\mu} ~dxdy.
\end{align*}
\item [(iii)] $(u-\e)^{+} \in X_0$ for all $\e>0$.
\end{enumerate}
\end{Definition}
\begin{Lemma}\label{lemsi20}
Let $u$ be a weak solution to $(P_\la)$ as it is defined in Definition \ref{defisi1}. Then for all compactly supported $0\leq v \in X_0\cap L^\infty(\Om)$, we have
\begin{align}\label{si83}
\ld u, v \rd - \int_{ \Om} u^{-q}v~dx -\iint_{\Om\times \Om}\frac{u^{2^*_\mu}u^{2^*_\mu-1}v}{|x-y|^{\mu}}~dxdy=0.
\end{align}
\end{Lemma}
\begin{proof}
Let $0\leq v \in X_0\cap L^\infty(\Om)$ be compactly supported function. Then there exists a sequence $v_n \in C_c^\infty(\Om)$ such that $v_n \geq 0, ~ K:= \cup_n \text{supp } v_n$ is contained in compact set of $\Om ,~\{ |v_n|_\infty\} $ is bounded sequence and $v_n \ra v$ strongly in $X_0$. Since $u$ is a weak solution to $(P_\la)$, we have
\begin{align}\label{si84}
\ld u, v_n \rd - \int_{ \Om} u^{-q}v_n~dx -\iint_{\Om\times \Om}\frac{u^{2^*_\mu}u^{2^*_\mu-1}v_n}{|x-y|^{\mu}}~dxdy=0.
\end{align}
Consider
\begin{align*}
\int_{Q} \frac{(u(x)-u(y))(v_n(x)-v_n(y))}{|x-y|^{N+2s}}~dxdy & = \int_{K}\int_{K} \frac{(u(x)-u(y))(v_n(x)-v_n(y))}{|x-y|^{N+2s}}~dxdy\\
& +2 \int_{\Om\setminus K}\int_{K} \frac{(u(x)-u(y))v_n(x)}{|x-y|^{N+2s}}~dxdy\\
& + 2 \int_{\R\setminus \Om}\int_{\Om} \frac{u(x)v_n(x)}{|x-y|^{N+2s}}~dxdy\\
&:= I+II+III.
\end{align*}
Now using the fact that $u \in H^s_{\text{loc}}(\Om)$ and the strong convergence of $v_n$, as $n \ra \infty$, we obtain
\begin{align*}
I\ra \int_{K}\int_{K} \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+2s}}~dxdy.
\end{align*}
Next taking into account $u \in L^{2^*_s}(\Om)$ and the fact that $ v_n $ is uniformly bounded in $L^\infty(\Om)$, by dominated convergence theorem, we deduce that
\begin{align*}
II\ra 2 \int_{\Om\setminus K}\int_{K} \frac{(u(x)-u(y))v(x)}{|x-y|^{N+2s}}~dxdy.
\end{align*}
Similarly, $III\ra 2 \int_{\R\setminus \Om}\int_{\Om} \frac{u(x)v(x)}{|x-y|^{N+2s}}~dxdy$. Hence $\ld u , v_n \rd \ra \ld u , v \rd $ as $n \ra \infty$. Trivially $\int_{ \Om} u^{-q}v_n~dx \ra \int_{ \Om} u^{-q}v~dx$ as $n \ra \infty$.
Also using the strong convergence of sequence $v_n$ and the fact that $u \in L^{2^*_s}(\Om)$, we infer that
\begin{align*}
\iint_{\Om\times \Om}\frac{u^{2^*_\mu}u^{2^*_\mu-1}v_n}{|x-y|^{\mu}}~dxdy\ra \iint_{\Om\times \Om}\frac{u^{2^*_\mu}u^{2^*_\mu-1}v}{|x-y|^{\mu}}~dxdy.
\end{align*}
It implies that passing the limit as $n \ra \infty$ in \eqref{si84}, we have \eqref{si83} for all compactly supported $0\leq v \in X_0\cap L^\infty(\Om)$. \QED
\end{proof}
In the direction of existence of solution to $(P_\la)$, we translate the problem $(P_\la)$ by the solution $\overline{u}$ of problem \eqref{si5}.
Consider the translated problem
\begin{equation*}
(\widetilde{P_\la})\;
\left\{\begin{array}{rllll}
(-\De)^s u +\overline{u}^{-q}-(u+\overline{u})^{-q}
&= \la \left(\ds \int_{\Om}\frac{(u+\overline{u})^{2^*_\mu}(y)}{|x-y|^{\mu}}dy\right) (u+\overline{u})^{2^*_\mu-1},\; u>0 \; \text{in}\;
\Om,\\
u&=0 \; \text{ in } \R \setminus \Om.
\end{array}
\right.
\end{equation*}
Observe that $u+\overline{u} $ is a solution to $(P_\la)$ if and only if $u\in X_0$ is a solution to $(\widetilde{P_\la})$. Define the function $g: \Om \times \mathbb{R}\ra \mathbb{R}\cap \{ \infty\}$ as
\begin{align*}
g(x,s)= \left\{
\begin{array}{ll}
\overline{u}^{-q}-(s+\overline{u})^{-q} & \text{ if } s+\overline{u}>0 , \\
-\infty & \text{ otherwise}
\end{array}
\right.
\end{align*}
and $G(x, y) = \int_{0}^y g(x,\tau)~d\tau$ for $(x,y ) \in \Om\times \mathbb{R}$.
\begin{Definition}
A function $u \in X$ is a subsolution (resp. supersolution) to $(\widetilde{P_\la})$ if the following holds
\begin{enumerate}
\item [(i)] $u^+ \in X_0$(resp. $u^- \in X_0$);
\item [(ii)] $g(\cdot, u) \in L^1_{\text{loc}}(\Om)$;
\item [(iii)] For all $\phi \in C_c^\infty(\Om),\; \phi \geq 0$, we have
\begin{align*}
\ld u, \phi \rd + \int_{ \Om} g(x,u)\phi~dx - \iint_{\Om\times \Om}\frac{(u+\overline{u})^{2^*_\mu}(u+\overline{u})^{2^*_\mu-1}\phi}{|x-y|^{\mu}}~dxdy\leq 0 \; (\text{resp.}\geq 0 ).
\end{align*}
\end{enumerate}
\end{Definition}
\begin{Definition}
A function $u \in X_0$ is a weak solution to $(\widetilde{P_\la})$ if it is both sub and supersolution and $
u \geq 0 $ in $\Om$.
\end{Definition}
\begin{Lemma}\label{lemsi18}
Let $u \in X_0$ be a weak solution to $(\widetilde{P_{\la}})$. Then for any $v \in X_0$, we have
\begin{align}\label{si81}
\ld u, v \rd + \int_{ \Om} g(x,u)v~dx - \iint_{\Om\times \Om}\frac{(u+\overline{u})^{2^*_\mu}(u+\overline{u})^{2^*_\mu-1}v}{|x-y|^{\mu}}~dxdy=0.
\end{align}
\end{Lemma}
\begin{proof}
Let $0\leq v \in X_0$ then by \cite[Lemma 3.1]{GMS}, there exists an increasing sequence $\{v_n \} \in X_0$ such that $v_n$ has a compact support and $v_n \ra v $ strongly in $X_0$. For each $n$, there exists a sequence $\phi_n^k \in C_c^\infty(\Om)$ such that $\phi_n^k \geq 0, ~ \cup_k \text{supp} \phi_n^k$ is contained in compact set of $\Om,~\{ |\phi_n^k|_\infty\} $ is bounded sequence and $\phi_n^k \ra v_n$ strongly in $X_0$. Since $u$ is a weak solution to $(\widetilde{P_{\la}})$, we have
\begin{align*}
\ld u, \phi_n^k \rd + \int_{ \Om} g(x,u)\phi_n^k~dx - \iint_{\Om\times \Om}\frac{(u+\overline{u})^{2^*_\mu}(u+\overline{u})^{2^*_\mu-1}\phi_n^k}{|x-y|^{\mu}}~dxdy=0.
\end{align*}
Using the fact that $\phi_n^k \ra v_n$ strongly in $X_0$ as $n \ra \infty$, we deduce that
\begin{align*}
\ld u, v_n \rd + \int_{ \Om} g(x,u)v_n~dx - \iint_{\Om\times \Om}\frac{(u+\overline{u})^{2^*_\mu}(u+\overline{u})^{2^*_\mu-1}v_n}{|x-y|^{\mu}}~dxdy=0.
\end{align*}
Now by using the dominated convergence theorem and the strong convergence of the sequence $v_n$ in $X_0$, we get $g(x,u)v \in L^1(\Om)$ and we have \eqref{si81} for any $0\leq v \in X_0$. For any $v \in X_0$, $v= v^+-v^-$. Employing the above procedure for $v^+$ and $v^-$ separately, we obtain the desired result. Hence the proof. \QED
\end{proof}
With this functional framework we record now the statement of our main Theorems.
\begin{Theorem}\label{thmsi1}
Let $\mu \leq \min\{ 4s,N\}$. There exists a $\La >0$ such that
\begin{enumerate}
\item For every $\lambda \in (0, \La)$ the problem $(P_\lambda)$ admits two solutions in $ C_{\phi_q}^+(\Om)\cap L^\infty(\Om)$.
\item For $\lambda = \La$ there exists a solution in $ C_{\phi_q}^+(\Om)\cap L^\infty(\Om)$.
\item For $\lambda > \La$, there exists no solution.
\end{enumerate}
Moreover, solution belongs to $ X_0$ if and only if $q(2s-1)<(2s+1)$.
\end{Theorem}
Concerning the H\"older and Sobolev regularity of solutions we have the following Theorem.
\begin{Theorem}\label{thmsi5}
Let $\mu \leq \min\{ 4s,N\}$. Let $q>0,\; \la \in (0,\La)$. Then
any weak solution in the sense of Definition \ref{defisi1} is classical and belongs to $C^\ga(\R)$ where $\ga$ is defined \eqref{si4}. Furthermore any weak solution satisfies the statements of Lemma \ref{lemsi16}.
\end{Theorem}
\begin{Remark}
We point out that regularity results contained in Theorem \ref{thmsi5} are much stronger
as compared to those obtained in \cite{adi, gts1} where regularity of continuous solutions are only investigated.
\end{Remark}
\subsection{Notions of nonsmooth Analysis}
In this subsection we record some basic definitions, observations and linking theorem to nonsmooth functionals. We remark that in case of $q(2s-1)<(2s+1)$, one can adapt the variational techniques of the \cite{hai,gts1} to prove the global multiplicity result as in Theorem \ref{thmsi1} but to incorporate the case of $q$ large we adopt the following notions of non-smooth analysis.
\begin{Definition}
Let $H$ be a Hilbert space and $J : H \ra (-\infty, \infty]$ be a proper (i.e. $J \not \equiv \infty$) lower semicontinuous
functional.
\begin{enumerate}
\item [(i)]Let $D(J) = \{u\in H\; : \; J(u) < \infty\}$ be the domain of $J$. For every $u \in D(J)$, we define the Fr\'echet sub-differential of $J$ at $u$ as the set
\begin{align*}
\pa^-J(u) = \left\{ z \in H \; :\; \varliminf_{v \ra u }\frac{ J(v)-J(u)- \ld z,v-u\rd}{\|v-u\|_{H}}\geq 0 \right\}.
\end{align*}
\item [(ii)] For each $u \in H$, we define
\begin{align*}
\vertiii{\pa^-J(u)} = \left\{
\begin{array}{ll}
\min \{ \|z\|_H\; :\; z \in \pa^- J(u) \} & \text{ if } \pa^- J(u) \not = \emptyset, \\
\infty & \text{ if } \text{ if } \pa^- J(u) = \emptyset. \\
\end{array}
\right.
\end{align*}
We know that $ \pa^-J(u)$ is a closed convex set which may be empty. If $u \in D(J)$ is a local minimizer for $J$, then it
can be seen that $0 \in \pa^-J(u)$.
\end{enumerate}
\end{Definition}
\begin{Remark}
We remark that if $J_0 : H \ra (-\infty, \infty]$ be a proper, lower semicontinuous, convex
functional, $J_1 : H \ra \mathbb{R}$ is a $C^1$ functional and $J= J_1 + J_0$, then
$\pa^{-}J(u) = \na J_1(u) + \pa J_0(u)$ for every $ u \in D(J) = D(J_0)$, where
$\pa J_0$ denotes the usual subdifferential of the convex functional $J_0$. Thus, $u$ is said to be a critical point of $J$ if
$ u \in D(J_0)$ and for every $ v \in H$, we have
$ \ld \na J_1(u), v-u\rd +J_0(v) -J_0(u)\geq 0$.
\end{Remark}
\begin{Definition}
For a proper, lower semicontinuous functional $J : H \ra (-\infty, \infty]$, we say that $J$ satisfies Cerami's
variant of the Palais-Smale condition at level $c$ (in short, $J$ satisfies $(\text{CPS})c$, if any sequence $\{z_n\} \subset D(J)$
such that $ J(z_n) \ra c $ and $(1 + z_n) \vertiii{\pa^-J(z_n)} \ra 0$ has a strongly convergent subsequence in $H$.
\end{Definition}
Analogous to the mountain pass theorem, we have the following linking theorem for non-smooth functionals.
\begin{Theorem}\label{thmsi4} \cite{sz}
Let $H$ be a Hilbert space. Assume $J= J_0+J_1$, where $J_0 : H \ra (-\infty, \infty]$ is
a proper, lower semicontinuous, convex functional and $J_1 : H \ra \mathbb{R}$ is a $C^1$-functional. Let $B^N , S^{N-1}$ denote
the closed unit ball and its boundary in $\R$ respectively. Let $\varphi : S^{N-1} \ra D(J)$ be a continuous function such
that
\begin{align*}
\Sigma = \{ \psi \in C(B^N, D(J))\; :\; \psi|_{S^{N-1}} = \varphi \} \not = \emptyset.
\end{align*}
Let $A$ be a relatively closed subset of $D(J)$ such that
\begin{align*}
A \cap \varphi(S^{N-1}) = \emptyset, \quad A \cap \varphi(B^N) \not = \emptyset \text{ for all } \psi \in \Sigma \text{ and } \quad \inf J(A)\geq \sup J(\varphi(S^{N-1})).
\end{align*}
Define $c = \inf_{\psi \in \Sigma} \sup_{x \in B^N} J(\psi(x))$.
Assume that $c$ is finite and that $J$ satisfies $(\text{CPS})c)$. Then there exists $u \in D(J)$ such that $J(u) = c$ and $ 0 \in \pa^-J(u)$. Furthermore, if $\inf J(A) = c$, then there exists $u \in A \cap D(J)$ such that $J(u) = c$ and $0 \in \pa^-J (u)$.
\end{Theorem}
\section{Very weak comparison principle}
Here we establish a new weak comparison principle that can be applied in the setting of $H^s_{loc}(\Om)$ sub and supersolutions to $(P_\lambda)$ and cover all $q>0$ (whereas \cite[Lemma 2.2]{gts1} $q(2s-1)<2s+1$ is required).
\begin{Lemma}\label{lemsi17}
Let $F \in X_0^*$ and let $z,w \in H^s_{\text{loc}}(\Om)$ be such that $z,w >0$ a.e in $\Om,~ z,w \geq 0 \in \R,~ z^{-q}, w^{-q} \in L^1_{\text{loc}} (\Om), ~ (z-\e)^+ \in X_0$ for all $\e>0,~ z \in L^1(\Om)$ and
\begin{align}\label{si71}
\ld z, \phi \rd \leq \int_{\Om} z^{-q} \phi ~dx + ( F,\phi ) , \quad \ld w,\phi \rd \geq \int_{\Om} w^{-q} \phi ~dx + ( F,\phi )
\end{align}
for all compactly supported $ \phi \in X_0 \cap L^\infty(\Om) $ with $\phi\geq 0 $. Then $z\leq w$ a.e in $\Om$.
\end{Lemma}
\begin{proof}
Let us denote that $\Psi_n : \mathbb{R} \ra \mathbb{R}$ the primitive of the function
\begin{align*}
s \mapsto
\left\{
\begin{array}{ll}
\max \{ -s^{-q}, -n\} & \text{ if } s>0 , \\
-n & \text{ if } s \leq 0
\end{array}
\right.
\end{align*}
such that $\Psi_n (1)=0$. Let us define a proper lower semicontinuous, strictly convex functional $\breve{H}_{0,n} : L^2(\Om) \ra \mathbb{R}$ given by
\begin{align*}
\breve{H}_{0,n}(u)=
\left\{
\begin{array}{ll}
\frac12 \|u\|^2 + \int_{\Om}\Psi_n(u)~dx & \text{ if } u \in X_0 , \\
\infty & \text{ if } u \in L^2(\Om) \setminus X_0.
\end{array}
\right.
\end{align*}
We define $H_{0,n}: L^2(\Om) \ra \mathbb{R}$ as
\begin{align*}
H_{0,n}(u) = \breve{H}_{0,n}(u) - \min \breve{H}_{0,n}= \breve{H}_{0,n}(u) - \breve{H}_{0,n}(u_{0,n})
\end{align*}
where $u_{0,n} \in X_0$ is the minimum of $ \breve{H}_{0,n}$. More generally, for $F \in X_0^*$ we set:
\begin{align*}
\breve{H}_{F,n}(u)=
\left\{
\begin{array}{ll}
\breve{H}_{0,n}(u) -( F, u- u_{0,n}) & \text{ if } u \in X_0 , \\
\infty & \text{ if } u \in L^2(\Om) \setminus X_0.
\end{array}
\right.
\end{align*}
Let $\e>0$ and $n > \e^{-q}$ and let $v$ be the minimum of the functional $\breve{H}_{F,n}$ on the convex set $K= \{ \varphi \in X_0: 0\leq \varphi \leq w \text{ a.e in } \Om \}$. Then for all $\varphi \in K$ we get
\begin{align}\label{si91}
\ld v, \varphi- v \rd \geq - \int_{\Om}\Psi_n^\prime(v) (\varphi - v )~dx + ( F, \varphi -v ).
\end{align}
Let $0\leq \varphi \in C_c^\infty(\Om), \; t>0$. Define $\varphi_t := \min \{ v+ t \varphi , w \} $. Now using the fact that $ w \in H^s_{\text{loc}} (\Om), \; v \in X_0,\; \varphi
\in C^\infty_c(\Om)$, we have $\varphi_t \in X_0$. Furthermore, $\varphi_t$ is uniformly bounded in $X_0$ for all $t<1$.
For the proof let $A= \text{supp}(\varphi)$. Since on $\Om \setminus A,~ \varphi_t = v$ and otherwise $v\leq \varphi_t \leq w$, we deduce that
\begin{equation}
\begin{aligned}\label{si89}
\int_{Q} \frac{(\varphi_t(x)-\varphi_t(y))^2}{|x-y|^{N+2s}}~dxdy& \leq [\varphi_t]_{H^s(A)} +\int_{ \Om \setminus A}\int_{\Om \setminus A}\frac{(v(x)- v(y))^2}{|x-y|^{N+2s}}~dxdy \\
&~+ 2 \int_{ \Om \setminus A}\int_{A}\frac{ (v(x)- v(y))^2+ 2t v(y)\varphi(y)+t^2\varphi^2(y)}{|x-y|^{N+2s}}~dxdy
\\
&\qquad + 2 \int_{ \R \setminus\Om}\int_{\Om}\frac{(v+t\varphi(y))^2}{|x-y|^{N+2s}}~dxdy <\infty.
\end{aligned}
\end{equation}
Employing the fact that for any $g:\R\ra \mathbb{R}$, $|g^+(x)-g^+(y)|^2\leq |g(x)-g(y)|^2$ for all $x,y \in \R$ coupled with $\varphi_t= v+t\varphi - (v+t\varphi-w)^+$, for all $t<1$, we conclude that
\begin{equation}
\begin{aligned}\label{si90}
[\varphi_t]_{H^s(A)} & \leq
2 \int_{ A}\int_{A}\frac{((v+t\varphi)(x)- (v+t\varphi)(y))^2}{|x-y|^{N+2s}}~dxdy\\
& \qquad + 2
\int_{ A}\int_{A}\frac{((v+t\varphi-w)(x)- (v+t\varphi-w))^2}{|x-y|^{N+2s}}~dxdy\\
& \quad \leq C\left([v]_{H^s(A)}+[\varphi]_{H^s(A)} + [w]_{H^s(A)} \right) .
\end{aligned}
\end{equation}
From \eqref{si89} and \eqref{si90} we obtain that $\varphi_t$ is uniformly bounded in $X_0$. Take the subsequence (still denoted by $\varphi_t$) such that $\varphi_t\rp v$ weakly in $X_0$ as $t\to 0^+$. Now test \eqref{si91} with $\varphi_t$, we get
\begin{align}\label{si72}
\ld v, \varphi_t- v \rd \geq - \int_{\Om}\Psi_n^\prime(v) (\varphi_t - v )~dx + ( F, \varphi_t -v ).
\end{align}
Using \eqref{si71} and the fact that $w^{-q}\geq -\Psi_n^\prime(w)$, we infer that $w$ satisfies
\begin{align}\label{si73}
\ld w,\phi \rd \geq - \int_{\Om}\Psi_n^\prime(w) \phi ~dx+ ( F,\phi )
\end{align}
Deploying the fact that $ \varphi_t \leq w$ coupled with $\varphi_t- v- t\varphi \leq 0 $ and if $\varphi_t = w$ then
$\varphi_t- v- t\varphi \not = 0$, we deduce that
\begin{equation}\label{si74}
\begin{aligned}
& \int_{Q} \frac{(\varphi_t(x)- \varphi_t(y))((\varphi_t-v-t\varphi ) (x)- (\varphi_t-v-t\varphi ) (y))}{|x-y|^{N+2s}}~dxdy\\
&\leq \int_{Q} \frac{w(x)(\varphi_t-v-t\varphi ) (x)}{|x-y|^{N+2s}}~dxdy + \int_{Q} \frac{w(y)(\varphi_t-v-t\varphi ) (y)}{|x-y|^{N+2s}}~dxdy \\
& \quad - \int_{Q} \frac{w(x)(\varphi_t-v-t\varphi ) (y)}{|x-y|^{N+2s}}~dxdy - \int_{Q} \frac{w(y)(\varphi_t-v-t\varphi ) (x)}{|x-y|^{N+2s}}~dxdy
= \ld w, \varphi_t-v-t\varphi \rd.
\end{aligned}
\end{equation}
Similarly, $\int_{\Om}(\Psi_n^\prime(\varphi_t)- \Psi_n^\prime(w)) (\varphi_t-v-t\varphi )~dx \leq 0$ and moreover $\Psi_n^\prime(w) \leq - w^{-q}$. Taking into account \eqref{si71}, \eqref{si72}, \eqref{si73}, \eqref{si74} and above observations, we deduce that
\begin{align*}
\|\varphi_t-v\|^2 & - \int_{\Om}(-\Psi_n^\prime(\varphi_t)+ \Psi_n^\prime(v)) (\varphi_t-v )~dx\\
& = \ld \varphi_t, \varphi_t-v \rd + \int_{\Om}\Psi_n^\prime(\varphi_t)(\varphi_t-v )~dx - \ld v , \varphi_t-v \rd - \int_{\Om} \Psi_n^\prime(v) (\varphi_t-v )~dx\\
& \leq \ld \varphi_t, \varphi_t-v \rd + \int_{\Om}\Psi_n^\prime(\varphi_t)(\varphi_t-v )~dx - ( F, \varphi_t -v )\\
& = \ld \varphi_t, \varphi_t-v -t\varphi \rd + \int_{\Om}\Psi_n^\prime(\varphi_t)(\varphi_t-v -t\varphi)~dx - ( F, \varphi_t -v-t\varphi )\\
& \hspace{4cm} + t\left( \ld \varphi_t, \varphi \rd + \int_{\Om}\Psi_n^\prime(\varphi_t)\varphi~dx - ( F, \varphi )\right)\\
& \leq \ld w, \varphi_t-v -t\varphi \rd + \int_{\Om}\Psi_n^\prime(w)(\varphi_t-v -t\varphi)~dx - ( F, \varphi_t -v-t\varphi )\\
& \hspace{4cm} + t\left( \ld \varphi_t, \varphi \rd + \int_{\Om}\Psi_n^\prime(\varphi_t)\varphi~dx - ( F, \varphi )\right)\\
& \leq t\left( \ld \varphi_t, \varphi \rd + \int_{\Om}\Psi_n^\prime(\varphi_t)\varphi~dx - ( F, \varphi )\right).
\end{align*}
Therefore, we obtain that
\begin{align*}
\ld \varphi_t, \varphi \rd + \int_{\Om}\Psi_n^\prime(\varphi_t)\varphi~dx- ( F, \varphi ) &\geq \frac1t\left(\|\varphi_t-v\|^2 - \int_{\Om}|\Psi_n^\prime(\varphi_t)- \Psi_n^\prime(v)| (\varphi_t-v )~dx \right)\\
&\geq -\int_{\Om}|\Psi_n^\prime(\varphi_t)- \Psi_n^\prime(v)| \varphi~dx .
\end{align*}
Now using the weak convergence of $\varphi_t$ and
monotone convergence theorem, and dominated convergence theorem, we obtain
\begin{align}\label{si75}
\ld v , \varphi \rd \geq - \int_{\Om}\Psi_n^\prime(v)\varphi~dx+ ( F, \varphi ) .
\end{align}
Using the density argument, one can easily show that \eqref{si75} is true for all $ \varphi \in X_0$ with $\varphi \geq 0$ a.e in $\Om$. Note that $v\geq 0$ implies $\text{supp}(z-\epsilon-v)^+ \subset \text{supp}(z-\epsilon)^+)$ that is, $(z-v-\e)^+ \in X_0$. So from \eqref{si75}, it implies that
\begin{align}\label{si79}
\ld v , (z-v-\e)^+ \rd \geq - \int_{\Om}\Psi_n^\prime(v)(z-v-\e)^+~dx+ ( F,(z-v-\e)^+ ) .
\end{align}
Let $(z-v-\e)^+:= \mathfrak{g} \in X_0$ such that $0\leq \mathfrak{g}\leq z$ a.e in $\Om$. Let $\{\hat{\mathfrak{g}}_m \}$ be a monotonically increasing sequence in $C_c^\infty(\Om)$ such that $\{\hat{\mathfrak{g}}_m\}$ converging to $\mathfrak{g}$ in $X_0$ and set $\mathfrak{g}_m = \min\{ \hat{\mathfrak{g}}_m^+, \mathfrak{g} \}$. Testing \eqref{si71} with $\mathfrak{g}_m$, we get
\begin{align}\label{si77}
\ld z,\mathfrak{g}_m\rd \leq \int_{ \Om}z^{-q} \mathfrak{g}_m ~dx +( F,\mathfrak{g}_m)
\end{align}
Observe that if $\mathfrak{g} >0$ then $z >\e$.
Now consider
\begin{equation}\label{si93}
\begin{aligned}
\int_{\{\mathfrak{g} >0\}} & \int_{\{\mathfrak{g} >0\}} \frac{(z(x)-z(y))((\mathfrak{g}_m-\mathfrak{g})(x)-(\mathfrak{g}_m-\mathfrak{g})(y))}{|x-y|^{N+2s}}~dxdy\\
& = \int_{\{\mathfrak{g} >0\}} \int_{\{\mathfrak{g} >0\}} \frac{((z-\e)^+(x)-(z-\e)^+(y))((\mathfrak{g}_m-\mathfrak{g})(x)-(\mathfrak{g}_m-\mathfrak{g})(y))}{|x-y|^{N+2s}}~dxdy\\
& \leq \|(z-\e)^+\|~ \|(\mathfrak{g}_m-\mathfrak{g})\|\ra 0 \text{ as } m \ra \infty.
\end{aligned}
\end{equation}
\begin{equation}\label{si94}
\begin{aligned}
\int_{\R\setminus\{\mathfrak{g}>0\}}& \int_{\{\mathfrak{g}>0\}}\frac{(z(x)-z(y))(\mathfrak{g}_m(x)-\mathfrak{g}_m(y))}{|x-y|^{N+2s}}dxdy\\
& =\int_{\R\setminus{\{\mathfrak{g}>0\}}}\int_{\{\mathfrak{g}>0\}}\frac{((z(x))(\mathfrak{g}_m(x))}{|x-y|^{N+2s}}dxdy -\int_{ \R\setminus\{\mathfrak{g}>0\}}\int_{\{\mathfrak{g}>0\}}\frac{(z(y))(\mathfrak{g}_m(x))}{|x-y|^{N+2s}}dxdy\\
& \geq\int_{ \R\setminus\{\mathfrak{g}>0\}}\int_{\{\mathfrak{g}>0\}}\frac{((z(x))(\mathfrak{g}_m(x))}{|x-y|^{N+2s}}dxdy -\int_{ \R\setminus\{\mathfrak{g}>0\}}\int_{\{\mathfrak{g}>0\}}\frac{(z(y))(\mathfrak{g}(x))}{|x-y|^{N+2s}}dxdy.
\end{aligned}
\end{equation}
Taking into account the fact that $z^{-q} \mathfrak{g}_m \leq z^{-q} \mathfrak{g} $, \eqref{si77}, \eqref{si93}, \eqref{si94} and monotone convergence theorem, if $z^{-q} \mathfrak{g} \in L^1(\Om)$ or $z^{-q} \mathfrak{g} \not \in L^1(\Om)$, we conclude that
\begin{align*}
\ld z,\mathfrak{g}\rd \leq \int_{ \Om}z^{-q} \mathfrak{g} ~dx +( F,\mathfrak{g}).
\end{align*}
That is,
\begin{align}\label{si80}
\ld z,(z-v-\e)^+\rd \leq \int_{ \Om}z^{-q} (z-v-\e)^+ ~dx +( F,(z-v-\e)^+)
\end{align}
Exploiting $n \geq \e^{-q}$, \eqref{si79}, \eqref{si80}, and the fact that for any measurable function $\mathfrak{h}$, $\ld \mathfrak{h}^+,\mathfrak{h}^+ \rd \leq \ld \mathfrak{h},\mathfrak{h}^+\rd$, we obtain that
\begin{align*}
\ld (z-v-\e)^+,(z-v-\e)^+\rd &\leq \ld z-v,(z-v-\e)^+\rd\\
& \leq \int_{ \Om}(z^{-q}+ \Psi_n^\prime(v)) (z-v-\e)^+ ~dx \\
& = \int_{ \Om}(-\Psi_n^\prime(z)+ \Psi_n^\prime(v)) (z-v-\e)^+ ~dx\leq 0.
\end{align*}
Thus, $z\leq v+\e\leq w+\e$. Since $\e$ was arbitrary chosen, hence proof follows. \QED
\end{proof}
\section{Regularity and Proof of Theorem \ref{thmsi5}}
In this section, we start by extending some regularity results contained in \cite{newpaper} and conclude the proof of Theorem \ref{thmsi5}.
\begin{Lemma}\label{lemsi15}
Any nonnegative solution to $(\widetilde{P_{\la}})$ belongs to $L^\infty(\Om)$.
\end{Lemma}
\begin{proof}
Let $u \in X_0$ be any non negative weak solution to \eqref{si15}. Let $u_\tau= \min\{ u,\tau \} $ for $ \tau>0$. Let $ \phi = u(u_\tau)^{r-2} \in X_0$ ($ r\geq 2$) be a test function to problem $(\widetilde{P_{\la}})$. Now from \cite[Lemma 3.5]{newpaper}, we have the following inequality
\begin{align}\label{si85}
\frac{4(r-1)}{r^2} \left( a|a_k|^{\frac{r}{2}-1} -b|b_k|^{\frac{r}{2}-1} \right)^2 \leq (a-b)(a_k|a_k|^{r-2}-b_k|b_k|^{r-2}).
\end{align}
where $a,b \in \mathbb{R}$ and $r \geq 2$. Using \eqref{si85}, we deduce that
\begin{equation}\label{si86}
\begin{aligned}
|u(u_\tau)^{\frac{r}{2}-1}|^2_{2^*_s}& \leq C \|u(u_\tau)^{\frac{r}{2}-1}\|^2 \leq \frac{Cr^2}{r-1} \int_{Q} \frac{(u(x)-u(y))(\phi(x)-\phi(y))}{|x-y|^{N+2s}}~dxdy\\
&= Cr \left(-g(x,u)u (u_\tau)^{r-2}~dx + \int_{ \Om}\int_{ \Om} \frac{(u+\overline{u})^{2^*_\mu}(u+\overline{u})^{2^*_\mu-1} u (u_\tau)^{r-2}}{|x-y|^{\mu}}~dxdy \right)\\
& \leq Cr \int_{ \Om}\int_{ \Om} \frac{(u+\overline{u})^{2^*_\mu}(u+\overline{u})^{2^*_\mu-1} u (u_\tau)^{r-2}}{|x-y|^{\mu}}~dxdy \\
& \leq Cr \left(\int_{ \Om}\int_{ \Om} \frac{u^{2^*_\mu}u^{2^*_\mu} (u_\tau)^{r-2}}{|x-y|^{\mu}}~dxdy + \int_{ \Om}\int_{ \Om} \frac{u^{2^*_\mu}\overline{u}^{2^*_\mu-1} u (u_\tau)^{r-2}}{|x-y|^{\mu}}~dxdy\right.\\
& \left. \qquad +\int_{ \Om}\int_{ \Om} \frac{\overline{u}^{2^*_\mu}u^{2^*_\mu} (u_\tau)^{r-2}}{|x-y|^{\mu}}~dxdy + \int_{ \Om}\int_{ \Om} \frac{\overline{u}^{2^*_\mu}\overline{u}^{2^*_\mu-1} u (u_\tau)^{r-2}}{|x-y|^{\mu}}~dxdy\right)\\
& \leq Cr \left(|u|_{2^*_s}^{2^*_\mu} \left(\int_{ \Om} (u^{2^*_\mu} u_\tau^{r-2})^{\frac{2^*_s}{2^*_\mu}} \right)^{\frac{2^*_\mu}{2^*_s}}+ |u|_{2^*_s}^{2^*_\mu} |\overline{u}|_\infty^{2^*_\mu-1}\left(\int_{ \Om} (u u_\tau^{r-2})^{\frac{2^*_s}{2^*_\mu}} \right)^{\frac{2^*_\mu}{2^*_s}} \right. \\
& \left. \quad + |\overline{u}|_{2^*_s}^{2^*_\mu} \left(\int_{ \Om} (u^{2^*_\mu} u_\tau^{r-2})^{\frac{2^*_s}{2^*_\mu}} \right)^{\frac{2^*_\mu}{2^*_s}} + |\overline{u}|_{2^*_s}^{2^*_\mu} |\overline{u}|_\infty^{2^*_\mu-1} \left(\int_{ \Om} ( uu_\tau^{r-2})^{\frac{2^*_s}{2^*_\mu}} \right)^{\frac{2^*_\mu}{2^*_s}} \right)
\end{aligned}
\end{equation}
\textbf{Claim:} Let $r_1= 2^*_s+1$. Then $ u \in L^{\frac{2^*_s r_1}{2}}(\Om)$.\\
In view of H\"older's inequality, we have
\begin{equation}\label{si87}
\begin{aligned}
\left(\int_{ \Om} (u^{2^*_\mu} u_\tau^{r-2})^{\frac{2^*_s}{2^*_\mu}} \right)^{\frac{2^*_\mu}{2^*_s}} & = \left(\int_{ u\leq R} (u^{2^*_\mu} u_\tau^{r_1-2})^{\frac{2^*_s}{2^*_\mu}} \right)^{\frac{2^*_\mu}{2^*_s}} +\left(\int_{ u>R} (|u|^{2^*_\mu} u_\tau^{r_1-2})^{\frac{2^*_s}{2^*_\mu}} \right)^{\frac{2^*_\mu}{2^*_s}} \\
& \leq R^{2^*_\mu} \left(\int_{ u\leq R} ( u_\tau^{r_1-2})^{\frac{2^*_s}{2^*_\mu}} \right)^{\frac{2^*_\mu}{2^*_s}} +\left(\int_{ u>R} (u u_\tau^{r_1-2})^{\frac{2^*_s}{2}} \right)^{\frac{2}{2^*_s}} \\
& \hspace{6cm}\left(\int_{ u>R} u^{2^*_s}\right)^{\frac{2^*_\mu-2}{2^*_s}}.
\end{aligned}
\end{equation}
Choose $R>0$ large enough such that
\begin{align}\label{si88}
\left( \int_{|u|>R} |u|^{2^*_s} ~dx\right)^{\frac{2^*_s-2}{2^*_s}} < \frac{1}{4Cr_1} \min\left\{ \frac{1}{|u|_{2^*_s}^{2^*_\mu}}, \frac{1}{|\overline{u}|_{2^*_s}^{2^*_\mu}}\right\}.
\end{align}
Taking into account \eqref{si86}, \eqref{si87} jointly with \eqref{si88}, we obtain
\begin{align*}
|u(u_\tau)^{\frac{r_1}{2}-1}|^2_{2^*_s}&
\leq Cr \left(R^{2^*_\mu}|u|_{2^*_s}^{2^*_\mu} \left(\int_{ u\leq R} ( u^{2^*_s-1})^{\frac{2^*_s}{2^*_\mu}} \right)^{\frac{2^*_\mu}{2^*_s}}+ |u|_{2^*_s}^{2^*_\mu} |\overline{u}|_\infty^{2^*_\mu-1}\left(\int_{ \Om} u^{2^*_s} \right)^{\frac{2^*_\mu}{2^*_s}} \right. \\
& \left. \qquad + |\overline{u}|_{2^*_s}^{2^*_\mu} R^{2^*_\mu} \left(\int_{ u\leq R} ( u^{2^*_s-1})^{\frac{2^*_s}{2^*_\mu}} \right)^{\frac{2^*_\mu}{2^*_s}} + |\overline{u}|_{2^*_s}^{2^*_\mu} |\overline{u}|_\infty^{2^*_\mu-1} \left(\int_{ \Om} u^{2^*_s} \right)^{\frac{2^*_\mu}{2^*_s}} \right).
\end{align*}
Appealing Fatou's Lemma as $\tau \ra \infty$, we obtain
\begin{align*}\label{ch41}
||u|^{\frac{r_1}{2}}|^2_{2^*_s}& \leq Cr_1\left(|u|_{2^*_s}^{2^*_\mu} + |\overline{u}|_{2^*_s}^{2^*_\mu}\right) \left(R^{2^*_\mu} \left(\int_{ u\leq R} ( u^{2^*_s-1})^{\frac{2^*_s}{2^*_\mu}} \right)^{\frac{2^*_\mu}{2^*_s}}+ |\overline{u}|_\infty^{2^*_\mu-1}\left(\int_{ \Om} u^{2^*_s} \right)^{\frac{2^*_\mu}{2^*_s}} \right) <\infty.
\end{align*}
This establishes the Claim. Now let $\tau \ra \infty$ in \eqref{si86} and using the inequality $x^p<1+x$ for $p<1$ and $x \geq 0$ we obtain
\begin{equation*}
\begin{aligned}
||u|^{\frac{r}{2}}|^2_{2^*_s}& \leq Cr \left(|u|_{2^*_s}^{2^*_\mu} + |\overline{u}|_{2^*_s}^{2^*_\mu}\right) \left(\left(\int_{ \Om} (u^{2^*_\mu+r-2})^{\frac{2^*_s}{2^*_\mu}} \right)^{\frac{2^*_\mu}{2^*_s}}+ |\overline{u}|_\infty^{2^*_\mu-1}\left(\int_{ \Om} (u^{r-1})^{\frac{2^*_s}{2^*_\mu}} \right)^{\frac{2^*_\mu}{2^*_s}} \right)\\
& \leq Cr \left(|u|_{2^*_s}^{2^*_\mu} + |\overline{u}|_{2^*_s}^{2^*_\mu}\right) \left(\left( 1+ \int_{ \Om} (u^{2^*_\mu+r-2})^{\frac{2^*_s}{2^*_\mu}} \right)+ |\overline{u}|_\infty^{2^*_\mu-1}\left(1+\int_{ \Om} (u^{r-1})^{\frac{2^*_s}{2^*_\mu}} \right) \right)\\
& \leq 2Cr(1+|\overline{u}|_\infty^{2^*_\mu-1}+ |\Om|)\left(|u|_{2^*_s}^{2^*_\mu} + |\overline{u}|_{2^*_s}^{2^*_\mu}\right) \left(1+ \int_{ \Om} (|u|^{r+2^*_\mu-2} )^{\frac{2^*_s}{2^*_\mu}}~dx \right).
\end{aligned}
\end{equation*}
It implies
\begin{align}\label{si2}
\left(1+ \int_{\Om } |u|^{ \frac{2^*_sr}{2} }~dx \right)^{\frac{2}{2^*(r-2)}} &
\leq C_{r}^{\frac{1}{(r-2)}} \left(1 + \int_{ \Om}(u^{2^*_\mu-2+r } ) ^{\frac{2^*_s}{2^*_\mu}}~dx\right)^{\frac{1}{(r-2)}}
\end{align}
where $C_r = 4C r(1+|u|_{2^*_s}^{2^*_\mu}+|\Om|)\left(|u|_{2^*_s}^{2^*_\mu} + |\overline{u}|_{2^*_s}^{2^*_\mu}\right)$. For $j\geq 1$ we define $r_{j+1}$ inductively as
\begin{align*}
(r_{j+1} + 2^*_\mu -2) \frac{2^*_s}{2^*_\mu}= \frac{2^*_sr_j}{2}.
\end{align*}
That is, $(r_{j+1}-2)= \left(\frac{2^*_\mu}{2}\right)^j (r_1-2)$. From \eqref{si2} with $ C_{r_{j+1}} = 4C r_{j+1}(1+|u|_{2^*_s}^{2^*_\mu}+|\Om|)$, it follows that
\begin{align*}
\left(1+ \int_{\Om }|u|^{ \frac{2^*_sr_{j+1}}{2} }~dx \right)^{\frac{2}{2^*_s(r_{j+1}-1)}} &
\leq C_{r_{j+1}}^{\frac{1}{(r_{j+1}-2)}} \left(1 + \int_{ \Om}(u^{2^*_\mu-2+r_{j} } ) ^{\frac{2^*_s}{2^*_\mu}}~dx\right)^{\frac{2}{2^*_s(r_{j}-2)}}.
\end{align*}
Defining $A_j:= \left(1 + \int_{ \Om}(u^{2^*_\mu-2+r_{j} } ) ^{\frac{2^*_s}{2^*_\mu}}~dx\right)^{\frac{2}{2^*_s(r_{j}-2)}}$. Then by Claim and limiting argument, there exists $C_0>0$ such that
\begin{align*}
A_{j+1}\leq \prod_{k=2}^{j+1} C_k^{(1/2(r_k-1))} A_1\leq C_0A_1.
\end{align*}
Hence $|u|_{\infty}\leq C_0A_1$. That is $ u \in L^\infty(\Om)$. \QED
\end{proof}
\begin{Remark}
We remark that if $u \in X_0$ be any weak solution of the following problem
\begin{equation}\label{si15}
(-\De)^s u
=f(x,u)+ \left(\ds \int_{\Om}\frac{|u|^{2^*_\mu}(y)}{|x-y|^{\mu}}dy\right) |u|^{2^*_\mu-1} \; \text{in}\;
\Om,
u=0 \; \text{ in } \R \setminus \Om,
\end{equation}
where $|f(x,u))|\leq C(1+ |u|^{2^*-1})$ and $\mu \leq \min\{ 4s,N\}$. Then by using the same assertions as in Lemma \ref{lemsi15}, we obtain that $ u \in L^\infty(\Om)$. This complements in the singular case previous results proved in \cite{newpaper}.
\end{Remark}
\begin{Lemma}\label{lemsi19}
Let $ z \in L^{2^*_s}(\Om)$ be a positive function, let $ h(x,z) = \left(\ds \int_{\Om}\frac{z^{2^*_\mu}(y)}{|x-y|^{\mu}}dy\right) z^{2^*_\mu-1}$. Assume $u \in X_0$ be a positive weak solution to
\begin{align}\label{si82}
(-\De)^s u +g(x,u)= h(x,z) \text{ in } \Om, \qquad u=0 \text{ in } \R\setminus \Om.
\end{align}
Then $(u+\overline{u} -\e)^+ \in X_0$ for every $\e>0$.
\end{Lemma}
\begin{proof}
Using the assertions and arguments used in \cite[Lemma 3.4]{GMS}, one can easily proof the result, we leave it for the readers. \QED
\end{proof}
\begin{Lemma}\label{lemsi21}
Let $\la>0$ and let $z \in H^s_{\text{loc}}(\Om)\cap L^{2^*_s}(\Om)$ be a weak solution to $(P_\la)$ as it is defined in definition \ref{defisi1}. Then $z- \overline{u}$ is a positive weak solution to $(\widetilde{P_{\la}})$ belonging to $L^\infty(\Om)$.
\end{Lemma}
\begin{proof}
Consider problem \eqref{si82} with $z$ given. Then $0$ is a strict subsolution to \eqref{si82}. Define the functional $I : X_0 \ra (-\infty, \infty]$ by
\begin{align*}
I(u)=
\left\{
\begin{array}{ll}
\frac12 \|u\|^2 + \int_{\Om}G(x,u)~dx - \frac{\la}{22^*_\mu} \iint_{\Om\times \Om}\frac{z^{2^*_\mu}z^{2^*_\mu-1}u }{|x-y|^{\mu}}~dxdy & \text{ if } G(\cdot, u ) \in L^1(\Om) , \\
\infty & \text{ otherwise. }
\end{array}
\right.
\end{align*}
Moreover for the closed convex set $K_0 = \{u \in X_0~:~ u \geq 0 \}$ we define $I_{K_0} : X_0 \ra (-\infty, \infty]$ by
\begin{align*}
I_{K_0}(u)=
\left\{
\begin{array}{ll}
I(u) & \text{ if } u \in K_0 \text{ and }G(\cdot, u ) \in L^1(\Om) , \\
\infty & \text{ otherwise. }
\end{array}
\right.
\end{align*}
we can easily prove that there exists $u \in K_0$ such that $I_{K_0}(u)= \inf I_{K_0}(K_0)$. It implies that $0 \in \pa^- I_{K_0} (u)$. Now from Proposition \ref{propsi2}, we obtain that $u$ is a non negative solution to \eqref{si82}. Using the Lemma \ref{lemsi19}, Lemma \ref{lemsi20} and assertions as in Lemma \ref{lemsi18}, we obtain that $(u+\overline{u}-\e)^+ \in X_0$ for every $\e>0$ and
\begin{align*}
& \ld u+\overline{u}, v \rd - \int_{ \Om} (u+\overline{u})^{-q}v~dx - \iint_{\Om\times \Om}\frac{z^{2^*_\mu}z^{2^*_\mu-1}v}{|x-y|^{\mu}}~dxdy=0\\
& \ld z, v \rd - \int_{ \Om} z^{-q}v~dx - \iint_{\Om\times \Om}\frac{z^{2^*_\mu}z^{2^*_\mu-1}v}{|x-y|^{\mu}}~dxdy=0
\end{align*}
for all compactly supported $0\leq v \in X_0 \cap L^\infty(\Om)$.
To prove the above equations for all compactly supported $0\leq v \in X_0 \cap L^\infty(\Om)$ one can use the fact that $u \in X_0,~ \overline{u} \in H^s_{\text{loc}}(\Om)$ (See Remark \ref{remsi1}) and the assertions as in Lemma \ref{lemsi20} and Lemma \ref{lemsi18}. Now using the Lemma \ref{lemsi17}, we get $z= u+\overline{u}$. That $u = z-\overline{u}$ is a solution to $(\widetilde{P_{\la}})$. And from Lemma \ref{lemsi15}, we have $ u \in L^\infty(\Om)$. \QED
\end{proof}
\begin{Lemma}\label{lemsi2}
Let $\mu \leq \min\{ 4s,N\}$. Let $u $ be any weak solution of problem $(P_\la)$. Then $ u \in L^\infty(\Om)\cap C^+_{\phi_q}(\Om)\cap C^\ga(\R) $ where $\ga$ is defined \eqref{si4}.
\end{Lemma}
\begin{proof}
Let $u$ be any weak solution of problem $(P_\la)$. Employing Lemma \ref{lemsi21}, $u-\overline{u} \in X_0$ is the solution to $(\widetilde{P_{\la}})$ and which on taking account Lemma \ref{lemsi5}, we have $u-\overline{u} \in L^\infty(\Om)$. Therefore, $u = (u-\overline{u})+ \overline{u} \in L^\infty(\Om)$.
Let $\hat{u}$ be a unique solution (See \cite[Theorem 1.2, Remark 1.5]{adi}) to the following problem
\begin{align*}
(-\De)^s \hat{u} = \hat{u}^{-q}+\la c, u>0 \text{ in } \Om, \hat{u} =0 \text{ in } \R\setminus \Om
\end{align*}
where $c= C^*|u|_\infty^{22^*_\mu-1} $ with $C^*= \bigg|\ds \int_{\Om}\frac{dy}{|x-y|^{\mu}}\bigg|_\infty$. Practising Lemma \ref{lemsi17}, one can easily show that $\overline{u}\leq u \leq \hat{u}$ a.e in $\Om$.
Now using the fact that $\overline{u}\leq u \leq \hat{u}$ a.e in $\Om$ and regularity of $\overline{u}$ and $\hat{u}$ we obtain $ u \in C^+_{\phi_q}(\Om)$. Observe that $u$ is a classical solution in sense of \cite[Definition 1]{adi} so by \cite[Theorem 1.2]{adi}, H\"older's regularity follows.
\QED
\end{proof}
{\bf Proof of Theorem \ref{thmsi5}}: It follows from the proof of Lemma \ref{lemsi2} and of Lemma \ref{lemsi16}.\QED
\section{Existence of first solution}
In this section, we have prove the existence of first solution and further establish that the first solution is actually a local minimizer of an appropriate functional. We start the section by defining the functional associated with $(\widetilde{P_{\la}})$.
Consider the functional $\mc J : X_0 \ra (-\infty, \infty]$ associated with
\begin{align*}
\mc J(u)=
\left\{
\begin{array}{ll}
\frac12 \|u\|^2 + \int_{\Om}G(x,u)~dx - \frac{\la}{22^*_\mu} \iint_{\Om\times \Om}\frac{|u|^{2^*_\mu}|u|^{2^*_\mu}}{|x-y|^{\mu}}~dxdy & \text{ if } G(\cdot, u ) \in L^1(\Om) , \\
\infty & \text{ otherwise. }
\end{array}
\right.
\end{align*}
For any convex subset $K \subset X_0$ we define the functional $\mc J_K : X_0 \ra (-\infty, \infty]$ by
\begin{align*}
\mc J_K(u)=
\left\{
\begin{array}{ll}
\mc J(u) & \text{ if } u \in K \text{ and }G(\cdot, u ) \in L^1(\Om) , \\
\infty & \text{ otherwise. }
\end{array}
\right.
\end{align*}
Define $\La:= \sup\{ \la>0\; : \; (P_\la) \text{ has a weak solution} \} $.
\begin{Lemma}\label{lemsi14}
Let $K$ be a convex subset of $X_0$ and let $w \in X_0$. Let $ u \in K $ with $G(\cdot, u) \in L^1(\Om)$. Then the following assertions are equivalent:
\begin{enumerate}
\item [(i)] $ \al \in \pa ^- \mc J_K (u)$.
\item [(ii)] For every $ w \in K$ with $G(\cdot , w) \in L^1(\Om)$, we have $g(\cdot,u )(w-u) \in L^1(\Om)$ and
\begin{align*}
\ld \al , w- u \rd & \leq \ld u, (w-u)\rd + \int_{\Om} g(x,u)(w-u)~dx\\
& -\la \iint_{\Om\times \Om}\frac{(u+\overline{u})^{2^*_\mu}(u+\overline{u})^{2^*_\mu-1}(w-u)}{|x-y|^{\mu}}~dxdy.
\end{align*}
\end{enumerate}
\end{Lemma}
\begin{proof}
(i) implies (ii). Let $ w \in K $ and $G(\cdot, w ) \in L^1(\Om) $. Define $z= w-u$. Then clearly since $g(x,u)$ is increasing in $u$, we have $g(x,u)z \leq G(x,w)- G(x,u)$. Moreover, $(g(\cdot,u)z)\vee 0 \in L^1(\Om)$ and $t \mapsto (G(x,u+tz)- G(x,u))/t, (0,1] \ra \mathbb{R}$, is increasing and
\begin{align*}
\frac{\mc J_K(u+tz)- \mc J_K(u)}{t} & = \ld u,w\rd +\frac{t\|z\|^2}{2} + \int_{ \Om}\frac{(G(x,u+tz)- G(x,u))}{t}\\
& \quad - \frac{1}{22^*_\mu t } \iint_{\Om\times \Om}\frac{(u+\overline{u}+tz)^{2^*_\mu}(u+\overline{u}+tz)^{2^*_\mu}}{|x-y|^{\mu}}~dxdy \\
& \quad + \frac{1}{22^*_\mu t } \iint_{\Om\times \Om}\frac{(u+\overline{u})^{2^*_\mu}(u+\overline{u})^{2^*_\mu}}{|x-y|^{\mu}}~dxdy.
\end{align*}
Passing to the limit as $t \ra 0$ and using the fact that $\al \in \pa^-\mc J_K(u)$, we deduce the required result.
(ii) implies (i). Let $z \in K$ and $G(\cdot, w ) \in L^1(\Om)$. Employing the fact that $G(x,s)$ is convex is $s$ and using (ii) we have that
\begin{align*}
\mc J_K(w)- \mc J_K(u)& = \frac{1}{2} \|z\|^2 +\int_{ \Om}(G(x,w)-G(x,u)- g(x,u)z)~dx + \ld \al, z\rd \\
& \quad - \frac{\la}{22^*_\mu} \iint_{\Om\times \Om}\frac{\left( (w+\overline{u})^{2^*_\mu}(w+\overline{u})^{2^*_\mu} - (u+\overline{u})^{2^*_\mu}(u+\overline{u})^{2^*_\mu}\right) }{|x-y|^{\mu}}~dxdy\\
& \quad + \la \iint_{\Om\times \Om}\frac{(u+\overline{u})^{2^*_\mu}(u+\overline{u})^{2^*_\mu-1}z}{|x-y|^{\mu}}~dxdy.
\end{align*}
It implies that $ \al \in \pa^-\mc J_K (u)$. \QED
\end{proof}
For any functions $\varphi, \psi : \Om \ra [-\infty,+\infty]$, we define the following subspaces
\begin{align*}
K_\varphi= \{ u \in X_0\; : \varphi \leq u \text{ a.e} \}, K^\psi= \{ u \in X_0\; : u \leq \psi \text{ a.e} \}, K_\varphi^\psi= \{ u \in X_0\; : \varphi \leq u \leq \psi\text{ a.e} \}.
\end{align*}
\begin{Proposition}\label{propsi2}
Assume one the following condition holds:
\begin{enumerate}
\item [(i)] $\phi_1$ is a subsolution to $(\widetilde{P_\la}), \; G(x,w(x)) \in L^1_{\text{loc}}(\Om)$ for all $ w \in K_{\phi_1}, u \in D(\mc J_{K_{\phi_1}})$ and $0 \in \pa^-\mc J_{K_{\phi_1}}(u)$.
\item [(ii)] $\phi_2$ is a supersolution to $(\widetilde{P_\la}), \; G(x,w(x)) \in L^1_{\text{loc}}(\Om)$ for all $ w \in K^{\phi_2}, u \in D(\mc J_{K^{\phi_2}})$ and $0 \in \pa^-\mc J_{K^{\phi_1}}(u)$.
\item [(iii)] $\phi_1, \phi_2$ are subsolution and supersolution to $(\widetilde{P_\la}),\; \phi_1\leq \phi_2, \; G(x, \phi_1), G(x, \phi_2) \in L^1_{\text{loc}}(\Om)$, $ u \in D(\mc J_{K_{\phi_1}^{\phi_2}})$ and $0 \in \pa^-\mc J_{K_{\phi_1}^{\phi_2}}(u)$.
\end{enumerate}
Then $u$ is weak solution to $(\widetilde{P_\la})$.
\end{Proposition}
\begin{proof}
Follow the \cite[Proposition 4.2]{GMS}, we have the required result. \QED
\end{proof}
Let $\vartheta \in C^s(\R)\cap X_0$ be the unique solution which satisfies $(-\De)^s\vartheta = 1/2$ in $\Om$ in the sense of distributions. By the definition of $g$ and $G$, we obtain the following properties
\begin{Lemma}\label{lemsi10}
\begin{enumerate}
\item[(i)]Let $ u \in L^1_{\text{loc}}(\Om)$ such that $ \ds \text{ess inf}_K
u >0 $ for any compact set $K \subset \Om$. Then $g(x,u(x)), G(x,u(x)) \in L^1_{\text{loc}}(\Om)$.
\item [(ii)] For all $x \in \Om$, the following holds
\begin{itemize}
\item [(a)] $G(x,st) \leq s^2G(x,t)$ for each $s \geq 1$ and $t\geq 0$.
\item [(b)] $G(x,s)-G(x,t) -(g(x,s)+g(x,t))(s-t)/2\geq 0 $ for each $s,t $ with $s\geq t > -\vartheta(x)$.
\item [(c)] $G(x,s)- g(x,s)s/2\geq 0 $ for each $s \geq 0 $.
\end{itemize}
\end{enumerate}
\end{Lemma}
\begin{Lemma}\label{lemsi12}
The following hold:
\begin{enumerate}
\item [(i)] $0$ is the strict subsolution to $(\widetilde{P_\la})$ for all $\la>0$.
\item [(ii)] $\vartheta$ is a strict supersolution to $(\widetilde{P_\la})$ for all sufficiently small $\la>0$.
\item [(iii)] Any positive weak solution $w$ to $(\widetilde{P_{\la_2}})$ is a strict supersolution to $(\widetilde{P_{\la_1}})$ for $0<\la_1<\la_2$.
\end{enumerate}
\end{Lemma}
\begin{proof}
(i) Trivial.\\
(ii) Choose $\la$ small enough such that $\la\left(\ds\int_{\Om}\frac{(\vartheta+\overline{u})^{2^*_\mu}}{|x-y|^{\mu}}~dy\right)(\vartheta+\overline{u})^{2^*_\mu-1}<1$ in $\Om$. From Lemma \ref{lemsi10}, $g(x,\vartheta), \; G(x,\vartheta) \in L^1_{\text{loc}}(\Om)$, for all $\psi \in X_0 \setminus \{0\}$, we deduce that
\begin{align*}
& \ld \vartheta, \psi \rd + \int_{ \Om} g(x,\vartheta)\psi ~dx -\la \int_{\Om}\int_{\Om}\frac{(\vartheta+\overline{u})^{2^*_\mu}(\vartheta+\overline{u})^{2^*_\mu-1}\psi}{|x-y|^{\mu}}~dxdy\\
& \geq \int\left(1- \la \left(\ds\int_{\Om}\frac{(\vartheta+\overline{u})^{2^*_\mu}}{|x-y|^{\mu}}~dy\right)(\vartheta+\overline{u})^{2^*_\mu-1}\right) \psi~dx >0.
\end{align*}
(iii) Let $0<\la_1<\la_2$ and $w$ be a positive weak solution to $(\widetilde{P_{\la_2}})$. Then for all $\psi \in X_0 \setminus \{0\}$, we have
\begin{align*}
& \ld w, \psi \rd + \int_{ \Om} g(x,w)\psi ~dx -\la_1 \int_{\Om}\int_{\Om}\frac{(w+\overline{u})^{2^*_\mu}(w+\overline{u})^{2^*_\mu-1}\psi}{|x-y|^{\mu}}~dxdy\\
& = (\la_2-\la_1) \int_{\Om}\int_{\Om}\frac{(w+\overline{u})^{2^*_\mu}(w+\overline{u})^{2^*_\mu-1}\psi}{|x-y|^{\mu}}~dxdy>0.
\end{align*}
The proof is now complete. \QED
\end{proof}
\begin{Theorem}\label{thmsi2}
Let $ w_1, w_2 :\Om \ra [-\infty, + \infty]$ with $w_1 \leq w_2$ such that $w_1 $ is a strict subsolution to $(\widetilde{P_\la})$ and $u \in D(\mc J_{K_{w_1}^{w_2}})$ be a minimizer for $\mc J_{K_{w_1}^{w_2}}$. Then $u$ is a local minimizer for $\mc J_{K_{w_1}}$.
\end{Theorem}
\begin{proof}
For each $v \in K_{w_1}$ and $0\leq \phi \in X_0$, we define $\sigma(v)= \min \{ v,w_2\}= v- (v-w_2)^+$ and
\begin{align*}
\varXi(\phi)= \ld w_2, \phi\rd + \int_{ \Om} g(x,w_2)\phi ~dx -\la \int_{\Om}\int_{\Om}\frac{(w_2+\overline{u})^{2^*_\mu}(w_2+\overline{u})^{2^*_\mu-1}\phi}{|x-y|^{\mu}}~dxdy.
\end{align*}
\textbf{Claim}: $\ld \sigma(v), v-\sigma(v)\rd\geq \ld w_2, v-\sigma(v)\rd$ and
\begin{align*}
\int_{\Om}\int_{\Om}\frac{\left( (\sigma(v)+\overline{u})^{2^*_\mu}(\sigma(v)+\overline{u})^{2^*_\mu-1} - (w_2+\overline{u})^{2^*_\mu}(w_2+\overline{u})^{2^*_\mu-1}\right) (v-\sigma(v))}{|x-y|^{\mu}}~dxdy \leq 0 .
\end{align*}
Notice that $v-\sigma(v)= (v-w_2)^+$. Let $\Om_1= \text{supp}((v-w_2)^+)$. Then on $\Om_1, \sigma(v)= w_2$ and using the fact that $\sigma(v) \leq w_2$ on $\Om$, we have
\begin{align*}
\ld \sigma(v), v-\sigma(v)\rd & = \left(
\int_{\Om_1}\int_{\Om_1} + 2\int_{\R\setminus \Om_1}\int_{\Om_1} + \int_{\Om\setminus \Om_1}\int_{\Om \setminus \Om_1} + 2\int_{\R\setminus \Om}\int_{\Om\setminus \Om_1} \right.\\
& \left. \quad \frac{(\sigma(v)(x)-\sigma(v)(y))((v-\sigma(v))(x)-(v-\sigma(v))(y))}{|x-y|^{N+2s}}~dxdy\right)
\\& \geq \ld w_2, v-\sigma(v)\rd .
\end{align*}
Second holds by using the fact that $\sigma(v) \leq w_2$ on $\Om$. It implies that the Claim holds.
Taking into account the fact that $u$ is a minimizer of for $\mc J_{K_{w_1}^{w_2}},\; \sigma(v) \in D(\mc J_{K_{w_1}^{w_2}})$, Lemma 2 of \cite{hirano} and the fact that $G(x,\cdot )$ is convex, we infer that
\begin{equation}\label{si41}
\begin{aligned}
& \mc J_{K_{w_1}}(v)- \mc J_{K_{w_1}}(u) \geq \mc J_{K_{w_1}}(v)- \mc J_{K_{w_1}}(\sigma(v))\\
& = \frac{\|v-\sigma(v)\|^2}{2} +\ld \sigma(v), v-\sigma(v)\rd + \int_{ \Om}(G(x,v)-G(x,\sigma(v)))~dx \\
& \qquad -\frac{\la}{22^*_\mu} \int_{\Om}\int_{\Om}\frac{ \left( (v+\overline{u})^{2^*_\mu}(v+\overline{u})^{2^*_\mu} - (\sigma(v)+\overline{u})^{2^*_\mu}(\sigma(v)+\overline{u})^{2^*_\mu}\right)}{|x-y|^{\mu}}~dxdy\\
& \geq \frac{\|v-\sigma(v)\|^2}{2} +\ld \sigma(v), v-\sigma(v)\rd + \int_{ \Om}g(x,\sigma(v))(v-\sigma(v))~dx \\
& \qquad -\frac{\la}{22^*_\mu} \int_{\Om}\int_{\Om}\frac{ \left( (v+\overline{u})^{2^*_\mu}(v+\overline{u})^{2^*_\mu} - (\sigma(v)+\overline{u})^{2^*_\mu}(\sigma(v)+\overline{u})^{2^*_\mu}\right)}{|x-y|^{\mu}}~dxdy\\
& \geq \frac{\|v-\sigma(v)\|^2}{2} +\ld w_2, v-\sigma(v)\rd + \int_{ \Om}g(x,w_2)(v-\sigma(v))~dx \\
& \qquad -\frac{\la}{22^*_\mu} \int_{\Om}\int_{\Om}\frac{ \left( (v+\overline{u})^{2^*_\mu}(v+\overline{u})^{2^*_\mu} - (\sigma(v)+\overline{u})^{2^*_\mu}(\sigma(v)+\overline{u})^{2^*_\mu}\right)}{|x-y|^{\mu}}~dxdy\\
&
\geq \frac{\|v-\sigma(v)\|^2}{2} + \varXi(v-\sigma(v)) -\frac{\la}{22^*_\mu} I
\end{aligned}
\end{equation}
where
\begin{align*}
I=& \int_{\Om}\int_{\Om}\frac{(v+\overline{u})^{2^*_\mu}(v+\overline{u})^{2^*_\mu}}{|x-y|^{\mu}}~dxdy - \int_{\Om}\int_{\Om}\frac{(\sigma(v)+\overline{u})^{2^*_\mu}(\sigma(v)+\overline{u})^{2^*_\mu}}{|x-y|^{\mu}}~dxdy \\ & \quad - 22^*_\mu \int_{\Om}\int_{\Om}\frac{(\sigma(v)+\overline{u})^{2^*_\mu}(\sigma(v)+\overline{u})^{2^*_\mu-1}(v-\sigma(v))}{|x-y|^{\mu}}~dxdy.
\end{align*}
Now we estimate $I$ from above. First observe that
\begin{equation}
\begin{aligned}\label{si42}
I& = 2^*_\mu \int_{\Om} \int_{\sigma(v)}^{v} \left( \int_{\Om}\frac{(v+\overline{u})^{2^*_\mu} + (\sigma(v)+\overline{u})^{2^*_\mu} }{|x-y|^{\mu}}~dy\right) \left( (t+\overline{u})^{2^*_\mu-1}- (\sigma(v)+\overline{u})^{2^*_\mu-1}\right)~dtdx\\
& \quad+ 2^*_\mu \int_{\Om} \int_{\sigma(v)}^{v} \left( \int_{\Om}\frac{(v+\overline{u})^{2^*_\mu} - (\sigma(v)+\overline{u})^{2^*_\mu} }{|x-y|^{\mu}}~dy\right) (\sigma(v)+\overline{u})^{2^*_\mu-1} ~dtdx.
\end{aligned}
\end{equation}
Using the mean value theorem, there exists $\theta\in [0,1]$ such that
\begin{align*}
\frac{(u+\overline{u})^{2^*_\mu-1}- (v+\overline{u})^{2^*_\mu-1}}{(u-v)}& = (2^*_\mu-1) (u+\overline{u} + \theta(v-u))^{2^*_\mu-2}(u-v) \\
&= (2^*_\mu-1) (\overline{u} + (1-\theta)u + \theta v)^{2^*_\mu-2}(u-v)\\
& \leq (2^*_\mu-1)2^{2^*_\mu-3} (\overline{u}^{2^*_\mu-2} +( (1-\theta)u + \theta v)^{^{2^*_\mu-2}})(u-v)\\
& \leq (2^*_\mu-1)2^{2^*_\mu-3} (\overline{u}^{2^*_\mu-2} +\max\{u,v \}^{^{2^*_\mu-2}})(u-v).
\end{align*}
For each $x \in \Om$ and $v \in D(\mc J_{K_{w_2}})$ define the functions
\begin{align*}
& k_v^1(x) = (2^*_\mu-1)2^{2^*_\mu-3} (\overline{u}^{2^*_\mu-2} +\max\{|w_2|,|v| \}^{^{2^*_\mu-2}})\chi_{\{v>w_2\}}, \\
& k_v^2(x) = 2^*_\mu2^{2^*_\mu-2} (\overline{u}^{2^*_\mu-1} +\max\{|w_2|,|v| \}^{^{2^*_\mu-1}}) \chi_{\{v>w_2\}}.
\end{align*}
Using the Hardy-Littlewood-Sobolev inequality, we have
\begin{equation}\label{si43}
\begin{aligned}
&\int_{\Om} \int_{\sigma(v)}^{v} \left( \int_{\Om}\frac{(v+\overline{u})^{2^*_\mu} + (\sigma(v)+\overline{u})^{2^*_\mu} }{|x-y|^{\mu}}~dy\right) \left( (t+\overline{u})^{2^*_\mu-1}- (\sigma(v)+\overline{u})^{2^*_\mu-1}\right)~dtdx\\
& \leq \frac12 \int_{\Om} \int_{\Om}\frac{((v+\overline{u})^{2^*_\mu} + (\sigma(v)+\overline{u})^{2^*_\mu}) k_v^1(x) (v-\sigma(v))^2 }{|x-y|^{\mu}}~dy dx\\
& \leq c_1 \left( | v+\overline{u}|_{2^*_s}^{2^*_\mu}+ | \sigma(v)+\overline{u}|_{2^*_s}^{2^*_\mu}\right)| k_v^1(x) (v-\sigma(v))^2|_{\frac{2^*_s}{2^*_\mu}}
\end{aligned}
\end{equation}
for some appropriate positive constant $c_1$. Similarly with the help of Hardy-Littlewood-Sobolev inequality, H\"older's inequality and the definition of $S$ we have
\begin{equation}\label{si44}
\begin{aligned}
& \int_{\Om} \int_{\sigma(v)}^{v} \left( \int_{\Om}\frac{(v+\overline{u})^{2^*_\mu} - (\sigma(v)+\overline{u})^{2^*_\mu} }{|x-y|^{\mu}}~dy\right) (\sigma(v)+\overline{u})^{2^*_\mu-1} ~dtdx \\
& \leq c_2 S^{-1/2} | k_v^2(x) (v-\sigma(v))|_{\frac{2^*_s}{2^*_\mu}} | \sigma(v)+\overline{u}|_{2^*_s}^{2^*_\mu-1} \|v-\sigma(v)\|
\end{aligned}
\end{equation}
for some appropriate positive constant $c_1$. Using \eqref{si42} jointly with \eqref{si43} and \eqref{si44}, we have
\begin{equation}\label{si45}
\begin{aligned}
I \leq & c_1 \left( | v+\overline{u}|_{2^*_s}^{2^*_\mu}+ | \sigma(v)+\overline{u}|_{2^*_s}^{2^*_\mu}\right)| k_v^1(x) (v-\sigma(v))^2|_{\frac{2^*_s}{2^*_\mu}}\\
& \quad +c_2 S^{-1/2} | k_v^2(x) (v-\sigma(v))|_{\frac{2^*_s}{2^*_\mu}} | \sigma(v)+\overline{u}|_{2^*_s}^{2^*_\mu-1} \|v-\sigma(v)\|.
\end{aligned}
\end{equation}
Let us suppose that the result is not true. This means that there exists a sequence $\{ v_n \} \subset X_0$ such that for any $v_n \in K_{w_1}$ and
\begin{align*}
\|v_n- u\| < \frac{1}{2^n},\; \mc J_{K_{w_1}} (v_n) < \mc J_{K_{w_1}} (u) \text{ for all } n.
\end{align*}
Define $ l:= u + \sum_{n=1}^{\infty} |v_n - u|$. By definition, $|v_n|\leq l $ a.e for all $n$. Now for each $v \in D(\mc J_{K_{w_1}})$, set
\begin{align*}
& \underline{k_v^1}(x) = (2^*_\mu-1)2^{2^*_\mu-3} (\overline{u}^{2^*_\mu-2} +\max\{|w_2|,|l| \}^{^{2^*_\mu-2}}) \chi_{\{v>w_2\}}\\
&\underline{k_v^2}(x) = 2^*_\mu2^{2^*_\mu-2} (\overline{u}^{2^*_\mu-1} +\max\{|w_2|,|l| \}^{^{2^*_\mu-1}}) \chi_{\{v>w_2\}}.
\end{align*}
Employing \eqref{si41} and \eqref{si45}, we deduce that
\begin{equation}\label{si46}
\begin{aligned}
0& > \mc J_{K_{w_1}}(v_n)- \mc J_{K_{w_1}}(u) \\
& \geq \mc J_{K_{w_1}}(v_n)- \mc J_{K_{w_1}}(\sigma(v_n))\\
& \geq \frac{\|v_n-\sigma(v_n)\|^2}{2} -\la\left( c_1 \left( | v_n+\overline{u}|_{2^*_s}^{2^*_\mu}+ | \sigma(v_n)+\overline{u}|_{2^*_s}^{2^*_\mu}\right)| \underline{k_{v_n}^1}(x) (v_n-\sigma(v_n))^2|_{\frac{2^*_s}{2^*_\mu}}\right. \\
& \left.\hspace{1cm}+c_2 S^{-1/2} | \underline{k_{v_n}^2} (x) (v_n-\sigma(v_n))|_{\frac{2^*_s}{2^*_\mu}} | \sigma(v_n)+\overline{u}|_{2^*_s}^{2^*_\mu-1} \|v_n-\sigma(v_n)\|\right) + \varXi(v_n-\sigma(v_n))\\
& \geq \frac{\|v_n-\sigma(v_n)\|^2}{2} + \varXi(v_n-\sigma(v_n)) -\left( \frac{C_1}{4} | \underline{k_{v_n}^1}(x) (v_n-\sigma(v_n))^2|_{\frac{2^*_s}{2^*_\mu}}\right. \\
& \left.\hspace{5cm}+\frac{C_2}{4} | \underline{k_{v_n}^2} (x) (v_n-\sigma(v_n))|_{\frac{2^*_s}{2^*_\mu}} \|v_n-\sigma(v_n)\|\right)
\end{aligned}
\end{equation}
where $C_1 = \sup_{n} 4\la c_1 \left( | v_n+\overline{u}|_{2^*_s}^{2^*_\mu}+ | \sigma(v_n)+\overline{u}|_{2^*_s}^{2^*_\mu}\right)$ and $C_2 = \sup_{n} 4\la c_2 S^{-1/2}| \sigma(v_n)+\overline{u}|_{2^*_s}^{2^*_\mu-1}$.
Consider
\begin{align*}
| \underline{k_{v_n}^1}(x)& (v_n-\sigma(v_n))^2|_{\frac{2^*_s}{2^*_\mu}}\leq | \underline{k_{v_n}^1}(x) |_{\frac{2^*_s}{2^*_\mu-2}} | (v_n-\sigma(v_n))^2|_{\frac{22^*_s}{2^*_\mu}}^2\\
& = \left( \left(\int_{ \{ \underline{k_{v_n}^1} \leq R_1\}} | \underline{k_{v_n}^1}(x) |^{\frac{2^*_s}{2^*_\mu-2}}\right)^{\frac{2^*_\mu-2}{2^*_s}} + \left(\int_{ \{ \underline{k_{v_n}^1} >R_1\}} | \underline{k_{v_n}^1}(x) |^{\frac{2^*_s}{2^*_\mu-2}}\right)^{\frac{2^*_\mu-2}{2^*_s}}\right)\\
& \hspace{8cm}| (v_n-\sigma(v_n))^2|_{\frac{22^*_s}{2^*_\mu}}^2.
\end{align*}
Choose $R_1, R_2>0$ such that, for all $n$,
\begin{align*}
C_1 S^{-1}\left(\int_{ \{ \underline{k_{v_n}^1} >R_1\}} | \underline{k_{v_n}^1}(x) |^{\frac{2^*_s}{2^*_\mu-2}}\right)^{\frac{2^*_\mu-2}{2^*_s}} < \frac12 \text{ and } C_2 S^{-1/2}\left(\int_{ \{ \underline{k_{v_n}^2} >R_2\}} | \underline{k_{v_n}^2}(x) |^{\frac{2^*_s}{2^*_\mu-1}}\right)^{\frac{2^*_\mu-1}{2^*_s}} < \frac12.
\end{align*}
Therefore, using the H\"older's inequality in \eqref{si46} with above estimates, we have
\begin{equation*}
\begin{aligned}
0>
& \frac{\|v_n-\sigma(v_n)\|^2}{4} + \varXi(v_n-\sigma(v_n)) -\left( \frac{C_1R_1}{4} \left( \int_{ \Om} (v_n-\sigma(v_n))^{\frac{22^*_s}{2^*_\mu}} ~dx \right)^{\frac{2^*_\mu}{2^*_s}} \right. \\
& \left.\hspace{5cm}+\frac{C_2R_2}{4} \left( \int_{ \Om} (v_n-\sigma(v_n))^{\frac{2^*_s}{2^*_\mu}} ~dx \right)^{\frac{2^*_\mu}{2^*_s}} \|v_n-\sigma(v_n)\|\right)\\
& \geq \frac{\|(v_n-w_2)^+\|^2}{4} + \varXi((v_n-w_2)^+)\\
& \quad -\left( \frac{C_1R_1}{4} |(v_n-w_2)^+|_{\frac{22^*_s}{2^*_\mu}}^2 +\frac{C_2R_2}{4} |(v_n-w_2)^+|_{\frac{22^*_s}{2^*_\mu}} \|v_n-\sigma(v_n)\|\right).
\end{aligned}
\end{equation*}
Let $C^*= \max \{ \frac{C_1R_1}{2}, \frac{C_2R_2}{2} \}$. Thus
\begin{equation}\label{si47}
\begin{aligned}
0>
& \frac{\|(v_n-w_2)^+\|^2}{4} + \varXi((v_n-w_2)^+)\\
&\qquad - \frac{C^*}{2}\left( |(v_n-w_2)^+|_{\frac{22^*_s}{2^*_\mu}}^2 +|(v_n-w_2)^+|_{\frac{22^*_s}{2^*_\mu}} \|(v_n-w_2)^+\|\right).
\end{aligned}
\end{equation}
Let $\nu = \inf \{ \varXi(\phi)\; :\;\phi \in \mathcal{A} \}$ where $\mathcal{A}= \{ \phi \in X_0 \; :\; \phi\geq 0, |\phi|_{\frac{22^*_s}{2^*_\mu}} =1, \|\phi\| \leq 2 C^* \}$.
Clearly, $\mathcal{A}$ is a weakly sequentially closed subset of $X_0$. Using Fatou's lemma and the fact that Riesz potential is a bounded linear functional, one can easily prove that $\varXi$ is a weakly lower semicontinuous on $\mathcal{A}$. Hence $\nu >0$. Indeed, let $z_n$ is a minimizing sequence of $\nu$ such that $z_n \rp z $ weakly in $X_0$ as $n \ra \infty$ then $\varXi(z)\leq \liminf \varXi(z_n)$. Now by the application of the fact that $w_2$ is a strict supersolution to $(\widetilde{P_{\la}})$ we get that $\varXi(z) >0$. Now notice that using the definition of $\nu$, \eqref{si47} can be rewritten as the following
\begin{equation}\label{si95}
\begin{aligned}
0>& \nu + \frac14\left( \left( \|(v_n-w_2)^+\| - C^* |(v_n-w_2)^+|_{\frac{22^*_s}{2^*_\mu}} \right)^2 - ((C^*)^2+2C^*)|(v_n-w_2)^+|_{\frac{22^*_s}{2^*_\mu}}^2 \right)\\
& > \nu - \frac14 ((C^*)^2+2C^*)|(v_n-w_2)^+|_{\frac{22^*_s}{2^*_\mu}}^2
\end{aligned}
\end{equation}
As $v_n$ is a sequence such that $v_n \ra u $ in $X_0$.
It implies that as $n \ra \infty$, $|(v_n-w_2)^+|_{\frac{22^*_s}{2^*_\mu}} \ra 0$.
So from \eqref{si95}, we get a contradiction to the fact that $\nu>0$. Hence the proof is complete.
\QED
\end{proof}
\begin{Lemma}\label{lemsi13}
$\La >0$.
\end{Lemma}
\begin{proof}
We will use the lower and upper solution method to prove the required result. From Lemma \ref{lemsi12}, $0$ and $\vartheta$ are the sub and supersolution respectively to $(\widetilde{P_\la})$. We define the closed convex set of $X_0$ as
\begin{align*}
W= \{ u \in X_0\; :\; 0 \leq u \leq \vartheta \}.
\end{align*}
Employing the definition of $W$, one can easily prove that
\begin{align*}
\mc J_W\geq \frac{\|u\|^2}{2}- c_1-c_2
\end{align*}
for appropriate positive constants $c_1$ and $c_2$. It implies $\mc J_W$ is coercive on $W$. $\mc J_W$ is weakly lower semi continuous on $W$. Indeed, let $\{ u_n\} \subset W$ such that $u_n \rp u$ weakly in $X_0$ as $n\ra \infty$. For each $n$,
\begin{align*}
& \int_{ \Om} G(x,u_n)~dx \leq \int_{ \Om} G(x,u)~dx< +\infty,\\
& \iint_{\Om\times \Om}\frac{(u_n+\overline{u})^{2^*_\mu}(u_n +\overline{u})^{2^*_\mu}}{|x-y|^{\mu}}~dxdy \leq \iint_{\Om\times \Om}\frac{(\vartheta+\overline{u})^{2^*_\mu}(\vartheta+\overline{u})^{2^*_\mu}}{|x-y|^{\mu}}~dxdy < +\infty.
\end{align*}
Now we may use the dominated convergence theorem and the weak lower semicontinuity of the norm to prove that $\mc J_W$ is weakly lower semi continuous on $W$. Thus, there exists $ u \in X_0$ such that
\begin{align*}
\inf_{v \in W} \mc J_W(v) = \mc J_W(u).
\end{align*}
Since $0 \in \pa^- \mc J_W(u)$, $u$ is a weak solution to $(\widetilde{P_{\la}})$. It implies $\La>0$. \QED
\end{proof}
\begin{Theorem}\label{thmsi3}
Let $\la \in (0, \La)$. Then there exists a positive weak solution $u_\la$ to $(\widetilde{P_{\la}})$ belonging to $ X_0$ such that $\mc J(u_\la)<0$ and $u_\la$ is a local minimizer for $\mc J_{K_0}$.
\end{Theorem}
\begin{proof}
Let $\la \in (0,\La)$ and $\la^\prime \in (\la, \La )$. Then by Lemma \ref{lemsi12}, $0$ and $u_{\la^\prime}$ are strict sub and supersolution to $(\widetilde{P_{\la}})$. The existence of $u_{\la^\prime}$ is clear by the definition of $\La$. Consider the convex set $ W= \{ u \in X_0\; :\; 0 \leq u \leq u_{\la^\prime} \}$. Using the same analysis as in Lemma \ref{lemsi13}, there exists a $ u_\la \in X_0$ such that$
\inf_{v \in W} \mc J_W(v) = \mc J_W(u_\la)$.
Notice that $0 \in W$ and $\mc J_W(0)<0$, it implies that $\mc J_W(u_\la)<0$. Let $\phi_1=0$ and $\phi_2 = u_{\la^\prime}$ in Theorem \ref{thmsi2} we have $u_{\la}$ is a local minimizer of $\mc J_{K_0}$. \QED
\end{proof}
\begin{Lemma}
$\La<\infty$.
\end{Lemma}
\begin{proof}
Assume by contradiction that $\La= +\infty$. Then there exists a sequence $\la_n \ra \infty$ as $n \ra \infty$. Let $u_{\la_n}$ be the corresponding solution to $(\widetilde{P_{\la}})$. Then by Theorem \ref{thmsi3}, $\mc J(u_{\la_n}) <0$ and $u_{\la_n}$ is a local minimizer for $\mc J_{K_0}$.
That is,
\begin{equation}\label{si51}
\begin{aligned}
& \frac12 \|u_{\la_n}\|^2 + \int_{\Om}G(x,u_{\la_n})~dx - \frac{\la_n}{22^*_\mu} \iint_{\Om\times \Om}\frac{(u_{\la_n}+\overline{u})^{2^*_\mu}(u_{\la_n}+\overline{u})^{2^*_\mu}}{|x-y|^{\mu}}~dxdy<0\\
\text{ and }& \|u_{\la_n}\|^2 + \int_{\Om}g(x,u_{\la_n})u_{\la_n}~dx - \la_n \iint_{\Om\times \Om}\frac{(u_{\la_n}+\overline{u})^{2^*_\mu}(u_{\la_n}+\overline{u})^{2^*_\mu-1}u_{\la_n}}{|x-y|^{\mu}}~dxdy=0.
\end{aligned}
\end{equation}
With the application of Lemma \ref{lemsi10}(ii) and statements, we have
\begin{align}\label{si49}
\frac12 \iint_{\Om\times \Om}\frac{(u_{\la_n}+\overline{u})^{2^*_\mu}(u_{\la_n}+\overline{u})^{2^*_\mu-1}u_{\la_n}}{|x-y|^{\mu}}~dxdy < \frac{1}{22^*_\mu} \iint_{\Om\times \Om}\frac{(u_{\la_n}+\overline{u})^{2^*_\mu}(u_{\la_n}+\overline{u})^{2^*_\mu}}{|x-y|^{\mu}}~dxdy.
\end{align}
Using the fact that $ \overline{u} \in L^\infty(\Om)$, for each $ x \in \Om$, we have
\begin{align*}
\lim_{t \ra\infty} \frac{\left( \int_{ \Om} \frac{|t+\overline{u}|^{2^*_\mu}}{|x-y|^{\mu }}~dy\right)|t+\overline{u}|^{2^*_\mu} }{\left( \int_{ \Om} \frac{|t+\overline{u}|^{2^*_\mu}}{|x-y|^{\mu }}~dy\right)|t+\overline{u}|^{2^*_\mu-1}t } =1.
\end{align*}
Therefore, it follows that for any small enough $\e>0$, there exists $M_\e >0$ such that, for all $n$
\begin{equation}
\begin{aligned}\label{si50}
\frac{1}{22^*_\mu} \iint_{\Om\times \Om}& \frac{(u_{\la_n}+\overline{u})^{2^*_\mu}(u_{\la_n}+\overline{u})^{2^*_\mu}}{|x-y|^{\mu}}~dxdy\\
& < \frac{1}{2+\e}\iint_{\Om\times \Om}\frac{(u_{\la_n}+\overline{u})^{2^*_\mu}(u_{\la_n}+\overline{u})^{2^*_\mu-1}u_{\la_n}}{|x-y|^{\mu}}~dxdy +M_\e .
\end{aligned}
\end{equation}
From \eqref{si49} and \eqref{si50}, we obtain
\begin{align*}
\iint_{\Om\times \Om}\frac{(u_{\la_n}+\overline{u})^{2^*_\mu}(u_{\la_n}+\overline{u})^{2^*_\mu-1}u_{\la_n}}{|x-y|^{\mu}}~dxdy <\infty \text{ for all } n .
\end{align*}
From \eqref{si51}, we have
\begin{align*}
\|u_{\la_n}\|^2 < \la_n \iint_{\Om\times \Om}\frac{(u_{\la_n}+\overline{u})^{2^*_\mu}(u_{\la_n}+\overline{u})^{2^*_\mu-1}u_{\la_n}}{|x-y|^{\mu}}~dxdy.
\end{align*}
Hence $\{\la_n^{-1/2}u_{\la_n}\}$ is uniformly bounded in $X_0$. Then there exists $z_0 \in X_0$ such that $z_n:= \la_n^{-1/2}u_{\la_n} \rp z_0$ weakly in $X_0$. Let $0\leq \psi \in C_c^\infty(\Om)$ be a non trivial function. Let $k>0$ such that $\overline{u}>k$ on $\text{supp}(\psi)$. Once again using \eqref{si51}, we deduce that
\begin{align*}
\sqrt{\la_n} \iint_{\Om\times \Om}\frac{k^{22^*_\mu-1}\psi}{|x-y|^{\mu}}~dxdy &
\leq \sqrt{\la_n} \iint_{\Om\times \Om}\frac{(u_{\la_n}+\overline{u})^{2^*_\mu}(u_{\la_n}+\overline{u})^{2^*_\mu-1}\psi}{|x-y|^{\mu}}~dxdy\\
& = \ld z_n, \psi\rd + \frac{1}{\sqrt{\la_n}} \int_{\Om}g(x,u_{\la_n})\psi~dx\\
&\leq \ld z_n, \psi\rd + \frac{1}{\sqrt{\la_n}} \int_{\Om}k^{-q}\psi~dx.
\end{align*}
Now passing the limit $ n \ra \infty$, we have $ \ld z_0, \psi\rd = \infty$, which is not true. Hence $\La<\infty$. \QED
\end{proof}
\section{Second solution}
In this section we will prove the existence of second solution to $(\widetilde{P_{\la}})$. Let $u_\la$ denotes the first solution to $(\widetilde{P_{\la}})$ obtained in Theorem \ref{thmsi3}.
\begin{Proposition}\label{propsi4}
The functional $\mc J_{K_{u_\la}}$ satisfies the $(CPS)_c$ for each $c$ satisfying
\begin{align*}
c < \mc J_{K_{u_\la}}(u_\la) + \frac12 \left(\frac{N-\mu+2s}{2N-\mu}\right) \left( \frac{S_{H,L}^{\frac{2N-\mu}{N-\mu+2s}}}{\la^{\frac{N-2s}{N-\mu+2s}}} \right).
\end{align*}
\end{Proposition}
\begin{proof}
Let $ c < \mc J_{K_{u_\la}}(u_\la) + \frac12 \left(\frac{N-\mu+2s}{2N-\mu}\right) \left( \frac{S_{H,L}^{\frac{2N-\mu}{N-\mu+2s}}}{\la^{\frac{N-2s}{N-\mu+2s}}} \right)$. Let $z_n$ be a sequence such that
\begin{align*}
\mc J_{K_{u_\la}}(z_n) \ra c \text{ and } (1+ \|z_n\|) |||\pa^-\mc J_{K_{u_\la}}(z_n)||| \ra 0 \text{ as } n \ra \infty.
\end{align*}
It implies there exists $\xi_n \in \pa^-\mc J_{K_{u_\la}}(z_n) $ such that $ \|\xi_n\|= |||\pa^-\mc J_{K_{u_\la}}(z_n)|||$ for every $ n$. From Lemma \ref{lemsi14}, for each $ v \in D(\mc J_{K_{u_\la}}) $ and for each $n$, $g(\cdot, z_n)(v-z_n) \in L^1(\Om)$ and
\begin{equation}\label{si52}
\begin{aligned}
\ld \xi_n, v-z_n\rd & \leq \ld z_n, v-z_n\rd + \int_{\Om} g(x,z_n)(v-z_n)~dx\\
& - \la \iint_{\Om\times \Om}\frac{(z_n+\overline{u})^{2^*_\mu}(z_n+\overline{u})^{2^*_\mu-1}(v-z_n)}{|x-y|^{\mu}}~dxdy.
\end{aligned}
\end{equation}
Using the fact that $G(\cdot, z_n ) \in L^1(\Om)$ and Lemma \ref{lemsi10}, we obtain that $G(\cdot, 2z_n) \in L^1(\Om)$. So $2z_n \in D(\mc J_{K_{u_\la}})$, now employing \eqref{si52}, we get
\begin{align*}
\ld \xi_n , z_n \rd \leq \|z_n\|^2 + \int_{\Om} g(x,z_n)z_n~dx - \la \iint_{\Om\times \Om}\frac{(z_n+\overline{u})^{2^*_\mu}(z_n+\overline{u})^{2^*_\mu-1}z_n}{|x-y|^{\mu}}~dxdy.
\end{align*}
With the help of Lemma \ref{lemsi10} and \eqref{si50}, for $\e>0$ small enough,
\begin{equation*}
\begin{aligned}
c+1 &\geq
\frac12 \|z_n\|^2 + \int_{\Om} G(x,z_n)~dx -\frac{\la}{22^*_\mu} \iint_{\Om\times \Om}\frac{(z_n+\overline{u})^{2^*_\mu}(z_n+\overline{u})^{2^*_\mu}}{|x-y|^{\mu}}~dxdy\\
& \geq \frac12 \|z_n\|^2 + \int_{\Om} G(x,z_n)~dx -\frac{1}{2+\e}\left(\ld \xi_n , z_n \rd - \|z_n\|^2 - \int_{\Om} g(x,z_n)z_n~dx \right)-\la M_\e\\
& \geq \frac12 \|z_n\|^2 -\frac{1}{2+\e}\left(\ld \xi_n , z_n \rd - \|z_n\|^2 \right)-\la M_\e.
\end{aligned}
\end{equation*}
It shows that $\{z_n\} $ is a bounded sequence in $X_0$. Hence, up to a subsequence, there exist $z_0 \in X_0$ such that $z_n \rp z_0 $ weakly in $X_0$ as $n \ra \infty$.
Let $\|z_n- z_0\| \ra a^2 $ and $ \iint_{\Om\times \Om}\frac{(z_n-z_0)^{2^*_\mu}(z_n-z_0)^{2^*_\mu}}{|x-y|^{\mu}}~dxdy \ra b^{22^*_\mu} $ as $n \ra \infty$.
By the mean value theorem, Brezis-Lieb Lemma (see \cite{ Leib,yang}) and \eqref{si52}, we deduce that
\begin{align*}
\int_{\Om}G(x,z_0)~dx& \geq \int_{\Om} G(x,z_n)~dx + \int_{\Om} g(x,z_n)(z_0-z_n)~dx \\
& \geq \int_{\Om} G(x,z_n)~dx - \la \iint_{\Om\times \Om}\frac{(z_n+\overline{u})^{2^*_\mu}(z_n+\overline{u})^{2^*_\mu-1}(z_n-z_0)}{|x-y|^{\mu}}~dxdy\\
& \quad - \ld \xi_n, z_n-z_0\rd + \ld z_n, z_n-z_0\rd \\
& = \int_{\Om} G(x,z_n)~dx - \ld \xi_n, z_n-z_0\rd + \ld z_n, z_n-z_0\rd \\
& - \la \iint_{\Om\times \Om}\frac{\left((z_n-z_0)^{2^*_\mu}(z_n-z_0)^{2^*_\mu} + (z_0+\overline{u})^{2^*_\mu}(z_0+\overline{u})^{2^*_\mu}\right) }{|x-y|^{\mu}}~dxdy\\
& \quad + \la \iint_{\Om\times \Om}\frac{(z_n+\overline{u})^{2^*_\mu}(z_n+\overline{u})^{2^*_\mu-1}(z_0+\overline{u})}{|x-y|^{\mu}}~dxdy.
\end{align*}
Now using the fact that $z_n$ converges to $z_0$ weakly in $X_0$, hence as $n \ra \infty$, we get
\begin{align*}
\int_{\Om}G(x,z_0)~dx& \geq \int_{\Om} G(x,z_0)~dx +a^2 - \la b^{22^*_\mu}.
\end{align*}
Thus
\begin{align}\label{si53}
\la b^{22^*_\mu}\geq a^2 .
\end{align}
On account of the fact that $u_\la$ is a weak positive solution to $(\widetilde{P_{\la}})$, for each $n$,
\begin{equation}
\begin{aligned}\label{si54}
0& = \ld u_\la,z_n-u_\la \rd + \int_{\Om} g(x,u_\la)(z_n-u_\la)~dx\\
& \qquad +\la \iint_{\Om\times \Om}\frac{(u_\la+\overline{u})^{2^*_\mu}(u_\la+\overline{u})^{2^*_\mu-1}(z_n-u_\la)}{|x-y|^{\mu}}~dxdy.
\end{aligned}
\end{equation}
In consideration of $G(\cdot, z_n), G(\cdot, 2z_n) \in L^1(\Om)$ and $ u_\la\leq 2z_n-u_\la\leq 2z_n$, we infer that $2z_n-u_\la \in D(\mc J_{K_{u_\la}})$. Testing \eqref{si52} with $2z_n-u_\la$, we obtain that
\begin{equation}\label{si55}
\begin{aligned}
\ld \xi_n, z_n- u_\la \rd & \leq \ld z_n, z_n- u_\la \rd + \int_{\Om} g(x,z_n)(z_n- u_\la)~dx\\
& - \la \iint_{\Om\times \Om}\frac{(z_n+\overline{u})^{2^*_\mu}(z_n+\overline{u})^{2^*_\mu-1}(z_n- u_\la)}{|x-y|^{\mu}}~dxdy.
\end{aligned}
\end{equation}
From Lemma \ref{lemsi10}, \eqref{si54} and \eqref{si55}, we have
\begin{equation}\label{si56}
\begin{aligned}
&\mc J_{K_{u_\la}}(z_n)- \mc J_{K_{u_\la}}(u_\la)\\
& = \frac12 \|z_n\|^2 + \int_{\Om}G(x,z_n)~dx - \frac{\la}{22^*_\mu} \iint_{\Om\times \Om}\frac{(z_n+\overline{u})^{2^*_\mu}(z_n+\overline{u})^{2^*_\mu}}{|x-y|^{\mu}}~dxdy \\
& \; - \frac12 \|u_\la\|^2 - \int_{\Om}G(x,u_\la)~dx + \frac{\la}{22^*_\mu} \iint_{\Om\times \Om}\frac{(u_\la+\overline{u})^{2^*_\mu}(u_\la+\overline{u})^{2^*_\mu}}{|x-y|^{\mu}}~dxdy \\
&\geq \int_{\Om}\!\!\left(G(x,z_n)- G(x,u_\la) - \frac12 \left(g(x,u_\la) +g(x,z_n)\right)(z_n-u_\la)\right)dx\\
& \; +\frac{\la}{22^*_\mu} \iint_{\Om\times \Om}\frac{(u_\la+\overline{u})^{2^*_\mu}(u_\la+\overline{u})^{2^*_\mu}- (z_n+\overline{u})^{2^*_\mu}(z_n+\overline{u})^{2^*_\mu}}{|x-y|^{\mu}}~dxdy + \frac12 \ld \xi_n, z_n- u_\la \rd \\
&\; + \frac{\la}{2} \iint_{\Om\times \Om}\frac{\left( (u_\la+\overline{u})^{2^*_\mu}(u_\la+\overline{u})^{2^*_\mu-1} - (z_n+\overline{u})^{2^*_\mu}(z_n+\overline{u})^{2^*_\mu-1} \right) (z_n-u_\la)}{|x-y|^{\mu}}~dxdy\\
& \geq \frac{\la}{22^*_\mu} \iint_{\Om\times \Om}\frac{(u_\la+\overline{u})^{2^*_\mu}(u_\la+\overline{u})^{2^*_\mu}- (z_n+\overline{u})^{2^*_\mu}(z_n+\overline{u})^{2^*_\mu}}{|x-y|^{\mu}}~dxdy \\
&\; + \frac{\la}{2} \iint_{\Om\times \Om}\frac{\left((u_\la+\overline{u})^{2^*_\mu}(u_\la+\overline{u})^{2^*_\mu-1}(z_n-u_\la) - (z_n+\overline{u})^{2^*_\mu}(z_n+\overline{u})^{2^*_\mu-1}( u_\la+\overline{u}) \right)}{|x-y|^{\mu}}~dxdy\\
& \quad +\frac{\la}{2} \iint_{\Om\times \Om}\frac{(z_n+\overline{u})^{2^*_\mu}(z_n+\overline{u})^{2^*_\mu}}{|x-y|^{\mu}}~dxdy + \frac12 \ld \xi_n, z_n- u_\la \rd\\
& =: \mc I + \frac12 \ld \xi_n, z_n- u_\la \rd.
\end{aligned}
\end{equation}
Using Brezis-Lieb Lemma (See \cite{yang}), we have
\begin{equation}\label{si57}
\begin{aligned}
\mc I& = \la \left(\frac12- \frac{1}{22^*_\mu} \right) \iint_{\Om\times \Om}\!\!\!\frac{ \left((z_n-z_0)^{2^*_\mu}(z_n-z_0)^{2^*_\mu}\right) + \left( (z_0+ \overline{u})^{2^*_\mu}(z_0+ \overline{u})^{2^*_\mu} \right)}{|x-y|^{\mu}}~dxdy
\\&
+ \frac{\la}{2} \iint_{\Om\times \Om}\frac{\left((u_\la+\overline{u})^{2^*_\mu}(u_\la+\overline{u})^{2^*_\mu-1}(z_n-u_\la)\right) -\left( (z_n+\overline{u})^{2^*_\mu}(z_n+\overline{u})^{2^*_\mu-1}( u_\la+\overline{u})\right)}{|x-y|^{\mu}}~dxdy\\ &\quad +\frac{\la}{22^*_\mu} \iint_{\Om\times \Om}\frac{(u_\la+\overline{u})^{2^*_\mu}(u_\la+\overline{u})^{2^*_\mu}}{|x-y|^{\mu}}~dxdy +o(1).
\end{aligned}
\end{equation}
Observe that by weak convergence of the sequence $\{ z_n \}$, we have
\begin{equation}\label{si58}
\begin{aligned}
& \iint_{\Om\times \Om}\!\!\!\frac{(u_\la+\overline{u})^{2^*_\mu}(u_\la+\overline{u})^{2^*_\mu-1}(z_n-z_0)}{|x-y|^{\mu}}~dxdy \ra 0\\
& \text{ and } \iint_{\Om\times \Om}\frac{\left( (z_n+\overline{u})^{2^*_\mu}(z_n+\overline{u})^{2^*_\mu-1} - (z_0+\overline{u})^{2^*_\mu}(z_0+\overline{u})^{2^*_\mu-1}\right)
( u_\la+\overline{u})}{|x-y|^{\mu}}~dxdy \ra 0.
\end{aligned}
\end{equation}
Taking into account \eqref{si56}, \eqref{si57}, \eqref{si58} and passing the limit as $n \ra \infty$, we obtain that
\begin{equation}\label{si59}
\begin{aligned}
c- \mc J_{K_{u_\la}}(u_\la) & \geq \frac{\la}{22^*_\mu} \iint_{\Om\times \Om}\frac{(u_\la+\overline{u})^{2^*_\mu}(u_\la+\overline{u})^{2^*_\mu}}{|x-y|^{\mu}}~dxdy + \la \left(\frac12- \frac{1}{22^*_\mu} \right) b^{22^*_\mu}
\\&
\qquad + \frac{\la}{2} \iint_{\Om\times \Om}\frac{(u_\la+\overline{u})^{2^*_\mu}(u_\la+\overline{u})^{2^*_\mu-1}(z_0-u_\la)}{|x-y|^{\mu}}~dxdy \\
& \qquad + \frac{\la}{2} \iint_{\Om\times \Om}\frac{(z_0+\overline{u})^{2^*_\mu}(z_0+\overline{u})^{2^*_\mu-1}( z_0- u_\la)}{|x-y|^{\mu}}~dxdy\\
&\qquad - \frac{\la}{22^*_\mu} \iint_{\Om\times \Om}\frac{ (z_0+ \overline{u})^{2^*_\mu}(z_0+ \overline{u})^{2^*_\mu}}{|x-y|^{\mu}}~dxdy \\
& \qquad := \mc I_1 (\text{say}) + \la \left(\frac12- \frac{1}{22^*_\mu} \right) b^{22^*_\mu}.
\end{aligned}
\end{equation}
Clearly, we infer
\begin{equation}\label{si60}
\begin{aligned}
\mc I_1& = \frac{\la}{2} \iint_{\Om\times \Om}\frac{(u_\la+\overline{u})^{2^*_\mu}\left( (u_\la+\overline{u})^{2^*_\mu-1}+ (z_0+\overline{u})^{2^*_\mu-1}\right) (z_0-u_\la) }{|x-y|^{\mu}}~dxdy \\
&\qquad - \frac{\la}{2} \iint_{\Om\times \Om}\frac{(u_\la+\overline{u})^{2^*_\mu} (z_0+\overline{u})^{2^*_\mu-1}(z_0-u_\la) }{|x-y|^{\mu}}~dxdy
\\&
\qquad + \frac{\la}{2} \iint_{\Om\times \Om}\frac{(z_0+\overline{u})^{2^*_\mu}\left( (z_0+\overline{u})^{2^*_\mu-1}+ (u_\la+\overline{u})^{2^*_\mu-1}\right) ( z_0- u_\la)}{|x-y|^{\mu}}~dxdy\\
& \qquad - \frac{\la}{2} \iint_{\Om\times \Om}\frac{(z_0+\overline{u})^{2^*_\mu} (u_\la+\overline{u})^{2^*_\mu-1} ( z_0- u_\la)}{|x-y|^{\mu}}~dxdy\\
& \qquad + \frac{\la}{22^*_\mu} \iint_{\Om\times \Om}\frac{(u_\la+\overline{u})^{2^*_\mu}\left( (u_\la+\overline{u})^{2^*_\mu}- (z_0+\overline{u})^{2^*_\mu}\right)}{|x-y|^{\mu}}~dxdy
\\&
\qquad + \frac{\la}{22^*_\mu} \iint_{\Om\times \Om}\frac{(z_0+\overline{u})^{2^*_\mu}\left( (u_\la+\overline{u})^{2^*_\mu}- (z_0+\overline{u})^{2^*_\mu}\right)}{|x-y|^{\mu}}~dxdy.
\end{aligned}
\end{equation}
Since
\begin{align*}
(u_\la+\overline{u})^{2^*_\mu}- (z_0+\overline{u})^{2^*_\mu} & = -2^*_\mu \int_{u_\la}^{z_0} (t+\overline{u})^{2^*_\mu-1}~dt \\
& \quad
\geq -2^*_\mu \left( \frac{(u_\la+\overline{u})^{2^*_\mu-1}+ (z_0+\overline{u})^{2^*_\mu-1}}{2} \right) (z_0-u_\la).
\end{align*}
It implies
\begin{equation}\label{si61}
\begin{aligned}
\frac{\la}{22^*_\mu}& \iint_{\Om\times \Om}\frac{(u_\la+\overline{u})^{2^*_\mu}\left( (u_\la+\overline{u})^{2^*_\mu}- (z_0+\overline{u})^{2^*_\mu}\right)}{|x-y|^{\mu}}~dxdy\\
&\geq - \frac{\la}{4} \iint_{\Om\times \Om}\frac{(u_\la+\overline{u})^{2^*_\mu}\left( (u_\la+\overline{u})^{2^*_\mu-1}+ (z_0+\overline{u})^{2^*_\mu-1}\right) (z_0-u_\la) }{|x-y|^{\mu}}~dxdy.
\end{aligned}
\end{equation}
\begin{equation}\label{si62}
\begin{aligned}
\text{Similarly, }& \frac{\la}{22^*_\mu}\iint_{\Om\times \Om}\frac{(z_0+\overline{u})^{2^*_\mu}\left( (u_\la+\overline{u})^{2^*_\mu}- (z_0+\overline{u})^{2^*_\mu}\right)}{|x-y|^{\mu}}~dxdy\\
&\geq - \frac{\la}{4} \iint_{\Om\times \Om}\frac{(z_0+\overline{u})^{2^*_\mu}\left( (u_\la+\overline{u})^{2^*_\mu-1}+ (z_0+\overline{u})^{2^*_\mu-1}\right) (z_0-u_\la) }{|x-y|^{\mu}}~dxdy.
\end{aligned}
\end{equation}
From \eqref{si60}, \eqref{si61} and \eqref{si62}, we deduce that
\begin{equation}\label{si63}
\begin{aligned}
\mc I_1& = \frac{\la}{4} \iint_{\Om\times \Om}\frac{(u_\la+\overline{u})^{2^*_\mu}\left( (u_\la+\overline{u})^{2^*_\mu-1}- (z_0+\overline{u})^{2^*_\mu-1}\right) (z_0-u_\la) }{|x-y|^{\mu}}~dxdy \\
&\qquad + \frac{\la}{4} \iint_{\Om\times \Om}\frac{(z_0+\overline{u})^{2^*_\mu}\left( (z_0+\overline{u})^{2^*_\mu-1}- (u_\la+\overline{u})^{2^*_\mu-1}\right) ( z_0- u_\la)}{|x-y|^{\mu}}~dxdy\\
& = \frac{\la}{4} \iint_{\Om\times \Om}\frac{\left((z_0+\overline{u})^{2^*_\mu} - (u_\la+\overline{u})^{2^*_\mu}\right) \left( (z_0+\overline{u})^{2^*_\mu-1}- (u_\la+\overline{u})^{2^*_\mu-1}\right) ( z_0- u_\la) }{|x-y|^{\mu}}~dxdy \\
& \geq 0.
\end{aligned}
\end{equation}
Hence from \eqref{si59} and \eqref{si63}, we obtain
\begin{equation}\label{si64}
\begin{aligned}
c- \mc J_{K_{u_\la}}(u_\la) & \geq \la \left(\frac12- \frac{1}{22^*_\mu} \right) b^{22^*_\mu}.
\end{aligned}
\end{equation}
Using definition of $S_{H,L}$ and \eqref{si53}, we have $ \la b^{22^*_\mu}\geq a^2$ and $a^2 \geq S_{H,L}b^2$, that is
\begin{align}\label{si65}
b\geq \left( \frac{S_{H,L}}{\la}\right)^{\frac{N-2s}{2(N-\mu+2s)}}.
\end{align}
Using \eqref{si64} and \eqref{si65}, we get
\begin{align*}
c- \mc J_{K_{u_\la}}(u_\la) & \geq \la \left(\frac12- \frac{1}{22^*_\mu} \right) \left( \frac{S_{H,L}}{\la}\right)^{\frac{2N-\mu}{N-\mu+2s}} = \frac12 \left(\frac{N-\mu+2s}{2N-\mu}\right) \left( \frac{S_{H,L}^{\frac{2N-\mu}{N-\mu+2s}}}{\la^{\frac{N-2s}{N-\mu+2s}}} \right).
\end{align*}
It contradicts the fact that $ c < \mc J_{K_{u_\la}}(u_\la) + \frac12 \left(\frac{N-\mu+2s}{2N-\mu}\right) \left( \frac{S_{H,L}^{\frac{2N-\mu}{N-\mu+2s}}}{\la^{\frac{N-2s}{N-\mu+2s}}} \right)$. Hence $a=0$. \QED
\end{proof}
Now consider the family of minimizers $\{U_\e\}_{\e>0}$ of $S$ defined as
\begin{align*}
U_\e = \e^{-\frac{(N-2s)}{2}} S^{\frac{(N-\mu)(2s-N)}{4(N-\mu+2s)}}(C(N,\mu))^{\frac{2s-N}{2(N-\mu+2s)}} u^*(x/\e)
\end{align*}
where $u^*(x)= \overline{u}(x/S^{1/2s}),\; \overline{u} (x)= \frac{\tilde{u}(x)}{|\tilde{u}|_{2^*_s}}$ and $\tilde{u}(x)= a(b^2+|x|^2)^{\frac{-(N-2s)}{2}}$ with $ \al \in \mathbb{R}\setminus\{0\}$ and $\ba >0$ are fixed constants. Then from Lemma \ref{lulem13}, for $\e>0, ~U_\e$ satisfies
\begin{align*}
(-\De)^s u = (|x|^{-\mu}* |u|^{2^*_\mu})|u|^{2^*_\mu-2}u \text{ in } \R.
\end{align*}
Let $\varrho >0$ such that $B_{4\varrho} \subset \Om$. Now define $\eta\in C_c^{\infty}(\mathbb{R}^N)$ such that $0\leq \eta\leq 1$ in $\mathbb{R}^N$, $\eta\equiv1$ in $B_\varrho(0)$ and $\eta\equiv 0 $ in $\mathbb{R}^N \setminus B_{2\varrho}(0)$. For each $\e>0 $ and $x \in \R$, we define $\varPsi_\e = \eta(x) U_\e(x) $.
\begin{Proposition}\label{propsi3}
Let $N> 2s,\; 0<\mu<N$ then the following holds:
\begin{enumerate}
\item [(i)] $\|\varPsi_\e\|^2 \leq S_{H,L}^{\frac{2N-\mu}{N-\mu+2s}}+O(\e^{N-2s})$.
\item [(ii)] $\|\varPsi_\e\|_{NL}^{2.2^*_{\mu}}\leq S_{H,L}^{\frac{2N-\mu}{N-\mu+2s}}+O(\e^N)$.
\item [(iii)] $\|\varPsi_\e\|_{NL}^{2.2^*_{\mu}}\geq S_{H,L}^{\frac{2N-\mu}{N-\mu+2s}}-O(\e^N)$.
\end{enumerate}
\end{Proposition}
\begin{proof}
Using the definition of $\varPsi_\e$ and \cite[Proposition 1]{sevadei1} part $(i)$ follows. For $(ii)$ and $(iii)$ see \cite[Proposition 2.8]{GMS1}.\QED
\end{proof}
\begin{Lemma}\label{lemsi5}
\cite{ds} The following holds:
\begin{enumerate}
\item[(i)]If $\mu<\min\{4s,N\}$ then for all $ \Theta <1 \nonumber$,
\begin{align*}
\|u_\la+t \varPsi_\e\|_{NL}^{2.2^*_{\mu}} &
\geq \|u_\la\|_{NL}^{2.2^*_{\mu}}+\|\varPsi_\e\|_{NL}^{2.2^*_{\mu}} +\widehat{C} t^{2.2^*_{\mu}-1} \int_{\Om}\int_{ \Om}\frac{(\varPsi_\e(x))^{2^*_{\mu}}(\varPsi_\e(y))^{2^*_{\mu}-1}u_\la(y)}{|x-y|^{\mu}}~dxdy\\& \quad + 2.2^*_{\mu} t \int_{\Om}\int_{ \Om}\frac{(u_\la(x))^{2^*_{\mu}}(u_\la(y))^{2^*_{\mu}-1}\varPsi_\e(y)}{|x-y|^{\mu}}~dxdy- O(\e^{(\frac{2N-\mu}{4}) \Theta}) .
\end{align*}
\item [(ii)]There exists a $R_0>0$ such that
$
\int_{\Om}\int_{\Om}\frac{(\varPsi_\e(x))^{2^{*}_{\mu}}(\varPsi_\e(y))^{2^{*}_{\mu}-1}u_\la(y)}{|x-y|^{\mu}}~dxdy\geq \widehat{C}R_0\e^{\frac{N-2s}{2}}$.
\end{enumerate}
\end{Lemma}
\begin{Lemma}\label{lemsi6}
$\sup\{ \mc J_{K_{u_\la}} (u_\la+ t \varPsi_\e): t\geq 0 \}< \mc J_{K_{u_\la}}(u_\la)+ \frac12\left( \frac{N-\mu +2s}{2N-\mu}\right) \frac{S_{H,L}^{\frac{2N-\mu}{N-\mu+2s}}}{\la^{\frac{N-2s}{N-\mu+2s}}}$ for any sufficiently small $\e>0$.
\end{Lemma}
\begin{proof}
Employing the fact that $u_\la$ is a weak solution to $(P_\la)$ and using Lemma \ref{lemsi5}, for all $\Theta<1$, we have
\begin{align*}
\mc J_{K_{u_\la}}(u_\la+ t\varPsi_\e) - \mc J_{K_{u_\la}}(u_\la)& \leq \frac12\|t\varPsi_\e\|^2- \frac{\la}{22^*_\mu} \|t \varPsi_\e\|_{NL}^{2.2^*_{\mu}}+ O(\e^{(\frac{2N-\mu}{4}) \Theta}) \\&\quad + \int_{ \Om}(G(u_\la+t\varPsi_\e)-G(x,u_\la) - g(x,u_\la)t\varPsi_\e)~dx
\\&\quad - \frac{\la\widehat{C} t^{2.2^*_{\mu}-1}}{22^*_\mu} \int_{\Om}\int_{ \Om}\frac{(\varPsi_\e(x))^{2^*_{\mu}}(\varPsi_\e(y))^{2^*_{\mu}-1}u_\la(y)}{|x-y|^{\mu}}~dxdy .
\end{align*}
From Proposition \ref{propsi3} and Lemma \ref{lemsi5}, we deduce that
\begin{equation}\label{si27}
\begin{aligned}
\mc J_{K_{u_\la}}(u_\la+ t\varPsi_\e) - \mc J_{K_{u_\la}}(u_\la)& \leq \frac{t^2}{2}(S_{H,L}^{\frac{2N-\mu}{N-\mu+2s}}+O(\e^{N-2s}))- \frac{\la t^{22^*_\mu}}{22^*_\mu} (S_{H,L}^{\frac{2N-\mu}{N-\mu+2s}}-O(\e^N))
\\&\quad + \int_{ \Om}(G(u_\la+t\varPsi_\e)-G(x,u_\la) - g(x,u_\la)t\varPsi_\e)~dx
\\&\quad - \frac{\widehat{C} t^{2.2^*_{\mu}-1}}{22^*_\mu} \widehat{C}R_0\e^{\frac{N-2s}{2}} + O(\e^{(\frac{2N-\mu}{4}) \Theta}).
\end{aligned}
\end{equation}
Observe that for any fix $1<\rho < \min \{2, \frac{2}{n-2s} \}$, there exists $R_1 >0$ such that
\begin{align*}
\int_{ \Om}|\varPsi_\e|^\rho ~dx \leq R_1 \e^{(n-2s)\rho/2}.
\end{align*}
Moreover, there exists $ R_2 >0 $ such that, for all $x \in \Om, r>m\text{ and } s \geq 0$,
\begin{align*}
G(x,r+s)- G(x,s)-g(x,r)s = \int_r^{r+s} (\tau^{-q}- r^{-q})~d\tau \leq R_2 s^{\rho} .
\end{align*}
Using last inequality and \eqref{si27} with $\Theta= \frac{2}{2^*_\mu}$, we obtain
\begin{equation*}\label{si28}
\begin{aligned}
\mc J_{K_{u_\la}}(u_\la+ t\varPsi_\e) - \mc J_{K_{u_\la}}(u_\la)& \leq \frac{t^2}{2}(S_{H,L}^{\frac{2N-\mu}{N-\mu+2s}}+O(\e^{N-2s}))- \frac{t^{22^*_\mu}}{22^*_\mu} (S_{H,L}^{\frac{2N-\mu}{N-\mu+2s}}-O(\e^N)) \\&\quad - \frac{\widehat{C} t^{2.2^*_{\mu}-1}}{22^*_\mu} \widehat{C}R_0\e^{\frac{N-2s}{2}} + R_1R_2t^\rho \e^{(n-2s)\rho/2} + o(\e^{\frac{N-2s}{2}})\\
& := K(t).
\end{aligned}
\end{equation*}
Clearly, one can check that $K(t)\ra -\infty, K(t)>0$ as $t \ra 0^+$ and there exists $t_\e >0$ such that $K^\prime(t_\e)=0$. Furthermore, there exist positive constants $T_1$ and $T_2$ such that $T_1\leq t_\e \leq T_2$ (for details see \cite{ds}). Hence,
\begin{align*}
K(t)& \leq \frac{t_\e^2}{2}(S_{H,L}^{\frac{2N-\mu}{N-\mu+2s}}+O(\e^{N-2s}))- \frac{t_\e^{22^*_\mu}}{22^*_\mu} (S_{H,L}^{\frac{2N-\mu}{N-\mu+2s}}-O(\e^N)) - \frac{\widehat{C} T_1^{2.2^*_{\mu}-1}}{22^*_\mu} \widehat{C}R_0\e^{\frac{N-2s}{2}}\\&\quad + R_1R_2T_2^\rho \e^{(n-2s)\rho/2} + o(\e^{\frac{N-2s}{2}})\\
& \leq \sup_{t\geq 0} K_1(t) - \frac{\widehat{C} T_1^{2.2^*_{\mu}-1}}{22^*_\mu} \widehat{C}R_0\e^{\frac{N-2s}{2}} + R_1R_2T_2^\rho \e^{(n-2s)\rho/2} + o(\e^{\frac{N-2s}{2}})
\end{align*}
where $K_1(t)=\frac{t^2}{2}(S_{H,L}^{\frac{2N-\mu}{N-\mu+2s}}+O(\e^{N-2s}))- \frac{t^{22^*_\mu}}{22^*_\mu} (S_{H,L}^{\frac{2N-\mu}{N-\mu+2s}}-O(\e^N))$. By trivial computations, we get
\begin{align*}
\mc J_{K_{u_\la}}(u_\la+ t\varPsi_\e) - \mc J_{K_{u_\la}}(u_\la)& \leq \frac12\left( \frac{N-\mu +2s}{2N-\mu}\right) \frac{S_{H,L}^{\frac{2N-\mu}{N-\mu+2s}}}{\la^{\frac{N-2s}{N-\mu+2s}}} +O(\e^{\frac{N-2s}{2}}) -C\e^{\frac{N-2s}{2}} + o(\e^{\frac{N-2s}{2}})
\end{align*}
for an appropriate constant $C>0$. Thus, for $\e$ sufficiently small,
\begin{align*}
\mc J_{K_{u_\la}}(u_\la+ t\varPsi_\e) - \mc J_{K_{u_\la}}(u_\la)& < \frac12\left( \frac{N-\mu +2s}{2N-\mu}\right) \frac{S_{H,L}^{\frac{2N-\mu}{N-\mu+2s}}}{\la^{\frac{N-2s}{N-\mu+2s}}} .
\end{align*}
Hence the proof follows. \QED
\end{proof}
\begin{Proposition}\label{propsi5}
For each $\la \in (0, \La)$ there exist a second positive solution to $(\widetilde{P_{\la}})$.
\end{Proposition}
\begin{proof}
From Theorem \ref{thmsi3}, $u_\la$ is a local minimizer of $ \mc J_{K_{u_\la}}$. It implies there exist $\varsigma >0$ such that $ \mc J_{K_{u_\la}}(z)\geq \mc J_{K_{u_\la}}(u_\la)$ for every $z \in K_{u_\la}$ with $\|z-u_\la\|\leq \varsigma$. Let $\varPsi= \varPsi_\e$ for $\e$ obtained in Lemma \ref{lemsi6}. Since $ \mc J_{K_{u_\la}}(u_\la+t\varPsi)\ra -\infty$ as $t \ra \infty$, so choose $t\geq \varsigma/\|\varPsi\| $ such that $ \mc J_{K_{u_\la}}(u_\la+t\varPsi)\leq \mc J_{K_{u_\la}}(u_\la)$. Define
\begin{align*}
& \varSigma= \{ \phi \in C([0,1], D( \mc J_{K_{u_\la}}))\; : \; \phi(0) = u_\la, \phi(1)= u_\la+t\varPsi \},\\
& A = \{ z \in D( \mc J_{K_{u_\la}})\;:\; \|z-u_\la\|= \al \} \text{ and } c= \inf_{\phi \in \varSigma}\sup_{r\in [0,1]} \mc J_{K_{u_\la}}(\phi(r)).
\end{align*}
With the help of Proposition \ref{propsi4} and Lemma \ref{lemsi6}, $ \mc J_{K_{u_\la}}$ satisfies $(\text{CPS})_c$ condition. If $c= \mc J_{K_{u_\la}}(u_\la)= \inf \mc J_{K_{u_\la}}(A)$ then $u_\la \not \in A,\; u_\la +t\varPsi \not \in A,
\; \inf \mc J_{K_{u_\la}}(A) \geq \mc J_{K_{u_\la}}(u_\la) \geq \mc J_{K_{u_\la}}(u_\la+t\varPsi)$, and for every $\phi \in \varSigma$, there exist $r \in [0,1]$ such that $\|\phi(r)-u_\la\|= \varsigma$. Thus by Theorem \ref{thmsi4}, we get there exists $z_\la \in D(\mc J_{K_{u_\la}})$ such that $z_\la \not = u_\la,\; \mc J_{K_{u_\la}}(z_\la) =c$ and $ 0 \in \pa^-\mc J_{K_{u_\la}}(z_\la)$. Using Proposition \ref{propsi2}, we obtain that $z_\la$ is positive weak solution to $(\widetilde{P_{\la}})$. \QED
\end{proof}
{\bf Proof of Theorem \ref{thmsi1}}: It follows from Theorem \ref{thmsi3}, Proposition \ref{propsi5} and Lemma \ref{lemsi2}. \QED
|
1,941,325,220,552 | arxiv |
\section*{Acknowledgments}
We thank Vaibhav Choudhary and Nikhil Gopalkrishnan for informative discussions.
\section{Conclusions and Future Work} \label{sec:conclusion}
In this paper, we have explored the impact of matching the likelihood function to the noise distribution on estimation accuracy in the context of the distance geometry problem (DGP). We have empirically found that matching the likelihood function to the noise distribution can offer a 3--4 dB compensation for the loss in estimation accuracy due to noisier measurements and smaller sample sizes. Moreover, matched estimation can be helpful in these scenarios, which often arise in practice.
There are several directions for future work that can be explored. For example, in practice, measurements may not be available for some edges, so the case where the numbers of edge measurements are unequal may also be considered. We may also explore the application of our approach to real-world measurements. Finally, a theoretical evaluation of the mismatch between the likelihood function and the noise distribution would be an insightful endeavor.
\section{Introduction} \label{sec:intro}
The distance geometry problem (DGP) \cite{liberti_distance_2017-1,voller_distance_2013,crippen_distance_1988} aims to determine the locations of points in Euclidean space given measurements of the distances between these points. The DGP arises in many important applications. In wireless networks with moving devices in two-dimensional (2D) planes, the positions of these devices can be estimated by solving the DGP~\cite{eren_rigidity_2004}. In computational biology, solving the DGP can allow us to determine the locations of atoms in molecules \cite{havel_evaluation_1991}. In robotics, a simple robotic movement can be modeled as a pair of rigid bars rotating along a joint. Whether these bars can flex along a joint to reach a specific point in 3D space can be determined by solving the DGP \cite{liberti_open_2018}.
The main challenge for the DGP is that pairwise distance measurements are noisy \cite{liberti_distance_2017-1}. Because of the noise, it is common to find a set of locations of points that minimizes the sum of squared errors (SSE) between the pairwise distances of a target structure and noisy measurements of these pairwise distances \cite{lavor_overview_2009}. When the noisy measurements of pairwise distances are Gaussian distributed, minimizing the SSE function yields a maximum likelihood (ML) estimate of a structure of points. However, in practice, it is unlikely that the measurements are exactly Gaussian distributed, and so the SSE function is unlikely to yield an ML estimate. In this work, we will provide better estimation quality by performing ML estimation using the correct noise distribution. We call our approach {\em matched DGP estimation}. In contrast, much of the prior art performed {\em mismatched} DGP estimation (details below).
\textbf{Problem formulation.} Let $\mathcal{X} = \{\matr{x}_1,\matr{x}_2,\dots,\matr{x}_N\}$ be an unobserved set of locations of $N$ points in $K$-dimensional Euclidean space $\mathbb{R}^K$. Additionally, let $\mathcal{E}(\X) = \{(i,j) \mid (i,j) \in \{1,\dots,N\}^2, i < j\}$ be the set of edges connecting every pair of points in $\mathcal{X}$. Note that $|\mathcal{E}(\X)| = \frac{1}{2}N(N-1)$, where $|\cdot|$ denotes the cardinality of a set. We define the length $\fnij{d}^{\X}$ of an edge $(i,j) \in \mathcal{E}(\X)$ as the $\ell_2$ norm of $\mathbf{x}_i - \mathbf{x}_j$, $\fnij{d}^{\X} = \left\| \mathbf{x}_i - \mathbf{x}_j \right\|_2$. We define $\fnij{M} \in \mathbb{N}$, where $\mathbb N$ is the set of nonnegative integers, as the number of noisy measurements of the length of edge $(i,j) \in \mathcal{E}(\X)$. An example structure $\mathcal{X}$ is shown in \autoref{fig:tri_example}. In the DGP, our objective is to determine $\mathcal{X}$ given noisy measurements of lengths of edges in $\mathcal{E}(\X)$.
\textbf{Estimating the locations of the points.} We estimate $\mathcal{X}$ using the method of ML. Let the random variable $\fnij{Y}^{[m]} \in \mathbb R^+$ represent the $m$th noisy measurement of the length of edge $(i,j)$, where a realization of $\fnij{Y}^{[m]}$ is denoted by $\fnij{y}^{[m]}$. For example, $y_{1,2}^{[8]}$ is the $8$th measurement of the length of edge $(1,2)$. We assume that $\fnij{M} = M$ for every $(i,j) \in \mathcal{E}(\X)$. Let the set of all $\fnij{Y}^{[m]}$ be, $\Y = \bigcup_{(i,j) \in \mathcal{E}(\X)} \fnij{\Y}$, where $\fnij{\Y} = \left\{\fnij{Y}^{[m]}\right\}_{m=1}^M$. The sets $\y$ and $\fnij{\y}$ are defined similarly for $\fnij{y}^{[m]}$. Our objective is to derive an expression for the likelihood function $f_{\Y} (\y \mid d_{\mathcal{E}(\X)})$, where $d_{\mathcal{E}(\X)} = \left\{\fnij{d}^{\X}\right\}_{(i,j) \in \mathcal{E}(\X)}$.
Suppose that the measurements for edge $(i_1,j_1)$ are independent of the measurements for edge $(i_2,j_2)$ for every $(i_1,j_1),(i_2,j_2) \in \mathcal{E}(\X)$, and that $\mathbb{E}\left[\fnij{Y}^{[m]}\right] = \fnij{d}^{\X}$ for every $m = 1,\dots,M$. Under these assumptions, the likelihood can be expressed as, $f_{\Y} (\y \mid d_{\mathcal{E}(\X)}) = \prod_{(i,j) \in \mathcal{E}(\X)} f_{\fnij{\Y}}\left(\fnij{\y} \bigm| \fnij{d}^{\X}\right)$. Additionally, suppose that all measurements associated with edge $(i,j)$ are independent and identically distributed (i.i.d.). The likelihood can be further simplified,
\begin{equation} \label{eq:ll}
f_{\Y} \left(\y \mid d_{\mathcal{E}(\X)}\right) = \prod_{(i,j) \in \mathcal{E}(\X)} \prod_{m = 1}^{M} f_{\fnij{Y}^{[m]}}\left(\fnij{y}^{[m]} \Bigm| \fnij{d}^{\X}\right),
\end{equation}
where $f_{\fnij{Y}^{[m]}}\left(\fnij{y}^{[m]} \Bigm| \fnij{d}^{\X}\right)$ is the \textit{true} marginal pdf of $\fnij{Y}^{[m]}$. In the rest of the paper, we will use $f_{\fnij{Y}^{[m]}}\left(\fnij{y}^{[m]} \Bigm| \fnij{d}^{\X},\fnij{\bm{\theta}}\right)$, where $\fnij{\bm{\theta}}$ is a vector of shape parameters, to denote the marginal pdf for $\fnij{Y}^{[m]}$. When a single shape parameter describes this marginal pdf, the scalar version $\fnij{\theta}$ will be used instead.
Given a set of noisy edge length measurements $\y$, an ML estimate of $\mathcal{X}$ can be obtained by maximizing \eqref{eq:ll}. This case will be referred to as matched DGP estimation. In practice, we often do not know what parametric form the marginal pdf of $\fnij{Y}^{[m]}$ takes. In this case, we would modify~ \!\eqref{eq:ll} by replacing the true marginal pdf of $\fnij{Y}^{[m]}$ with an \textit{assumed} version, $f_{\fnij{Y}^{[m]}}^*\left(\fnij{y}^{[m]} \Bigm| \fnij{d}^{\X}\right)$. If $f_{\fnij{Y}^{[m]}}^*\left(\fnij{y}^{[m]} \Bigm| \fnij{d}^{\X}\right) \neq f_{\fnij{Y}^{[m]}}\left(\fnij{y}^{[m]} \Bigm| \fnij{d}^{\X}\right)$, then we refer to the modified version of the likelihood as the \textit{mismatched likelihood}. Moreover, an estimate of $\mathcal{X}$ that is obtained by maximizing the mismatched likelihood will not necessarily be an ML estimate. We refer to this case as mismatched DGP estimation. We are interested in the difference in quality of estimates of $\mathcal{X}$ in the matched and mismatched cases.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{sections/intro/figures/simple_example.pdf}
\caption{Example structure $\mathcal{X}$ comprised of 3 points, $\mathbf{x}_1,\mathbf{x}_2,\mathbf{x}_3 \in \mathbb R^2$.}
\label{fig:tri_example}
\end{figure}
\textbf{Limitations of least-squares.} Suppose that $\fnij{Y}^{[m]}$ follows a Gaussian distribution with mean $\fnij{d}^{\X}$ and variance $\fnij{\theta}$, namely, \[f_{\fnij{Y}^{[m]}}\left(\fnij{y}^{[m]} \Bigm| \fnij{d}^{\X}, \fnij{\theta}\right) = \frac{1}{\sqrt{2\pi\fnij{\theta}}} \exp\left(-\frac{(\fnij{y}^{[m]}-\fnij{d}^{\X})^2}{2\fnij{\theta}}\right).\] Then, maximizing the likelihood $f_{\Y} \left(\y \mid d_{\mathcal{E}(\X)}\right)$ with respect to (w.r.t.) $\mathcal{X}$ is equivalent to minimizing the SSE w.r.t. $\mathcal{X}$. Several approaches to solve the DGP in the case of noisy measurements use variants of the SSE function \cite{crippen_distance_1988,hendrickson_molecule_1995,more_global_1997-2}. A survey of prior art appears in \autoref{sec:related_work}.
The aforementioned equivalence between minimizing the SSE function and maximizing the likelihood derived in \eqref{eq:ll} is only true when $\fnij{Y}^{[m]}$ is Gaussian. In practice, depending on the measurement technique, it is unlikely that the noisy measurement $\fnij{Y}^{[m]}$ is Gaussian distributed. Therefore, it is unlikely that minimizing the SSE function will yield an ML estimate of a structure.
\textbf{Contributions.} In this paper, we empirically show that, compared to mismatched DGP estimation, matched DGP estimation can offer an improvement of several dB. Moreover, our contribution is that, to the best of our knowledge, and in the context of the DGP, we are the first to focus on ML estimation beyond Gaussian-distributed measurements. The rest of the paper is structured as follows: we review the prior art in \autoref{sec:related_work}. The different marginal pdfs of $\fnij{Y}^{[m]}$ that will be considered, and how estimates of $\mathcal{X}$ will be evaluated, are discussed in \autoref{sec:proposed_method}. Experimental results are presented in \autoref{sec:results}. Finally, we summarize our results and describe future work in \autoref{sec:conclusion}\!.
\section{Noise Distributions and Evaluation} \label{sec:proposed_method}
We will consider different forms of the marginal pdf of $\fnij{Y}^{[m]}$, as included in \eqref{eq:ll}, that result in a likelihood $f_{\Y} \left(\y \mid d_{\mathcal{E}(\X)}\right)$ that, when maximized w.r.t. $\mathcal{X}$, yields ML estimates. It was previously shown that if $\fnij{Y}^{[m]}$ follows a Gaussian distribution, then minimizing the SSE function is equivalent to maximizing the likelihood in \eqref{eq:ll}. We consider the cases when $\fnij{Y}^{[m]}$ follows either a Laplace or non-standardized Student's $t$ (NSST) distribution.
If $\fnij{Y}^{[m]}$ follows a Laplace distribution with mean $\fnij{d}^{\X}$ and variance $2\fnij{\theta}^2$, then
\begin{equation} \label{eq:laplace_pdf}
f_{\fnij{Y}^{[m]}}\left(\fnij{y}^{[m]} \Bigm| \fnij{d}^{\X}, \fnij{\theta}\right) = \frac{1}{2\fnij{\theta}} e^{-\left\lvert \fnij{y}^{[m]}-\fnij{d}^{\X}\right\rvert \big/ \fnij{\theta}}.
\end{equation}
If $\fnij{Y}^{[m]}$ follows an NSST distribution with mean $\fnij{d}^{\X}$ and variance $\fnij{b}^2\fnij{\nu}/(\fnij{\nu}-2)$, assuming $\fnij{\nu} > 2$, then an expression for the marginal pdf of $\fnij{Y}^{[m]}$ can be derived as follows. Let $W$ follow a Student's $t$ distribution with parameter $\fnij{\nu}$, which is a positive integer. The pdf of $W$ is
\begin{equation*}
f_W(w \mid \fnij{\nu}) = \frac{\Gamma\left((\fnij{\nu} + 1)/2\right)}{\Gamma\left(\fnij{\nu}/2\right)\sqrt{\fnij{\nu} \pi}} \left(1 + (w^2 / \fnij{\nu})\right)^{-(\fnij{\nu} + 1)/2},
\end{equation*}
where $\Gamma(z) = \int_0^\infty t^{z - 1}e^{-t} \, dt$ for every $0 < z \in \mathbb R$. If $\fnij{Y}^{[m]} = \fnij{b}W + \fnij{d}^{\X}$, then
\begin{equation} \label{eq:nsst_pdf}
f_{\fnij{Y}^{[m]}}\left(\fnij{y}^{[m]} \Bigm| \fnij{d}^{\X}, \fnij{\bm{\theta}}\right) = \frac{1}{\fnij{b}}f_W\left(\frac{\fnij{y}^{[m]} - \fnij{d}^{\X}}{\fnij{b}} \Biggm| \fnij{\nu}\right),
\end{equation}
where $\fnij{\bm{\theta}} = (\fnij{\nu},\fnij{b})$.
\textbf{Evaluation.} Given an estimate $\widehat{\mathcal{X}}$ of $\mathcal{X}$, if we rotate, reflect, and/or translate the points in $\widehat{\mathcal{X}}$ by an equal amount, then $\widehat{\mathcal{X}}$ will still have the same edge lengths as $\mathcal{X}$. Therefore, a unique solution to the DGP cannot exist without additional constraints. Moreover, the proximity of $\widehat{\mathcal{X}}$ to $\mathcal{X}$ will need to be evaluated up to rotation, translation, and reflection. Let $\mathbf{X}$ and $\widehat{\mathbf{X}}$ be $K \times N$ matrices, where the $n$th columns of $\matr{X}$ and $\widehat{\matr{X}}$ correspond to the $n$th points in $\mathcal{X}$ and $\widehat{\mathcal{X}}$, respectively. We first translate the column vectors in $\mathbf{X}$ and $\widehat{\mathbf{X}}$ so that they are centered at the origin by computing $\mathbf{X}_c = \mathbf{X} - \mathbf{X}\matr{1} \big/ N$ and $\widehat{\mathbf{X}}_c = \widehat{\mathbf{X}} - \widehat{\mathbf{X}}\matr{1}\big/ N$, where $\matr{1}$ is an $N \times N$ matrix of ones.
Next, how close $\widehat{\mathcal{X}}$ is to $\mathcal{X}$ is equivalent to the solution of the orthogonal Procrustes problem (OPP) \cite{schonemann_generalized_1996},
\begin{equation} \label{eq:orth_procrustes}
\begin{aligned}
\min_{\matr{R}}
\quad & \left\| \mathbf{R}\widehat{\mathbf{X}}_c - \mathbf{X}_c \right\|_F, \\
\textrm{s.t.} \quad & \mathbf{R}^T \mathbf{R} = \matr{I},
\end{aligned}
\end{equation}
where $\left\| \cdot \right\|_F$ is the Frobenius norm, which for a $K \times N$ matrix, $\matr{A} = (a_{ij})$, is defined as, $\left\| \matr{A} \right\|_F = \left(\sum_{k=1}^{K} \sum_{n=1}^{N} |a_{ij}|^2\right)^{1/2}$. The objective of the OPP is to determine the orthonormal matrix $\mathbf{R}$ that most closely maps $\widehat{\mathbf{X}}_c$ to $\mathbf{X}_c$. We will refer to the value obtained by solving the OPP for an estimate $\widehat{\mathcal{X}}$, divided by the number of points $N$, as the \emph{OPP loss} associated with $\widehat{\mathcal{X}}$. As our objective is to show that an ML estimate $\widehat{\mathcal{X}}_{\text{ML}}$ of $\mathcal{X}$ more closely approximates $\mathcal{X}$ than a non-ML estimate $\widehat{\mathcal{X}}$, we will compare the OPP losses of $\widehat{\mathcal{X}}_{\text{ML}}$ and $\widehat{\mathcal{X}}$.
\section{Related Work} \label{sec:related_work}
One of the earliest discussions of cost functions related to the solution of the DGP was due to Crippen and Havel \cite{crippen_distance_1988}. In their work, they used a variant of the SSE function, which we refer to as SSE-CH, to estimate molecular structures with physical constraints. Although others have empirically shown that SSE-CH yields acceptable results \cite{crippen_distance_1988}, the resulting estimate of a structure need not be an ML estimate.
Subsequent papers implement modifications to the SSE-CH cost function. For example, Mor\'e and Wu \cite{more_global_1997-2} adopted a continuation approach: the SSE-CH cost function is repeatedly smoothed via a convolution with a Gaussian kernel. After each successive smoothing stage, a minimizer of the smoothed SSE-CH is obtained. Eventually, the minimizers are traced back to the original function \cite[p. 827]{more_global_1997-2}. Although this approach yielded promising results, the underlying cost function that is smoothed is still the SSE function. Moreover, there is no guarantee that the resulting estimate of $\mathcal{X}$ is an ML estimate.
Souza et al. \cite{souza_hyperbolic_2011} attempt to make SSE-CH differentiable by approximating certain parts using hyperbolic functions. Souza et al. \cite{souza_solving_2013} made further improvements by applying these approximations to the \textit{geometric buildup} algorithm proposed by Luo and Wu \cite{luo_least-squares_2011}. However, the underlying cost function is still a variant of the SSE function, and so there is no guarantee of obtaining ML estimates. For a detailed review of approaches to solve the DGP, see \cite{lavor_overview_2009}. More recently, Gopalkrishnan et al. \cite{gopalkrishnan_dna_2020-1} optimize a variation of SSE-CH to reconstruct DNA molecules from pairwise distance measurements. Giamou et al. \cite{giamou_convex_2022} also optimize a variation of SSE-CH to determine the joint angles of a robot given the robot's forward kinematics and desired poses. However, because variations of SSE-CH are used, it is unlikely that estimates obtained in these applications are ML estimates.
\section{Numerical Results} \label{sec:results}
\begin{figure}[!t]
\centering
\subfloat[]{\includegraphics[width=0.9\columnwidth]{sections/results/figures/3pts1.pdf}
\label{fig:tri1}}
\\
\subfloat[]{\includegraphics[width=0.9\columnwidth]{sections/results/figures/3pts2.pdf}
\label{fig:tri2}}
\caption{We illustrate 2 of the 8 triangles that were considered. Ground truth structures are labeled $\mathcal{X}$, while example estimates, together with their associated OPP losses, are also shown. We observe that estimates with lower OPP losses more closely approximate the structure $\mathcal{X}$.}
\label{fig:tris}
\end{figure}
We begin our discussion with numerical results on triangles, which are structures consisting of 3 points (see \autoref{fig:tris}). Later, we demonstrate analogous numerical results for structures with 10 points (see \autoref{fig:10pts}). In our experiments, we let $\fnij{Y}^{[m]}$ follow either a Laplace \eqref{eq:laplace_pdf} or NSST \eqref{eq:nsst_pdf} distribution. If $\fnij{Y}^{[m]}$ follows a Laplace (respectively, NSST) distribution, then the marginal pdf of $\fnij{Y}^{[m]}$ in the likelihood \eqref{eq:ll} is varied between the Laplace (respectively, NSST) pdf in the matched case and the Gaussian pdf in the mismatched case.
The variance of the noisy measurements was changed as a function of different signal-to-noise ratio (SNR) values. If the distribution from which the i.i.d. noisy measurements associated with edge $(i,j)$ were obtained has variance $\fnij{\sigma}^2$, and if we let $\sigma_\mathcal{X}$ be the average edge length of a structure $\mathcal{X}$, then $\text{SNR} = 10 \log_{10}\left(\sigma_\mathcal{X}^2 / \fnij{\sigma}^2 \right)$, where SNR is quantified in decibel (dB) units. We leave the case where $\fnij{\sigma}^2$ depends on edge $(i,j)$ for future work. The SNR was varied from --20 dB to 20 dB in steps of 5 dB. For each SNR value, we sample a set of noisy measurements, maximize the likelihood\footnote{An implementation of the L-BFGS algorithm \cite{nocedal_quasi-newton_2006} that is available in PyTorch \cite{paszke_pytorch_2019-1} was used to maximize the likelihood.} in \eqref{eq:ll} for the Laplace (or NSST) cases, and then report the OPP loss associated with an estimated structure $\widehat{\mathcal{X}}$ for each case.
\begin{figure*}[!t]
\centering
\subfloat[]{\includegraphics[width=0.66\columnwidth]{sections/results/figures/3pts_M10_laplace.pdf}
\label{fig:3pts_M10_laplace}}
\hfil
\subfloat[]{\includegraphics[width=0.66\columnwidth]{sections/results/figures/3pts_M10_nsst.pdf}
\label{fig:3pts_M10_nsst}}
\hfil
\subfloat[]{\includegraphics[width=0.67\columnwidth]{sections/results/figures/3pts_M10_diff.pdf}
\label{fig:3pts_M10_diff}}
\caption{Distributions of OPP losses for the 8 triangles when $\fnij{Y}^{[m]}$ follows a \protect\subref{fig:3pts_M10_laplace} Laplace or \protect\subref{fig:3pts_M10_nsst} NSST distribution, and when matched and mismatched (Gaussian) likelihood (LL) functions are used. Each box represents the percentiles (bottom to top): 10, 25, 50, 75, and 90. Distributions of pairwise differences in OPP loss when $\fnij{Y}^{[m]}$ follows a Laplace or NSST distribution are shown in \protect\subref{fig:3pts_M10_diff}. Pairwise differences are computed by subtracting the OPP loss for the matched estimate from the corresponding OPP loss for the mismatched (Gaussian LL function) estimate.}
\label{fig:3pts_M10}
\end{figure*}
\textbf{Triangles.} We first considered 8 triangles in 2D. Two of these triangles, together with example estimates and their associated OPP losses, are shown in \autoref{fig:tris}. For each SNR value, and for each triangle, we sampled $M = 10$ noisy measurements per edge 100 times. For each sample, we computed an estimate $\widehat{\mathcal{X}}$ using different likelihoods and then computed the OPP loss associated with $\widehat{\mathcal{X}}$. The distributions of these OPP losses are shown in \autoref{fig:3pts_M10_laplace} and \autoref{fig:3pts_M10_nsst}. We also computed the pairwise difference in OPP losses resulting from mismatched and matched estimation. The distributions of these differences are shown in \autoref{fig:3pts_M10_diff}.
We observe that, in general, the distribution of OPP losses associated with the likelihood that is matched to the noise distribution is shifted lower than the distribution associated with the likelihood that is not matched to the noise distribution. Moreover, in several instances, the increase in median OPP loss due to a 3--4 dB decrease in SNR can approximately be compensated for by choosing to use the likelihood that is matched to the noise distribution.
\begin{figure}[!t]
\centering
\subfloat[]{\includegraphics[trim={2.65mm 0mm 0mm 2.5mm},clip,
width=0.532\columnwidth]{sections/results/figures/3pts_Ms_laplace.pdf}
\label{fig:3pts_Ms_laplace}}
\hspace{-2.3mm}
\subfloat[]{\includegraphics[trim={0cm 1.2mm 2.3mm 2mm},clip,
width=0.441\columnwidth]{sections/results/figures/3pts_Ms_nsst.pdf}
\label{fig:3pts_Ms_nsst}}
\caption{Median OPP losses for the 8 triangles and for different $M$ values when $\fnij{Y}^{[m]}$ follows a \protect\subref{fig:3pts_Ms_laplace} Laplace or \protect\subref{fig:3pts_Ms_nsst} NSST distribution, and when matched or mismatched (Gaussian) likelihood functions are used.}
\label{fig:3pts_Ms}
\end{figure}
We then repeated this experiment for different values of $M$, the number of noisy measurements per edge. The median OPP losses for $M = 10,25,50,100$ are shown in \autoref{fig:3pts_Ms}. We observe that, in general, maximizing the likelihood that is matched to the noise distribution offers an improvement over mismatched DGP estimation comparable to the improvement obtained by doubling the number of noisy measurements per edge.
\begin{figure}[t]
\centering
\includegraphics[trim={0cm 2.5mm 0cm 0.25cm},clip,
width=0.7\columnwidth]{sections/results/figures/10pts.pdf} \vspace{-2mm}
\caption{One of the 30 10-point structures that were considered. Example estimates are omitted for clarity.}
\label{fig:10pts}
\end{figure}
\begin{figure*}[!t]
\centering
\subfloat[]{\includegraphics[width=0.66\columnwidth]{sections/results/figures/10pts_M10_laplace.pdf}
\label{fig:10pts_M10_laplace}}
\hfil
\subfloat[]{\includegraphics[width=0.66\columnwidth]{sections/results/figures/10pts_M10_nsst.pdf}
\label{fig:10pts_M10_nsst}}
\hfil
\subfloat[]{\includegraphics[width=0.67\columnwidth]{sections/results/figures/10pts_M10_diff.pdf}
\label{fig:10pts_M10_diff}}
\caption{Distributions of OPP losses for the 30 10-point structures when $\fnij{Y}^{[m]}$ follows a \protect\subref{fig:10pts_M10_laplace} Laplace or \protect\subref{fig:10pts_M10_nsst} NSST distribution, and when matched and mismatched (Gaussian) likelihood (LL) functions are used. Percentiles are marked as in \autoref{fig:3pts_M10}. Distributions of pairwise differences in OPP loss when $\fnij{Y}^{[m]}$ follows a Laplace or NSST distribution are shown in \protect\subref{fig:10pts_M10_diff}. Pairwise differences are computed by subtracting the OPP loss for the matched estimate from the corresponding OPP loss for the mismatched (Gaussian LL function) estimate.}
\label{fig:10pts_M10}
\end{figure*}
\textbf{10 points.} To further support our results, we repeated our experiments for 30 10-point structures in 2D. One of these structures is shown in \autoref{fig:10pts}. The distributions of OPP losses are shown in \autoref{fig:10pts_M10_laplace} and \autoref{fig:10pts_M10_nsst}, and the distributions of pairwise differences in OPP losses for the mismatched and matched cases are shown in \autoref{fig:10pts_M10_diff}. Similar to the case of triangles, we observe that, in general, the distribution of OPP losses associated with the likelihood that is matched to the noise distribution is lower than the distribution associated with the likelihood that is not matched to the noise distribution.
\section{Conclusions and Future Work} \label{sec:conclusion}
In this paper, we have explored the impact of matching the likelihood function to the noise distribution on estimation accuracy in the context of the distance geometry problem (DGP). We have empirically found that matching the likelihood function to the noise distribution can offer a 3--4 dB compensation for the loss in estimation accuracy due to noisier measurements and smaller sample sizes. Moreover, matched estimation can be helpful in these scenarios, which often arise in practice.
There are several directions for future work that can be explored. For example, in practice, measurements may not be available for some edges, so the case where the numbers of edge measurements are unequal may also be considered. We may also explore the application of our approach to real-world measurements. Finally, a theoretical evaluation of the mismatch between the likelihood function and the noise distribution would be an insightful endeavor.
\section{Introduction} \label{sec:intro}
The distance geometry problem (DGP) \cite{liberti_distance_2017-1,voller_distance_2013,crippen_distance_1988} aims to determine the locations of points in Euclidean space given measurements of the distances between these points. The DGP arises in many important applications. In wireless networks with moving devices in two-dimensional (2D) planes, the positions of these devices can be estimated by solving the DGP~\cite{eren_rigidity_2004}. In computational biology, solving the DGP can allow us to determine the locations of atoms in molecules \cite{havel_evaluation_1991}. In robotics, a simple robotic movement can be modeled as a pair of rigid bars rotating along a joint. Whether these bars can flex along a joint to reach a specific point in 3D space can be determined by solving the DGP \cite{liberti_open_2018}.
The main challenge for the DGP is that pairwise distance measurements are noisy \cite{liberti_distance_2017-1}. Because of the noise, it is common to find a set of locations of points that minimizes the sum of squared errors (SSE) between the pairwise distances of a target structure and noisy measurements of these pairwise distances \cite{lavor_overview_2009}. When the noisy measurements of pairwise distances are Gaussian distributed, minimizing the SSE function yields a maximum likelihood (ML) estimate of a structure of points. However, in practice, it is unlikely that the measurements are exactly Gaussian distributed, and so the SSE function is unlikely to yield an ML estimate. In this work, we will provide better estimation quality by performing ML estimation using the correct noise distribution. We call our approach {\em matched DGP estimation}. In contrast, much of the prior art performed {\em mismatched} DGP estimation (details below).
\textbf{Problem formulation.} Let $\mathcal{X} = \{\matr{x}_1,\matr{x}_2,\dots,\matr{x}_N\}$ be an unobserved set of locations of $N$ points in $K$-dimensional Euclidean space $\mathbb{R}^K$. Additionally, let $\mathcal{E}(\X) = \{(i,j) \mid (i,j) \in \{1,\dots,N\}^2, i < j\}$ be the set of edges connecting every pair of points in $\mathcal{X}$. Note that $|\mathcal{E}(\X)| = \frac{1}{2}N(N-1)$, where $|\cdot|$ denotes the cardinality of a set. We define the length $\fnij{d}^{\X}$ of an edge $(i,j) \in \mathcal{E}(\X)$ as the $\ell_2$ norm of $\mathbf{x}_i - \mathbf{x}_j$, $\fnij{d}^{\X} = \left\| \mathbf{x}_i - \mathbf{x}_j \right\|_2$. We define $\fnij{M} \in \mathbb{N}$, where $\mathbb N$ is the set of nonnegative integers, as the number of noisy measurements of the length of edge $(i,j) \in \mathcal{E}(\X)$. An example structure $\mathcal{X}$ is shown in \autoref{fig:tri_example}. In the DGP, our objective is to determine $\mathcal{X}$ given noisy measurements of lengths of edges in $\mathcal{E}(\X)$.
\textbf{Estimating the locations of the points.} We estimate $\mathcal{X}$ using the method of ML. Let the random variable $\fnij{Y}^{[m]} \in \mathbb R^+$ represent the $m$th noisy measurement of the length of edge $(i,j)$, where a realization of $\fnij{Y}^{[m]}$ is denoted by $\fnij{y}^{[m]}$. For example, $y_{1,2}^{[8]}$ is the $8$th measurement of the length of edge $(1,2)$. We assume that $\fnij{M} = M$ for every $(i,j) \in \mathcal{E}(\X)$. Let the set of all $\fnij{Y}^{[m]}$ be, $\Y = \bigcup_{(i,j) \in \mathcal{E}(\X)} \fnij{\Y}$, where $\fnij{\Y} = \left\{\fnij{Y}^{[m]}\right\}_{m=1}^M$. The sets $\y$ and $\fnij{\y}$ are defined similarly for $\fnij{y}^{[m]}$. Our objective is to derive an expression for the likelihood function $f_{\Y} (\y \mid d_{\mathcal{E}(\X)})$, where $d_{\mathcal{E}(\X)} = \left\{\fnij{d}^{\X}\right\}_{(i,j) \in \mathcal{E}(\X)}$.
Suppose that the measurements for edge $(i_1,j_1)$ are independent of the measurements for edge $(i_2,j_2)$ for every $(i_1,j_1),(i_2,j_2) \in \mathcal{E}(\X)$, and that $\mathbb{E}\left[\fnij{Y}^{[m]}\right] = \fnij{d}^{\X}$ for every $m = 1,\dots,M$. Under these assumptions, the likelihood can be expressed as, $f_{\Y} (\y \mid d_{\mathcal{E}(\X)}) = \prod_{(i,j) \in \mathcal{E}(\X)} f_{\fnij{\Y}}\left(\fnij{\y} \bigm| \fnij{d}^{\X}\right)$. Additionally, suppose that all measurements associated with edge $(i,j)$ are independent and identically distributed (i.i.d.). The likelihood can be further simplified,
\begin{equation} \label{eq:ll}
f_{\Y} \left(\y \mid d_{\mathcal{E}(\X)}\right) = \prod_{(i,j) \in \mathcal{E}(\X)} \prod_{m = 1}^{M} f_{\fnij{Y}^{[m]}}\left(\fnij{y}^{[m]} \Bigm| \fnij{d}^{\X}\right),
\end{equation}
where $f_{\fnij{Y}^{[m]}}\left(\fnij{y}^{[m]} \Bigm| \fnij{d}^{\X}\right)$ is the \textit{true} marginal pdf of $\fnij{Y}^{[m]}$. In the rest of the paper, we will use $f_{\fnij{Y}^{[m]}}\left(\fnij{y}^{[m]} \Bigm| \fnij{d}^{\X},\fnij{\bm{\theta}}\right)$, where $\fnij{\bm{\theta}}$ is a vector of shape parameters, to denote the marginal pdf for $\fnij{Y}^{[m]}$. When a single shape parameter describes this marginal pdf, the scalar version $\fnij{\theta}$ will be used instead.
Given a set of noisy edge length measurements $\y$, an ML estimate of $\mathcal{X}$ can be obtained by maximizing \eqref{eq:ll}. This case will be referred to as matched DGP estimation. In practice, we often do not know what parametric form the marginal pdf of $\fnij{Y}^{[m]}$ takes. In this case, we would modify~ \!\eqref{eq:ll} by replacing the true marginal pdf of $\fnij{Y}^{[m]}$ with an \textit{assumed} version, $f_{\fnij{Y}^{[m]}}^*\left(\fnij{y}^{[m]} \Bigm| \fnij{d}^{\X}\right)$. If $f_{\fnij{Y}^{[m]}}^*\left(\fnij{y}^{[m]} \Bigm| \fnij{d}^{\X}\right) \neq f_{\fnij{Y}^{[m]}}\left(\fnij{y}^{[m]} \Bigm| \fnij{d}^{\X}\right)$, then we refer to the modified version of the likelihood as the \textit{mismatched likelihood}. Moreover, an estimate of $\mathcal{X}$ that is obtained by maximizing the mismatched likelihood will not necessarily be an ML estimate. We refer to this case as mismatched DGP estimation. We are interested in the difference in quality of estimates of $\mathcal{X}$ in the matched and mismatched cases.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{sections/intro/figures/simple_example.pdf}
\caption{Example structure $\mathcal{X}$ comprised of 3 points, $\mathbf{x}_1,\mathbf{x}_2,\mathbf{x}_3 \in \mathbb R^2$.}
\label{fig:tri_example}
\end{figure}
\textbf{Limitations of least-squares.} Suppose that $\fnij{Y}^{[m]}$ follows a Gaussian distribution with mean $\fnij{d}^{\X}$ and variance $\fnij{\theta}$, namely, \[f_{\fnij{Y}^{[m]}}\left(\fnij{y}^{[m]} \Bigm| \fnij{d}^{\X}, \fnij{\theta}\right) = \frac{1}{\sqrt{2\pi\fnij{\theta}}} \exp\left(-\frac{(\fnij{y}^{[m]}-\fnij{d}^{\X})^2}{2\fnij{\theta}}\right).\] Then, maximizing the likelihood $f_{\Y} \left(\y \mid d_{\mathcal{E}(\X)}\right)$ with respect to (w.r.t.) $\mathcal{X}$ is equivalent to minimizing the SSE w.r.t. $\mathcal{X}$. Several approaches to solve the DGP in the case of noisy measurements use variants of the SSE function \cite{crippen_distance_1988,hendrickson_molecule_1995,more_global_1997-2}. A survey of prior art appears in \autoref{sec:related_work}.
The aforementioned equivalence between minimizing the SSE function and maximizing the likelihood derived in \eqref{eq:ll} is only true when $\fnij{Y}^{[m]}$ is Gaussian. In practice, depending on the measurement technique, it is unlikely that the noisy measurement $\fnij{Y}^{[m]}$ is Gaussian distributed. Therefore, it is unlikely that minimizing the SSE function will yield an ML estimate of a structure.
\textbf{Contributions.} In this paper, we empirically show that, compared to mismatched DGP estimation, matched DGP estimation can offer an improvement of several dB. Moreover, our contribution is that, to the best of our knowledge, and in the context of the DGP, we are the first to focus on ML estimation beyond Gaussian-distributed measurements. The rest of the paper is structured as follows: we review the prior art in \autoref{sec:related_work}. The different marginal pdfs of $\fnij{Y}^{[m]}$ that will be considered, and how estimates of $\mathcal{X}$ will be evaluated, are discussed in \autoref{sec:proposed_method}. Experimental results are presented in \autoref{sec:results}. Finally, we summarize our results and describe future work in \autoref{sec:conclusion}\!.
\section{Noise Distributions and Evaluation} \label{sec:proposed_method}
We will consider different forms of the marginal pdf of $\fnij{Y}^{[m]}$, as included in \eqref{eq:ll}, that result in a likelihood $f_{\Y} \left(\y \mid d_{\mathcal{E}(\X)}\right)$ that, when maximized w.r.t. $\mathcal{X}$, yields ML estimates. It was previously shown that if $\fnij{Y}^{[m]}$ follows a Gaussian distribution, then minimizing the SSE function is equivalent to maximizing the likelihood in \eqref{eq:ll}. We consider the cases when $\fnij{Y}^{[m]}$ follows either a Laplace or non-standardized Student's $t$ (NSST) distribution.
If $\fnij{Y}^{[m]}$ follows a Laplace distribution with mean $\fnij{d}^{\X}$ and variance $2\fnij{\theta}^2$, then
\begin{equation} \label{eq:laplace_pdf}
f_{\fnij{Y}^{[m]}}\left(\fnij{y}^{[m]} \Bigm| \fnij{d}^{\X}, \fnij{\theta}\right) = \frac{1}{2\fnij{\theta}} e^{-\left\lvert \fnij{y}^{[m]}-\fnij{d}^{\X}\right\rvert \big/ \fnij{\theta}}.
\end{equation}
If $\fnij{Y}^{[m]}$ follows an NSST distribution with mean $\fnij{d}^{\X}$ and variance $\fnij{b}^2\fnij{\nu}/(\fnij{\nu}-2)$, assuming $\fnij{\nu} > 2$, then an expression for the marginal pdf of $\fnij{Y}^{[m]}$ can be derived as follows. Let $W$ follow a Student's $t$ distribution with parameter $\fnij{\nu}$, which is a positive integer. The pdf of $W$ is
\begin{equation*}
f_W(w \mid \fnij{\nu}) = \frac{\Gamma\left((\fnij{\nu} + 1)/2\right)}{\Gamma\left(\fnij{\nu}/2\right)\sqrt{\fnij{\nu} \pi}} \left(1 + (w^2 / \fnij{\nu})\right)^{-(\fnij{\nu} + 1)/2},
\end{equation*}
where $\Gamma(z) = \int_0^\infty t^{z - 1}e^{-t} \, dt$ for every $0 < z \in \mathbb R$. If $\fnij{Y}^{[m]} = \fnij{b}W + \fnij{d}^{\X}$, then
\begin{equation} \label{eq:nsst_pdf}
f_{\fnij{Y}^{[m]}}\left(\fnij{y}^{[m]} \Bigm| \fnij{d}^{\X}, \fnij{\bm{\theta}}\right) = \frac{1}{\fnij{b}}f_W\left(\frac{\fnij{y}^{[m]} - \fnij{d}^{\X}}{\fnij{b}} \Biggm| \fnij{\nu}\right),
\end{equation}
where $\fnij{\bm{\theta}} = (\fnij{\nu},\fnij{b})$.
\textbf{Evaluation.} Given an estimate $\widehat{\mathcal{X}}$ of $\mathcal{X}$, if we rotate, reflect, and/or translate the points in $\widehat{\mathcal{X}}$ by an equal amount, then $\widehat{\mathcal{X}}$ will still have the same edge lengths as $\mathcal{X}$. Therefore, a unique solution to the DGP cannot exist without additional constraints. Moreover, the proximity of $\widehat{\mathcal{X}}$ to $\mathcal{X}$ will need to be evaluated up to rotation, translation, and reflection. Let $\mathbf{X}$ and $\widehat{\mathbf{X}}$ be $K \times N$ matrices, where the $n$th columns of $\matr{X}$ and $\widehat{\matr{X}}$ correspond to the $n$th points in $\mathcal{X}$ and $\widehat{\mathcal{X}}$, respectively. We first translate the column vectors in $\mathbf{X}$ and $\widehat{\mathbf{X}}$ so that they are centered at the origin by computing $\mathbf{X}_c = \mathbf{X} - \mathbf{X}\matr{1} \big/ N$ and $\widehat{\mathbf{X}}_c = \widehat{\mathbf{X}} - \widehat{\mathbf{X}}\matr{1}\big/ N$, where $\matr{1}$ is an $N \times N$ matrix of ones.
Next, how close $\widehat{\mathcal{X}}$ is to $\mathcal{X}$ is equivalent to the solution of the orthogonal Procrustes problem (OPP) \cite{schonemann_generalized_1996},
\begin{equation} \label{eq:orth_procrustes}
\begin{aligned}
\min_{\matr{R}}
\quad & \left\| \mathbf{R}\widehat{\mathbf{X}}_c - \mathbf{X}_c \right\|_F, \\
\textrm{s.t.} \quad & \mathbf{R}^T \mathbf{R} = \matr{I},
\end{aligned}
\end{equation}
where $\left\| \cdot \right\|_F$ is the Frobenius norm, which for a $K \times N$ matrix, $\matr{A} = (a_{ij})$, is defined as, $\left\| \matr{A} \right\|_F = \left(\sum_{k=1}^{K} \sum_{n=1}^{N} |a_{ij}|^2\right)^{1/2}$. The objective of the OPP is to determine the orthonormal matrix $\mathbf{R}$ that most closely maps $\widehat{\mathbf{X}}_c$ to $\mathbf{X}_c$. We will refer to the value obtained by solving the OPP for an estimate $\widehat{\mathcal{X}}$, divided by the number of points $N$, as the \emph{OPP loss} associated with $\widehat{\mathcal{X}}$. As our objective is to show that an ML estimate $\widehat{\mathcal{X}}_{\text{ML}}$ of $\mathcal{X}$ more closely approximates $\mathcal{X}$ than a non-ML estimate $\widehat{\mathcal{X}}$, we will compare the OPP losses of $\widehat{\mathcal{X}}_{\text{ML}}$ and $\widehat{\mathcal{X}}$.
\section{Numerical Results} \label{sec:results}
\begin{figure}[!t]
\centering
\subfloat[]{\includegraphics[width=0.9\columnwidth]{sections/results/figures/3pts1.pdf}
\label{fig:tri1}}
\\
\subfloat[]{\includegraphics[width=0.9\columnwidth]{sections/results/figures/3pts2.pdf}
\label{fig:tri2}}
\caption{We illustrate 2 of the 8 triangles that were considered. Ground truth structures are labeled $\mathcal{X}$, while example estimates, together with their associated OPP losses, are also shown. We observe that estimates with lower OPP losses more closely approximate the structure $\mathcal{X}$.}
\label{fig:tris}
\end{figure}
We begin our discussion with numerical results on triangles, which are structures consisting of 3 points (see \autoref{fig:tris}). Later, we demonstrate analogous numerical results for structures with 10 points (see \autoref{fig:10pts}). In our experiments, we let $\fnij{Y}^{[m]}$ follow either a Laplace \eqref{eq:laplace_pdf} or NSST \eqref{eq:nsst_pdf} distribution. If $\fnij{Y}^{[m]}$ follows a Laplace (respectively, NSST) distribution, then the marginal pdf of $\fnij{Y}^{[m]}$ in the likelihood \eqref{eq:ll} is varied between the Laplace (respectively, NSST) pdf in the matched case and the Gaussian pdf in the mismatched case.
The variance of the noisy measurements was changed as a function of different signal-to-noise ratio (SNR) values. If the distribution from which the i.i.d. noisy measurements associated with edge $(i,j)$ were obtained has variance $\fnij{\sigma}^2$, and if we let $\sigma_\mathcal{X}$ be the average edge length of a structure $\mathcal{X}$, then $\text{SNR} = 10 \log_{10}\left(\sigma_\mathcal{X}^2 / \fnij{\sigma}^2 \right)$, where SNR is quantified in decibel (dB) units. We leave the case where $\fnij{\sigma}^2$ depends on edge $(i,j)$ for future work. The SNR was varied from --20 dB to 20 dB in steps of 5 dB. For each SNR value, we sample a set of noisy measurements, maximize the likelihood\footnote{An implementation of the L-BFGS algorithm \cite{nocedal_quasi-newton_2006} that is available in PyTorch \cite{paszke_pytorch_2019-1} was used to maximize the likelihood.} in \eqref{eq:ll} for the Laplace (or NSST) cases, and then report the OPP loss associated with an estimated structure $\widehat{\mathcal{X}}$ for each case.
\begin{figure*}[!t]
\centering
\subfloat[]{\includegraphics[width=0.66\columnwidth]{sections/results/figures/3pts_M10_laplace.pdf}
\label{fig:3pts_M10_laplace}}
\hfil
\subfloat[]{\includegraphics[width=0.66\columnwidth]{sections/results/figures/3pts_M10_nsst.pdf}
\label{fig:3pts_M10_nsst}}
\hfil
\subfloat[]{\includegraphics[width=0.67\columnwidth]{sections/results/figures/3pts_M10_diff.pdf}
\label{fig:3pts_M10_diff}}
\caption{Distributions of OPP losses for the 8 triangles when $\fnij{Y}^{[m]}$ follows a \protect\subref{fig:3pts_M10_laplace} Laplace or \protect\subref{fig:3pts_M10_nsst} NSST distribution, and when matched and mismatched (Gaussian) likelihood (LL) functions are used. Each box represents the percentiles (bottom to top): 10, 25, 50, 75, and 90. Distributions of pairwise differences in OPP loss when $\fnij{Y}^{[m]}$ follows a Laplace or NSST distribution are shown in \protect\subref{fig:3pts_M10_diff}. Pairwise differences are computed by subtracting the OPP loss for the matched estimate from the corresponding OPP loss for the mismatched (Gaussian LL function) estimate.}
\label{fig:3pts_M10}
\end{figure*}
\textbf{Triangles.} We first considered 8 triangles in 2D. Two of these triangles, together with example estimates and their associated OPP losses, are shown in \autoref{fig:tris}. For each SNR value, and for each triangle, we sampled $M = 10$ noisy measurements per edge 100 times. For each sample, we computed an estimate $\widehat{\mathcal{X}}$ using different likelihoods and then computed the OPP loss associated with $\widehat{\mathcal{X}}$. The distributions of these OPP losses are shown in \autoref{fig:3pts_M10_laplace} and \autoref{fig:3pts_M10_nsst}. We also computed the pairwise difference in OPP losses resulting from mismatched and matched estimation. The distributions of these differences are shown in \autoref{fig:3pts_M10_diff}.
We observe that, in general, the distribution of OPP losses associated with the likelihood that is matched to the noise distribution is shifted lower than the distribution associated with the likelihood that is not matched to the noise distribution. Moreover, in several instances, the increase in median OPP loss due to a 3--4 dB decrease in SNR can approximately be compensated for by choosing to use the likelihood that is matched to the noise distribution.
\begin{figure}[!t]
\centering
\subfloat[]{\includegraphics[trim={2.65mm 0mm 0mm 2.5mm},clip,
width=0.532\columnwidth]{sections/results/figures/3pts_Ms_laplace.pdf}
\label{fig:3pts_Ms_laplace}}
\hspace{-2.3mm}
\subfloat[]{\includegraphics[trim={0cm 1.2mm 2.3mm 2mm},clip,
width=0.441\columnwidth]{sections/results/figures/3pts_Ms_nsst.pdf}
\label{fig:3pts_Ms_nsst}}
\caption{Median OPP losses for the 8 triangles and for different $M$ values when $\fnij{Y}^{[m]}$ follows a \protect\subref{fig:3pts_Ms_laplace} Laplace or \protect\subref{fig:3pts_Ms_nsst} NSST distribution, and when matched or mismatched (Gaussian) likelihood functions are used.}
\label{fig:3pts_Ms}
\end{figure}
We then repeated this experiment for different values of $M$, the number of noisy measurements per edge. The median OPP losses for $M = 10,25,50,100$ are shown in \autoref{fig:3pts_Ms}. We observe that, in general, maximizing the likelihood that is matched to the noise distribution offers an improvement over mismatched DGP estimation comparable to the improvement obtained by doubling the number of noisy measurements per edge.
\begin{figure}[t]
\centering
\includegraphics[trim={0cm 2.5mm 0cm 0.25cm},clip,
width=0.7\columnwidth]{sections/results/figures/10pts.pdf} \vspace{-2mm}
\caption{One of the 30 10-point structures that were considered. Example estimates are omitted for clarity.}
\label{fig:10pts}
\end{figure}
\begin{figure*}[!t]
\centering
\subfloat[]{\includegraphics[width=0.66\columnwidth]{sections/results/figures/10pts_M10_laplace.pdf}
\label{fig:10pts_M10_laplace}}
\hfil
\subfloat[]{\includegraphics[width=0.66\columnwidth]{sections/results/figures/10pts_M10_nsst.pdf}
\label{fig:10pts_M10_nsst}}
\hfil
\subfloat[]{\includegraphics[width=0.67\columnwidth]{sections/results/figures/10pts_M10_diff.pdf}
\label{fig:10pts_M10_diff}}
\caption{Distributions of OPP losses for the 30 10-point structures when $\fnij{Y}^{[m]}$ follows a \protect\subref{fig:10pts_M10_laplace} Laplace or \protect\subref{fig:10pts_M10_nsst} NSST distribution, and when matched and mismatched (Gaussian) likelihood (LL) functions are used. Percentiles are marked as in \autoref{fig:3pts_M10}. Distributions of pairwise differences in OPP loss when $\fnij{Y}^{[m]}$ follows a Laplace or NSST distribution are shown in \protect\subref{fig:10pts_M10_diff}. Pairwise differences are computed by subtracting the OPP loss for the matched estimate from the corresponding OPP loss for the mismatched (Gaussian LL function) estimate.}
\label{fig:10pts_M10}
\end{figure*}
\textbf{10 points.} To further support our results, we repeated our experiments for 30 10-point structures in 2D. One of these structures is shown in \autoref{fig:10pts}. The distributions of OPP losses are shown in \autoref{fig:10pts_M10_laplace} and \autoref{fig:10pts_M10_nsst}, and the distributions of pairwise differences in OPP losses for the mismatched and matched cases are shown in \autoref{fig:10pts_M10_diff}. Similar to the case of triangles, we observe that, in general, the distribution of OPP losses associated with the likelihood that is matched to the noise distribution is lower than the distribution associated with the likelihood that is not matched to the noise distribution.
\section{Related Work} \label{sec:related_work}
One of the earliest discussions of cost functions related to the solution of the DGP was due to Crippen and Havel \cite{crippen_distance_1988}. In their work, they used a variant of the SSE function, which we refer to as SSE-CH, to estimate molecular structures with physical constraints. Although others have empirically shown that SSE-CH yields acceptable results \cite{crippen_distance_1988}, the resulting estimate of a structure need not be an ML estimate.
Subsequent papers implement modifications to the SSE-CH cost function. For example, Mor\'e and Wu \cite{more_global_1997-2} adopted a continuation approach: the SSE-CH cost function is repeatedly smoothed via a convolution with a Gaussian kernel. After each successive smoothing stage, a minimizer of the smoothed SSE-CH is obtained. Eventually, the minimizers are traced back to the original function \cite[p. 827]{more_global_1997-2}. Although this approach yielded promising results, the underlying cost function that is smoothed is still the SSE function. Moreover, there is no guarantee that the resulting estimate of $\mathcal{X}$ is an ML estimate.
Souza et al. \cite{souza_hyperbolic_2011} attempt to make SSE-CH differentiable by approximating certain parts using hyperbolic functions. Souza et al. \cite{souza_solving_2013} made further improvements by applying these approximations to the \textit{geometric buildup} algorithm proposed by Luo and Wu \cite{luo_least-squares_2011}. However, the underlying cost function is still a variant of the SSE function, and so there is no guarantee of obtaining ML estimates. For a detailed review of approaches to solve the DGP, see \cite{lavor_overview_2009}. More recently, Gopalkrishnan et al. \cite{gopalkrishnan_dna_2020-1} optimize a variation of SSE-CH to reconstruct DNA molecules from pairwise distance measurements. Giamou et al. \cite{giamou_convex_2022} also optimize a variation of SSE-CH to determine the joint angles of a robot given the robot's forward kinematics and desired poses. However, because variations of SSE-CH are used, it is unlikely that estimates obtained in these applications are ML estimates.
\section*{Acknowledgments}
We thank Vaibhav Choudhary and Nikhil Gopalkrishnan for informative discussions.
|
1,941,325,220,553 | arxiv | \section{Introduction}
We consider only finite, non-oriented graphs without loops and multiple edges. We call a $p$-clique
of the graph G a set of $p$ vertices, each two of which are adjacent. The largest positive integer
$p$, such that the graph G contains a $p$-clique is denoted by $cl(G)$ . We denote by $V(G)$ and
$E(G)$ the vertex set and the edge set of the graph $G $ respectively. The symbol $K_n$ denotes
the complete graph on $n$ vertices.
Let $G_1$ and $G_2$ be two graphs without common vertices. We denote
by $G_1 + G_2$ the graph $G$ for which $V(G)=V(G_1) \cup V (G_2)$ and
$E(G)=E(G_1)\cup E(G_2) \cup E'$ , where $ E' = \{[x , y ] \mid x \in
V(G_1) , y \in V(G_2) \}$.
\begin{Definition}
Let $ a_1 , \ldots , a_r $ be positive integers. We say that the
$r$-coloring
$$
V(G)= V_1 \cup \ldots \cup V_r \text{ , } V_i \cap V_j = \emptyset
\text{ , } i \neq j,
$$
of the vertices of the graph $G$ is $(a_1 ,\ldots , a_r)$-free, if
$V_i$ does not contain an $a_i$-clique for each $i \in \{ 1 , \ldots ,
r \}$. The symbol $G\rightarrow (a_1 , \ldots , a_r)$ means that
there is not an $(a_1 , \ldots , a_r)$-free coloring of the vertices
of $G.$
\end{Definition}
We consider for arbitrary natural numbers $a_1 , \ldots , a_r $ and
$q$
$$
H(a_1,\ldots , a_r ; q) = \{G : G\rightarrow (a_1,\ldots ,a_r) \text
{
and } cl(G)<q \}.
$$
The vertex Folkman numbers are defined by the equality
$$ F(a_1, \ldots , a_r ;q) = \min\{ | V(G)| : G\in H(a_1, \ldots ,
a_r; q)\}.
$$
It is clear that $G\rightarrow (a_1,\ldots ,a_r)$ implies $cl(G)\geq
\max\{a_1,\ldots , a_r\}$. Folkman [1] proved that there exists a
graph $G$ such that $G \rightarrow (a_1, \ldots , a_r) $ and $cl(G)
= \max\{a_1,\ldots , a_r\}$. Therefore
\begin{equation}
\label{1} F(a_1 ,\ldots , a_r ;q) \text{ exists
if and only if }
q> \mbox{max} \{ a_1,\ldots , a_r \}.
\end{equation}
If $a_1 , \ldots ,a_r$ are positive integers, $r\geq2$ and $a_i=1$
then it is easy to see that
\begin{equation}
G\rightarrow(a_1 , \ldots , a_r) \Leftrightarrow G \rightarrow(a_1 ,
\ldots , a_{i-1} , a_{i+1} , \ldots ,a_r).
\end{equation}
It is also easy to see that for an arbitrary permutation $\varphi \in
S_r$ we have
$$
G\rightarrow(a_1, \ldots ,a_r) \Leftrightarrow
G\rightarrow(a_{\varphi(1)} , \ldots ,a_{\varphi(r)}).
$$
That is why
\begin{equation}
F(a_1 , \ldots ,a_r;q)=F(a_{\varphi(1)} , \ldots , a_{\varphi(r)})
,\text{ for each } \varphi \in S_r
\end{equation}
According to (2) and (3) it is enough to consider just such numbers $F(a_1, \ldots, a_r;q)$ for
which
\begin {equation}
2 \leq a_1 \leq \ldots \leq a_r .
\end{equation}
For arbitrary positive integers $a_1, \ldots , a_r$ define:
\begin {gather}
p=p(a_1 , \ldots ,a_r)= \max\{ a_1 ,\ldots , a_r \};\\
m= 1+ \sum_{i=1}^{r} (a_i-1)
\end{gather}
It is easy to see that $ K_m\rightarrow(a_1 ,\ldots ,a_r)$ and
$K_{m-1}\nrightarrow(a_1 , \ldots ,a_r).$ Therefore
$$F(a_1 , \ldots , a_r;q)=m \text {, if } q>m . $$
In [4] it was proved that $F(a_1 ,\ldots , a_r;m)=m+p,$ where $m$ and $p$ are defined by the
equalities (5) and (6). About the numbers $F(a_1 , \ldots , a_r;m-1)$ we know that $F(a_1, \ldots ,
a_r;m-1) \geq m+p+2 \text { , } p \geq 2$ and
\begin{equation}
F(a_1 , \ldots , a_r;m-1) \leq m+3p \text{ , [3]}.
\end{equation}
The exact values of all numbers $F(a_1 , \ldots , a_r;m-1)$ for
which $\max \{a_1 , \ldots , a_r \} \leq 4 $ are known. A detailed
exposition of these results was given in [8]. We must add the
equality $F(2,2,3;4)=14$ obtained in [2] to this exposition. We do
not know any exact values of $F(a_1 , \ldots , a_r;m-1)$ in the case
when $ \max \{a_1 , \ldots , a_r) \geq 5.$
According to (1), $F(a_1 , \ldots , a_r;m-1)$ exists exactly when $m \geq p+2.$ In this paper we
shall improve inequality (7) in the boundary case when $ m=p+2$ , $p \geq 5.$ From the equality
$m=p+2$ and (4) it easily follows that there are two such numbers only: $F(2,2,p;p+1)$ and
$F(3,p;p+1).$ It is clear that from $G\rightarrow (3,p)$ it follows $G\rightarrow(2,2,p).$
Therefore
\begin{equation}
F(2,2,p;p+1) \leq F(3,p;p+1).
\end{equation}
The inequality (7) gives us that:
\begin{gather}
F(3,p;p+1) \leq 4p+2;\\
F(2,2,p:p+1) \leq 4p+2.
\end{gather}
Our goal is to improve the inequalities (9) and (10). We shall need
the following
\begin{Lemma}
Let $G_1$ and $G_2$ be two graphs such that
\begin{equation}
G_1 \rightarrow (a_1 , \ldots , a_{i-1}, a'_i , a_{i+1} , \ldots
,a_r)
\end{equation}
and
\begin{equation}
G_2 \rightarrow (a_1, \ldots ,a_{i-1} , a''_i , a_{i+1} , \ldots ,
a_r).
\end{equation}
Then
\begin{equation}
G_1+G_2 \rightarrow (a_1 , \ldots , a_{i-1} , a'_i + a''_i , a_{i+1} , \ldots , a_r).
\end{equation}
\end{Lemma}
\begin{proof}
Assume that (13) is wrong and let
$$V_1 \cup \ldots \cup V_r \text { , } V_i \cap V_j = \emptyset
\text{ , } i \neq j$$
be a $(a_1 , \ldots ,a_{i-1} , a'_i + a''_i ,a_{i+1}, \ldots , a_r)-$free $r-$coloring of
$V(G_1+G_2)$.Let $V'_i = V_i \cap V(G_1) $ and $V''_i=V_i \cap V(G_2),$ for $i=1, \ldots , r.$ Then
$V'_1 \cup \ldots \cup V'_r$ is an $r-$coloring of $V(G_1),$ such that $V_j$ does not contain an
$a_j-$clique, $j \neq i.$ Thus from (11) it follows that $V'_i$ contains an $a'_i-$clique.
Analoguously from the $r-$colouring $V''_1 \cup \ldots \cup V''_r$ of $V(G_2)$ it follows that
$V''_i$ contains an $a''_i$-clique.Therefore $V_i=V'_i \cup V''_i$ contains a $(a'_i
+a''_i)-$clique,which contradicts the assumption that $V_1 \cup \ldots \cup V_r$ is a $(a_1, \ldots
, a_{i-1} , a'_i +a''_i , a_{i+1} , \ldots , a_r)-$ free $r-$coloring of $V(G_1+G_2).$ This
contradiction proves the Lemma.
\end{proof}
\section{Results}
The main result in this paper is the following
\begin{Theorem}
Let $a_1 \leq \ldots \leq a_r $ , $r \geq 2 $ be positive integers
and $ a_r = b_1 + \ldots + b_s,$ where $b_i$ are positive integers
,too and $b_i \geq a_{r-1}, \ $ $i=1 , \ldots , s.$ Then
\begin{equation}
F(a_1 , \ldots , a_r; a_r+1) \leq \sum_{i=1}^{s}
F(a_1 , \ldots , a_{r-1} , b_i;b_i +1).
\end{equation}
\end{Theorem}
\begin{proof}
We shall prove the Theorem by induction on $s.$ As
the inductive step is trivial we shall just prove the inductive base
$s=2.$ Let $G_1$ and $G_2$ be two graphs such that $cl(G_1)=b_1$ and
$ cl(G_2)=b_2,$ $a_r = b_1 +b_2 , \ b_1 \geq a_{r-1}, \ b_2 \geq
a_{r-1} $ and
\begin{gather*}
G_1 \rightarrow (a_1 , \ldots , a_{r-1} , b_1) \text{ , }
|V(G_1)|= F(a_1 , \ldots , a_{r-1},b_1;b_1 + 1)\\
G_2 \rightarrow (a_1 , \ldots , a_{r-1}, b_2) \text{ , } |V(G_2)|
= F(a_1 , \ldots , a_{r-1} , b_2 ;b_2 + 1).
\end{gather*}
According to the Lemma, $G_1+G_2 \rightarrow (a_1, \ldots , a_{r-1} , a_r).$ As $cl(G_1 + G_2)
=cl(G_1) + cl(G_2) =b_1+ b_2 = a_r, $ we have
$$F( a_1 , \ldots , a_r;a_r +1) \leq |V(G_1 + G_2)| = |V(G_1)| +
|V(G_2)|.$$
From this inequality (14) trivially follows when $s=2$ and hence , for arbitrary $s,$ as explained
above. The Theorem is proved.
\end{proof}
We shall derive some corollaries from the Theorem. Let $p \geq 4$
and $p=4k+l, \ $ $0 \leq l \leq 3.$ Then from (14) it easily
follows that
\begin{gather}
F(3,p;p+1) \leq (k-1) F(3,4,;5) + F(3, 4+l;5+l) \\
F(2,2,p; p+1) \leq (k-1) F(2,2,4;5) + F(2,2,4 +l ;5+l).
\end{gather}
From (15), (9) $(p=5 , 6 , 7)$ and the equality $F(3,4;5) = 13,$ [6]
we obtain
\begin{corollary}
Let $p \geq 4$. Then:
\begin{gather*}
F(3,p;p+1) \leq \frac{13p}{4} \ \text{ for } \ p \equiv 0 \mod 4; \\
F(3,p;p+1) \leq \frac {13p+23}{4} \ \text { for } \ p \equiv 1 \mod 4;\\
F(3,p;p+1) \leq \frac {13p+26}{4} \ \text { for } \ p \equiv 2 \mod 4;\\
F(3,p;p+1) \leq \frac {13p+29}{4} \ \text { for } \ p \equiv 3 \mod 4.
\end{gather*}
\end{corollary}
From (16), the equality $F(2, 2 ,4;5)=13,$ [7], the inequality (10)
$(p=5)$ and the inequalities $F(2, 2 , 6;7 ) \leq 22,$ [9];
$F(2,2,7;8) \leq 28 ,$ [9] we obtain
\begin{corollary}
Let $p \geq 4$. Then
\begin{gather*}
F(2,2,p;p+1) \leq \frac {13p}{4} \ \text { for } \ p \equiv 0 \mod 4;\\
F(2,2,p;p+1) \leq \frac {13p+23}{4} \ \text { for } \ p \equiv 1 \mod 4;\\
F(2,2,p;p+1) \leq \frac {13p+10}{4} \ \text { for } \ p \equiv 2 \mod 4;\\
F(2,2,p;p+1) \leq \frac {13p+21}{4} \ \text { for } \ p \equiv 3 \mod 4.
\end{gather*}
\end{corollary}
We conjecture that the following inequalities hold:
\begin{gather}
F(3,p;p+1) \leq \frac {13p}{4} \ \text { for } p \geq 4 ;\\
F(2,2,p;p+1) \leq \frac {13p}{4} \ \text { for } p \geq 4 .
\end{gather}
From the Theorem it follows that
\begin {equation}
F(3,p;p+1) \leq F(3,p-4 ; p-3) + F(3,4;5) ,p \geq 8 ;
\end{equation}
\begin {equation}
F(2,2,p;p+1) \leq F(2,2,p-4 ;p-3) + F(2,2,4;5) , p \geq 8.
\end{equation}
From $F(3,4;5) = 13$ (see [6]) and (19) we obtain
\begin{corollary}
If the inequality (17) holds for $p=5$, $6$ and $7$, then (17) is true for
every $p \geq 4$.
\end{corollary}
From $F(2 , 2 , 4 ;5) = 13,$ (see [7]) and from (20) it follows
\begin{corollary}
If the inequality (18) holds for $p=5$, $6$
and $7$ then (18) is true for every $p \geq 4$.
\end{corollary}
At the end in regard with (8) we shall pose the following
\begin{Problem}
Is there a positive integer $p,$ for which $F(2,2,p;p+1) \neq
F(3,p;p+1)$?
\end{Problem}
|
1,941,325,220,554 | arxiv | \section{Introduction}
\label{sec:intro}
Building Automatic Speech Recognition (ASR) systems historically was an expertise-intensive task and involved a complex pipeline \cite{soltau2005ibm, dahl2012context}, which consists of phonetic decision trees and multiple stages of alignments and model updating.
Recently, there are increasing interests in developing end-to-end ASR systems \cite{graves2006connectionist, graves2012sequence, chorowski2014end, xiang2019crf} to reduce expert efforts and simplify the system.
The progress largely relies on utilizing deep neural networks (DNNs) of various architectures, which is generally known as deep learning.
The success of deep learning is largely due to its automation of the feature engineering process: hierarchical feature extractors are automatically learned from data rather than manually designed.
This success has been accompanied, however, by a rising demand for architecture engineering.
Various neural architectures, e.g., 2D Convolutional Neural Networks (CNNs) \cite{simonyan2014very}, which is also known as (a.k.a.) VGG-Net, 1D dilated CNN \cite{peddinti2015time} (a.k.a. TDNN), ResNet \cite{he2016deep} and so on, are manually designed by experts through intuitions plus laborious trial and error experiments.
Hyper-parameters for an architecture (e.g., kernel size, stride, and dilation of CNN) are set empirically, which may not be optimal for the specific task at hand, since naive grid-search is highly expensive.
Neural Architecture Search (NAS) \cite{zoph2016neural}, the process of automating architecture engineering, is thus an appealing next step to advancing end-to-end ASR.
Early NAS methods are computationally demanding despite their remarkable performance. For example, it takes 2000 GPU days of reinforcement learning and 3150 GPU days of evolution to obtain a state-of-the-art (SOTA) architecture for image classification over CIFAR-10 \cite{zoph2018learning} and ImageNet \cite{real2019regularized} respectively .
Some recent NAS methods, e.g., DARTS (Differentiable ARchiTecture Search) \cite{liu2018darts}, SNAS (Stochastic NAS) \cite{xie2018snas} and ProxylessNAS \cite{cai2018proxelessnas}, significantly improve the NAS efficiency and can obtain SOTA architecture over CIFAR-10 within one GPU day.
Note that NAS methods roughly can be categorized according to three dimensions - search space, search method, and performance evaluation method \cite{elsken2019neural}.
Some common features shared by these efficient NAS methods are: representing the search space as a weighted directed acyclic graph (DAG)\footnote{When used to represent the entire architecture, the DAG is often called the super-network \cite{veniat2018learning,wu2019fbnet} or the over-parameterized network \cite{cai2018proxelessnas}.
When the search space is defined over smaller building blocks \cite{pham2018efficient, liu2018darts}, the DAG is called the cell block or the cell, that could be stacked in some way via a preset or learned meta-architecture to form the entire architecture.}, using gradient-based search methods to learn the edge weights, and weight sharing in performance evaluation of candidate architectures (i.e., sub-graphs of the DAG).
The final architecture is derived from the learned edge weights.
Notably, DARTS uses the Softmax trick to relax the search space to be continuous and performs gradient search over the whole super-network, while SNAS uses the Gumbel-Softmax trick \cite{Jang2016Categorical}.
These are less efficient in both memory and computation than ProxylessNAS, which uses discrete search spaces and does sub-graph sampling.
To back-propagate gradients through the discrete variables, which index the sampled edges, ProxylessNAS uses an ad-hoc trick, analogous to BinaryConnect \cite{courbariaux2015binaryconnect}.
In this paper, we make two contributions. First, we observe that ProxylessNAS essentially uses the Straight-Through (ST) estimator \cite{bengio2013estimating} for gradient approximation, which is missed in the ProxylessNAS paper.
To back-propagate gradients through discrete variables, the basic idea of ST is that the sampled discrete index is used for forward computation, and the continuous Softmax probability of the sampled index is used for backward gradient calculation.
Using ST gradients to support sub-graph sampling is a core element to achieve efficient NAS beyond DARTS and SNAS.
Based on ProxylessNAS and also the above observation, we develop an efficient NAS method via ST gradients, called ST-NAS.
In contrast to ProxylessNAS whose NAS objective definition is not clearly shown, ST-NAS uses the NAS objective definition from SNAS and is more rigorous.
Compared to SNAS, ST-NAS does not use the Gumbel-Softmax trick and uses the more efficient ST gradients.
The development of the ST-NAS method is the first contribution of this paper.
Second, we apply ST-NAS to end-to-end ASR.
Compared to the application of NAS techniques in computer vision tasks, there are limited studies in applying NAS to speech recognition tasks.
In \cite{veniat2019stochastic}, NAS via policy gradient based reinforcement learning \cite{veniat2018learning} is applied to keyword spotting.
In \cite{mazzawi2019improving}, evolution-based NAS is applied to keyword spotting and language identification.
DARTS are used in \cite{ding2020autospeech} and \cite{chen2020darts} for speaker recognition and Connectionist Temporal Classification (CTC) based speech recognition respectively.
The NAS methods used in these previous studies are not as efficient as NAS via ST gradients.
This work represents the first to introduce ST gradient based NAS into end-to-end speech recognition, which is the second contribution of this paper.
We evaluate the ST-NAS method in end-to-end ASR experiments over the widely benchmarked 80-hour WSJ and 300-hour Switchboard datasets and show that the ST-NAS induced architectures significantly outperform the human-designed architecture (TDNN-D in \cite{peddinti2018low}) across the two datasets.
Notably, our ST-NAS induced model obtains the lowest word error rate (WER) of 2.77\%/5.68\% on WSJ eval92/dev93 among all published end-to-end ASR results, to the best of our knowledge.
Our NAS implementation is based on the CAT toolkit \cite{an2020cat} - a PyTorch-based ASR toolkit, which offers two benefits. First, it enables us to seamlessly integrate the NAS code with the ASR code to flexibly use PyTorch functionalities. Second, the CAT toolkit supports the end-to-end CTC-CRF loss, which is defined by a CRF (conditional random field) with CTC topology and has been shown to perform significantly better than CTC \cite{xiang2019crf, an2020cat}.
Additionally, we show that the architectures learned by the CTC loss based ST-NAS are transferable to be retrained under the CTC-CRF loss, i.e., these architectures achieve close performance to the architectures searched under the CTC-CRF loss. This enables us to reduce the cost of running NAS to search the architecture, since CTC-CRF is somewhat more expensive than CTC.
We also show that the model transferred from the WSJ experiment performs close to the model searched over Switchboard and better than the comparably-sized TDNN-D.
We release the code\footnote{https://github.com/thu-spmi/ST-NAS} for reproducible NAS study, as it is found in \cite{Li2019Random} that reproducibility is crucial to foster NAS research.
\begin{figure}[t]
\centering
\includegraphics[width=8.5cm]{toplevelsystem}
\vspace{-3mm}
\caption{An overview of our ST-NAS method, which illustrates the concepts of macro-DAG, candidate operations (OP$_1$, OP$_2$, OP$_3$), super-network, and sub-graph sampling.
Here we plot the macro-DAG used in our ASR experiments, which has a serial structure. More complicated macro-DAGs are possible in general.
}
\label{fig:toplevelsystem}
\end{figure}
\section{Background and Related Work}
\label{sec:relatedwork}
Recent gradient-based NAS methods such as DARTS \cite{liu2018darts}, SNAS \cite{xie2018snas} and ProxylessNAS \cite{cai2018proxelessnas} significantly improve over previous reinforcement learning and evolution based NAS methods with reduced computational cost.
In these methods, the search space is represented as a DAG, which consists of nodes (numbering by $0,1,\cdots,N$) and directed edges (pointing from lower-numbered nodes to higher-numbered). The $i$-th node, denoted by node$_i$, represents a latent representation $x_i$ (e.g., a feature map in CNNs). A directed edge leaving node$_i$ is associated with an operation that transforms $x_i$. An intermediate node (i.e., after the input node $x_0$) is computed as follows:
\begin{equation}
x_j = \sum_{i \in \mathcal{A}_j}{\Omega_{ij}(x_i)}
\label{eq:x_j}
\end{equation}
where $\mathcal{A}_j$ is the set of parent nodes of node$_j$, $\Omega_{ij}$ denotes the operation that connects node$_i$ to node$_j$ in the computation flow.
Suppose that $\Omega_{ij}$ can take from $K$ different candidate operations $\{o_{ij}^{(k)},k=1,\cdots,K\}$ (e.g., different convolutions), where each candidate operation is associated a weight $\alpha_{ij}^{(k)}$, called architecture weight.
It can be easily seen that by sampling one of the $K$ candidate operations for each connected pair of nodes, we obtain a candidate architecture.
The task of NAS therefore is reduced to learning the architecture weights. The final architecture is derived from the learned architecture weights, e.g., by selecting the largest weighted candidate operation for each connected pair of nodes $(i,j)$.
Different NAS methods mainly differ in how to learn the architecture weights $\alpha = \{\alpha_{ij}^{(k)}\}$, together with the operation parameters $\theta$, which are used to define the operations $\{o_{ij}^{(k)}\}$.
Note that it is often a practice to plot all the candidate operations $\{o_{ij}^{(k)},k=1,\cdots,K\}$ between each connected node$_i$ and node$_j$ in the DAG and call the resulting expanded DAG a super-network, e.g., as in \cite{liu2018darts,xie2018snas,cai2018proxelessnas} and also shown in Fig.~\ref{fig:toplevelsystem}.
Then each edge in the super-network represents a candidate operation, and a candidate architecture corresponds to a sub-graph in the super-network.
Each connected pair of nodes together with the $K$ edges between them is called a searching block.
To be differentiated from the super-network, the initial DAG is referred to as the macro-DAG.
\subsection{DARTS}
\label{sec:relatedwork:DARTS}
Instead of searching over a discrete set of sub-graphs, DARTS relaxes the categorical choice of a particular operation $o_{ij}^{(k)}$ on edge $(i,j)$ to a mixture of all possible $K$ candidate operations, by defining the following mixed operation:
\begin{equation}
\Omega_{ij}^{\text{DARTS}}(x_i) = \sum_{k=1}^K \pi_{ij}^{(k)} o_{ij}^{(k)}(x_i)
\end{equation}
where $\pi_{ij}=(\pi_{ij}^{(1)}, \cdots, \pi_{ij}^{(K)})$ denotes the mixing probabilities of the $K$ operations defined by a Softmax over the architecture weights:
\begin{equation}
\pi_{ij}^{(k)}=\frac{\exp(\alpha_{ij}^{(k)})}{\sum_{k'=1}^{K} \exp(\alpha_{ij}^{(k')})}
\label{eq:pi}
\end{equation}
Here we add DARTS as the superscript to denote the particular operation $\Omega_{ij}$ used in DARTS, which is plugged in Eq.~\eqref{eq:x_j} to define the computation flow.
In this manner, DARTS relaxes the search space to be continuous and directly incorporates the architecture weights $\alpha$ into the super-network forward computation to define the loss $\mathcal{L}^{\text{DARTS}}(\alpha,\theta)$, together with the operation parameters $\theta$. Thus $\alpha$ and $\theta$ can be jointly trained via the standard backpropagation algorithm by optimizing $\mathcal{L}^{\text{DARTS}}(\alpha,\theta)$\footnote{In fact, $\mathcal{L}^{\text{DARTS}}_{\text{train}}(\alpha,\theta)$ with respect to (w.r.t.) $\theta$ and $\mathcal{L}^{\text{DARTS}}_{\text{val}}(\alpha,\theta)$ w.r.t.~$\alpha$ are alternately optimized, which are defined over training data and validation data respectively.}.
A limitation of DARTS is its expensive computation in both memory and time, which is around $K$ times of training a single model.
The mixed operation of DARTS requires to store the whole super-network in memory, and all edges are involved in the forward and backward computation.
\vspace{-2mm}
\subsection{SNAS}
\label{sec:relatedwork:SNAS}
Note that DARTS uses the super-network in NAS training, but after NAS training, uses the extracted sub-graph in inference. This incurs an inconsistency, which harms the performance.
SNAS uses the same super-network setup as in DARTS, but proposes to optimize the expected performance of all sub-graphs sampled with $p_\alpha(z)$:
\begin{equation}
\mathcal{L}(\alpha,\theta) = \mathbb{E}_{z\sim p_{\alpha}(z)}[\mathcal{L}_\theta(z)]
\label{eq:SNAS-loss}
\end{equation}
where $z = \{z_{ij}\}$ denotes the collection of the independent one-hot random variable vectors\footnote{As usual, categorical variables are encoded as $K$-dimensional one-hot vectors lying on the corners of the ($K-1$)-dimensional simplex.} for all pairs of connected nodes in the super-network, indexing, which edge is sampled. Thus a sample $z$ represents a sampled sub-graph, and $\mathcal{L}_\theta(z)$ denotes the loss evaluated under the sampled sub-graph $z$.
The one-hot vector $z_{ij}=(z_{ij}^{(1)}, \cdots, z_{ij}^{(K)})$ for connected node$_i$ and node$_j$ is assumed to follow the categorical distribution $Cat(\pi_{ij}^{(1)}, \cdots, \pi_{ij}^{(K)})$.
The loss $\mathcal{L}(\alpha,\theta)$ is not directly differentiable w.r.t.~the architecture weights $\alpha$, since we cannot pass the gradient through the discrete random variable $z_{ij}$ to $\alpha_{ij}=(\alpha_{ij}^{(1)}, \cdots, \alpha_{ij}^{(K)})$.
To sidestep this, SNAS relaxes the discrete one-hot variable $z_{ij}$ to be a continuous random variable computed by the Gumbel-Softmax (G-S) function \cite{Jang2016Categorical}:
\begin{equation}
y_{ij}^{(k)}=\frac{\exp((\alpha_{ij}^{(k)}+g_{ij}^{(k)})/\tau)}{\sum_{k'=1}^{K} \exp((\alpha_{ij}^{(k')}+g_{ij}^{(k')})/\tau)}
\end{equation}
where $g_{ij}^{(k)}=-\log(-\log(u_{ij}^{(k)}))$, and $\{u_{ij}^{(k)}\}$ are independent and identical distributed (i.i.d.) samples from $u\sim \text{Uniform}(0,1)$. $\tau$ is the temperature, which is gradually annealed to be close to zero.
The soften one-hot variable $y_{ij}=(y_{ij}^{(1)}, \cdots, y_{ij}^{(K)})$ follows the G-S distribution.
It is shown in \cite{Jang2016Categorical} that as the temperature $\tau$ approaches $0$, $y_{ij}$ from the G-S distribution become one-hot, and the G-S distribution becomes identical to the categorical distribution $Cat(\pi_{ij}^{(1)}, \cdots, \pi_{ij}^{(K)})$. Thus, SNAS uses the following surrogate loss to approximate the loss defined in Eq.~\eqref{eq:SNAS-loss}:
\begin{equation}
\tilde{\mathcal{L}}(\alpha,\theta) = \mathbb{E}_{y\sim p_{\alpha}(y)}[\mathcal{L}_\theta(y)]
\end{equation}
which becomes directly differentiable w.r.t.~$\alpha$.
Here $\mathcal{L}_\theta(y)$ is defined by the computation flow still according to Eq.~\eqref{eq:x_j} but with the following soften operation:
\begin{equation}
\Omega_{ij}^{\text{SNAS}}(x_i) = \sum_{k=1}^K y_{ij}^{(k)} o_{ij}^{(k)}(x_i)
\end{equation}
It can be seen that the computation cost of SNAS in both memory and time is identical to that of DARTS.
The difference is that while DARTS uses the Softmax trick for continuous relaxation, SNAS uses the Gumbel-Softmax trick with annealed temperature. When $\tau$ approaches $0$, the objectives in NAS training and in model inference becomes consistent. The Gumbel-Softmax trick is also used in \cite{wu2019fbnet, dong2019searching} for NAS.
\subsection{ProxylessNAS}
\label{sec:relatedwork:pNAS}
Using the same super-network setup as in DARTS, ProxylessNAS proposes to use binary gates (essentially a one-hot vector) $z_{ij}$ for each edge to define the operation to reduce the memory footprint:
\begin{equation}
\Omega_{ij}(x_i) = \sum_{k=1}^K z_{ij}^{(k)} o_{ij}^{(k)}(x_i),
\label{eq:z-operation}
\end{equation}
Here we use the same $z_{ij}$ from SNAS as it carries the same meaning - indexing, which edge is sampled.
In this manner, the architecture weights $\alpha_{ij}$ are not directly involved in the computation flow as defined in Eq.~\eqref{eq:x_j} and thereby we cannot directly calculate the gradient of $\alpha_{ij}$.
Motivated by BinaryConnect \cite{courbariaux2015binaryconnect}, ProxylessNAS proposes the following gradient approximation:
\begin{equation}
\begin{aligned}
\frac{\partial{\mathcal{L}}}{\partial{\alpha_{ij}^{(k)}}}
= \sum_{k'=1}^K \frac{\partial{\mathcal{L}}}{\partial{\pi_{ij}^{(k')}}} \frac{\partial{\pi_{ij}^{(k')}}}{\partial{\alpha_{ij}^{(k)}}}
\approx \sum_{k'=1}^K \frac{\partial{\mathcal{L}}}{\partial{z_{ij}^{(k')}}} \frac{\partial{\pi_{ij}^{(k')}}}{\partial{\alpha_{ij}^{(k)}}}
\end{aligned}
\label{eq:proxyless}
\end{equation}
By using the sampled sub-graph $z$, ProxylessNAS reduces the memory footprint and backward computation cost, compared to DARTS and SNAS.
We leave the detailed comparison of SNAS, ProxylessNAS and our method to Section~\ref{sec:method}.
\section{method}
\label{sec:method}
Our NAS method is based on the following two observations from the review of existing gradient-based NAS methods.
First, ProxylessNAS is an interesting method, but the loss $\mathcal{L}$ used in ProxylessNAS is not explicitly shown in the original paper \cite{cai2018proxelessnas}. We observe that the ProxylessNAS loss $\mathcal{L}$ in Eq.~\eqref{eq:proxyless} is in fact the loss $\mathcal{L}_\theta(z)$ as defined in SNAS.
Second, we observe that ProxylessNAS essentially uses the Straight-Through (ST) estimator \cite{bengio2013estimating} for gradient approximation - a simple yet effective technique to back-propagate gradients through discrete variables. This point is also missed in the ProxylessNAS paper.
In the following, we first introduce the ST gradient estimator, and then present our NAS method via ST gradients, called ST-NAS. Basically, ST-NAS uses the loss from SNAS but optimizes the loss using the ST gradients.
\subsection{Straight-Through gradient}
To optimize the loss $\mathcal{L}(\alpha,\theta)$, we sample a sub-graph, represented by $z$, according to the architecture weights $\alpha$.
Specifically, for each edge $(i,j)$, $z_{ij}$ is independently sampled from $Cat(\pi_{ij}^{(1)}, \cdots, \pi_{ij}^{(K)})$, which is defined by the architecture weights $\alpha_{ij}$ as in Eq.~\eqref{eq:pi}.
Then the forward computation is conducted over the sampled sub-graph to obtain the loss $\mathcal{L}_\theta(z)$ as a Monte Carlo estimate of $\mathcal{L}(\alpha,\theta)$, by executing the operation as defined in Eq.~\eqref{eq:z-operation}.
During the backward computation, we encounter the problem that we cannot pass the gradient through the one-hot sample $z_{ij}$ to $\alpha_{ij}$.
The basic idea of ST is that the sampled discrete $z_{ij}$ is used for forward computation, but the continuous probability $\pi_{ij}$ is used for backward gradient calculation, namely approximating $\partial{z_{ij}} \approx \partial{\pi_{ij}}$\footnote{Clearly, this is the approximation used by Eq.~\eqref{eq:proxyless} in ProxylessNAS.}. Specifically, we compute the gradient w.r.t.~$\alpha_{ij}$ as follows:
\begin{equation}
\begin{aligned}
\frac{\partial{\mathcal{L}_\theta(z)}}{\partial{\alpha_{ij}^{(k)}}}
& =\frac{\partial{\mathcal{L}_\theta(z)}}{\partial{x_j}} \frac{\partial{x_j}}{\partial{\alpha_{ij}^{(k)}}} =
\frac{\partial{\mathcal{L}_\theta(z)}}{\partial{x_j}} \sum_{i\in\mathcal{A}_j} \frac{\partial{\Omega_{ij}(x_i)}}{\partial{\alpha_{ij}^{(k)}}} \\
%
\frac{\partial{\Omega_{ij}(x_i)}}{\partial{\alpha_{ij}^{(k)}}}
& =\sum_{k'=1}^K \frac{\partial{z_{ij}^{(k')}}}{\partial{\alpha_{ij}^{(k)}}} o_{ij}^{(k')}(x_i) \approx \sum_{k'=1}^K \frac{\partial{\pi_{ij}^{(k')}}}{\partial{\alpha_{ij}^{(k)}}} o_{ij}^{(k')}(x_i)
\end{aligned}
\label{eq:ST-grad}
\end{equation}
\begin{figure}[t]
\begin{minipage}[b]{0.32\linewidth}
\centering
\includegraphics[width=2.8cm]{dartsflow}
\centerline{(a) DARTS}\medskip
\end{minipage}
\begin{minipage}[b]{0.32\linewidth}
\centering
\includegraphics[width=2.8cm]{snasflow}
\centerline{(b) SNAS}\medskip
\end{minipage}
\begin{minipage}[b]{0.32\linewidth}
\centering
\includegraphics[width=2.8cm]{stgflow}
\centerline{(c) ST-NAS}\medskip
\end{minipage}
%
\vspace{-4mm}
\caption{Computation flow of different NAS methods locally between connected node$_i$ and node$_j$ when $K=4$. Solid and dashed lines denote the forward and backward computations respectively. (a) For DARTS and SNAS, the forward is fully continuous and the backward is fully differentiable. (b) For ProxylessNAS and ST-NAS, the forward from $\pi_{ij}$ to $z_{ij}$ involves sampling and the backward uses the ST gradient to flow from $z_{ij}$ to $\alpha_{ij}$.}
\label{fig:compute-flow}
\end{figure}
An illustration of the forward and backward computation performed locally between connected node$_i$ and node$_j$ is shown in Fig.~\ref{fig:compute-flow}.
We compare different gradient-based NAS methods in Table~\ref{table:tab1}.
Compared to training a single model, both DARTS and SNAS are around $K$ times more expensive in both memory and time.
Note that as shown in Eq.~\eqref{eq:ST-grad}, the gradient w.r.t.~$\alpha_{ij}$ involves all the $K$ features $\{o_{ij}^{(k')}(x_i), k'=1,\cdots,K\}$.
Regarding this, the backward computation complexity in ProxylessNAS and ST-NAS can be reduced to be $O(1)$, but the forward computation complexity is still $O(K)$.
As shown in Table~\ref{table:tab1}, the memory cost in ProxylessNAS and ST-NAS is far less than $K C_1$, where $C_1$ denotes the memory size for training a single model.
\begin{table}[t]
\centering
\vspace{-4mm}
\caption{Comparison of different gradient-based NAS methods.\tnote{1}}
\vspace{-2mm}
\resizebox{0.48\textwidth}{!}{
\begin{threeparttable}
\label{table:tab1}
\begin{tabular}{ccccc}
\toprule
Methods & Loss & $\alpha$ gradient & Memory & \makecell{Backward \\computation} \\
\midrule
DARTS & $\mathcal{L}^{\text{DARTS}}(\alpha, \theta)$ & continuous & $K C_1$ & $O(K)$ \\
SNAS & $\tilde{\mathcal{L}}(\alpha, \theta)$ & continuous & $K C_1$ & $O(K)$ \\
Proxyless & $\mathcal{L}_\theta(z)$ & ST & $C_1+(K-1) C_2$ & $O(1)$ \\
ST-NAS & $\mathcal{L}(\alpha, \theta)$ & ST & $C_1+(K-1) C_2$ & $O(1)$ \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\footnotesize
\item[1] Computation costs are estimated relative to training a single model. $K$ denotes the number of possible operations for each connected pair of nodes. The forward computation complexity of all methods is $O(K)$.
$C_1$ denotes the memory size for training a single model.
$C_2$ denotes the average memory size for storing the output features for all connected pairs of nodes in a sub-graph.
Usually we have $C_2\ll C_1$ (see numerics in Section~\ref{sec:exp:resource}).
\end{tablenotes}
\end{threeparttable}}
\end{table}
\vspace{-2mm}
\subsection{The NAS procedure}
\label{sec:method:nasprocedure}
Here we present the whole NAS procedure, which uses the ST gradients and is illustrated in Fig.~\ref{fig:toplevelsystem}.
First, we need to choose the macro-DAG and the set of possible candidate operations, which together define the super-network, as the search space.
The setup of macro-DAG, candidate operations and the super-network is flexible and depends on the target task.
One example setup for the end-to-end ASR task is described in Section~\ref{sec:exp}.
Given this setup, we divide the NAS procedure into three stages: super-network initialization (also called warm-up), architecture search and retraining.
There are two sets of learnable parameters: the architecture weights and operation parameters, denoted by $\alpha$ and $\theta$ respectively.
We split the original dataset into a training set and a validation set.
\textbf{Super-network initialization.}
Note that the single set of operation parameters $\theta$ is shared for all sampled sub-graphs, and thus plays a critical role in later architecture search.
It is found in our experiments that an initialization stage to warm up $\theta$ is beneficial.
Basically, this initialization stage is similar to the following architecture search stage, except that only $\theta$ is trained.
Specifically, for each minibatch from the training data, we uniformly sample a sub-graph, update $\theta$ using the standard gradient descent.
After each training epoch, we evaluate the super-network over the validation data (which is to be detailed below) to monitor the convergence.
The effect of this initialization stage on the performance of the searched architecture is detailed in Section~\ref{sec:method:effectSI}.
\textbf{Architecture search.}
After completing the super-network initialization, we hold $\theta$, reset the optimizer and run the architecture search stage, which involves a bi-level optimization problem \cite{Colson2007an}:
\begin{equation}
\begin{aligned}
& \min_{\alpha} \mathcal{L}_{val}(\alpha, \theta^*(\alpha)) \\
\text{s.t.}\ \ & \theta^*(\alpha) = \mathop{\arg\min}_{\theta} \mathcal{L}_{train}(\alpha, \theta)
\end{aligned}
\end{equation}
where $\mathcal{L}_{train}$ and $\mathcal{L}_{val}$ are the losses over the training and validation data respectively.
Solving the above bi-level optimization problem is difficult. In practice, updating $\alpha$ and $\theta$ alternately by stochastically optimizing $\mathcal{L}_{val}$ and $\mathcal{L}_{train}$ over minibatches respectively is found to work reasonably well, as shown in many previous NAS methods and given in Algorithm \ref{alg:AS}.
Note that as the validation set is usually smaller than the
training set, we cyclically use the validation set in drawing validation minibatches in \emph{Step 1} of Algorithm \ref{alg:AS}.
For evaluating the super-network over the validation data to monitor convergence, we average the validation losses over the minibatches in the whole validation set (not cyclically). Specifically, for each validation minibatch, we sample a sub-graph and calculate the validation loss over the sampled sub-graph.
In this manner, we evaluate the expected performance of the current super-network.
\begin{algorithm}[t]
\caption{Architecture Search}
\label{alg:AS}
\begin{algorithmic}
\WHILE{\text{not converged}}
\FOR{each training minibatch in an epoch}
\STATE \emph{Step 1.} Freeze $\theta$, draw a validation minibatch, sample a sub-graph, run the forward computation\footnotemark over the super-network under the validation minibatch, and update $\alpha$ with the ST gradients;
\STATE \emph{Step 2.} Freeze $\alpha$, sample a sub-graph, run the forward computation over the sampled sub-graph under the training minibatch, and update $\theta$ with the standard gradients;
\ENDFOR
\STATE Evaluate the super-network over the validation data to monitor convergence.
\ENDWHILE
\end{algorithmic}
\end{algorithm}
\footnotetext{In this forward computation, we calculate $\{o_{ij}^{(k')}(x_i), k'=1,\cdots,K\}$, but use $\Omega_{ij}(x_i)$ as defined in Eq.~\eqref{eq:z-operation}.}
\textbf{Retraining.}
After the convergence of the architecture search, we derive a single model by selecting the top-$1$ edge between connected nodes and prune others from the super-network.
The derived single model is trained from scratch to yield the final model.
\vspace{-3mm}
\section{Experiments}
\label{sec:exp}
\vspace{-1mm}
Experiments are conducted on the 80-hour WSJ and the 300-hour Switchboard datasets. Input features are 40 dimension fbank with delta and delta-delta features (120 dimensions in total). The features are augmented by 3-fold speed perturbation, and are mean and variance normalized.
We use the CAT toolkit \cite{an2020cat} for all experiments.
\begin{figure*}[t]
\begin{minipage}[t]{1.0\linewidth}
\centering
\includegraphics[width=17cm]{supernetandbest}
\centerline{(a)}\medskip
\end{minipage}
%
\begin{minipage}[b]{1.0\linewidth}
\centering
\includegraphics[width=17cm]{fig3b.pdf}
\centerline{(b)}\medskip
\end{minipage}
%
\vspace{-8mm}
\caption{(a) shows the super-network for WSJ experiments. Labels of candidate operations over edges are formatted in ``-\{half of context\}-\{dilation\}''. In Switchboard experiments, we add extra ``TDNN-3-1'' and ``TDNN-3-2'' candidate operations to each searching blocks. The solid lines in (a) indicate one of the derived single model from the 5 runs of NAS on WSJ.
(b) shows the evolution of architecture probabilities (i.e., $\pi_{ij}^{(k)}$) for the searching blocks in the NAS run that yields the derived single model in (a).}
\label{fig:fig3}
%
\vspace{-6mm}
\end{figure*}
\vspace{-2mm}
\subsection{NAS settings and hyper-parameters}
\label{sec:exp:settings}
We conduct NAS for the acoustic model.
Unless otherwise stated, we follow the basic settings in \cite{xiang2019crf}.
The denominator graph uses a phone-based 4-gram language model, and the language model in decoding is a word-based 4-gram language model.
For NAS settings, we design a super-network by serially connecting 3 searching blocks, 1 subsampling TDNN layer, 3 searching blocks, and 1 fully-connected layer, as shown in Fig.~\ref{fig:fig3}(a). Although more complex super-networks can be designed, this super-network is inspired by the ``TDNN-D'' of \cite{peddinti2018low}. It can be easily seen that the ``TDNN-D'' is a sub-graph in the super-network, thus lies in our search space.
The subsampling layer is a TDNN layer with convolution kernel size 3, dilation 1, and stride 3, which leads to a subsampling of factor 3.
The candidate operations in each searching block are TDNNs with different configurations.
The number of hidden units in all TDNN layers is 640.
Convolution strides of candidate TDNNs are all set to 1.
Layer normalization and dropout with probability 0.5 are applied after each TDNN layer.
By default, during the warm-up and architecture search stages, the super-network is trained with CTC. This is called NAS with CTC.
The derived single model is then retrained with CTC-CRF.
We run the 3-stage NAS procedure as described in Section~\ref{sec:method:nasprocedure} with the following hyper-parameters. We use the Adam optimizer (in PyTorch) with its default arguments unless otherwise stated. It can be seen that for simplicity and reproducibility, we only introduce a few extra hyper-parameters compared to training a single model.
\begin{itemize}
\item Super-network initialization:
minibatch size is 128, learning rate is fixed to $10^{-3}$. Iterations stop if no improvement of the validation loss for 3 training epochs.
\item Architecture search:
minibatch size is 64, learning rate decays from $10^{-3}$ to $10^{-4}$ with 0.1 decay rate if no improvement of the validation loss for 3 training epochs. Iterations stop if learning rate smaller than $10^{-4}$.
\item Retraining:
minibatch size is 128, learning rate decays from $10^{-3}$ to $10^{-5}$ with 0.1 decay rate if no improvement of the validation loss for 1 training epoch. Iterations stop if learning rate smaller than $10^{-5}$.
\end{itemize}
\vspace{-2mm}
\subsection{WSJ}
\label{sec:exp:wsj}
\vspace{-1mm}
The candidate operations of WSJ experiments are shown in Fig.~\ref{fig:fig3}(a), with which the search space contains 4096 sub-graphs in total.
Eval92 and dev93 sets are both for test and excluded from training set and validation set.
In warm-up and architecture search, the original training data are split into 90\%:10\% proportions for training and validation. For retraining, 5\% sentences of the original training set are for validation and the others for training.
The experiments run on 4 NVIDIA GTX1080 GPUs. To reduce the randomness, we conduct NAS with CTC for 5 times with different random seeds. The retraining uses the CTC-CRF loss with the CTC loss (weighted by 0.01). For comparison, NAS with fully CTC-CRF (namely the warm-up and architecture are trained with CTC-CRF) is conducted 3 times due to the expensive computation of CTC-CRF. Random search is conducted 5 times.
\begin{table}[t]
\centering
\vspace{-2mm}
\caption{WERs on the 80-hour WSJ.}
\resizebox{0.435\textwidth}{!}{
\begin{threeparttable}
\label{table:tab2}
\begin{tabular}{cccc}
\toprule
\multicolumn{2}{c}{Methods} & eval92 & dev93 \\
\midrule
\multicolumn{2}{c}{EE-Policy-CTC \cite{Zhou2018Improving}} & 5.53 & 9.21 \\
\multicolumn{2}{c}{SS-LF-MMI \cite{Hadian2018flatstart}} & 3.0 & 6.0 \\
\multicolumn{2}{c}{EE-LF-MMI \cite{Hadian2018e2e}} & 3.0 & - \\
\multicolumn{2}{c}{FC-SR \cite{Zeghidour2018fully}\tnote{1}} & 3.5 & 6.8 \\
\multicolumn{2}{c}{ESPRESSO \cite{Wang2019Espresso}} & 3.4 & 5.9 \\
\midrule
\multicolumn{1}{c}{\multirow{2}{*}{CTC}} & BLSTM & 4.93 & 8.57 \\
~
~ & ST-NAS & 4.72$\pm$0.03 & 8.82$\pm$0.07 \\
\midrule
\multicolumn{1}{c}{\multirow{5}{*}{CTC-CRF}} & BLSTM \cite{xiang2019crf} & 3.79 & 6.23 \\
~ & VGG-BLSTM \cite{an2020cat} & 3.2 & 5.7 \\
\cmidrule{2-4}
~ & TDNN-D\tnote{2} \cite{peddinti2018low} & 2.91 & 6.24 \\
~ & Random search\tnote{3} & 2.82$\pm$0.01 & 5.71$\pm$0.03 \\
~ & ST-NAS & {\bf 2.77}$\pm$0.00 & {\bf 5.68}$\pm$0.01 \\
\midrule
\multicolumn{2}{c}{ST-NAS with fully CTC-CRF} & 2.81$\pm$0.01 & 5.74$\pm$0.02 \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\footnotesize
\item[1] FC-SR uses dev93 as validation set and eval92 for test.
\item[2] Obtained based on our implementation of the ``TDNN-D'' in \cite{peddinti2018low}.
\item[3] Random search is a competitive baseline for NAS \cite{Li2019Random} but still inferior to our ST-NAS.
\end{tablenotes}
\end{threeparttable}}
\end{table}
We compare our searched models to various human-designed DNN architectures in end-to-end ASR, trained with CTC, CTC-CRF or attention-based losses.
As shown in Table~\ref{table:tab2}, the model searched by ST-NAS with CTC and retrained with CTC-CRF achieves the lowest WER of 2.77\%/5.68\% on WSJ eval92/dev93, outperforming all other end-to-end ASR models.
This model obtains significant improvement over both BLSTM and VGG-BLSTM with CTC-CRF, with 26.9\% and 13.4\% relative WER reductions respectively on eval92.
On dev93, this model shows an 8.8\% relative improvement over BLSTM and is close to VGG-BLSTM.
Notably, the number of parameters in our searched models on average is around 11.9 million, 11.8\% less than BLSTM and 25.6\% less than VGG-BLSTM.
There are some ablation results in Table~\ref{table:tab2}.
First, for the models searched by NAS with CTC, retraining with CTC-CRF loss achieves improvement over retraining with CTC loss.
Second, compared to the models searched and retrained all with CTC-CRF (namely NAS with fully CTC-CRF), the models searched with CTC but retrained with CTC-CRF perform equally well.
In summary, these results show that the architectures searched by NAS with CTC are transferable to be retrained with CTC-CRF.
This enables us to reduce the cost of running NAS to search the architecture, since CTC-CRF is somewhat expensive than CTC.
\vspace{-3mm}
\subsection{Switchboard}
\vspace{-1mm}
In the Switchboard experiment, we add extra ``TDNN-3-1'' and ``TDNN-3-2'' candidate operations, in addition to those shown in Fig.~\ref{fig:fig3}(a) for WSJ. The entire search space contains 46656 sub-graphs.
The retraining uses the CTC-CRF loss with the CTC loss (weighted by 0.1). In warm-up and architecture search, the original training data are split into 95\%:5\% proportions for training and validation.
For retraining, we take around 5 hours of speech as validation set and the rest for training, following the setting in CAT \cite{an2020cat}.
The Eval2000 data, including the Switchboard evaluation dataset (SW) and Callhome evaluation dataset (CH), are for testing.
We conduct a single run of ST-NAS on 4 NVIDIA P100 GPUs, and do not conduct multiple random searches due to time limitation.
As shown in Table~\ref{table:tab3}, our ST-NAS models outperform both the TDNN-D-Small and TDNN-D-Large.
Notably, the model transferred from the WSJ experiment performs close to the model searched over Switchboard; and compared to the comparably-sized TDNN-D-Large, it obtains 13.7\% and 8.6\% relative WER reductions on SW and CH respectively.
\begin{table}[t]
\centering
\caption{WERs on the 300-hour Switchboard\tnote{1}.}
\vspace{-2mm}
\resizebox{0.44\textwidth}{!}{
\begin{threeparttable}
\label{table:tab3}
\begin{tabular}{ccccc}
\toprule
\multicolumn{2}{c}{Methods} & SW & CH & Params \\
\midrule
\multicolumn{2}{c}{TDNN-D-Small} & 15.2 & 26.8 & 7.64M \\
\multicolumn{2}{c}{TDNN-D-Large} & 14.6 & 25.5 & 11.85M \\
\midrule
\multicolumn{1}{c}{\multirow{2}{*}{ST-NAS}} & Transferred from WSJ\tnote{2} & 12.5 & 23.2 & 11.89M \\
~ & Searched on Switchboard & 12.6 & 23.2 & 15.98M \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\footnotesize
\item[1] All experiments are trained with CTC-CRF. TDNN-D-Small is with the hidden size of 640, which is the same as that of our searched models. TDNN-D-Large is with the hidden size of 800.
\item[2] Randomly taken from one of the 5 runs of NAS with CTC over WSJ, and retrained on Switchboard.
\end{tablenotes}
\end{threeparttable}}
\end{table}
\vspace{-3mm}
\subsection{Complexity analysis}
\label{sec:exp:resource}
\vspace{-1mm}
To complement Table~\ref{table:tab1}, this section provides numerical complexity analysis in memory and time for running ST-NAS.
In the WSJ experiment, there are 6 searching blocks in our super-network and we run on 4 GPUs with data parallel in PyTorch.
Training a single model requires around $C_1=3.5\text{GB}$ memory on each GPU.
The quantity $C_2$ in Table~\ref{table:tab1} can be calculated as follows:
\begin{equation}
\begin{aligned}
C_2 & = 6 \times \text{MinibatchSize} \times \text{SequenceLen} \\
& \times \text{HiddenUnitsNum}\times 4\text{~Byte} \div \text{GPUNum} \\
& =6\times 64 \times 850 \times 640 \times 4\text{~Byte} \div 4 \approx 209\text{MB}
\end{aligned}
\end{equation}
which is far less than $C_1$. $\text{SequenceLen}$ denotes the average length of sequences, which is around 850.
For time complexity, warm-up and architecture search are trained with CTC, which is much faster than CTC-CRF in retraining. Thus the extra time cost is limited.
As shown in Table~\ref{table:tab4}, the total time of running the 3-stage NAS procedure is less than 3 times of training a single model from scratch.
\begin{table}[t]
\centering
\setlength{\tabcolsep}{3mm}{
\begin{threeparttable}
\caption{The estimated run-time for the three stages in the ST-NAS procedure, averaged over the 5 runs in WSJ.}
\label{table:tab4}
\begin{tabular}{cccc}
\toprule
Stage & \makecell[c]{warm-up} & \makecell[c]{architecture \\search} & retraining \\
\midrule
Epochs & 65.2 & 22 & 28.6 \\
Minutes/epoch & 11 & 31 & 27.2 \\
\midrule
Total time (minute) & 717.2 & 682 & 777.9 \\
\bottomrule
\end{tabular}
\end{threeparttable}}
\end{table}
\begin{figure}[t]
\begin{minipage}[b]{0.48\linewidth}
\centering
\includegraphics[width=4.2cm]{fig4a.pdf}
\centerline{(a)}
\end{minipage}
%
\begin{minipage}[b]{0.48\linewidth}
\centering
\includegraphics[width=4cm]{fig4b.pdf}
\centerline{(b)}
\end{minipage}
\vspace{-2mm}
\caption{(a) The two points A and B in the loss curve during warm-up, which represent differently initialized super-networks. (b) The curves of validation losses for retraining the two models, obtained by running architecture search starting from A and B respectively.}
\label{fig:method:effectSPI}
\end{figure}
\vspace{-2mm}
\subsection{Effect of super-network initialization}
\label{sec:method:effectSI}
\vspace{-1mm}
The effect of the super-network initialization (i.e., warm-up) seems to be overlooked in previous NAS studies.
We run architecture search from two differently initialized super-networks (A and B), retrain the two searched models, and compare the performance of the two retrained models, as shown in Fig.~\ref{fig:method:effectSPI}.
Experiments are conducted under the same settings as in Section~\ref{sec:exp:wsj} on the WSJ dataset.
Super-network A represents a randomly-initialized super-network, without any warm-up training, and super-network B is obtained when the warm-up converges.
It can be seen that sufficient warm-up helps the architecture search stage to find a better final model, which achieves lower validation loss in retraining.
\section{Conclusion}
\label{sec:conclusion}
NAS is an appealing next step to advancing end-to-end ASR.
In this paper, we review existing gradient-based NAS methods and develop an efficient NAS method via Straight-Through (ST) gradients, called ST-NAS.
Basically, ST-NAS uses the loss from SNAS but optimizes the loss using the ST gradients.
We successfully apply ST-NAS to end-to-end ASR.
Experiments over WSJ and Switchboard show that the ST-NAS induced architectures significantly outperform the human-designed architecture across the two datasets. Strengths of ST-NAS such as architecture transferability and low computation cost in memory and time are also reported.
Remarkably, the ST-NAS method is flexible and can be further explored by using different macro-DAGs, candidate operations and super-networks for ASR, not limited to the example setup in this paper.
\bibliographystyle{IEEEbib}
|
1,941,325,220,555 | arxiv | \section{Introduction}
Public digital randomness is an essential building component for a large spectrum of applications and protocols. For example, a reliable source of continuous randomness providing high entropy, also known as \textit{random beacon}, is crucial for many security applications. A notion of \textit{coin tossing protocol}~\cite{blum1983coin} was proposed by Blum in 1983 that addressed the question of generating the trustworthy random value in a network of mutually distrustful participants. Further, Rabin~\cite{rabin1983transaction} formalized the notion of the random beacon. Since then, randomness generation has been advanced significantly due to the underlying modern cryptography.
Lately, coin tossing protocols became more appealing in Proof-of-Work (PoW) or Proof-of-Stake (PoS)~\cite{gilad2017algorand} consensus. Random beacon has a range of applications that includes cryptographic parameter generation~\cite{lenstra2015random}, design of byzantine fault tolerant (BFT) protocols~\cite{cachin2005random,gilad2017algorand}, privacy-preserving message services~\cite{goel2003herbivore}, e-voting protocols~\cite{adida2008helios}, online gaming~\cite{bonneau2015bitcoin}, publicly auditable selections~\cite{bonneau2015bitcoin}, anonymous browsing~\cite{goulet2015random}, sharded blockchains~\cite{croman2016scaling} and smart contracts~\cite{kosba2016hawk}.
Due to the applicability of shared randomness in a variety of applications, a rich body of literature has emerged that proposes many DRB protocols differing in their designs and underlying cryptography. Nevertheless, the system models and design challenges in these DRB protocols are highly disparate. Therefore, to address these challenges and to provide a general definition of a DRB protocol, we present a Systematization of Knowledge (SoK). The purpose of this SoK is to provide a systematic overview of existing DRB protocols that can help researchers and practitioners to find suitable solutions for randomness generation.
\paragraph{\textbf{Background.}}
An easy approach to achieve continuous randomness is through a single node or a trusted third party such as NIST~\cite{kelsey2019reference}, Random.org~\cite{haahr2010random} or Oraclize.it~\cite{Oraclize}. The NIST beacon continuously outputs hardware-generated random values from a quantum-mechanical effect. Since these beacon services are centralized, they can be unreliable. Moreover, in the past, they suffered a significant public trust deterioration after the revealed backdoor in the NSA-designed Dual elliptic curve pseudorandom number generator~\cite{shumow2007possibility}. Due to these problems, these centralized beacon services are undesirable for secure applications.
As a consequence, \textit{Decentralized Randomness Beacon} (DRB) protocols were proposed and constructed where trust is distributed across multiple nodes that jointly generate random values at a regular interval. More concretely, a consortium of organizations launched a distributed publicly verifiable randomness beacon that periodically provides unbiasable and unpredictable random outputs. The deployment is known as League of Entropy (LoE))~\cite{LoE}
that aims to provide collaborative governance for protection of its random beacon.
The consortium believes that their beacon can become a fundamental service on the internet.
DRB protocols can be constructed by employing different cryptographic primitives e.g. Publicly Verifiable Secret Sharing schemes (PVSS)~\cite{kiayias2017ouroboros,syta2017scalable,schindler2020hydrand,cascudo2017scrape,cascudo2020albatross,bhat2020randpiper,das2021spurt}, Threshold Crypto-Systems~\cite{cachin2005random,hanke2018dfinity,Drand,cherniaeva2019homomorphic,nguyen2019scalable}, Verifiable Random Functions (VRF)~\cite{david2018ouroboros,gilad2017algorand,galindo2020fully,wang2020randchain,Corestar,DAOBet}, Verifiable Delay Functions (VDF)~\cite{lenstra2015random,drake2018minimal,ephraim2020continuous,schindler2021randrunner,han2020randchain}. Randomness can also be extracted from external data sources such as~\cite{bentov2016bitcoin,clark2010use,bonneau2015bitcoin} or from the blockchain schemes having their own random beacon~\cite{kiayias2017ouroboros,galindo2020fully,hanke2018dfinity}. These DRB protocols are not equally-suited in all applications or use-cases due to the diversity in their designs, characteristics, and underlying assumptions.
DRB protocols differ significantly due to their underlying techniques. A DRB protocol should have a list of desirable beacon properties along with low communication complexity, low computational cost, and low trust requirement (e.g., setup assumptions). Additionally, the DRB protocol should be efficient in practical settings. Therefore, despite having many constructions of DRB protocols, a few problems such as scalability, trust, and network assumptions need to be addressed to construct a desirable DRB protocol for practical applications.
Motivated by the above, the contributions of this SoK are as follows:
\begin{enumerate}
\item We provide a formal definition of a Decentralized Randomness Beacon (DRB) with a brief description of its security properties (Sect.~\ref{Sec:DRB}).
\item We present a classification of DRB protocols in Interactive and Non-interactive DRB protocols and we describe these protocols in detail (Sect.~\ref{Sec:DRB-classification}).
\item We give a brief discussion on several crucial issues related to DRB protocols, including complexity, scalability, and assumptions. We also identify a few efficient building components to construct efficient DRB protocols (Sect.~\ref{Sec:Discussion}).
\end{enumerate}
\section{Decentralized Randomness Beacon (DRB)} \label{Sec:DRB}
A DRB allows a group of participants to collaboratively produce random values without the need of a central party. A DRB consists of $n$ participants~\footnote{We use node and participant interchangeably in protocols throughout the paper.} $\mathcal{P} = (P_1, P_2, \ldots, P_n)$. These participants are connected in a distributed manner. Without loss of generality, we assume that a DRB protocol works in rounds and maintains a beacon state $st$ for each round. For every round $e \in \{1, 2, \ldots\}$, given the current state $st_{e-1}$, the DRB protocol collectively produces a random output $v_e$; the state $st_0$ is jointly computed and agreed from the protocol participants during the bootstrapping of the DRB protocol. Following, we present a formalization of DRB and we formally define the required security properties of a DRB. Additionally, we define a secure DRB protocol in Appendix~\ref{appendix:secure-DRB}.
\begin{definition}{(Decentralized Randomness Beacon (DRB))} \label{Def:DRB}
A DRB on a set of participants $\mathcal{P} = (P_1, \ldots, P_n)$ is defined as a tuple $\mathcal{B}$ of polynomial algorithms:\\
$\mathcal{B} = \mathsf{(Setup, LocalRand, GlobalRand, VerifyRand, UpdateSt)}$
\begin{itemize}[leftmargin=*]
\item $\mathsf{Setup}(1^{\lambda},n)$: Given input security parameter $\lambda$, and $n$ participants, it generates public parameter $pp$ and keypair for each participant $(pk_i,sk_i)$. All participants agree on public parameter $pp$ and $\{pk_i\}$.
\item $\mathsf{LocalRand}(st_{e-1},pp,sk_i,s_{e,i})$: Given input state $st_{e-1}$ from round $e-1$, public parameter $pp$, and input seed $s_{e,i}$, a participant $P_i$ computes a local output value $v_{e,i}$ with a proof ${\pi}_{e,i}$ using $sk_i$ and $s_{e,i}$ for round $e$. Output $(i,v_{e,i},{\pi}_{e,i})$.
\item $\mathsf{GlobalRand}(st_{e-1},pp,\mathcal{S} = \{(i,v_{e,i},{\pi}_{e,i})\},m)$: Given input state $st_{e-1}$, public parameter $pp$, a set $\mathcal{S}$ of local output values from $|\mathcal{S}|$ participants, if $|\mathcal{S}| \geq m$, where $m$ is the minimum number of required local output values, the algorithm computes the beacon output $v_e$ for round $e$ by executing a function $f$ on $\{v_{e,i}\}$ from set $\mathcal{S}$. It also computes proof of correctness ${\pi}_e$ using $\{ {\pi}_{e,i} \}$ from set $\mathcal{S}$. Output $(v_e,{\pi}_e)$ or $\perp$.
\item $\mathsf{VerifyRand}(st_{e-1},pp,v_e,{\pi}_e)$: Given input state $st_{e-1}$, public parameter $pp$, a beacon output $v_e$, and a proof ${\pi}_e$, the algorithm verifies the beacon value $v_e$ and the corresponding proof ${\pi}_e$. Output $\mathsf{0}$ or $\mathsf{1}$.
\item $\mathsf{UpdateSt}(st_{e-1},pp,v_e,{\pi}_e)$: Given input state $st_{e-1}$, public parameter $pp$, a beacon output $v_e$, and a proof ${\pi}_e$ generated at the round $e$, the algorithm updates the state from $st_{e-1}$ to $st_e$ for round $e$. Output $st_e$ or $\perp$.
\end{itemize}
\end{definition}
The security properties of a DRB corresponds to: \textit{Unpredictability}: An adversary should not be able to predict (precompute) future beacon outcomes; \textit{Bias-Resistance}: A single participant or a colluding adversary cannot bias the future beacon values; \textit{Availability (or Liveness)}: A single participant or a colluding adversary can not prevent the generation of the new beacon value; \textit{Public Verifiability}: Any third party can verify the correctness of the new beacon value. \textbf{Note:} \textit{We use DRB protocols and DRBs interchangeably throughout the paper}.
These formal security guarantees of a DRB protocol are evolved during the time. Initial proposals lack the formal definitions and mathematical proofs of their DRB protocols. Nevertheless, the recent proposals put an emphasis on the security of their protocols. These protocols define and prove the security properties of their DRB using the mathematical properties of the underlying cryptographic primitives. Due to different designs, setting up a formal provability framework for DRBs should define the least common security requirements, therefore, we formulate the desiderata of a DRB protocol as follows where $\lambda$ is a security parameter and $\mathsf{negl}(\lambda)$ is a negligible function of $\lambda$.
\begin{definition}{(Unpredictability)}
Let $\mathcal{A}(v_1, \ldots,v_{e}, st_e)$ be a probabilistic polynomial time algorithm that receives the values $v_1, \ldots,v_{e}$ and the current state $st_e$ as the input values. Let $\mathcal{A}$ outputs a value $v_{e+f}$ for any value (future rounds) $f \geq 2$, and for all rounds $e \geq 1$. Then
\begin{equation}
\Pr [\mathcal{A}(v_1, \ldots,v_{e},st_e) = v_{e+f}] \leq \mathsf{negl}(\lambda)
\end{equation}
\end{definition}
\begin{definition}(Bias-Resistance)
Let $\mathsf{bit}_{i}(v_e) $ denotes the $i$-th bit in the binary representation of $v_e$, let $b = |v_e|$ is the number of bits of $v_e$, and let $\mathcal{A}_i(v_1, \ldots,v_{e-1}, \\st_{e-1})$ for $i=1,\ldots,b$, be $b$ probabilistic polynomial-time algorithms that receive the values $v_1, \ldots,v_{e-1}$ and the current state $st_{e-1}$ as input and output one bit: $0$ or $1$. Then for every round $e \geq 1$, every $\mathcal{A}_i(\ )$ and for all $i=1,\ldots,b$
\begin{align}
\Pr [\mathsf{bit}_{i}(v_e) & = \mathcal{A}_i(v_1, \ldots,v_{e-1}, st_{e-1})] \leq \frac{1}{2} + \mathsf{negl}(\lambda)\\
\Pr [\mathsf{bit}_{i}(v_e) & = 0] \leq \frac{1}{2} + \mathsf{negl}(\lambda)
\end{align}
\end{definition}
More concretely, we say that a DRB protocol is Bias-resistant if predicting any single bit of the random beacon output $v_e$ has only a non-negligible advantage over the trivial guessing strategy that has a probability of 1/2.
\begin{definition}(Availability)
Let $\mathcal{A}$ be an adversary controlling a fraction of participants and ${\mathcal{P}}^h \subseteq{} \mathcal{P}$ be a set of honest participants in the DRB protocol. Given $v_e, {\pi}_e, pp$ and $st_{e-1}$, for every round $e \geq 1$ and for every participant $P_i \in {\mathcal{P}}^h$
\begin{equation}
\Pr [\mathsf{UpdateSt}(st_{e-1},pp,v_e,{\pi}_e) \neq st_e] \leq \mathsf{negl}(\lambda)
\end{equation}
\end{definition}
\begin{definition}(Public Verifiability)
Given $\mathsf{VerifyRand}(\ )$ as a public probabilistic polynomial-time algorithm run by an external verifier $P_x \notin \mathcal{P}$ that receives $v_e, {\pi}_e$ and the state $st_{e-1}$ at the end of round $e$ as input values and outputs a bit $\mathsf{0}$ or $\mathsf{1}$ based on the verification of $v_e$ using ${\pi}_e$. Then for every round $e \geq 1$
\begin{equation}
\Pr [\mathsf{VerifyRand}(v_e,{\pi}_e,st_{e-1}) \neq \mathsf{1}] \leq \mathsf{negl}(\lambda)
\end{equation}
\end{definition}
\section{DRB Classification} \label{Sec:DRB-classification}
We classify DRB protocols in two ways: \textit{Interactive} and \textit{Non-Interactive}. \textit{Interactive} DRB protocols generate a beacon output in an interactive manner which involves multiple rounds of communication among participants. However, \textit{Non-Interactive} DRB protocols do not involve interactions among participants to produce a random beacon value for each round. Therefore, non-interactive DRBs are preferable for decentralized applications. Nevertheless, the setup for the public parameter generation can be interactive for both types of DRBs.
\subsection{Interactive Decentralized Randomness Beacon Protocols} \label{Sec:I-DRB}
Interactive DRB protocols employ multiple rounds of interaction among participants in order to produce one beacon output. These protocols are constructed using interactive cryptographic primitives such as Publicly Verifiable Secret Sharing (PVSS) or Interactive Threshold Signature Scheme. The existing interactive DRB protocols are based on PVSS involving two logical rounds of coin-tossing wherein the first round, the participants broadcast commitments to their shares, and further, these commitments are revealed in another round. Constructions of DRBs with other interactive cryptographic primitives, we left as open problems.
\begin{researchproblem}
Construct a DRB protocol based on interactive threshold signature scheme with better complexity compared to existing interactive DRBs.
\end{researchproblem}
The main advantage of PVSS-based DRBs is that the generated randomness is indistinguishable from uniform. Nevertheless, due to the interaction and broadcast, interactive DRBs incur high communication cost. Some of the PVSS-based DRBs improve upon the general PVSS scheme to reduce the communication complexity by utilizing a threshold version of PVSS or electing a committee to perform PVSS or introducing a leader to relay the messages. Hence, these optimized versions of DRB protocols can be used to obtain periodic fresh randomness in real-world applications. A PVSS scheme consists of a tuple of algorithms $\mathsf{(PVSS.Setup, PVSS.Share, PVSS.Verify, PVSS.Recon)}$ described in Appendix~\ref{appendix:PVSS}.
PVSS-based DRB protocols are mainly of two types: with leader~\cite{schindler2020hydrand,bhat2020randpiper,das2021spurt} and without leader~\cite{kiayias2017ouroboros,cascudo2017scrape,cascudo2020albatross}. In a leader-based protocol, a leader $L_e$ is elected in each round $e$ which is responsible for performing the distribution of the secret shares of the PVSS scheme. A further illustration can be found in Appendix~\ref{appendix:PVSS}. Following we present a description of PVSS-based interactive DRB protocols.
\begin{itemize}[leftmargin=*]
\item Ouroboros~\cite{kiayias2017ouroboros}: Ouroboros is a PoS-based blockchain where a set of elected participants run the DRB protocol to fetch the randomness for the leader election. It operates in two phases \textit{commit} and \textit{reveal}. In \textit{commit} phase, participants encrypt the shares for all other participants by running $\mathsf{PVSS.Share}$ and submit the shares on the blockchain. In \textit{reveal} phase, each participant decrypts all the encrypted shares that are encrypted using his public key. Then, each participant computes a local random value using all the decrypted shares and posts it in the blockchain. Finally, a beacon output is computed by performing an XOR operation on all the published local random values.
\item RandHound, RandHerd~\cite{syta2017scalable}: Syta et al. constructed scalable randomness generation protocols by following client-server architecture and threshold cryptography. RandHound is a one-shot on-demand protocol to generate single randomness. However, RandHerd is a beacon protocol that emits continuous random values. RandHound divides the servers into groups, and each group is responsible for running PVSS among the group members. RandHound employs the commit-reveal technique as defined in Ouroboros for each group. Finally, to produce global randomness in RandHound, a client operates on all the received valid local randomness from each server group. RandHerd improves upon the complexity of RandHound by leveraging communication trees among the server groups and collective signing to produce beacon outputs.
\item SCRAPE~\cite{cascudo2017scrape}: Cascudo et al. constructed an honest majority coin-tossing protocol SCRAPE with guaranteed output delivery. It constructs a threshold PVSS scheme where sharing, verification, and reconstruction take only a linear number of exponentiations compared to quadratic in basic PVSS scheme~\cite{schoenmakers1999simple}. In SCRAPE, all participants have access to a ledger where messages are posted similar to Ouroboros. Cascudo et al. constructed an efficient share verification procedure with linear complexity by observing the fact that sharing a secret using PVSS is equivalent to encoding the secret with a Reed Solomon error correcting code~\cite{reed1960polynomial}. The dealer in the PVSS scheme~\cite{schoenmakers1999simple} not only encrypts the shares but also commits to the shares. Therefore, to prove that shares in encrypted shares are the same as shares in commitments, the efficient share verification procedure involving error-correcting code is applied. SCRAPE improves the computation and verification cost compared to Ouroboros.
\item HydRand~\cite{schindler2020hydrand}: HydRand improves upon the complexity of SCRAPE's PVSS protocol. HydRand works in rounds consisting of three phases: \textit{propose}, \textit{acknowledge} and \textit{vote}. In each round, a leader is selected deterministically from the set of potential leaders and by using the last round randomness. In \textit{propose} phase, the leader reveals his previously committed value which is acknowledged, signed and further broadcast by the other participants in \textit{acknowledge} phase. In \textit{vote} phase, each participant performs some checks, including the checks on the number of received acknowledgments. If the leader does not reveal his secret, the secret is reconstructed using $\mathsf{PVSS.Recon}$. The beacon value is computed using the revealed secret and the last round randomness.
\item ALBATROSS~\cite{cascudo2020albatross}: ALBATROSS significantly improves, amortizes the computation complexity of SCRAPE and provides a universal composability (UC)-secure model. It shows efficiency gain through the packed Shamir secret sharing scheme in PVSS or by using a linear t-resilient function to extract randomness as a vector of random group elements. It utilizes Cooley-Tukey fast Fourier transformation to amortize the complexity and for further improvement, it uses $\sum$-protocol to prove that the published sharing is correct. ALBATROSS provides two variants of UC security: 1) First variant uses UC-Non-Interactive Zero-Knowledge (NIZK) proofs for discrete logarithm, 2) Second variant introduces and uses a new primitive named ``designated verifier'' homomorphic commitments where a sender can open a commitment for one specific receiver. Later, the receiver can prove the same opening to a third party.
\item RandPiper~\cite{bhat2020randpiper}: Bhat et al. constructed a reconfiguration-friendly DRB protocol RandPiper with strong security guarantees and quadratic communication complexity. It combines PVSS with State-Machine Replication protocol and presents two protocols: GRandPiper and BRandPiper. GRandPiper is a communication optimal DRB with strong unpredictability in the presence of a static adversary. However, BRandPiper shows the best communication complexity and the best possible unpredictability in case of a dynamic adversary.
\item SPURT~\cite{das2021spurt}: SPURT protocol constructs a new PVSS scheme using pairing to produce beacon output and involves a leader. The new PVSS scheme relies on Decisional Bilinear Diffie-Hellman (DBDH) assumption~\cite{boneh2001short}. In addition, SPURT uses State Machine Replication to lower the communication complexity compared to the broadcast channel used by other DRBs e.g., HydRand. SPURT operates in a semi-synchronous network and has no trusted setup.
\end{itemize}
\subsection{Non-Interactive Decentralized Randomness Beacon Protocols} \label{Sec:NI-DRB}
We categorize Non-Interactive DRB (NI-DRB) protocols based on the main constituent cryptographic primitive, further, we illustrate these protocols in Table~\ref{tab:NI-DRB}.
\vspace{-0.09cm}
\subsubsection{VDF-based}
These DRBs are based on stand-alone Verifiable Delay Function $\mathsf{VDF = (VDF.Setup, VDF.Eval, VDF.Verify)}$ described in Appendix~\ref{appendix:VDF}. A VDF is a function $f:\mathcal{X} \rightarrow \mathcal{Y}$ that takes a prescribed number of sequential steps to compute the output and provides exponentially easy verification of the output. In a VDF-based DRB, the participants evaluate an Iteratively Sequential Function (ISF) to generate their local random values. The verification of these values can be efficiently done using $\mathsf{VDF.Verify}$. Due to the non-parallelizable property of VDF, an adversary cannot bias the output of the random beacon.
Lenstra and Wesolowski~\cite{lenstra2015random} constructed a DRB protocol, Unicorn, using a slow-time hash function named \textit{sloth}. This function takes inputs from a set of distrusting participants and outputs a random value. Keeping Unicorn protocol as a successor to VDF, the following VDF-based DRB protocols are constructed.
\begin{itemize}[leftmargin=*]
\item Minimal VDF Randomness Beacon~\cite{drake2018minimal}: Justin Drake constructed a minimal randomness beacon using RANDAO~\cite{RANDAO} and VDF. RANDAO is a smart contract based DRB where participants submit their local entropy to the smart contract, and further, the smart contract produces a single global entropy. RANDAO biasable entropy is used as input to the VDF to produce unbiasable randomness. Nevertheless, there is no formal security analysis of this protocol.
\item Continuous VDF~\cite{ephraim2020continuous}: Ephraim et al. presented a new notion of Continuous Verifiable Delay Function (cVDF) by adapting Pietrzak scheme~\cite{pietrzak:LIPIcs:2018}. A cVDF $f$ provides the output computation of each intermediate steps (i.e. $f(t)$ for $t < T$) with an efficient proof $\pi^t$ (used for public verification of the output). A cVDF can be used to construct a DRB protocol where beacon outputs are generated by applying a suitable hash to the intermediate outputs of each step. The drawback with this protocol is that the nodes having the most efficient (fastest) processors can always learn the beacon outputs before the other participating nodes. A similar argument goes for the Unicorn protocol.
\item RandRunner~\cite{schindler2021randrunner}: RandRunner leverages trapdoor VDF with strong uniqueness to construct a DRB protocol. Each participant $P_i$ of RandRunner initializes its public parameter ${pp}_i$ with a corresponding trapdoor $sk_i$. The participants exchange their public parameters and verify the received ones. RandRunner executes in consecutive rounds where in each round, a leader is elected. Further, the leader tries to solve the VDF using its trapdoor, and other participants attempt to solve the VDF using the common $\mathsf{VDF.Eval}$ algorithm. The drawback with the RandRunner protocol is that once a powerful adversary becomes a leader, it can keep corrupting the round leaders (e.g., via DoS), withhold its output computed via trapdoor, and keep working on for the next outputs for many subsequent rounds hence breaking unpredictability.
\item RANDCHAIN~\cite{han2020randchain}: RANDCHAIN is a competitive DRB where in each round, nodes compete to be a leader which solely produces the beacon output. RANDCHAIN constructs a non-parallelizable Sequential Proof-of-Work (SeqPoW) puzzle by employing VDF or Sloth. A node solves the SeqPoW puzzle by incrementing an ISF for a randomized time. RANDCHAIN works as a Nakamoto-based blockchain where nodes synchronize their local blockchains and keep solving the puzzle to mine new blocks to the main blockchain. RANDCHAIN mimics a blockchain structure, so it can suffer from front-running (block withholding) attacks and can also have forks due to problems with blocks' finality.
\end{itemize}
\subsubsection{VRF-based}
These DRBs compute randomness using Verifiable Random Function $\mathsf{VRF = (VRF.KeyGen, VRF.Eval, VRF.Verify)}$ described in Appendix~\ref{appendix:VRF}. A VRF is a pseudorandom function that produces pseudorandom output along with proof about the correctness of the output. Participants in these DRBs apply VRF on an input seed to generate their local entropy which is used to compute the beacon output. VRF-based DRBs are explained as follows:
\begin{itemize}[leftmargin=*]
\item Blockchain Protocol Designs: Ouroboros Praos~\cite{david2018ouroboros}, Algorand~\cite{gilad2017algorand} and Erlond~\cite{elrond2019highly} blockchains have their DRB as a byproduct. In these DRBs, each participant $P_i$ runs $\mathsf{VDF.Eval}$ on a seed (e.g., previous output or state) using its secret key $sk_i$ and the DRB output is computed from the participants' VRF outputs. These DRBs do not guarantee generation of uniformly random values and do not have strong bias-resistance as an adversary can include/exclude the corrupted participants' VRF outputs used for DRB output computation.
\item Distributed VRF-based DRBs: A distributed VRF (DVRF)~\cite{dodis2003efficient} based DRB was first introduced by DFINITY~\cite{hanke2018dfinity}. Later DRBs~\cite{Corestar,DAOBet} employed DFINITY-DVRF along with BLS cryptography. Nevertheless, these DRBs do not provide formal security analysis. A recent paper~\cite{galindo2020fully} provides two new constructions of DVRF: 1) DDH-DVRF based on elliptic curve cryptography; 2) GLOW-DVRF based on cryptographic pairings. These constructions also formalize a security model with proper security analysis. DRBs based on DDH-DVRF, and GLOW-DVRF show strong bias resistance and strong pseudorandomness.
\item RandChain~\cite{wang2020randchain}: RandChain follows \textit{commit-and-reveal} strategy by building a sub-routine \textit{RandGene} using VRF. RandChain has a two-layer hierarchical blockchain structure where nodes form distinct committees. Each committee has a local blockchain and generates local entropy through the RandGene protocol, further, global randomness is computed from these local entropy by forming a RandChain block. RandChain security depends on a secure sharding process, followed by a leader election for each shard (committee). However, both processes can be influenced by an adversary to obstruct DRB properties.
\end{itemize}
\vspace{-0.6cm}
\subsubsection{HE-based}
These DRBs utilize homomorphic encryption scheme $\mathsf{HE} = (\mathsf{HE.Setup}$,\\ $\mathsf{HE.KeyGen, HE.Enc, HE.Dec, HE.Eval)}$. Homomorphic encryption allows performing arithmetic operations on ciphertext directly without decryption (details in Appendix~\ref{appendix:HE}). Following DRBs employ ElGamal encryption~\cite{elgamal1985public} as partial HE.
\begin{itemize}[leftmargin=*]
\item Nguyen-Van et al.~\cite{nguyen2019scalable}: Their DRB has three components: a Requester, a Public Distributed Ledger (PDL), and a Core Layer. The protocol works in rounds where, first, the Requester sends a nonce to the PDL that computes a ticket $T$ and publishes it. Further, participants of the core layer run a VRF using the ticket $T$ to check if they are selected as a contributor. Each contributor publishes a ciphertext computed on a random value using the Requester's public key. Later, the Requester performs a homomorphic operation on the published ciphertexts and computes a single ciphertext. Finally, the Requester publishes the decrypted value as DRB output with a proof of correct decryption. There are two drawbacks: 1) A malleable ElGamal encryption, 2) The Requester can collude with contributors or refuse to decrypt the resulting ciphertext.
\item HERB~\cite{cherniaeva2019homomorphic}: Homomorphic Encryption Random Beacon (HERB) DRB uses threshold ElGamal encryption scheme with a distributed key generation (DKG) protocol. DKG is used to generate a common public key and secret key shares for participants. Each participant publishes a ciphertext share with proof of correct encryption (NIZK Proof) on a public bulletin board. These shares generate an aggregated ciphertext through ElGamal aggregation which is subsequently decrypted by a threshold of participants to produce the DRB output.
\end{itemize}
\vspace{-0.6cm}
\subsubsection{External Source-based}
In these DRBs, participants extract the randomness from an external entropy source, i.e., real-world entropy. These entropy sources can be public blockchains~\cite{bunz2017proofs,bentov2016bitcoin}, real-time financial data~\cite{clark2010use} or national lottery results~\cite{baigneres2015trap}. PoW-based blockchains are promising sources but an adversarial miner can manipulate the generated randomness. Therefore, to achieve most of the beacon properties, the following DRBs apply different defense mechanisms.
\begin{table*}[bth]
\resizebox{\textwidth}{!}{
\centering
\begin{tabular}{|l|c|c|}
\hline
\textbf{Scheme} & \textbf{Advantages} & \textbf{Disadvantages}\\
\hline
VDF-based & \makecell[l]{1. These DRBs achieve liveness under the period of full asynchrony. \\2. These DRBs avoid byzantine agreement consensus hence have \\less communication complexity. \\3. It shows strong bias-resistance as long as there is an honest node.} & \makecell[l]{1. Front-running attack can hinder some DRB properties.\\ 2. In most of these DRBs, the significant powerful adversary\\ can learn the output of DRB earlier than other nodes. \\ 3. These protocols rely on the new assumptions of VDF.} \\[0.1cm]
\hline
VRF-based & \makecell[l]{1. Most of these DRBs do not have any trusted setup and achieve \\strong notion of pseudo-randomness and bias-resistance~\cite{galindo2020fully}. \\ 2. These DRBs incur less computation and communication cost.} & \makecell[l]{1. In some of these DRBs, leader uniqueness is not \\guaranteed that introduces additional consensus protocol \\to agree on the beacon output.}\\[0.1cm]
\hline
HE-based & \makecell[l]{1. The output of these DRBs for a round $e$ does not depend on the\\ output of the previous round $e-1$. \\ 2. Partial homomorphic encryption schemes used in these DRBs \\can be replaced by a lattice-based fully homomorphic scheme to\\ ensure the post-quantum security.} & \makecell[l]{1. Scalability issue due to the homomorphic evaluation of \\multiple ciphertexts. \\ 2. The existing DRBs use public ledger to publish the local\\ and global entropy. But distributing the local entropy in\\ DRBs using a consensus incur a high communication cost.}\\[0.1cm]
\hline
\makecell[l]{External\\Source\\-based} & \makecell[l]{1. These DRBs do not incur communication cost as the DRB \\output is published in a public bulletin board.\\ 2. These DRBs work perfectly even in the asynchronous network.} & \makecell[l]{1. Most of these DRBs do not provide public verifiability. \\ 2. Proof-of-Work based beacons are not energy efficient and\\ nodes with better hardware can outperform other nodes in\\ producing the beacon output.}\\[0.1cm]
\hline
\makecell[l]{Threshold\\Signature\\-based} & \makecell[l]{1. These DRBs provide strong bias resistance and unpredictability. \\ 2. Consortium of organizations can participate to construct\\ such beacon due to threshold property (e.g. Drand~\cite{Drand}).} & \makecell[l]{1. These DRBs require either a trusted setup or DKG, hence \\ do not offer a reconfiguration-friendly setup. \\ 2. Security of the DRBs depend on the security assumptions \\ of elliptic curve pairings due to the use of BLS-signature.}\\[0.1cm]
\hline
\hline
\end{tabular}}
\caption{Advantages and Disadvantages of different Non-Interactive DRB protocols}
\label{tab:NI-DRB}
\vspace{-0.82cm}
\end{table*}
\begin{itemize}[leftmargin=*]
\item Rand Extractor~\cite{clark2010use}: Clark et al.~\cite{clark2010use} created a model to generate randomness by combining the information theory with computational finance. They used the closing prices of the stock market to compute a random output. During the market's closing in the day, one entity publishes this random output in the protocol. This entity can also induce its own local entropy to transparently construct a publicly verifiable final randomness, but liveness is hard to achieve.
\item Proofs of Delay~\cite{bunz2017proofs}: In this DRB, a beacon smart contract (BC) publishes the random beacon values on a public blockchain. The DRB is built on \textit{Proof-of-Delay} which uses an ISF such as sloth~\cite{lenstra2015random}. In this DRB, a beacon maintainer executes this ISF and publishes the result to BC with queryable access to the beacon output using a \textit{refereed delegation of computation} protocol. To show the honest behavior, the maintainer is incentivized; otherwise punished.
\item Bitcoin Beacon~\cite{bonneau2015bitcoin}~\cite{bentov2016bitcoin}: These DRBs extract randomness from the bitcoin blockchain~\cite{Nakamoto_bitcoin} and follows the security of the bitcoin. In~\cite{bonneau2015bitcoin}, an extractor fetches the randomness from the block headers. As each block contains several transactions involving ECDSA signatures~\cite{gennaro2016threshold} that rely on strong randomness for security hence, the extractor gives good public randomness as a beacon output. Bentov et al.~\cite{bentov2016bitcoin} constructed a bitcoin beacon protocol that fetches $m$ consecutive blocks $B_1,B_2,\ldots,B_m$ such that the block $B_m$ already have $l$ subsequent blocks. Further, the protocol acquires a bit $b_i$ from each block and runs a $\mathsf{majority}$ function on all these bits as input to get the DRB output.
\end{itemize}
\vspace{-0.6cm}
\subsubsection{Threshold Signature-based}
These DRBs are based on a non-interactive threshold signature scheme that requires a single round of communication among participants to produce the unique group signatures from a threshold number of participants' signature shares. Most of the existing threshold signature-based DRBs employ threshold BLS signature. These DRBs require a setup to generate the secret shares for the participants. Additionally, the complexity of unique signature construction comply with DRB protocol for practical use.
\begin{itemize}[leftmargin=*]
\item Cachin et al.~\cite{cachin2005random}, Drand~\cite{Drand}: Cachin et al. presented a common coin protocol using threshold signature along with a random-access coin-tossing scheme. In this DRB, a trusted dealer distributes the secret key shares to the participants. The DRB output is a unique signature on the hash of a counter (epoch number). Drand~\cite{Drand} follows a similar idea, but it replaces the threshold secret to the threshold BLS key. Drand can be considered as an implementation of the Cachin et al. scheme. Drand utilizes the DKG protocol of Gennaro et al.~\cite{gennaro1999secure} during the setup phase that yields a high communication complexity.
\item DFINITY~\cite{hanke2018dfinity}: It also employs a threshold BLS signatures scheme but the selection of the best initialization vector in the scheme creates a challenge. The protocol works well even in the partial synchronous network model. It employs a non-interactive DKG setup and achieves better communication complexity than Drand. The DRB acts as a VRF that produces unbiasable output.
\end{itemize}
\textbf{Note:} We present Hybrid DRB protocols in Appendix~\ref{appendix:other-DRBs}.
\section{Discussion} \label{Sec:Discussion}
\subsection{Security Assumptions}
The security of all these DRBs depends on well-defined security assumptions. These assumptions can be assumptions about the underlying network, adversary, setup, or cryptographic primitives. If these assumptions are failed in some cases, then the DRB using these assumptions will break its security properties.
\begin{itemize}[leftmargin=*]
\item \textit{Cryptographic Assumptions (Primitive)}: As the above described DRBs are based on cryptographic primitives such as PVSS, VDF, VRF, these DRBs inherit the security assumptions from the primitives. These assumptions are well-known hard problems of cryptography such as standard decisional or computational Diffie-Hellman assumptions~\cite{boneh1998decision} (or their variants) depending on the underlying cryptographic scheme (e.g., PVSS, DVRF). VDF-based DRBs depend on the new security assumptions on sequential computation (e.g., iterated squaring over groups of unknown order~\cite{rivest1996time}) that are not well studied and understood in the current literature. Modeling of the hash function as a random oracle~\cite{bellare1993random} is also considered in security assumptions in some DRBs.
\item \textit{Network Assumptions (Model)}: Most of the PVSS-based and VRF-based DRBs assume a strong \textit{synchronous} network which can be an unrealistic setting in the real world. Hence, these DRBs require a lock-step synchronous network where the messages are delivered before the end of each round. In case of no lock-step synchrony, participants might employ round synchronization protocols~\cite{naor2019cogsworth,yin2019hotstuff}. Some of the DRBs work well in \textit{semi-synchronous} network where the messages are delivered within a known finite time-bound. VDF-based and external-source-based DRBs work well in an \textit{asynchronous} network where messages are delivered without a known time-bound. However, the trust of these models depends on the underlying setup assumptions or on the public blockchain, where the local entropy of the participants are posted.
\item \textit{Setup Assumptions}: Many DRB protocols~\cite{Drand,cherniaeva2019homomorphic,bhat2020randpiper,syta2017scalable,hanke2018dfinity} require an initial trusted setup assumption where private keys for the participants and uniformly random public parameters are generated by a trusted third party (dealer) or by a distributed key generation (DKG) protocol. The Security of DRBs with a trusted third party crucially depends on the action and ability of the trusted party. Nevertheless, DKG incurs a considerable setup cost (high communication complexity) with its limitation of adding or replacing the participants. Therefore, DKG-based DRBs are preferred when the participants are fixed. Hence, many recent DRB protocols~\cite{schindler2020hydrand,cascudo2017scrape,david2018ouroboros,azouvi2018winning,gilad2017algorand} have a transparent setup where the public parameters are trapdoor free.
\end{itemize}
Following the above security assumption, most of the DRB protocols perform well in permissioned systems. However, permissionless systems have a highly dynamic set of nodes that maintain the system state. Due to the dynamically changing participants, integrating an existing DRB with the system is challenging. Moreover, setting the assumption on a number of adversarial nodes is hard.
\begin{researchproblem}
Study the hardness of embedding the existing DRB protocols in permissionless systems, based on Proof-of -Work (PoW) or -Stake (PoS).
\end{researchproblem}
\subsection{Complexity}
DRB protocols following different approaches exhibit different complexity. Finding a good balance between computation and communication complexity in a DRB protocol is a challenging task. Therefore, an extensive amount of work has been devoted to reduce the complexity of DRB protocols.
\begin{itemize}[leftmargin=*]
\item \textit{Communication Complexity}: Most of the interactive DRB protocols assume a broadcast channel. Therefore, Ouroboros~\cite{kiayias2017ouroboros}, RandShare~\cite{syta2017scalable}, and SCRAPE \cite{cascudo2017scrape} have a communication complexity of $\mathcal{O}(n^3)$ due to the broadcasting of $\mathcal{O}(n)$ size message. HydRand~\cite{schindler2020hydrand} improves upon the communication complexity to $\mathcal{O}(n^2)$ by having a leader-based approach where a leader node performs the PVSS share distribution. Relaying the messages through a single node to reduce the communication complexity is also embraced by ALBATROSS~\cite{cascudo2020albatross}, GLOW~\cite{galindo2020fully}. RandHound, RandHerd~\cite{syta2017scalable}, DFINITY~\cite{hanke2018dfinity} employ sharding to sample a committee for output generation that results in lower communication complexity. But such a procedure can be immediately subject to attacks by an adaptive adversary who can corrupt the committee once it is determined.
Most of the non-interactive DRBs~\cite{david2018ouroboros,gilad2017algorand,bunz2017proofs,nguyen2019scalable} have less communication complexity as a successful participant (e.g., leader) usually need to perform one broadcast. Therefore, it incurs the communication complexity in $\mathcal{O}(n)$. Moreover, most of the NI-DRBs involving blockchain~\cite{RANDAO,han2020randchain} to publish shared local and global randomness also have lower communication complexity.
DKG setup based DRBs~\cite{cherniaeva2019homomorphic,hanke2018dfinity,Drand,cachin2005random} suffer from additional communication cost. Complexity of DRBs can be improved using asynchronous data dissemination (ADD)~\cite{dasasynchronous} or using hbACSS~\cite{yurek2021hbacss}(for PVSS-based DRBs).
\begin{researchproblem}
Design a DRB protocol with $\textit{sub-quadratic}$ communication complexity together with optimal fault-tolerance.
\end{researchproblem}
\item \textit{Computation Complexity}: It is defined as the number of operations needed to be performed by a participant during one round of DRB protocol. PVSS-based protocols such as RandShare~\cite{syta2017scalable} and Ouroboros~\cite{kiayias2017ouroboros} requires a computation complexity of $\mathcal{O}(n^3)$. An improved version of PVSS further reduces this cost in SCRAPE~\cite{cascudo2017scrape}. Puzzle-based DRBs ~\cite{bonneau2015bitcoin,bunz2017proofs} have a high computational cost due to the involved puzzle. VDF-based DRBs also have the drawback of high computational complexity due to the repeated squaring. On the contrary, VRF-based DRBs incur a minimum computation cost.
\begin{researchproblem}
Design a puzzle-based DRB protocol incurring low computation complexity.
\end{researchproblem}
\item \textit{Verification Complexity} Verification cost refers to the number of operations performed by an external participant to verify the output of a beacon protocol. Although VDF-based DRBs have high computational costs, they do provide efficient verification hence incur less verification cost. The most efficient DRB protocols with regard to computation and verification complexity are based on VRF~\cite{david2018ouroboros,gilad2017algorand,wang2020randchain} or threshold crypto-systems~\cite{cachin2005random,hanke2018dfinity,Drand,nguyen2019scalable}.
\begin{researchproblem}
Design a PVSS-based DRB protocol with a constant verification complexity, linear communication cost and no trusted setup.
\end{researchproblem}
\end{itemize}
\subsection{Scalability}
Despite a decade of research on DRB protocols, only quite a few recent DRBs emphasize the scalability of their DRB. Scalability in a DRB protocol refers to the number of participants it can support. Many of the described DRB protocols do not offer good scalability. Especially, DRBs involving DKG setup provide poor scalability as DKG does not support frequent modification in the set of key holders. In addition, the high complexity along with the underlying network model in many of these DRBs significantly affect the scalability of the DRBs.
A general approach for achieving good scalability is ``\textit{sharding}'' which is considered in recent DRBs, including RandHerd~\cite{syta2017scalable}, DFINITY~\cite{hanke2018dfinity} and Algorand~\cite{gilad2017algorand} with the cost of slightly degrading the fault-tolerance. RandHerd shows a direct consequence of the sharding where nodes are split into smaller groups. Each group produces local entropy and each group's entropy is combined to produce the DRB output. Algorand and DFINITY show selection of a committee to generate the DRB output. Therefore, this orthogonal technique of randomly sampling a committee for protocol execution can improve scalability.
Another way for improving scalability is using a \textit{leader-based approach} where a leader relays the messages to the participants. Moreover, having a \textit{public ledger} where participants post their local entropy messages also improves scalability.
\textit{Reconfiguration Friendliness} directly impact scalability. A protocol is reconfiguration friendly when the parameters and list of participants can be changed dynamically without affecting the current execution. When there is no binding between the setup and the system, the reconfiguration becomes easier. DRBs involving DKG setup are not reconfiguration-friendly, hence poor scalability. On the contrary, non-interactive DRBs (with no DKG) show better scalability.
\begin{researchproblem}
Study the (im)possibility of designing a reconfiguration-friendly sub-quadratic DRB protocol that do not employ committee sampling.
\end{researchproblem}
\begin{table*}[htbp]
\resizebox{\textwidth}{!}{
\centering
\begin{threeparttable}
\caption{Comparison of Existing Decentralized Randomness Beacon Protocol}
\label{tab:comparison}
\begin{tabular}{l c c c c c c c c c c c}
\hline
& & & & & & & & & & &\\
\rotatebox{90}{Protocol} & \rotatebox{90}{Network Model} & \rotatebox{90}{Adaptive Adversary} & \rotatebox{90}{Liveness} & \rotatebox{90}{Unpredictability} & \rotatebox{90}{Bias-Resistance} & \rotatebox{90}{Fault-tolerance} & \rotatebox{90}{\makecell{Communication \\ Complexity}} & \rotatebox{90}{\makecell{Computation \\ Complexity}} & \rotatebox{90}{\makecell{Verification \\ Complexity}} & \rotatebox{90}{\makecell{Cryptographic \\ Primitive}} & \rotatebox{90}{\makecell{No Trusted Dealer \\ or DKG required}}
\\[0.1cm]
\hline
ALBATROSS~\cite{cascudo2020albatross} & syn. & \ding{55} & \ding{51} & \ding{51} & \ding{51} & 1/2 & $\mathcal{O}(n)$ & $\mathcal{O}(\log n)$ & $\mathcal{O}(n)$ & PVSS & \ding{51} \\[0.1cm]
Algorand~\cite{gilad2017algorand} & semi-syn. & \ding{55} & \ding{51} & \ding{51}$^{\ddagger}$ & \ding{55} & $1/3^{\diamond}$ & $\mathcal{O}(cn)$ & $\mathcal{O}(c)$ & $\mathcal{O}(1)$ & VRF & \ding{51} \\[0.1cm]
BRandPiper~\cite{bhat2020randpiper} & syn. & \ding{51} & \ding{51} & \ding{51} & \ding{51} & 1/2 & $\mathcal{O}(n^3)$ & $\mathcal{O}(n^2)$ & $\mathcal{O}(n^2)$ & PVSS & \ding{51} \\[0.1cm]
Cachin et. al~\cite{cachin2005random} & asyn. & \ding{55} & \ding{51} & \ding{51} & \ding{51} & 1/3 & $\mathcal{O}(n^2)$ & $\mathcal{O}(n)$ & $\mathcal{O}(1)$ & Uniq. thr-sig. & \ding{55} \\[0.1cm]
Caucus~\cite{azouvi2018winning} & syn. & \ding{55} & \ding{51} & \ding{51}$^{\ddagger}$ & \ding{55} & 1/2 & $\mathcal{O}(n)$ & $\mathcal{O}(1)$ & $\mathcal{O}(1)$ & Hash func. & \ding{51} \\[0.1cm]
Continuous VDF~\cite{ephraim2020continuous} & asyn. & \ding{55} & \ding{55}$^{\dagger}$ & \ding{51} & \ding{51} & 1/2 & $\mathcal{O}(1)$ & VDF & $\mathcal{O}(1)$ & VDF & \ding{51} \\[0.1cm]
DFINITY~\cite{hanke2018dfinity} & semi-syn. & \ding{55} & \ding{51} & \ding{51} & \ding{51} & 1/3 & $\mathcal{O}(n^2)$ & $\mathcal{O}(n)$ & $\mathcal{O}(1)$ & BLS thr-sig. & \ding{55} \\[0.1cm]
Drand~\cite{Drand} & syn. & \ding{55} & \ding{51} & \ding{51} & \ding{51} & 1/2 & $\mathcal{O}(n^2)$ & $\mathcal{O}(n)$ & $\mathcal{O}(1)$ & Uniq. thr-sig. & \ding{55} \\[0.1cm]
GLOW~\cite{galindo2020fully} & syn. & \ding{55} & \ding{51} & \ding{51} & \ding{51} & 1/3 & $\mathcal{O}(n)$ & $\mathcal{O}(n)$ & $\mathcal{O}(1)$ & DVRF & \ding{55} \\[0.1cm]
GRandPiper~\cite{bhat2020randpiper} & syn. & \ding{55} & \ding{51} & \ding{51}$^{\ddagger}$ & \ding{51} & 1/2 & $\mathcal{O}(n^2)$ & $\mathcal{O}(n^2)$ & $\mathcal{O}(n^2)$ & PVSS & \ding{51} \\[0.1cm]
HERB~\cite{cherniaeva2019homomorphic} & syn. & \ding{55} & \ding{51} & \ding{51} & \ding{51} & 1/3 & $\mathcal{O}(n^2)^{\ast}$ & $\mathcal{O}(n)$ & $\mathcal{O}(n)$ & PHE & \ding{55} \\[0.1cm]
HydRand~\cite{schindler2020hydrand} & syn. & \ding{55} & \ding{51} & \ding{51}$^{\ddagger}$ & \ding{51} & 1/3 & $\mathcal{O}(n^2)$ & $\mathcal{O}(n)$ & $\mathcal{O}(n)$ & PVSS & \ding{51} \\[0.1cm]
Nguyen-Van et. al~\cite{nguyen2019scalable} & syn & \ding{55} & \ding{55} & \ding{51} & \ding{55} & 1/2 & $\mathcal{O}(n)$ & $\mathcal{O}(1)$ & $\mathcal{O}(n)$ & PHE, VRF & \ding{51} \\[0.1cm]
Ouroboros~\cite{kiayias2017ouroboros} & syn. & \ding{55} & \ding{51} & \ding{51} & \ding{51} & 1/2 & $\mathcal{O}(n^3)$ & $\mathcal{O}(n^3)$ & $\mathcal{O}(n^3)$ & PVSS & \ding{51} \\[0.1cm]
Ouroboros Praos~\cite{david2018ouroboros} & semi-syn. & \ding{51} & \ding{51} & \ding{51}$^{\ddagger}$ & \ding{55} & 1/2 & $\mathcal{O}(n)^{\star}$ & $\mathcal{O}(1)^{\star}$ & $\mathcal{O}(1)^{\star}$ & VRF & \ding{51} \\[0.1cm]
Proof-of-Delay~\cite{bunz2017proofs} & syn. & \ding{55} & \ding{51} & \ding{51} & \ding{51} & 1/2 & $\mathcal{O}(n)$ & very high & $\mathcal{O}(\log{\Delta})^{\circ}$ & Hash func. & \ding{51} \\[0.1cm]
Proof-of-Work~\cite{Nakamoto_bitcoin} & syn. & \ding{55} & \ding{51} & \ding{51}$^{\ddagger}$ & \ding{55} & 1/2 & $\mathcal{O}(n)$ & very high & $\mathcal{O}(1)$ & Hash func. & \ding{51} \\[0.1cm]
RandChain~\cite{wang2020randchain} & syn. & \ding{55} & \ding{51} & \ding{51} & \ding{51} & 1/3 & $\mathcal{O}(cn)$ & $\mathcal{O}(cn)$ & $\mathcal{O}(n)$ & VRF & \ding{51} \\[0.1cm]
RANDCHAIN~\cite{han2020randchain} & syn. & \ding{51} & \ding{51} & \ding{51} & \ding{51} & 1/3 & $\mathcal{O}(n)$ & VDF & $\mathcal{O}(1)$ & VDF & \ding{51} \\[0.1cm]
RANDAO~\cite{RANDAO} & asyn. & \ding{55} & \ding{51} & \ding{55} & \ding{55} & 1/2 & $\mathcal{O}(n)$ & VDF & $\mathcal{O}(1)$ & VDF & \ding{51} \\[0.1cm]
RandHerd~\cite{syta2017scalable} & syn. & \ding{55} & \ding{51} & \ding{51} & \ding{51} & 1/3 & $\mathcal{O}(c^{2}\log n)$ & $\mathcal{O}(c^{2}\log n)$ & $\mathcal{O}(1)$ & PVSS, CoSi & \ding{55} \\[0.1cm]
RandHound~\cite{syta2017scalable} & syn. & \ding{55} & \ding{51} & \ding{51} & \ding{51} & 1/3 & $\mathcal{O}(c^{2}n)$ & $\mathcal{O}(c^{2}n)$ & $\mathcal{O}(c^{2}n)$ & PVSS & \ding{51} \\[0.1cm]
RandRunner~\cite{schindler2021randrunner} & syn. & \ding{51} & \ding{51} & \ding{51}$^{\ddagger}$ & \ding{51} & 1/2 & $\mathcal{O}(n^2)$ & VDF & $\mathcal{O}(1)$ & VDF & \ding{51} \\[0.1cm]
RandShare~\cite{syta2017scalable} & asyn. & \ding{51} & \ding{55}$^{\odot}$ & \ding{51} & \ding{51} & 1/3 & $\mathcal{O}(n^3)$ & $\mathcal{O}(n^3)$ & $\mathcal{O}(n^3)$ & VSS & \ding{51} \\[0.1cm]
Rand Extractor~\cite{clark2010use,bonneau2015bitcoin} & asyn.$^{\pm}$ & \ding{51} & \ding{51}$^{\amalg}$ & \ding{51} & \ding{51} & 1/2 & $\mathcal{O}(1)$ & $\mathcal{O}(1)$ & $\mathcal{O}(1)$ & Hash func. & \ding{51} \\[0.1cm]
SCRAPE~\cite{cascudo2017scrape} & syn. & \ding{55} & \ding{51} & \ding{51} & \ding{51} & 1/2 & $\mathcal{O}(n^3)$ & $\mathcal{O}(n^2)$ & $\mathcal{O}(n^2)$ & PVSS & \ding{51} \\[0.1cm]
SPURT~\cite{das2021spurt} & semi-syn & \ding{55} & \ding{51} & \ding{51} & \ding{51} & 1/3 & $\mathcal{O}(n^2)$ & $\mathcal{O}(n)$ & $\mathcal{O}(n)$ & PVSS, Pairing & \ding{51} \\[0.1cm]
Unicorn~\cite{lenstra2015random} & asyn. & \ding{55} & \ding{51}$^{\dagger}$ & \ding{51} & \ding{51} & 1/2 & $\mathcal{O}(1)$ & high & $\mathcal{O}(1)$ & Sloth & \ding{51} \\[0.1cm]
\hline
\end{tabular}
\small{
\begin{tablenotes}
\item Fault-tolerance refers to number of byzantine faults a DRB can tolerate and $c$ is average committee size.
\item $\ddagger$ refers to probabilistic guarantees for unpredictability, and has a bound on the number of future rounds an adaptive rushing adversary can predict the beacon output.
\item $\diamond$ Due to the randomly sampling a committee of size c in Algorand, the fault-tolerance reduces slightly.
\item $\dagger$ The node with more computational power learns the beacon output earlier than others.
\item $\ast$ HERB achieves communication complexity of $\mathcal{O}(n^2)$ when nodes use Avalanche algorithm or public blockchain to share their ciphertexts.
\item $\star$ Ouroboros Praos is not a stand-alone DRB and does not describe randomness generation approach, so the presented complexity does not account the additional complexity for communication or verification.
\item $\circ$ The verification in beacon smart contract has complexity $\mathcal{O}(\log{\Delta})$ in the security parameter $\Delta$.
\item ${\odot}$ An additional synchrony assumption is needed to provide liveness in RandShare.
\item $\pm$ DRB protocols built on public blockchain also follow the network structure of the respective blockchain. Therefore, ~\cite{bonneau2015bitcoin} uses a synchronous network as it relies on the bitcoin blockchain.
\item $\amalg$ Liveness can be hindered due to the limited availability of financial data caused by closed exchanges in~\cite{clark2010use} or due to the fork situation in the blockchain in~\cite{bonneau2015bitcoin}.
\end{tablenotes}}
\end{threeparttable}}
\end{table*}
\vspace{-0.4cm}
\subsection{Adversarial Model}
Most of these DRBs consider a fixed set of $n$ nodes; out of these nodes, $f$ nodes may exhibit byzantine behavior. An adversary in these DRBs can be defined as:
\textit{Active vs. Passive} An active adversary actively modifies the messages (e.g., public shares, DRB output) in DRB; a passive adversary observes the transcript (i.e. messages) of an honest run of the DRB and predicts the DRB's next output.
\textit{Adaptive vs. Static} An adaptive adversary corrupts the nodes during the protocol execution, while a static adversary does corruption before the execution.
An adversary can affect the security guarantees of a DRB in many ways, such as by trying to bias the produced random output, withholding the output, predicting the future output, or tricking an outsider (third party) into accepting invalid beacon output. Leader-based DRBs suffer from targeted attacks, however, blockchain-based DRBs suffer from blockchain-specific attacks. Moreover, unpredictability can be affected by network model.
\begin{researchproblem}
Choose a static secure DRB protocol and transform it to an adaptively secure DRB that retains the efficiency standard of the static one.
\end{researchproblem}
\begin{figure}[th]
\centering
\includegraphics[width = 0.8\textwidth, height = 0.35\textwidth]{Throughput-Evaluation.png}
\vspace{-0.4cm}
\caption{Overview of throughput for various DRB protocols}
\label{fig:DRB}
\vspace{-0.4cm}
\end{figure}
\subsection{Throughput Evaluation} We report throughput of state-of-the-art public implementations of various DRBs in Figure~\ref{fig:DRB}. DFINITY and SPURT operate on a semi-synchronous network but BRandPiper, Drand, GLOW, and HydRand assume synchronous networks. The network delay parameter in these DRBs directly affects the throughput of DRBs.
Drand is a practically deployed DRB protocol. However, when the number of nodes increases to more than 64, nodes in the DRand abort the DKG step of the protocol, and yet it suffers from significant network delay. For HydRand, we chose the public implementation~\cite{HydRand-code} available on Github. For BRandPiper, we depict throughput for the Merkle-tree-based implementation, which is quantitatively practical for real-world scenarios. For SPURT, we used the throughput values directly from their paper. For DFINITY and GLOW, we followed the public implementation~\cite{DVRF-code} of DVRFs to get the throughput while assuming no failure.
\vspace{-0.3cm}
\subsection{Others}
\begin{itemize}[leftmargin=*]
\item \textit{Incentive} Some DRBs~\cite{bunz2017proofs,RANDAO} involve incentivizing or punishing the participants to enforce fairness against rational adversaries. In particular, these (dis)incentivizing approaches (e.g.,~\cite{baum2020insured}) are considered mostly in smart contract-based DRBs. Putting an economic incentivization scheme to reward the participants of beacon enforces the honest behavior from the participants. An incentivization scheme can also reward the right computation or verification. Interesting research would be to create an \textit{incentive structure} for a DRB.
\item \textit{Output Uniqueness} It states that the DRB produces a unique output even in the presence of an adversary having the trapdoor information of honest participants. It implies strong bias-resistance in DRBs. Therefore, DRBs such as RandRunner~\cite{schindler2021randrunner}, DFINITY~\cite{hanke2018dfinity} and GLOW~\cite{galindo2020fully} provide strong bias-resistance due to their output uniqueness. Having this property also prevents an adversary from manipulating the beacon output for any financial gain.
\item \textit{Universal Composability} (UC) It is arguably one of the strongest security guarantees. A UC-secure protocol ensures that the protocol can be employed as a building block in more complex systems while preserving its security. The earlier UC-secure DRBs~\cite{kiayias2017ouroboros} do not provide bias-resistance. The first UC-secure DRB ALBATROSS~\cite{cascudo2020albatross} leverages UC-secure NIZK proofs. UC-secure time-lock puzzles (TLP)~\cite{baum2021tardis} can be scrutinized to construct a UC-secure DRB.
\end{itemize}
\section{New Components for Construction of DRB Protocols}
There have been many new efficient constructions of cryptographic primitives in recent years. These primitives can be embedded as new building blocks or replace old ones in the DRB protocols to improve the performance of DRBs.
\begin{itemize}[leftmargin=*]
\item \textit{Using New Verifiable Functions}:
Gurkan et al.~\cite{gurkan2021aggregatable} constructed a new aggregatable DKG scheme that leverages gossip instead of broadcast communication to reduce the communication complexity. Further, they introduced an efficient Verifiable Unpredictable Function (VUF) and combined it with DKG. This threshold VUF can be utilized to construct a DRB protocol.
VDF-based DRBs can benefit from the recent work~\cite{rotem2021simple} about batch verification of VDF in which the verification of beacon outputs during the last several rounds can be batched and verified efficiently. Work~\cite{loe2021practical} on VDF can be investigated and applied to construct a practical VDF-based DRB protocol.
Current DRBs are not Post-Quantum (PQ) secure (except DRB~\cite{li2021postquantum} that does not depend on any third party to construct a quantum-safe beacon). Recent constructions of Post-Quantum VRF~\cite{buser2021post} and Post-Quantum VDF~\cite{PQ-VDF} can be carefully studied and applied to construct practical PQ-secure DRB protocols.
\item \textit{Using New Threshold Signatures}: Tomescu et al.~\cite{tomescu2020towards} designed a fast BLS-based threshold signature scheme (BLS-TSS). Their scheme has fast signature aggregation and verification. There are some DRBs that use the BLS signature scheme. The new BLS-TSS scheme can be directly applied to these DRBs to improve their performance. Otherwise, a new large-scale, simple DRB protocol can be designed and implemented using this new BLS-TSS scheme.
\item \textit{Using New Erasure Codes}: All known Verifiable Secret Sharing (VSS) schemes published so far in the open literature (without exception) use the well-known Reed-Solomon codes \cite{reed1960polynomial}. Reed-Solomon codes are Maximum Distance Separable (MDS) erasure codes of type $(t, n)$, where the original message is equally split in $t$ parts and is encoded to $n$ (where $n>t$) parts. In the recent decade, the coding theory community constructed new MDS erasure codes. The most significant line of work was done by Dimakis et al., in~\cite{5550492} where they constructed Minimum Bandwidth Regenerating (MBR) codes (optimal in terms of the repair bandwidth) and Minimum Storage Regenerating (MSR) codes (optimal in terms of the storage). Soon after that, those MDS codes were practically employed in Facebook data centers~\cite{179255}, and new variants of MDS codes e.g.~\cite{ye2017explicit} were proposed. Therefore, it would be interesting to research the potential replacement of the Reed-Solomon code in VSS schemes with another MDS (MSR) code to improve the performance of VSS-based DRBs.
\end{itemize}
\section{Conclusion}
Within recent years, there has been a dramatic surge in the construction of new Decentralized Randomness Beacon (DRB) protocols due to its emergence in cryptographic protocols. We present the first systematization of knowledge (SoK) for the existing efforts on DRB protocols. This SoK provides a comprehensive review of the design paradigms and approaches of DRB protocols. This SoK can serve as a starting point to explore DRB protocols and can help researchers or practitioners to pick a DRB protocol well-suited for their application.
In this SoK, we presented basic standard definitions of a DRB protocol and its required properties. We discussed the key components and the most significant features of DRB protocols and summarized the existing DRB protocols in Table~\ref{tab:comparison}. We identified several research challenges related to the complexity, scalability, and security of DRB protocols. We highlighted respective solutions to encounter some of the challenges. Finally, we proposed promising research directions for the future design of DRB protocols by employing the new cryptographic components that can help to advance the state-of-the-art of DRB protocols.
\bibliographystyle{splncs04}
|
1,941,325,220,556 | arxiv | \section{Introduction}
Machine learning algorithms have been increasingly used as diagnostic tools in biomedical research\cite{gulshan2016,esteva2017,golden2017}. The widespread availability of smartphones and other health tracking devices generates high volumes of sensor data, and makes machine learning uniquely well posed to impact clinical research using digital health tools. In clinical applications, gender, age, and other demographic characteristics of the study participants often play the role of confounders. Confounding is particularly prevalent in mobile health studies run under uncontrolled conditions outside clinical and laboratory settings, where we have little control over the demographic and clinical characteristics of the cohort of participants that self-select to participate in a study.
In the context of predictive modeling, we define a confounder as a variable that causes spurious associations between the features and response variable. In machine learning applications, the presence of confounding can lead to ambiguous inference and poor generalizability of models. Confounding is usually present when the joint probability distribution of the confounder and response variables is different in the data available to develop the learner (which we from now on denote as the “development dataset”) relative to the population where the learner will be applied (denoted as the “target population”)\cite{rao2017}. For example, consider a diagnostic application where most cases are old aged while most controls are young, but where age is not associated with disease status in the target population (e.g., the target population is composed of older patients only). If the classifier can more efficiently detect age-related signals than disease-related signals, then it will likely perform poorly when deployed in the target population.
Confounding adjustment is an active area of research in machine learning. The goal is to prevent an algorithm from learning the confounding signal. Since any variable that confounds the feature-response relationship has to be associated with both the features and the response, most of the methods proposed in the literature can be divided into two approaches: (i) methods that remove the association between the confounder and the response; or (ii) methods that remove the association between the confounder and the features. A standard example of the first approach is to match subjects from the development data in order to obtain a subsample that more closely resembles the target population. This strategy, however, results in a smaller number of participants to train and evaluate the machine learning algorithm, and, in highly unbalanced situations, might lead to the exclusion of most of the participants from the analyses. Alternative methods that make more efficient use of the data include inverse probability weighting approaches\cite{linn2016,rao2017}, which weight the training samples in order to make model training better tailored to the target population. A canonical example of the second approach is to separately regress each feature on the confounders, and use the residuals as the predictors in the machine learning algorithm. Other approaches that do not fall into categories (i) or (ii) include penalized learners\cite{li2011} and backdoor adjustment\cite{landeiro2016}.
In this paper, we present statistical methods to detect and quantify the influence of observed confounders, and to estimate the actual (i.e., unconfounded) predictive performance of a learner. We use a large Parkinsons digital health study cohort to illustrate how our methods can be used to evaluate the effectiveness of standard confounding adjustment methods.
\section{Methods}
We adopt restricted permutations\cite{good2000,rao2017} to isolate the contribution of the confounder from the predictive performance of a learner. The key idea is to shuffle the response data within the levels of a categorical/ordinal confounder (as illustrated in the Supplementary Figure \ref{fig:stratifiedperm}) in order to destroy the direct association between the response and the features while still preserving the indirect association due to the confounder. Algorithm \ref{alg:stratifiedShuffling} describes the procedure for an arbitrary performance metric, $m$ (such as the area under the receiver operating characteristic curve, AUC, or root mean square error).
\begin{algorithm}
\caption{Restricted Monte Carlo permutation null distribution for performance metric $m$}\label{alg:stratifiedShuffling}
\begin{algorithmic}[1]
\State \textbf{Input}: Number of permutations, $b$. Development data set feature matrix, response vector, and confounder vector, $\boldsymbol{X}$, $\boldsymbol{y}$, $\boldsymbol{c}$. Training and test set indexes, $i_{train}$, $i_{test}$
\State Split $\boldsymbol{X}$, $\boldsymbol{y}$ and $\boldsymbol{c}$ into training and test sets
\For{$i = 1, 2, \ldots, b$}
\State $\boldsymbol{y}^{\ast}_{train} \leftarrow \mbox{RestrictedShuffle}(\boldsymbol{y}_{train}, \boldsymbol{c}_{train})$, and $\boldsymbol{y}^{\ast}_{test} \leftarrow \mbox{RestrictedShuffle}(\boldsymbol{y}_{test}, \boldsymbol{c}_{test})$
\State Train a machine learning algorithm on the $\boldsymbol{X}_{train}$ and $\boldsymbol{y}_{train}^{\ast}$ data
\State Evaluate the algorithm on the $\boldsymbol{X}_{test}$ and $\boldsymbol{y}_{test}^{\ast}$ data
\State Record the value of the performance metric, $m^\ast_i$, on the shuffled data
\EndFor
\State \textbf{Output}: $m^\ast_1$, $m^\ast_2$, \ldots, $m^\ast_b$
\end{algorithmic}
\end{algorithm}
Building upon the restricted permutation null distribution, we developed two statistical tools to deal with confounding. First, we estimate the ``unconfounded” predictive performance of a learner by building a mapping from the restricted permutation null to the standard permutation null (where the standard permutation null distribution is generated by shuffling the labels in the usual unconstrained manner). As fully described in the Supplement, for any performance metric that can be expressed as a (generalized) U-statistic\cite{hoeffding1948,lehmann1951,serfling1980,delong1988} (e.g., AUC), or expressed as a simple average (e.g., mean square error, mean absolute error, and classification accuracy), we have that an asymptotic estimate of the unconfounded performance metric is given by,
\begin{equation}
\hat{m}_u \, = \, (m_o - a_{\hat{\pi}^{\ast}}) (s_{\hat{\pi}^{\ast\ast}}/s_{\hat{\pi}^{\ast}}) + a_{\hat{\pi}^{\ast\ast}}~,
\label{eq:correctedmetric}
\end{equation}
where $m_o$ represents the uncorrected metric value; $a_{\hat{\pi}^{\ast}}$ and $s^2_{\hat{\pi}^{\ast}}$ represent the sample average and variance of the restricted permutation null; and $a_{\hat{\pi}^{\ast\ast}}$ and $s^2_{\hat{\pi}^{\ast\ast}}$ represent the analogous quantities for the standard permutation null. It is important to point out that we don’t view the unconfounded metric estimation as an adjustment method (in the sense that it does not prevent an algorithm from learning the confounding signal in the first place). It simply quantifies the amount of response signal learned by the algorithm, after the algorithm has had a chance to learn both confounding and response signals.
Second, by noticing that the location of the restricted permutation null provides a natural measure of the amount of confounding signal learned by the algorithm, we adopt the average of the restricted permutation null as a test statistic, and develop a statistical test to compare the hypotheses,
\begin{align}
H_0^c &: \mbox{the algorithm has not learned the confounding signal~,} \\
H_1^c &: \mbox{the algorithm has learned the confounding signal~,} \nonumber
\end{align}
and detect confounding learning per se. In the Supplement we show that, under $H_0^c$, the average of the restricted permutation null distribution is asymptotically distributed as a $N(a_{\hat{\pi}^{\ast\ast}}, s^2_{\hat{\pi}^{\ast\ast}}/n)$ distribution, where $n$ represents the test set sample size.
For the AUC metric, additional analytical results are available. It is well known\cite{bamber1975,mason2002} that the standard permutation null can be approximated by,
\begin{equation}
N(0.5 \, , \, (n_n + n_p + 1)/(12 \, n_n \, n_p))~,
\label{eq:standnullapprox}
\end{equation}
where $n_n$ and $n_p$ represent the number of negative and positive labels in the test set. Hence, $a_{\hat{\pi}^{\ast\ast}} \approx 0.5$ and $s^2_{\hat{\pi}^{\ast\ast}} \approx (n_n + n_p + 1)/(12 \, n_n \, n_p)$ and the estimator in (\ref{eq:correctedmetric}) becomes,
\begin{equation}
auc_u = (auc_o - a_{\hat{\pi}^{\ast}})(n_n + n_p + 1)/(12 \, n_n \, n_p \, s_{\hat{\pi}^\ast}) + 0.5~,
\label{eq:auccorrectionformula}
\end{equation}
the null distribution under $H_0^c$ reduces to $N(0.5, (n_n + n_p + 1)/(12 \, n_n \, n_p \, n))$, and,
\begin{equation}
p = 1 - \Phi\big((a_{\hat{\pi}^{\ast}} - 0.5)/[(n_n + n_p + 1)/(12 \, n_n \, n_p \, n)]^{1/2}\big)~,
\label{eq:confoundingpval}
\end{equation}
corresponds to the confounding p-value, where $\Phi(.)$ represents the c.d.f. of standard normal variable.
\section{Real data illustrations}
A key practical application of our tools is to evaluate if an adjustment method is working as expected. This is important in practice since most of these methods rely on assumptions, and it is generally unclear how robust they are to violations of these assumptions. Here we illustrate the application of our tools to two confounding adjustment methods: sample matching, and approximate inverse probability weighting (IPW) based on the propensity score\cite{propensityscore1983}.
Our development data was collected in a digital health study on Parkinsons disease\cite{bot2016,trister2016} and consist of features generated from 30 second inertial sensor readings captured during walking. We focused on walking, as walking patterns are influenced by age and gender\cite{ko2011} in addition to Parkinson’s disease. The development data was split into training and test sets with similar joint distributions for the age, gender, and disease status (Supplementary Figure \ref{sfig:agegenderassociations}). We apply the adjustment methods to both training and test sets, and the analyses are based on a combined gender/discretized age\footnote{While, in theory, we can only perform restricted permutations using categorical/ordinal confounders, in practice we can discretize and evaluate continuous confounders as well. Clearly, if the discretization is too coarse the discretized confounder might not be able to fully capture the association between the confounder and the response, and we might end up underestimating the amount of confounding learned by the algorithm. In practice, one should experiment with distinct discretizations, as illustrated in Supplementary Figure \ref{sfig:agediscretization}.} confounder with levels: young male, young female, middle age male, middle age female, senior male, and senior female.
Figure \ref{fig:adjustmentcomparison} shows the results based on logistic regression (top panels) and random forest (bottom panels) classifiers. In all panels, the blue histograms represent the restricted permutation null distributions generated by Algorithm 1, the red curves represent the normal approximation for the standard permutation null distribution presented in equation 3, the orange line shows the unconfounded estimate of AUC computed using equation \ref{eq:auccorrectionformula}, and the cyan line represents the observed AUC.
For the sake of comparison, panels a and d report the results when no confounding adjustment is performed. Both logistic regression and random forest classifiers are clearly learning confounding signal since the restricted permutation nulls are centered around 0.7, and the confounding test p-values (equation \ref{eq:confoundingpval}) are highly significant ($p<10^{-16}$). Hence, the high AUC scores (cyan lines above 0.81) reflect the classifiers' ability to detect both disease and confounding signals, while the unconfounded estimates (orange scores around 0.66) are considerably more modest.
Panels b and e show the results based on a matched subset of participants. The fact that the restricted permutation nulls are centered around 0.5, and closely match the standard permutation null density (red curve), suggests that matching effectively prevented the classifier from learning the confounding signal ($p<0.51$ and $p<0.58$, respectively) and that the classifiers are only learning the disease signal. As expected, the observed and unconfounded AUC scores match each other closely in this situation. Finally, note that the much larger spread of the null distributions (in comparison to panels a and d) is due to the smaller test set available after matching.
Panels c and f report the results for the approximate IPW approach. This method makes use of the entire data set and attempts to prevent confounding learning by weighting the samples according to the inverse of their estimated propensity scores (i.e., the conditional probability that a participant has the disease given its gender and age). While several approaches have been proposed in the literature for the estimation of propensity scores\cite{lee2010,pirracchio2014}, here, we adopt the most commonly used method based on logistic regression. The panels show that the approximate IPW approach managed to reduce the amount of confounding (the blue histograms are closer to 0.5 compared to panels a and d). However, it didn’t remove it completely ($p < 10^{-16}$). This suggests that the estimated inverse probability weights did not generate a well balanced augmented data set. (Supplementary Figure \ref{sfig:failuretobalance} confirms this is indeed the case.) Most likely, the reason for this suboptimal performance is that propensity score estimation using logistic regression makes the strong assumption that the observed associations between the confounders and disease labels can be well described by the logistic function. This example illustrates how the violation of a parametric modeling assumption can lead to an inefficient confounding adjustment.
\begin{figure}[h]
\centering
\centerline{\includegraphics[width=\linewidth, bb = 0 20 720 210]{adjustment_checks_glm_and_rf3.pdf}}
\caption{Comparison of confounding adjustment methods.}
\label{fig:adjustmentcomparison}
\end{figure}
\section{Final remarks}
Digital health enabled diagnostic systems have the potential to provide low cost remote diagnostic tools to underserved communities that lack easy access to medical care. However this opportunity cannot be fully realized without (i) efficient approaches to combat confounding (without which we run the risk of making spurious inferences from the data) and (ii) rigorous methods to evaluate these adjustment methods. The tools proposed in this paper address the second need.
To the best of our knowledge, the use of restricted permutations in the context of predictive modeling has only been leveraged by\cite{rao2017}. These authors, however, use restricted permutations to test if a machine learning algorithm has learned the response signal in the presence (or absence) of confounders, but not to detect and quantify confounding learning per se, as proposed in this paper.
For the sake of clarity, our illustrations focused on the case where confounder and response are associated in the development set but not in the target population. We can, however, still apply our methodology when response and confounder are known to be associated in the target population but have a different joint probability distribution compared to the development data. Section 9 of the Supplement provides an illustrative example based on synthetic data.
We also conducted a simulation study to evaluate the statistical properties of the confounding test (Section 10 of the Supplement). Our simulations show reasonable statistical power for the range of parameters investigated in our experiments, and well-controlled type I error rates.
Finally, we point out that while this paper has focused on digital health applications, the proposed tools can be more generally applied to any other areas impacted by confounders.
\clearpage
|
1,941,325,220,557 | arxiv | \section{Introduction}
Let $R$ be a commutative ring with identity and a fixed invertible element $A$. For a 3-manifold $M$, the {\it (Kauffman bracket) skein module} of $M$ over $R$, denoted by $\mathcal{S}(M;R)$, is defined as the quotient of the free $R$-module generated by framed links (which may be empty) embedded in $M$ modulo the following local relations:
\begin{figure}[h]
\centering
\includegraphics[width=8.5cm]{local}\\
\end{figure}
As a convention, $R$ is identified with $R{\O}\subset\mathcal{S}(M;R)$ via $\lambda\mapsto\lambda{\O}$.
In the case $M=\Sigma\times[0,1]$ where $\Sigma$ is a surface, $\mathcal{S}(M;R)$ is usually denoted as $\mathcal{S}(\Sigma;R)$. With $L_1L_2$ defined by stacking $L_1$ over $L_2$ in the $[0,1]$ direction, $\mathcal{S}(\Sigma;R)$ becomes a $R$-algebra, called the {\it skein algebra} of $\Sigma$ over $R$. Through the local relations, each element of $\mathcal{S}(\Sigma;R)$ can be written as a $R$-linear combination of multi-curves on $\Sigma$, with the vertical framing understood.
When $R=\mathbb{C}$ and $A=-1$, it was shown in \cite{Bu97,PS00} that $\mathcal{S}(M;R)$ is isomorphic to the coordinate ring of the ${\rm SL}(2,\mathbb{C})$-character variety $\mathcal{X}_{{\rm SL}(2,\mathbb{C})}(\pi_1(M))$. To be explicit, the isomorphism sends a framed link $L$ to the function
$$\mathcal{X}_{{\rm SL}(2,\mathbb{C})}(\pi_1(M))\to\mathbb{C}, \qquad \chi\mapsto{\prod}_{i=1}^m-\chi([K_i]),$$
where $K_1,\ldots, K_m$ are the components of $L$, and $[K_i]$ stands for the conjugacy class in $\pi_1(M)$ represented by $K_i$.
In particular, $\mathcal{S}(\Sigma;R)$ is isomorphic to $\mathbb{C}[\mathcal{X}_{{\rm SL}(2,\mathbb{C})}(\pi_1(\Sigma))]$.
The description of the structure of $\mathcal{S}_{g,n}:=\mathcal{S}(\Sigma_{g,n};\mathbb{Z}[A^{\pm1}])$ was raised as \cite{Ki97} Problem 1.92 (J) (proposed by Bullock and Przytycki), and also \cite{Oh02} Problem 4.5. Bullock \cite{Bu99} gave a finite set of generators for $\mathcal{S}_{g,n}$.
The structure of $\mathcal{S}_{g,n}$ for $(g,n)\in\{(0,2),(0,3),(1,0),(1,1)\}$ was early known.
Recently, Cooke and Lacabanne \cite{CL22} obtained a presentation for $\mathcal{S}_{0,4}$.
It remains to be a difficult problem to find all relations of $\mathcal{S}_{g,n}$ for general $g$ and $n$.
The main purpose of this paper is to determine the structure of $\mathcal{S}(\Sigma_{0,n+1};R)$ for $R$ in the abstract. Actually, it is for $\mathcal{S}'_n=\mathcal{S}\big(\Sigma_{0,n+1};\mathbb{Z}[A^{\pm1},(A^6+A^{-6})^{-1}]\big)$ that we find $n+{n\choose 2}+{n\choose 3}$ generators and a family $\mathfrak{I}$ of relations, and show that for each relation $\mathfrak{u}$ of $\mathcal{S}'_n$, there exists a polynomial $\gamma\in\mathbb{Z}[A]$ with $A^4-1\mid \gamma(A)-1$ such that $\gamma\mathfrak{u}$ belongs to the two-sided ideal generated by $\mathfrak{I}$.
The content is organized as follows. In Section 2 we recall some facts on $\mathbb{C}[\mathcal{X}_{{\rm SL}(2,\mathbb{C})}(F_n)]$, including its generators and defining relations. In Section 3 we introduce basic notions and develop some useful techniques for computation. In Section 4 we find a set of generators and determine the commuting relations among them. In Section 5 we find several families of relations, which together quantize the classical defining relations. In Section 6 we state and prove the main result. The proofs for several formulas are delayed to the final section.
\section{The classical case}
For generic $\mathbf{x}_i\in{\rm SL}(2,\mathbb{C}), i=1,\ldots,n$, let $t_i=-{\rm tr}(\mathbf{x}_i)$, let $\overline{\mathbf{x}}_i=\mathbf{x}_i+(1/2)t_i\mathbf{e}$ (where $\mathbf{e}$ denotes the $2\times 2$ identity matrix), let $s_{ij}=-{\rm tr}(\overline{\mathbf{x}}_i\overline{\mathbf{x}}_j)$, and let $s_{ijk}=-{\rm tr}(\overline{\mathbf{x}}_i\overline{\mathbf{x}}_j\overline{\mathbf{x}}_k)$. These notations are adopted to be consistent with those used in later sections. Note that $s_{ijk}+s_{ikj}=0$.
Let
\begin{align*}
g^{a_1a_2a_3}_{b_1b_2b_3}&=2s_{a_1a_2a_3}s_{b_1b_2b_3}-\theta^{a_1a_2a_3}_{b_1b_2b_3}, \\
f^b_c(a_1a_2a_3)&=s_{bc}s_{a_1a_2a_3}-s_{a_1c}s_{ba_2a_3}+s_{a_2c}s_{ba_1a_3}-s_{a_3c}s_{ba_1a_2}, \\
f^{b_1b_2}_{c_1c_2}(a_1a_2a_3)&=-\theta^{b_1b_2}_{c_1c_2}s_{a_1a_2a_3}+\theta^{b_1a_2}_{c_1c_2}s_{a_1b_2a_3}
-\theta^{b_1a_3}_{c_1c_2}s_{a_1b_2a_2}-\theta^{b_2a_2}_{c_1c_2}s_{a_1b_1a_3} \nonumber \\
&\ \ \ +\theta^{b_2a_3}_{c_1c_2}s_{a_1b_1a_2}-\theta^{a_2a_3}_{c_1c_2}s_{a_1b_1b_2}, \\
f^{b_1b_2b_3}_{c_1c_2c_3}(a_1a_2a_3)&=\theta^{b_1b_2b_3}_{c_1c_2c_3}s_{a_1a_2a_3}-\theta^{b_1b_2a_3}_{c_1c_2c_3}s_{a_1a_2b_3}
+\theta^{b_1b_3a_3}_{c_1c_2c_3}s_{a_1a_2b_2}-\theta^{b_2b_3a_3}_{c_1c_2c_3}s_{a_1a_2b_1},
\end{align*}
where
$\theta^{b_1\cdots b_k}_{c_1\cdots c_k}=\det\big[(s_{b_ic_j})_{i,j=1}^k\big]$.
As easy to verify,
\begin{align}
f^{b_1b_2}_{c_1c_2}(a_1a_2a_3)&=s_{b_2c_2}f^{b_1}_{c_1}(a_1a_2a_3)+s_{b_1c_2}f^{a_1}_{c_1}(b_2a_2a_3)-s_{a_1c_1}f^{b_1}_{c_2}(b_2a_2a_3) \nonumber \\
&\ \ \ +s_{a_2c_2}f^{a_1}_{c_1}(b_1b_2a_3)-s_{a_3c_2}f^{a_1}_{c_1}(b_1b_2a_2), \label{eq:f2} \\
f^{b_1b_2b_3}_{c_1c_2c_3}(a_1a_2a_3)&=s_{a_2c_3}f^{b_1b_2}_{c_1c_2}(a_1b_3a_3)-s_{a_1c_3}f^{b_1b_2}_{c_1c_2}(a_2b_3a_3) \nonumber \\
&\ \ \ +\theta^{b_1b_2}_{c_1c_2}f^{a_1}_{c_3}(a_2b_3a_3)-\theta^{b_1b_3}_{c_1c_2}f^{a_1}_{c_3}(a_2b_2a_3)
+\theta^{b_1a_3}_{c_1c_2}f^{a_1}_{c_3}(a_2b_2b_3) \nonumber \\
&\ \ \ +\theta^{b_2b_3}_{c_1c_2}f^{a_1}_{c_3}(a_2b_1a_3)-\theta^{b_2a_3}_{c_1c_2}f^{a_1}_{c_3}(a_2b_1b_3)
+\theta^{b_3a_3}_{c_1c_2}f^{a_1}_{c_3}(a_2b_1b_2), \label{eq:f3} \\
\theta^{a_1a_2a_3a_4}_{b_1b_2b_3b_4}&=s_{a_4b_1}g^{a_1a_2a_3}_{b_2b_3b_4}-s_{a_3b_1}g^{a_1a_2a_4}_{b_2b_3b_4}+s_{a_2b_1}g^{a_1a_3a_4}_{b_2b_3b_4}
-s_{a_1b_1}g^{a_2a_3a_4}_{b_2b_3b_4} \nonumber \\
&\ \ \ +2s_{b_2b_3b_4}f^{a_1}_{b_1}(a_2,a_3,a_4). \label{eq:theta}
\end{align}
It is known (see \cite{ABL18,Do01,Dr03}) that the coordinate ring of the ${\rm SL}(2,\mathbb{C})$-character variety of the free group $F_n$ (i.e. the ring of invariants of the diagonal conjugate action on ${\rm GL}(2,\mathbb{C})$ on ${\rm SL}(2,\mathbb{C})^{\times n}$) is generated by
$$\{t_i\colon 1\le i\le n\}\cup\{s_{ij}\colon 1\le i<j\le n\}\cup \{s_{ijk}\colon1\le i<j<k\le n\},$$
with two families of defining relations. The so-called type I relations are
\begin{align}
g^{a_1a_2a_3}_{b_1b_2b_3}=0, \qquad 1\le a_1<a_2<a_3\le n, \quad 1\le b_1<b_2<b_3\le n; \label{eq:typeI}
\end{align}
the type II relations are
\begin{align}
f^b_c(a_1a_2a_3)=0, \qquad 1\le b\le n, \quad 1\le c<a_1<a_2<a_3\le n. \label{eq:typeII}
\end{align}
Let $H$ denote the set of strictly increasing sequences of integers from $\{1,\ldots,n\}$. Introduce on $H$ the partial order: $\underline{a}=(a_1,\ldots,a_u)\preceq (b_1,\ldots,b_v)=\underline{b}$ if $u\ge v$ and $a_i\le b_i$ for $1\le i\le v$.
By \cite{Do01} Theorem 3.1, with respect to the pure lexicographic monomial order induced by the ordering $\prec_0$ to be introduced in a moment, $\mathcal{G}=\mathcal{G}_0\cup\mathcal{G}_1\cup\mathcal{G}_2\cup\mathcal{G}_3\cup\mathcal{G}_4$ is a Gr\"obner basis, where
\begin{align*}
\mathcal{G}_0&=\{g^{a_1a_2a_3}_{b_1b_2b_3}\colon \underline{a},\underline{b}\in H\}, \\
\mathcal{G}_1&=\{f^b_c(a_1a_2a_3)\colon \underline{b},\underline{c}\in H,\ b\le c, \ b<a_1\}, \\
\mathcal{G}_2&=\{f^{b_1b_2}_{c_1c_2}(a_1a_2a_3)\colon \underline{a},\underline{b},\underline{c}\in H,\ \underline{b}\preceq\underline{c}, \ a_1\le b_1<b_2<a_2\}, \\
\mathcal{G}_3&=\{f^{b_1b_2b_3}_{c_1c_2c_3}(a_1a_2a_3)\colon \underline{a},\underline{b},\underline{c}\in H,\ \underline{b}\preceq\underline{c}, \ \underline{a}\preceq(b_1,b_2), \ b_3<a_3\}, \\
\mathcal{G}_4&=\{\theta^{a_1a_2a_3a_4}_{b_1b_2b_3b_4}\colon \underline{a},\underline{b}\in H,\ \underline{a}\preceq\underline{b}\}.
\end{align*}
Here the ordering $\prec_0$ is defined as
\begin{itemize}
\item $t_1\prec_0\cdots\prec_0t_n$;
\item $s_{i_1i_2i_3}\prec_0s_{j_1j_2j_3}$ if $i_k<j_k$ holds for the largest $k\in\{1,2,3\}$ with $i_k\ne j_k$;
\item $s_{i_1j_1}\prec_0s_{i_2j_2}$ if $i_1<i_2$, and $s_{ij}\prec_0s_{ij'}$ if $j<j'$;
\item $t_k\prec_0s_{ij}\prec_0s_{i_1i_2i_3}$ for any $k,i,j,i_1,i_2,i_3$.
\end{itemize}
\begin{rmk}\label{rmk:classical-ideal}
\rm Let $\mathcal{I}^0$ denote the set of integer-coefficient defining relations such that each monomial contains at most one element from $\{s_{ijk}\colon1\le i<j<k\le n\}$. The knowledge of Gr\"obner basis implies that actually $\mathcal{I}^0$ is just the $\mathbb{Z}[t_1,\ldots,t_n,s_{12},\ldots,s_{n-1,n}]$-module generated by $\mathcal{G}_1\cup\mathcal{G}_4$.
\end{rmk}
\section{Basic notions and identities}
Fix two points $b_0,b_1$ on $\partial_0\Sigma_{0,n+1}$, the outer boundary of $\Sigma_{0,n+1}$. An {\it open framed link} $L\sqcup C\subset\Sigma_{0,n+1}\times[0,1]$ is the disjoint union of framed link $L$ and a framed arc $C$ such that $\partial C=\{b_0,b_1\}$.
Let $\mathcal{S}^{\circ}$ denote the $\mathbb{Z}[A^{\pm1}]$-module generated by framed links and open framed links, modulo the same local relations as $\mathcal{S}_{0,n}$. In the notation of \cite{PS19}, $\mathcal{S}^\circ=\mathcal{S}(\Sigma_{0,n+1},\{b_0,b_1\})$.
Let $\mathbf{e}\in\mathcal{S}^\circ$ denote the short arc connecting $b_0$ to $b_1$.
For an open framed link $\mathbf{x}=L\sqcup C$, let ${\rm tr}(\mathbf{x})=L\sqcup C^\vee$, where $C^\vee$ denotes the framed link obtained by connecting $b_0,b_1$ using $\mathbf{e}$, call it the {\it trace} of $\mathbf{x}$. Extend this map by linearity to ${\rm tr}:\mathcal{S}^\circ\to\mathcal{S}$.
For $\mathbf{x}\in\mathcal{S}^\circ$ and $\mathbf{y}\in\mathcal{S}$, define $\mathbf{x}\mathbf{y}$ (resp. $\mathbf{y}\mathbf{x}$) by stacking $\mathbf{x}$ over $\mathbf{y}$ (resp. stacking $\mathbf{y}$ over $\mathbf{x}$).
We display $\Sigma_{0,n+1}$ as a disk with $n$ punctures numerated from left to right. Let $\mathbf{x}_i$ denote the arc enclosing precisely the $i$-th puncture.
\begin{figure}[h]
\centering
\includegraphics[width=12cm]{Xbar}\\
\caption{$\overline{\mathbf{x}}_4=\mathbf{x}_4+\beta t_4\mathbf{e}$, in the case $n=5$.}\label{fig:Xbar}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=11.5cm]{X-s}\\
\caption{Left: $\overline{\mathbf{x}}_2\overline{\mathbf{x}}_4\overline{\mathbf{x}}_5$; middle: $s_{13\lrcorner(45)}$; right: $s_{1\check{4}\lrcorner3}$.}\label{fig:X-s}
\end{figure}
Let $\alpha=A^2+A^{-2}=-{\rm tr}(\mathbf{e})$, and let $\beta=\alpha^{-1}$.
Let $t_i={\rm tr}(\mathbf{x}_i)$, the small circle enclosing precisely the $i$-th puncture. Let $\overline{\mathbf{x}}_i=\mathbf{x}_i+\beta t_i\mathbf{e}$, displayed by filling the $i$-th puncture in black; see Figure \ref{fig:Xbar} for one example. Note that ${\rm tr}(\overline{\mathbf{x}}_i)=0$.
Let $s_{ii}=\alpha-\beta t_i^2$.
Although the ``horizontal product" for general $\mathbf{x}, \mathbf{y}\in\mathcal{S}^\circ$ cannot be defined, it still makes sense to define $\mathbf{x}_i\mathbf{x}_j$ and $\overline{\mathbf{x}}_i\overline{\mathbf{x}}_j$ for $i<j$, in an obvious way.
For $i_1<\cdots<i_m$, let $t_{i_1\cdots i_m}={\rm tr}(\mathbf{x}_{i_1}\cdots\mathbf{x}_{i_m})$ and $s_{i_1\cdots i_m}={\rm tr}(\overline{\mathbf{x}}_{i_1}\cdots\overline{\mathbf{x}}_{i_m})$.
In particular,
\begin{align}
s_{i_1i_2}&=t_{i_1i_2}+\beta t_{i_1}t_{i_2}, \label{eq:sij} \\
s_{i_1i_2i_3}&=t_{i_1i_2i_3}+\beta(t_{i_1}s_{i_2i_3}+t_{i_2}s_{i_1i_3}+t_{i_3}s_{i_1i_2})-\beta^2t_{i_1}t_{i_2}t_{i_3}. \label{eq:sijk}
\end{align}
Let $s_{\hat{i}_1i_2\cdots i_m}={\rm tr}(\mathbf{x}_{i_1}\overline{\mathbf{x}}_{i_2}\cdots\overline{\mathbf{x}}_{i_m})=s_{i_1\cdots i_m}-\beta t_{i_1}s_{i_2\cdots i_m}$. Note that $s_{\hat{i}j}=s_{ij}$.
To use formulas instead of figures as much as possible, we further introduce the notations $s_{ij\lrcorner k}$, $s_{i\check{j}\lrcorner k}$, and so on; see Figure \ref{fig:X-s}.
\begin{figure}[h]
\centering
\hspace{5mm} \includegraphics[width=12cm]{basic}
\caption{Some useful local relations.}\label{fig:basic}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=11cm]{overlap}\\
\caption{Expanding when the $i$-th puncture is overlapped.}\label{fig:overlap}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=11cm]{sij-times-xj}\\
\caption{Computing $s_{i\hat{j}}\overline{\mathbf{x}}_j$ for $i<j$.}\label{fig:sij-times-xj}
\end{figure}
Some useful local relations are depicted in Figure \ref{fig:basic}.
Furthermore, if there is an ``overlapping" in $s_{i_1\cdots i_m}s_{j_1\cdots j_r}$, by which we mean $\{i_1,\ldots,i_m\}\cap\{j_1,\ldots,j_r\}\ne\emptyset$ and label the overlapping via a tightly enclosing circle, then the product can be computed using the rule given in Figure \ref{fig:overlap}.
An application is shown in Figure \ref{fig:sij-times-xj}.
As illustrated in the bottom of Figure \ref{fig:basic}, for $i_1<i_2<i_3$,
\begin{align}
s_{i_1i_3}\overline{\mathbf{x}}_{i_2}=\ &A^4s_{i_1i_2}\overline{\mathbf{x}}_{i_3}+A^{-4}s_{i_2i_3}\overline{\mathbf{x}}_{i_1}
+\alpha\overline{\mathbf{x}}_{i_1}\overline{\mathbf{x}}_{i_2}\overline{\mathbf{x}}_{i_3}+s_{i_1i_2i_3}\mathbf{e}. \label{eq:basic-2}
\end{align}
Multiplying by $\overline{\mathbf{x}}_{i_4}$ (for $i_4>i_3$) and taking traces, we obtain
\begin{align}
s_{i_1i_3}s_{i_2i_4}=A^4s_{i_1i_2}s_{i_3i_4}+A^{-4}s_{i_2i_3}s_{i_1i_4}+\alpha s_{i_1i_2i_3i_4}, \label{eq:product-22-0}
\end{align}
which implies
\begin{align}
s_{i_1i_2i_3i_4}=\beta(s_{i_1i_3}s_{i_2i_4}-A^4s_{i_1i_2}s_{i_3i_4}-A^{-4}s_{i_2i_3}s_{i_1i_4}).
\end{align}
From (\ref{eq:basic-2}) we obtain
\begin{align}
\alpha\overline{\mathbf{x}}_i\overline{\mathbf{x}}_j\overline{\mathbf{x}}_k
&=-A^{-4}s_{jk}\overline{\mathbf{x}}_i+s_{ik}\overline{\mathbf{x}}_j-A^{4}s_{ij}\overline{\mathbf{x}}_k-s_{ijk}\mathbf{e}. \label{eq:basic-3-product-1}
\end{align}
Hence for $i_1<i_2<i_3<i_4$,
\begin{align}
\alpha\overline{\mathbf{x}}_{i_1}\overline{\mathbf{x}}_{i_2}\overline{\mathbf{x}}_{i_3}\overline{\mathbf{x}}_{i_4}
&=-A^{-4}s_{i_2i_3}\overline{\mathbf{x}}_{i_1}\overline{\mathbf{x}}_{i_4}+s_{i_1i_3}\overline{\mathbf{x}}_{i_2}\overline{\mathbf{x}}_{i_4}
-A^4s_{i_1i_2}\overline{\mathbf{x}}_{i_3}\overline{\mathbf{x}}_{i_4}-s_{i_1i_2i_3}\overline{\mathbf{x}}_{i_4}. \label{eq:x1x2x3x4}
\end{align}
For $i<j<k$,
\begin{align}
s_{ij}s_{ik}&=A^{-2}s_{i\lrcorner jk}+(s_{ii}-A^{-2})s_{jk}+t_is_{ijk}, \label{eq:product-22-1} \\
s_{ij}s_{jk}&=A^2s_{ij\lrcorner k}+(s_{jj}-A^2)s_{ik}+t_js_{ijk}, \label{eq:product-22-2} \\
s_{ik}s_{jk}&=A^{-2}s_{i\check{k}\lrcorner j}+(s_{kk}-A^{-2})s_{ij}+t_ks_{ijk}. \label{eq:product-22-3}
\end{align}
For $i<j<k<\ell$,
\begin{align}
s_{ij}s_{jk\ell}&=A^2s_{ij\lrcorner(k\ell)}+(s_{jj}-A^2)s_{ik\ell}+t_j(s_{ijk\ell}+\beta s_{ij}s_{k\ell}), \label{eq:product-23-1} \\
s_{ij}s_{ik\ell}&=A^{-2}s_{i\lrcorner jk\ell}+(s_{ii}-A^{-2})s_{jk\ell}+t_i(s_{ijk\ell}+\beta s_{ij}s_{k\ell}), \label{eq:product-23-2} \\
s_{i\ell}s_{ijk}&=A^{2}s_{i\lrcorner(jk)\ell}+(s_{ii}-A^{2})s_{jk\ell}+t_i(s_{ijk\ell}+\beta s_{i\ell}s_{jk}), \label{eq:product-23-3} \\
s_{ik}s_{ij\ell}&=A^{2}s_{i\lrcorner jk\ell}+A^{-2}s_{i\lrcorner(jk)\ell}+t_i(s_{ijk\ell}+\beta s_{ik}s_{j\ell})-\beta t_i^2s_{ijk}. \label{eq:product-23-4}
\end{align}
To show (\ref{eq:product-23-1}), note that
$$s_{ij}s_{\hat{j}k\ell}=s_{i\hat{j}}s_{\hat{j}k\ell}=A^2s_{ij\lrcorner(k\ell)}+A^{-2}s_{ik\ell}+t_js_{i\hat{j}k\ell}.$$
Similarly for (\ref{eq:product-23-2}).
Acting on (\ref{eq:product-23-1}) via the permutation $(i\ell kj)$ yields (\ref{eq:product-23-3}).
Finally, (\ref{eq:product-23-4}) is obtained via
$$s_{ik}s_{ij\ell}=s_{\hat{i}k}s_{\hat{i}j\ell}+\beta t_is_{ik}s_{j\ell}=A^{2}s_{i\lrcorner jk\ell}+A^{-2}s_{i\lrcorner(jk)\ell}
+t_is_{\hat{i}jk\ell}+\beta t_is_{ik}s_{j\ell}.$$
\begin{rmk}
\rm If $\mathfrak{a}=a(A)x_1\cdots x_r\in\mathcal{S}^\circ$ with $a(A)\in\mathbb{Z}[A^{\pm1}]$ such that each $x_i$ has no crossing, then we say that $\mathfrak{a}$ is {\it simple}, and call $\mathfrak{a}^{\rm op}:=a(A^{-1})x_r\cdots x_1$ the {\it mirror} of $\mathfrak{a}$.
Observe if $\mathfrak{a}_1+\cdots+\mathfrak{a}_r=0$ holds in $\mathcal{S}^\circ$, with each $\mathfrak{a}_i$ being simple, then $\mathfrak{a}^{\rm op}_1+\cdots+\mathfrak{a}^{\rm op}_r=0$ is also true.
\end{rmk}
\section{Generators and commuting relations}
Let $\mathfrak{T}=\mathfrak{T}_1\sqcup\mathfrak{T}_2\sqcup\mathfrak{T}_3$, with
\begin{align*}
\mathfrak{T}_1&=\{t_i\colon 1\le i\le n\}, \\
\mathfrak{T}_2&=\{s_{i_1i_2}\colon 1\le i_1<i_2\le n\}, \\
\mathfrak{T}_3&=\{s_{i_1i_2i_3}\colon 1\le i_1<i_2<i_3\le n\}.
\end{align*}
\begin{prop}
The skein algebra $\mathcal{S}(\Sigma_{0,n+1};\mathbb{Z}[A^{\pm1},\beta])$ is generated by $\mathfrak{T}$.
\end{prop}
\begin{proof}
Let $\mathcal{T}$ denote the ring generated by $\mathfrak{T}$.
Since $\mathcal{S}(\Sigma_{0,n+1};\mathbb{Z}[A^{\pm1}])$ as a $\mathbb{Z}[A^{\pm1}]$-module is generated by multi-curves, it suffices to show that each non-boundary-parallel simple curve belongs to $\mathcal{T}$.
\begin{enumerate}
\item Generalizing (\ref{eq:x1x2x3x4}), by repeatedly using (\ref{eq:basic-3-product-1}), we may write $\overline{\mathbf{x}}_1\cdots\overline{\mathbf{x}}_k$ as a
$\mathbb{Z}[A^{\pm1},\beta]$-linear combination of terms of the form $\mathfrak{a}\overline{\mathbf{y}}$, where $\mathfrak{a}$ is a product of elements of $\mathfrak{T}$, and
$\overline{\mathbf{y}}\in \{\mathbf{e},\overline{\mathbf{x}}_1,\ldots,\overline{\mathbf{x}}_k,\overline{\mathbf{x}}_1\overline{\mathbf{x}}_2,\ldots,
\overline{\mathbf{x}}_{k-1}\overline{\mathbf{x}}_k\}.$
Hence ${\rm tr}(\mathbf{x}_1\cdots\mathbf{x}_k)\in\mathcal{T}$.
\item Let $B_n$ denote the $n$-strand braid group, acting on $\Sigma_{0,n+1}$ in a natural way.
For each generator $\sigma_j$ of $B_n$, which counter-clockwise rotates by $\pi$ interchanging
the $j$- and $(j+1)$-th punctures,
\begin{align*}
\sigma_j(\mathbf{x}_i)=\begin{cases} \mathbf{x}_{j+1},&i=j, \\
A^{-2}(t_{j,j+1}\mathbf{x}_{j+1}-t_{j+1}\mathbf{x}_j\mathbf{x}_{j+1}-t_j\mathbf{e})-A^{-4}\mathbf{x}_j,&i=j+1, \\
\mathbf{x}_i, &i\ne j,j+1.
\end{cases}
\end{align*}
This implies $\sigma_j^{\pm 1}(t_{i_1i_2})\in\mathcal{T}$ for all $i_1<i_2$ and $\sigma_j^{\pm 1}(t_{i_1i_2i_3})\in\mathcal{T}$ for all $i_1<i_2<i_3$. Consequently, $\sigma_j^{\pm 1}(x)\in\mathcal{T}$ for each $x\in\mathfrak{T}$, due to (\ref{eq:sij}), (\ref{eq:sijk}).
\item For each non-boundary-parallel simple closed curve $\gamma$, there exists $k\in\{2,\ldots,n-1\}$ and $\sigma\in B_n$ such
that $\gamma=\sigma({\rm tr}(\mathbf{x}_1\cdots\mathbf{x}_k))$. By the previous two steps, $\gamma\in\mathcal{T}$.
\end{enumerate}
Therefore, $\mathcal{S}(\Sigma_{0,n+1};\mathbb{Z}[A^{\pm1},\beta])=\mathcal{T}$.
\end{proof}
\begin{rmk}
\rm If $\mathfrak{f}=0$ is an identity in $\mathcal{S}_{0,n}$ involving subscripts $i_1<\cdots<i_m$, then clearly it remains true under the cyclic permutations $(i_1,\ldots,i_m)^r,r=1,\ldots,m-1$. In this sense we say ``for $i_1,\ldots,i_m$ in cyclic order, $\mathfrak{f}=0$".
\end{rmk}
\begin{prop}\label{prop:commutator-22}
For $(i_1,i_2,i_3)$ in cyclic order,
\begin{align}
A^2s_{i_2i_3}s_{i_1i_2}-A^{-2}s_{i_1i_2}s_{i_2i_3}&=(A^{2}-A^{-2})(s_{i_2i_2}s_{i_1i_3}+t_{i_2}s_{i_1i_2i_3}). \label{eq:commutator-22-1}
\end{align}
For $(i_1,i_2,i_3,i_4)$ in cyclic order,
\begin{align}
s_{i_2i_4}s_{i_1i_3}-s_{i_1i_3}s_{i_2i_4}=(A^{-4}-A^4)(s_{i_1i_2}s_{i_3i_4}-s_{i_1i_4}s_{i_2i_3}). \label{eq:commutator-22-2}
\end{align}
\end{prop}
\begin{proof}
The first identity follows from (\ref{eq:product-22-1})--(\ref{eq:product-22-3}) and their mirrors. The second identity is obtained from taking the difference between (\ref{eq:product-22-0}) and its mirror.
\end{proof}
\begin{prop}\label{prop:commutator-23}
For $(i_1,i_2,i_3)$ in cyclic order,
\begin{align}
s_{i_1i_2i_3}s_{i_1i_2}-s_{i_1i_2}s_{i_1i_2i_3}
&=(A^4-A^{-4})\big(A^2t_{i_2}(s_{i_1i_2}s_{i_1i_3}-s_{i_1i_1}s_{i_2i_3}-t_{i_1}s_{i_1i_2i_3}) \nonumber \\
&\ \ \ \ \ -A^{-2}t_{i_1}(s_{i_1i_2}s_{i_2i_3}-s_{i_2i_2}s_{i_1i_3}-t_{i_2}s_{i_1i_2i_3})\big). \label{eq:commutator-23-1}
\end{align}
For $(i_1,i_2,i_3,i_4)$ in cyclic order,
\begin{align}
&A^2s_{i_2i_3i_4}s_{i_1i_2}-A^{-2}s_{i_1i_2}s_{i_2i_3i_4} \nonumber \\
=\ &(A^2-A^{-2})\big(s_{i_2i_2}s_{i_1i_3i_4}+t_{i_2}(s_{i_1i_2i_3i_4}+\beta s_{i_1i_2}s_{i_3i_4})\big), \label{eq:commutator-23-2} \\
&A^{-2}s_{i_1i_3i_4}s_{i_1i_2}-A^2s_{i_1i_2}s_{i_1i_3i_4} \nonumber \\
=\ &(A^{-2}-A^2)\big(s_{i_1i_1}s_{i_2i_3i_4}+t_{i_1}(s_{i_1i_2i_3i_4}+\beta s_{i_1i_2}s_{i_3i_4})\big), \label{eq:commutator-23-3} \\
&s_{i_1i_2i_4}s_{i_1i_3}-s_{i_1i_3}s_{i_1i_2i_4} \nonumber \\
=\ &(A^2-A^{-2})\big(A^{-2}s_{i_1i_4}s_{i_1i_2i_3}-A^2s_{i_1i_2}s_{i_1i_3i_4} \nonumber \\
&\ +(A^2-A^{-2})(s_{i_1i_1}s_{i_2i_3i_4}+t_{i_1}s_{i_1i_2i_3i_4})-\beta t_{i_1}(A^2s_{i_1i_4}s_{i_2i_3}+A^{-2}s_{i_1i_2}s_{i_3i_4})\big). \label{eq:commutator-23-4}
\end{align}
For $(i_1,i_2,i_3,i_4,i_5)$ in cyclic order,
\begin{align}
s_{i_2i_4i_5}s_{i_1i_3}-s_{i_1i_3}s_{i_2i_4i_5}=(A^{-4}-A^4)(s_{i_1i_2}s_{i_3i_4i_5}-s_{i_2i_3}s_{i_1i_4i_5}). \label{eq:commutator-23-5}
\end{align}
\end{prop}
\begin{proof}
Using (\ref{eq:sijk}) to expand $s_{i_1i_2i_3}$, noting that $t_{i_1i_2i_3}$ commutes with $s_{i_1i_2}$, and applying (\ref{eq:commutator-22-1}), we can deduce (\ref{eq:commutator-23-1}).
The identities (\ref{eq:commutator-23-2}), (\ref{eq:commutator-23-3}) respectively follow from (\ref{eq:product-23-1}), (\ref{eq:product-23-2}).
Combining (\ref{eq:product-23-3}) with (\ref{eq:product-23-4}) leads to (\ref{eq:commutator-23-4}).
To derive (\ref{eq:commutator-23-5}), we multiply (\ref{eq:basic-2}) on the right by $\overline{\mathbf{x}}_{i_4}\overline{\mathbf{x}}_{i_5}$, take traces to get
$$s_{i_1i_3}s_{i_2i_4i_5}=A^4s_{i_1i_2}s_{i_3i_4i_5}+A^{-4}s_{i_2i_3}s_{i_1i_4i_5}+s_{i_1i_2i_3}s_{i_4i_5}+\alpha s_{i_1i_2i_3i_4i_5},$$
and then take the difference between this and its mirror.
\end{proof}
Call the identities given in Proposition \ref{prop:commutator-22} {\it commuting relations of type $[2,2]$}, and those in Proposition \ref{prop:commutator-23} {\it commuting relations of type $[2,3]$}.
\begin{rmk}
\rm We do not investigate commuting relations of type $[3,3]$ (whose meanings are self-evident), because they follow from the established commuting relations together with type I quantized relations given in the next section.
\end{rmk}
\section{Quantization of defining relations}
Let $\eta=A^6+A^{-6}$, so that $\beta\eta=A^4+A^{-4}-1$.
\subsection{Relations of type II}\label{sec:typeII}
\begin{prop}\label{prop:relation-II}
For $i_0,i_1,i_2,i_3,i_4$ in cyclic order,
\begin{align*}
&A^{4}s_{i_0i_1}s_{i_2i_3i_4}-s_{i_0i_2}s_{i_1i_3i_4}+s_{i_0i_3}s_{i_1i_2i_4}-A^{-4}s_{i_0i_4}s_{i_1i_2i_3} \\
=\ &(A^{4}-1)(A^{-4}s_{i_1i_2}s_{i_0i_3i_4}+s_{i_3i_4}s_{i_0i_1i_2}).
\end{align*}
For $i_0,i_1,i_2,i_3$ in cyclic order,
\begin{align*}
&A^4s_{i_0i_1}s_{i_0i_2i_3}-s_{i_0i_2}s_{i_0i_1i_3}+A^{-4}s_{i_0i_3}s_{i_0i_1i_2}-\beta\eta s_{i_0i_0}s_{i_1i_2i_3} \\
=\ &(A^2-A^{-2})^2\beta t_{i_0}(s_{i_0i_2}s_{i_1i_3}-A^4s_{i_0i_1}s_{i_2i_3}-A^{-4}s_{i_0i_3}s_{i_1i_2}).
\end{align*}
\end{prop}
\begin{proof}
We have
\begin{align*}
s_{i_0i_2}s_{i_1i_3i_4}&=A^4s_{i_0i_1}s_{i_2i_3i_4}+A^{-4}s_{i_1i_2}s_{i_0i_3i_4}+\alpha s_{i_0i_1i_2i_3i_4}+s_{i_3i_4}s_{i_0i_1i_2}, \\
s_{i_0i_3}s_{i_1i_2i_4}&=A^4s_{i_3i_4}s_{i_0i_1i_2}+A^{-4}s_{i_0i_4}s_{i_1i_2i_3}+\alpha s_{i_0i_1i_2i_3i_4}+s_{i_1i_2}s_{i_0i_3i_4},
\end{align*}
from which the first identity follows.
For the second identity, we use
\begin{align*}
s_{i_0i_1}s_{i_0i_2i_3}&=(s_{i_0i_0}-A^{-2})s_{i_1i_2i_3}+A^{-2}s_{i_0\lrcorner i_1i_2i_3}+t_{i_0}s_{i_0i_1i_2i_3}+\beta t_{i_0}s_{i_0i_1}s_{i_2i_3}, \\
s_{i_0i_2}s_{i_0i_1i_3}&=A^2s_{i_0\lrcorner i_1i_2i_3}+A^{-2}s_{i_0\lrcorner(i_1i_2)i_3}+t_{i_0}s_{i_0i_1i_2i_3}-\beta t_{i_0}^2s_{i_1i_2i_3}+\beta t_{i_0}s_{i_0i_2}s_{i_1i_3}, \\
s_{i_0i_3}s_{i_0i_1i_2}&=A^2s_{i_0\lrcorner(i_1i_2)i_3}+(s_{i_0i_0}-A^{2})s_{i_1i_2i_3}+t_{i_0}s_{i_0i_1i_2i_3}+\beta t_{i_0}s_{i_0i_3}s_{i_1i_2}.
\end{align*}
Note that $s_{i_0i_2}s_{i_1i_3}=\alpha s_{i_0i_1i_2i_3}+A^4s_{i_0i_1}s_{i_2i_3}+A^{-4}s_{i_0i_3}s_{i_1i_2}$.
\end{proof}
\begin{rmk}\label{rmk:quantization-II}
\rm Setting $A^4=1$ recovers classical type II relations (\ref{eq:typeII}). We call the identities given in Proposition \ref{prop:relation-II}
{\it type II quantized relations}.
\end{rmk}
\subsection{Relations of type I} \label{sec:typeI}
\begin{prop}\label{prop:intersection0}
Let $\kappa=A^8+A^{-4}-1$ and $\overline{\kappa}=A^{-8}+A^4-1$. For $i_1,\ldots,i_6$ in cyclic order,
\begin{align*}
\eta s_{i_1i_2i_3}s_{i_4i_5i_6}&=s_{i_1i_4}s_{i_2i_5}s_{i_3i_6}-A^{-4}(s_{i_1i_4}s_{i_2i_6}s_{i_3i_5}+s_{i_1i_5}s_{i_2i_4}s_{i_3i_6}) \\
&\ \ \ +A^{-8}(s_{i_1i_5}s_{i_2i_6}s_{i_3i_4}+s_{i_1i_6}s_{i_2i_4}s_{i_3i_5})-A^{-12}s_{i_1i_6}s_{i_2i_5}s_{i_3i_4} \\
&\ \ \ +(1-A^4)\Big(A^4s_{i_1i_2}s_{i_3i_4}s_{i_5i_6}-s_{i_1i_2}s_{i_3i_5}s_{i_4i_6}-s_{i_1i_3}s_{i_2i_4}s_{i_5i_6} \\
&\ \ \ \ \ +A^{-4}(s_{i_1i_2}s_{i_3i_6}s_{i_4i_5}+s_{i_1i_3}s_{i_2i_5}s_{i_4i_6}+s_{i_1i_4}s_{i_2i_3}s_{i_5i_6}) \\
&\ \ \ \ \ -A^{-8}(s_{i_1i_3}s_{i_2i_6}s_{i_4i_5}+s_{i_1i_5}s_{i_2i_3}s_{i_4i_6})+A^{-12}s_{i_1i_6}s_{i_2i_3}s_{i_4i_5}\Big), \\
\eta s_{i_1i_2i_4}s_{i_3i_5i_6}&=s_{i_1i_3}s_{i_2i_5}s_{i_4i_6}+s_{i_1i_5}s_{i_2i_6}s_{i_3i_4}+A^{-8}s_{i_1i_6}s_{i_2i_3}s_{i_4i_5} \\
&\ \ \ -A^{-4}(s_{i_1i_3}s_{i_2i_6}s_{i_4i_5}+s_{i_1i_5}s_{i_2i_3}s_{i_4i_6}+s_{i_1i_6}s_{i_2i_5}s_{i_3i_4}) \\
&\ \ \ +(1-A^{4})\Big(\overline{\kappa}s_{i_1i_2}s_{i_3i_4}s_{i_5i_6}+A^{-4}(s_{i_1i_2}s_{i_3i_5}s_{i_4i_6}+s_{i_1i_3}s_{i_2i_4}s_{i_5i_6} \\
&\ \ \ \ \ +s_{i_1i_4}s_{i_2i_6}s_{i_3i_5}+s_{i_1i_5}s_{i_2i_4}s_{i_3i_6})-s_{i_1i_4}s_{i_2i_5}s_{i_3i_6} \\
&\ \ \ \ \ -A^{-8}(s_{i_1i_2}s_{i_3i_6}s_{i_4i_5}+s_{i_1i_4}s_{i_2i_3}s_{i_5i_6}+s_{i_1i_6}s_{i_2i_4}s_{i_3i_5})\Big), \\
\eta s_{i_1i_3i_5}s_{i_2i_4i_6}&=s_{i_1i_2}s_{i_3i_4}s_{i_5i_6}-A^4s_{i_1i_2}s_{i_3i_6}s_{i_4i_5}-A^4s_{i_1i_4}s_{i_2i_3}s_{i_5i_6} \\
&\ \ \ +s_{i_1i_4}s_{i_2i_5}s_{i_3i_6}+A^{-4}\kappa s_{i_1i_6}s_{i_2i_3}s_{i_4i_5}-\overline{\kappa}s_{i_1i_6}s_{i_2i_5}s_{i_3i_4} \\
&\ \ \ +(1-A^{4})\Big(A^{-4}(s_{i_1i_3}s_{i_2i_5}s_{i_4i_6}+s_{i_1i_6}s_{i_2i_4}s_{i_3i_5})-s_{i_1i_2}s_{i_3i_5}s_{i_4i_6} \\
&\ \ \ \ \ -s_{i_1i_3}s_{i_2i_4}s_{i_5i_6}-s_{i_1i_4}s_{i_2i_6}s_{i_3i_5}-s_{i_1i_5}s_{i_2i_4}s_{i_3i_6} \\
&\ \ \ \ \ +A^4(s_{i_1i_3}s_{i_2i_6}s_{i_4i_5}+s_{i_1i_5}s_{i_2i_3}s_{i_4i_6})-A^{-4}\kappa s_{i_1i_5}s_{i_2i_6}s_{i_3i_4}\Big).
\end{align*}
\end{prop}
When there is one overlapped puncture, up to rotation and mirror there are essentially three cases.
\begin{prop}\label{prop:intersection1}
For $i_1,\ldots,i_5$ in cyclic order,
\begin{align*}
\alpha s_{i_1i_2i_3}s_{i_3i_4i_5}&=s_{i_1i_3}s_{i_2i_4}s_{i_3i_5}+A^{-4}(s_{i_1i_4}s_{i_2i_5}s_{i_3i_3}-s_{i_1i_3}s_{i_2i_5}s_{i_3i_4}
-s_{i_1i_4}s_{i_2i_3}s_{i_3i_5}) \\
&\ \ \ +A^{-8}(s_{i_1i_5}s_{i_2i_3}s_{i_3i_4}-s_{i_1i_5}s_{i_2i_4}s_{i_3i_3})+(1-A^4)s_{i_3i_3}s_{i_1i_2}s_{i_4i_5} \\
&\ \ \ +(A^{-4}-1)t_{i_3}\big(s_{i_1i_3}s_{i_2i_4i_5}-A^{-4}s_{i_2i_3}s_{i_1i_4i_5}+(A^4-1)s_{i_4i_5}s_{i_1i_2i_3}\big), \\
\alpha s_{i_1i_3i_5}s_{i_2i_3i_4}&=s_{i_1i_3}s_{i_2i_5}s_{i_3i_4}-s_{i_2i_5}s_{i_1i_4}s_{i_3i_3}+s_{i_3i_5}s_{i_1i_4}s_{i_2i_3}
-A^{4}s_{i_3i_5}s_{i_1i_2}s_{i_3i_4} \\
&\ \ \ +A^{-4}(s_{i_4i_5}s_{i_1i_2}s_{i_3i_3}-s_{i_4i_5}s_{i_1i_3}s_{i_2i_3})+(1-A^{-4})s_{i_3i_3}s_{i_1i_5}s_{i_2i_4} \\
&\ \ \ +(A^4-1)t_{i_3}\big(s_{i_3i_4}s_{i_1i_2i_5}-A^{-4}s_{i_2i_3}s_{i_1i_4i_5}+(A^{-4}-1)s_{i_1i_5}s_{i_2i_3i_4}\big), \\
\alpha s_{i_1i_3i_4}s_{i_2i_3i_5}&=s_{i_1i_3}s_{i_2i_4}s_{i_3i_5}+A^4(s_{i_3i_3}s_{i_1i_2}s_{i_4i_5}-s_{i_1i_2}s_{i_3i_4}s_{i_3i_5}
-s_{i_1i_3}s_{i_2i_3}s_{i_4i_5}) \\
&\ \ \ +A^{-4}(s_{i_1i_5}s_{i_2i_3}s_{i_3i_4}-s_{i_3i_3}s_{i_1i_5}s_{i_2i_4})+(1-A^{-4})s_{i_3i_3}s_{i_1i_4}s_{i_2i_5} \\
&\ \ \ +(1-A^{-4})t_{i_3}\big(s_{i_2i_3}s_{i_1i_4i_5}-A^4s_{i_4i_5}s_{i_1i_2i_3}+s_{i_1i_3i_4}s_{i_2i_5}\big).
\end{align*}
\end{prop}
When there are two overlapped punctures, up to rotation and mirror there are essentially two cases.
\begin{prop}\label{prop:intersection2}
For $i_1,\ldots,i_4$ in cyclic order,
\begin{align*}
\alpha s_{i_1i_2i_3}s_{i_2i_3i_4}&=A^{-4}(s_{i_1i_2}s_{i_2i_3}s_{i_3i_4}-s_{i_1i_4}s_{i_2i_3}^2+s_{i_2i_2}s_{i_3i_3}s_{i_1i_4}
-s_{i_3i_3}s_{i_1i_2}s_{i_2i_4}) \\
&\ \ \ +s_{i_2i_3}s_{i_1i_3}s_{i_2i_4}-\beta\eta s_{i_2i_2}s_{i_1i_3}s_{i_3i_4} \\
&\ \ \ +(A^4-1)\Big(t_{i_2}(s_{i_3i_4}s_{i_1i_2i_3}-s_{i_1i_3}s_{i_2i_3i_4}-s_{i_3i_3}s_{i_1i_2i_4}) \\
&\ \ \ \ \ +A^{-4}t_{i_3}(s_{i_1i_2}s_{i_2i_3i_4}-s_{i_2i_2}s_{i_1i_3i_4})\Big)+(A^2-A^{-2})^2t_{i_2}t_{i_3}s_{i_1i_2i_3i_4} \\
&\ \ \ +(A^4-1)\beta t_{i_2}t_{i_3}\big((A^4-A^{-4}-1)s_{i_1i_2}s_{i_3i_4}+A^{-4}s_{i_1i_4}s_{i_2i_3}\big), \\
\alpha s_{i_1i_2i_3}s_{i_1i_3i_4}&=A^4\beta\eta s_{i_1i_1}s_{i_2i_3}s_{i_3i_4}-s_{i_1i_1}s_{i_2i_4}s_{i_3i_3}+s_{i_1i_3}^2s_{i_2i_4}-A^8s_{i_1i_2}s_{i_1i_3}s_{i_3i_4} \\
&\ \ \ +s_{i_1i_2}s_{i_1i_4}s_{i_3i_3}-A^{-4}s_{i_1i_3}s_{i_1i_4}s_{i_2i_3} \\
&\ \ \ +(A^4-1)\Big(t_{i_1}(A^{-4}s_{i_2i_3}s_{i_1i_3i_4}+A^8s_{i_3i_4}s_{i_1i_2i_3}-A^{-4}s_{i_3i_3}s_{i_1i_2i_4}) \\
&\ \ \ \ \ +t_{i_3}(s_{i_1i_1}s_{i_2i_3i_4}-s_{i_1i_2}s_{i_1i_3i_4})\Big) \\
&\ \ \ -(A^2-A^{-2})^2t_{i_1}t_{i_3}\big(A^8\beta s_{i_1i_2}s_{i_3i_4}+(A^8+3)s_{i_1i_2i_3i_4}\big).
\end{align*}
\end{prop}
The last case is the one with three overlapped punctures.
\begin{prop}\label{prop:intersection3}
For $i_1,i_2,i_3$ in cyclic order,
\begin{align*}
\alpha s_{i_1i_2i_3}^2&=A^2\alpha s_{i_1i_2}s_{i_1i_3}s_{i_2i_3}+s_{i_1i_1}s_{i_2i_2}s_{i_3i_3}-A^4s_{i_1i_1}s_{i_2i_3}^2-A^{-4}s_{i_2i_2}s_{i_1i_3}^2-A^4s_{i_3i_3}s_{i_1i_2}^2 \\
&\ \ \ +(1-A^4)\big(t_{i_1}s_{i_2i_3}-A^{-4}t_{i_2}s_{i_1i_3}+t_{i_3}s_{i_1i_2}+(A^2-A^{-2})^2\beta t_{i_1}t_{i_2}t_{i_3}\big)s_{i_1i_2i_3} \\
&\ \ \ +(A^2-A^{-2})^2\beta\big(s_{i_1i_1}t_{i_2}t_{i_3}s_{i_2i_3}+s_{i_2i_2}t_{i_1}t_{i_3}s_{i_1i_3}-A^4s_{i_3i_3}t_{i_1}t_{i_2}s_{i_1i_2} \\
&\ \ \ \ \ +A^2\alpha t_{i_1}t_{i_2}s_{i_1i_3}s_{i_2i_3}\big).
\end{align*}
\end{prop}
\begin{rmk}\label{rmk:quantization-I}
\rm It is straightforward to verify that setting $A^2=1$ recovers the type I relations. Call the identities given in Proposition \ref{prop:intersection0}--\ref{prop:intersection3} and their mirrors {\it type I quantized relations}.
\end{rmk}
\section{The presentation}
Let $\mathcal{S}_1=\mathcal{S}(\Sigma_{0,n+1};R_1)$, with $R_1=\mathbb{Z}[A^{\pm1},\eta^{-1}])$.
Recall $\mathfrak{T}=\mathfrak{T}_1\cup\mathfrak{T}_2\cup\mathfrak{T}_3$, with
$\mathfrak{T}_1=\{t_i\colon 1\le i\le n\}$, $\mathfrak{T}_2=\{s_{ij}\colon 1\le i<j\le n\}$, $\mathfrak{T}_3=\{s_{ijk}\colon 1\le i<j<k\le n\}$.
Set $|t_i|=1$, $|s_{ij}|=2$, and $|s_{ijk}|=3$. For a monomial $\mathfrak{a}=x_1\cdots x_k$ with $x_i\in\mathfrak{T}$, define its {\it degree} as $|\mathfrak{a}|=|x_1|+\cdots+|x_k|$.
Let $\mathfrak{M}_0$ denote the set of monomials of the form
\begin{align}
\mathfrak{a}=t_{i_1}\cdots t_{i_m}s_{i_1j_1}\cdots s_{i_rj_r}. \label{eq:monomial}
\end{align}
Let $\mathfrak{M}_1=\{\mathfrak{a}x\colon \mathfrak{a}\in\mathfrak{M}_0, x\in\mathfrak{T}_3\}$.
\begin{lem}\label{lem:reduce-T3}
$\mathcal{S}_1$ is generated by $\mathfrak{M}_0\cup\mathfrak{M}_1$.
\end{lem}
\begin{proof}
For each monomial $\mathfrak{a}=x_1\cdots x_k$ with $x_i\in\mathfrak{T}$, let $\mu(\mathfrak{a})=\#\{i\colon |x_i|=3\}$, let $\nu(\mathfrak{a})$ be the number of pairs $(i,j)$ with $i<j$ and $|x_i|=3, |x_j|=2$.
If $\nu(\mathfrak{a})>0$, then there exists $i$ such that $|x_i|=3$ and $|x_{i+1}|=2$. Let $\mathfrak{b}$ be the monomial obtained from $\mathfrak{a}$ by interchanging $x_i$ and $x_{i+1}$. According to Proposition \ref{prop:commutator-23}, there exists $a\in\{1,A^4,A^{-4}\}$ with $\mathfrak{a}-a\mathfrak{b}$ being equal to a linear combination of monomials $\mathfrak{c}_i$ such that for each $i$, one of the following occurs: $\mu(\mathfrak{c}_i)<\mu(\mathfrak{a})$; $\mu(\mathfrak{c}_i)=\mu(\mathfrak{a})$ and $\nu(\mathfrak{c}_i)<\nu(\mathfrak{a})$.
It follows by induction on $(\mu(\mathfrak{a}),\nu(\mathfrak{a}))$ in the lexicographical order that $\mathfrak{a}$ can be written as a linear combination of monomials $\mathfrak{a}_i$ with $\nu(\mathfrak{a}_i)=0$.
Using type I relations, we may express each of such $\mathfrak{a}_i$'s as a linear combination of monomials in $\mathfrak{M}_0\cup\mathfrak{M}_1$.
\end{proof}
For $\mathfrak{a}\in\mathfrak{M}_0$ in the form (\ref{eq:monomial}), let $\varrho(\mathfrak{a})=m$, let $\delta_h(\mathfrak{a})=\#\{v\colon j_v-i_v=h\}$ for each $h$, and set
$e(\mathfrak{a})=(\varrho(\mathfrak{a}),\delta_1(\mathfrak{a}),\ldots,\delta_{n-1}(\mathfrak{a}))$;
for $x\in\mathfrak{T}_3$, set $e(\mathfrak{a}x)=e(\mathfrak{a})$.
Declare $(\varrho,\delta_1,\ldots,\delta_{n-1})<(\varrho',\delta'_1,\ldots,\delta'_{n-1})$ if one of the following holds: $\varrho>\varrho'$; $\varrho=\varrho'$ and there exists $u$ with $\delta_u>\delta'_u$ and $\delta_i=\delta'_i$ for $1\le i<u$.
Let $\prec$ be a linear ordering on $\mathfrak{T}_2$. Call $\mathfrak{a}$ of the form (\ref{eq:monomial}) {\it normal} with respect to $\prec$ if $s_{i_vj_v}\preceq s_{i_{v+1}j_{v+1}}$ for all $v$. For each $\mathfrak{a}=x_1\cdots x_k\in\mathcal{M}_0$, let $N^\prec(\mathfrak{a})$ denote the normal monomial obtained by permuting the $x_v$'s.
For $\mathfrak{a}x\in\mathfrak{M}_1$ (with $\mathfrak{a}\in\mathfrak{M}_0$, $|x|=3$), call $\mathfrak{a}x$ normal if $\mathfrak{a}$ is normal; let $N^\prec(\mathfrak{a}x)=N^\prec(\mathfrak{a})x$.
Let $\mathcal{R}^\prec$ denote the $R_1$-module generated by elements of $\mathfrak{M}_0\cup\mathfrak{M}_1$ that are normal with respect to $\prec$.
\begin{lem}
For each $\mathfrak{a}\in\mathfrak{M}_0\cup\mathfrak{M}_1$, there exists an integer $h_\mathfrak{a}$ such that $\mathfrak{a}=A^{4h_\mathfrak{a}}N^\prec(\mathfrak{a})+\sum_ib_i\mathfrak{b}_i\in\mathcal{R}^\prec$ for some $b_i\in R_1$ with $b_i|_{A^4=1}=0$ and normal $\mathfrak{b}_i$ with $e(\mathfrak{b}_i)<e(\mathfrak{a})$.
\end{lem}
\begin{proof}
Use induction on $(e(\mathfrak{a}),\pi(\mathfrak{a}))$ in the lexicographical order, where
$$\pi(\mathfrak{a}):=\#\{(u,v)\colon u<v,\ s_{i_{v}j_{v}}\prec s_{i_uj_u}\}.$$
Suppose $\pi(\mathfrak{a})>0$, so that there exists $v$ with $s_{i_{v+1}j_{v+1}}\prec s_{i_vj_v}$. Let $a=i_v,b=j_v,c=i_{v+1},d=j_{v+1}$.
If $s_{ab}s_{cd}=s_{cd}s_{ab}$, then switching $s_{ab}$ and $s_{cd}$ will reduce $\pi(\mathfrak{a})$. Let us assume $s_{ab}s_{cd}\ne s_{cd}s_{ab}$.
If $b=c$, then by (\ref{eq:commutator-22-1}),
$$s_{ab}s_{cd}=A^{4}s_{cd}s_{ab}+(1-A^{4})(s_{bb}s_{ad}+t_{b}s_{abd}).$$
Let $\mathfrak{a}',\mathfrak{a}_1,\mathfrak{a}'_2$ denote the monomial obtained from $\mathfrak{a}$ by replacing $s_{ab}s_{cd}$ by $s_{cd}s_{ab}$, $s_{bb}s_{ad}$, $t_{b}s_{abd}$, respectively. If necessary, applying the process in the proof of Lemma \ref{lem:reduce-T3} to convert $\mathfrak{a}'_2$ into a linear combination $\mathfrak{a}_2=\sum_ic_i\mathfrak{c}_i$ with $\mathfrak{c}_i\in\mathfrak{M}_0\cup\mathfrak{M}_1$ and $e(\mathfrak{c}_i)<e(\mathfrak{a})$; otherwise, let $\mathfrak{a}_2=\mathfrak{a}'_2$. Then $\mathfrak{a}=A^{4}\mathfrak{a}'+(1-A^{4})(\mathfrak{a}_1+\mathfrak{a}_2)$, and $e(\mathfrak{a}_1),e(\mathfrak{a}_2)<e(\mathfrak{a}')=e(\mathfrak{a})$, $\pi(\mathfrak{a}')<\pi(\mathfrak{a})$.
The situation for $b=d$ or $a=c$ or $a=d$ is similar.
If $c<a<d<b$, then by (\ref{eq:commutator-22-2}),
$$s_{ab}s_{cd}=s_{cd}s_{ab}+(A^{-4}-A^4)(s_{ca}s_{db}-s_{cb}s_{ad}).$$
Let $\mathfrak{a}',\mathfrak{a}_1,\mathfrak{a}_2$ denote the monomial obtained from $\mathfrak{a}$ by replacing $s_{ab}s_{cd}$ by $s_{cd}s_{ab}$, $s_{ca}s_{db}$, $s_{cb}s_{ad}$, respectively. Then $\mathfrak{a}=\mathfrak{a}'+(A^{-4}-A^4)(\mathfrak{a}_1-\mathfrak{a}_2)$, $e(\mathfrak{a}')=e(\mathfrak{a})$ and $\pi(\mathfrak{a}')<\pi(\mathfrak{a})$. Since $\min\{a-c,b-d\}<\min\{d-c,b-a\}$, we have $e(\mathfrak{a}_1)<e(\mathfrak{a})$; since $d-a<\min\{d-c,b-a\}$, we have $e(\mathfrak{a}_2)<e(\mathfrak{a})$.
The situation for $a<c<b<d$ is similar.
\end{proof}
For a classical defining relation $f\in\mathcal{I}^0$ (referring to Remark \ref{rmk:classical-ideal}), a {\it quantization} of $f$ is an element $\mathfrak{f}\in\mathcal{R}^\prec$ with $\mathfrak{f}|_{A^2=1}=f$.
\begin{lem}
Each element of $\mathcal{G}_1\cup\mathcal{G}_2\cup\mathcal{G}_3\cup\mathcal{G}_4$ can be quantized.
\end{lem}
\begin{proof}
We prove this statement into four steps.
\begin{enumerate}
\item For each $f^b_c(a_1a_2a_3)\in\mathcal{G}_1$, we can find a unique monic $\mathfrak{f}^b_c(a_1a_2a_3)\in\mathcal{R}^\prec$
quantizing $f^b_c(a_1a_2a_3)$ such that $\mathfrak{f}^b_c(a_1a_2a_3)=0$ is equivalent to the corresponding identity in Section \ref{sec:typeII}.
\item From (\ref{eq:f2}), (\ref{eq:f3}) it is clear that elements of $\mathcal{G}_2\cup\mathcal{G}_3$ all can be quantized.
\item For each $g^{a_1a_2a_3}_{b_1b_2b_3}\in\mathcal{G}_0$, we can find a unique $\mathfrak{g}^{a_1a_2a_3}_{b_1b_2b_3}\in\mathcal{S}_1$ with leading coefficient being $\eta$ or $\alpha$ such that $\mathfrak{g}^{a_1a_2a_3}_{b_1b_2b_3}|_{A^2=1}=g^{a_1a_2a_3}_{b_1b_2b_3}$ and $\mathfrak{g}^{a_1a_2a_3}_{b_1b_2b_3}=0$ is equivalent to the corresponding identity in Section \ref{sec:typeI}.
\item To quantize $\theta=\theta^{a_1a_2a_3a_4}_{b_1b_2b_3b_4}$, we set
\begin{align*}
\mathfrak{j}=\mathfrak{j}^{a_1a_2a_3a_4}_{b_1b_2b_3b_4}&=s_{a_4b_1}\mathfrak{g}^{a_1a_2a_3}_{b_2b_3b_4}-s_{a_3b_1}\mathfrak{g}^{a_1a_2a_4}_{b_2b_3b_4}
+s_{a_2b_1}\mathfrak{g}^{a_1a_3a_4}_{b_2b_3b_4}-s_{a_1b_1}\mathfrak{g}^{a_2a_3a_4}_{b_2b_3b_4} \nonumber \\
&\ \ \ +\eta\mathfrak{f}^{a_1}_{b_1}(a_2,a_3,a_4)s_{b_2b_3b_4}.
\end{align*}
From (\ref{eq:theta}) we see that $\mathfrak{j}|_{A^2=1}=\theta$. In the expression of $\mathfrak{j}$ there remains some terms, each of which is a multiple of $(A^4-1)s_{i_1i_2i_3}s_{j_1j_2j_3}$; replace these terms with multiples of $\mathfrak{M}_0$ via quantized type I relations, to obtain $\mathfrak{h}\in\mathcal{R}^{\prec_0}$. Then $\mathfrak{h}$ is a quantization of $\theta$.
\end{enumerate}
\end{proof}
\begin{rmk}
\rm We tend to eliminate products $s_{i_1i_2i_3}s_{j_1j_2j_3}$ to bypass commuting relations of type $[3,3]$ which we never consider.
\end{rmk}
\begin{thm}\label{thm:main}
Let $R$ be the localization of $\mathbb{Z}[A^{\pm1}]$ at the multiplicative subset generated by $A^{6}+A^{-6}$ and polynomials
$\gamma\in\mathbb{Z}[A]$ with $A^4-1\mid \gamma(A)-1$. The skein algebra $\mathcal{S}(\Sigma_{0,n+1};R)$ is presented by the generating set $\mathfrak{T}$ and quantized relations of type I and II given in Section 5.
\end{thm}
\begin{proof}
As shown in the previous lemmas, modulo commuting relations of type $[2,2]$ and $[2,3]$ and type I quantized relations, each element of $\mathcal{S}(\Sigma_{0,n+1};R)$ is equal to some $\mathfrak{a}\in\mathcal{R}^{\prec_0}$.
Let $S_1$ denote the set of polynomials $\gamma\in\mathbb{Z}[A]$ such that $A^4-1\mid \gamma(A)-1$.
Fix $d\in\mathbb{N}$. Let $\{u_1,\ldots,u_N\}$ be the finite set of monomials of degree $d$ that are divisible by the leading term of some element of $\mathcal{G}_1\cup\mathcal{G}_2\cup\mathcal{G}_3\cup\mathcal{G}_4$, and suppose $u_N\prec_0\cdots\prec_0u_1$. For each $i$, take a normal $\mathfrak{u}_i$ with the same letters as $u_i$; find $g_i$ whose leading term is $u_i$, and take a quantization $\mathfrak{g}_i$ for $g_i$. By induction on $i$, (starting from $i=1$), we can ensure that $\mathfrak{u}_i$ does not appear in $\mathfrak{g}_j$ for all $j>i$, by replacing $\mathfrak{g}_j$ with $\gamma\mathfrak{g}_j-\mathfrak{g}_i$ for some $\gamma\in S_1$ if necessary. Next, by induction on $i$, (starting from $i=1$), we can further ensure that $\mathfrak{u}_i$ does not appear in $\mathfrak{g}_j$ for all $j<i$, in a similar way. Thus, actually we may assume that $\mathfrak{u}_i$ appears in $\mathfrak{g}_j$ exactly for $i=j$, and the coefficient of $u_i$ is in $S_1$.
By \cite{SW07} Corollary 4.1, multi-curves form a free basis for the $\mathbb{Z}[A^{\pm1}]$-module $\mathcal{S}(\Sigma_{0,n+1};R)$. Consequently, the monomials in any relation have equal degree, called the degree of the relation.
Suppose $\mathfrak{f}=0$ is a relation of degree $d$. For each $1\le i\le N$, now that elements of $S_1$ are invertible in $R$, by subtracting a multiple of $\mathfrak{g}_i$ if necessary, we may assume that $\mathfrak{u}_i$ does appear in $\mathfrak{f}$. Assume $\mathfrak{f}=0$, then by \cite{PS19} Theorem 2, we can divide by $A^2-1$ whenever necessary to ensure that at least one of the coefficients of monomials in $\mathfrak{f}$ is coprime to $A^2-1$. By Remark \ref{rmk:quantization-II} and \ref{rmk:quantization-I}, $\mathfrak{f}|_{A^2=1}=0$ is a classical relation. This contradicts the assumption that the leading term of $\mathfrak{f}|_{A^2=1}$ is not divisible by that of any element of the Gr\"obner basis $\mathcal{G}$.
\end{proof}
\begin{que}
What is the smallest multiplicative set $S$ such that the skein algebra over the localization of $\mathbb{Z}[A^{\pm1}]$ at $S$ has a presentation given in the theorem?
\end{que}
\newpage
\section{Delayed proofs of some results}
To simplify the writing, without loss of generality we just assume $i_k=k$.
Let
\begin{align*}
&\mathfrak{u}_1=s_{12}s_{34}s_{56}, &&\mathfrak{u}_2=s_{12}s_{35}s_{46}, &&\mathfrak{u}_3=s_{12}s_{36}s_{45}, \\
&\mathfrak{u}_4=s_{13}s_{24}s_{56}, &&\mathfrak{u}_5=s_{13}s_{25}s_{46}, &&\mathfrak{u}_6=s_{13}s_{26}s_{45}, \\
&\mathfrak{u}_7=s_{14}s_{23}s_{56}, &&\mathfrak{u}_8=s_{14}s_{25}s_{36}, &&\mathfrak{u}_9=s_{14}s_{26}s_{35}, \\
&\mathfrak{u}_{10}=s_{15}s_{23}s_{46}, &&\mathfrak{u}_{11}=s_{15}s_{24}s_{36}, &&\mathfrak{u}_{12}=s_{15}s_{26}s_{34}, \\
&\mathfrak{u}_{13}=s_{16}s_{23}s_{45}, &&\mathfrak{u}_{14}=s_{16}s_{24}s_{35}, &&\mathfrak{u}_{15}=s_{16}s_{25}s_{34}, \\
&\mathfrak{a}_1=s_{123}s_{456}, &&\mathfrak{a}_2=s_{234}s_{156}, &&\mathfrak{a}_3=s_{345}s_{126}; \\
&\mathfrak{b}_1=s_{12}s_{3456}, &&\mathfrak{b}_2=s_{23}s_{1456}, &&\mathfrak{b}_3=s_{34}s_{1256}, \\
&\mathfrak{b}_4=s_{45}s_{1236}, &&\mathfrak{b}_5=s_{56}s_{1234}, &&\mathfrak{b}_6=s_{16}s_{2345}; \\
&\ &&\mathfrak{e}=s_{123456}.
\end{align*}
Given variables or parameters $a_1,\ldots,a_n$, abbreviate $a_{k_1}+\cdots+a_{k_r}$ to $a_{k_1,\ldots,k_r}$.
\begin{lem}
We have
\begin{align*}
s_{13}s_{25}s_{46}&=\alpha^2\mathfrak{e}+\alpha(\mathfrak{a}_1+A^4\mathfrak{b}_{1,5}+A^{-4}\mathfrak{b}_{2,4})
+A^8\mathfrak{u}_1+\mathfrak{u}_{3,7}+A^{-8}\mathfrak{u}_{13}, \\
s_{14}s_{26}s_{35}&=\alpha^2\mathfrak{e}+\alpha(\mathfrak{a}_3+A^4\mathfrak{b}_{1,4}+A^{-4}\mathfrak{b}_{3,6})
+\mathfrak{u}_{1,13}+A^8\mathfrak{u}_3+A^{-8}\mathfrak{u}_{15}, \\
s_{15}s_{24}s_{36}&=\alpha^2\mathfrak{e}+\alpha(\mathfrak{a}_2+A^4\mathfrak{b}_{2,5}+A^{-4}\mathfrak{b}_{3,6})
+\mathfrak{u}_{1,13}+A^{8}\mathfrak{u}_7+A^{-8}\mathfrak{u}_{15}, \\
s_{14}s_{25}s_{36}&=A^{-2}\alpha^3\mathfrak{e}+\alpha(A^4\mathfrak{a}_1+A^{-4}\mathfrak{a}_{2,3}
+\mathfrak{b}_{1,2,4,5}+A^{-8}\mathfrak{b}_{3,6}) \\
&\ \ \ +A^{-4}\mathfrak{u}_{1,13}+A^4\mathfrak{u}_{3,7}+A^{-12}\mathfrak{u}_{15}.
\end{align*}
\end{lem}
\begin{proof}
Applying (\ref{eq:product-22-0}), we compute
\begin{align*}
&s_{13}s_{25}s_{46}=(A^4s_{12}s_{35}+A^{-4}s_{23}s_{15}+\alpha s_{1235})s_{46} \\
=\ &A^4s_{12}(A^{4}s_{34}s_{56}+A^{-4}s_{36}s_{45}+\alpha s_{3456})+A^{-4}s_{23}(A^4s_{14}s_{56}+A^{-4}s_{16}s_{45}+\alpha s_{1456}) \\
&+\alpha(A^{-4}s_{45}s_{1236}+A^{4}s_{56}s_{1234}+\alpha s_{123456}+s_{123}s_{456}), \\
&s_{14}s_{26}s_{35}=(A^4s_{12}s_{46}+A^{-4}s_{24}s_{16}+\alpha s_{1246})s_{35} \\
=\ &A^4s_{12}(A^{-4}s_{34}s_{56}+A^4s_{45}s_{36}+\alpha s_{3456})+A^{-4}s_{16}(A^4s_{23}s_{45}+A^{-4}s_{34}s_{25}+\alpha s_{2345}) \\
&+\alpha(A^{-4}s_{34}s_{1256}+A^{4}s_{45}s_{1236}+\alpha s_{123456}+s_{126}s_{345}),
\end{align*}
respectively establishing the first and the second identities.
We have
\begin{align*}
&s_{15}s_{24}s_{36}=s_{15}(A^{4}s_{46}s_{23}+A^{-4}s_{26}s_{34}+\alpha s_{2346}) \\
=\ &A^4(A^4s_{14}s_{56}+A^{-4}s_{16}s_{45}+\alpha s_{1456})s_{23}+A^{-4}(A^4s_{12}s_{56}+A^{-4}s_{16}s_{25}+\alpha s_{1256})s_{34} \\
&+\alpha(A^4s_{1234}s_{56}+A^{-4}s_{16}s_{2345}+s_{234}s_{156}+\alpha s_{123456}),
\end{align*}
where $s_{15}s_{2346}$ is computed by applying (\ref{eq:basic-2}) as follow:
\begin{align*}
s_{15}s_{2346}&={\rm tr}(\overline{\mathbf{x}_2}\overline{\mathbf{x}_3}\overline{\mathbf{x}_4}s_{51}\overline{\mathbf{x}_6}) \\
&={\rm tr}\big(\overline{\mathbf{x}_2}\overline{\mathbf{x}_3}\overline{\mathbf{x}_4}(A^4s_{56}\overline{\mathbf{x}_1}
+A^{-4}s_{61}\overline{\mathbf{x}_5}+\alpha\overline{\mathbf{x}_5}\overline{\mathbf{x}_6}\overline{\mathbf{x}_1}+s_{561}\mathbf{e})\big) \\
&=A^4s_{56}s_{1234}+A^{-4}s_{16}s_{2345}+\alpha s_{123456}+s_{234}s_{156}.
\end{align*}
Then the third identity is established.
We have
\begin{align*}
s_{14}s_{25}s_{36}&=(A^4s_{12}s_{45}+A^{-4}s_{24}s_{15}+\alpha s_{1245})s_{36} \\
&=A^4\mathfrak{d}_3+A^{-4}s_{24}(A^{-4}s_{35}s_{16}+A^4s_{13}s_{56}+\alpha s_{1356})+\alpha s_{1245}s_{36} \\
&=A^4\mathfrak{d}_3+A^{-8}s_{24}s_{35}\cdot s_{16}+s_{24}s_{13}\cdot s_{56}+A^{-4}\alpha s_{24}s_{1356}+\alpha s_{1245}s_{36} \\
&=A^4\mathfrak{d}_3+A^{-8}(A^4s_{23}s_{45}+A^{-4}s_{34}s_{25}+\alpha s_{2345})s_{16} \\
&\ \ \ +(A^{-4}s_{12}s_{34}+A^4s_{23}s_{14}+\alpha s_{1234})s_{56}+A^{-4}\alpha s_{24}s_{1356}+\alpha s_{1245}s_{36}.
\end{align*}
By (\ref{eq:basic-2}), $s_{24}\overline{\mathbf{x}}_3=A^4s_{23}\overline{\mathbf{x}}_4+A^{-4}s_{34}\overline{\mathbf{x}}_2+s_{234}\mathbf{e}+\alpha\overline{\mathbf{x}}_2\overline{\mathbf{x}}_3\overline{\mathbf{x}}_4$,
so that
$$s_{24}s_{1356}={\rm tr}(\overline{\mathbf{x}}_1s_{24}\overline{\mathbf{x}}_3\overline{\mathbf{x}}_5\overline{\mathbf{x}}_6)=A^4\mathfrak{b}_2+A^{-4}\mathfrak{b}_3+\mathfrak{a}_2+\alpha\mathfrak{e}.$$
By the mirror of the formula in the middle of Figure \ref{fig:basic},
$$\overline{\mathbf{x}}_4\overline{\mathbf{x}}_5s_{36}=A^{-4}s_{345}\overline{\mathbf{x}}_6+A^4s_{456}\overline{\mathbf{x}}_3+s_{3456}\mathbf{e}+s_{45}\overline{\mathbf{x}}_3\overline{\mathbf{x}}_6
+\alpha\overline{\mathbf{x}}_3\overline{\mathbf{x}}_4\overline{\mathbf{x}}_5\overline{\mathbf{x}}_6.$$
hence
\begin{align*}
s_{1245}s_{36}={\rm tr}(\overline{\mathbf{x}}_1\overline{\mathbf{x}}_2\overline{\mathbf{x}}_4\overline{\mathbf{x}}_5s_{36})=A^{-4}\mathfrak{a}_3+A^4\mathfrak{a}_1+\mathfrak{b}_1+\mathfrak{b}_4+\alpha\mathfrak{e}.
\end{align*}
Then the fourth identity follows.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:intersection0}]
By the previous Lemma, for any $\mu_0,\ldots,\mu_3$, $\nu_1,\ldots,\nu_6$, $\lambda_1,\ldots,\lambda_{15}$, from
\begin{align*}
&\alpha\eta\mu_0\mathfrak{e}+\eta\sum\limits_{i=1}^{3}\mu_i\mathfrak{a}_i
+\alpha\sum\limits_{i=1}^{6}\nu_i\mathfrak{b}_i=\sum\limits_{i=1}^{15}\lambda_i\mathfrak{u}_i \\
=\ &\alpha^2(\lambda_{5,9,11}+A^{-2}\alpha\lambda_8)\mathfrak{e}+\alpha(\lambda_{5}+A^{4}\lambda_8)\mathfrak{a}_1
+\alpha(A^{-4}\lambda_{8}+\lambda_{11})\mathfrak{a}_2 \\
&+\alpha(A^{-4}\lambda_{8}+\lambda_9)\mathfrak{a}_3+\alpha(\lambda_{2,8}+A^{4}\lambda_{5,9})\mathfrak{b}_1+\alpha(A^{-4}\lambda_{5}+\lambda_{8,10}+A^{4}\lambda_{11})\mathfrak{b}_2 \\
&+\alpha(A^{-8}\lambda_{8}+A^{-4}\lambda_{9,11}+\lambda_{12})\mathfrak{b}_3+\alpha(A^{-4}\lambda_{5}+\lambda_{6,8}+A^{4}\lambda_9)\mathfrak{b}_4 \\
&+\alpha(\lambda_{4,8}+A^{4}\lambda_{5,11})\mathfrak{b}_5+\alpha(A^{-8}\lambda_{8}+A^{-4}\lambda_{9,11}+\lambda_{14})\mathfrak{b}_6 \\
&+(\lambda_{1,9,11}+A^4\lambda_{2,4,12}+A^{8}\lambda_5+A^{-4}\lambda_8)\mathfrak{u}_1 \\
&+(A^{-4}\lambda_2+\lambda_{3,5}+A^4\lambda_{6,8}+A^{8}\lambda_{9})\mathfrak{u}_3 \\
&+(A^{-4}\lambda_4+\lambda_{5,7}+A^4\lambda_{8,10}+A^{8}\lambda_{11})\mathfrak{u}_7 \\
&+(A^{-8}\lambda_5+A^{-4}\lambda_{6,8,10}+\lambda_{9,11,13}+A^{4}\lambda_{14})\mathfrak{u}_{13} \\
&+(A^{-12}\lambda_8+A^{-8}\lambda_{9,11}+A^{-4}\lambda_{12,14}+\lambda_{15})\mathfrak{u}_{15}
\end{align*}
we can deduce
\begin{align*}
\lambda_{1}&=A^{12}\mu_0+(A^4-A^8)\mu_{1,2,3}-A^4\nu_{1,3,5}, \\
\lambda_2&=A^8(\mu_2-\mu_0)+(A^4-1)\mu_{1,3}+\nu_1, \\
\lambda_{3}&=A^4(\mu_0-\mu_2)+(A^{-4}-1)\mu_{1,3}-A^{-4}\nu_1-A^4\nu_4, \\
\lambda_{4}&=A^8(\mu_3-\mu_0)+(A^4-1)\mu_{1,2}+\nu_5, \\
\lambda_5&=A^4(\mu_0-\mu_{2,3})+(A^{-4}-1)\mu_1, \\
\lambda_{6}&=\mu_3-\mu_0+(A^{-4}-A^{-8})\mu_1+(A^4-A^8)\mu_2+\nu_4, \\
\lambda_{7}&=A^4(\mu_0-\mu_3)+(A^{-4}-1)\mu_{1,2}-A^4\nu_2-A^{-4}\nu_5, \\
\lambda_8&=\mu_{1,2,3}-\mu_0, \\
\lambda_9&=A^{-4}(\mu_0-\mu_{1,2})+(A^{4}-1)\mu_3, \\
\lambda_{10}&=\mu_3-\mu_0+(A^{-4}-A^{-8})\mu_1+(A^4-A^8)\mu_2+\nu_2, \\
\lambda_{11}&=A^{-4}(\mu_0-\mu_{1,3})+(A^{4}-1)\mu_2, \\
\lambda_{12}&=A^{-8}(\mu_1-\mu_0)+(A^{-4}-1)\mu_{2,3}+\nu_3, \\
\lambda_{13}&=A^{-4}\mu_0+(A^{-12}-A^{-8})\mu_1+(A^4-1)\mu_{2,3}-A^{-4}\nu_{2,4}-A^4\nu_6, \\
\lambda_{14}&=A^{-8}(\mu_1-\mu_0)+(A^{-4}-1)\mu_{2,3}+\nu_6, \\
\lambda_{15}&=A^{-12}(\mu_0-\mu_1)+(A^{-4}-A^{-8})\mu_{2,3}-A^{-4}\nu_{3,6}.
\end{align*}
The first identity is obtained by setting $\mu_1=1$ and $\mu_0=\mu_2=\mu_3=\nu_1=\cdots=\nu_6=0.$
According to the local relation given at the top of Figure \ref{fig:basic},
\begin{align*}
s_{124}s_{356}=\alpha\mathfrak{e}+A^4\mathfrak{a}_1+\mathfrak{b}_1+A^{-4}\mathfrak{b}_3+\mathfrak{b}_5+A^{-2}\mathfrak{u}_1.
\end{align*}
The second identity is obtained by setting $\mu_0=1,\mu_1=A^4$, $\nu_1=\nu_5=\beta\eta$, $\nu_3=A^{-4}\beta\eta$.
\begin{figure}[h]
\centering
\includegraphics[width=10cm]{135-246}\\
\caption{The product $s_{135}s_{246}$.}\label{fig:135-246}
\end{figure}
From Figure \ref{fig:135-246} it is clear that
\begin{align*}
s_{135}s_{246}=\ &A^6s_{12}s_{34}s_{56}+A^4s_{12}s_{3456}+A^2s_{123456}+A^4s_{56}s_{1234}+s_{123}s_{456} \\
&+s_{35}s_{1246}+A^{-2}(A^2s_{156}s_{234}+s_{123456}+A^{-2}s_{23}s_{1456}) \\
&+A^{-5}(As_{16}s_{2345}+A^{-1}s_{16}s_{23}s_{45}) \\
=\ &2\alpha\mathfrak{e}+\mathfrak{a}_{1,2,3}+A^4\mathfrak{b}_{1,3,5}+A^{-4}\mathfrak{b}_{2,4,6}+A^6\mathfrak{u}_1+A^{-6}\mathfrak{u}_{13},
\end{align*}
where we have used (\ref{eq:basic-2}) to compute
$$s_{35}s_{1246}=A^4s_{34}s_{1256}+A^{-4}s_{45}s_{1236}+\alpha s_{123456}+s_{126}s_{345}.$$
The third identity is obtained by setting $\mu_0=2,\mu_1=\mu_2=\mu_3=1$, $\nu_1=\nu_3=\nu_5=A^4\beta\eta$, $\nu_2=\nu_4=\nu_6=A^{-4}\beta\eta$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:intersection1}]
We have
\begin{align*}
s_{123}s_{345}&=A^2s_{123\lrcorner(45)}+t_3s_{12345}+(s_{33}-A^2)s_{1245}+\beta s_{33}s_{12}s_{45} \\
&\ \ \ +\beta t_3(s_{12}s_{345}+s_{45}s_{123}), \\
s_{13}s_{24}s_{35}&=\alpha s_{1234}s_{35}+A^4s_{12}s_{34}s_{35}+A^{-4}s_{14}s_{23}s_{35}, \\
s_{13}s_{25}s_{34}&=\alpha s_{1235}s_{34}+A^4s_{12}s_{35}s_{34}+A^{-4}s_{15}s_{23}s_{34}.
\end{align*}
Hence
\begin{align*}
&s_{13}s_{24}s_{35}-A^{-4}s_{13}s_{25}s_{34}-A^{-4}s_{14}s_{23}s_{35}+A^{-8}s_{15}s_{23}s_{34} \\
=\ &\alpha(s_{1234}s_{35}-A^{-4}s_{1235}s_{34})+s_{12}(A^{4}s_{34}s_{35}-s_{35}s_{34}) \\
=\ &\alpha\big(A^2s_{123\lrcorner(45)}+(2-A^{-4})t_3s_{12345}+(A^{-2}-A^{-4}s_{33}-\beta t_3^2)s_{1235}\big) \\
&+t_3(A^4s_{45}s_{123}+s_{12}s_{345})+(A^4-1)s_{12}(s_{33}s_{45}+t_3s_{345}),
\end{align*}
using
\begin{align*}
s_{1234}s_{35}&=s_{12\hat{3}4}s_{\hat{3}5}+\beta t_3s_{124}s_{35} \\
&=A^{-2}s_{123\lrcorner45}+A^{2}s_{123\lrcorner(45)}+2t_3s_{12345}-\beta t_3^2s_{1245} \nonumber \\
&\ \ \ +\beta t_3(A^{-4}s_{34}s_{125}+A^{4}s_{45}s_{123}+s_{12}s_{345}), \\
s_{1235}s_{34}&=A^{2}s_{123\lrcorner45}+(s_{33}-A^2)s_{1245}+t_3s_{12345}+\beta t_3s_{34}s_{125}.
\end{align*}
Then the first identity follows.
For the second identity, we compute
\begin{align*}
s_{135}s_{234}&=s_{1\hat{3}5}s_{2\hat{3}4}+\beta t_3s_{15}s_{234}+\beta t_3s_{135}s_{24}-\beta^2t_3^2s_{15}s_{24} \\
&=A^2s_{123\lrcorner 45}+s_{15}s_{23\lrcorner4}+t_3(s_{12345}-\beta t_3s_{1245})+A^{-2}s_{1\check{3}\lrcorner245}
-\beta^2t_3^2s_{15}s_{24} \\
&\ \ \ +\beta t_3s_{15}s_{234}+\beta t_3(A^{-4}s_{23}s_{145}+A^4s_{34}s_{125}+s_{15}s_{234}+\alpha s_{12345}) \\ &=2t_3s_{12345}+A^2s_{123\lrcorner45}+A^{-2}s_{1\check{3}\lrcorner245}+s_{15}s_{23\lrcorner4} \\
&\ \ \ +\beta t_3(A^{-4}s_{23}s_{145}+A^4s_{125}s_{34}+2s_{15}s_{234})-\beta t_3^2(s_{1245}+\beta s_{15}s_{24}), \\
s_{13}s_{25}s_{34}&=\alpha s_{1235}s_{34}+A^{-4}s_{15}s_{23}s_{34}+A^{4}s_{12}s_{35}s_{34} \\
&=\alpha\big(A^2s_{123\lrcorner45}+(s_{33}-A^{2})s_{1245}+t_3(s_{12345}+\beta s_{125}s_{34})\big) \\
&\ \ \ +A^{-4}s_{15}\big(A^2s_{23\lrcorner4}+(s_{33}-A^2)s_{24}+t_3s_{234}\big)+A^{4}s_{12}s_{35}s_{34}, \\
s_{35}s_{14}s_{23}&=\alpha s_{1345}s_{23}+A^{4}s_{34}s_{23}s_{15}+A^{-4}s_{13}s_{23}s_{45} \\
&=\alpha\big(A^{-2}s_{1\check{3}\lrcorner245}+(s_{33}-A^{-2})s_{1245}+t_3(s_{12345}+\beta s_{145}s_{23})\big) \\
&\ \ \ +A^4s_{15}\big(A^{-2}s_{23\lrcorner4}+(s_{33}-A^{-2})s_{24}+t_3s_{234}\big)+A^{-4}s_{45}s_{13}s_{23},
\end{align*}
where the following has been used:
\begin{align*}
s_{1235}s_{34}&=A^2s_{123\lrcorner45}+(s_{33}-A^{2})s_{1245}+t_3(s_{12345}+\beta s_{125}s_{34}), \\
s_{1345}s_{23}&=A^{-2}s_{1\check{2}\lrcorner245}+(s_{33}-A^{-2})s_{1245}+t_3(s_{12345}+\beta s_{145}s_{23}).
\end{align*}
Then the identity follows.
For the third, since
$s_{134}=s_{1\hat{3}4}+\beta t_3s_{14}$ and $s_{235}=s_{2\hat{3}5}+\beta t_3s_{25}$, we have
\begin{align*}
s_{134}s_{235}&=s_{1\hat{3}4}s_{\hat{2}35}+\beta t_3(s_{14}s_{235}+s_{134}s_{25})-\beta^2t_3^2s_{14}s_{25} \\
&=A^4s_{12}s_{45}+A^2s_{1245}+s_{12\hat{3}4}s_{\hat{3}5}+A^{-2}s_{1\check{3}\lrcorner245}+A^{-4}s_{15}s_{23\lrcorner 4}-\beta^2t_3^2s_{14}s_{25} \\
&\ \ \ +\beta t_3(A^4s_{45}s_{123}+A^{-4}s_{15}s_{234}+\alpha s_{12345}+s_{23}s_{145}+s_{134}s_{25}).
\end{align*}
We also have
\begin{align*}
s_{13}s_{24}s_{35}&=A^{4}s_{12}s_{34}s_{35}+A^{-4}s_{23}s_{14}s_{35}+\alpha s_{1234}s_{35} \\
&=A^{4}s_{12}s_{34}s_{35}+A^{-4}s_{23}(A^4s_{13}s_{45}+A^{-4}s_{15}s_{34}+\alpha s_{1345})+\alpha s_{1234}s_{35} \\
&=A^{4}s_{12}s_{34}s_{35}+s_{23}s_{13}s_{45}+A^{-8}s_{15}s_{23}s_{34}+\alpha(A^{-4}s_{23}s_{1345}+s_{1234}s_{35}) \\
&=A^{4}s_{12}s_{34}s_{35}+s_{23}s_{13}s_{45}+A^{-8}s_{15}s_{23}s_{34}+A^{-2}\alpha^2t_3s_{12345} \\
&\ \ \ +\alpha\big(A^{-2}s_{1\check{3}\lrcorner 245}+s_{12\hat{3}4}s_{\hat{3}5}+A^{-4}(s_{33}-A^2)s_{1245}\big) \\
&\ \ \ +t_3(s_{12}s_{345}+A^{-4}s_{23}s_{145}+A^{-4}s_{34}s_{125}+A^4s_{45}s_{123}),
\end{align*}
where we have used
\begin{align*}
s_{23}s_{1345}&=A^{2}s_{1\check{3}\lrcorner 245}+(s_{33}-A^{2})s_{1245}+t_3(s_{12345}+\beta s_{23}s_{145}), \\
s_{1234}s_{35}&=s_{12\hat{3}4}s_{\hat{3}5}+\beta t_3s_{124}s_{35} \\
&=s_{12\hat{3}4}s_{\hat{3}5}+\beta t_3(A^{-4}s_{34}s_{125}+A^4s_{45}s_{123}+s_{12}s_{345}+\alpha s_{12345}).
\end{align*}
Then, noting $s_{134}s_{25}=A^4s_{12}s_{345}+A^{-4}s_{15}s_{234}+s_{34}s_{125}+\alpha s_{12345}$, we have
\begin{align*}
\alpha s_{134}s_{235}-s_{13}s_{24}s_{35}+A^4s_{12}s_{34}s_{35}+s_{23}s_{13}s_{45}-A^{-4}s_{15}s_{23}s_{34} \\
-s_{33}s_{12}s_{45}+A^{-4}s_{33}s_{15}s_{24}=(1-A^{-4})(s_{33}s_{14}s_{25}+t_3(s_{23}s_{145}+s_{134}s_{25})).
\end{align*}
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:intersection2}]
For the first identity, note that
\begin{align*}
s_{123}s_{234}&=(s_{1\hat{2}\hat{3}}+\beta t_2s_{13}+\beta t_3s_{12})(s_{\hat{2}\hat{3}4}+\beta t_2s_{34}+\beta t_3s_{24}) \\
&=s_{1\hat{2}\hat{3}}s_{\hat{2}\hat{3}4}+\beta(t_2s_{13}+t_3s_{12})s_{234}+\beta s_{123}(t_2s_{34}+t_3s_{24}) \\
&\ \ \ -\beta^2(t_2s_{13}+t_3s_{12})(t_2s_{34}+t_3s_{24}) \\
&=A^2s_{1(23)\lrcorner 4}+A^{-2}s_{14}+t_{23}(s_{1234}-\beta t_2s_{134}-\beta t_3s_{124}+\beta^2t_2t_3s_{14}) \\
&\ \ \ +\beta(t_2s_{13}+t_3s_{12})s_{234}+\beta s_{123}(t_2s_{34}+t_3s_{24}) \\
&\ \ \ -\beta^2(t_2s_{13}+t_3s_{12})(t_2s_{34}+t_3s_{24}), \\
s_{12}s_{23}s_{34}&=(A^2s_{12\lrcorner 3}+(s_{22}-A^2)s_{13}+t_2s_{123})s_{34} \\
&=A^4s_{1(23)\lrcorner 4}+A^2(s_{33}-A^2)s_{12\lrcorner4}+A^2t_3s_{12\lrcorner(34)} \\
&\ \ \ +(s_{22}-A^2)s_{13}s_{34}+t_2s_{123}s_{34} \\
s_{23}s_{13}s_{24}&=s_{23}(A^4s_{12}s_{34}+A^{-4}s_{14}s_{23}+\alpha s_{1234}) \\
&=(s_{12}s_{23}+(A^4-1)(s_{22}s_{13}+t_2s_{123}))s_{34}+A^{-4}s_{14}s_{23}^2+\alpha s_{23}s_{1234}.
\end{align*}
Hence
\begin{align*}
&\alpha s_{123}s_{234}-A^{-4}s_{12}s_{23}s_{34}-s_{23}s_{13}s_{24}+A^{-4}s_{14}s_{23}^2 \\
=\ &\alpha s_{123}s_{234}-A^{-2}\alpha s_{12}s_{23}s_{34}+(1-A^4)(s_{22}s_{13}+t_2s_{123})s_{34}-\alpha s_{23}s_{1234} \\
=\ &t_2(s_{123}s_{34}+s_{13}s_{234}-s_{23}s_{134})+t_3(s_{123}s_{24}+s_{12}s_{234}-s_{23}s_{124}) \\
&+\beta t_2t_3(s_{14}s_{23}-\alpha s_{1234}-s_{12}s_{34}-s_{13}s_{24})+\beta t_2^2t_3s_{134}+\beta t_2t_3^2s_{124} \\
&+(A^{-2}\alpha-\beta^2t_2^2t_3^2)s_{14}-\beta t_2^2s_{13}s_{34}-\beta t_3^2s_{12}s_{24} \\
&+A^{-2}\alpha(A^2-s_{33})(s_{12}s_{24}+(A^2-s_{22})s_{14}-t_2s_{124}) \\
&-\alpha t_3s_{12\lrcorner(34)}+A^{-2}\alpha(A^2-s_{22})s_{13}s_{34}-A^{-2}\alpha t_2s_{123}s_{34} \\
&+(1-A^4)(s_{22}s_{13}+t_2s_{123})s_{34} \\
=\ &t_2\big((2A^2-A^6-A^{-2})s_{123\lrcorner4}+(1-A^4)(s_{33}-A^2)s_{124}+(A^{-2}-A^2)s_{1\check{3}\lrcorner24}\big) \\
&+(2-A^4-A^{-4})t_2t_3s_{1234}+(1-A^{-4})\beta t_2t_3(s_{12}s_{34}-s_{14}s_{23})+(A^{-2}-A^2)t_3s_{134} \\
&+(A^2-A^{-2})t_3s_{12\lrcorner(34)}+A^{-4}s_{22}s_{33}s_{14}-A^{-4}s_{33}s_{12}s_{24}+(1-A^4-A^{-4})s_{22}s_{13}s_{34} \\
=\ &(A^4-1)\big(t_2(s_{34}s_{123}-s_{13}s_{234}-s_{33}s_{124})+A^{-4}t_3(s_{12}s_{234}-s_{22}s_{134})\big) \nonumber \\
&+(A^2-A^{-2})^2t_2t_3s_{1234}+(A^4-1)\beta t_2t_3((A^4-A^{-4}-1)s_{12}s_{34}+A^{-4}s_{14}s_{23}) \\
&+A^{-4}s_{22}s_{33}s_{14}-A^{-4}s_{33}s_{12}s_{24}+(1-A^4-A^{-4})s_{22}s_{13}s_{34}.
\end{align*}
For the second identity, let $\mathfrak{h}_1=s_{11}s_{23}s_{34}$, $\mathfrak{h}_2=s_{11}s_{24}s_{33}$, $\mathfrak{h}_3=s_{13}^2s_{24}$, $\mathfrak{h}_4=s_{12}s_{13}s_{34}$, $\mathfrak{h}_5=s_{12}s_{14}s_{33}$, $\mathfrak{h}_6=s_{13}s_{14}s_{23}$.
We have
\begin{align*}
s_{123}s_{134}=\ &(s_{\hat{1}2\hat{3}}+\beta t_1s_{23}+\beta t_3s_{12})(s_{\hat{1}\hat{3}4}+\beta t_1s_{34}+\beta t_3s_{14}) \\
=\ &s_{\hat{1}2\hat{3}}s_{\hat{1}\hat{3}4}+\beta t_1(s_{23}s_{1\hat{3}4}+s_{12\hat{3}}s_{34})+\beta t_3(s_{\hat{1}23}s_{14}+s_{12}s_{\hat{1}34}) \\
&-\beta^{2}t_1^2s_{23}s_{34}-\beta^{2}t_3^2s_{12}s_{14}+\beta^{2}t_1t_3(s_{12}s_{34}+s_{14}s_{23}) \\
=\ &A^2s_{23\lrcorner4}+t_{13}s_{\hat{1}2\hat{3}4}+A^{-2}s_{1\lrcorner24}-\beta^2t_1t_3(s_{12}s_{34}+s_{14}s_{23}) \\
&+\beta t_1(s_{23}s_{134}+s_{123}s_{34})+\beta t_3(s_{123}s_{14}+s_{12}s_{134})-\beta^{2}t_1^2s_{23}s_{34}-\beta^{2}t_3^2s_{12}s_{14}, \\
\mathfrak{h}_3=\ &s_{13}(\alpha s_{1234}+A^4s_{12}s_{34}+A^{-4}s_{14}s_{23}) \\
=\ &\alpha s_{13}s_{1234}+A^4(A^4s_{12}s_{13}+(1-A^4)(s_{11}s_{23}+t_1s_{123}))s_{34}+A^{-4}s_{13}s_{14}s_{23} \\
=\ &\alpha s_{13}(s_{\hat{1}2\hat{3}4}+\beta t_1s_{234}+\beta t_3s_{124}-\beta^2t_1t_3s_{24}) \\
\ &+A^8\mathfrak{h}_4+(A^4-A^8)(\mathfrak{h}_1+t_1s_{123}s_{34})+A^{-4}\mathfrak{h}_6.
\end{align*}
Hence
\begin{align*}
&\alpha s_{123}s_{134}-(1-A^4+A^8)\mathfrak{h}_1+\mathfrak{h}_2-\mathfrak{h}_3+A^8\mathfrak{h}_4-\mathfrak{h}_5+A^{-4}\mathfrak{h}_6 \\
=\ &t_1(s_{23}s_{134}-s_{13}s_{234}+(A^8-A^4+1)s_{123}s_{34}-s_{33}s_{124}) \\
&+t_3(s_{12}s_{134}-s_{13}s_{124}+s_{123}s_{14}-s_{11}s_{234})+(A^4-1)\beta t_1t_3(s_{12}s_{34}-A^{-4}s_{14}s_{23}) \\
=\ &(A^4-1)t_1(A^{-4}s_{23}s_{134}+A^8s_{34}s_{123}-A^{-4}s_{33}s_{124}) \\
&+(A^4-1)t_1t_3\big((A^4-A^8-1)\beta s_{12}s_{34}+(A^4+A^{-4}-A^8-2)s_{1234}\big) \\
&+t_3(s_{12}s_{134}-s_{13}s_{124}+s_{123}s_{14}-s_{11}s_{234}) \\
=\ &(A^4-1)\big(t_1(A^{-4}s_{23}s_{134}+A^8s_{34}s_{123}-A^{-4}s_{33}s_{124})+t_3(s_{11}s_{234}-s_{12}s_{134})\big) \\
&-(A^2-A^{-2})^2t_1t_3(A^8\beta s_{12}s_{34}+(A^8+3)s_{1234}).
\end{align*}
\end{proof}
\begin{figure}[h]
\centering
\includegraphics[width=10cm]{three}\\
\caption{Computing $s_{12}s_{1\check{3}\lrcorner2}$.}\label{fig:three}
\end{figure}
\begin{proof}[Proof of Proposition \ref{prop:intersection3}]
We have
\begin{align*}
A^2s_{12}s_{13}s_{23}&=A^2s_{12}(A^{-2}s_{1\check{3}\lrcorner2}+(s_{33}-A^{-2})s_{12}+t_3s_{123}) \\
&=s_{12}s_{1\check{3}\lrcorner2}+(A^2s_{33}-1)s_{12}^2+A^2t_3s_{12}s_{123}.
\end{align*}
Since $t_{123}=s_{123}-\beta(t_1s_{23}+t_2s_{13}+t_3s_{12})+\beta^2t_1t_2t_3$ and $s_{123}$ commutes with $t_1s_{23}+t_2s_{13}+t_3s_{12}$,
we have
\begin{align*}
&A^2s_{12}s_{13}s_{23} \\
=\ &t_{123}^2+A^4t_{23}^2-A^2s_{11}+A^2t_1t_{23}t_{123}+A^{-4}t_{13}^2+t_3^2-A^{-2}s_{11}+A^{-2}t_1t_3t_{13} \\
&+t_2(A^{-2}t_{13}t_{123}+A^2t_3t_{23}+\beta s_{11}t_2+t_1t_3t_{123}+\beta t_1s_{12}+\beta t_1s_{1\check{3}\lrcorner 2}) \\
&+(A^2s_{33}-1)s_{12}^2+A^2t_3s_{12}s_{123} \\
=\ &s_{123}^2-\beta(t_1(s_{123}s_{23}+s_{23}s_{123})+t_2(s_{123}s_{13}+s_{13}s_{123})+t_3(s_{123}s_{12}+s_{12}s_{123})) \\
&+2\beta^2t_1t_2t_3s_{123}+\beta^2(t_1^2s_{23}^2+t_2^2s_{13}^2+t_3^2s_{12}^2+t_1t_2(s_{23}s_{13}+s_{13}s_{23}) \\
&+t_1t_3(s_{23}s_{12}+s_{12}s_{23})+t_2t_3(s_{12}s_{13}+s_{13}s_{12})) \\
&-2\beta^3t_1t_2t_3(t_1s_{23}+t_2s_{13}+t_3s_{12})+\beta^4t_1^2t_2^2t_3^2 \\
&+A^4t_{23}^2-s_{11}s_{22}+(A^2t_1s_{23}+A^{-2}t_2s_{13})t_{123}+A^{-4}t_{13}^2+t_3^2+A^{-2}t_1t_3t_{13} \\
&+t_2(A^2t_3t_{23}+\beta t_1s_{12}+\beta t_1s_{1\check{3}\lrcorner 2})+(A^2s_{33}-1)s_{12}^2+A^2t_3s_{12}s_{123} \\
=\ &s_{123}^2-2\beta(t_1s_{23}+t_2s_{13}+t_3s_{12})s_{123}+2\beta^2t_1t_2t_3s_{123}+\beta^2(t_1^2s_{23}^2+t_2^2s_{13}^2+t_3^2s_{12}^2) \\
&+\beta^2t_1t_2(s_{23}s_{13}+s_{13}s_{23})+\beta^2t_1t_3(s_{23}s_{12}+s_{12}s_{23})+\beta^2t_2t_3(s_{12}s_{13}+s_{13}s_{12}) \\
&-2\beta^3t_1t_2t_3(t_1s_{23}+t_2s_{13}+t_3s_{12})+\beta^4t_1^2t_2^2t_3^2 \\
&+A^4(s_{23}^2-2\beta t_2t_3s_{23}+\beta^2t_2^2t_3^2)-s_{11}s_{22}+A^{-4}(s_{13}^2-2\beta t_1t_3s_{13}+\beta^2t_1^2t_3^2) \\
&+(A^2t_1s_{23}+A^{-2}t_2s_{13})(s_{123}-\beta(t_1s_{23}+t_2s_{13}+t_3s_{12})+\beta^2t_1t_2t_3) \\
&+t_3^2+A^{-2}t_1t_3(s_{13}-\beta t_1t_3)+A^2t_2t_3(s_{23}-\beta t_2t_3)+\beta t_1t_2s_{12} \\
&+\beta t_1t_2(A^2s_{13}s_{23}+(1-A^2s_{33})s_{12}-A^2t_3s_{123})+(A^2s_{33}-1)s_{12}^2+A^2t_3s_{12}s_{123} \\
=\ &s_{123}^2+(A^4-1)\beta\big(t_1s_{23}-A^{-4}t_2s_{13}+t_3s_{12}-\beta t_1t_2t_3\big)s_{123} \\
&+\beta(A^4s_{11}s_{23}^2+A^{-4}s_{22}s_{13}^2+A^4s_{33}s_{12}^2)-\beta s_{11}s_{22}s_{33} \\
&+(1-A^4)\beta^2(s_{11}t_2t_3s_{23}-A^{-4}s_{22}t_1t_3s_{13}+s_{33}t_1t_2s_{12}) \\
&+\beta^2t_1t_2((A^4+1-A^{-4})s_{13}s_{23}-A^4s_{23}s_{13})+\beta^2t_1t_3(s_{12}s_{23}-A^4s_{23}s_{12}) \\
&+\beta^2t_2t_3(s_{12}s_{13}-A^{-4}s_{13}s_{12}) \\
=\ &s_{123}^2+(A^4-1)\beta\big(t_1s_{23}-A^{-4}t_2s_{13}+t_3s_{12}+(A^2-A^{-2})^2\beta t_1t_2t_3\big)s_{123} \\
&+\beta(A^4s_{11}s_{23}^2+A^{-4}s_{22}s_{13}^2+A^4s_{33}s_{12}^2-s_{11}s_{22}s_{33}) \\
&-(A^2-A^{-2})^2\beta^2\big(s_{11}t_2t_3s_{23}+s_{22}t_1t_3s_{13}-A^4s_{33}t_1t_2s_{12}+A^2\alpha t_1t_2s_{13}s_{23}\big),
\end{align*}
where in the last equality we use
\begin{align*}
s_{23}s_{12}&=A^{-4}s_{12}s_{23}+(1-A^{-4})(s_{22}s_{13}+t_2s_{123}), \\
s_{13}s_{12}&=A^{4}s_{12}s_{13}+(1-A^{4})(s_{11}s_{23}+t_1s_{123}), \\
s_{23}s_{13}&=A^{4}s_{13}s_{23}+(1-A^{4})(s_{33}s_{12}+t_3s_{123}).
\end{align*}
\end{proof}
|
1,941,325,220,558 | arxiv | \section{Introduction}
Quantum mechanics was originally developed to account for a number of otherwise
unexplained phenomena in the microscopic domain. The scope of the theory
was not thought to extend beyond the microscopic. Indeed, the existence of an
external, macroscopic, classical domain was felt to be necessary for the
theory's interpretation. This view of quantum mechanics has persisted for a
very long time, with not one shred of experimental evidence against it.
Today, however, more ambitious views of quantum mechanics are entertained. The
experimental possibilities afforded by SQUIDS suggest that quantum coherence
may act on macroscopic scales (Leggett, 1980). Furthermore, the separation of
classical and quantum domains is often felt unnatural from a foundational point
of view. The domain of applicability of quantum mechanics may therefore need to
be extended. But in extrapolating quantum mechanics to the macroscopic scale,
where do we stop? At the scale of large molecules? At the laboratory scale? At
the planetary scale? There is only one natural place -- at the scale of the
entire universe. One rapidly arrives, therefore, at the subject of quantum
cosmology, in which quantum mechanics is applied to the entire universe. It
might only be through this subject that the form of quantum mechanics as we
know it may be fully understood.
Yet quantum cosmology was not originally developed with the foundational
problems of quantum mechanics in mind. Rather, it was perhaps viewed as a
natural area to investigate as part of a more general drive to understand
quantum gravity (DeWitt, 1967; Misner, 1969; Wheeler, 1963, 1968). More
recently, the success of models like inflation (Guth, 1981) in early universe
cosmology has led to a greater desire to understand the initial conditions with
which the universe began. On very general grounds, the universe appears to have
emerged from an era in which quantum gravitational effects were important.
Quantum cosmology, in which both matter and gravitational fields are taken to
be quantized, is therefore the natural framework in which to address questions
of initial conditions. Indeed, there has been a considerable amount of
activity in this subject in
recent years.\footnote{$^{\dag}$}{\eightpoint
For general reviews of quantum cosmology, see Fang and Ruffini (1987), Fang and
Wu (1986), Halliwell (1990, 1991), Hartle (1985, 1986, 1990), Hawking (1984b),
Hu (1989), Laflamme (1991), Page (1986, 1991).}
Much of this recent attention on quantum cosmology has been focused on
obtaining a crude idea of the physical predictions made by certain theories of
initial conditions. However, little attention has been devoted to understanding
or even clarifying the principles involved in extracting these predictions from
the mathematical formalism. This contribution represents a small step
towards filling this gap. That is, it is concerned with the {\it
interpretation} of quantum cosmology. In particular, I shall be addressing the
question, ``How are predictions extracted from a given wave function of the
universe?''. I will state immediately that there is no wholly satisfactory
answer to this question. The aim of this contribution, therefore, is to
describe the difficulties involved and the attempts to overcome them, the
procedures that are actually used in practice and the motivation behind them.
The interpretation of quantum cosmology raises two particular sets of issues
which may be discussed separately. The first set concerns the fact that, unlike
ordinary quantum mechanics, the system under scrutiny, the universe, is a
closed and isolated system. The special features of (non-relativistic) quantum
mechanics applied to such systems are discussed in the accompanying paper by
Hartle (Hartle, 1992a). For completeness, some of these features are also
covered here, but from a somewhat different angle (namely from the wave
function point of view, rather than that of the decoherent histories approach).
The second set of issues concerns the fact that the wave function for the
system is described not by a time-dependent Schr\"odinger equation, but by the
Wheeler-DeWitt equation. Associated with this is the so-called ``problem of
time'', and the absence of the usual machinery of projection operators, unitary
evolution, Hilbert space {\it etc.} It is this second set of issues that will
form the primary focus of this paper.
To focus the discussion, consider the sort of features of the universe
one might hope to predict in quantum cosmology. Some of the most important
are:
\item{$\bullet$} {\bf Spacetime is Classical.} One of the crudest and most
obvious observations about the universe we can make, is that the structure of
spacetime is described very accurately by classical laws. The very first
requirement of a quantum theory of the universe, therefore, is that it predict
this. The question of how classical behaviour emerges in quantum systems is a
difficult one, that has not been completely solved even in non-relativistic
quantum mechanics. Still, it ought be possible, at least in some crude sense,
to see whether a given wave function for the universe is consistent with a
prediction of classical spacetime.
\item{$\bullet$} {\bf Initial Inflationary Phase.} Many current observational
features of the universe, such as the horizon, flatness and monopole problems,
are explained by postulating an inflationary phase at very early times. In
models of the universe which admit inflationary solutions, the occurrence and
amount of inflation are generally dependent on initial conditions. A correct
quantum theory of initial conditions should supply the initial conditions for
an inflationary phase to take place.
\item{$\bullet$} {\bf Spectra of Density Fluctuations and Gravitational Waves}.
In inflationary models, it is possible to generate gravitational waves and
density fluctuations consistent with the observed isotropy of the microwave
background, yet sufficiently large for the formation of large scale structure.
These results are obtained by considering the quantum field theory of the
fluctuations during the inflationary phase. Again they are initial conditions
dependent, and one might hope that quantum cosmology will supply the requisite
initial conditions.
\item{$\bullet$} {\bf Low Entropy Initial State.} One of the most striking
features of the universe is its time asymmetry. It is very far from
equilibrium, indicating that it started out in a very special, low entropy
initial state. One would therefore hope to predict this very smooth beginning.
This is closely related to the spectra of fluctuations.
\noindent In what follows, we will not be concerned with detailed predictions
of particular theories of initial conditions. Rather, we will assume that we
are given a wave function for the universe (perhaps determined by {\it some}
theory of initial conditions), and discuss the extraction of predictions from
it. In particular, we will focus on the emergence of a {\it semiclassical
domain} -- an approximately classical spacetime background with quantum fields
in it\footnote{$^{\dag}$}{\eightpoint This is, loosely speaking, the same
thing as the {\it quasi-classical} domain defined by Gell-Mann and Hartle
(1990) in the context of the decoherent histories approach, but I give it a
slightly different name since the wave function approach considered here
does not permit it to be defined in quite the same way.}. This is
appropriate for two reasons. First, all observational features of the universe
are semiclassical in nature. Second, even if the theory could make observable
quantum gravitational predictions, as one would certainly hope of a quantum
theory of cosmology, observation of them would be through their correlation
with the semiclassical domain. Quite generally, therefore, the emergence of the
semiclassical domain is the appropriate thing to look for.
\section{Canonical Quantization}
In the interests of conciseness, I shall assume that the formalism of quantum
cosmology is known, and give only the briefest account here (see for example
Halliwell (1990), and the general
references cited earlier). For definiteness, we consider the canonical
quantization of Einstein gravity coupled to matter for closed universes. The
quantum state of the system, the universe, is represented by a wave
functional, $\Psi[h_{ij}, \phi]$, a functional on superspace, the space of
three-metrics $h_{ij}$ and matter fields $\phi$ on a three-surface. The wave
function has no explicit dependence on time. There is therefore no
Schr\"odinger equation, only the constraints of the Dirac quantization
procedure,
$$
\hat {\fam 2 H} \Psi[h_{ij}, \phi]= 0, \quad \quad \quad \hat {\fam 2
H}_i \Psi [h_{ij}, \phi]= 0
\eqno(2.1)
$$
where $\hat {\fam 2 H}$ and $\hat {\fam 2 H}_i$ are the operator versions of the
classical constraints,
$$
\eqalignno{
{\fam 2 H} &= h^{-{\frac 12}} \left(\pi^{ij} \pi_{ij} - {\frac 12} \pi^i_i \pi^j_j \right)
- h^{{\frac 12}} \left( \ ^3R- 2 \Lambda \right) + {\fam 2 H}^{matter} = 0 &(2.2) \cr
{\fam 2 H}_i &= -2 \pi_{ij}^{\ |j} + {\fam 2 H}_i^{matter} = 0 &(2.3) \cr }
$$
(in the usual notation). Eqs.(2.1) are of course the Wheeler-DeWitt equation
and the momentum constraints.
Wave functions satisfying the constraints (2.1) may also be generated using a
(complex) path integral representation (Halliwell and Hartle, 1990, 1991). This
is used, for example, in the specification of the no-boundary wave function
(Hawking, 1982, 1984a; Hartle and Hawking, 1983). The sum-over-histories
approach is more general in that it may be used to construct more complicated
amplitudes ({\it e.g.}, ones depending on the three-metric and matter fields on
many three-surfaces) and in fact the latter type of amplitude is likely to be
the most useful for interpreting the theory. Here we are concerned with
describing what has actually been done, and this means discussing single
surface amplitudes.
There are other very different ways of quantizing the classical system
described by the constraints, (2.2), (2.3). In particular, one could
contemplate solving the constraints classically, prior to quantization, and
then quantizing an unconstrained theory. The Dirac quantization scheme outlined
here is the one most commonly used in practical quantum cosmology, and it is
this that we shall use in what follows.
Of course, all known approaches to quantum gravity are fraught with severe
technical difficulties, and the formal framework outlined above is no
exception. Moreover, a proper theory of quantum gravity, should it exist, might
involve substantial departures from this general framework ({\it e.g.}, string
theory). However, there are a number of reasons why it might nevertheless make
sense to persevere with this approach to quantum cosmology. Perhaps the most
important is that in quantum cosmology one is frequently concerned with {\it
issues}. Many issues, and in particular the interpretational issues considered
here, are not very sensitive to the resolution of technical difficulties.
Furthermore, they are likely to be equally present in any approach to
quantum gravity. No generality is lost, therefore, in employing the formal
scheme envisaged in Eq.(2.1).
\section{Interpretation}
The question of interpretation is that of extracting physical statements about
the universe from the wave function, $\Psi[h_{ij}, \phi]$. In non-relativistic
quantum mechanics there is a well-defined procedure for achieving this. It is
generally known as the Copenhagen interpretation. Although it is frequently
regarded as problematic from a foundational point of view, it has been very
successful in its physical predictions.\footnote{$^{\dag}$}{\eightpoint
Many of the key papers on the Copenhagen interpretation may be found in Wheeler
and Zurek (1983).} The Copenhagen interpretation involves a number of features
and assumptions that should be highlighted for the purposes of this account:
\item{\bf C1.} It assumes the existence of an {\it a priori} split of the world
into two parts: a (usually microscopic) quantum system, and an external
(usually macroscopic) classical agency.
\item{\bf C2.} It concerns systems that are not genuinely closed, because they
are occasionally subject to intervention by the external agency.
\item{\bf C3.} The process of prediction places heavy emphasis on the notion of
{\it measurement} by the external agency.
\item{\bf C4.} Predictions are probabilistic in nature, and generally only have
meaning when measurements are performed on either a large ensemble of identical
systems, or on the same system many times, each time prepared in the same
state.
\item{\bf C5.} Time plays a very distinguished and central role.
\noindent Quantum cosmology, by contrast, involves a number of corresponding
features and assumptions that render the Copenhagen interpretation woefully
inadequate:
\item{\bf QC1.} It is assumed that quantum mechanics is universal, applying to
microscopic and macroscopic systems alike, up to and including the entire
universe. There can therefore be no {\it a priori} split of the universe into
quantum and classical parts.
\item{\bf QC2.} The system under scrutiny is the entire universe. It is a
genuinely closed and isolated system without exterior.
\item{\bf QC3.} Measurements cannot play a fundamental role, because there can
be no external measuring apparatus. Even internal measuring apparatus should
not play a role, because the conditions in the early universe were so extreme
that they could not exist.
\item{\bf QC4.} The universe is a unique entity. It does not belong to an
ensemble of identical systems; nor is it possible to make repeated measurements
on it prepared in the same state.
\item{\bf QC5.} The problem of time: general relativity does not obviously
supply the time parameter so central to the formulation and interpretation of
quantum theory.
\noindent Of these features, only (QC5) is specific to quantum gravity. The
rest would be true of any closed and isolated system described by
non-relativistic quantum mechanics, and may be discussed in this context.
The problem of interpretation is that of finding a scheme which confronts
features (QC1)--(QC5), yet which may be shown to be consistent with or to
reduce to (C1)--(C5) under suitable approximations and restrictions.
Probably the best currently available approach to these difficulties is the
decoherent histories approach, which employs not the wave function, but the
decoherence functional as its central tool.\footnote{$^{\dag}$}{\eightpoint The
consistent histories approach to quantum mechanics was introduced by Griffiths
(1984), and later developed by Omn\`es (reviewed in Omn\'es, 1990, 1992). Much
of it was developed independently under the name decoherent histories, by
Gell-Mann and Hartle (1990). See also Hartle (1990, 1992a). For applications
and generalizations, see Blencowe (1991), Albrecht (1990) and Dowker and
Halliwell (1992).} It has a number of features which strongly recommend it for
quantum cosmology: it specifically applies to closed systems; it assumes no
{\it a priori} separation of classical and quantum domains; it does not rely on
the notion of measurement or observation; and its focus on histories rather
than events at a single moment of time might sidestep (or at least alleviate)
the problem of time in quantum gravity. Unfortunately, a detailed application
of this approach to quantum cosmology has not yet been carried out (although
efforts in this direction are currently being made (Hartle, 1992b)).
Furthermore, my task here is to describe what has actually been done, and for
that reason I will describe the somewhat cruder approaches to interpretation
based on the wave function. However, I find it useful to remain close to the
spirit of the decoherent histories approach, and to think of the wave function
approach as an approximation to it (although in a sense yet to be explained).
In particular, it is convenient to have as one's aim the assignment of
probabilities to histories of the universe.
In order to deal with the issues noted above, conventional wave function-based
approaches invoke the Everett (or ``Many Worlds'')
interpretation.\footnote{$^{\dag\dag}$}{\eightpoint Everett (1957). A useful
collection of papers on the Everett interpretation may be found in DeWitt and
Graham (1973). See also Bell (1981), Deutsch (1985), Kent (1990), Smolin (1984)
and Tipler (1986).} Above all, the Everett interpretation is a scheme
specifically designed for quantum mechanical systems that are closed and
isolated. Everett asserted that quantum mechanics should be applicable to the
entire universe, and there should be no separation into quantum and classical
domains. These features of the Everett interpretation are therefore consistent
with features (QC1) and (QC2) of quantum cosmology. Furthermore, the state of
the entire system should evolve solely according to a wave equation, such as
the Schr\"odinger equation, or in quantum cosmology the Wheeler-DeWitt
equation, and there should be no discontinuous changes (collapse of the wave
function).
Everett went on to model the measurement process by considering a world divided
into a large number of subsystems, and showed how the conventional Copenhagen
view of quantum mechanics can emerge. It is this part of the Everett
interpretation that leads to its ``Many Worlds'' feature -- the idea that the
universe splits into many copies of itself whenever a measurement is performed.
It is at this stage, however, that the original version of the Everett
interpretation departs in its usefulness from practical quantum cosmology. For
the sort of models one is often interested in interpreting in quantum cosmology
are minisuperspace models, which are typically very simple, and do not contain
a large number of subsystems. Furthermore, the many worlds aspect of the
interpretation has, I believe, been rather over-emphasized, perhaps at the
expense of undermining the credibility of the overall set of ideas. The Everett
interpretation, especially as it applies to practical quantum cosmology, is not
so much about many worlds, but rather, about how one might make sense of
quantum mechanics applied to genuinely closed systems.
It is, therefore, convenient to pass to a restatement of the Everett
interpretation due to Geroch (1984). Geroch translated Everett's modeling of
the measurement process using a large number of subsystems into statements
about the wave function of the entire system. He argued that predictions for
closed quantum systems essentially boil down to statements about ``precluded''
regions -- regions of configuration space in which the wave function for the
entire system is very small. From here, it is a small step to a slightly more
comprehensive statement given by Hartle, specifically for quantum cosmology
(Hartle, 1986). It is the following:
\item{} {\it If the wave function for the closed system is strongly peaked
about a particular region of configuration space, then we predict the
correlations associated with that region; if it is very small, we predict the
lack of the corresponding correlations; if it is neither strongly peaked, nor
very small, we make no prediction}.
\noindent This basic idea appears to have been adopted,
perhaps provisionally, in most, if not all, attempts to interpret the wave
function in quantum cosmology (see also Wada (1988)).
The statement is admittedly rather vague and a number of qualifying remarks are
in order. Firstly, to say that a wave function is ``strongly peaked''
necessarily involves some notion of a measure, and this has to be specified.
For example, do we use $ | \Psi|^2 $? The Klein-Gordon current constructed from
$\Psi$? We will return to this below. Second, the interesting correlations in
the wave function are often not evident in the configuration space form. It is
therefore appropriate to interpret the above statement rather broadly, as
refering to not just the wave function, but any distribution constructed from
the wave function. For example, phase space distributions such as the Wigner
function, and related functions, have been the focus of attention in a number
of papers, and these have sometimes proved quite useful in identifying
classical correlations (Anderson, 1990; Calzetta and Hu, 1989; Habib, 1990;
Habib and Laflamme, 1990; Halliwell, 1987, 1992; Kodama, 1988; Singh and
Padmanabhan, 1989).
For practical quantum cosmology, the implication of the above interpretational
scheme is that rather than find the probability distributions for quantities
felt to be of interest, as in ordinary quantum mechanics, it is necessary to
determine those quantities for which the theory gives probabilities close to
zero or one, and hence, for which it makes predictions. On the face of it, this
may seem to suggest that the predictive power of quantum cosmology is very
limited in comparison to ordinary quantum mechanics. However, by studying
isolated systems consisting of a large number of identical subsystems, it may
be shown that this interpretation implies the usual statistical interpretation
of ordinary quantum mechanics, in which subsystem probabilities are given by
$|\psi|^2$ where $\psi$ is the subsystem wave function (Farhi, Goldstone and
Gutmann, 1989; Finkelstein, 1963; Graham, 1973; Hartle, 1968). It is in this
sense that feature (QC4) of quantum cosmology is reconciled with (C4) of the
Copenhagen interpretation.
Given a measure on a set of possible histories for the universe, it is often
not peaked about a particular history or family of histories. To obtain
probabilities close to one or zero, it is often necessary to restrict attention
to a certain subset of the possible histories of the universe, and make
predictions within that subset. That is, one looks at {\it conditional
probabilities}. The motivation behind this is anthropic reasoning -- we as
observers do not look out into a generic universe, but to one in which the
conditions for our own existence have necessarily been realized (see, for
example, Barrow and Tipler, 1986). Obviously the detailed conditions for our
existence could be very complicated and difficult to work out. However, it is
possible to get away with exploiting only the weakest of anthropic assumptions
in order to make useful and interesting predictions about the universe. It is,
for example, extremely plausible that the existence of life requires the
existence of stars like our sun. It therefore makes sense to restrict attention
only to those histories of the universe which exist long enough for stars to
form before recollapsing.
This completes our brief survey of the special features of closed quantum
systems. We have discussed features (QC1), (QC2) and (QC4) of quantum
cosmology. The status of measurements, (QC3), in the above discussion is
admittedly somewhat vague, although it is clear that they do not play a
significant role. Their status is perhaps clarified more fully in the
decoherent histories approach. Finally, although it is an important issue for
the interpretation of quantum cosmology, I have not discussed the problem of
time, (QC5). This will be briefly mentioned below, but it would take a lot of
space to do it justice. The interested reader is referred to the reviews by
Kucha{\v r} (1989, 1992), which cover the problem of time and its connections
with the interpretation of quantum cosmology. See also Unruh and Wald (1989).
\section{Interpretation of Solutions to the Wheeler-DeWitt Equation}
We now come to the central part of this contribution, which is to describe how
in practice predictions are actually extracted from solutions to the
Wheeler-DeWitt equation. As mentioned earlier, the primary aim is to determine
the location and features of the semiclassical domain. Different papers in the
literature treat the process of prediction in different ways, but it seems to
boil down to four distinct steps, which I now describe in turn.
\vskip 0.15in
\noindent {\bf A. Restriction to Perturbative Minisuperspace.}
\vskip 0.05in
\noindent The full theory described by Eq.(2.1) is not only difficult to handle
technically, it is not even properly defined. This problem is normally avoided
by artificially restricting the fields to lie in the region of superspace in
the neighbourhood of homogeneity (and often isotropy). That is, one restricts
attention to the finite dimensional subspace of superspace called
``minisuperspace'', and considers small but completely general inhomogeneous
perturbations about it.\footnote{$^{\dag}$}{\eightpoint This general approach
has been the topic of many papers, including Banks (1985), Banks,
Fischler and Susskind (1985), Fischler, Ratra and Susskind (1986), Halliwell
and Hawking (1985), Lapchinsky and Rubakov (1979), Shirai and Wada (1988),
Vachaspati and Vilenkin (1988), Wada (1986).}
In more detail, the sort of restrictions entailed are as follows. One restricts
the three-metric and matter fields to be of the form,
$$
h_{ij}({\bf x},t) = h^{(0)}_{ij} (t) + \delta h_{ij} ({\bf x},t ), \quad \Phi({\bf x}, t) =
\phi(t) + \delta \phi ({\bf x}, t)
\eqno(4.1)
$$
Here, the minisuperspace background
is described by the homogeneous fields $ h^{(0)}_{ij} $ and $\phi(t)$. For
example, the three-metric could be restricted to be homogeneous and isotropic,
described by a single scale factor $a$. We will denote the minisuperspace
background coordinates by the finite set of functions $q^{\alpha}(t)$, where
$\alpha=1,\cdots n $. $\delta h_{ij}$ and $\delta \phi $ are small inhomogeneous
perturbations about the minisuperspace background, describing gravitational
waves and scalar field density perturbations. They are retained only up to
second order in the action and Hamiltonian (and therefore to first order in the
field equations). For convenience, we will denote the perturbation modes simply
by $\delta \phi$.
With these restrictions on the class of fields considered, the Wheeler-DeWitt
equation, after integration over the three-surface, takes the form $$ \left[ -
{1 \over 2 m_p^2 } \nabla^2 + m_p^2 U(q) + H_2 ( q, \delta \phi ) \right] \Psi
(q, \delta \phi ) = 0 \eqno(4.2) $$ Here, $ \nabla^2 $ is the Laplacian
operator in the minisuperspace modes $q$ and we have explicitly included the
Planck mass $m_p$, since it is to be used as a large parameter in a
perturbative
expansion. $H_2$ is the Hamiltonian of the perturbation modes, $\delta \phi$,
and is quadratic in them. There are more constraint equations associated with
the remaining parts of the Wheeler-DeWitt equation, and with the momentum
constraints. These are all linear in the perturbations and can be solved,
after gauge-fixing. When this is done, only Eq.(4.2) remains, in which $\delta
\phi$ may be thought of as a gauge-invariant perturbation variable.
The restriction to perturbative minisuperspace is of course very difficult to
justify from the point of view of the full theory. A number of attempts have
been made to understand the sense in which minisuperspace models might be part
of a systematic approximation to the full theory, but the answer seems to be
that their connection is at best tenuous (Kucha{\v r} and Ryan, 1986, 1989).
What one can say, however, is that solutions to the minisuperspace field
equations, including perturbations about them, will (with a little care) be
solutions to the full field equations, and thus the lowest order semiclassical
approximation to perturbative minisuperspace quantum cosmological models will
agree with the lowest order semiclassical approximation to the full theory.
These models may therefore be thought of as useful models in which a number of
issues can be profitably investigated, but which may also give some crude
predictions about the physical universe.
The essential ideas of the practical interpretational scheme described here
will very probably be applicable to situations more general than perturbative
minisuperspace, but very little work on such situations has been carried out.
We will not go into that here.
\vskip 0.15in
\noindent {\bf B. Identification of the Semiclassical Regime.}
\vskip 0.05in
\noindent The next step involves inspecting the wave function $\Psi(q^{\alpha},
\delta \phi)$, asking how it behaves as a function of the minisuperspace
variables $q$, and in particular, identifying the regions in which the wave
function is exponentially growing or decaying in $q$, or oscillatory in $q$.
The regions in which the wave function is rapidly oscillating in $q$ are
regarded as the semiclassical domain, in which the modes $q$, are approximately
classical, whilst the perturbation modes $\delta \phi$ need not be. This
interpretation comes partly from analogy with ordinary quantum mechanics. But
also one can often argue that certain distributions constructed from a rapidly
oscillating wave function are peaked about classical configurations. For
example, the Wigner function, mentioned earlier, is often used to support this
interpretation.
The other regions, in which the wave function tends to be predominantly
exponential in behaviour are regarded as non-classical, like the
under-the-barrier wave function in tunneling situations. Were this ordinary
quantum mechanics, then the wave function would typically be exponentially
small in the tunneling regions. One could then invoke Geroch's version of the
Everett interpretation, and just that the system will not be found in this
region, because it is ``precluded''. However, a peculiar feature of gravity
(readily traced back to the indefiniteness of the action) is that the wave
function may be either exponentially small or exponentially large in the
regions where it is of exponential form. Nevertheless, one still says that the
system is not approximately classical in this regime, even when the wave
function is exponentially large. To support this claim, one can argue that, in
contrast to the oscillatory regions, a predominantly exponential wave function,
either growing or decaying, is {\it not peaked} about classical configurations.
\vskip 0.15in \noindent {\bf C. WKB Solution in the Oscillatory Regime.} \vskip
0.05in
\noindent The next stage of the scheme involves solving in more detail in the
region in which the wave function is rapidly oscillating in the minisuperspace
variables. This involves focusing on a particular type of state, namely the WKB
state, $$ \Psi(q, \delta \phi)= C (q) \ \exp\left(im_p^2 S_0(q) \right)\
\psi(q,\delta \phi) + O(m_p^{-2}) \eqno(4.3) $$ $S_0(q)$ is real, but $\psi$
may be complex. Many models also involve a slowly varying exponential prefactor
contributing at order $m_p^2$, but for the purposes of the present discussion
it can be assumed that this is absorbed into $C$. Also, a possible phase at
order $m_p^0$ depending only on $q$ may be absorbed into $\psi$, so $C$ may be
taken to be real.
Eq.(4.3), it must be stressed, is an {\it ansatz} for the solution, and many of
the predictions subsequently derived depend on assuming this particular form.
We will return later to the question of the validity and usefulness of this
particular ansatz.
The Wheeler-DeWitt equation may now be solved perturbatively, by inserting the
ansatz (4.3), and using the Planck mass $m_p$ as a large parameter. Since this
parameter is not dimensionless, the expansion is meaningful only on length
scales much greater than the Planck length. Note also that a double expansion
of the full Wheeler-DeWitt equation is involved: a WKB expansion in the Planck
mass, and a perturbation expansion in small inhomogeneities about
minisuperspace.
Equating powers of the Planck mass, one obtains the following. At lowest order,
one gets the Hamilton-Jacobi equation for $ S_0 $, $$ {1\over 2} (\nabla
S_0)^2+U(q)=0. \eqno(4.4) $$ To next order one obtains a conservation equation
for $C$ $$ 2 \nabla S_0 \cdot \nabla C - C \nabla^2 S_0 = 0 \eqno(4.5) $$ and a
Schr\"odinger equation for $\psi$, $$ i \nabla S_0 \cdot \nabla \psi = H_2 \psi
\eqno(4.6) $$
Consider now (4.4). As indicated above, it may be argued that a wave function
predominantly of the form ${\e}^{im_p^2 S_0}$ indicates a strong correlation
between coordinates and momenta of the form $$ p_{\alpha} = m_p^2 { \partial S_0
\over \partial q^{\alpha} } \eqno(4.7) $$ Since $ \dot q^{\alpha} = p^{\alpha}$, (4.7) is a
set of $n$ first order differential equations (where, recall, $\alpha=1,\cdots n$).
Furthermore, since by (4.4), $S_0$ is a solution to the Hamilton Jacobi
equation, it may be shown that these equations define a first integral to the
classical field equations. The first integral (4.7) may be solved to yield an
$n$-parameter set of classical solutions. It is for this reason that one says
that the wave function (4.3) to leading order, corresponds to an ensemble of
classical solutions to the field equations.
In ordinary quantum mechanics, this interpretation may be substantiated by
subjecting an initial wave function of the form (4.3) to a sequence of
approximate position samplings, and showing that the resulting probability
distribution for histories is peaked about the set of classical solutions
satisfying the first integral (4.7). See Halliwell and Dowker (1992), for
example, for efforts in this direction.
The tangent vector to this congruence of classical paths is $$ \nabla S_0 \cdot
\nabla \ \equiv \ {\partial \over \partial t} \eqno(4.8) $$ (4.6) is therefore
the functional Schr\"odinger equation for the perturbation modes along the
family of minisuperspace trajectories. This indicates that the perturbation
modes are described by quantum field theory (in the functional Schr\"odinger
picture) along the family of classical backgrounds described by (4.7).
\vfill\eject
\vskip 0.15in \noindent {\bf D. The Measure on the Set of Classical
Trajectories} \vskip 0.05in
\noindent The final stage is to put a measure on the congruence of classical
paths, and to find an inner product for solutions to the Schr\"odinger equation
(4.8). Suppose one chooses some $(n-1)$-dimensional surface in minisuperspace
as the beginning of classical evolution. Through (4.7), the wave function then
effectively fixes the initial velocities on that surface, in terms of the
coordinates. However, the wave function should also provide a probability
measure on the set of classical trajectories about which the wave function is
peaked. How is this measure constructed from the wave function? The
construction of a satisfactory non-negative measure remains an outstanding
problem of quantum cosmology, but perhaps the most successful attempts so far
involve the Klein-Gordon current,
$$
J = {i \over 2} \left( \Psi \nabla \Psi^*
- \Psi^* \nabla \Psi \right)
\eqno(4.9)
$$
It is conserved by virtue of the
Wheeler-DeWitt equation (4.2), $$ \nabla \cdot J = 0 \eqno(4.10) $$ Choose a
family of surfaces $\{ \Sigma_{\lambda} \}$, parametrized by $\lambda$, cutting
across the flow of $J$. Then (4.10) suggests that for each $\lambda$, a
probability measure on the congruence of trajectories is the flux of $J$ across
the surface: $$ dP = J \cdot d \Sigma \eqno(4.11) $$ This measure is conserved
along the flow of $J$, as is readily shown from (4.10).
The problem with (4.11), however, is that it is not always positive. For
example, if the surfaces $\Sigma$ are taken to be surfaces of constant scale
factor, the flow of $J$ typically cuts these surfaces more than once, because
of the possibility of expanding and collapsing universes, leading to negative
values for (4.11). Furthermore, $J$ vanishes when $\Psi$ is real. Still, some
sense may be made out of (4.11) by restricting to the WKB wave functions,
(4.3). For these wave functions, the current is $$ J \approx m_p^2 \ | C|^2 \ |
\psi|^2 \ \nabla S_0 + O (m_p^0) \eqno(4.12) $$ For reasonably large regions of
minisuperspace (but not globally), it is usually possible to choose a set of
surfaces $\{ \Sigma_{\lambda} \} $ for which $ \nabla S_0\cdot d \Sigma \ge 0
$, and thus the probability measure will be positive. Furthermore, this measure
implies the standard $|\psi|^2$ measure for the perturbation wave functions,
completing the demonstration that quantum field theory for the perturbations
emerges in the semiclassical limit.
This approach was described a long time ago by Misner (1972), and developed
more recently by Vilenkin (1989). It is problematic for a number of reasons:
one is the global problem of choosing the surfaces $\{\Sigma_{\lambda}\}$;
another is that is very tied to the WKB wave functions (4.3). More will be said
about this in the next section. Still, it allows predictions to be made in a
number of situations of interest.
The ``naive'' Schr\"odinger measure is also sometimes proposed in place of
(4.11) ({\it e.g.}, Hawking and Page, 1986, 1988). This is the assertion that
the probability of finding the system in a region of superspace of volume $dV$
is $| \Psi |^2 dV $, where $\Psi$ is the wave function of the whole system.
This does at least have the advantage that is is everywhere positive.
Furthermore, it can be argued to reduce to (4.11) for WKB wave functions, in
the limit in which the volume of superspace $dV$ is taken to be hypersurface of
codimension one slightly thickened along the direction of the flow of $J$. As
argued by Kucha{\v r}, however, this measure is problematic for other reasons
(Kucha{\v r}, 1992).
At this stage, it is appropriate to comment on the problem
of time. In the Wheeler-DeWitt equation (4.2) (or (2.1)), there is no
distinguished variable that plays the role of time. This is
the problem of time. In the scheme described above, however, a time parameter
has emerged. It is the parameter $\lambda$ labeling the family
of surfaces $\{\Sigma_{\lambda} \}$, and may be chosen to be the same as the
parameter $t$ defined in Eq.(4.8), the affine parameter along the integral
curves of $\nabla S_0$. The point to be made is that this parameter has emerged
only in the region where the wave function is oscillatory, and in particular,
as a consequence of the assumed semiclassical form of the wave function, (4.3).
This is therefore an explicit illustration of the point of view, not uncommonly
expressed, that time, and indeed spacetime, need not exist at the most
fundamental level but may emerge as approximate features under some suitable
set of conditions. It is in this sense that feature (QC5) of quantum cosmology
may be reconciled with (C5) of ordinary quantum mechanics.
Modulo the above difficulties, Eq.(4.11) is the desired probability measure on
possible histories of the universe. It is commonly not normalizable over the
entire surface $\Sigma$; but this need not matter, because it is {\it
conditional} probabilities that one is typically interested in. Suppose, for
example, one is given that the history of the universe passed through a subset
$s_1$ of a surface $\Sigma$, and one wants the probability that the universe
passed through a subset $s_0$ of $s_1$. The relevant conditional probability
is, $$ p(s_0|s_1) = { \int_{s_0} J \cdot d \Sigma \over \int_{s_1} J \cdot d
\Sigma } \eqno(4.13) $$ The set of all histories intersecting $\Sigma$ could
include universes which recollapse after a very short time. A reasonable choice
for $s_1$, therefore, might be universes that exist long enough for stars to
form before recollapsing, as discussed earlier. $s_0$ could then be taken to be
the subset of such universes which possess certain features resembling our
universe. If the resulting conditional probability turned out to be close to
one or zero, this would then constitute a definite prediction.
Steps (A)--(D) above constitute the general interpretational scheme implicit or
explicit in most attempts to extract predictions from the wave function of the
universe. It is not by any means a consistent interpretational scheme, but is
almost a list of rules of thumb inspired by the Everett interpretation, and
built on analogies with ordinary quantum mechanics. It has many difficulties,
some of which we now discuss.
\section{Problems and Objections}
We have argued that the WKB wave functions (4.3) correspond to an ensemble of
classical paths defined by the first integral (4.7). Strictly speaking, what
this means is that the WKB wave function is really some kind of {\it
superposition} of wave functions, each of which corresponds to an individual
classical history (like a superposition of coherent
states).\footnote{$^{\dag}$}{\eightpoint For the explicit construction of wave
packets in quantum cosmology, see Kazama and Nakayama (1985) and Kiefer
(1988).} A closely related point is the question of why one should be allowed
to study an interpretation based on the WKB form, (4.3): one would expect a
more general wave function to be expressed as a {\it sum} of such terms. In
each of these cases, we are acting as if we had a classical statistical
ensemble, when what we really have is a superposition of interfering states.
Why should it be permissible to treat each term in the sum separately, when
strictly they are interfering?
This point concerns the general question of why or when it is permissable to
ignore the interference terms in a superposition, and treat each term as if it
were the member of a statistical ensemble. Technically, the destruction of
interference is generally referred to as decoherence.
Two notions of decoherence have been employed. The most precise is that of the
decoherent histories approach, where it enters at a very fundamental level:
Interference is most generally and properly thought of as the failure of the
probability sum rules for quantum-mechanical histories. Decoherence, as
destruction of interference, is therefore best regarded as the recovery of
these rules (Gell-Mann and Hartle, 1990).
By contrast, in the wave function approach to quantum cosmology, decoherence
appears to have been added as an afterthought, using a different and somewhat
vaguer definition: decoherence is held to be related to the tendancy of the
density matrix towards diagonality (Joos and Zeh, 1985). It is also associated
with the establishment of correlations of a given system with its environment,
and with the stability of certain states under evolution in the presence of an
environment (Zurek, 1981, 1982; Unruh and Zurek, 1989). These definitions are
problematic for a number of reasons. One is that (in ordinary quantum
mechanics) the density matrix refers only to a single moment of time, yet the
proper definition of interference -- the effect one is trying to destroy -- is
in terms of histories. Another is the question of the basis in which the
density matrix should be diagonal.\footnote{$^{\dag}$}{\eightpoint The basis
issue is discussed, for example, in Barvinsky and Kamenshchik (1990),
Deutsch (1985), and Markov and Mukhanov (1988). Also, see Zurek (1992) for a
possible reconciliation of the above two differing views of decoherence.}
Despite the difficulties, the density matrix approach has been the topic of a
number of papers on decoherence in quantum cosmology, perhaps because it is
technically much simpler (Fukuyama and Morikawa, 1989; Habib and Laflamme,
1990; Halliwell, 1989; Kiefer, 1987; Laflamme and Louko, 1991; Mellor, 1989;
Morikawa, 1989; Padmanabhan, 1989; Paz, 1991; Paz and Sinha, 1992). Models are
considered in which the variables of interest are coupled to a wider
environment, and a coarse-graining is carried out, in which the states of the
environment are traced over. In the case of the whole universe, which strictly
has no environment, this is achieved quite simply by postulating a sufficiently
complex universe with a suitably large number of subsystems, and ignoring
some of the subsystems. The typical result of such models is that the
interference terms are suppressed very effectively. Furthermore, one and the
same mechanism of coarse-graining also causes decoherence in the
histories-based definition of the process. It is therefore plausible that a
more sophisticated analysis using the decoherent histories approach will lead
to the same conclusions. One way or another, one finds some amount of
justification for treating the terms in a superposition separately, and
treating the set of paths to which a WKB wave function correspond as a
statistical ensemble. These arguments have not gained universal acceptance,
however (see for example, Kucha{\v r} (1992)).
After the presentation of this contribution, J.Ehlers asked about the
observational status of quantum cosmology. Since quantum cosmology aspires to
make observable predictions, this is obviously a very important question.
Interest in quantum cosmology arose partly as a result of the realization that
conventional classical cosmological scenarios relied on certain (possibly
tacit) assumptions about initial conditions. As is well known, the inflationary
universe scenario alleviates the hot big bang model of extreme dependence on
initial conditions; but it does not release it from {\it all} dependence. The
amount of inflation, the details of the density fluctuations generated, and
indeed, the very occurrence of inflation are initial conditions-dependent. One
of the main successes of quantum cosmology (modulo the objections described in
this paper) has been to demonstrate that the desired initial conditions for
inflation are implied by certain simple boundary condition proposals for the
wave function of the universe. This might therefore be regarded as an
observational prediction, although it is admittedly a very indirect one.
More direct tests will clearly be very difficult. The universe has gone through
many stages of evolution, each of which is modeled separately. In observing the
universe today, it is difficult to distinguish between effects largely due to
initial conditions and those largely due to dynamical evolution or to the
modeling of a particular stage. What is needed is an effect produced very early
in the history of the universe, but that is largely insensitive to subsequent
evolution. Grishchuk (1987) has argued that gravitational waves might be the
sought-after probe of the very early universe. Boundary condition proposals for
the wave function of the universe typically make definite predictions about the
initial state of the graviton field ({\it e.g.}, Halliwell and Hawking, 1985).
Memory of this initial state could well be preserved throughout the subsequent
evolution of the universe, because gravitational waves interact so weakly.
Parts of their spectra observable today might therefore contain signatures of
the initial state, leading to the exciting possibility of distinguishing
observationally between different boundary condition proposals. These ideas are
of course rather speculative, and gravitational wave astronomy is still in a
very primitive state. Still, quantum cosmology suffers from an acute lack of
connections observational cosmology, and any potentially observable effect
deserves further study.
L.Smolin commented that the scheme described here is very semiclassical in
nature, and suggested that it is perhaps not much more than a classical
statistical theory. It is certainly true that it is very semiclassical in
nature, that its predictive output has the form of classical statistical
theory, and perhaps causes one to wonder how much of it is really
quantum-mechanical in nature. However, there {\it are} some genuinely
quantum-mechanical aspects to this predictive scheme. Perhaps the principle one
is the prediction of regions in which classical laws are not valid (the regions
of superspace in which the wave function is exponential). Determination of the
existence and location of these regions requires the quantum theory -- their
existence and location cannot be anticipated by inspection of the classical
theory alone. Furthermore, the existence of these regions underscores the
necessity of discussing the issue of initial or boundary conditions from within
the context of the quantum theory. For in classical theories of initial
conditions (including classical statistical ones), one might be attempting to
impose classical initial conditions in a region in which classical laws are
quite simply not valid.
Finally, a critical appraisal of the quantum cosmology program (which
inspired some of the remarks made here) may be found in Isham (1991).
\section{Conclusions}
In this talk I have described the heuristic set of rules that have been used so
far to make crude but plausible predictions in quantum cosmology. These rules
are, however, rather heuristic and semiclassical in nature. The interpretation
of the wave function seems to proceed on a case by case basis, and no
satisfactory scheme for a completely general wave function is available. I am
therefore forced to conclude that quantum cosmology does not yet possess an
entirely satisfactory scheme for the extraction of predictions from
the wave function.
At the present time, the decoherent histories approach appears to offer the
most promising hope of improving the situation. A reasonable expectation is
that the heuristic interpretational techniques described here will emerge from
this more sophisticated approach. Detailed demonstration of this assertion is
very much a matter of current research.
\ack
I thank Jim Hartle and Raymond Laflamme for very useful discussions on the
material of this talk. I am very grateful to the organizers of GR13, and in
particular to Carlos Kozameh and Bob Wald, for inviting me to take part in this
stimulating meeting. I would also like to thank J\"urgen Ehlers for initiating
the session on Interpretational Issues in Quantum Cosmology, and for his
thought-provoking questions. Finally, I am very grateful to Mario Castagnino,
and his colleagues in Buenos Aires and Rosario, for their warm hospitality
during my stay in Argentina.
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\end
|
1,941,325,220,559 | arxiv | \section*{Introduction}
Since Poincare's pioneering work, we have had over a hundred years of the profound theory of dynamical systems that allows analyzing tractable models. With the advancement of technology, our ability to record rich and complex data, and forecast it has increased profoundly. Often, machine learning algorithms can accomplish a prediction task rather than build a model to explain the dynamics behind the data, and hence have poor long-term consistency. From a philosophical outlook of science, finding meeting points between rationalism (dynamical systems theory) and empiricism (machine learning) could yield important results in modeling.
When observed data originates from a dynamical system, we solve a theoretical problem in data-driven forecasting that entails equations from data and exceptionally long-term consistent models.
Performing experimental measurements
on biological, physical, and artificial systems to obtain
a more informative dynamical model has been well established in modern-day science. The practical purpose of obtaining high fidelity models is not just to minimize the point-wise prediction error. Although better forecasts do help better manage service interruptions and resource management, a model with long-term consistency concerning time-averaged characteristics is useful for understanding the response to changes in the statistical properties of the perturbations of models that have potentially distinguished applications, for instance in climate science (e.g., \cite{majda2010high,ghil2020physics} and references therein). Attempts made to build model equations from complex data such as sunspot cycles date back to the 1920s in \cite{yule1927vii}, but a major breakthrough came through the Takens embedding theorem \cite{takens1981detecting} that
guaranteed an equivalent state-space reconstruction, a valuable tool for any kind of analysis. Often, experimental measurements of dynamical systems are not directly the system's states, but a univariate time-series whose span is smaller than the underlying system's dynamics. The Takens embedding theorem and their various generalizations (e.g., \cite{sauer1991embedology,stark1999delay,robinson2010dimensions,gutman2018embedding,gutman2018embedding}), under some generic conditions, establish the learnability of a system that is created out of concatenating sufficiently large previous observations of a dynamical system into a vector (called delay coordinates). The system determined by such vectors is equivalent (topologically conjugate) (e.g., \cite{de2013elements}) to the system from which the observed time-series was derived. While topological conjugacy ensures an alternate representation of the underlying system, the quality of the representation remains sensitive to various parameters making learning the dynamics unreliable (e.g., \cite{principe1992prediction}). One of the reasons for this fragility is that the embedded attractor in the reconstruction space is, without doubt, an invariant set, but not necessarily an attractor. For learning the map on the invariant set, it is desirable to have locally conservative but globally dissipative dynamics i.e., there is an attractor containing the invariant set so that small errors do not lead to the future iterates completely veering off from the invariant set before predictions totally fail \cite[Fig. 1 and Fig. 2]{Supp}.
Theoretical conditions under which the geometry of the attractor can be preserved (e.g.,\cite{eftekhari2018stabilizing}) are not adequate for attractor reconstruction in practice.
Moreover, global approximation techniques that find a single map to fit the data often work well only when the data can be fit by functions with low functional complexity, i.e., functions that have relatively fewer oscillatory graphs.
Recently, machine learning algorithms like reservoir computing \cite{jaeger2004harnessing, lu2018attractor} and data-driven algorithms based on Koopman Operator (e.g.,\cite{koopman1932dynamical,budivsic2012applied}) have shown greater quantitative accuracy in forecasting dynamical systems.
While the reservoir computing methods fail to establish a learnable map often yielding poor consistency \cite{herteux2020breaking,Manju_IEEE,wu2018statistical, faranda2019boosting}, the Koopman theory and other data-driven based approaches suffer from either lack of determining the right observables \cite{schmid2010dynamic,williams2015data} or demand full knowledge of the underlying state space variables \cite{brunton2016discovering,champion2019data} that govern the underlying system or data obtained at high sampling-rates. Attempts of combining principles of such data-driven methods with delay embedding \cite{brunton2017chaos,champion2019discovery} are yet to theoretically guarantee exact reconstruction.
Since studies are suggestive that with large enough time-lagged observations
\cite{eftekhari2018stabilizing, yair2017reconstruction} one gets closer to isometric embeddings,
here we determine observables for the Koopman operator of the inverse-limit system \cite{ingram2011inverse} instead of the actual system. Remarkably, such observables need not be changed when the underlying dynamical system is changed. The methodology provides exact reconstruction (or finite-faithful representation \cite{mezic2020koopman}), global dissipativity while forecasting, and equations from data. We demonstrate long-term topological
consistency (attractor learning)
and statistical consistency (density of the orbits) of the models obtained through data from standard benchmark chaotic systems and also on systems that show intermittency -- Type I intermittent systems exhibit bursts, and signify great sensitivity to the distant past much more than the immediate past and is a feature of transition to turbulence and convection in confined spaces \cite{pomeau1980intermittent}, seismic data \cite{bottiglieri2007off} and anomalous diffusion in biology \cite{klages2013weak}, and are extremely difficult to model from data. Our approach is to learn the action of the Koopman operator on observables of the inverse-limit system of the underlying dynamical system so that the observables depend on past history. We determine these observables by a specific kind of driven dynamical system and using the given data from the underlying system.
\section*{Causal Embedding and Learning the Koopman Operator}
A dynamical system in this work is a tuple $(U,T)$ where $U$ is a compact metric space and $T: U \to U$ is a surjective function (that is not necessarily continuous). In fact, if $A$ is any invariant set, i.e., if $T(A):=\cup_{u\in A} Tu = A$, then $T$ is surjective on $A$, so if $T$ is not surjective to start with we can restrict the non-transient dynamics to an invariant set. We call a bi-infinite sequence $\bar{u} = \{u_n\}_{n\in \mathbb{Z}}$ that obeys the update equation, $u_{n+1}=Tu_n$ where $n \in \mathbb{Z}$ as an orbit/trajectory of $T$, and since $T$ is surjective there is at least one orbit through each point in $U$. Throughout, we say that a space $Z$ is embedded in $X$ if there exists $Y\subset X$ and a function $f$ so that $f:Z \to Y$ is a homeomorphism.
The problem of forecasting a dynamical system involves predicting $u_{m+1},u_{m+2},\ldots$ given the finite segment of an orbit $u_{0},u_{1},\ldots,u_{m}$ of an unknown $T$. By a problem of learning a dynamical system from data, we mean learning model equations that can be used to regenerate and forecast the data. The model equations could involve other hidden or auxiliary variables. A core concept in dynamical systems theory is the notion of equivalent dynamical systems.
Finding an equivalent dynamical system to $(U,T)$ means finding another dynamical system $(V,S)$ so that there exists a homeomorphism $\phi:U \to V$ with the property that $\phi \circ T=S\circ \phi$. Such a map $\phi$ is called a conjugacy and we say that $(V,S)$ is conjugate to $(U,T)$ or simply $S$ is conjugate to $T$. If we relax the condition on $\phi$ where, instead of having a homeomorphism, we only require $\phi$ to be continuous, then we call $\phi$ a semi-conjugacy, and say that $S$ is semi-conjugate to $T$. When $S$ is conjugate to $T$ it means that there is a one-to-one correspondence (e.g., \cite{de2013elements}) in the dynamics between the two systems, whereas when $S$ is semi-conjugate to $T$ with $\phi$ a many-to-one mapping then $(V,S)$ provides a coarse-grained description of the $(U,T)$. When $S$ is semi-conjugate to $T$ then it is customary to say that $S$ is a factor of $T$ or that $T$ is an extension of $S$. In essence, an extension is a larger system that captures all the important dynamics of its factor (e.g., \cite{de2013elements}).
Given a segment of an orbit $(U,T)$, our approach is to build model equations of a system that is topologically conjugate to an extension of $(U,T)$.
To define such an extension,
we first let the left-infinite countable cartesian product of $U$ be denoted by $\overleftarrow{U} := \cdots \times U \times U$ (and equip it with the product topology). Any dynamical system $(U,T)$ (since we assume $T$ is surjective) determines a nonempty subspace $\widehat{U}_T$ of $\overleftarrow{U}$ comprising the left-infinite orbits, i.e., $$
\widehat{U}_T: = \{(\ldots,u_{-2},u_{-1}) : Tu_{n} = u_{n+1} \},
$$
and $\widehat{U}_T$ is called the inverse-limit space of $(U,T)$ in the literature (e.g., \cite{ingram2011inverse}); in the literature, the elements of $\widehat{U}_T$ are rather written as right-infinite sequences. The map $T$ also induces a self-map $\widehat{T}$ on
$\widehat{U}_T$ defined by $\widehat{T}: (\ldots,u_{-2},u_{-1}) \mapsto
(\ldots,u_{-2},u_{-1},T(u_{-1}))$.
It is straightforward (see \cite[Eqn. 11]{Supp}) to find that the dynamical system $(U,T)$ is semi-conjugate to $(\widehat{U}_T,\widehat{T})$ and whenever, $(U,T)$ is invertible, $(\widehat{U}_T,\widehat{T})$ is conjugate to $(U,T)$.
Towards the end of constructing a dynamical system conjugate to $(\widehat{U}_T,\widehat{T})$, we consider the notion of a driven (dynamical) system.
A driven system comprises of two compact metric spaces $U$ and $X$, and a {\bf continuous function} $g: U \times X \to X$.
We call $U$ and $X$, respectively the input and state space of the driven system.
For brevity, we refer to $g$ as a driven system with all underlying entities quietly understood.
Since $X$ is assumed to be compact, it follows (e.g., \cite{kloeden2011nonautonomous,manjunath2014dynamics,manjunath2013echo}) that for every chosen (bi-infinite) input $\bar{u}= \{u_n\} \subset U$ there exists at least one sequence $\{x_n\}$ that satisfies $x_{n+1} = g(u_n,x_n)$ for all $n \in \mathbb{Z}$. Any such bi-infinite sequence $\{x_n\}$ is called a solution of $g$ for the input $\bar{u}$. In particular, if we denote a bi-infinite sequence $\{u_n\}_{n\in \mathbb{Z}}\subset U$ by $\bar{u}$, and a left-infinite part by $\cev{u}^n:= (\ldots,u_{n-2},u_{n-1})$, we note that $\cev{u}^n$ belongs to $\overleftarrow{U}$ regardless of $n\in \mathbb{Z}$.
We next identify a subspace $X_U$ of $X$ that contains all possible solutions of $g$. To realize such a subspace of a driven system $g$, we again discard $T$ and define
the \textit{reachable set} of the driven system $g$ to be the union of all the elements of all the solutions, i.e.,
$$X_U :=\Big \{x \in X: x = x_k \mbox{ where $\{x_n\}$ is a solution for some $\bar{u}$} \Big \}.$$
For example, when $U=[0,1]$ and $X=[0,1]$, for the driven system
$g(u,x) := \frac{ux}{2}$ regardless of any sequence in $U$, $x_n\equiv 0$ for $n \in \mathbb{Z}$ is the only solution of $g$, and hence the reachable set $X_U=\{0\}$. In this case, it is not possible to relate (this will be made precise later) the temporal variation in the input to the reachable set because it is singleton subset.
We would like the reachable sets of driven systems to be such that the inverse-limit space $\widehat{U}_T$ of $(U,T)$ can be topologically embedded into the reachable set or embedded into a finite self-product of the reachable set. To achieve such a desired embedding, we consider driven systems that are also state-input invertible (SI-invertible) that we define next. We say $g$ is SI-invertible if $g(\cdot,x):U \to X$ is invertible for all $x$, or equivalently if $x_n$ and the $x_{n+1}$ are related by $x_{n+1}=g(u_n,x_n)$, then $u_n$ can be uniquely determined from $x_n$ and $x_{n+1}$. A recurrent neural network (RNN) with $N$ artificial neurons e.g., \cite{jaeger2001echo}) is an example of $g$ with $U \subset \mathbb{R}^N$
and $X=[-1,1]^N$ (the cartesian product of $N$ copies of $[-1,1]$) defined by
\begin{equation} \label{eq_RNN}
g(u,x) = (1-a)x + a\overline{\tanh}(Au + \alpha Bx),
\end{equation}
where $A$ is a $N \times N$ matrix that represents input connections
to the neurons called the input matrix. The matrix $B$ is also of dimension $N \times N$ representing the strength of the interconnections between the neurons (known as the reservoir matrix in reservoir computing), and $a$ and $\alpha$ are real-valued parameter customarily called the leak rate and scaling of the reservoir
$B$ respectively and $\overline{\tanh}(*)$ is (the nonlinear activation) $\tanh$ performed component-wise on $*$. The RNN accepts inputs as sequence of vectors with $N$ elements, and it can be readily verified that when both $A$ and $B$ are invertible, then $g$ has SI-invertibility (see \cite[Eqn. 4]{Supp}).
We also note that if a real vector $v_n$ is of dimension $K<N$, SI-invertibility can still be realized by (isometrically) embedding $v_n$ into $\mathbb{R}^N$, for instance by padding $N-K$ zeroes to obtain an input of dimension $N$ i.e., $v_n \mapsto (v^1_n,v^2_n,\ldots,v^K_n,0,0,\ldots 0) =: u_n$.
Given a driven system $g$ that is SI-invertible and a dynamical system $(U,T)$, we define a relation on $X_U$ (a relation on $X_U$ is a subset defined on $X_U \times X_U$) by $$Y_T:=\{(x_{n-1},x_n): \{x_k\}_{k\in \mathbb{Z}} \mbox{ is a solution for some orbit of } T \mbox{ and } n \in \mathbb{Z}\}, $$ and call $Y_T$ relation induced by $(U,T)$. It easily follows \cite[Theorem 3]{Supp} that there exists a map $G_T: Y_T \to Y_T$ defined by $(x_{n-1},x_n) \mapsto (x_n,x_{n+1})$ that describes the the single-delay lag dynamics on the driven system's states.
In general, there could be no well-founded relationship like a topological conjugacy or a semi-conjugacy between the maps $T$ and $G_T$ since the dynamics of the driven system could be more complex than in the input (\cite[Fig. 3]{Supp}). However, it is possible to identify a class of driven systems for which we can relate $T$ and $G_T$. First, we pause to ask a question: is the complexity of the solution of $g$ exclusively determined by the complexity in
the input? To make this precise, we first consider
the mapping that defines the evolution of the input by appending a new input value $v\in U$ at time $n$, $\sigma_v:
\cev{u}^{n} \mapsto \cev{u}^{n}v$,
where symbolically $\cev{u}^{n}v:=(\ldots,u_{n-2} ,u_{n-1}, v)$, and then ask if the dynamics of the driven system is topologically semi-conjugate to $\sigma_v$? This is equivalent to asking if there exists a continuous surjective map $h: \overleftarrow{U} \to X_U$ so that the following diagram commutes:
\begin{equation} \label{comm_h}
\psset{arrows=->, arrowinset=0.25, linewidth=0.6pt, nodesep=3pt, labelsep=2pt, rowsep=0.7cm, colsep = 1.1cm, shortput =tablr}
\everypsbox{\scriptstyle}
\begin{psmatrix}
\overleftarrow{U} & \overleftarrow{U}\\%
X_U & X_U
\ncline{1,1}{1,2}^{\sigma_v} \ncline{1,1}{2,1} <{h}
\ncline{1,2}{2,2} > {h}
\ncline{2,1}{2,2}^{g(v,\cdot)}.
\end{psmatrix}
\end{equation}
If the driven system $g$ is such that the above commutativity holds then we call $h$ a universal semi-conjugacy. Remarkably, $g$ satisfies the above commutativity (see \cite[Theorem 2]{Supp}) when $g$ satisfies the unique solution property: we say a driven system $g$ has the unique solution property (USP) if for each input $\bar{u}$ there exists exactly one solution. In other words, $g$ has the USP if there exists a well-defined solution-map $\Psi$ so that $\Psi(\bar{u})$ denotes the unique solution obtained from the input $\bar{u}$. Also in this context, the USP is equivalent to saying $g$ is a topological contraction \cite[Eqn. 6]{Supp}, and this notion is independent of SI-invertibility. This topological contraction is often also called the echo state property \cite{Manju_ESP}, and systems with this property have other important useful properties \cite{grigoryeva2018echo,manjunath2020stability} for information processing. Attempts have been made in the literature to algorithmically construct a semi-conjugacy between the driven dynamics in the special case of periodic inputs \cite{sauer2009detection}, but we remark that we are considering a very general problem by considering all possible left-infinite sequences in \eqref{comm_h}. For more illumination, we recall that the diagram in \eqref{comm_h} commutes if and only if the system $g$ has the USP (see \cite[Lemma 5]{Manju_Nonlinearity}). Furthermore, when \eqref{comm_h} holds, $h(\cev{u}^k) = x_k$, where $x_k$ is the value of the solution at the $k^{\mathit{th}}$ instant for any input $\bar{u}$ whose left-infinite segment is $\cev{u}^k$.
With inputs to $g$ being restricted to be orbits of $T$, in the context of left-infinite spaces as in \eqref{comm_h}, the system on the top in \eqref{comm_h} turns out to be the inverse-limit system
$(\widehat{U}_T, \widehat{T})$. This is since if we restrict the map $\sigma_v$ to $\widehat{U}_T$, and select $v=Tu_{-1}$, then we have
$\sigma_v(\ldots,u_{-2},u_{-1})=\sigma_{Tu_{-1}}(\ldots,u_{-2},u_{-1})=\widehat{T}(\ldots,u_{-2},u_{-1})$.
Next, we denote $r:\overleftarrow{U} \to \overleftarrow{U}$ to be the right-shift map, i.e., $r: (\cdots, u_{-2},u_{-1}) \mapsto
(\cdots, u_{-3},u_{-2})$. We then can establish that the two maps $G_T$ and $\widehat{T}$ are related (a formal proof is in \cite[Theorem 4]{Supp}) through the function $H_2(\cev{u}):=(h(r\cev{u}),h(\cev{u}))$ (strictly speaking, it is $H_2$ restricted to $\widehat{U}_T$) by
the following commutativity diagram printed in black:
\begin{equation} \label{comm_imp}
\psset{arrows=->, arrowinset=0.25, linewidth=0.6pt, nodesep=3pt, labelsep=2pt, rowsep=1.3cm, colsep = 1.1cm, shortput =tablr}
\everypsbox{\scriptstyle}
\begin{psmatrix}
\widehat{U}_T & \widehat{U}_T\\%
Y_T & Y_T\\
& \textcolor{red}{U \times X}
\ncline{1,1}{1,2}^{\widehat{T}} \ncline{1,1}{2,1} <{H_2}
\ncline{1,2}{2,2} > {H_2}
\ncline{2,1}{2,2}^{G_T}
\psset{linecolor=red,linewidth=1pt}
\textcolor{red}{\ncline{2,1}{3,2}<{(\Gamma,\pi_2)}}
\ncline{3,2}{2,2}>{\textcolor{red}{(\pi_2,g)}}.
\end{psmatrix}
\end{equation}
Note that $H_2$ maps an entire-left infinite sequence into an element in $X \times X$. Specifically if $\{x_n\}$ is the solution obtained with the input $\{u_n\}$, then $H_2(\cev{u}^k) = (x_{k-1},x_{k})$ for all $k \in \mathbb{Z}$ since $h(\cev{u}^k)=x_k$. To qualify $H_2$ further, we say a driven system $g$ \emph{causally embeds} a dynamical system $(U,T)$ if it satisfies the two properties: (i) a universal semi-conjugacy exists, i.e., the diagram in \eqref{comm_h} commutes (and thus, \eqref{comm_imp} also commutes) (ii) $H_2(\cev{u}) := (h(r\cev{u}),h(\cev{u}))$ embeds the inverse-limit space $\widehat{U}_T$ in $X \times X$.
Now we list the consequences of our definitions and the commutativity in \eqref{comm_imp}. When $g$ is SI-invertible and $\{u_n\} \subset U$ is an orbit of an dynamical system $(U,T)$, the following results (formal proofs in \cite[Theorem 3, Theorem 4]{Supp}) hold: (i) there exists a map $G_T: Y_T \to Y_T$ defined by $(x_{n-1},x_n) \mapsto (x_n,x_{n+1})$ \cite[Theorem 3]{Supp} (ii) further, if $g$ has the USP then $(Y_T, G_T)$ is topologically semi-conjugate to $(\widehat{U}_T,\widehat{T})$ (iii) furthermore, if $T:U\to U$ is a homeomorphism, then $(Y_T,G_T)$ is topologically conjugate to $(\widehat{U}_T,\widehat{T})$ (and hence also conjugate to $(U,T)$) and hence $g$ causally embeds $(U,T)$. Since two successive points $(x_{n-1},x_n)$ of a solution of $g$ with an input $\bar{u}$ satisfy $(x_{n-1},x_n) = H_2(\cev{u}^n)$ and lie in $Y_T$, one can learn the single-delay lag dynamics of the driven states through the map $G_T: (x_{n-1},x_{n}) \mapsto (x_n,x_{n+1})$ from sufficiently large finite set of data points $(x_0,x_1)$ , $(x_1,x_2), \ldots, (x_{m-1},x_m)$. Once $G_T$ is learnt, then applying the iterates of $G_T$, all future successive points on a solution can be forecasted.
Forthwith, one can also predict the input value $u_n$, since two successive states $x_n$ and $x_{n+1}$ determine $u_n$ since $g$ is SI-invertible \cite[Eqn. 14]{Supp}. In particular if $G_T$ is learnt without errors, prediction would be exact whenever $H_2$ causally embeds $(U,T)$, or else the predicted value of $u_n$ would be an approximation to it obtained from the system $(Y_T,G_T)$ that is semi-conjugate to $(\widehat{U}_T,\widehat{T})$. Synoptically, restricting $H_2$ to inverse-limit spaces of different dynamical systems contained in $\overleftarrow{U}$ establishes semi-conjugacies/conjugacies between $G_T$ and those inverse-limit spaces.
Thus one does not need to change $g$ to learn $G_T$ when the dynamical system that drives it changes. Bringing out this fact is a theoretical advancement that drives us to obtain a universal set of observables for learning the Koopman operator \cite{koopman1932dynamical} of any $(\widehat{U}_T, \widehat{T})$ as long as $\widehat{U}_T$ is contained in $\overleftarrow{U}$.
Given a dynamical system $(U,T)$ and some vector space $V$ of functions $f$ whose domain is $U$, the (linear) operator $\mathcal{K}: V \to V$ so that
$\mathcal{K} f = f \circ T$ holds is called the Koopman operator of $T$. If one knows the action of the Kooopman operator on an observable $f$, then since
$\mathcal{K} f(u) = f(T(u))$ and likewise $\mathcal{K} f(Tu) = f(T^{(2)}u)$, one can forecast the observed values $(f(Tx), f(T^{(2)}u), f(T^{(3)}u),\ldots)$, where $T^{(n)}$ denote the $n$-fold composition with itself. In Koopman's theory (e.g., \cite{koopman1932dynamical,klages2013weak,budivsic2012applied}) the dynamics of $(U,T)$ is inferred by the study of the action of the linear operator $\mathcal{K}$ on functions in $V$.
It is customary to consider a class of observables where $V$ is some Hilbert space or some $L^p$ space, i.e., an observable $f$ represents an equivalence class of measurable functions. One of the central ideas in using Koopman's theory for forecasting dynamical systems is to obtain a collection of observables $\mathbf{f}=\{f_1,\ldots,f_N\}$ so that the resultant dynamics from $\mathbf{f}(u) \mapsto \mathbf{f}(Tu)$ can be approximated by a map, say $F$ with a lower functional complexity exploiting the fact that $\mathcal{K}$ is linear on $V$. The choice of the $L^p$ space considered determines the spectrum \cite{eisner2015operator} of the Koopman operator $\mathcal{K}$, and when it is a Hilbert space, and the span of the observables is invariant under $\mathcal{K}$, one can capture a portion of the spectrum of $\mathcal{K}$ through a linear map $F$ (e.g., \cite{williams2015data,korda2020data}).
In our setup, when $X \subset \mathbb{R}^N$, then $Y_T \subset X \times X$ and $H_2(\cev{u})$ takes values in $\mathbb{R}^N \times \mathbb{R}^N$. If $\pi_i$ picks the $i^{\mathit{th}}$ component function of $H_2 :\widehat{U}_T \to X_U \times X_U$, and we set $f_i = \pi_i \circ H_2$ then each $f_i$ is a $\mathbb{R}^2$-valued observable. The fact $H_2 \circ \widehat{T} =G_T \circ H_2 $ (due to \eqref{comm_imp}) would imply $f_i \circ \widehat{T} = \pi_i \circ G_T \circ H_2$. Thus, $\mathcal{K} f_i = \pi_i \circ G_T \circ H_2$ when $\mathcal{K}$ is the Koopman operator of $\widehat{T}$ (formal proof in \cite[Corollary 1]{Supp}). Thus if $G_T$ is learnt, we would have learnt the
action of the Koopman operator of $\widehat{T}$ on each of the observables $f_i$, and remarkably these $f_i$'s are universal, i.e., they do not need to be changed as long as the inverse limit space of the underlying system is contained in $\overleftarrow{U}_T$.
We remark that in some recent applied Koopmanism language, $\mathbf{f}$ the collection of component functions of $H_2$ and the map $G_T$ together form a finite faithful representation \cite{mezic2020koopman} of $(\widehat{U}_T,\widehat{T})$.
In reality, the setup does not give an analytical expression for $H_2$ or the observables $f_i$ but gives only the values they attain, however, this is inconsequential for learning the action of the Koopman operator of the inverse-limit system.
Besides being able to learn the action of the Koopman operator exactly, our setup has other advantages over selecting or optimizing the choice of observables as in the Extended Dynamic Mode Decomposition (EDMD) algorithm (e.g., \cite{schmid2010dynamic,williams2015data}). We can alter the functional complexity of $G_T$ by changing the number of observables or equivalently by altering the dimension of $X$ -- in the case of a RNN implementation of $g$, by simply changing the number of neurons in the network. Empirically, increasing the dimension of $X$ is found to increase the linear relationship (or intuitively reduces the functional complexity of $G_T)$ that is measured as a generalization of the Pearson correlation coefficient to random vectors (e.g., \cite{puccetti2019measuring}) between
$(x_{n-1},x_n)$ and $G_T(x_{n-1},x_n)$, and such numerical evidence is tabulated in \cite[Table 1]{Supp}. In contrast, expanding the set of observables while using EDMD to learn a map with a lower functional complexity is a highly involved task, not least because so much that one often does not know how to guess a new observable, as well as the fact that adding observables does not necessarily retain the invariance of the span of the observables under the Koopman operator that is needed \cite{williams2015data,korda2020data} to capture a portion of the spectrum of the operator. Also, it is a fact that the spectrum of the Koopman operator of conjugate systems are identical while the spectrum of the Koopman operator of a factor is contained in its extension (e.g. \cite{eisner2015operator}). These spectra are equal when $T$ is a homeomorphism. In any case the spectrum of the Koopman operator of $G_T$ contains that of the Koopman operator of $T$, and thus we have presented a methodology that avoids capturing only a portion of the spectrum of $T$ for forecasting.
\section*{Equations from Data and Forecasting}
We learn single-delay lag dynamics defined by $G_T$ indirectly by learning $\Gamma: (x_{k-1},x_{k}) \mapsto u_k$ -- the map $\Gamma$ always exists when $G_T$ exists (see \eqref{comm_imp} and \cite[Theorem 3]{Supp}). The map $G_T$ entails another map (see the vertical red line and the diagonal line in red in that order in Eqn. \eqref{comm_imp} or in Fig.~\ref{Fig_1}A), $S: (u_k,x_k) \mapsto (u_{k+1},x_{k+1})$ defined by
\begin{eqnarray}
u_{k+1} &=& \pi_1 \circ (\Gamma, \pi_2) \circ (\pi_2,g) (u_k,x_k) \label{eqn_u}\\
x_{k+1} &=& \pi_2 \circ (\Gamma, \pi_2) \circ (\pi_2,g) (u_k,x_k), \label{eqn_x}
\end{eqnarray}
where $\pi_i : (a_1,a_2) \mapsto a_i$. The expressions \eqref{eqn_u}-\eqref{eqn_x} are equations constructed from data! Although the equations are coupled, \eqref{eqn_u} per se represents (possibly nonlinear) difference equation of infinite order since we can replace $x_k$ by $h(\cev{u}^k)$, where $h$ is the universal semi-conjugacy. While realizing \eqref{eqn_u}-\eqref{eqn_x} in practice, $x_k$ acts as a vector with several auxiliary variables as its components.
Learning the map $\Gamma$ instead of $G_T$ not only saves computational resources since $u_n$ lies in a low-dimensional subspace due to the zero-padding, but owing to the USP of the driven system $g$ one gains input-related, parameter-related stability \cite{manjunath2020stability}, and importantly it prevents large numerical errors due to input noise due to global dissipativity (\cite[Theorem 5]{Supp}) that is not guaranteed for learning $G_T$ (see \cite[Section 7]{Supp} for details). Also, in practice, one does not need the entire-left infinite history of the input; by initializing the driven system with an arbitrary initial value $y_m \in X$, then the sequence $y_{m+1}, y_{m+2}, y_{m+3},...$ satisfying $y_{k+1}= g(u_k,y_k)$ for $k \geq m$ approximates the corresponding elements of the actual solution $\{x_n\}$ uniformly thanks to the uniform attraction property (see \cite[Theorem 1]{Manju_Nonlinearity} or \cite[Eqn. 18]{Supp}).
When only observations $\theta(w_{0}),\theta(w_{1}),\ldots,\theta(w_{m})$ from a dynamical system $(W,T)$ where $\theta:W \to \mathbb{R}$ is an observable are available, the delay-coordinates $\Phi_{\theta,2d}(\theta(w_{n})) := (\theta(w_{n-2d}),\ldots,\theta(w_{n-1}),\theta(w_{n}))$ is fed into the driven system as $u_n$. Assuming $(W,T)$ and $\theta$ are such that Takens embedding holds, i.e., there exists a map $F:\Phi_{\theta,2d}(\theta(w_{n})) \mapsto \Phi_{\theta,2d}(\theta(w_{n+1}))$, then the single-delay lag dynamics obtained through $g$ is topologically conjugate to the inverse-limit system $(\Phi_{\theta,2d}(W), F)$, and forecasting can be carried out as before. The dimension of the input-vector is increased since delay-coordinates are fed into $g$, but the major gain is global dissipativity \cite[Fig. 6]{Supp}.
One of the perplexities of employing the delay coordinate map is that the required delay to embed the attractor is not known immediately. In this case, by feeding the observations directly into the same driven system $g$ instead of the delay coordinates, ceteris paribus, we demonstrate appealing numerical results where the theory is based on a conjecture (see \cite[Conjecture 1]{Supp}). When $H_2$ has at least $2m+1$ generic observables so that Whitney's embedding theorem holds and $H_2$ embeds $\Theta(\widehat{W}_T) :=
\{(\ldots, \theta(w_{-2}),\theta(w_{-1})) : w_{n+1} = T w_n\}$ in $X \times X$ then the dynamics of the single-delay lag dynamics becomes conjugate to that of $T$ and forecasting can be carried out as before (see \cite[Section 8]{Supp} for more details).
\vspace{-1cm}
\section*{Methods and Forecasting Results}
We realize $g$ through a RNN of the type \eqref{eq_RNN} that is also SI-invertible and has the USP. Such RNNs fall into the reservoir computing framework and are amenable for hardware implementations (e.g., \cite{appeltant2011information, kudithipudi2016design, vandoorne2014experimental}).
We call a forecasting system described by \eqref{eqn_u}--\eqref{eqn_x} a \emph{recurrent conjugate network}(RCN), regardless of how the map $\Gamma$ is implemented as long as $g$ is a RNN. A schematic of a RCN is shown in \cite[Fig. 8]{Supp}. We set the spectral radius of the matrix $\alpha \,B$ to be $\alpha$ by using a matrix $B$ with spectral radius $1$. For $\alpha$ sufficiently small, and usually in $(0,1)$, RNNs have the USP that can be verified empirically (e.g., the parameter-stability plot in \cite{manjunath2020stability}).
\begin{figure
\greybox{
\centering
\includegraphics[width=17.8cm]{Figure_1.eps}
\caption{Forecasting results using a RCN. In all experiments, matrices $A$ and $B$ of the RNN in Eqn. \eqref{eq_RNN} were initialized randomly, with $B$ having unit spectral radius; without explicitly mentioning the input is always padded with the required number of zeroes to match the row number of $A$.
We set the parameters $a=0.5$ and $\alpha=0.99$ and a 1000 dimensional RNN, i.e. $X=[-1,1]^{1000}$. Training to learn $\Gamma$ was accomplished with 2000 data points after 500 of these are discarded to allow the network to forget its initial state. {\bf A.} The inner workings of forecasting.
{\bf (B)}
Data $(w_n)$ is generated from the Lorenz system \cite[Eqn. 22]{Supp}, sampled every 0.1 time-steps. The input into the network is scaled down to fit inside $[-1,1]^3$, and perturbed with noise. Specifically, the input $(u_n)$ is given by $u_n=\frac{1}{100}w_n + \varepsilon_n$ where $\varepsilon_n$ is normally distributed with 0 mean and standard deviation equal to 0.01. This translates to a signal to noise ratio of roughly 18dB. 50,000 points of actual and predicted data. {\bf C,D.} A sequence of observations of the input in {\bf B} was fed into the system. Specifically $u_n$ is given by $u_n=\frac{1}{10}\left( \sin\left(\gamma w_x^{(n)}\right)+\sin\left(\gamma w_y^{(n)}\right)+\sin\left(\gamma w_z^{(n)}\right)\right)$, with $\gamma=0.1$ intended to be a parameter to control the complexity of the observation. {\bf (C)} Two different epochs of the actual and forecasted time-series in {\bf (C)}.
{\bf (D)} Full states of the Lorenz system with the input in {\bf C} is reconstructed by learning a map $\Gamma_{\mbox{full}}:(x_{n-1},x_n)\mapsto \frac{1}{100}w_n$. {\bf (E,F)} Data $(w_n)$ is generated from the full Logistic Map \cite[Eqn. 23]{Supp}, then normalized to have zero mean, and perturbed with noise. Specifically, the input $(u_n)$ is given by $u_n=w_n - \mbox{mean}(w) + \varepsilon_n$, where $\varepsilon_n$ is normally distributed of zero mean and standard deviation equal to 0.01. This translates to a signal-to-noise ratio of roughly 30dB. Plotted is the phase portrait in {\bf (E)} and the smoothened invariant density {\bf F} for actual (red) and predicted (blue) data, over 10000 prediction steps. }
\label{Fig_1}
}
\end{figure}
In our forecasting experiments using RCNs, we learn $\Gamma: (x_{n-1},x_{n}) \mapsto u_n$ by learning a feedforward network (NN) on the principal components of states $x_n$ of $g$. The principal components are optional and used only for an efficient state representation possibly to reduce the errors while learning, and is not intended for a lossy approximation since all principal components are used. To be explicit, we denote the matrix with the first $N$ states of $g$ as row vectors by $X_{1:N}$. If $X_{1:N}=U\Sigma P^T$ denotes the singular value decomposition of $X_{1:N}$, then the principal component matrix is given by $P$, and the principal components are given by
$Z_{1:N}=X_{1:N}P.$
These principal components are used to train a feedforward neural network implemented in {\it Python} using the {\it Keras} library built on {\it Tensorflow} (\cite{chollet2015keras2}). Training is accomplished using the {\it Adam optimizer}, optimizing the mean squared error loss function. If we denote the row vectors of $Z_{1:N}$ by
$z_i^T, i=1,2,\cdots,N$ and the neural network by $NN$, then we learn an approximation of the map
$NN: (z_{n-1},z_n) \mapsto u_n.
$
Hence $\Gamma$ that we learn is composed as $$
NN\left( \begin{bmatrix}
P^Tx_{n-1} \\
P^Tx_n
\end{bmatrix}
\right) = NN \circ
\begin{bmatrix}
P^T & 0 \\
0 & P^T
\end{bmatrix}\begin{bmatrix}
x_{n-1}\\
x_n
\end{bmatrix} = u_n = \Gamma(x_{n-1},x_n).
$$
The dynamical systems chosen are a Lorenz system that exhibits turbulence, the full logistic map, a chaotic system without statistical stability as described by the linear response (e.g., \cite{baladi2014linear}), a map in the H\`enon family that exhibits intermittency and chaos \cite{kaplan1993type}, a map in the Pomeau-Manneville family \cite{pomeau1980intermittent} that exhibits weak chaos and intermittency. Intermittency is a feature of transition to turbulence and convection in confined spaces \cite{pomeau1980intermittent}, earthquake occurrence \cite{bottiglieri2007off}, and anomalous diffusion in biology \cite{klages2013weak}. We note that the logistic map and the maps from the Pomeau-Manneville are both non-invertible, and hence the single-delay lag dynamics of the RCN could possibly be just semi-conjugate to the inverse-limit space. For the precise description of these dynamical systems we refer to \cite[Eqns. 22, 23, 24, 25]{Supp}. Even negligible deviations due to computational noise would make it hard to minimize the long-term pointwise prediction error for such chaotic systems, but long-term topological consistency, that is the requirement that the orbit lies on the attractor can be achieved. The ability to reconstruct or forecast a dynamical system also relies on how the statistical properties like an invariant density \cite{lasota2013chaos} are retained by the forecasted data.
We use the same driven system, a RNN in our RCN for all the experiments, and demonstrate the forecasting results in Fig.~\ref{Fig_1} and Fig.~\ref{Fig_2}.
A notable feature is that despite the very low sampling rate in Fig.~\ref{Fig_1}B (step size of 0.1) in the discretization of the Lorenz system, the RCN predictions help reconstruct the attractor. To test our conjecture \cite[Conjecture 1]{Supp}, we also feed a univariate time-series $\{\theta(w_n)\}$ obtained from an orbit $w_n$ of the Lorenz system, and observe that the hidden states in the entire-attractor can be recovered by training the states in $Y_T$ by $(x_{n-1},x_{n}) \mapsto w_n$. We note that without relying on any such conjecture if the input is fed as delay-coordinates the attractor is learnt (see \cite[Fig. 10]{Supp}) as per the theory.
In Fig.~\ref{Fig_1}F, despite the logistic map not having the linear response \cite{baladi2014linear}, the forecasted invariant density shows a good resemblance to the actual invariant density.
Since each node in a RNN receives a weighted sum of the external inputs and states as its input, heuristically such summing results in some noise cancellation, and it is found that the reconstructed attractor has lesser noise than the noisy attractor of the original. This for instance is evident in Fig.~\ref{Fig_2}A.
Obtaining models that capture the statistical behavior for the data from the Pomeau-Manneville map as illustrated in Fig.~\ref{Fig_2}F is a hallmark of the methodology used here since the length of a slow drifting phase in the intermittency is unpredictable (details in \cite[Section 9]{Supp}), and to the best of our knowledge, no machine learning algorithm has succeeded so far on such maps. Also, we remark that we have not explored the parameters of the RCN that achieves the best forecasting results, and instead have used the same RNN to demonstrate the universality of the observables rendered by $H_2$. As an example to show that the results could be improved, we show that with different choices of $\alpha$ in the RCN we can forecast more pronounced on-off intermittency of maps in the Pomeau-Manneville family in \cite[Section 9]{Supp}. The interested reader may refer to the effectiveness of a RCN in accurately reproducing seasonal variations in the temperatures from physical data of the South African climate dynamics in \cite[Section 9]{Supp}.
\begin{figure
\greybox{
\centering
\includegraphics[width=11.4cm]{Figure_2.eps}
\caption{More Forecasting results with the same RCN setup as in Fig.~ \ref{Fig_1}. However, to capture intermittency in noisy data, longer training data was used; 1000 samples were discarded to allow the network to forget its initial conditions, and 5000 were used for training. {\bf (A,B,C)} Data $(w_n)$ comes from an intermittent realisation from the H\`enon family of maps \cite[Eqn. 24]{Supp}. Input $(u_n)$ was normalized to have mean zero, scaled to fit inside $[-1,1]^2$, and perturbed with noise. To be explicit, $u_n=\frac{1}{10}(w_n - \mbox{mean}(w))+\varepsilon_n$ with $\varepsilon_n$ normally distributed with zero mean and standard deviation equal to 0.001, resulting in a signal to noise ratio of 40dB. {\bf (A)} Actual (red) and learnt (blue) attractors; predicted attractor is observed to have less noise especially along ``hairpin bend".
{\bf (B)} represents the learnt (blue) and actual (red) densities of the first coordinates of the data.
{\bf (C)} Time-series of actual and predicted the $v_y$ coordinates in an epoch.
{\bf (D,E,F)} Analogous plots for data $(w_n)$ coming from the Pomeau-Manneville family of maps \cite[Eqn. 25]{Supp}. Input $(u_n)$ is given by $u_n=w_n-\mbox{mean}(w)+\varepsilon_n$ with $\varepsilon_n$ distributed normally with zero mean and standard deviation equal to 0.01, roughly resulting in a signal to noise ratio of 26dB. } \label{Fig_2}
}
\end{figure}
\vspace{-1cm}
\section*{Discussion}
Finding the set of observables of data that determine a less functionally complex learnable map so that its dynamics in the observed space gets closer to that of the action of the Koopman operator on the observables has been a pursuit for the last decade. In practice, however, the Koopman operators of complex and chaotic systems tend to have significantly more complicated spectral properties (e.g., non-isolated eigenvalues and/or continuous spectra) hindering the performance of data-driven approximation techniques that capture only a portion of the spectrum (e.g., \cite{budivsic2012applied,korda2020data}). Also, in the temporal domain, approximations of the Koopman operators of the underlying dynamical systems are amnesiacs regarding their distant past.
Here, we show a driven dynamical system that can causally embed dynamical systems is capable of determining a set of observables of the inverse-limit system of the underlying dynamical system and learning the Koopman operator exactly.
In particular, we produce a topological conjugacy (semi-conjugacy) between a map that describes the single-delay lag dynamics and the underlying homeomorphism (non-invertible map that can be discontinuous) dynamical system. Also, the map that describes the single-delay lag dynamics is not found in other reservoir computing algorithms.
With our methodology, one does not need expert human intuition or physical insights to decide on the observables and library functions that are employed in other data-driven approaches. This is very useful in high-dimensional forecasting tasks where such insights are rare. Besides exact reconstruction when conjugacy holds, we have the universality of observables, equations from data, and robustness to input noise. All this helps in obtaining high-fidelity models that give exceptional forecasting results with long-term topological and statistical consistency on chaotic data and even on data showing intermittency.
{\bf Acknowledgements} We wish to thank Herbert Jaeger, Markus Haase, Igor Mezi\'{c} for their helpful comments during the course of our work. The research was supported by the National Research Foundation of South Africa under Grant UID 115940.
\vspace{-1cm}
\bibliographystyle{pnas-new}
|
1,941,325,220,560 | arxiv | \section{Introduction}\label{sec1}
Image steganography is the main content of information hiding. The sender conceal a secret message into a cover image, then get the container image called stego, and finish the secret message's transmission on the public channel by transferring the stego image. Then the receiver part of the transmission can reveal the secret message out. Steganalysis is an attack to the steganography algorithm. The listener on the public channel intercept the image and analyze whether the image contains secret information. Since their proposed, steganography and steganalysis promote each other's progress.
Image steganography can be used into the transmission of secret information, watermark, copyright certification and many other applications. In general, we can measure a steganography algorithm by capacity, invisibility and security. The capacity is measured by bits-per-pixel (bpp) which means the average number of bits concealed into each pixel of the cover image. With the capacity becomes larger, the security and the invisibility become worse. The invisibility is measured by the similarity of the stego image and its corresponding cover image. The invisibility becomes better as the similarity going higher. The security is measured by whether the stego image can be recognized out from natural images by steganalysis algorithms. Correspondingly, there are two focused challenges constraining the steganography performance. The amount of hidden message alters the quality of stego images. The more message in it, the easier the stego image can be checked out. Another keypoint is the cover image itself. Concealing message into noisy, rich semantic region of the cover image yields less detectable perturbations than hiding into smooth region.
Nowadays, traditional steganography algorithms, such as S-UNIWARD~\cite{s-uniward}, J-UNIWARD~\cite{s-uniward}, conceal the secret information into cover images' spatial domain or transform domains by hand-crafted embedding algorithms successfully and get excellent invisibility and security. With the rise of deep learning in recent years, deep learning has become the hottest research method in computer vision and has been introduced into information hiding domain. Volkhonskiy et al.~\cite{SGAN} proposed a steganography enhancement algorithm based on GAN, they concealed secret message into generated images with conventional algorithms and enhanced the security. But their generated images are warping in semantic, which will be drawn attention easily. Tang et al.~\cite{ASDL-GAN} proposed an automatic steganographic distortion learning framework, their generator can find pixels which are suitable for embedding and conceal message into them, their discriminator is trained as a steganalyzer. With the adversarial training, the model can finish the steganography process. But this kind of method has low capacity and is less secure than conventional algorithms. Baluja~\cite{nips} proposed a convolutional neural network based on the structure of encoder-decoder. The encoder network can conceal a secret image into a same size cover image successfully and the decoder network can reveal out the secret image completely. This method is different from other deep learning based models and conventional steganography algorithms, it has large capacity and strong invisibility. But stego images generated by this model is distorted in color and its security is bad. Inspired by Baluja's work, we proposed an invisible steganography via generative adversarial network named ISGAN\xspace. Our model can conceal a gray secret image into a color cover image with the same size, and our model has large capacity, strong invisibility and high security. Comparing with previous works, the main contributions of our work are as below:
1. In order to suppress the distortion of stego images, we select a new steganography position. We only embed and extract secret information in the Y channel of the cover image. The color information is all in Cr and Cb channels of the cover image and can be saved completely into stego images, so the invisibility is strengthened.
2. From the aspect of mathematics, the difference between the empirical probability distributions of stego images and natural images can be measured by the divergence. So we introduce the generative adverasial networks to increase the security throughout minimizing the divergence. In addition, we introduce several architectures from classic computer vision tasks to fuse the cover image and the secret image together better and get faster training speed.
3. In order to fit the human visual system (HVS) better, we introduce the structure similarity index (SSIM)~\cite{SSIM} and its variant to construct a mixed loss function. The mixed loss function helps to generate more realistic stego images and reveal out better secret images. This point is never considered by any previous deep-learning-based works in information hiding domain.
The rest of the paper is organized as follows. \secref{2} discusses related works, \secref{3} introduces architecture details of ISGAN\xspace and the mixed loss function. \secref{4} gives details of different datasets, parameter settings, our experiment processes and results. Finally, \secref{5} concludes the paper with relevant discussion.
\section{Related Works}\label{sec:2}
\paragraph{Steganalysis} There have been plenty of works using deep learning to do image steganalysis and got excellent performance. Qian et al.~\cite{GNCNN} proposed a CNN-based steganalysis model GNCNN, the model introduced the hand-crafted KV filter to extract residual noise and used the gaussian activation function to get more useful features. The performance of the GNCNN is inferior to the state-of-the-art hand-crafted feature set spatial rich model (SRM)~\cite{SRM} slightly. Based on GNCNN, Xu et al.~\cite{XuNet} presented Batch Normalization~\cite{BN} in to prevent the network falling into the local minima. XuNet was equipped with Tanh, $1\times1$ convolution, global average pooling, and got comparable performance to SRM~\cite{SRM}. Ye et al.~\cite{YeNet} put forward YeNet which surpassed SRM and its several variants. YeNet used 30 hand-crafted filters from SRM to prepropose images, applied well-designed activation function named TLU and selection-channel module to strengthen features from rich texture region where is more suitable for hiding information. Zeng et al.~\cite{Zeng} proposed a JPEG steganalysis model with less parameters than XuNet and got better performance than XuNet. These works have applied deep learning to steganalysis successfully, but there is still space for improvement.
\paragraph{Steganography} Since its introduction, generative adversarial networks\cite{GAN} have received more and more attention, achieved the state-of-art performance on tasks such as image generation, style transfer, speech synthesis and so on. The earliest application of deep learning to steganography was based on GAN. Volkhonskiy et al.~\cite{SGAN} proposed a DCGAN-based~\cite{DCGAN} model SGAN. SGAN consists of a generator network for generating cover images, a discriminator network for discriminating generated images from real images and a steganalyzer network for steganalysis. Hiding information in cover images generated by SGAN is securer than in natural images. Shi et al.~\cite{SSGAN} proposed SSGAN based on WGAN~\cite{WGAN}, their work was similar to SGAN and got better outcome. However, stego images generated by models similar to SGAN and SSGAN are warping in semantic and are more easily to draw attention than natural images, although these models reduce the detection rate of steganalysis algorithms. Tang et al.~\cite{ASDL-GAN} proposed an automatic steganographic distortion learning framework named as ASDL-GAN. The generator can translate a cover image into an embedding change probability matrix and the discriminator incorporates the XuNet architecture. In order to fit the optimal embedding simulator as well as propagate the gradient in back propagation, they proposed a ternary embedding simulator (TES) activation function. ASDL-GAN can learn steganographic distortions automatically, but its performance is inferior to S-UNIWARD. Yang et al.~\cite{Yang} improved ASDL-GAN and achieved better performance than S-UNIWARD. They used Selection-Channel-Aware (SCA)~\cite{YeNet} in generator as well as the U-Net framework~\cite{unet} which is introduced from the medical images segmentation. However, ASDL-GAN still refers too many prior knowledge from conventional steganography algorithms and its capacity is small. Hayes~\cite{Hayes} proposed a GAN-based model to hide a secret message into a cover image, and could reveal the secret message by his decoder successfully, but the invisibility is weak.
Baluja~\cite{nips} designed a CNN model to conceal a color secret image into a color cover image yielding state-of-art performance. Atique et al.~\cite{Atique} proposed another encoder-decoder based model to finish the same steganography task (their secret images are gray images). This is a novel steganography method which gets rid of hand-crafted algorithms. It can learn how to merge the cover image and the secret image together automatically. But stego images generated by their models are distorted in color. As shown in Fig.~\ref{fig:LFWshow}, Atique's stego images are yellowing when compared with the corresponding cover images. And their stego images are easily recognized by well trained CNN-based steganalyzer~\cite{nips} because of the large capacity. Inspired by works of Baluja and Atique, we improve each shortcoming and get ISGAN\xspace.
\section{Our Approach}\label{sec:3}
The complete architecture of our model is shown in Fig.~\ref{fig:overall-architecture}. In this section, the new steganography position is introduced firstly. Then we discuss about our design considerations on the basic model and show specfic details of the encoder and the decoder. Thirdly, we present why the generative adversarial networks can improve the security and details of the discriminator. Finally, we explain the motivation to construct the mixed loss function.
\begin{figure*}[!ht]
\begin{center}
\includegraphics[width=\textwidth]{overall-architecture.jpg}
\caption{The overall architecture. The encoder network conceals a gray secret image into the Y channel of a same size cover image, then the Y channel output by the encoder net and the U/V channels constitute the stego image. The decoder network reveals the secret image from the Y channel of the stego image. The steganalyzer network tries to distinguish stego images from cover images thus improving the overall architecture's security.}
\label{fig:overall-architecture}
\end{center}
\end{figure*}
\subsection{New Steganography Position}\label{sec:3.1}
Works of Baluja~\cite{nips} and Atique~\cite{Atique} have implemented the entire hiding and revealing procedure, while their stego images' color is distorted as shown in Fig.~\ref{fig:LFWshow}. To against this weakness, we select a new steganography position. As shown in Fig.~\ref{fig:rgb-yuv}, a color image in the RGB color space can be divided into R, G and B channels, and each channel contains both semantic information and color information. When converted to the YCrCb color space, a color image can be divided into Y, Cr and Cb channels. The Y channel only contains part of semantic information, luminance information and no color information, Cr and Cb channels contain part of semantic information and all color information. To guarantee no color distortion, we conceal the secret image only in the Y channel and all color information are saved into the stego image. In addition, we select gray images as our secret images thus decreasing the secret information by $\frac{2}{3}$.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{Barn-show}
\caption{Three images in the first column are original RGB color images. Three images in the right of the first row are R channel, G channel and B channel of the original image respectively saved as gray images, three channels all constitutes the luminance information and color information. Three images in the right of the second row are Y channel, Cr channel and Cb channel respectively saved as gray images, and three images in the right of the third row are also Y channel, Cr channel and Cb channel respectively from Wikipedia. We can see that, the Y channel constitutes only the luminance information and semantic information, and the color information about chrominance and chroma are all in the Cr channel and the Cb channel.}
\label{fig:rgb-yuv}
\end{figure}
When embedding, the color image is converted to the YCrCb color space, then the Y channel and the gray secret image are concatenated together and then are input to the encoder network. After hiding, the encoder's output and the cover image's CrCb channels constitute the color stego image. When revealing, we get the revealed secret image through decoding the Y channel of the stego image. Besides, the transformation between the RGB color space and the YCrCb color space is just the weighted computation of three channels and doesn't affect the backpropagation. So we can finish this tranformation during the entire hiding and revealing process. The encoder-decoder architecture can be trained end-to-end, which is called as the basic model.
\subsection{Basic Model}\label{sec:3.2}
Conventional or classic image stegnography are usually designed in a heuristic way. Generally, these algorithms decide whether to conceal information into a pixel of the cover image and how to conceal 1 bit information into a pixel. So the key of the classic steganography methods is well hand-crafted algorithms, but all of these algorithms need lots of expertise and this is very difficult for us. The best solution is to mix the secret image with the cover image very well without too much expertise. Deep learning, represented by convolutional neural networks, is a good way to achieve this exactly. What we need to do is to design the structure of the encoder and the decoder as described below.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{residual-inception.jpg}
\caption{The inception module with residual shortcut we use in our work.}
\label{fig:inception}
\end{figure}
\begin{table}
\caption{Architecture details of the encoder network: ConvBlock1 represents $3\times3$ Conv+BN+LeakyReLU, ConvBlock2 represents $1\times1$ Conv+Tanh, InceptionBlock represents the inception module with residual shortcut as shown in Fig.~\ref{fig:inception}.}
\label{tab:Encoder-Network}
\begin{tabular}{lcc}
\hline\noalign{\smallskip}
Layers & process & Output size \\
\noalign{\smallskip}\hline\noalign{\smallskip}
Input & / & $2\times256\times256$ \\
Layer 1 & ConvBlock1 & $16\times256\times256$ \\
Layer 2 & InceptionBlock & $32\times256\times256$ \\
Layer 3 & InceptionBlock & $64\times256\times256$ \\
Layer 4 & InceptionBlock & $128\times256\times256$ \\
Layer 5 & InceptionBlock & $256\times256\times256$ \\
Layer 6 & InceptionBlock & $128\times256\times256$ \\
Layer 7 & InceptionBlock & $64\times256\times256$ \\
Layer 8 & InceptionBlock & $32\times256\times256$ \\
Layer 9 & ConvBlock1 & $16\times256\times256$ \\
Output & ConvBlock2 & $1\times256\times256$ \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
Based on such a starting point, we introduce the inception module~\cite{inception} in our encoder network. The inception module has excellent performance on the ImageNet classification task, which contains several convolution kernels with different kernel sizes as shown in Fig.~\ref{fig:inception}. Such a model structure can fuse feature maps with different receptive field sizes very well. As shown in both residual networks~\cite{resnet} and batch normalization~\cite{BN}, a model with these modifications can achieve the performance with significantly fewer training steps comparing to its original version. So we introduce both residual module and batch normalization into the encoder network to speed up the training procedure. The detail structure of the encoder is described in Tab.~\ref{tab:Encoder-Network}. When using MSE as the metric on LFW dataset, we use our model to train for 30 epochs to get the performance Atique's model can achieve while training for 50 epochs.
\begin{table}
\caption{Architecture details of the decoder network: ConvBlock1 represents $3\times3$ Conv+BN+LeakyReLU, ConvBlock2 represents $1\times1$ Conv+Sigmoid.}
\label{tab:Decoder-Network}
\begin{tabular}{lcc}
\hline\noalign{\smallskip}
Layers & process & Output size \\
\noalign{\smallskip}\hline\noalign{\smallskip}
Input & / & $1\times256\times256$ \\
Layer 1 & ConvBlock1 & $32\times256\times256$ \\
Layer 2 & ConvBlock1 & $64\times256\times256$ \\
Layer 3 & ConvBlock1 & $128\times256\times256$ \\
Layer 4 & ConvBlock1 & $64\times256\times256$ \\
Layer 5 & ConvBlock1 & $32\times256\times256$ \\
Output & ConvBlock2 & $1\times256\times256$ \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
On the other hand, we need a structure to reveal the secret image out automatically. So we use a fully convolutional network as the decoder network. Feature maps output by each convolutional layer have the same size. To speed up training, we add a batch normalization layer after each convolutional layer other than the last layer. Details of the decoder network are described in Tab.~\ref{tab:Decoder-Network}.
\subsection{Our steganalyzer}\label{sec:3.3}
Works of Baluja and Atique didn't consider the security problem, while the security is the keypoint in steganography. In our work, we want to take the steganalysis into account automatically throughout training the basic model.
Denoting $\mathcal{C}$ as the set of all cover images $c$, the selection of cover images from $\mathcal{C}$ can be described by a random variable $c$ on $\mathcal{C}$ with probability distribution function (pdf) $P$. Assuming the cover images are selected with pdf $P$ and embedded with a secret image which is chosen from its corresponding set, the set of all stego images is again a random variable $s$ on $\mathcal{C}$ with pdf $Q$. The statistical detectability can be measured by the Kullback-Leibler divergence~\cite{KL} shown in \eqref{kl} or the Jensen-Shannon divergence shown in \eqref{js}.
\begin{equation}
KL(P||Q) = \sum_{c\in \mathcal{C}} P(c) log\frac{P(c)}{Q(c)} \label{kl}
\end{equation}
\begin{equation}
JS(P||Q) = \frac{1}{2}KL(P\|\frac{P+Q}{2}) + \frac{1}{2}KL(Q\|\frac{P+Q}{2}) \label{js}
\end{equation}
The KL divergence or the JS divergence is a very fundamental quantity because it provides bounds on the best possible steganalyzer one can build ~\cite{information theory}. So the keypoint for us is how to decrease the divergence. The generative adversarial networks (GAN) are well-designed in theory to achieve this exactly. The objective of the original GAN is to minimize the JS divergence \eqref{js}, a variant of the GAN is to minimize the KL divergence \eqref{kl}. The generator network $G$, which input is a noise $z$, tries to transform the input to a data sample which is similar to the real sample. The discriminator network $D$, which input is the real data or the fake data generated by the generator network, determines the difference between the real and fake samples. $D$ and $G$ play a two-player minmax game with the value function \eqref{V1}.
\begin{equation}
\min_{G}\max_{D}=E_{x\sim p_{data}(x)}[logD(x)] + E_{z\sim p_z(z)}[log(1-D(G(z)))] \label{V1}
\end{equation}
Now we introduce the generative adversarial networks into our architecture. The basic model can finish the entire hiding and revealing process, so we use the basic model as the generator, and introduce a CNN-based steganalysis model as the discriminator and the steganalyzer. So the value function in our work becomes \eqref{V2}, where $D$ represents the steganalyzer network, $G$ represents the basic model, $x$, $s$ and $G(x, s)$ represent the cover image, the secret image and the generated stego image respectively.
\begin{equation}
\min_{G}\max_{D}=E_{x\sim P(x)}[logD(x)] + E_{x\sim P(x), s\sim P(s)}[log(1-D(G(x, s)))] \label{V2}
\end{equation}
\begin{table}
\caption{Architecture details of the steganalyzer network: ConvBlock1 represents $3\times3$ Conv+BN+LeakyReLU+AvgPool, ConvBlock2 represents $1\times1$ Conv+BN and ConvBlock3 represents $1\times1$ Conv+BN+LeakyReLU. SPPBlock contains a SPP modeule and the FC represents a fully connected layer.}
\label{tab:Steganalyzer-Network}
\begin{tabular}{lcc}
\hline\noalign{\smallskip}
Layers & process & Output size \\
\noalign{\smallskip}\hline\noalign{\smallskip}
Input & / & $3\times256\times256$ \\
Layer 1 & ConvBlock1 & $8\times128\times128$ \\
Layer 2 & ConvBlock1 & $16\times64\times64$ \\
Layer 3 & ConvBlock2 & $32\times32\times32$ \\
Layer 4 & ConvBlock2 & $64\times16\times16$ \\
Layer 5 & ConvBlock3 & $128\times8\times8$ \\
Layer 6 & SPPBlock & $2688\times1$ \\
Layer 7 & FC & $128\times1$ \\
Layer 8 & FC & $2\times1$ \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
Xu et al.\cite{XuNet} studied the design of CNN structure specific for image steganalysis applications and proposed XuNet. XuNet embeds an absolute activation (ABS) in the first convolutional layer to improve the statistical modeling, applies the TanH activation function in early stages of networks to prevent overfitting, and adds batch normalization (BN) before each nonlinear activation layer. This well-designed CNN provides excellent detection performance in steganalysis. So we design our steganalyzer based on XuNet and adapt it to fit our stego images. In addition, we use the spatial pyramid pooling (SPP) module to replace the global average pooling layer. The spatial pyramid pooling (SPP) module\cite{spp} and its variants, which play a huge role in models for objection detection and semantic segmentation, break the limit of fully connected layers, so that images with arbitrary sizes can be input into convolutional networks with fully connected layers. On the other hand, the SPP module can extract more features from different receptive fields, thus improving the performance. Our steganalyzer's detail architecture is shown in Tab.~\ref{tab:Steganalyzer-Network}.
\subsection{Mixed Loss Function}\label{sec:3.4}
In previous works, Baluja~\cite{nips} used the mean square error (MSE\xspace) between the pixels of original images and the pixels of reconstructed images as the metric \eqref{eq1}. Where c and s are the cover and secret images respectively, $c^{'}$ and $s^{'}$ are the stego and revealed secret images respectively, and $\beta$ is how to weight their reconstruction errors. In particular, we should note that the error term $||c-c'||$ doesn't apply to the weights of the decoder network. On the other hand, both the encoder network and the decoder network receive the error signal $\beta||s-s'||$ for reconstructing the secret image.
\begin{equation}
L(c, c', s, s')=\| c - c'\| + \beta \| s - s'\|\label{eq1}
\end{equation}
However, the MSE\xspace just penalizes large error of two images' corresponding pixels but disregards the underlying structure in images. The human visual system (HVS) is more sensitive to luminance and color variations in texture-less regions. Zhao et al.~\cite{Zhao} analyzed the importance of perceptually-motivated losses when the resulting image of image restoration tasks is evaluated by a human observer. They compared the performance of several losses and proposed a novel, differentiable error function. Inspired by their work, we introduce the structure similarity index (SSIM\xspace)~\cite{SSIM} and its variant, the multi-scale structure similarity index (MS-SSIM\xspace)~\cite{MS-SSIM} into our metric.
The SSIM\xspace index separates the task of similarity measurement into three comparisons: luminance, contrast and structure. The luminance, contrast and structure similarity of two images are measured by \eqref{eq2}, \eqref{eq3} and \eqref{eq4} respectively. Where $\mu_x$ and $\mu_y$ are pixel average of image x and image y, $\theta_x$ and $\theta_y$ are pixel deviation of image x and image y, and $\theta_{xy}$ is the standard variance of image x and y. In addition, $C_1$, $C_2$ and $C_3$ are constants included to avoid instability when denominators are close to zero. The total calculation method of SSIM\xspace is shown in \eqref{eq5}, where $l>0, m>0, n>0$ are parameters used to adjust the relative importance of three components. More detail introduction to SSIM\xspace can be found in ~\cite{SSIM}. The value range of the SSIM\xspace index is $[0, 1]$. The higher the index is, the more similar the two images are. So we use $1 - SSIM(x, y)$ in our loss function to measure the difference of two images. And the MS-SSIM\xspace~\cite{MS-SSIM} is an enhanced variant of the SSIM\xspace index, so we also introduce it into our loss function (We use $MSSIM$ in functions to represent MS-SSIM).
\begin{equation}
L(x, y) = \frac{2\mu_x\mu_y+C_1}{\mu_x^2+\mu_y^2+C_1}\label{eq2}
\end{equation}
\begin{equation}
C(x, y) = \frac{2\theta_x\theta_y+C_2}{\theta_x^2+\theta_y^2+C_2}\label{eq3}
\end{equation}
\begin{equation}
S(x, y) = \frac{\theta_{xy}+C_3}{\theta_x\theta_y + C_3}\label{eq4}
\end{equation}
\begin{equation}
SSIM(x, y) = [L(x, y)]^{l} \cdot [C(x, y)]^{m} \cdot [S(x, y)]^{n}\label{eq5}
\end{equation}
Considering pixel value differences and structure differences simultaneously, we put MSE\xspace, SSIM\xspace and MS-SSIM\xspace together. So, the metric for the basic steganography network in our framework is as below:
\begin{equation}
\begin{aligned}
L(c, c') = \alpha (1 - SSIM(c, c')) \\
+ (1 - \alpha)(1 - MSSIM(c, c')) \\
+ \beta MSE(c, c')\label{eq6}
\end{aligned}
\end{equation}
\begin{equation}
\begin{aligned}
L(s, s') = \alpha (1 - SSIM(s, s')) \\
+ (1 - \alpha)(1 - MSSIM(s, s')) \\
+ \beta MSE(s, s')\label{eq7}
\end{aligned}
\end{equation}
\begin{equation}
L(c, c', s, s') = L(c, c') + \gamma L(s, s')\label{eq8}
\end{equation}
Where $\alpha$ and $\beta$ are hyperparameters to weigh influences of three metrics and $\gamma$ is a hyperparameter to trade off the quality of stego images and revealed secret images. Experiment results in \secref{4} will compare the performance of different loss functions.
\section{Experiments and Results}\label{sec:4}
In this section, we'll introduce our experiment details and results. Firstly, the datasets we used are LFW~\cite{lfw}, Pascal VOC 2012~\cite{voc} and ImageNet~\cite{imagenet}. The Labeled Faces in the Wild (LFW) contains more than 13000 face images belonging to 1680 people collected from the web. 10k images were selected from LFW and constituted 5k cover-secret image pairs as our training set, others of LFW were as our validation set. Pascal VOC 2012 is a dataset designed for object detection and semantic segmentation, we selected 16k images randomly to constitute 8k cover-secret image pairs as our training set and selected 5k images from the remaining part as our validation set. To further verify our model's performance on the big dataset, we did similar experiments on a subset of the ImageNet. Limited by the computing power, we only used the validation set of ImageNet as our training set which contains 50k images, these images constituted 25k cover-secret image pairs randomly. Then we selected 30k images from the test set of ImageNet as our validation set.
We used SSIM\xspace~\cite{SSIM}, Peak Signal to Noise Ration (PSNR\xspace) as metrics to measure our model's performance. It is widely accepted that the PSNR\xspace doesn't correlate well with the human's perception of image quality~\cite{Zhang}, so we just used it as a reference. In addition, we designed a CNN-based steganalyzer specially to measure our model's security.
All settings of our model on three datasets were the same. All parameters of our model were initialized by the Xavier initialization~\cite{xavier} and the initial learning rate was set as 1e-4 and was descended during training after 20 epochs. The batch size was set as 4 limited by the computing power, and we used Adam to optimize our basic model. After several attempts, we set $\alpha$, $\beta$ and $\gamma$ of the loss function as 0.5, 0.3 and 0.85 respectively, which can trade off the quality of stego images and revealed secret images very well. Because our secret message is an image, so we don't need to reveal out the secret image completely. Certainly, you can set $\gamma$ higher if you want better revealed secret images. The size of all images we used is $256 \times 256$, and the capacity of our model is 8bpp\xspace (it is equivalent to that we hide a pixel (8 bits) in a pixel).
\begin{table}
\centering
\caption{We use several loss functions to train our basic model on LFW for 50 epochs. $MSE\xspace+SSIM\xspace$ represents a mix loss of MSE\xspace and SSIM\xspace, others are similar, and $revealed$ represents revealed secret images. We can see that the mix loss of MSE\xspace, SSIM\xspace and MS-SSIM\xspace is superior than others.}
\label{tab:loss-compare}
\resizebox{0.5\textwidth}{!}{
\begin{tabular}{lllllll}
\hline\noalign{\smallskip}
Loss Function & \begin{tabular}[c]{@{}l@{}}Stego-Cover\\ PSNR\xspace (db)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Revealed-Secret\\ PSNR\xspace (db)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Stego-Cover\\ SSIM\xspace \end{tabular} & \begin{tabular}[c]{@{}l@{}}Revealed-Secret\\ SSIM\xspace \end{tabular} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
MSE\xspace & 27.97 & 26.30 & 0.8592 & 0.8391 \\% & 5e-3 & 6e-3\\
SSIM\xspace & 21.71 & 22.76 & 0.8877 & 0.8466 \\% & 7e-3 & 8e-3\\
MSE\xspace+SSIM\xspace & 27.12 & 26.71 & 0.8921 & 0.8805 \\% & 5e-3 & 6e-3\\
MSE\xspace+MS-SSIM\xspace & 23.92 & 25.97 & 0.8287 & 0.8832 \\% & 5e-3 & 5e-3\\
MSE\xspace+SSIM\xspace+MS-SSIM\xspace & 26.72 & 25.97 & \textbf{0.9305} & \textbf{0.9160} \\% & 5e-3 & 5e-3\\
\noalign{\smallskip}\hline
\end{tabular}}
\end{table}
\begin{table}
\centering
\caption{We can see that the SSIM\xspace index between stego images and their corresponding cover images of ISGAN\xspace is higher than our basic model and Atique's work\cite{Atique}.}
\label{tab:dataresults}
\resizebox{0.5\textwidth}{!}{
\begin{tabular}{lllllllll}
\hline\noalign{\smallskip}
Model & \begin{tabular}[c]{@{}l@{}}Cover\\ Image\end{tabular} & \begin{tabular}[c]{@{}l@{}}Secret\\ Image\end{tabular} & \begin{tabular}[c]{@{}l@{}}Stego-Cover\\ PSNR\xspace (db)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Revealed-Secret\\ PSNR\xspace (db)\end{tabular} & \begin{tabular}[c]{@{}l@{}}Stego-Cover\\ SSIM\xspace \end{tabular} & \begin{tabular}[c]{@{}l@{}}Revealed-Secret\\ SSIM\xspace \end{tabular} \\% & \begin{tabular}[c]{@{}l@{}}Stego-Cover \\ MSE\end{tabular} & \begin{tabular}[c]{@{}l@{}}Revealed-Secret \\ MSE\end{tabular} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
Atique's model & LFW\xspace & LFW\xspace & 33.7 & 39.9 & 0.95 \\% & \textbf{0.96} & - & - \\
Basic model & LFW\xspace & LFW\xspace & \textbf{34.28} & 33.53 & \textbf{0.9529} & 0.9453 \\% & 4.5e-3 & 4.2e-3 \\
ISGAN & LFW\xspace & LFW\xspace & \textbf{34.63} & 33.63 & \textbf{0.9573} & 0.9429 \\% & 4.2e-3 & 4.4e-3 \\
Atique's model & ImageNet\xspace & ImageNet\xspace & 32.9 & 36.6 & 0.96 & \textbf{0.96} \\% & - & - \\
Basic model & ImageNet\xspace & ImageNet\xspace & \textbf{34.57} & 33.53 & \textbf{0.9634} & 0.9510 \\% & 3.8e-3 & 4.1e-3 \\
ISGAN & ImageNet\xspace & ImageNet\xspace & \textbf{34.89} & 33.42 & \textbf{0.9681} & 0.9474 \\% & 3.8e-3 & 4.2e-3 \\
Atique's model & PASCAL-VOC12\xspace & PASCAL-VOC12\xspace & 33.7 & 35.9 & 0.96 & \textbf{0.95} \\% & - & - \\
Basic model & PASCAL-VOC12\xspace & PASCAL-VOC12\xspace & \textbf{33.79} & 33.47 & \textbf{0.9617} & 0.9475 \\% & 4.1e-3 & 4.1e-3 \\
ISGAN & PASCAL-VOC12\xspace & PASCAL-VOC12\xspace & \textbf{34.49} & 33.31 & \textbf{0.9661} & 0.9467 \\% & 4.0e-3 & 4.2e-3 \\
Atique's model & PASCAL-VOC12\xspace & LFW\xspace & 33.8 & 37.7 & 0.96 & 0.95 \\% & - & - \\
Basic model & PASCAL-VOC12\xspace & LFW\xspace & \textbf{33.85} & 37.68 & \textbf{0.9612} & 0.9503 \\% & 4.2e-3 & 4.5e-3 \\
ISGAN & PASCAL-VOC12\xspace & LFW\xspace & \textbf{34.45} & 37.59 & \textbf{0.9647} & 0.9495 \\% & 4.1e-3 & 4.4e-3 \\
ISGAN & ImageNet\xspace & PASCAL-VOC12\xspace & \textbf{34.57} & 36.58 & \textbf{0.9652} & 0.9495 \\% & 4.1e-3 & 4.4e-3 \\
\noalign{\smallskip}\hline
\end{tabular}}
\end{table}
As shown in Tab.~\ref{tab:loss-compare}, we do several experiments with different loss functions on the LFW, the result demonstrates that our proposed mixed loss function is superior to others. Tab.~\ref{tab:dataresults} describes final results of our model on three datasets, we can see that the invisibility of our model get a little improvement, while our model's performance is superior to Atique's work intuitively as shown in Fig.~\ref{fig:LFWshow}, ~\ref{fig:VOCshow} and ~\ref{fig:ImageNetshow}. Stego images generated by our model are complete similar to corresponding cover images in semantic and color, this is not reflected by SSIM\xspace. On the training set, the average SSIM\xspace index between stego images generated by our model and their corresponding cover images is more than 0.985, and the average SSIM\xspace index between revealed images and their corresponding secret images is more than 0.97. In practice, we can use several cover images to conceal one secret image and choose the best stego image to transfer on the Internet.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{lfw-revision.jpg}
\caption{Two examples on LFW. We can see that our stego images are almost same as cover images, while Atique's stego images are yellowing. By analyzing residuals between stego images and cover images, we can see that our stego images are more similar to cover images than Atique's results.}
\label{fig:LFWshow}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{pascal-revision.jpg}
\caption{Two examples on Pascal VOC12. We can see that our stego images are almost same as cover images, while Atique's stego images are yellowing. By analyzing residuals between stego images and cover images, we can even distinguish the outline of secret images from Atique's residual images, while our residual images are blurrier.}
\label{fig:VOCshow}
\end{figure}
On the other hand, by analyzing the detail difference between cover images and stego images, we can see that our residual images are darker than Atique’s, which means that our stego images are more similar to cover images and ISGAN\xspace has stronger invisibility. Additionally, from Atique’s residual images we can even distinguish secret images’ outline, while our residual images are blurrier. So these residual images can also prove that our ISGAN\xspace is securer.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{imagenet-revision.jpg}
\caption{Two examples on ImageNet. We can see that our stego images are almost same as cover images, while Atique's stego images are yellowing. Residual images between stego images and cover images show that our stego images are more similar to cover images than Atique's results.}
\label{fig:ImageNetshow}
\end{figure}
When training ISGAN\xspace, we referred some tricks from previous works~\cite{tricks}. We flipped labels when training our basic model, replaced the ReLU activation function by the LeakyReLU function, optimized the generator by Adam, optimized the steganalyzer by SGD and applied the L2 normalization to inhibit overfitting. These tricks helped us to speed up training and get better results.
To prove the improvement of the security produced by generative adversarial networks, we designed a new experiment. We used a well-trained basic model to generate 5000 stego images on LFW. These 5000 stego images and their corresponding cover images constituted a tiny dataset. We designed a new CNN-based model as a binary classifier to train on the tiny dataset. After training, we used this model to recognize stego images out from another tiny dataset which contains 2000 stego images generated by ISGAN\xspace and their corresponding cover images. Similar experiments were done on the other two datasets. The results can be seen from Tab.~\ref{tab:security-results}. ISGAN\xspace strengthens indeed the security of our basic model. And with the training going, we can see that the security of ISGAN\xspace is improving slowly.
\begin{table}[tbp]
\begin{center}
\resizebox{0.5\textwidth}{!}{
\begin{tabular}{lllll}
\hline\noalign{\smallskip}
Dataset & Basic Model & ISGAN (50) & ISGAN (100) & ISGAN (150) \\
\noalign{\smallskip}\hline\noalign{\smallskip}
LFW & 0.8305 & 0.8059 & 0.7887 & \textbf{0.7825} \\
Pascal-VOC12 & 0.7953 & 0.769 & 0.756 & \textbf{0.7438} \\
ImageNet & 0.7814 & 0.7655 & 0.7462 & \textbf{0.7360} \\
\noalign{\smallskip}\hline
\end{tabular}
}
\end{center}
\caption{Accuracy of CNN-based steganalysis model on tiny-datasets generated by basic model and ISGAN\xspace training for different epochs. Along with the training going, we can see that the security of ISGAN\xspace is improving slowly.}
\label{tab:security-results}
\end{table}
\section{Discussion and Conclusion}\label{sec:5}
Fig.~\ref{fig:secret-residual} shows the difference between revealed images and their corresponding secret images. It shows that this kind of model cannot reveal out secret images completely. This is accepted as the information in the secret image is very redundant. However, it is unsuitable for tasks which need to reveal the secret information out completely.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{secret-residual.png}
\caption{Secret images' residual image on three datasets. We can see that there are differences between original secret images and our revealed secret images, which means that ISGAN\xspace is a lossy steganography.}
\label{fig:secret-residual}
\end{figure}
As we described before, ISGAN\xspace can conceal a gray secret image into a color cover image with the same size excellently and generate stego images which are almost the same as cover images in semantic and color. By means of the adversarial training, the security is improved. In addition, experiment results demonstrate that our mixed loss function based on SSIM can achieve the state-of-art performance on the steganography task.
In addition, our steganography is done in the spatial domain and stego images must be lossless, otherwise some parts of the secret image will be lost. There may be methods to address this problem. It doesn't matter if the stego image is sightly lossy since the secret image is inherently redundant. Some noise can be added into the stego images to simulate the image loss caused by the transmission during training. Then our decoder network should be modified to fit both the revealing process and the image enhancement process together. In our future work, we'll try to figure out this problem and improve our model's robustness.
|
1,941,325,220,561 | arxiv |
\section{Discussion of Bilinear Pooling}
\label{appendix:bilinear}
In Eq.~\eqref{eq:interaction_embedding}, $g(d_i, m_{i,j})$ uses a form of bilinear pooling to explicitly capture the interaction between the Dx code and the treatment code.
The original bilinear pooling~\cite{tenenbaum12emph} derives a scalar feature $f_i$ between two embeddings $\mathbf{x}, \mathbf{y}$ such that $f_i = \mathbf{x}^T \mathbf{W}_i \mathbf{y}$ where $\mathbf{W}_i$ is a trainable weight matrix.
Since we typically extract many features $f_0, \ldots, f_i$, to capture the interaction between two embeddings, bilinear pooling requires us to train multiple weight matrices (\textit{i.e.} weight tensor).
Due to this requirement, researchers developed more efficient methods such as compact bilinear pooling~\cite{gao2016compact,fukui2016multimodal} and low-rank bilinear pooling~\cite{kim2016hadamard}, which is used in this work.
\section{Heart Failure Case-Control Selection Criteria}
\label{appendix:hf_criteria}
\begin{figure*}
\centering
\captionof{table}{Qualifying ICD-9 codes for heart failure}
\includegraphics[scale=0.6]{./figs/hf_code_table}
\label{table:hf_codes}
\end{figure*}
Case patients were 40 to 85 years of age at the time of HF diagnosis. HF diagnosis (HFDx) is defined as: 1) Qualifying ICD-9 codes for HF appeared in the encounter records or medication orders. Qualifying ICD-9 codes are displayed in Table \ref{table:hf_codes}. 2) a minimum of three clinical encounters with qualifying ICD-9 codes had to occur within 12 months of each other, where the date of diagnosis was assigned to the earliest of the three dates. If the time span between the first and second appearances of the HF diagnostic code was greater than 12 months, the date of the second encounter was used as the first qualifying encounter. The date at which HF diagnosis was given to the case is denoted as HFDx.
Up to ten eligible controls (in terms of sex, age, location) were selected for each case, yielding an overall ratio of 9 controls per case. Each control was also assigned an index date, which is the HFDx of the matched case. Controls are selected such that they did not meet the operational criteria for HF diagnosis prior to the HFDx plus 182 days of their corresponding case. Control subjects were required to have their first office encounter within one year of the matching HF case patient’s first office visit, and have at least one office encounter 30 days before or any time after the case’s HF diagnosis date to ensure similar duration of observations among cases and controls.
\section{Training Details}
\label{appendix:implementation}
All models were implemented in TensorFlow 1.4~\cite{abadi2016tensorflow}, and trained with a system equipped with Intel Xeon E5-2620, 512TB memories and 8 Nvidia Pascal Titan X's.
We used Adam~\cite{kingma2014adam} for optimization, with the learning rate $1e-3$.
In all experiments, the reported results are averaged over 5-fold random data splits: training (70\%), validation (10\%) and test (20\%).
All models were trained with the minibatch of 20 patients for 20,000 iterations to guarantee convergence.
At every 100 iterations, we evaluated the loss value of the validation set for early stopping.
For the non-linear activation functions in \texttt{MiME}\xspace, we used ReLU in all places except for the one in Eq.~\eqref{eq:visit_embedding} where we used sigmoid to benefit from its regularization effect.
We avoid the vanishing gradient problem by using the skip connections.
Note that simply adding skip connections to \textbf{sigmoid}$_{mlp}$ did not improve performance.
For the first experiment in section~\ref{ssec:exp1},
size of the visit vector $\mathbf{v}$ was 128 in all baseline models except \textbf{raw}.
We ran a number of preliminary experiments with values 64, 128, 256 and 512, and we concluded that 128 was sufficient for all models to obtain optimal performance, as the datasets $D_1, D_2$ and $D_3$ were rather small.
For $\texttt{MiME}\xspace$, we adjusted the size of the embeddings $z$ to match the number of parameters to the baselines.
\textbf{Med2Vec} was also trained to obtain 128 dimensional visit vectors.
Note that \textbf{sigmoid$_{mlp}$}, \textbf{tanh$_{mlp}$}, \textbf{relu$_{mlp}$} and \textbf{GRAM} used $128\times128$ more parameters than other models.
We used $L_2$ regularization with the coefficient $1e-4$ for all models. We did not use any dropout technique.
All models used GRU for the function $h(\mathbf{v}^{(1)}, \ldots, \mathbf{v}^{(T)})$ as described in section~\ref{ssec:exp_tasks}, the cell size of which was 128.
For the second experiment in section~\ref{ssec:exp2}, where the models were trained on gradually larger datasets $E_1, E_2, E_3$ and $E_4$, the size of $\mathbf{v}$ was set to 256 for all baseline models except \textbf{raw}. The same adjustments were made to \texttt{MiME}\xspace as before, and the cell size of GRU was also set to 256.
\section{Heart Failure Prediction Performance on Datasets $D_1, D_2$ and $D_3$, Full Version}
\label{appendix:exp1_full}
\input{tables/exp1_table_full.tex}
Table~\ref{table:exp1_full} shows the performance of all models on datasets $D_1, D_2$ and $D_3$.
An interesting finding is that both \textbf{sigmoid} and \textbf{tanh} mostly outperform \textbf{relu} in both measures in $D_1, D_2$ and $D_3$, although \textit{ReLU} is the preferred nonlinear activation for hidden layers in many studies .
This seems due to the regularizing effect of \textit{sigmoid} and \textit{tanh} functions.
Whereas \textit{ReLU} can produce outputs as high as infinity, \textit{sigmoid} and \textit{tanh} have bounded outputs.
Considering that \textbf{sigmoid}, \textbf{tanh} and \textbf{relu} all sum up the code embeddings in a visit $\mathcal{V}^{(t)}$ before applying the nonlinear activation, constraining the output of the nonlinear activation seems to work favorably, especially in $D_3$ where there are more codes per visit.
This regularization benefit, however, diminishes as the dataset grows, which can be confirmed by Table~\ref{table:exp2_full} in section~\ref{appendix:exp2_full}.
In addition, as can be seen by the performance of \textbf{sigmoid}$_{mlp}$, \textit{sigmoid} clearly suffers from the vanishing gradient problem as opposed to \textit{tanh} or \textit{ReLU} that have larger gradient values.
\section{ROC-AUC of Heart Failure Prediction on Datasets $D_1, D_2$ and $D_3$}
\label{appendix:exp1_auc}
\input{tables/appendix_exp1.tex}
Table~\ref{table:appendix_exp1_auc} shows ROC-AUC of all models on datasets $D_1, D_2$ and $D_3$.
Except for $D_1$ where patients have low visit complexity, \texttt{MiME}\xspace again consistently outperforms all baseline models.
However, the ROC-AUC gap between \texttt{MiME}\xspace and baselines is not as great as PR-AUC.
This is because ROC-AUC is determined by sensitivity (\textit{i.e.} recall, or true positive rate) and specificity (\textit{i.e.} true negative rate).
A model achieves a high specificity if it can correctly identify as many negative samples as possible, which is easier for problems with many negative samples and few positive samples.
PR-AUC, on the other hand, is determined by precision and recall.
Therefore, for a model to achieve a high PR-AUC, it must correctly retrieve as many positive samples as possible while ignoring negative samples, which is harder for problems with few positive samples.
For heart failure (HF) prediction, achieving high specificity is relatively easy as there are way more
controls (\textit{i.e.} negative samples) than cases (\textit{i.e.} positive samples).
However, correctly identifying cases while ignoring controls requires a model to recognize what differentiates cases from controls.
This means paying attention to the details of the patient records, such as the relationship between the diagnosis codes and treatment codes.
That is why \texttt{MiME}\xspace shows significant improvement in PR-AUC while showing moderate improvement in ROC-AUC.
Also, this also explains why \textbf{Med2Vec} shows very poor PR-AUC as opposed to its competitive ROC-AUC.
Med2Vec only pays attention to the co-occurrence of codes within a single visit, and not the interaction between diagnosis codes and treatment codes.
It can work as a very efficient code grouper (codes that often appear in the same visit end up having similar code embeddings), leading to a increased ROC-AUC.
But it cannot achieve a high PR-AUC, as that code grouping loses much of the subtle interaction between diagnosis codes and medication codes.
\section{Test PR-AUC on Datasets $E_1, E_2, E_3$ and $E_4$, Full Version}
\label{appendix:exp2_full}
\input{tables/exp2_table_full.tex}
Table~\ref{table:exp2_full} shows the PR-AUC of all models on datasets $E_1, E_2, E_3$ and $E_4$.
It is notable that some baseline models show fluctuating performance as dataset grows.
For example, \textbf{tanh}$_{mlp}$ showed competitive performance in small datasets, but weaker performance in large datasets.
\textbf{relu}$_{mlp}$, on the other hand, did not stand out in small datasets, but became the best baseline in large datasets.
Such behaviors, along with the finding in Appendix~\ref{appendix:exp1_full} regarding the regularization effect, suggest that we should carefully choose activation functions of our model depending on the dataset size.
\section{Test Loss and Test ROC-AUC on Datasets $E_1, E_2, E_3$ and $E_4$}
\label{appendix:exp2}
Table~\ref{table:appendix_exp2_loss} and Table~\ref{table:appendix_exp2_auc} respectively shows the test loss and test ROC-AUC of all models on datasets of varying sizes $E_1, E_2, E_3$ and $E_4$.
Both \texttt{MiME}\xspace and \texttt{MiME}\xspace$_{aux}$ consistently outperformed all baselines in terms of both test loss and test ROC-AUC, except \textbf{Med2Vec}.
Moreover, \texttt{MiME}\xspace$_{aux}$ always showed better performance than \texttt{MiME}\xspace except test loss in $E_4$, especially for the smallest dataset $E_1$, confirming our assumption that auxiliary tasks can train a robust model when large datasets are unavailable.
\textbf{tanh}$_{mlp}$ consistently showed good performance in terms of ROC-AUC across all datasets, as opposed to showing fluctuating PR-AUC in Table~\ref{table:exp2_full}.
\textbf{Med2Vec} again showed a competitive ROC-AUC in all datasets, even outperforming \texttt{MiME}\xspace$_{aux}$ in $E_3$.
This suggests that initializing \texttt{MiME}\xspace's code embeddings with Med2Vec can be an interesting future direction as it may lead to an even better performance.
\input{tables/appendix_exp2_loss.tex}
\input{tables/appendix_exp2_auc.tex}
\section{Sequential Disease Prediction}
\label{appendix:sdp_task}
\noindent\underline{\textit{Sequential disease prediction}}
In order to test if leveraging EHR's inherent structure is a strategy generalizable beyond heart failure prediction, we test \texttt{MiME}\xspace's prediction performance in another context, namely sequential disease prediction.
The objective is to predict the diagnosis codes occurring in visit $\mathcal{V}^{(t+1)}$, given all past visits $\mathcal{V}^{(1)}, \mathcal{V}^{(2)}, \ldots, \mathcal{V}^{(t)}$.
The input features are diagnosis codes $\mathcal{A}$ and treatment codes $\mathcal{B}$, while the output space only consists of diagnosis codes $\mathcal{A}$.
This task is useful for preemptively assessing the patient's potential future risk~\cite{choi2016doctor}, but is also appropriate for assessing how well a model captures the progression of the patient status over time.
We used GRU as the mapping function $h(\cdot)$, and hidden vectors from all timesteps were fed to the softmax function with $|\mathcal{A}|$ output classes to perform sequential prediction.
\section{Experiment Results for Sequential Disease Prediction}
\label{appendix:sdp_results}
\input{tables/appendix_sdp.tex}
\input{tables/accuracy_per_freq.tex}
After training all models until convergence, performance was measured by sorting the predicted diagnosis codes for $\mathcal{V}^{(t+1)}$ by their prediction values, and calculating $Recall@k$ using the true diagnosis codes of $\mathcal{V}^{(t+1)}$.
Table~\ref{table:sdp} shows the performance of all models for sequential disease prediction.
\texttt{MiME}\xspace demonstrated the best performance in all metrics, showing that \texttt{MiME}\xspace can properly capture the temporal progression of the patient status.
It is noteworthy that \textbf{linear} displayed very competitive performance compared to the best performing models.
This is due to the fact that chronic conditions such as hypertension or diabetes persist over a long period of time, and sequentially predicting them becomes an easy task that does not require an expressive model.
This was also reported in \cite{choi2016doctor} where a strategy to choose the most frequent diagnosis code as the prediction showed competitive performance in a similar task.
In order to study whether explicitly incorporating the structure of EHR helps when there are small data volume, we calculated the test performance in terms of $Precision@5$ for predicting each diagnosis (Dx) code of $\mathcal{A}$.
In Table~\ref{table:accuracy_per_freq}, we report average $Precision@5$ for four different groups of Dx codes, where the groups were formed by the rarity/frequency of the Dx codes in the training data.
For example, the first column represents
the Dx codes that appear in the 0.01\%-0.05\% of the entire visits (433407) in the training data, which are very rare diseases.
On the other hand, the Dx codes in the last column appear in maximum 13.39\% of the visits, indicating high-prevalence diseases.
We selected the best performing activation function \textbf{tanh} among the three.
As can be seen from Table~\ref{table:accuracy_per_freq}, except for the rarest Dx codes, \texttt{MiME}\xspace outperforms all other baseline models, as much as 11.6\% relative gain over \textbf{tanh}$_{mlp}$.
It is notable that \textbf{Med2Vec} demonstrated the greatest performance for the rarest Dx code group. However, the benefit of using pre-trained embedding vectors quickly diminishes to the point of degrading the performance when there are at least several hundred training samples.
Overall, \texttt{MiME}\xspace demonstrated good performance in prediction tasks in diverse settings, and it is notable that they significantly outperformed the baseline models in the more complex task, namely HF prediction, where the relationship between the label and the features (\textit{i.e.} codes) from the data was more than straightforward.
\subsection{Source of Data}
\label{ssec:exp_data}
We conducted all our experiments using EHR data provided by Sutter Health\xspace.
The dataset was constructed for a study designed to predict a future diagnosis of heart failure, and included EHR data from 30,764 senior patients 50 to 85 years of age.
We extracted the diagnosis codes, medication codes and the procedure codes from encounter records, and related orders.
We used Clinical Classification Software for ICD9-CM\footnote{https://www.hcup-us.ahrq.gov/toolssoftware/ccs/ccs.jsp} to group the ICD9 diagnosis codes into 388 categories.
Generic Product Identifier Drug Group\footnote{http://www.wolterskluwercdi.com/drug-data/medi-span-electronic-drug-file/} was used to group the medication codes into 99 categories.
Clinical Classifications Software for Services and Procedures\footnote{https://www.hcup-us.ahrq.gov/toolssoftware /ccs\_svcsproc/ccssvcproc.jsp} was used to group the CPT procedure codes into 1,824 categories.
Any code that did not fit into the grouper formed its own category.
Table~\ref{table:data_stat} summarizes data statistics.
\input{tables/data_stat.tex}
\subsection{Baseline Models}
\label{ssec:exp_baselines}
First, we use Gated Recurrent Units (GRU)~\cite{cho2014learning} with different embedding strategies to map visit embedding sequence $\mathbf{v}^{(1)}, \ldots, \mathbf{v}^{(T)}$ to a patient representation $\mathbf{h}$:
\begin{itemize}[leftmargin=5.5mm]
\item \textbf{raw}: A single visit $\mathcal{V}^{(t)}$ is represented by a binary vector $\mathbf{x}^{(t)} \in \{0,1\}^{|\mathcal{A}| + |\mathcal{B}|}$.
Only the dimensions corresponding to the codes occurring in that visit is set to 1, and the rest are 0.
\item \textbf{linear}: The binary vector $\mathbf{x}^{(t)}$ is linearly transformed to a lower-dimensional vector $\mathbf{v}^{(t)} = \mathbf{W}_x \mathbf{x}^{(t)}$ where $\mathbf{W}_x \in \mathbb{R}^{b \times (|\mathcal{A}| + |\mathcal{B}|)}$ is the embedding matrix.
This is equivalent to taking the vector representations of the codes (\textit{i.e.} columns of the embedding matrix $\mathbf{W}_x$) in the visit $\mathcal{V}^{(t)}$, and summing them up to derive a single vector $\mathbf{v}^{(t)} \in \mathbb{R}^{b}$.
\item \textbf{sigmoid, tanh, relu}: The binary vector $\mathbf{x}^{(t)}$ is transformed to a lower-dimensional vector $\mathbf{v}^{(t)} = \sigma(\mathbf{W}_x \mathbf{x}^{(t)})$ where we use either \textit{sigmoid}, \textit{tanh}, or \textit{ReLU} for $\sigma(\cdot)$ to add non-linearity to \textbf{linear}.
\item \textbf{sigmoid$_{mlp}$, tanh$_{mlp}$, relu$_{mlp}$}: We add one more layer to \textbf{sigmoid}, \textbf{tanh} and \textbf{relu} to increase their expressivity.
The visit embedding is now $\mathbf{v}^{(t)} = \sigma(\mathbf{W}_{x_2} \sigma(\mathbf{W}_{x_1} \mathbf{x}^{(t)}))$ where $\sigma$ is either sigmoid, tanh or ReLU. We do not test \textbf{linear$_{mlp}$} since two consecutive linear layers can be collapsed to a single linear layer.
\end{itemize}
Second, we also compare with two advanced embedding methods that are specific designed for modeling EHR data.
\begin{itemize}[leftmargin=5.5mm]
\item \textbf{Med2Vec}: We use Med2Vec~\cite{choi2016multi} to learn visit representations, and use those fixed vectors as input to the prediction model. We test this model as a representative case of unsupervised embedding approach using EHR data.
\item \textbf{GRAM}: We use GRAM~\cite{choi2017gram}, which is equivalent to injecting domain knowledge (ICD9 Dx code tree) to \textbf{tanh} via attention mechanism.
We test this model as a representative case of incorporating external domain knowledge.
\end{itemize}
\subsection{Prediction Tasks}
\label{ssec:exp_tasks}
\noindent\underline{\textit{Heart failure prediction}}
The objective is to predict the first diagnosis of heart failure (HF), given an 18-months observation records discussed in section~\ref{ssec:exp_data}.
Among 30,764 patients, 3,414 were case patients who were diagnosed with HF within a 1-year window after the 18-months observation.
The remaining 27,350 patients were controls.
The case-control selection criteria are detailed in~\cite{vijayakrishnan2014prevalence} and summarized in Appendix~\ref{appendix:hf_criteria}.
While an accurate prediction of HF can save a large amount of costs and lives~\cite{roger2004trends}, this task is also suitable for assessing how well a model can learn the relationship between the external label (\textit{i.e.} the label information is not inherent in the EHR data) and the features (\textit{i.e.} codes).
We applied logistic regression to the patient representation $\mathbf{h}$ to obtain a value between $0$ (no HF onset) and $1$ (HF onset).
All models were trained end-to-end except \textbf{Med2Vec}.
We report Area under the Precision-Recall Curve (PR-AUC) in the experiment and Area under the Receiver Operating Characteristic (ROC-AUC) in the appendix, as PR-AUC is considered a better measure for imbalanced data like ours~\cite{saito2015precision,davis2006relationship}.
Implementation and training configurations are described in Appendix~\ref{appendix:implementation}.
We also performed \textit{sequential disease prediction (SDP)} (predicting all diagnoses of the next visit at every timestep) where \texttt{MiME}\xspace demonstrated superior performance over all baseline models. The detailed description and results of SDP are provided in Appendix~\ref{appendix:sdp_task} and Appendix~\ref{appendix:sdp_results} respectively.
\subsection{Experiment 1: Varying the Data Size}
\label{ssec:exp2}
\begin{figure}[t]
\centering
\includegraphics[width=.6\textwidth]{figs/exp2_plot.png}
\caption{
Test PR-AUC of HF prediction for increasing data size. A table with the results of all baseline models is provided in Appendix~\ref{appendix:exp2_full}
}
\label{fig:exp2}
\end{figure}
To evaluate \texttt{MiME}\xspace's performance in another perspective, we created four datasets $E_1, E_2, E_3, E_4$ from the original data such that each dataset consisted of patients with varying maximum sequence length $T_{max}$ (\textit{i.e.} maximum number of visits).
In order to simulate a new hospital collecting patient records over time, we increased $T_{max}$ for each dataset such that $10, 20, 30, 150$ for $E_1, E_2, E_3, E_4$ respectively.
Each dataset had 6299 (414 cases), 15794 (1177 cases), 21128 (1848 cases), 27428 (3173 cases) patients respectively.
For \texttt{MiME}\xspace$_{aux}$, we used the same $0.015$ for the auxiliary loss coefficient $\lambda_{aux}$.
Figure~\ref{fig:exp2} shows the test PR-AUC for HF prediction across all datasets (loss and ROC-AUC are described in Appendix~\ref{appendix:exp2}).
Again we show the strongest activation functions \textbf{tanh} and \textbf{tanh}$_{mlp}$ here and provide the full table in Appendix~\ref{appendix:exp2_full}.
We can readily see that \texttt{MiME}\xspace outperforms all baseline models across all datasets.
However, the performance gap between \texttt{MiME}\xspace and the baselines are larger in datasets $E_1, E_2$ than in datasets $E_3, E_4$, confirming our assumption that exploiting the inherent structure of EHR can alleviate the data insufficiency problem.
Especially for the smallest dataset $E_1$, \texttt{MiME}\xspace$_{aux}$ (0.2831 PR-AUC) demonstrated significantly better performance than the best baseline \textbf{tanh}$_{mlp}$ (0.2462 PR-AUC), showing $15\%$ relative improvement.
It is notable that \texttt{MiME}\xspace consistently outperformed \textbf{GRAM} in both Table~\ref{table:exp1} and Figure~\ref{fig:exp2} in terms of test loss and test PR-AUC.
To be fair, GRAM was only using Dx code hierarchy (thus ungrouped 5814 Dx codes were used), and no additional domain knowledge regarding treatment codes.
However, the experiment results tell us that even without resorting to external domain knowledge, we can still gain improved predictive performance by carefully studying the EHR data and leveraging its inherent structure.
\subsection{Experiment 2: Varying Visit Complexity}
\label{ssec:exp1}
\input{tables/exp1_table_short.tex}
Next, we conducted a series of experiments to confirm that \texttt{MiME}\xspace can indeed capture the relationship between Dx codes and treatment codes, thus producing robust performance in small datasets.
Specifically, we created three small datasets $D_1, D_2, D_3$ from the original data such that each dataset consisted of patients with varying degree of Dx-treatment interactions (\textit{i.e.} visit complexity).
We defined \textit{visit complexity} as below to calculate for a patient the percentage of visits that have at least two diagnosis codes associated with different sets of treatment codes,
\small
\begin{equation*}
\mbox{visit complexity} = \frac{\# \mathcal{V}^{(t)} \mbox{ where } |set(\mathcal{M}^{(t)}_1, \ldots, \mathcal{M}^{(t)}_{|\mathcal{V}^{(t)}|} )| \ge 2}{T}
\end{equation*}
\normalsize
where $T$ denotes the total number of visits.
For example, in Figure~\ref{fig:ehr_graph}, the $t$-th visit $\mathcal{V}^{(t)}$ has \textit{Fever} associated with no treatments, and \textit{Cough} associated with two treatments.
Therefore $\mathcal{V}^{(t)}$ qualifies as a complex visit.
From the original dataset, we selected patients with a short sequence (less than 20 visits) to simulate a hospital newly equipped with a EHR system, and there aren't much data collected yet.
Among the patients with less than 20 visits, we used visit complexity ranges $0-15\%, 15-30\%, 30-100\%$ to create $D_1, D_2, D_3$ consisting of 5608 (464 HF cases), 5180 (341 HF cases), 5231 (383 HF cases) patients respectively.
For training $\texttt{MiME}\xspace$ with auxiliary tasks, we explored various $\lambda_{aux}$ values between $0.01-0.1$, and found $0.015$ to provide the best performance, although other values also improved the performance in varying degrees.
Table~\ref{table:exp1} shows the HF prediction performance for the dataset $D_1, D_2$ and $D_3$.
To enhance readability, we show here the results of the strongest activation function \textbf{tanh} and \textbf{tanh}$_{mlp}$, and we report test loss and test PR-AUC.
The results of other activation functions and the test ROC-AUC are provided in Appendix~\ref{appendix:exp1_full} and Appendix~\ref{appendix:exp1_auc}.
Table~\ref{table:exp1} provides two important messages.
First of all, both \texttt{MiME}\xspace and \texttt{MiME}\xspace$_{aux}$ show close to the best performance in all datasets $D_1, D_2$ and $D_3$, especially high complexity dataset $D_3$
This confirms that \texttt{MiME}\xspace indeed draws its power from the interactions between Dx codes and treatment codes, with or without the auxiliary tasks.
In $D_1$, patients' visits do not have much structure, that it makes little difference whether we use \texttt{MiME}\xspace or not, and its performance is more or less similar to many baselines.
Second, auxiliary tasks indeed help \texttt{MiME}\xspace generalize better to patients unseen during training.
In all datasets $D_1, D_2$ and $D_3$, \texttt{MiME}\xspace$_{aux}$ outperforms \texttt{MiME}\xspace in all measures, especially in $D_3$ where it shows PR-AUC $0.4787$ ($8.4\%$ relative improvement over the best baseline \textbf{tanh}).
\subsection{Notations of \texttt{MiME}\xspace}
\input{tables/notation.tex}
Assume a patient has a sequence of visits $\mathcal{V}^{(1)}, \ldots, \mathcal{V}^{(t)}$ over time, where each visit $\mathcal{V}^{(t)}$ contains a varying number of diagnosis (Dx) objects $\mathcal{O}_{1}^{(t)}, \ldots, \mathcal{O}_{|\mathcal{V}^{(t)}|}^{(t)}$.
Each $\mathcal{O}_{i}^{(t)}$ consists of a single Dx code $d_{i}^{(t)} \in \mathcal{A}$ and a set of associated treatments (medications or procedures) $\mathcal{M}_{i}^{(t)}$.
Similarly, each $\mathcal{M}_{i}^{(t)}$ consists of varying number of treatment codes $m_{i,1}^{(t)}, \ldots, m_{i,|\mathcal{M}_{i}^{(t)}|}^{(t)} \in \mathcal{B}$.
To reduce clutter, we omit the superscript $(t)$ indicating $t$-th visit, when we are discussing a single visit.
Table~\ref{table:notation} summarizes notations we will use throughout the paper.
In Figure~\ref{fig:ehr_graph}, there are five Dx codes, hence five Dx objects $\mathcal{O}_{1}^{(t)}, \ldots, \mathcal{O}_{5}^{(t)}$. More specifically, the first Dx object $\mathcal{O}_{1}$ has $d_1^{(t)} = \textit{Fatigue}$ as the Dx code, but no treatment codes. $\mathcal{O}_{2}$, on the other hand, has Dx code $d_2^{(t)} = \textit{Cough}$ and two associated treatment codes $m_{2,1}^{(t)} = \textit{Benzonatate}$ and $m_{2,2}^{(t)} = \textit{Acetaminophen}$. In this case, we can use $g(d_2^{(t)}, m_{2,1}^{(t)})$ to capture the interaction between Dx code {\it Cough} and treatment code {\it Benzonatate}, which will be fed to $f(d_2^{(t)}, \mathcal{M}_{2}^{(t)})$ to obtain the vector representation of Dx object $\mathbf{o}^{(t)}_{2}$.
Using the five Dx object embeddings $\mathbf{o}^{(t)}_{1}, \ldots, \mathbf{o}^{(t)}_{5}$, we can obtain a visit embedding $\mathbf{v}^{(t)}$.
In addition, some treatment codes (\textit{e.g. Acetaminophen}) can be shared by two or more Dx codes (\textit{e.g. Cough, Fever}), if the doctor ordered a single medication for more than one diagnosis.
Then each Dx object will have its own copy of the treatment code attached to it, in this case denoted, $m_{2,2}^{(t)}$ and $m_{3,1}^{(t)}$, respectively.
\subsection{Description of \texttt{MiME}\xspace}
\label{ssec:model_description}
\noindent\textbf{Multilevel Embedding}
As discussed earlier, previous approaches often flatten a single visit such that Dx codes and treatment codes are packed together so that a single visit $\mathcal{V}^{(t)}$ can be expressed as a binary vector $\mathbf{x}^{(t)} \in \{0,1\}^{|\mathcal{A}| + |\mathcal{B}|}$ where each dimension corresponds to a specific Dx and treatment code.
Then a patient's visit sequence is encoded as:
\begin{align*}
\mathbf{v}^{(t)} & = \sigma(\mathbf{W}_x \mathbf{x}^{(t)} + \mathbf{b}_x) \\
\mathbf{h} & = h(\mathbf{v}^{(1)}, \mathbf{v}^{(2)}, \ldots, \mathbf{v}^{(t)})
\end{align*}
where $\mathbf{W}_x$ is the embedding matrix that converts the binary vector $\mathbf{x}$ to a lower-dimensional visit representation\footnote{We omit bias variables throughout the paper to reduce clutter.}, $\sigma$ a non-linear activation function such as sigmoid or rectified linear unit (ReLU), $h(\cdot)$ a function that maps a sequence of visit representations $\mathbf{v}^{(0)}, \ldots, \mathbf{v}^{(t)}$ to a patient representation $\mathbf{h}$.
In contrast, \texttt{MiME}\xspace effectively derives a visit representation $\mathbf{v}^{(t)}$, than can be plugged into any $h(\cdot)$ for the downstream prediction task.
$h(\cdot)$ can simply be an RNN or a combination of RNNs and CNN and attention mechanisms~\cite{bahdanau2014neural}.
\begin{figure}[t]
\centering
\includegraphics[width=0.85\textwidth, height=7cm]{figs/model_architecture}
\caption{
Prediction model using \texttt{MiME}\xspace.
Codes are embedded into multiple levels: diagnosis-level, visit-level, and patient-level.
Final prediction $p(\mathbf{y}|\mathbf{h})$ is based on the patient representation $\mathbf{h}$, which is derived from visit representations $\mathbf{v}^{(0)}, \mathbf{v}^{(1)}, \ldots$, where each $\mathbf{v}^{(t)}$ is generated using \texttt{MiME}\xspace framework.
As shown in the \textit{Treatment level}, \texttt{MiME}\xspace explicitly captures the interactions between a diagnosis code and the associated treatment codes.
\texttt{MiME}\xspace also uses those codes as auxiliary prediction targets to improve generalizability when large training data are not available.
}
\label{fig:model}
\end{figure}
\texttt{MiME}\xspace explicitly captures the hierarchy between Dx codes and treatment codes depicted in Figure~\ref{fig:ehr_graph}.
Figure~\ref{fig:model} illustrates how \texttt{MiME}\xspace builds the representation of $\mathcal{V}$ (omitting the superscript $(t)$) in a bottom-up fashion via multilevel embedding.
In a single Dx object $\mathcal{O}_i$, a Dx code $d_i$ and its associated treatment codes $\mathcal{M}_i$ are used to obtain a vector representation of $\mathcal{O}_i$, $\mathbf{o}_i$.
Then multiple Dx object embeddings $\mathbf{o}_0, \ldots, \mathbf{o}_{|\mathcal{V}|}$ in a single visit are used to obtain a visit embedding $\mathbf{v}$, which in turn forms a patient embedding $\mathbf{h}$ with other visit embeddings.
The formulation of \texttt{MiME}\xspace is as follows:
\begin{align}
& \mathbf{v} = \sigma \bigg( \mathbf{W}_v \Big( \underbrace{\sum_{i}^{|\mathcal{V}|} f ( d_i, \mathcal{M}_i ) }_{\text{F: used for skip-connection}}\Big) \bigg) + F \label{eq:visit_embedding} \\
& f(d_i, \mathcal{M}_i) = \mathbf{o}_i = \sigma \bigg( \mathbf{W}_{o} \Big( \underbrace{r(d_i) + \sum_{j}^{|\mathcal{M}_i|} g(d_i, m_{i,j})}_{\text{G: used for skip-connection}} \Big) \bigg) + G \label{eq:dx_object_embedding} \\
& g(d_i, m_{i,j}) = \sigma \big( \mathbf{W}_{m} r(d_i) \big) \odot r(m_{i,j}) \label{eq:interaction_embedding}
\end{align}
where Eq.~\eqref{eq:visit_embedding}, Eq.~\eqref{eq:dx_object_embedding} and Eq.~\eqref{eq:interaction_embedding} describe \texttt{MiME}\xspace in a top-down fashion, respectively corresponding to \textit{Visit level}, \textit{Diagnosis level} and \textit{Treatment level} in Figure~\ref{fig:model}.
In Eq.~\eqref{eq:visit_embedding}, a visit embedding $\mathbf{v}$ is obtained by summing Dx object embeddings $\mathbf{o}_1, \ldots, \mathbf{o}_{|\mathcal{V}|}$, which are then transformed with $\mathbf{W}_v \in \mathbb{R}^{z \times z}$.
$\sigma$ is a non-linear activation function such as sigmoid or rectified linear unit (ReLU).
In Eq.~\eqref{eq:dx_object_embedding}, $\mathbf{o}_i$ is obtained by summing $r(d_i) \in \mathbb{R}^{z}$, the vector representation of the Dx code $d_i$, and the effect of the interactions between $d_i$ and its associated treatments $\mathcal{M}_i$, which are then transformed with $\mathbf{W}_o \in \mathbb{R}^{z \times z}$.
The interactions captured by $g(d_i, m_{i,j})$ are added to the $r(d_i)$, which can be interpreted as adjusting the diagnosis representation according to its associated treatments (medications and procedures).
Note that in both Eq.~\eqref{eq:visit_embedding} and Eq.~\eqref{eq:dx_object_embedding}, $F$ and $G$ are used to denote skip-connections~\cite{he2016deep}.
In Eq.~\eqref{eq:interaction_embedding}, the interaction between a Dx code embedding $r(d_i)$ and a treatment code embedding $r(m_{i,j})$ is captured by element-wise multiplication $\odot$.
Weight matrix $\mathbf{W}_{m} \in \mathbb{R}^{z \times z}$ sends the Dx code embedding $r(d_i)$ into another latent space, where the interaction between $d_i$ and the corresponding $m_{i,j}$ can be effectively captured.
The formulation of Eq.~\eqref{eq:interaction_embedding} was inspired by recent developments in bilinear pooling technique~\cite{tenenbaum12emph,gao2016compact,fukui2016multimodal,kim2016hadamard}, which we discuss in more detail in Appendix~\ref{appendix:bilinear}.
With Eq.~\eqref{eq:interaction_embedding} in mind, $G$ in Eq.~\eqref{eq:dx_object_embedding} can also be interpreted as $r(d_i)$ being skip-connected to the sum of interactions $g(d_i, m_{i,j})$.
\noindent\textbf{Joint Training with Auxiliary Tasks}
Patient embedding $\mathbf{h}$ is often used for specific prediction tasks, such as heart failure prediction or mortality.
The representation power of $\mathbf{h}$ comes from properly capturing each visit $\mathcal{V}^{(t)}$, and modeling the longitudinal aspect with the function $h(\mathbf{v}_0, \ldots, \mathbf{v}_t)$.
Since the focus of this work is on modeling a single visit $\mathcal{V}^{(t)}$, we perform auxiliary predictions as follows:
\begin{align}
& \hat{d}_{i}^{(t)} = p(d_{i}^{(t)} | \mathbf{o}_{i}^{(t)}) = \mbox{softmax}(\mathbf{U}_{d} \mathbf{o}_{i}^{(t)}) \label{eq:aux_diagnosis} \\
& \hat{m}_{i,j}^{(t)} = p(m_{i,j}^{(t)} | \mathbf{o}_{i}^{(t)}) = \sigma (\mathbf{U}_{m} \mathbf{o}_{i}^{(t)}) \label{eq:aux_treatment} \\
& L_{aux} = -\lambda_{aux} \sum_{t}^{T} \bigg( \sum_{i}^{|\mathcal{V}^{(t)}|} \Big( CE(d_{i}^{(t)}, \hat{d}_{i}^{(t)}) + \sum_{j}^{|\mathcal{M}^{(t)}_{i}|} CE(m_{i,j}^{(t)}, \hat{m}_{i,j}^{(t)}) \Big) \bigg) \label{eq:aux_loss}
\end{align}
Given Dx object embeddings $\mathbf{o}_{1}^{(t)}, \ldots, \mathbf{o}_{|\mathcal{V}^{(t)}|}^{(t)}$, while aggregating them to obtain $\mathbf{v}^{(t)}$ as in Eq.~\eqref{eq:visit_embedding}, \texttt{MiME}\xspace predicts the Dx code $d_{i}^{(t)}$, and the associated treatment code $m_{i,j}^{(t)}$ as depicted by Figure~\ref{fig:model}.
In Eq.~\eqref{eq:aux_diagnosis} and Eq.~\eqref{eq:aux_treatment},
$\mathbf{U}_{d} \in \mathbb{R}^{|\mathcal{A}| \times z}$ and $\mathbf{U}_{m} \in \mathbb{R}^{|\mathcal{B}| \times z}$ are weight matrices used to compute the the prediction of Dx code $\hat{d}_{i}^{(t)}$ and the prediction of the treatment code $\hat{m}_{i,j}^{(t)}$, respectively.
In Eq.~\eqref{eq:aux_loss}, $T$ denotes the total number of visits the patient made, $CE(\cdot, \cdot)$ the cross-entropy function
and $\lambda_{aux}$ the coefficient for the auxiliary loss term.
We used the softmax function for predicting $d_{i}^{(t)}$ since in a single Dx object $\mathcal{O}_{i}^{(t)}$, there is only one Dx code involved.
However, there could be no (or many) treatment codes associated with $\mathcal{O}_{i}^{(t)}$, and therefore we used $|\mathcal{B}|$ number of sigmoid functions for predicting each treatment code.
Auxiliary tasks are based on the inherent structure of the EHR data, and require no additional labeling effort.
These auxiliary tasks guide the model to learn Dx object embeddings $\mathbf{o}_{i}^{(t)}$ that are representative of the specific codes involved with it.
Correctly capturing the events within a visit is the basis of all downstream prediction tasks, and these general-purpose auxiliary tasks, combined with the specific target task, encourage the model to learn visit embeddings $\mathbf{v}^{(t)}$ that are not only tuned for the target prediction task, but also grounded in general-purpose foundational knowledge.
\section{Introduction}
\label{sec:intro}
\input{intro.tex}
\section{Method}
\label{sec:method}
\input{method.tex}
\section{Experiments}
\label{sec:exp}
\input{exp.tex}
\section{Related Work}
\label{sec:background}
\input{background.tex}
\section{Conclusion}
\label{sec:conclusion}
\input{conclusion.tex}
\subsubsection*{Acknowledgments}
This work was supported by the National Science Foundation, award IIS-\#1418511 and CCF-
\#1533768, the National Institute of Health award 1R01MD011682-01 and R56HL138415, and Samsung Scholarship.
We would also like to thank Sherry Yan for her helpful comments on the original manuscript.
\subsection{EHR Data Modeling}
In most EHR systems, EHR data can be seen as a collection of individual patient records, where each patient has a sequence of encounters over time.
Within an encounter, different types of events are intertwined via the clinical decision making process.
Typically, a patient is first diagnosed by a physician, which generates a set of diagnosis codes.
Then the physician can order medications or procedures based on the diagnosis, which generates a set of medication codes and procedure codes.
Therefore, there is an explicit, perhaps causal, relationship between diagnosis codes and medication/procedure codes, as depicted by Figure~\ref{fig:ehr_graph}.
\begin{figure}[t]
\centering
\includegraphics[width=.7\textwidth]{figs/ehr_graph}
\caption{A visit record of a single patient. Red denotes diagnosis codes, and blue denotes medication/procedure codes. A visit is not merely a set of codes, but rather a hierarchically structured codes with heterogeneous relations.
For example, while both \textit{Acetaminophen} and \textit{IV fluid} form an explicit relationship with \textit{Fever}, they also are correlated with each other as children of \textit{Fever}.}
\label{fig:ehr_graph}
\end{figure}
In order to leverage deep learning's ability to learn complex features from input data, the EHR data must be transformed into a vector.
A patient can be seen as a sequence of visits (\textit{i.e.} encounters), and the visit sequence can be transformed into a vector using recurrent neural networks (RNN)~\cite{elman1990} or convolutional neural networks (CNN)~\cite{LeCun:2004:LMG:1896300.1896315,krizhevsky2012imagenet}.
However, transforming a visit to a vector leaves some room for imagination as it has a multilevel structure with inter-code relationships.
A typical approach adopted by a number of previous works~\cite{choi2016using, choi2016doctor, choi2016retain, choi2017gram, ma2017dipole, bajor2016predicting} flattens all events within a visit and represents it as a vector of individual features, losing the structural information embedded in the visit.
In this work, \texttt{MiME}\xspace focuses on the internal structure information of EHR data, and explicitly captures the heterogeneous relationships between the diagnosis codes and the medication/procedure codes.
Although we focus on diagnosis codes and medication/procedures codes in this work, the proposed methodology \texttt{MiME}\xspace can be applied to model other relations in EHR data as well.
\subsection{Notations of \texttt{MiME}\xspace}
\begin{table}[t]
\caption{Notations for \texttt{MiME}\xspace}
\label{table:notation}
\centering
\begin{tabular}{c|l}
\hline
\bf Notation & \bf Definition \\
\hline
\hline
$\mathcal{A}$ & Set of all diagnosis codes.\\
$\mathcal{B}$ & Set of all treatment (medications and procedures) codes.\\
$\mathcal{V}^{{(t)}}$ & A patient's $t$-th hospital visit. It is a set of diagnosis objects $\mathcal{O}^{(t)}_0, \mathcal{O}^{(t)}_1, \ldots$.\\
$\mathbf{v}^{(t)}$ & A vector representation of $\mathcal{V}^{(t)}$.\\
$\mathcal{O}^{(t)}_{i}$ & $i$-th diagnosis object of $t$-th hospital visit.\\
& It contains a single diagnosis code $d_{i}^{(t)} \in \mathcal{A}$ and the associated treatment codes $\mathcal{M}_{i}^{(t)}$.\\
$\mathbf{o}^{(t)}_{i}$ & A vector representation of $\mathcal{O}^{(t)}_{i}$.\\
$m_{i,j}^{(t)}$ & $j$-th treatment code of $\mathcal{M}_{i}^{(t)}$. It is an element of $\mathcal{B}$.\\
$g(d_{i}^{(t)}, m_{i,j}^{(t)})$ & A function that captures the interaction between $d_{i}^{(t)}$ and $m_{i,j}^{(t)}$.\\
$f(d_{i}^{(t)}, \mathcal{M}_{i}^{(t)})$ & A function that derives $\mathbf{o}^{(t)}_{i}$.\\
$r(\cdot)$ & A function that retrieves $d_{i}^{(t)}$ or $m_{i,j}^{(t)}$'s embedding vector.\\
\hline
\end{tabular}
\end{table}
Notations we will use throughout the paper are described in Table~\ref{table:notation}.
Let's assume a patient makes a sequence of visits $\mathcal{V}^{(1)}, \mathcal{V}^{(2)}, \ldots, \mathcal{V}^{(t)}$ over time.
Each visit $\mathcal{V}^{(t)}$ contains a varying number of diagnosis (Dx) objects $\mathcal{O}_{1}^{(t)}, \mathcal{O}_{2}^{(t)}, \ldots, \mathcal{O}_{|\mathcal{V}^{(t)}|}^{(t)}$, where each $\mathcal{O}_{i}^{(t)}$ consists of a single Dx code $d_{i}^{(t)} \in \mathcal{A}$ and a set of associated treatments (medications or procedures) $\mathcal{M}_{i}^{(t)}$, which in turn consists of treatment codes $m_{i,1}^{(t)}, m_{i,2}^{(t)}, \ldots, m_{i,|\mathcal{M}_{i}^{(t)}|}^{(t)} \in \mathcal{B}$.
To reduce clutter, we omit the superscript $(t)$ indicating $t$-th visit, when we are discussing a single visit.
In Figure~\ref{fig:ehr_graph}, there are five Dx codes, hence five Dx objects.
If we count from leftmost Dx (\textit{Fatigue}) to the rightmost Dx (\textit{Chest paint}), the first Dx object $\mathcal{O}_{1}$ has \textit{Fatigue} as the Dx code, but no treatment codes.
$\mathcal{O}_{2}$, on the other hand, has \textit{Cough} as the Dx code and two associated treatment codes, \textit{Benzonatate} and \textit{Acetaminophen}.
In such cases, we can use $g(\cdot, \cdot)$ to capture the interaction between a Dx code and a treatment code, which will be fed to $f(\cdot)$ to obtain the vector representation of a Dx object.
Figure~\ref{fig:model} depicts the overall process of our approach, further details will be described in the following section.
As shown by Figure~\ref{fig:ehr_graph}, some treatment codes (\textit{e.g. Acetaminophen}) can be shared by two or more Dx codes (\textit{e.g. Cough, Fever}), if the doctor ordered a single medication for more than one diagnosis.
In that case, each Dx object will have its own copy of the treatment code attached to it.
\subsection{Description of \texttt{MiME}\xspace}
\label{ssec:model_description}
\begin{figure}[ht]
\centering
\includegraphics[width=0.85\textwidth]{figs/FLOW2.png}
\caption{Prediction model using \texttt{MiME}\xspace. Medical codes are embedded into multiple levels: diagnosis-level, visit-level, and patient-level. Final prediction $p(\mathbf{y}|\mathbf{h})$ is based on the patient representation $\mathbf{h}$, which is derived from visit representations $\mathbf{v}^{(0)}, \mathbf{v}^{(1)}, \ldots$, where each $\mathbf{v}^{(t)}$ is generated using \texttt{MiME}\xspace framework. \ecedit{Talk about capturing interaction and auxiliary task.}
}
\label{fig:model}
\end{figure}
As discussed earlier, previous approaches often flatten the EHR structure such that Dx codes and treatment codes are packed in the same vector.
Then a single visit $\mathcal{V}^{(t)}$ can be expressed as a binary vector $\mathbf{x}^{(t)} \in \{0,1\}^{|\mathcal{A}| + |\mathcal{B}|}$ where each dimension corresponds to a specific Dx and treatment code.
Therefore encoding a sequence of visits made by a patient was achieved as follows:
\begin{align*}
\mathbf{v}^{(t)} & = \sigma(\mathbf{W}_x \mathbf{x}^{(t)}) \\
\mathbf{h} & = h(\mathbf{v}^{(1)}, \mathbf{v}^{(2)}, \ldots, \mathbf{v}^{(t)})
\end{align*}
where $\mathbf{W}_x$ is the embedding matrix that converts the binary vector $\mathbf{x}$ to a lower-dimensional visit representation\footnote{We omit bias variables throughout the paper to reduce clutter.}, $\sigma$ a non-linear activation function such as sigmoid or rectified linear unit (ReLU), $h(\cdot)$ a function that maps a sequence of visit representations $\mathbf{v}^{(0)}, \ldots, \mathbf{v}^{(t)}$ to a patient representation $\mathbf{h}$. $h(\cdot)$ can be simply an RNN or a complex combination of RNNs and CNN and attention mechanisms~\cite{bahdanau2014neural}.\js{maybe we can experiment with all 3 RNN, CNN and attention}
\ecedit{Justification for using RNN in the experiments: We want to minimize the expressivity of $h(\cdot)$ so that the effect of visit embedding can be properly evaluated.}
\texttt{MiME}\xspace treats the Dx codes $\mathcal{A}$ and the treatment codes $\mathcal{B}$ separately, in order to capture the relationship between $\mathcal{A}$ and $\mathcal{B}$ depicted by Figure~\ref{fig:ehr_graph}.
Then a single visit $\mathcal{V}$ (omitting the index $(t)$ of the visit) consists of two levels: the diagnosis level, and the treatment level.
Figure~\ref{fig:model} illustrates how the proposed model \texttt{MiME}\xspace builds the representation of $\mathcal{V}$ in a bottom-up fashion and how we capture both co-occurrence and parent-child relationship among the codes in Figure~\ref{fig:ehr_graph} via multilevel embedding.
In a single Dx object $\mathcal{O}_i$, a Dx code $d_i$ and its associated treatment codes $\mathcal{M}_i$ are used to obtain a vector representation of $\mathcal{O}_i$, $\mathbf{o}_i$.
Then multiple Dx object embeddings $\mathbf{o}_0, \ldots, \mathbf{o}_{|\mathcal{V}|}$ in a single visit are used to obtain a visit embedding $\mathbf{v}$, which in turn forms a patient embedding $\mathbf{h}$ with other visit embeddings.
The skeletal form of \texttt{MiME}\xspace is as follows,
\begin{align}
\mathbf{v} & = \sigma \Big( \sum_{i}^{|\mathcal{V}|} \underbrace{f ( d_i, \mathcal{M}_i ) \Big)}_{\text{Diagnosis object embedding $\mathbf{o}_i$}} \label{eq:skelenton} \\
f(d_i, \mathcal{M}_i) & = \mathbf{o}_i = \mathbf{U} \bigg( r(d_i) + \sigma \Big( \sum_{j}^{|\mathcal{M}_i|} g(d_i, m_{i,j}) \Big) \bigg) \\
g(d_i, m_{i,j}) & = \mathbf{W} r(d_i) \odot r(m_{i,j}))
\end{align}
where $f_{dx}(\cdot), f_{rx}(\cdot)$ and $f_{pr}(\cdot)$ respectively denote the embedding functions that map given diagnosis, medication, and procedure codes to their corresponding vector representations.
In \texttt{MiME}\xspace$_{sum}$, the size of all code embeddings are the same as they are summed to obtain the diagnosis embedding $\mathbf{d}_i$.
$\mathbf{W}_d$ is the weight matrix used to obtain the representation of a single diagnosis object, which is applied to the sum of diagnosis code embedding $f_{dx}(d_i)$ and its associated medication/procedure code embeddings $f_{rx}(m_{i,j})$ and $f_{pr}(p_{i,k})$. $\sigma$ is the non-linear activation function, for which we used ReLU in this work.
In $\texttt{MiME}\xspace_{sum}$, the functions $f_{d,m}(\cdot, \cdot)$ and $f_{d,m}(\cdot, \cdot)$, which aim to capture the interactions between a diagnosis code and medication/procedure codes are simply implemented as a medication code embedding $f_{rx}(m_{i,j})$ and a procedure code embedding $f_{pr}(p_{i,k})$.
In $\texttt{MiME}\xspace_{sum}$, it is assumed that the code interactions are implicitly captured by the summation inside the diagnosis embedding generator $f_d(\cdot, \cdot, \cdot)$.
Although this is a rather straightforward implementation of Eq.~\ref{eq:skelenton}, it shows robust performance in various prediction tasks as we will demonstrate in the next section.
The second implementation of Eq.~\ref{eq:skelenton}, which we name $\texttt{MiME}\xspace_{bp}$, captures the relationship between the diagnosis code and medication/procedure orders more explicitly as follows:
\small
\begin{align}
f_{d,m}(d_i, m_{i,j}) & = \sigma(\mathbf{W}_{dm} f_{dx}(d_i) \odot f_{rx}(m_{i,j})) \nonumber \\
f_{d,p}(d_i, p_{i,k}) & = \sigma(\mathbf{W}_{dp} f_{dx}(d_i) \odot f_{pr}(p_{i,k})) \nonumber
\\
f_{d}(\cdot, \cdot, \cdot) = \mathbf{d}_i & = \sigma \bigg( \mathbf{W}_d \Big( f_{dx}(d_i) + \sum_{j}^{|m_i|} f_{d,m} (d_i, m_j) + \sum_{k}^{|p_i|} f_{d,p} (d_i, p_j) \Big) \bigg) \nonumber
\end{align}
\normalsize
Both $f_{d,m}$ and $f_{d,p}$ use a form of bilinear pooling to explicitly capture the interactions between the diagnosis code and medication/procedure codes.
Bilinear pooling~\cite{tenenbaum12emph} derives a scalar feature $f_i$ between two embeddings $\mathbf{x}, \mathbf{y}$ such that $f_i = \mathbf{x}^T \mathbf{W}_i \mathbf{y}$ where $\mathbf{W}_i$ is a trainable weight matrix.
Since we typically extract many features $f_0, \ldots, f_i$, to capture the interaction between two embeddings, bilinear pooling requires us to train a number of weight matrices (\textit{i.e.} weight tensor).
Due to the large number of parameters required, researchers developed more efficient methods such as compact bilinear pooling~\cite{gao2016compact,fukui2016multimodal} and low-rank bilinear pooling~\cite{kim2016hadamard}.
In this work, we used a simplified version of low-rank bilinear pooling with Hadamard product~\cite{kim2016hadamard}, which showed impressive performance in Visual Question Answering (VQA), where it is important to effectively capture the interaction between two different domains:text-based question and image.
Weight matrices $\mathbf{W}_{dm}$ and $\mathbf{W}_{dp}$ are used to send the diagnosis code embedding $f_{dx}(d_i)$ into another latent space, where the interaction between diagnosis $d_i$ and the corresponding $m_{i,j}$ (or procedure $p_{i,k}$) can be captured effectively.
In $\texttt{MiME}\xspace_{bp}$ the interactions captured by $f_{d,m}$ and $f_{d,p}$ are added to the diagnosis code embedding $f_{dx}(d_i)$, which can be interpreted as adjusting the diagnosis representation according to its associated medication and procedure orders
\footnote{For both $\texttt{MiME}\xspace_{sum}$ and $\texttt{MiME}\xspace_{bp}$, we also implemented $f_d$ as a concatenation of $f_{dx}(d_i), \sum f_{d,m}$ and $\sum f_{d,p}$ instead of a summation, but it showed slightly weaker performance.}.
\subsection{Joint Training with Auxiliary Tasks}
\label{ssec:method_auxiliary}
\js{I doubt reviewers have read this section as they are all confused with the notations, let's make sure we clean up this as well}
Since the expressive power of deep learning models comes from their large number of parameters in multiple layers, they typically require a large amount of training data. This poses a challenge in deep embedding tasks on smaller datasets, which are often the case in new EHR systems, with a small number of patients each with a relatively fewer number of visits.
In such cases, to prevent the deep models from over-fitting, we adopt a multi-task learning (MTL) approach that can inductively transfer the knowledge contained in multiple auxiliary prediction tasks to improve the model's generalization performance on the original prediction task.
MTL was shown to improve model robustness in medical concept embedding.
For example, in ~\cite{suo_amia_mtl}, multiple severity status of a disease were seen as multiple tasks for improving accuracy in disease prediction.
For deep learning models which are typically composed of layers of neurons, an MTL strategy can require certain neurons to be shared among all tasks, and certain neurons to be specialized for specific tasks ~\cite{caruana1998multitask}.
Specifically, in longitudinal EHR data based health analytics, the patient embedding $\mathbf{h}$ is typically used for specific prediction tasks, such as heart failure prediction or mortality prediction.
And the representation power of $\mathbf{h}$ comes from properly capturing each visit $\mathcal{V}_t$, and modeling the longitudinal aspect with the function $f(\mathbf{v}_0, \ldots, \mathbf{v}_t)$.
Since the focus of this work is on modeling a single visit $\mathcal{V}_t$, our MTL strategy performs auxiliary tasks while obtaining the $t$-th visit embedding $\mathbf{v}_t$ as follows.
\small
\begin{align*}
& \mathbf{v}_t = \sum_{i}^{|d_t|} \mathbf{d}_{t,i} \\
& \hat{d}_{t,i} = p(d_{t,i} | \mathbf{d}_{t,i}) = \mbox{softmax}(\mathbf{W}_{ad} \mathbf{d}_{t,i}) \\
& \hat{\mathbf{m}}_{t,i} = p(\mathbf{m}_{t,i} | \mathbf{d}_{t,i}) = \sigma (\mathbf{W}_{am} \mathbf{d}_{t,i}) \\
& \hat{\mathbf{p}}_{t,i} = p(\mathbf{p}_{t,i} | \mathbf{d}_{t,i}) = \sigma (\mathbf{W}_{ap} \mathbf{d}_{t,i})\\
& L_{aux} = -\lambda_{aux} \sum_{t}^{T} \Big( CE(d_{t,i}, \hat{d}_{t,i}) + CE(\mathbf{m}_{t,i}, \hat{\mathbf{m}}_{t,i}) + CE(\mathbf{p}_{t,i}, \hat{\mathbf{p}}_{t,i}) \Big)
\end{align*}
\normalsize
Given diagnosis embeddings $\mathbf{d}_{t,0}, \ldots, \mathbf{d}_{t,|d_t|}$, before aggregating them to obtain $\mathbf{v}_t$, the model predicts the diagnosis code $d_{t,i}$, and the associated medication codes $\mathbf{m}_{t,i} = m_{t,i,0}, \ldots$, $m_{t,i,|m_{t,i}|}$ and procedure codes $\mathbf{p}_{t,i} = p_{t,i,0}, \ldots, p_{t,i,|p_{t,i}|}$, involved with $\mathbf{d}_{t,i}$.
$\mathbf{W}_{ad}, \mathbf{W}_{am}$ and $\mathbf{W}_{ap}$ are weight matrices used for predicting the diagnosis, medication and procedure code, respectively.
$CE(\cdot, \cdot)$ denotes the cross-entropy function, and $\lambda_{aux}$ is the coefficient for the auxiliary loss term.
We used the softmax function for predicting $d_{t,i}$ since in a single diagnosis embedding, there is only one diagnosis code involved.
However, there could be no medication/procedure codes associated with $d_{t,i}$, and therefore we used the sigmoid function for predicting medication and procedure codes.
These auxiliary tasks guide the model to learn diagnosis embeddings $\mathbf{d}_{t,i}$ that are representative of the specific codes involved with it.
Correctly capturing the events within an encounter is a foundation of all downstream prediction tasks, and these general-purpose auxiliary tasks, combined with the specific target task, encourage the model to learn visit embeddings $\mathbf{v}_t$ that are not only tuned for the target prediction task, but also grounded in general-purpose foundational knowledge.
\subsection{Power game between prediction and ELBO}
\begin{figure}
\centering
\includegraphics[scale=.5]{figs/model_1}
\caption{(Left) Maximizing $p(\mathbf{y} | \mathbf{z})$. (Right) Maximizing ELBO using VAE}
\label{fig:model1}
\end{figure}
When maximizing $p(\mathbf{y} | \mathbf{z})$, we can actually let the error signal flow from $\mathbf{y}$ all the way to $\mathbf{x}$, because we use the reparametrization trick between $\mathbf{x}$ and $\mathbf{z}$. Figure~\ref{fig:model1} shows the model architecture of both $p(\mathbf{y} | \mathbf{z})$ and ELBO (VAE). Maximizing $p(\mathbf{y} | \mathbf{z})$ in such a way will certainly shape $\mathbf{z}$ in a way that is useful for prediction. However, it will destroy the distribution $q(\mathbf{z} | \mathbf{x})$, thus destroying Eq.\eqref{eq:obj}. Therefore we must repeatedly maximize Eq.\eqref{eq:elbo} every now and then.
\subsection{Double VAE}
|
1,941,325,220,562 | arxiv | \section{Introduction}
\label{sec:intro}
Coronal mass ejections (CMEs) are huge clouds of magnetized plasma that erupt from the solar corona into the interplanetary space. They propagate in the heliosphere with velocities ranging from $20$ to $3200$~km\,s$^{-1}$ with an average speed of $489$~km\,s$^{-1}$, based on SOHO/LASCO coronagraph (Brueckner \cite{brueckner1995}) measurements between 1996 and 2003.
CMEs are associated with enormous changes and disturbances in the coronal magnetic field. The \emph{magnetic breakout}, the \emph{torus instability}, and the \emph{kink instability\/} are the main physical mechanisms that initiate and drive solar eruptions (see, e.g., Aulanier \cite{aulanier2014}, Schmieder et al.\ \cite{schmieder2013}; Chandra et al.\ \cite{chandra2011}; Aulanier et al.\ \cite{aulanier2010}; Forbes et al.\ \cite{forbes2006}; T{\"o}r{\"o}k \& Kliem \cite{torok2005}; Sakurai \cite{sakurai1976}, and references therein). CMEs are a key aspect of coronal and interplanetary dynamics and are known to be the major contributor to severe space weather at the Earth. Observations from SOHO, TRACE, Wind, ACE, STEREO, and SDO spacecrafts, along with ground-based instruments, have improved our knowledge of the origins and development of CMEs at the Sun (Webb \& Howard \cite{webb2012}).
It is well-known that the solar atmosphere is magnetically structured and in many cases the plasma in so-called magnetic flux tubes is flowing. Tangential velocity discontinuity at a tube surface due to the stream motion of the tube itself with respect to the environment leads to the Kelvin-Helmholtz (KH) instability which via KH vortices may trigger enhanced magnetohydrodynamic (MHD) turbulence. While in hydrodynamic flows KH instability can develop for small velocity shears (Drazin \& Reid \cite{drazin1981}), a flow-aligned magnetic field stabilizes plasma flows (Chandrasekhar \cite{chandrasekhar1961}) and the instability occurrence may require larger velocity discontinuities. For the KH instability in various solar structures of flowing plasmas (chromospheric jets, spicules, soft X-ray jets, solar wind) see, e.g., Zaqarashvili et al.\ (\cite{zaqarashvili2014a}) and references therein.
When CMEs start to rise from the lower corona upwards, then the velocity discontinuity at their boundary may trigger KH instability. Recently, Foullon et al.\ (\cite{foullon2011}) reported the observation of KH vortices at the surface of CME based on unprecedented high-resolution images (less than $150$~Mm above the solar surface in the inner corona) taken with the Atmospheric Imaging Assembly (AIA) (Lemen et al.\ \cite{lemen2012}) on board the Solar Dynamics Observatory (SDO) (Dean Pesnell et al.\ \cite{pesnell2012}). The CME event they studied occurred on 2010 November 3, following a C4.9 GOES class flare (peaking at 12:15:09 UT from active region NOAA 11121, located near the southeast solar limb). The instability was detected in the highest AIA temperature channel only, centered on the $131$~\AA~EUV bandpass at $11$~MK. In this temperature range, the ejecta lifting off from the solar surface forms a bubble of enhanced emission against the lower density coronal background (see Fig.~1 in Foullon et al.\ (\cite{foullon2011})). Along the northern flank of the ejecta, a train of three to four substructures forms a regular pattern in the intensity contrast. An updated and detailed study of this event was done by Foullon et al.\ (\cite{foullon2013}). A similar pattern was reported by Ofman \& Thompson (\cite{ofman2011}) -- the authors presented observations of the formation, propagation, and decay of vortex-shaped features in coronal images from the SDO associated with an eruption starting at about 2:30 UT on 2010 April 8. The series of vortices were formed along the interface between an erupting (dimming) region and the surrounding corona. They ranged in size from several to $10$~arc sec and traveled along the interface at $6$--$14$~km\,s$^{-1}$. The features were clearly visible in six out of the seven different EUV wave bands of the AIA. Later, using the AIA on board the SDO, M\"{o}stl et al.\ (\cite{meostl2013}) observed a CME with an embedded filament on 2011 February 24, revealing quasi-periodic vortex-like structures at the northern side of the filament boundary with a wavelength of approximately $14.4$~Mm and a propagation speed of about $310 \pm 20$~km\,s$^{-1}$. These structures, according to authors, could result from the KH instability occurring on the boundary.
The aim of this paper is to model the imaged KH instability by Foullon et al.\ (\cite{foullon2013}) via investigating
the stability/instability status of tangential velocity discontinuity at the boundary of the ejecta. The paper is organized as follows. In Sec.~\ref{sec:obs}, we present a brief description about the observations on 2010 November, 3. In Sec.~\ref{sec:basic} we specify the geometry of the problem, governing equations and the derivation of the wave dispersion relation. Sec.~\ref{sec:dispers} deals with its numerical solutions for specified, according to observational data, values of the input parameters, as well as the extracting KH instability parameters from computed dispersion curves and growth rates of the unstable $m = 1$ and $m = -3$ harmonics. The last Sec.~\ref{sec:concl} summarises the results derived in this paper.
\section{Observational description of 2010 November 3 event}
\label{sec:obs}
\begin{figure}[ht]
\centering
\begin{minipage}{\columnwidth}
\centering
\includegraphics[width=7.0cm]{aa-izh-fg1a.ps} \\
\vspace{0mm}
\includegraphics[width=7.0cm]{aa-izh-fg1b.ps}
\end{minipage}
\caption{(\emph{Top panel}) GOES time profiles of C4.9 flare in two wavelengths ($1$--$8$ and $0.5$--$4.0$~\AA). (\emph{Bottom panel}) RHESSI time profiles of same flare at different energy bands.}
\label{fig:fig1}
\end{figure}
We present here the observations of GOES C4.9 class flare on 2010 November 3 associated with CME from the NOAA active region 11121. The active region was located at S21E88 on that day on the solar disk. According to GOES observations the flare was initiated at 12:07 UT, peaked 12:15 UT and ended around 12:30 UT. The flare/eruption was well observed by SDO/AIA at different wavelengths with high spatial ($0.6$~arcsec) and temporal evolution ($12$~s) as well as by Reuven Ramaty High-Energy Solar Spectroscopic Imager (RHESSI, Lin et al.\ \cite{lin2002}) in X-ray $3$--$6$~keV, $6$--$12$~keV, $12$--$25$~keV, and $25$--$50$~keV energy bands, respectively. The temporal evolution of flare in X-rays observed by GOES and RHESSI is presented in Fig.~\ref{fig:fig1}.
The SDO/AIA observations on 2010 November 3 at different wavelengths indicate the evidence of KH instability around 12:15 UT at $131$~\AA~(see also Foullon et al.\ \cite{foullon2011} and Foullon et al.\ \cite{foullon2013}). Figure~\ref{fig:fig2} (top panel) shows the SDO/AIA $131$~\AA~at 12:15 UT, where we can see the KH instability features (see the white rectangular box). The image is overlaid by the RHESSI $6$--$12$~keV contours. The RHESSI image is constructed from seven collimators (3F to 9F) using the CLEAN algorithm, which yields a spatial resolution of ${\sim}7$~arcsec (Hurford et al.\ \cite{hurford2002}).
\begin{figure}[ht]
\centering
\begin{minipage}{\columnwidth}
\centering
\includegraphics[width=9.35cm]{aa-izh-fg2a.ps} \\
\vspace{-5mm}
\includegraphics[width=7.0cm]{aa-izh-fg2b.eps}
\end{minipage}
\vspace{-10mm}
\caption{(\emph{Top panel}) SDO/AIA $131$~\AA~image overlaid by RHESSI X-ray contours (contour level: $40\%$, $70\%$, and
$95\%$ of the peak intensity). The rectangular white box represents where the KH instability is visible. (\emph{Bottom panel})
The associated difference image of LASCO CME overlaid on the SDO/AIA image.}
\vspace*{-1mm}
\label{fig:fig2}
\end{figure}
LASCO observed a CME at 12:36 UT in C2 field-of-view with speed of $241$~km\,s$^{-1}$ and angular width of $66^{\circ}$ associated with the flare. The difference LASCO C2 images overlaid on SDO $193$~\AA~difference image is presented in the bottom panel of Fig.~\ref{fig:fig2}. In this image, some step-like structure is also visible, which corresponds to KH instability features visible in the the SDO $131$~\AA~wavelengths. This CME image indicates that we can see the KH instability features up to a ${\sim}3$\,$R_{\odot}$ height.
The study by Foullon et al.\ (\cite{foullon2013}) of the dynamics and origin of the CME on 2010 November 3 by means of the Solar TErrestrial RElations Observatory Behind (STEREO-B) located eastward of SDO\ by $82^{\circ}$ of heliolongitude, and used in conjunction with SDO give some indication of the magnetic field topology and flow pattern. At the time of the event, Extreme Ultraviolet Imager (EUVI) from STEREO's Sun--Earth Connection Coronal and Heliospheric Investigation (SECCHI) instrument suite (Howard et al.\ \cite{howard2008}) achieved the highest temporal resolution in the $195$~\AA~bandpass: EUVI's images of the active region on the disk were taken every $5$ minutes in this bandpass. The authors applied the Differential Emission Measure (DEM) techniques on the edge of the ejecta to determine the basic plasma parameters -- they obtained electron temperature of $11.6 \pm 3.8$~MK and electron density $n = (7.1 \pm 1.6)\times 10^8$~cm$^{-3}$, together with a layer width $\Delta L = 4.1 \pm 0.7$~Mm. Density estimates of the ejecta environment (quiet corona), according to Aschwanden \& Acton (\cite{aschwanden2001}), vary from ($2$ to $1$)$\times10^8$~cm$^{-3}$ between $0.05$ and $0.15$~$R_{\odot}$ ($40$--$100$~Mm), at heights where Foullon et al.\ (\cite{foullon2013}) started to see the KH waves developing. The final estimation based on a maximum height of $250$~Mm and the highest DEM value on the northern flank of the ejecta yields electron density of $(7.1 \pm 0.8)\times 10^8$~cm$^{-3}$. The adopted electron temperature in the ambient corona is $T = 4.5 \pm 1.5$~MK. The other important parameters derived on using the pressure balance equation assuming a benchmark value for the magnetic field $B$ in the environment of $10$~G are summarized in Table~2. The main features of the imaged KH instability presented in Table~3 include (in their notation) the speed of $131$~\AA~CME leading edge, $V_{\rm LE} = 687$~km\,s$^{-1}$, flow shear on the $131$~\AA~CME flank, $V_1 - V_2 = 680 \pm 92$~km\,s$^{-1}$, KH group velocity, $v_{\rm g} = 429 \pm 8$~km\,s$^{-1}$, KH wavelength, $\lambda = 18.5 \pm 0.5$~Mm, and exponential linear growth rate, $\gamma_{\rm KH} = 0.033 \pm 0.012$~s$^{-1}$.
\section{Geometry, the governing equations, and the wave dispersion relation}
\label{sec:basic}
\begin{figure}[ht]
\centering
\includegraphics[height=.25\textheight]{aa-izh-fg3}
\caption{Magnetic field geometry of a coronal mass ejection. \emph{Picture credit to: R.~Erd\'elyi \& V.~Fedun 2010.}}
\label{fig:fig3}
\end{figure}
As we already said, coronal mass ejections are ejections of magnetized plasma from the solar corona. Recently, Vourlidas (\cite{vourlidas2014}) presented observational evidence for the existence of magnetic flux ropes within CMEs. The observations detect the formation of the flux rope in the low corona, reveal its sometimes extremely fast evolution and follow it into interplanetary space. The results validate many of the expectations of the CME initiation theories. The idea for formation of magnetic flux ropes associated with the CMEs is not new. Manchester et al.\ (\cite{manchester2005}), in studying coronal mass ejection shocks and sheath structures relevant to particle acceleration in the solar wind, by means of a three-dimensional numerical ideal MHD model, drive a CME to erupt by introducing a Gibson-Low magnetic flux rope that is embedded in the helmet
streamer in an initial state of force imbalance. The flux rope rapidly expands and is ejected from the corona with maximum speeds in
excess of $1000$~km\,s$^{-1}$, driving a fast-mode shock from the inner corona to a distance of $1$~AU. Kumar et al.\ (\cite{kumar2012}) presenting multi wavelength observations of helical kink instability as a trigger of a CME which occurred in active region NOAA 11163 on 2011 February 24, state that the high-resolution observations from the SDO/AIA suggest the development of helical kink instability in the erupting prominence, which implies a flux rope structure of the magnetic field (see also Srivastava et al.\ \cite{srivastava2010} for observed kink instability in the solar corona). A brightening starts below the apex of the prominence with its slow rising motion (${\sim}100$~km\,s$^{-1}$) during the activation phase. In his review of recent studies on coronal dynamics: streamers, coronal mass ejections, and their interactions, Chen (\cite{chen2013}) claims that the energy release mechanism of CMEs can be explained through a flux rope magnetohydrodynamic model. According to him, the rope is formed through a long-term reconnection process driven by the shear, twist, and rotation of magnetic footpoints; moreover, recent SDO studies discovered solar tornadoes providing a natural mechanism of rope formation (see, e.g., Zhang \& Liu \cite{zhang2011}; Li et al.\ \cite{li2012}; Wedemeyer-B\"{o}hm et al.\ \cite{wedemeyer2012}). Recent \emph{in-situ\/} measurements also suggest that the twisted magnetic tubes could be common in the solar wind (Zaqarashvili et al.\ \cite{zaqarashvili2014b}). Thus, we are convinced to accept the expanded definition of the CME suggested in Vourlidas et al.\ (\cite{vourlidas2013}): ``A CME is the eruption of a coherent magnetic, twist-carrying coronal structure with angular width of at least $40^{\circ}$ and able to reach beyond $10$~$R_{\odot}$ which occurs on a time scale of a few minutes to several hours.'' Accordingly, the most appropriate model of the CME under consideration is a twisted magnetic flux tube of radius $a$ (${=}\Delta L/2$) and density $\rho_{\rm i}$ embedded in a uniform field environment with density $\rho_{\rm e}$ (see Fig.~\ref{fig:fig3}). The magnetic field inside the tube is
helicoidal, $\vec{B}_{\rm i} = (0, B_{{\rm i} \varphi}(r), B_{{\rm i} z}(r))$, while outside it is uniform and directed along the $z$-axis, $\vec{B}_{\rm e} = (0, 0, B_{\rm e})$. The tube is moving along its axis with speed of $\vec{v}_0$ with regards to the
surrounding medium. The jump of tangential velocity at the tube boundary then triggers the
magnetic KH instability when the jump exceeds a critical value.
In cylindrical equilibrium the magnetic field and plasma pressure satisfy the equilibrium condition in the radial direction
\begin{equation}
\label{eq:equilib}
\frac{\mathrm{d}}{\mathrm{d}r}\left( p_{\rm i} + \frac{B_{\rm i}^2}{2\mu}
\right) = -\frac{B_{{\rm i} \varphi}^2}{\mu r}.
\end{equation}
Here, $B_{\rm i}(r) = \left( B_{{\rm i} \varphi}^2 + B_{{\rm i} z}^2 \right)^{1/2} = \left| \vec{B}_{\rm i} \right|$ denotes the strength of the equilibrium magnetic field, and $\mu$ is the magnetic permeability. We note that in Eq.~(\ref{eq:equilib}) the total (thermal plus magnetic) pressure gradient is balanced by the tension force (the right-hand side of Eq.~(\ref{eq:equilib}))
in the twisted field. We consider the special case of an equilibrium with uniform twist, i.e., the one for which $B_{{\rm i} \varphi}(r)/r B_{{\rm i} z}(r)$ is a constant. Thus the background magnetic field is assumed to be
\begin{equation}
\label{eq:magnfield}
\vec{B}(r) = \left\{ \begin{array}{lc}
(0, Ar, B_{{\rm i} z}) & \mbox{for $r \leqslant a$}, \\
(0, 0, B_{\rm e}) & \mbox{for $r > a$},
\end{array}
\right.
\end{equation}
where $A$, $B_{{\rm i} z}$, and $B_{\rm e}$ are constant. Then the equilibrium condition (\ref{eq:equilib}) gives the equilibrium plasma pressure $p_{\rm i}(r)$ as
\[
p_{\rm i}(r) = p_0 - \frac{A^2 r^2}{\mu},
\]
where $p_0$ is the plasma pressure at the center of the tube. We note that the jump in the value of $B_{\varphi}(r)$ across $r = a$ implies a surface current there.
Prior to starting with governing MHD equations, we have to specify what kind of plasma is each medium (the moving tube and its environment). One sees from CME parameters listed in Table~2 in Foullon et al.\ (\cite{foullon2013}) that the plasma beta inside the flux tube might be equal to $1.5 \pm 1.01$, while that of the coronal plasma is $0.21 \pm 0.05$. Hence, we can consider the ejecta as an incompressible medium and treat its environment as a cool plasma ($\beta_{\rm e} = 0$).
The plasma motion inside the twisted flux tube is governed by the set of linearised MHD equations for an ideal incompressible plasma:
\begin{eqnarray}
\label{eq:moment}
\rho_0 \frac{\partial}{\partial t}\vec{v}_1 + \rho_0 \left(
\vec{v}_0 \cdot \nabla \right) \vec{v}_1 + \nabla \left( p_1
+ \frac{\vec{B}_0 \cdot \vec{B}_1}{\mu} \right) \nonumber \\
\nonumber \\
{}-\frac{1}{\mu} ( \vec{B}_0 \cdot \nabla ) \vec{B}_1 -
\frac{1}{\mu} ( \vec{B}_1 \cdot \nabla ) \vec{B}_0 = 0,
\end{eqnarray}
\begin{equation}
\label{eq:induct}
\frac{\partial}{\partial t}\vec{B}_1 - \nabla \times \left( \vec{v}_1
\times \vec{B}_0 \right) - \nabla \times \left( \vec{v}_0 \times \vec{B}_1
\right) = 0,
\end{equation}
\begin{equation}
\label{eq:divv}
\nabla \cdot \vec{v}_1 = 0,
\end{equation}
\begin{equation}
\label{eq:divb}
\nabla \cdot \vec{B}_1 = 0.
\vspace{3mm}
\end{equation}
Here, the index `$0$' denotes equilibrium values of the fluid velocity and the medium magnetic field, and the index `$1$' their perturbations. Below, the sum $p_1 + \vec{B}_0 \cdot \vec{B}_1/\mu$ in Eq.~(\ref{eq:moment}) will be replaced by $p_{\rm 1 tot}$, which represents the total pressure perturbation.
As the unperturbed parameters depend on the $r$ coordinate only, the perturbations can be Fourier analyzed with $\exp \left[ \mathrm{i} \left( -\omega t + m \varphi + k_z z \right) \right]$. Bearing in mind that in cylindrical coordinates the
nabla operator has the form
\[
\nabla \equiv \frac{\partial}{\partial r}\hat{r} + \frac{1}{r}
\frac{\partial}{\partial \phi}\hat{\varphi} + \frac{\partial}{\partial
z}\hat{z},
\]
from the above set of equations one can obtain a second-order differential equation for the total pressure perturbation $p_{\rm 1 tot}$
\begin{equation}
\label{eq:diffeq}
\left[ \frac{\mathrm{d}^2}{\mathrm{d}r^2} + \frac{1}{r}
\frac{\mathrm{d}}{\mathrm{d} r} - \left( \kappa_{\rm i}^2 + \frac{m^2}{r^2}
\right) \right] p_{\rm 1 tot} = 0,
\end{equation}
as well as an expression for the radial component $v_{1r}$ of the fluid
velocity perturbation $\vec{v}_1$ in terms of $p_{\rm 1 tot}$ and its
first derivative
\begin{equation}
\label{eq:v1r}
{v}_{1r} = -\mathrm{i}\frac{1}{\rho}_{\rm i} \frac{1}{Y} \frac{\omega -
\vec{k} \cdot \vec{v}_0}{\left( \omega - \vec{k} \cdot
\vec{v}_0 \right)^2 - \omega_{\rm Ai}^2} \left(
\frac{\mathrm{d}}{\mathrm{d} r} p_{\rm 1 tot} - Z \frac{m}{r}
p_{\rm 1 tot} \right).
\end{equation}
In Eq.~(\ref{eq:diffeq}), $\kappa_{\rm i}$ is the so-called \emph{wave attenuation coefficient}, which characterises the space structure of the wave and whose squared magnitude is given by the expression
\begin{equation}
\label{eq:kappa}
\kappa_{\rm i}^2 = k_z^2 \left( 1 - \frac{4 A^2 \omega_{\rm Ai}^2}
{\mu \rho_{\rm i} \left[ \left( \omega - \vec{k} \cdot
\vec{v}_0 \right)^2 - \omega_{\rm Ai}^2\right]^2} \right),
\end{equation}
where
\begin{equation}
\label{eq:alfvenfrq}
\omega_{\rm Ai} = \frac{\vec{k}\cdot \vec{B}_{\rm i}}{\sqrt{\mu \rho}_{\rm i}} = \frac{1}{\sqrt{\mu \rho}_{\rm i}}\left( mA + k_z B_{{\rm i}z} \right)
\end{equation}
is the so-called \emph{local Alfv\'en frequency\/} (Bennett et al.\ \cite{bennett1999}). The numerical coefficients $Z$ and $Y$ in the expression of $v_{1r}$ (see Eq.~(\ref{eq:v1r})) are respectively
\[
Z = \frac{2A \omega_{\rm Ai}}{\sqrt{\mu \rho_{\rm i}}\left[ \left( \omega -
\vec{k} \cdot \vec{v}_0 \right)^2 - \omega_{\rm Ai}^2\right]}
\qquad \mbox{and} \qquad Y = 1 - Z^2.
\]
The solution to Eq.~(\ref{eq:diffeq}) bounded at the tube axis is
\begin{equation}
\label{eq:solution}
p_{\rm tot}(r \leqslant a) = \alpha_{\rm i}I_m(\kappa_{\rm i}r),
\end{equation}
where $I_m$ is the modified Bessel function of order $m$ and $\alpha_{\rm i}$ is a constant.
For the cool environment (thermal pressure $p_{\rm e} = 0$) with straight-line magnetic field $B_{{\rm e}z} = B_{\rm e}$ and homogeneous density $\rho_{\rm e}$, from governing Eqs.~(\ref{eq:moment}), (\ref{eq:induct}), and (\ref{eq:divb}), in a similar way one obtains the same modified Bessel equation (\ref{eq:diffeq}), but $\kappa_{\rm i}^2$ is replaced by
\begin{equation}
\label{eq:kappae}
\kappa_{\rm e}^2 = k_z^2 \left( 1 - \omega^2/\omega_{\rm Ae}^2 \right),
\end{equation}
where
\begin{equation}
\label{eq:omegaae}
\omega_{\rm Ae} = \frac{k_z B_{{\rm e}z}}{\sqrt{\mu \rho_{\rm e}}} = k_z v_{\rm Ae},
\end{equation}
and $v_{\rm Ae} = B_{\rm e}/\sqrt{\mu \rho_{\rm e}}$ is the Alfv\'en speed in the corona.
The solution to modified Bessel equation bounded at infinity is
\begin{equation}
\label{eq:pmag}
p_{\rm tot}(r > a) = \alpha_{\rm e}K_m(\kappa_{\rm e}r),
\end{equation}
where $K_m$ is the modified Bessel function of order $m$ and $\alpha_{\rm e}$ is a constant.
From Eq.~(\ref{eq:moment}), written down for the radial components of all perturbations, we obtain that
\[
v_{1r} = -\mathrm{i}\frac{1}{\rho}_{\rm e} \frac{\omega}{\omega^2 - \omega_{\rm Ae}^2}\frac{\mathrm{d}}{\mathrm{d} r} p_{\rm 1 tot},
\]
or
\begin{equation}
\label{eq:v1er}
v_{1r}(r > a) = -\mathrm{i}\frac{1}{\rho}_{\rm e} \frac{\omega}{\omega^2 - \omega_{\rm Ae}^2} \alpha_{\rm e} \kappa_{\rm e}K_m^{\prime}(\kappa_{\rm e}r),
\end{equation}
where the prime sign means a differentiation by the Bessel function argument.
To obtain the dispersion relation of MHD modes, we have to merge the solutions for $p_{\rm tot}$ and $v_{1r}$ inside and outside the tube through boundary conditions. The boundary conditions have to ensure that the normal component of the interface perturbation
\[
\xi_r = -\frac{v_{1r}}{\mathrm{i}\left( \omega - \vec{k} \cdot
\vec{v}_0 \right)}
\]
remains continuous across the unperturbed tube boundary $r = a$, and also that
the total Lagrangian pressure is conserved across the perturbed boundary. This
leads to the conditions (Bennett et al.\ \cite{bennett1999})
\begin{equation}
\label{eq:contxi}
\xi_{{\rm i}r}|_{r=a} = \xi_{{\rm e}r}|_{r=a}
\end{equation}
and
\begin{equation}
\label{eq:contp1tot}
\left.p_{\rm tot\,i} - \frac{B_{{\rm i}\varphi}^2}{\mu a}\xi_{{\rm i}r}\right\vert_{r=a} = p_{\rm tot\,e}|_{r=a},
\end{equation}
where total pressure perturbations $p_{\rm tot\,i}$ and $p_{\rm tot\,e}$ are given by Eqs.~(\ref{eq:solution}) and (\ref{eq:pmag}), respectively. Applying boundary conditions (\ref{eq:contxi}) and (\ref{eq:contp1tot}) to our solutions of $p_{\rm 1 tot}$ and $v_{1r}$ (and accordingly $\xi_r$), we obtain after some algebra the dispersion relation of the normal modes propagating along a twisted magnetic tube with axial mass flow $\vec{v}_0$
\begin{eqnarray}
\label{eq:dispeq}
\frac{\left[ \left( \omega - \vec{k}\cdot \vec{v}_0 \right)^2 -
\omega_{\rm Ai}^2 \right]F_m(\kappa_{\rm i}a) - 2mA \omega_{\rm Ai}/\sqrt{\mu \rho_{\rm i}}}
{\left[ \left( \omega - \vec{k}\cdot \vec{v}_0 \right)^2 -
\omega_{\rm Ai}^2 \right]^2 - 4A^2\omega_{\rm Ai}^2/\mu \rho_{\rm i} } \nonumber \\
\nonumber \\
{} = \frac{P_m(\kappa_{\rm e} a)}
{{\displaystyle \frac{\rho_{\rm e}}{\rho_{\rm i}}} \left( \omega^2 - \omega_{\rm Ae}^2
\right) + A^2 P_m(\kappa_{\rm e} a)/\mu \rho_{\rm i}},
\end{eqnarray}
where, $\omega - \vec{k}\cdot \vec{v}_0$ is the Doppler-shifted wave frequency in the moving medium, and
\[
F_m(\kappa_{\rm i}a) = \frac{\kappa_{\rm i}a I_m^{\prime}(\kappa_{\rm i}a)}{I_m(\kappa_{\rm i}a)} \quad \mbox{and} \quad P_m(\kappa_{\rm e}a) = \frac{\kappa_{\rm e}a K_m^{\prime}(\kappa_{\rm e}a)}{K_m(\kappa_{\rm e}a)}.
\]
This dispersion equation is similar to the dispersion equation of normal MHD modes in a twisted flux tube surrounded by incompressible plasma (Zhelyazkov \& Zaqarashvili \cite{zhelyazkov2012}) -- there, in Eq.~(13), $\kappa_{\rm e} = k_z$, and to the dispersion equation for a twisted tube with non-magnetized environment, i.e., with $\omega_{\rm Ae} = 0$ (Zaqarashvili et al.\ \cite{zaqarashvili2010}).
\section{Numerical solutions and wave dispersion diagrams}
\label{sec:dispers}
The main goal of our study is to see under which conditions the propagating along the moving flux tube MHD waves can become unstable. To conduct such an investigation it is necessary to assume that the wave frequency $\omega$ is a complex quantity, i.e., $\omega \to \omega + \mathrm{i}\gamma$, where $\gamma$ is the instability growth rate, while the longitudinal wave number $k_z$ is a real variable in the wave dispersion relation. Since the occurrence of the expected KH instability is determined primarily by the jet velocity, in looking for a critical/threshold value of it, we will gradually change its magnitude from zero to that critical value (and beyond). Thus, we have to solve dispersion relations in complex variables obtaining the real and imaginary parts of the wave frequency, or as it is normally accepted, of the wave phase velocity $v_{\rm ph} = \omega/k_z$, as functions of $k_z$ at various values of the velocity shear between the surge and its environment, $v_0$.
We focus our study first on the propagation of the kink mode, i.e., for $m = 1$. It is obvious that Eq.~(\ref{eq:dispeq}) can be
solved only numerically. The necessary step is to define the input parameters which characterise the moving twisted magnetic flux tube, and also to normalise all variables in the dispersion equation. The density contrast between the tube and its environment is characterised by the parameter $\eta = \rho_{\rm e}/\rho_{\rm i}$, and the twisted magnetic field by the ratio of the two magnetic field components, $B_{{\rm i}\varphi}$ and $B_{{\rm i}z}$, evaluated at the inner boundary of the tube, $r = a$, i.e., via $\varepsilon = B_{{\rm i}\varphi}/B_{{\rm i}z}$, where $B_{{\rm i}\varphi} = Aa$. As usual, we normalise the velocities to the Alfv\'en speed $v_{\rm Ai} = B_{{\rm i}z}/(\mu \rho_{\rm i})^{1/2}$. Thus, we introduce the dimensionless wave phase velocity $v_{\rm ph}/v_{\rm Ai}$ and the \emph{Alfv\'en Mach number\/} $M_{\rm A} = v_0/v_{\rm Ai}$, the latter characterising the axial motion of the tube. The wavelength, $\lambda = 2\pi/k_z$, is normalised to the tube radius $a$ which implies that the dimensionless wave number is $k_z a$. The natural way of normalising the local Alfv\'en frequency $\omega_{\rm Ai}$ is to multiply it by the tube radius, $a$, and after that divide by the Alfv\'en speed $v_{\rm Ai}$, i.e.,
\[
a \omega_{\rm Ai} = \frac{B_{{\rm i}z}}{\sqrt{\mu \rho_{\rm i}}}\left( m\frac{B_{{\rm i}\varphi}}{B_{{\rm i}z}} + k_z a \right) \quad \mbox{or} \quad \frac{a \omega_{\rm Ai}}{v_{\rm Ai}} = m \varepsilon + k_z a,
\]
where $B_{{\rm i}\varphi}/B_{{\rm i}z} \equiv Aa/B_{{\rm i}z} = \varepsilon$ is the above defined twist parameter. We note that the normalisation of Alfv\'en frequency outside the jet, $\omega_{\rm Ae}$, requires (see Eq.~(\ref{eq:omegaae})) except the tube radius, $a$, and the density contrast, $\eta$, the ratio of the two axial magnetic fields, $b = B_{\rm e}/B_{{\rm i}z}$.
Before starting the numerical job, we have to specify the values of the input parameters. Our choice for the density contrast is $\eta = 0.88$, which corresponds to electron densities $n_{\rm i} = 8.7 \times 10^8$~cm$^{-3}$ and $n_{\rm e} = 7.67 \times 10^8$~cm$^{-3}$, respectively. With $\beta_{\rm i} = 1.5$ and $\beta_{\rm e} = 0$, the ratio of axial magnetic fields is $b = 1.58$. If we fix the Alfv\'en speed in the environment to be $v_{\rm Ae} \cong 787$~km\,s$^{-1}$ (i.e., the value corresponding to $n_{\rm e} = 7.67 \times 10^8$~cm$^{-3}$ and $B_{\rm e} = 10$~G), the total pressure balance equation at $\eta = 0.88$ requires a sound speed inside the jet $c_{\rm si} \cong 523$~km\,s$^{-1}$ and Alfv\'en speed $v_{\rm Ai} \cong 467$~km\,s$^{-1}$ (more exactly, $467.44$~km\,s$^{-1}$), which corresponds to a magnetic field in the flux tube $B_{{\rm i}z} = 6.32$~G. Following Ruderman (\cite{ruderman2007}), to satisfy the Shafranov-Kruskal stability criterion for a kink instability we assume that the azimuthal component of the magnetic field $\vec{B}_{\rm i}$ is smaller than its axial component, i.e., we will choose our twist parameter $\varepsilon$ to be always less than $1$. In particular, we will study the dispersion diagrams of kink, $m = 1$ and $m = -3$ MHD modes and their growth rates (when the modes are unstable) for three fixed values of $\varepsilon$: $0.025$, $0.1$, and $0.2$.
\begin{figure}[ht]
\centering
\begin{minipage}{\columnwidth}
\centering
\includegraphics[width=7.5cm]{aa-izh-fg4a} \\
\vspace{0mm}
\includegraphics[width=7.5cm]{aa-izh-fg4b}
\end{minipage}
\caption{(\emph{Top panel}) Dispersion curves of unstable kink ($m = 1$) mode propagating on a moving twisted magnetic flux tube with twist parameter $\varepsilon = 0.025$ at $\eta = 0.88$, $b = 1.58$ and Alfv\'en Mach numbers equal to $2.95$, $2.975$, $3$, and $3.025$, respectively. (\emph{Bottom panel}) Growth rates of the unstable kink mode. Note that the marginal red curve reaches values smaller than $0.001$ beyond $k_z a = 53$.}
\label{fig:fig4}
\end{figure}
\begin{figure}[ht]
\centering
\begin{minipage}{\columnwidth}
\centering
\includegraphics[width=7.5cm]{aa-izh-fg5a} \\
\vspace{0mm}
\includegraphics[width=7.5cm]{aa-izh-fg5b}
\end{minipage}
\caption{(\emph{Top panel}) Growth rates of unstable $m = -3$ MHD mode in three instability windows. Adequate numerical values consistent with the observational data one obtains at $k_z a = 0.696245$, located in the third instability window, for which value of $k_z a$ the normalized mode growth rate is equal to $0.232117$. (\emph{Bottom panel}) Dispersion curves of unstable $m = -3$ MHD mode for three values of the twist parameter $\varepsilon = 0.025$, $0.1$, and $0.2$. The normalized phase velocity at $k_z a = 0.696245$ is equal to $1.45$.}
\label{fig:fig5}
\end{figure}
With these input parameters ($M_{\rm A}$ will be varied from zero (no flow) to reasonable values during computations), the solutions to Eq.~(\ref{eq:dispeq}) yield the dependence of the normalized wave phase velocity $\omega/k_z v_{\rm Ai}$ on $k_z a$.
In Fig.~\ref{fig:fig4} we display the dispersion curves of the kink mode at $\varepsilon = 0.025$ for various values of $M_{\rm A}$ -- the red curve is in fact a \emph{marginal\/} dispersion curve corresponding to the critical $M_{\rm A}^{\rm cr} = 2.95$. The critical jet speed, $M_{\rm A}^{\rm cr}v_{\rm Ai} \cong 1380$~km\,s$^{-1}$, being in principal accessible for CMEs, is, in fact, more than $2$ times higher than the registered by Foullon et al.\ (\cite{foullon2013}) threshold speed of $680$~km\,s$^{-1}$ for observing KH instability. For the next higher values of $\varepsilon$ one obtains threshold Alfv\'en Mach numbers around $3$. Hence, detected KH instability cannot be associated with the kink mode. But situation distinctly changes for the $m = -3$ MHD mode. As seen from Fig.~\ref{fig:fig5}, one can observe the appearance of three instability windows on the $k_z a$-axis. The width of each instability window depends upon the value of twist parameter $\varepsilon$ -- the narrowest window corresponds to $\varepsilon = 0.025$, and the widest to $\varepsilon = 0.2$. It is worth noticing that the phase velocities of unstable $m = -3$ MHD modes coincides with the magnetic flux tube speeds (one sees in the bottom panel of Fig.~\ref{fig:fig5} that the normalized wave velocity on given dispersion curve is equal to its label $\mathsf{M_{\sf A}}$). Therefore, unstable perturbations are frozen in the flow and consequently they are vortices rather than waves. It has a firm physical ground as KH instability in hydrodynamics deals with unstable vortices. All critical Alfv\'en Mach numbers yield acceptable threshold speeds of the ejecta which ensure the occurrence of KH instability -- these speeds are equal to $701$~km\,s$^{-1}$, $689$~km\,s$^{-1}$, and $678$~km\,s$^{-1}$, respectively, in very good agreement with the speed of $680$~km\,s$^{-1}$ found by Foullon et al.\ (\cite{foullon2013}). The observationally detected KH instability wavelength $\lambda_{\rm KH} = 18.5$~Mm and ejecta width $\Delta L = 4.1$~Mm define the corresponding instability dimensionless wave number, $k_z a = \pi \Delta L/\lambda$, to be equal to $0.696245$. As seen from Fig.~\ref{fig:fig5}, that $k_z a = 0.696245$ lies in the third instability window and accordingly determines a value of the dimensionless growth rate Im$(v_{\rm ph}/v_{\rm Ai}) = 0.232117$ (see the top panel of Fig.~\ref{fig:fig5}), which implies a computed wave growth rate $\gamma_{\rm KH} = 0.037$~s$^{-1}$, being in good agreement with the deduced from observations $\gamma_{\rm KH} = 0.033$~s$^{-1}$. We note also that the estimated from Fig.~\ref{fig:fig5} (bottom panel) wave phase velocity of $678$~km\,s$^{-1}$ is rather close to the speed of the $131$~\AA~CME leading edge equal to $687$~km\,s$^{-1}$. It is worth pointing out that the position of a given instability window, at fixed input parameters $\eta$ and $b$, is determined chiefly by the magnetic field twist in the moving flux tube. This circumstance allows us by slightly shifting the third instability window to the right, to tune the vertical purple line (see the top panel of Fig.~\ref{fig:fig5}) to cross the growth rate curve at a value, which would yield $\gamma_{\rm KH} = 0.033$~s$^{-1}$. The necessary shift of only $0.023625$ can be achieved by taking the magnetic field twist parameter, $\varepsilon$, to be equal to $0.20739$ -- in that case the normalised Im$(v_{\rm ph}/v_{\rm Ai}) = 0.207999$ gives the registered KH instability growth rate of $0.033$~s$^{-1}$. (That very small instability window shift does not change noticeably the critical ejecta speed.) In this way, we demonstrate the flexibility of our model allowing the derivation of numerical KH instability characteristics in very good agreement with observational data.
\section{Discussion and conclusion}
\label{sec:concl}
In this paper, as we have shown, the imaged by Foullon et al.\ (\cite{foullon2013}) KH vortices on 2010 November 3 CME can be explained as KH instability of the $m = -3$ harmonic in a twisted flux tube moving in external cool magnetized plasma embedded in homogeneous untwisted magnetic field. We have assumed the wave vector $\vec{k}$ aligned with the $\vec{v}_0 = \vec{V}_{\rm i} - \vec{V}_{\rm e}$ vector. We would like to point out that the results of the numerical modelling crucially depend on the input parameters. Any small change in the density contrast, $\eta$, or the magnetic fields ratio, $b$, can dramatically change the picture. In our case, the input parameters for solving the MHD mode dispersion relation in complex variables (the mode frequency $\omega$ and, respectively, the mode phase velocity $v_{\rm ph} = \omega/k_z$, were considered as complex quantities) were chosen to be consistent with the plasma and magnetic field parameters listed in Table~2 in Foullon et al.\ (\cite{foullon2013}). It was found that with a twist parameter of the background magnetic filed $\vec{B}_{\rm i}$, $\varepsilon = 0.2$, the critical jet's speed is $v_0^{\rm cr} = 678$~km\,s$^{-1}$, and at wavelength of the unstable $m = -3$ mode $\lambda_{\rm KH} = 18.5$~Mm and ejecta width $\Delta L = 4.1$~Mm, its growth rate is $\gamma_{\rm KH} = 0.037$~s$^{-1}$. These values of $v_0^{\rm cr}$ and $\gamma_{\rm KH}$ are in a good agreement with the data listed in Table~3 of Foullon et al.\ (\cite{foullon2013}). We have also shown that the numerically obtained instability growth rate can be a little reduced to coincide with observational one of $0.033$~s$^{-1}$ through slightly shifting to the right the appropriate instability window -- this can be done by performing the calculations with a new value of the magnetic field twist parameter $\varepsilon$, equal to $0.20739$. Thus, our model is flexible enough to allow us numerically get KH instability characteristics very close to the observed ones. The two ``cross points'' in Fig.~\ref{fig:fig5} can be considered as a `computational portrait' of the imaged on the 2010 November 3 coronal mass ejecta KH instability. Critical ejecta speed and KH instability growth rate values, in good agreement with those derived by Foullon et al.\ (\cite{foullon2013}), can also be obtained by exploring the flute-like, $m = -2$, MHD mode (Zhelyazkov \& Chandra \cite{zhelyazkov2014}). In that case, a better agreement with the observational data one achieves at $\varepsilon = 0.188$ -- the computed values of $v_0^{\rm cr}$ and $\gamma_{\rm KH}$ are exactly the same as those for the $m = -3$ mode.
It is necessary to stress that each CME is a unique event and its successful modelling requires a full set of observational data for the plasma densities, magnetic fields, and temperatures of both media along with the detected ejecta speeds. Concerning values of the flux tube speeds at which the instability starts, they can vary from a few kilometers per second, $6$--$10$~km\,s$^{-1}$, as observed by Ofman \& Thompson (\cite{ofman2011}), through $310 \pm 20$~km\,s$^{-1}$ of M\"{o}stl et al.\ (\cite{meostl2013}), to $680 \pm 92$~km\,s$^{-1}$ deduced from Foullon et al.\ (\cite{foullon2013}). It is curious to see whether the $13$ fast flareless CMEs (with velocities of $1000$~km\,s$^{-1}$ and higher) observed from 1998 January 3 to 2005 January 4 (see Table~1 in Song et al.\ (\cite{song2013})), are subject to the KH instability.
In spite of the fact that KH instability characteristics of both $m = -2$ and $m = -3$ MHD modes are in a good agreement with observational data, only the instability of the $m = -3$ harmonic may explain why the KH vortices are seen only at one side of rising CME (Foullon et al.\ (\cite{foullon2011,foullon2013}), see also Fig.~\ref{fig:fig2} in the present paper). This harmonic yields that the unstable vortices have $3$ maxima around the magnetic tube with a $360/3 = 120$ degree interval. Therefore, if one maximum is located in the plane perpendicular to the line of sight (as it is clearly seen by observations), then one cannot detect two other maxima in imaging observations as they will be largely directed along the line of sight.
Finally we would like to comment on whether there is a required fields orientations $\phi = \widehat{(\vec{k}, \vec{B})}$. Analysing a flat (semi-infinite) geometry in their study, Foullon et al.\ (\cite{foullon2013}) conclude that it is most likely to observe a quasi-perpendicular to the magnetic field $\vec{B}_{\rm e}$ wave propagation. But this restriction on the ejecta magnetic field tilt angle drops out in our numerical investigation because the inequality $|V_{\rm i} - V_{\rm e}| \geqslant \sqrt{2}v_{\rm Ai} = 661$~km\,s$^{-1}$, required for wave parallel propagation, is satisfied. Thus, the adopted magnetic flux rope nature of a CME and its (ejecta) consideration as a moving twisted magnetic flux tube allow us to explain the emerging instability as a manifestation of the KH instability of a suitable MHD mode.
\begin{acknowledgements}
The work of two of us, IZh and RC, was supported by the Bulgarian Science Fund and the Department of Science \& Technology, Government of India Fund under Indo-Bulgarian bilateral project CSTC/INDIA 01/7, /Int/Bulgaria/P-2/12. The work of TZ was supported by the European FP7-PEOPLE-2010-IRSES-269299 project-SOLSPANET, by the Austrian Fonds zur F\"{o}rderung der Wissenschaftlichen Forschung (project P26181-N27), by Shota Rustaveli National Science Foundation grant DI/14/6-310/12 and by Austrian Scientific-technology collaboration (WTZ) grant IN 10/2013. We would like to thank the referee for the very useful comments and suggestions which help us improve the quality of our paper, and are indebted to Dr.~Snezhana Yordanova for drawing one figure.
\end{acknowledgements}
|
1,941,325,220,563 | arxiv | \section*{Introduction}
The magnitude of genetic effects on phenotype depends strongly on genetic background, the effects of the same mutation can be larger in one genetic background and smaller in another. The idea that wild-type genotypes are mutationally robust, i.e.\ show invariance in the face of mutations (more generally heritable perturbations), goes back to Waddington \cite{waddington}, who originally introduced the concept as canalization. While genetic robustness has been found across different levels of organization from individual genes, through simple genetic circuits to entire organisms (approximately $80\%$ of yeast single knockouts have no obvious effect in rich medium \cite{hillenmeyer}), the origin of the observed robustness has remained a source of contention. The three main hypotheses regarding the potential origin of genetic robustness predate the concept itself, and fall along the lines of the famous debate between members of the modern synthesis (in particular Wright, Haldane and Fisher) surrounding the origin of dominance (dominance can be understood as a simple case of robustness, a dominant phenotype being more robust against mutations) \cite{devisser,burger}: (i) the most straightforward explanation, favoured by Wright, was that robustness evolves \emph{directly}, through natural selection \cite{fisher}; (ii) an alternative \emph{congruent} hypotheses, put forward in the context of dominance by Haldane, proposes that the evolution of genetic robustness is a correlated byproduct of selection for environmental robustness, i.e.\ invariance in the face of nonheritable perturbations, e.g.\ temperature, salinity or internal factors such as fluctuations in the concentration of gene products during development \cite{haldane}; (iii) while a third view holds that genetic robustness is \emph{intrinsic}, arising simply because the buffering of a character with respect to mutations is the necessary or likely consequence of character adaptation, in the context of dominance Wright \cite{wright} and later Kacser and Burns \cite{kacserburns} argued that it arises as an inevitable, passive consequence of enzyme biochemistry and selection for increased metabolic flux.
Recently robustness has been a subject of renewed interest.
Several theoretical and simulation studies have addressed robustness in a wide range of contexts ranging from gene redundancy \cite{krakauerPNAS} to model regulatory networks \cite{siegalPNAS,azevedoNAT,wagnerPLOS,wagnerPNAS,crombachPLOS}. {By in the first case building on evidence of excess mutational robustness present in RNA secondary structure \cite{wagnerJexp} and in the second case on the expectation that high mutation rates present among RNA viruses should favour mutational robustness \cite{wilkeadami}} two pioneering studies, by Montville et al.\ \cite{montville} and Borenstein and Ruppin \cite{ruppin} have managed to step beyond computer simulations and through using, respectively, \emph{in vitro} evolution experiments \cite{montville} and microRNA sequences from diverse taxa found evidence to support the hypotheses that genetic robustness can evolve directly. {\color{black}Further work on \emph{in vitro} evolution experiments has provided additional evidence showing that if a population is highly polymorphic robustness can evolve directly \cite{sanjuan,bloom}. }
The theoretical underpinnings of these studies is provided by the results of van Nimwegen et al.\ \cite{nimwegen}, who through solving the quasispecies equations describing the evolution of a population on a network of phenotypically neutral sequences, were able to demonstrate, that provided a sufficiently polymorphic population, mutational robustness can evolve directly. The necessary mutation rates and/or population sizes were found to be very large in simulation studies using RNA secondary structure as a genotype-phenotype map \cite{nimwegen,adamiJTB,szollosiMBS}, direct evolution of increased neutrality requiring the product of the effective population size $N_e $ and the mutation rate per nucleotide $u$ to be well in excess of one. Such high mutation rates can only readily be found among RNA viruses, are extraordinary even among unicellular organisms (\emph{Prochlorococcus} $2N_e u\approx2.$, \emph{E.\ coli} $2N_e u\approx0.2$, \emph{S.\ cerevisiae} $4N_e u\approx0.09$) and completely unheard of among multicellular eukaryotes possessing RNA silencing mechanisms and microRNA genes (\emph{A.\ thaliana} $4N_e u\approx0.012$, \emph{D.\ melanogaster} $4N_e u\approx0.015$, \emph{C.\ elegans} $4N_e u\approx0.013$, \emph{C.\ intestinalis} $4N_e u\approx0.012$, \emph{M.\ musculus} $4N_e u\approx0.001$, \emph{H.\ sapiens} $4N_e u\approx0.001$) \cite{lynch}.
In their study Borenstein and Ruppin examined microRNA (miRNA) precursor sequences from several eukaryotic species. miRNA are small endogenous noncoding RNAs that regulate the expression of protein coding genes through the RNA interference (RNAi) pathway \cite{lagosQSCI,lauSCI,leeSCI,bartelCELL}. Functionally relevant short ($\approx22$nt) mature miRNA sequences are excised from longer precursor sequences that fold into a stem-loop hairpin structure. The hairpin like secondary structure of precursor stem-loops plays a crucial role in the maturation process \cite{bartelCELL} and is under evolutionary constraint to conserve its structure. Borenstein and Ruppin used the novel and ingenuous method of generating for each miRNA sequence a random sample of sequences with identical minimum free-energy (MFE) structure to uncover traces of adaptation. To compare the mutational robustness of miRNA precursor sequences to random sample sequences with identical MFE structure they compared the single mutant neighborhood of a given miRNA precursor sequence to the single mutant neighborhood of the sample sequences. Calculating the average distance of the MFE structure of each single mutant sequence to the MFE structure of the original sequence for both stem-loop and sample sequences {\color{black} (details on secondary structure calculations are presented below)} they demonstrated that miRNA precursor sequences have single mutant neighborhoods with sequences that fold into more similar MFE structures compared to sequences in the single mutant neighborhoods of sample sequences with identical MFE structure. While a similar comparison of the folding minimum free-energy showed a comparable, but lower bias, the finding that the two were only weakly correlated allowed the authors to conclude that the observed bias is a result of direct selection for mutational robustness. {\color{black} Their results were reexamined by Shu et al.\ \cite{Shu} who argued that mutational robustness among miRNA precursors may be the correlated byproduct of selection for environmental robustness, but found only a moderately higher correlation using a different measure of mutational robustness. }
In light of the consistently low value of $uN_e$ among multicellular eukaryotes the results of Borenstein and Ruppin are highly surprising. There is no known mechanism which can explain the direct evolution of robustness that they observe. According to the classic results of Kimura and Maruyama the average fitness of an asexually reproducing population (in the limit of very large populations) depends only on the mutation rate and is independent of the details of the fitness landscape \cite{kimura}. This result, however, only holds under the assumption that the fittest genotype does not have any neutral sites. While, the extension of these results to more general fitness landscapes by van Nimwegen et al.\ demonstrates that the presence of neutral genotypes can lead to selective pressure to evolve mutational robustness simulation studies using genotype-phenotype maps induced by RNA secondary structure have demonstrated that $uN_e>1$ is a necessary condition \cite{adamiJTB} even in the presence of recombination \cite{szollosiMBS}. The case for the direct evolution of genetic robustness rests on the parallel findings that a stronger bias for mutational robustness is present in miRNA precursor sequences than for environmental robustness and that the two are only weakly correlated. Introducing a new measure of environmental robustness in this paper we endeavor to demonstrate that, indeed as previously also suggested by Shu et al.\ \cite{Shu}, the exact opposite is true: the bias for environmental robustness is stronger and it is highly correlated with mutational robustness. {\color{black} The correlated evolution of environmental and mutational robustness in RNA sequences under selection to retain secondary} {\color{black}structure is expected as a corollary of a general casual link between environmental robustness and genetic robustness} {\color{black}in RNA sequences proposed by Ancel and Fontana \cite{ancelfontana}. They argue that \emph{plastogenetic congruence}, i.e.\ the correlation between the set of structures thermally accessible to a sequence, its \emph{plastic repertoire}, and the MFE structures of its \emph{genetic neighbourhood} will lead to the emergence of mutational robustness in the presence of selection for some predefined structure. }
\begin{figure}
\centerline{\includegraphics[width=0.5\textwidth]{fig1}}
\caption{{\bf a} Generating a random sample of sequences with a desired MFE structure by stochastic minimization of the free-energy of the desired fold (the method employed the RNAinverse program used by Borenstein and Ruppin \cite{ruppin} as well as Shu et al.\ \cite{Shu} ) results in a biased sample in which sequences with lower than average neutrality (higher than average number single mutant neighbors) are overrepresented. This can be avoided if after finding a sequence with the desired MFE structure a random walk is performed among sequences with the desired MFE structure. {\color{black} This random walk on the neutral network associated with the MFE structure mimics the sequnce drift of a sequence evolving under the constraint to fold in to the desired MFE structure.}
{\bf b} Rank score distributions for two measures of mutational robustness ($\eta_s$ and $\eta_n$ see text). Comparing the distributions derived from sampling using only stochastic optimization (top, $\bar r^{\rm biased}_s=0.25,R^{\rm biased}_s=0.83,\bar r^{\rm biased}_n=0.37,R^{\rm biased}_n=0.66$) to that derived from sampling with subsequent randomization (bottom, $\bar r_s=0.29,R_s=0.78,\bar r_n=0.44,R_n=0.59$) shows that increased neutrality is predominately an artifact of biased sampling, while the lower than average distance of MFE structures in the mutational neighborhood to the wild-type MFE structure becomes somewhat less pronounced, but is still significant. }\label{fig1}
\end{figure}
\section*{Materials and Methods}
\subsection*{microRNA sequences and sampling }
miRNA precursor sequences were downloaded from miRBase version $9.0$ \cite{mirbase}.
All $4361$ miRNA genes were used, yielding $3641$ unique miRNA precursor sequences. For each miRNA precursor sequence we produced a sample of random sequences by (i) using the stochastic optimization routine from the Vienna RNA package \cite{vienna} to produce a sequence with MFE structure identical to that of the native sequence that is stored (ii) and subsequently randomizing this sequence by attempting $4L$ random nucleotide substitutions in a miRNA precursor sequence of length $L$, accepting a substitution if the resulting sequence's MFE structure remains unchanged. For each miRNA precursor sequence on average $>800$ sample sequences with identical MFE structure was generated. Supplemental material accompanying our paper contains the robustness values for all $4361$ genes associated with $3641$ unique sequences we considered.
\subsection*{Measuring thermodynamic robustness}
In order to calculate {\color{black} the thermodynamic robustness measure $\eta_t$, defined below,} we sampled the equilibrium thermodynamic ensemble of stem-loop and sample sequences using the stochastic backtracking routine from the Vienna RNA package producing $10^6$ suboptimal structures per sequence, using the default temperature of $310\ K$. The average distance from the MFE structure in the thermodynamic ensemble can be calculated exactly with the help of base-pairing probabilities, which are available as a byproduct of partition function folding in the Vienna package, and were used to validate the sampling.
\subsection*{Statistics}
Given a rank score $r$ and sample size $N$ a good estimate for the probability of observing an equal or lower rank score by chance is given by $(r N)/(N+1)\approx r $. Following \cite{ruppin} rank scores of $r<0.05$ are considered significantly robust. To determine if the robustness of miRNA precursor sequences according to some measure $\eta$ has the same distribution as the robustness of sample sequences $\eta'$ for a group of sequences, following \cite{ruppin} we test against the null hypothesis that they are drawn from identical distributions using the nonparametric Wilcoxon signed rank test. In contrast to \cite{ruppin}, however, we do not consider as paired values $\eta$ and the average of $\eta'$ over all $N$ sample sequences, $\langle \eta' \rangle$, as we found this to result in spuriously low $p$-values, but instead calculate the $p$-values for a given group of sequences by averaging over $1000$ different sets of $\{ \eta,\eta' \}$ pairs where in each set the $\eta'$ values belong to a random sample sequence. As a complementary approach we also tested the hypothesis that the distribution of rank scores of a group of sequences for a given robustness measure is uniform -- as we would expect if miRNA precursor sequences were randomly sampled from the set of sequences with identical MFE structure -- using a standard Kolmogorov-Smirnov goodness of fit test. We found the two significance analyses to be in good agreement indicating highly significant bias for higher values of $\eta_t(d_{\rm th.})$ and $\eta_s$, but mostly no or only nonsignificant bias for higher $\eta_n$. The supplemental information accompanying our paper contains species level statistics and significance analyses.
\begin{figure}
\begin{center}
\centerline{\includegraphics[width=0.5\textwidth]{fig2}}
\caption{In order to examin the robustness of miRNA precursor sequences to thermal fluctuations we sampled the equilibrium thermodynamic ensemble of structures. Sampling $10^6$ structures for each miRNA precursor sequence and each member of the random sequence sample, we binned structures according to their distance from the MFE structure. {\bf a} For, e.g. the \emph{Monodelphis domestica} miRNA precursor sequence \emph{mdo-mir-1} examining the distribution of structures as a function of the base-pair distance shows that the averaged random sequence sample distribution (white bars) has a much larger fraction of structures that are drastically different from the MFE structure, compared to the distribution of structures for the original miRNA precursor sequence (black bars). {\bf b} Examining the averaged distribution of stem-loop (black bars) and random corresponding random sequence sample distributions (white bars) shows that there is a general tendency among miRNA precursor sequences for increased thermodynamic robustness, i.e.\ of avoiding structures that are highly dissimilar to the MFE structure. A strikingly similar effect can be observed if we examine the distribution of structures in the mutational neighborhood. Analogous to {\bf a}, in {\bf c} we binned, according to their distance from the MFE structure of the wild type, the MFE structures of all ($3L$) single point mutants for the \emph{Monodelphis domestica} miRNA precursor sequence \emph{mdo-mir-1} (black bars) as well as the MFE structures for each sequence in the single mutant neighborhood of sample sequences (white bars). The distribution of structures in both the thermodynamic ensemble ({\bf a}) and the mutational neighborhood ({\bf c}) of the \emph{mdo-mir-1} miRNA precursor sequence have a significantly smaller fraction of structures that are highly dissimilar then sample sequences with identical MFE structure. Comparing the the averaged distribution of stem-loop (black bars) and random corresponding random sequence sample distributions (white bars) in the mutational neighborhood ({\bf d}) to similar averaged distributions in the thermodynamic ensembles of the same sequences ({\bf b}) shows that the tendency among miRNA precursor sequences for increased robustness is present both in the mutational neighborhood and the thermodynamic ensemble, i.e. miRNA precursor sequences show excess robustness in the face of both thermal and mutational perturbation.}\label{fig2}
\end{center}
\end{figure}
\begin{table*}
\caption{ Phylogenetic breakdown of different measures of robustness }
\begin{tabular}{@{\vrule height 10.5pt depth4pt width0pt}l|ccc|ccc|ccc|cc}
group / species &$\bar r_n$ & $R_n$ & $S_n$ & $\bar r_s$ & $R_s$ & $S_s$ & $\bar r_t(25)$ & $R_t(25)$ & $S_t(25)$ & $c(r_s,r_t(25))$ & \# of seqs. \\
\hline\hline
all
& {\bf 0.44}& 0.59 & 0.08 & {\bf 0.29}& 0.78 & 0.17&{\bf 0.31}& 0.74 & 0.28 & 0.73 & 3641 \\
\hline
vertebrate
& 0.46 & 0.55 & 0.06 &{\bf 0.31}& 0.78 & 0.13&{\bf 0.29}& 0.75 & 0.30 & 0.76 & 2215 \\
invertebrate
&{\bf 0.37}& 0.68 & 0.10 &{\bf 0.21}& 0.88 & 0.27&{\bf 0.22}& 0.84 & 0.36 & 0.73 & 488 \\
landplant
&{\bf 0.41}& 0.63 & 0.11 &{\bf 0.31}& 0.75 & 0.21&{\bf 0.40}& 0.63 & 0.19 & 0.68 & 848 \\
virus
& 0.38 & 0.66 & 0.09 &{\bf 0.23}& 0.84 & 0.18&{\bf 0.21}& 0.85 & 0.32 & 0.65 & 82 \\
\hline
\emph{Homo sapiens}
& 0.48& 0.53 & 0.04 &{\bf 0.32}& 0.75 & 0.11&{\bf 0.28}& 0.76 & 0.33 & 0.74 & 471 \\
\emph{Mus musculus}
& 0.46& 0.56 & 0.06 &{\bf 0.33}& 0.76 & 0.12&{\bf 0.31}& 0.74 & 0.27 & 0.79 & 373 \\
\emph{Drosophila melanogaster}
& 0.40 & 0.64 & 0.08 &{\bf 0.22}& 0.88 & 0.24&{\bf 0.23}& 0.82 & 0.35 & 0.74 & 78 \\
\emph{Caenorhabditis elegans}
&{\bf 0.30}& 0.78 & 0.18 &{\bf 0.20}& 0.89 & 0.37 &{\bf 0.23}& 0.82& 0.34 & 0.75 & 114 \\
\emph{Arabidopsis thaliana}
& 0.39& 0.62 & 0.11 &{\bf 0.29}& 0.78 & 0.19 & 0.43& 0.60 & 0.15 & 0.75 & 131 \\
Epstein-Barr virus
& 0.31& 0.78 & 0.00 &{\bf 0.16}& 0.96 & 0.22 & 0.16& 0.87 & 0.48 & 0.81 & 23 \\
\end{tabular}
\begin{center}
{\small
Average rank-scores that indicate significantly increased according to both measures discussed in the Materials and Methods section ($p$-value $<10^{-3}$) are given in bold.}
\end{center}
\label{table1}
\end{table*}
\begin{figure}
\begin{center}
\centerline{\includegraphics[width=0.5\textwidth]{fig3}}
\caption{
To quantify the extent of thermodynamic robustness present in miRNA precursor sequences we examined the rank statistics of the threshold thermodynamic robustness $\eta_t(d_{\rm th.})$ which we defined as the cumulative frequency of structures in the thermodynamic ensemble that are equal to or less than a distance threshold $d_{\rm th.}$. {\bf a} For threshold values lager than $d_{\rm th.}\approx20$ the average rank of miRNA precursor sequences with respect to $\eta_t(d_{\rm th.})$, denoted by $\bar r_t(d_{\rm th.})$, becomes larger than $\bar r_s$, while remaining highly correlated with it. For thresholds $d_{\rm th.}>25$ the $\eta_t(d_{\rm th.})$ values for an increasingly larger fraction of random sample sequences becomes indistinguishable from the $\eta_t(d_{\rm th.})$ value of the original miRNA precursor sequence {\color{black} due to a lack of structures with $d>d_{th.}$ in our finite resolution sample of the equilibrium ensemble. The fraction of sequences which remain distinguishable are labeled \emph{fraction resolved}}. {\bf b} Among the majority of the random sample sequences that are resolved, however, the threshold robustness of large number of the miRNA precursor sequences becomes highly significant (black and white bars) compared to the mutation robustness as measured by $\eta_s$ (grey bars), whilst retaining a high correlation among $r_s$ and $r_t(d_{\rm th.})$.}\label{fig3}
\end{center}
\end{figure}
\section*{Results}
We assessed the environmental and mutational robustness of $3641$ unique miRNA precursor sequences and for each sequence compared them to a random sample of sequences with the same MFE structure. The idea of looking for signs of adaptation for increased robustness among miRNA precursor sequences by comparing the robustness of naturally occurring sequences to that of random sequences with the same secondary structure is conceptually similar to the approach used to support the argument that the genetic code has evolved to minimize mutational load \cite{digulio,haighurst,szathmarybook}. In the case of the genetic code the authors took the common genetic code, and, for each codon, calculated the change in polarity of the encoded amino acid caused by replacing each of the three nucleotides, one after the other. In order to determine whether the genetic code is adapted to minimize mutational load they proceeded by comparing the mean squared change caused by the replacement of a single nucleotide in the common genetic code to $10000$ randomly generated codes with the same redundancies. They found that only two of the random codes were more conservative than the common code with respect to polarity distances between neighbouring amino acids.
We undertook a similar program in the case of miRNA precursor sequences. Each miRNA gene encodes a short $\approx22$ nucleotide sequence that is partially complementary to the mRNA of proteins regulated by the particular miRNA gene. For the proper short sequence to be excised by the protein Dicer, and hence for the miRNA gene to be functional, a larger part of the miRNA sequence, called the miRNA precursor sequence, must fold into the proper secondary structure. In order to determine whether a miRNA precursor sequence is adapted to minimize the effects of mutational and/or environmental perturbations, i.e.\ to maximize mutational and/or environmental robustness, we compared the mutational and environmental robustness of each miRNA precursor sequence {\color{black}(robustness measures we used are defined below)} to the mutational and environmental robustness of a random sample of sequences with identical structural phenotype (i.e.\ identical MFE structure).
To generate a random sample of sequences with given MFE structure we first used, starting from a random sequence, stochastic minimization of the free-energy of the target structure to find a sequence with the desired MFE structure. This method by itself, however, yields a \emph{biased} sample of sequences (see Fig.\ \ref{fig1}a and b) and must be supplemented by an additional randomization step (see Materials and Methods). To measure the mutational robustness of a given sequence we used the measures introduced by Borenstein and Ruppin \cite{ruppin}: (i) the structural distance based mutational robustness measure $\eta_s$ of an RNA sequence of length $L$ is defined by $\eta_s= 1/(3L) \sum_{i=0}^{3L} (L-d_i)/L $, where $d_i$ is the base-pair distance between the secondary structure of mutant $i$ and the native sequence ({\color{black} given by the number of base pairs present in one structure, but not the other}), and the sum goes over all $3L$ single mutant neighbours and (ii) the more stringent measure $\eta_n$ is simply defined as the fraction of neutral single mutant neighbours, i.e.\ those that have identical MFE structure to the original sequence. In order to quantify the level of excess mutational robustness among miRNA precursor sequences we counted, for each miRNA precursor sequence, the number of sample sequences that have higher mutational robustness according to a given measure (see Materials and Methods) and used this to calculate the rank scores $r_s$ and $r_n$, defined as the fraction of sample sequences with identical or higher robustness according to $\eta_s$ and $\eta_n$, respectively. To facilitate an overview of the extent of excess mutational robustness we also calculated the average of the rank scores over all miRNA precursor sequences $\bar r_s$ and $\bar r_n$ as well as the fraction of miRNA precursor sequences with higher than average robustness (i.e.\ rank-scores $<0.5$) $R_s$ and $R_n$ and the fraction of sequences with statistically significant increased robustness (i.e.\ rank-scores $<0.05$, see Materials and Methods) $S_s$ and $S_n$, respectively, according to a give measure. The statistical significance of both rank scores for individual miRNA precursor sequences as well as that of the finding a given fraction of robust sequences among a group of sequences was determined as detailed in the Materials and Methods section.
Reexamining the mutational robustness of miRNA precursor sequences in comparison to an \emph{unbiased} sample of sequences with identical MFE structure we found that miRNA precursor sequences -- in contrast to the results of Borenstein and Ruppin -- do not have significantly more neutral single mutant neighbours than sample sequences, but do show a statistically significant increase in robustness measured according to $\eta_s$ (see Fig.\ \ref{fig1}b and Table \ref{table1}). In other words, native miRNA precursor sequences have on average the same number of single mutant neighbours with MFE structures identical to their own, as random sample sequences with the same structure. The MFE structure of those single mutant neighbours that are not identical to their own are, on the other hand, significantly more similar than in the case of sample sequences.
The presence of excess mutational robustness is, by itself, insufficient to determine whether mutational robustness has evolved as a result of direct selection or in congruence with selection for environmental robustness. As established previously \cite{ruppin} there is evidence for excess thermodynamic robustness, robustness to thermal fluctuations, as evidenced by a significantly lower than chance minimum folding energy among miRNA precursor sequences. Defining the environmental robustness measure $\eta_E$ simply as minus the minimum folding energy we also find $\bar r_E=0.278$, $R_E=0.796$, $S_E=0.220$ using unbiased sampling. The correlation between $r_s$ and $r_E$ across miRNA precursor sequences is, however, rather low with a Pearson's correlation coefficient of $c(r_s,r_E)=0.217$ and $c(r_n,r_E)=0.071$. The minimum folding energy is a somewhat crude measure of thermodynamic robustness and does not even reflect the excess mutational robustness according to the measure $\eta_s$. {\color{black} There is no good reason to assume that a low MFE in itself confers environmental robustness, as even for relatively high free energies a given sequence may none the less with high probability fold into structures sufficiently similar to the MFE structure to remain functional.} The large number of miRNA precursor sequences that exhibit excess mutational robustness as measured by the structural similarity based measure $\eta_s$ suggests that a strict adherence to the MFE structure is not necessary to retain functionality -- a sufficiently similar, but not necessarily identical, secondary structure is enough to guarantee the excision of the proper subsequence. {\color{black} This is further supported by the fact that folding free energy alone is not sufficient to discriminate miRNA precursors, as well as recent evidence that a diverse set of structural features are needed for successful cleavage of a miRNA precursor sequence \cite{ritchie}. }
To construct an appropriate measure of thermodynamic robustness that also reflects this observation we would need to know the extent of similarity that is required to retain functionality -- indeed we would require detailed knowledge of the interaction between the RNA substrate and the enzyme Dicer to establish an appropriate measure of structure similarity. As such information is not at present available we chose to use the most simple and widely employed structure similarity measure, the base-pair distance used above. In order to determine the extent of similarity required to retain functionality we defined the threshold thermodynamic robustness measure $\eta_t(d_{\rm th.})$ as a function of the threshold distance $d_{\rm th.}$, by equating it with the probability in the equilibrium thermodynamic ensemble of structures that have base-pair distances equal to or less than a threshold $d_{\rm th.}$ with respect to the MFE structure, i.e.
\begin{equation}
\eta_t(d_{\rm th.}) = \sum_{i \in \Omega} H(d_{\rm th.}-d_i)\ \frac{{\rm e}^{-E_i/kT}}{Z},
\end{equation}
where the sum goes over the set of all possible structures $\Omega$, $d_i$ denotes the base-pair distance of structure $i$ to the MFE structure, $Z= \sum_{i \in \Omega}{\rm e}^{-E_i/kT} $ is the partition sum and $H(x)$ is the unit step function, i.e.\ $H(x)=0$ if $x<0$ and $H(x)=1$ if $x \geq 0$.
Examining the thermodynamic robustness of miRNA precursor sequences in comparison to an unbiased sample of sequences with identical MFE structure we found that miRNA precursor sequences have significantly more structures in their equilibrium thermodynamic ensemble that are similar to the MFE structure than sample sequences (see Fig.\ \ref{fig2}a,b and Table \ref{table1}). In other words miRNA precursor sequences tend to adapt more similar structures as a result of thermal fluctuations than random sample sequences with the same structure. Calculating the average rank score $\bar r_t(d_{\rm th.})$ and the fraction of robust $R_t(d_{\rm th.})$ and significantly robust $S_t(d_{\rm th.})$ miRNA precursor sequences, with respect to the measure $\eta_t(d_{\rm th.})$ (Fig. \ref{fig3}a) and examining the distribution of structures as a function of the base-pair distance for individual miRNA precursor sequences (see e.g.\ Fig.\ \ref{fig2}a) indicates that above a threshold distance $d_{\rm th.}\approx20$ the measures start to saturate, yielding an estimate of the required similarity to retain function. The correlation between the rank-score of miRNA precursor sequences according to the distance similarity based mutational robustness measure and the threshold thermodynamics measure is high for all threshold values. This is the direct result of the high degree of similarity between the distribution of structures in the thermodynamic ensemble and the mutational neighborhood (Fig.\ \ref{fig2}). The average rank score $\bar r_t(d_{\rm th.})$ and the fraction of robust $R_t(d_{\rm th.})$ and significantly robust $S_t(d_{\rm th.})$ miRNA precursor sequences with respect to the threshold thermodynamic robustness measure indicate a markedly larger extent of excess robustness than their counterparts for mutation robustness, i.e.\ $\bar r_s$, $R_s$ and $S_s$ (see Fig.\ \ref{fig3}a,b and Table \ref{table1}) above $d_{\rm th.}>20$.
\section*{Discussion}
The results presented above demonstrate the correlated presence of excess environmental (thermodynamic) and genetic (mutational) robustness among miRNA precursor sequences as measured according to, respectively, $\eta_s$ and $\eta_t(d_{\rm th.})$. A rather general causality between environmental and genetic robustness in the context of RNA secondary structure has been suggested by Ancel and Fontana \cite{ancelfontana}, who studied the dynamics of an \emph{in silico} population of RNA sequences evolving towards a predefined target shape. They found that a correlation exists between the set of shapes in the plastic repertoire of a sequence and the set of dominant (minimum free energy) shapes in its genetic neighborhood. They argue that this statistical property of the RNA genotype-phenotype map, which they call plastogenetic congruence, traps populations in regions where most genetic variation is phenotypically neutral. In other words RNA sequences explore a similar repertoire of suboptimal structures as a result of perturbations due to mutations and perturbations resulting from thermal fluctuations, and selection for a given target structure favours sequences with higher robustness to perturbations of both type.
Since, in contrast to genetic robustness, environmental robustness does not require high values of $uN_e$, as it is a property of the sequence and not its mutational neighborhood, we contend that the observed bias in mutational robustness is in fact the result of the \emph{congruent} evolution of environmental and genetic robustness.
The correlation between the response to heritable (mutational) and nonheritable (thermodynamic) perturbation, and hence the congruent evolution of genetic and environmental robustness may extend to other systems with genotype-phenotype maps different from RNA secondary structure. In particular, Xia and Levitt \cite{xia}, have found compelling evidence of the correlated evolution of increased thermodynamic stability and the number of neutral neighbours in lattice protein models. Understanding the relationship between sequence, structure, and function is, and will remain to be in the foreseeable future, a central theme in both molecular and evolutionary biology. A comprehensive view of how the relationship between sequence, structure, and function is shaped during the course of evolution must take into consideration both the potential correlations that arise from the physics of the structure-sequence relationship as well as the relevant population genetic conditions in the context of which it takes on the role of a genotype-phenotype map. {\color{black} In the context of computational miRNA gene discovery our thermodynamic robustness measure potentially offers an improved structural feature that may perform better than the free energy score of the hairpin or its ensemble diversity (which have proved uninformative \cite{freyhult}).}
\begin{acknowledgments}
This work was partially supported by the Hungarian Science Foundation (K60665).
\end{acknowledgments}
|
1,941,325,220,564 | arxiv | \section{Introduction}
\label{intro}
This paper is a follow-up of the study initiated in \cite{DPTVjsp}, \cite{DPTVpro}, where current reservoirs in the context of stochastic interacting particle systems have been proposed as a method to investigate stationary non-equilibrium states with steady currents produced
by action at the boundary.
Due to the particular difficulties in implementing this new method, we consider the simplest possible particle system.
The bulk dynamics is the symmetric simple exclusion process (SSEP) in the interval $\La_N=[-N,N]\cap \mathbb Z$ ($N$ a positive integer and $N\to \infty$ eventually), namely the state space is $\{0,1\}^{\La_N}$ (at most one particle per site): independently each particle tries to jump at rate $N^2/2$ to each one of its nearest neighbor (n.n.) sites,
the jump then takes place if and only if the chosen site is empty, jumps outside $\La_N$ are suppressed.
To induce a current we send in
particles from the right and take them out from the left, and would like this to happen at rate $Nj/2$,
$j>0$ a fixed parameter independent of $N$. Due to the restrictions imposed by the configurational space, we have to be more precise when defining this dynamics. For this we fix a parameter $K\ge 1$ (an integer) and two intervals
$I_{\pm}$ of length $K$ at the boundaries: $I_+\equiv [N-K+1,N]$ and $I_-\equiv [-N,-N+K-1]$.
At rate $Nj/2$, when $I_+$ is not totally occupied, we create a particle at its rightmost empty site; with the same rate, unless $I_-$ is empty, we take out a particle from its leftmost occupied site. In case $I_+$ is already full, or $I_-$ empty, the corresponding mechanism aborts.
\smallskip
In \cite{DPTVjsp}, \cite{DPTVpro} we have proved that at any time $t>0$ propagation of chaos holds and that in the limit $N\to \infty$ the hydrodynamical equation is the linear heat equation:
\begin{eqnarray}
\nn&&\frac {\partial}{\partial t} \rho(r,t)= \frac 12 \frac {\partial^2}{\partial r^2} \rho(r,t), \qquad r\in (-1,1),\, t>0,
\\&&\rho(r,0)=\rho_0(r),\qquad \rho(\pm 1,t)= u_{\pm}(t),
\label{1.1}
\end{eqnarray}
where $\rho_0(\cdot)$ is given but $u_{\pm}(t)$ are solutions of a nonlinear system of two integral equations, see \eqref{dptv2.4} below.
\smallskip
The goal of this paper is to investigate the limiting density profile (as $N\to \infty$) of the (unique) invariant measure of the process. The main result is Theorem \ref{thm2.2}, which shows that this rescaled limiting profile coincides with the unique stationary solution of \eqref{1.1}. In particular, taking into account the validity of the Fourier law, proven as Theorem 2 in \cite{DPTVjsp}, we see that the effective current in the stationary regime is strictly smaller than its desired maximum value which is $\min\{j/2,1/4\}$, but this value is indeed approached by letting $K\to \infty$.
\vskip1cm
\section{Model and main results}
Particle configurations are elements $\eta$ of $\{0,1\}^{\La_N}$, $\eta(x)=0,1$ being the occupation number at $x\in\La_N$. We consider the Markov process on $\{0,1\}^{\La_N}$ defined via the generator
$$
L_N:=N^{2}\Big(L_0+\frac 1N L_b\Big),
$$
where $L_b= L_{b,+}+ L_{b,-}$ and
\begin{eqnarray}
\nn
&&
L_0 f(\eta):=\frac 12\sum_{x=-N}^{N-1} [f(\eta^{(x,x+1)})-f(\eta)],
\\ \label{1}
\\&& L_{b,\pm} f(\eta):= \frac{j}{2}\sum_{x\in I_\pm}D_{\pm}\eta(x) [f(\eta^{(x)})-f(\eta)\Big],
\nn
\end{eqnarray}
$\eta^{(x)}$ being the configuration obtained from $\eta$ by changing the occupation number at $x$, $\eta^{(x,x+1)}$ by exchanging the occupation numbers at $x,x+1$; for any $u:\La_N\to [0,1]$
\begin{eqnarray}
&& D_+u (x)= [1-u(x)]u(x+1)u(x+2)\dots u(N), \quad x\in I_+
\nn\\
&& D_-u (x)= u(x)[1-u(x-1)][1-u(x-2)]\dots[1-u(-N)], \quad x\in I_-.
\label{6}
\end{eqnarray}
\vskip 0.5cm
Given $\rho_0\in C([-1,1],[0,1])$, let $\nu^{(N)}$ be the product probability measure on $\{0,1\}^{\La_N}$ such that $\dis{\nu^{(N)}(\eta(x))=\rho_0(N^{-1} x)}$ for all $x\in\La_N$. Let $\mathbb{P}_{\nu^{(N)}}$ denote the law of the process with initial distribution $\nu^{(N)}$ and $\E_{\nu^{(N)}}$ the corresponding expectation.\footnote{Omitting the initial profile to avoid too heavy notation.}
\vskip.5cm
The following theorem has been proven (in a stronger form) in \cite{DPTVjsp}, \cite{DPTVpro}. The statement below contains all what is needed in the present paper. In the following, for $n$ a positive integer we write $\Lambda_N^{n,\neq}$ for the set of all sequences $(x_1,...,x_n)$ in $\La_N^n$ such that $x_i\ne x_j$ whenever $i\neq j$.
\vskip.5cm
\begin{thm}
\label{thm2.1}
There exists $\tau>0$ so that for any $\rho_0$ as above and any $n\ge 1$,
\begin{equation}
\label{2.0}
\lim_{N\to \infty} \Big|\E_{\nu^{(N)}}\big(\prod_{i=1}^n \eta(x_i,t)\big)-\prod_{i=1}^n\E_{\nu^{(N)}}\big( \eta(x_i,t)\big)\Big|=0,\qquad \text{for any } t\le \tau\log N.
\end{equation}
Furthermore
\begin{equation}
\label{dptv2.3}
\lim_{N\to \infty} \sup_{x\in \La_N}\sup_{ t \le \tau\log N} \big|\E_{\nu^{(N)}}\big( \eta(x,t) )-\rho( N^{-1}x,t)\big|=0,
\end{equation}
where the function $\rho(r,t)$ solves the heat equation $\frac{\partial \rho}{\partial t}=\frac 12 \frac{\partial^2 \rho}{\partial r^2}$, $r\in (-1,1), t>0$ with initial datum $\rho_0$ and boundary conditions $\rho(\pm 1,t)=u_\pm(t)$, the pair $(u_+(t),u_-(t))$ being the unique solution of the non linear system
\begin{eqnarray}
\nn
&&
\hskip-.3cm
u_\pm(t)= \int_{[-1,1]} P_t(\pm 1,r) \rho_0(r) dr+ \frac j2\int_0^t \Big\{ P_{s}(\pm 1,1)\left(1-u_+(t-s)^K\right)
\\&&\hskip3.8cm
- P_{s}(\pm 1,-1)\left(1- (1-u_-(t-s))^K\right)\Big\}ds,
\label{dptv2.4}
\end{eqnarray}
where $P_t(r,r')$ is the density kernel of the semigroup (also denoted as ${P_t}$) with generator $\Delta/2$, $\Delta$ the laplacian in $[-1,1]$ with reflecting, Neumann, boundary conditions.
The function $\rho(r,t)$ satisfies
\begin{equation}
\label{4.2}
\frac{\partial \rho(r,t)}{\partial r}|_{r=1}=j (1-u_+(t)^K),\quad \frac{\partial \rho(r,t)}{\partial r}|_{r=-1}=j (1-(1-u_-(t))^K).
\end{equation}
\end{thm}
\vskip.5cm
\noindent {\bf Remark.} The following is the integral form of the macroscopic equation:
\begin{eqnarray}
\label{dptv2.4bis}
&& \rho(r,t)= \int_{[-1,1]} P_t(r,r') \rho(r',0) dr' + \frac j2\int_0^t \Big\{ P_{s}(r,1)\left(1-\rho(1,t-s)^K\right)\nn
\\&&\hskip3cm
- P_{s}(r,-1)\left(1- (1-\rho(-1,t-s))^K\right)\Big\}ds.
\end{eqnarray}
It will be convenient to recall the expression for the density kernel $P_t(r,r^\prime)$ in terms of the Gaussian kernel
\begin{equation}
\label{N4.3}
G_t (r,r') = \frac{e^{-(r-r')^2/(2t)}}{\sqrt {2\pi t}},\quad r,r' \in \mathbb{R},
\end{equation}
as
\begin{eqnarray}
\label{N4.10}
P_t (r,r') &=& \sum_{r'':\psi(r'')=r'} G_t(r,r'') \quad \text{for} \; r'\neq \pm 1\nn\\
P_t(r,\pm 1) &=& \sum_{r'':\psi(r'')=\pm 1} 2G_t(r,r''),
\end{eqnarray}
where $\psi:\mathbb R\to [-1,1]$ denotes the usual reflection map:
$\psi (x)= x $ for $x \in [-1,1]$, $\psi(x)=2-x $ for $x \in [1,3]$, $\psi$ extended to the whole line
as periodic of period 4.
\vskip .5cm
\noindent {\bf Notation.} $P_t g(r)=\int P_t(r,r^\prime) g(r^\prime) dr^\prime$, for $g$ a bounded continuous function, $t>0$.
\vskip1cm
\noindent
The main result of this paper is about the density profile of the unique invariant measure $\mu_N$.
\begin{thm}
\label{thm2.2}
For any integer $k\ge 1$ we have
\begin{equation}
\label{2.1-a}
\lim_{N\to\infty} \max_{(x_1,..,x_k)\in \La_N^{k,\ne}}\Big|\mu_N\big(\eta(x_1)\cdots \eta(x_k)\big)-\rho^*(x_1/N)\cdots \rho^*(x_k/N)\Big| = 0
\end{equation}
where $\rho^*(r)$ is the unique stationary solution of the macroscopic equation.
Namely $\rho^*(r)=J\,r +\frac 12$,
\begin{equation}
\label{2.1}
J=j(1-\alpha^K),\quad \text{ with $\alpha$ the solution of}\quad \alpha(1+j\alpha^{K-1})=j+\frac 12.
\end{equation}
\end{thm}
\vskip.5cm
By Theorem \ref{thm2.2} it follows that $\mu_N$ concentrates on a $L^1$-neighborhood of the
limit profile $\rho^*$: let $r\in (0,1)$ and
\begin{equation*}
\rho^{(\ell)}(r;\eta) = \frac1{2\ell+1} \sum_{x\in \La_N: |x-rN| \le \ell} \eta(x)
\end{equation*}
Then for any $a\in (0,1)$
\begin{equation*}
\lim_{N\to \infty} \mu_N \Big( \int_{-1}^1 | \rho^{(N^a)}(r;\eta) -\rho^*(r)|dr\Big)=0
\end{equation*}
Theorem \ref{thm2.2} will follow from $\bullet$\; uniformly on the initial datum $\rho_0$ the solution $\rho(r,t|\rho_0)$ of the macroscopic equation \eqref{dptv2.4bis} converges in sup norm to $\rho^*$ exponentially fast, see Theorem \ref{thm3.3} below; $\bullet$\;
for any integer
$k\ge 1$,
\begin{equation}
\label{2.1-b}
\lim_{t\to \infty}\lim_{N\to\infty}\max_{\eta \in \{0,1\}^{\La_N}}\max_{(x_1,..,x_k)\in \La_N^{k,\ne}}\Big|\mathbb E_\eta \Big(\prod_{i=1}^k\eta(x_i,t)\Big)-\prod_{i=1}^k\rho^*(x_i/N)\Big| = 0
\end{equation}
We are also working on an extension of the theorem where we prove exponential convergence in time to $\mu_N$ uniformly in $N$.
\vskip1cm
\section{Monotonicity properties}
\vskip.5cm
We consider the space $\{0,1\}^{\La_N}$ endowed with the usual partial order, namely we say that $\eta\le \xi$ iff $\eta(x)\le \xi(x)$ for all $x\in \La_N$.
The following proposition is an immediate consequence of general facts on attractive systems, see e.g. \cite{li} (chs. II and III).
\begin{prop}
\label{thm3.1}
Let $\eta_0$ and $\xi_0$ be two particle configurations such that $\eta_0\le \xi_0$, and let $\mathbb{P}_{\eta_0}$, respectively
$\mathbb{P}_{\xi_0}$, be the law of the process starting from $\eta_0$, respectively $\xi_0$. Then there is a coupling $\mathbb{Q}$ of $\mathbb{P}_{\eta_0}$ and $\mathbb{P}_{\xi_0}$ (i.e. $\mathbb{Q}$ is a measure on the product space, with $\mathbb{P}_{\eta_0}$ as its
first marginal, and $\mathbb{P}_{\xi_0}$ as the second one) such that
\begin{equation}
\label{3.1}
\mathbb{Q}\{(\eta,\xi)\colon \eta_t \,\le \,\xi_t\,, \forall t\}=1
\end{equation}
\end{prop}
\noindent {\bf Proof}. Being well known that the process corresponding to $L_0$ is attractive, it suffices to observe that the flip rates $c(x,\eta):=D_\pm \eta (x)$ in $I_\pm$ are attractive in the sense that if $\eta(x)=\xi(x)=0$ and $\eta \le \xi$ then $c(x,\eta) \le c(x,\xi)$, while if $\eta(x)=\xi(x)=1$ and $\eta \le \xi$ then $c(x,\xi) \le c(x,\eta)$. \qed
\vskip.5cm
The analogous monotonicity property holds for the macroscopic equation.
Instead of a direct proof
we derive the result as a consequence of the monotonicity of the particle system and that it converges
to the macroscopic equation.
\vskip.5cm
\begin{thm}
\label{thm3.2}
Let $ \rho_0, \tilde\rho_0$ be bounded measurable functions from $[-1,1]$ to $[0,1]$ such that $\rho_0(r)\le \tilde\rho_0(r)$ for all $r\in[-1,1]$, and let $\rho(r,t)$, respectively $\tilde\rho(r,t)$, be the corresponding solution of \eqref{dptv2.4bis} with initial datum $\rho_0$, respectively $\tilde\rho_0$. Then $\rho(r,t)\le \tilde\rho(r,t)$ for all $r\in[-1,1]$ and $t\ge 0$.
\end{thm}
\noindent {\bf Proof}. Let $\nu^{(N)}$ and $\tilde\nu^{(N)}$ be the product probability measures on $\{0,1\}^{\La_N}$ such that $\nu^{(N)}(\eta(x))=\rho_0(N^{-1} x)$ and $\tilde\nu^{(N)}(\eta(x))=\tilde\rho_0(N^{-1} x)$ for all $x\in\La_N$.
It is well known that a coupling $\la^{(N)}$ of $\nu^{(N)}$ and $\tilde\nu^{(N)}$ such that $\la^{(N)}\{(\eta,\tilde \eta)\colon \eta\le \tilde\eta\}=1$ exists. Using Proposition \ref{thm3.1} and the notation of Theorem \ref{thm2.1} we have
\begin{equation}
\label{3.2-a}
\mathbb{E}_{\nu^{(N)}}(\eta(x,t))\le \mathbb{E}_{\tilde\nu^{(N)}}(\eta(x,t)),\qquad \forall x\in \La_N,\quad \forall t\ge 0.
\end{equation}
From \eqref{dptv2.3} we then have that for all $t\ge 0$ and for all $r\in[-1,1]$, (below $[\cdot]$ denotes the integer part)
\begin{equation}
\label{3.2}
\rho(r,t)=\lim_{N\to\infty}\mathbb{E}_{\nu^{(N)}}\big(\eta([Nr],t)\big)\le \lim_{N\to\infty}\mathbb{E}_{\tilde\nu^{(N)}}\big(\eta([Nr],t)\big)=\tilde\rho(r,t).
\end{equation}
\qed
\vskip1cm
\section{The macroscopic profile}
\vskip.5cm
\nopagebreak
We first prove that the function $\rho^*$ in the statement of Theorem \ref{thm2.2} is a stationary solution to the Dirichlet problem \eqref{1.1} with boundary condition \eqref{dptv2.4} or, equivalently, of the integral equation \eqref{dptv2.4bis}. In fact by requiring that a stationary solution is a linear function we get, due to \eqref{4.2}, that the values of this function at $\pm 1$, denoted with $u_\pm$, must satisfy
\begin{equation*}
j (1-u_+^K) =j (1-(1-u_-)^K).
\end{equation*}
This implies
\begin{equation*}
u_+=(1-u_-),\qquad \text{and }\quad \frac {2u_+-1}2=j(1-u_+^K),\quad u_+=\frac 12 +j(1-u_+^K).
\end{equation*}
Solving we get
\begin{equation*}
u_+(1+ju_+^{K-1}) =j +\frac 12
\end{equation*}
in agreement with \eqref{2.1}.
On the other hand, since $\frac{\partial}{\partial t}P_t(r,r')=\frac12\frac{\partial^2}{\partial(r')^2}P_t(r,r')$ and
it satisfies Neumann boundary conditions at $\pm 1$ we easily see that
\begin{equation*}
\frac{d}{dt} \int_{[-1,1]} P_t(r,r')r'dr'=\frac 12\left(P_t(r,-1)-P_t(r,1)\right).
\end{equation*}
Recalling (from \eqref{2.1}) that $J=j(1-(\rho^*(1))^K)=j(1-(1-\rho^*(-1))^K)$
we see at once that $\rho^*$ satisfies \eqref{dptv2.4bis}, which in this case can be written as:
\begin{equation}
\rho^*(r)= P_t\rho^*(r)+\frac j2 (1-(\rho^*(1))^K)\int_0^t \Big\{ P_{s}(r,1)
- P_{s}(r,-1)\Big\}ds,
\label{4.1}
\end{equation}
for all $t\ge 0$.
\vskip.5cm
We now prove that any solution to the Dirichlet problem converges exponentially fast to $\rho^*$ as $t\to \infty$. In particular, one has uniqueness of the stationary solution.
\vskip.5cm
\begin{thm}
\label{thm3.3}
There exist positive constants $c,c^\prime$ so that for any function $\rho_0$ $\in L^\infty([-1,1],[0,1])$ the solution $\rho(r,t|\rho_0)$ of the
macroscopic equation \eqref{dptv2.4bis} with initial datum $\rho(r,0)=\rho_0(r)$ satisfies
\begin{equation}
\label{t4.1}
\sup_{r\in[-1,1]}|\rho(r,t|\rho_0)-\rho^*(r)|\le c' e^{-ct}.
\end{equation}
\end{thm}
\noindent {\bf Proof}.
Let $\bar \rho(r,t)$ denote the solution with initial datum $\rho\equiv 1$, and $\und \rho(r,t)$ that
corresponding to initial datum $\rho\equiv 0$. From Theorem \ref{thm3.2} we know that
$\und \rho(r,t)\le \rho(r,t|\rho_0)\le \bar \rho (r,t)$, for any initial $\rho_0$. Hence, calling
\begin{equation*}
w(r,t):=\bar\rho(r,t)-\und\rho(r,t) \ge 0,\qquad w(t)=\sup_{r\in[-1,1]}w(r,t)
\end{equation*}
it suffices to show that $w(t)\le c^\prime e^{-ct}$ for suitable positive constants $c,c^\prime$ and all $t>0$.
In the proof below $c, \bar c, \tilde c$ will denote suitable positive constants (that might depend on the model parameter $j$)
whose value may change from line to line.
Let
\begin{equation*}
\bar u_\pm(t):=\bar\rho(\pm 1,t),\quad \und u_\pm(t):=\und\rho(\pm 1,t), \qquad w_\pm(t):=\bar u_\pm(t)-\und u_\pm (t)\ge 0.
\end{equation*}
>From \eqref{dptv2.4bis} we see that for all $r\in[-1,1]$, and all $t\ge t_0\ge 0$,
\begin{equation}
\label{4.3a}
w(r,t)=(P_{t-t_0}w(\cdot,t_0))(r)- \frac j2\int_{t_0}^t f(r,s,t-s) ds,
\end{equation}
where
\begin{eqnarray}
\nn
&&\hskip-.8cm
f(r, s,t-s):= P_{s}(r,1)\left\{\bar u_+(t-s)^K-\und u_+(t-s)^K\right\}
\\&&\hskip1cm +P_{s}( r,-1)\left\{(1-\und u_-(t-s))^K-(1-\bar u_-(t-s))^K\right\}.
\label{4.0a}
\end{eqnarray}
Interchanging particles and holes, one can couple at once the evolutions starting from the configurations $\bar\eta=\und 1$ ({\it all occupied sites}) and $\und \eta =\und 0$ ({\it all empty sites}) so that $\bar \eta(x,t)=1-\und \eta(-x,t)$. Therefore, by the
same argument as in the proof of Theorem \ref{thm3.2} one has $\bar \rho(r,t)=1-\und\rho(-r,t)$ for all $r$ and all $t$. In particular $w(-r,t)=w(r,t)$, $\bar u_{\pm}(t)=1-\und u_{\mp}(t)$ and $w_+(t)=w_-(t)$ for all $t$. (Still from Theorem \ref{thm3.2} we see that
$w(r,t)$ and so also $w(t)$ decrease in $t$.) Of course $w(r,0)=1$ for all $r$.
In particular, we may rewrite \eqref{4.3a} with $t_0=0$ as
\begin{equation}
w(r,t)=1- \frac j2\int_0^{t} [P_{s}(r,1)+P_s(r,-1)]w(1,t-s)h(t-s)ds\label{4.15}
\end{equation}
where
\begin{equation}
h(t-s):=\sum_{\ell=0}^{K-1}\bar u_+(t-s)^{K-1-\ell}\und u_+(t-s)^{\ell}
\label{4.16}
\end{equation}
and where we have used that for any integer $K\ge 1$,
\begin{equation}
\label{4.3a.1}
a^K-b^K=(a-b)\sum_{\ell=0}^{K-1} b^\ell a^{K-1-\ell},\qquad a \ge b \ge 0.
\end{equation}
Also, from \eqref{4.16} and the monotonicity properties we see that
\begin{equation}
\label{ck}
b:={\rho^*(1)}^{K-1}\le h(t) \le b+K-1=: c_K.
\end{equation}
The proof will use local times. To this end we introduce
the kernel operators $K^{(\eps)}_{s}$, $\eps>0$:
\begin{equation*}
K^{(\eps)}_{s} f(r)=\frac 1{\eps}\int_{[-1,-1+\eps]\cup[1-\eps,1]} P_s(r,r')f(r')dr', \quad f \in C([-1,1],\mathbb R).
\end{equation*}
In particular $K^{(\eps)}_{s} f(r)=K^{(\eps)}_{s} f(-r)$ for all $r \in [-1,1]$. Let $w^{(\eps)}$ be the solution to the following
integral equation:
\begin{eqnarray}
\nn
&&\hskip-1cm
w^{(\eps)}(r,t)=1
- \frac j2\int_0^{t} (K^{(\eps)}_{s} w^{(\eps)}(\cdot,t-s))(r)h(t-s)ds.
\label{4.15b}
\end{eqnarray}
We shall next prove that for all $T>0$,
\begin{equation}
\label{4.16'}
\lim_{\eps\to 0}\,\sup_{r\in [-1,1]}\,\sup_{0\le t\le T}\big|w(r,t)- w^{(\eps)}(r,t)\big|=0.
\end{equation}
Calling
\begin{equation}
\label{4.16a}
\psi(r,t)=\big|w(r,t)- w^{(\eps)}(r,t)\big|,\qquad \Psi(t)=\sup_{r\in [-1,1]}\psi(r,t)
\end{equation}
and using \eqref{ck}, we can write
\begin{eqnarray}
\nn
&&\Big|\int_0^{t} \left\{(K^{(\eps)}_{s} w^{(\eps)}(\cdot,t-s))(r)- \{P_{s}(r,1)+P_s(r,-1)\}w(1,t-s)\right\}h(t-s)ds
\Big|\le c\eps\\
&&\nn +c_K
\int_\eps^{t}\left\{\frac 1{\eps}\int_{1-\eps}^1\big|P_s(r,y)-P_s(r, 1)+P_s(-r,y)-P_s(-r,1)\big|dy\right\} \big|w^{(\eps)}(y,t-s)\big|ds \\
&& +c_K\int_\eps^{t} \{P_{ s}(r, 1)+P_s(-r,1)\} \Psi(t-s) ds.
\label{4.17}
\end{eqnarray}
Using that for all $y\in[1-\eps,1], \, r\in [-1,1]$
\begin{equation}
\label{4.18}
|P_s(r,y)-P_s(r, 1)|\le c \frac {1-y}{\sqrt {s^3}}, \qquad \forall s\in [\eps,t]
\end{equation}
we see that the second term on the r.h.s. of \eqref{4.17} is bounded above by
\begin{equation*}
\tilde c \int_\eps^{t} \frac 1{\sqrt {s^3}}ds\frac 1{\eps}\int_{1-\eps}^1(1-y)dy\le c' \sqrt\eps
\end{equation*}
for suitable constants $\tilde c, c'$. We then easily get
\begin{equation}
\psi(r,t)\le c_1 \sqrt\eps+c_2 \int_{0}^t \Psi(s)ds
\label{4.20}
\end{equation}
for suitable constants $c_1,c_2$. By the Gronwall inequality we conclude \eqref{4.16'}.
\vskip.5cm
We now estimate $w^{(\eps)}$. Let $\{B_t\}$ be a standard Brownian motion with reflecting b.c. at $\pm 1$,
with $\mathbb P_r$ denoting its law when $B_0=r$ (and corresponding expectations denoted by $\mathbb E_r$). Then
\begin{equation}
w^{(\eps)}(r,t)= \mathbb E_r\Big( e^{-\int_0^{t} \varphi_\eps(B_s,t-s)ds} w^{(\eps)}(B_{t},0)\Big)
\label{4.21}
\end{equation}
where
\begin{eqnarray}
\varphi_\eps(B,t-s)= \phi_\eps(B)h (t-s),\quad \phi_\eps(r)=\frac {j}{2\eps}\mathbf 1_{[1-\eps,1]}(|r|), \; r\in [-1,1].
\label{4.22}
\end{eqnarray}
By \eqref{ck}
\begin{equation}
w^{(\eps)}(r,t) \le \mathbb E_r\Big( e^{-b \int_0^{t} \phi_\eps(B_s)ds} \Big).
\label{4.21.1}
\end{equation}
For $0<\bar t<t$ we write
\begin{equation}
w^{(\eps)}(r,t) \le \mathbb E_r\Bigg( e^{-b\int_{0}^{t-{\bar t}} \phi_\eps(B_s)ds}\,
\mathbb E_{B_{t-\bar t}}\Big( e^{-b\int_{t-\bar t}^t \phi_\eps(B_s)ds}\Big)\Bigg)
\label{4.23}
\end{equation}
We shall prove below that taking $\bar t$ sufficiently small, we can take $\alpha<1$ so that for all $\eps>0$
\begin{equation}
\sup_{r\in[-1,1]}
\mathbb E_{r}\Big( e^{-b\int_{0}^{\bar t} \phi_\eps(B_s)ds} \Big)\le 1-\alpha
\label{4.24}
\end{equation}
From \eqref{4.24} and \eqref{4.23} we then get
\begin{equation}
|w^{(\eps)}(r,t)|\le (1-\alpha)^{[t/\bar t]}
\label{4.25}
\end{equation}
($[a]$ the integer part of $a$) which then concludes the proof of the theorem.
\vskip.5cm
{\bf Proof of \eqref{4.24}}. Let $T=\inf\{t \ge 0: |B_t|=1\}$.
We then have
\begin{eqnarray}
\label{4.26}
\mathbb E_{r}\Big( e^{-b\int_{0}^{\bar t} \phi_\eps(B_s)ds} \Big)&\le& \mathbb E_{r}\Big( \mathbf 1_{\{T\le \bar t/2\}}e^{-b\int_{T}^{\bar t} \phi_\eps(B_s)ds} \Big)+\mathbb P_r(T> {\bar t}/2)
\end{eqnarray}
and write
\begin{eqnarray}
\mathbb E_{r}\Big( \mathbf 1_{\{T\le {\bar t}/2\}}e^{-b\int_{T}^{\bar t} \phi_\eps(B_s)ds} \Big) &\le& \mathbb E_{r}\Big(\mathbf 1_{\{T \le {\bar t}/2\}}\mathbb E_{B_T}\big( e^{-b\int_0^{{\bar t}/2} \phi_\eps(B_s)ds} \big)\Big) \nn
\\ &\le& \mathbb{P}_r(T \le {\bar t}/2)\;
\mathbb E_{1}\big( e^{-b\int_0^{{\bar t}/2} \phi_\eps(B_s)ds} \big)\Big)
\label{a4.26b}
\end{eqnarray}
where we also used that $\mathbb E_{1}\big( e^{-b\int_0^{{\bar t}/2} \phi_\eps(B_s)ds} \big)=\mathbb E_{-1}\big( e^{-b\int_0^{{\bar t}/2} \phi_\eps(B_s)ds} \big)$ by symmetry.
By Taylor expansion
\begin{equation}
\mathbb E_{1}\big( e^{-b\int_0^{{\bar t}/2} \phi_\eps(B_s) ds}\big)
\le 1-b\mathbb E_{1}\Big(\int_0^{{\bar t}/2} \phi_\eps (B_s)ds \Big)
+\xi_2
\label{4.27}
\end{equation}
where
\begin{equation}
\xi_2=(\frac{jb}{2\eps})^2\int_0^{{\bar t}/2} dt_1\int_0^{t_1}dt_2
\int_{|y_1|\in [1-\eps,1], |y_2|\in [1-\eps,1]} P_{t_1}(1,y_1)P_{t_2}(y_1,y_2) dy_1dy_2
\label{4.28}
\end{equation}
But, from \eqref{N4.3}--\eqref{N4.10} we see that
\begin{equation}
\label{4.29}
\sup_{x,y\in [-1,1]}P_s(x,y)\le c \frac 1{\sqrt s}
\end{equation}
so that for $\bar t$ small enough we get
\begin{equation}
\label{4.30}
\xi_2\le \bar c {\bar t}/2
\end{equation}
for suitable constant $\bar c$.
Using again \eqref{N4.3}--\eqref{N4.10}, we see at once that a positive constant $c$ can be taken so that for all $\bar t$ small,
and all $\eps>0$
\begin{equation}
\label{4.31}
b\mathbb E_{1}\Big(\int_0^{{\bar t}/2} \phi_\eps(B_s)ds \Big)\ge
c\sqrt {\bar t/2}.
\end{equation}
From \eqref{4.27}, \eqref{4.30} and \eqref{4.31} we then get
for $\bar t$ small (with possibly different constant $c$),
\begin{equation}
\label{4.32}
\mathbb E_{1}\big( e^{-b\int_0^{{\bar t}/2} \phi_\eps(B_s)ds} \big)\le 1-c\sqrt{{\bar t}/2}.
\end{equation}
By \eqref{4.26} and \eqref{a4.26b} we then have
\begin{equation}
\label{4.41}
\mathbb E_{r}\Big( e^{-b\int_{0}^{\bar t} \phi_\eps(B_s)ds} \Big)\le \big[1-c\sqrt{{\bar t} /2}\big]\mathbb{P}_r(T\le {\bar t}/2)+\mathbb{P}_r(T>{\bar t}/2)\le 1-\alpha
\end{equation}
with
\begin{equation}
\alpha=\inf_{r\in [-1,1]} \mathbb{P}_r(T\le {\bar t}/2)c\sqrt{{\bar t} /2}.
\end{equation}
\vskip1cm
\section{Proof of Theorem \ref{thm2.2} }
\vskip.5cm
\nopagebreak
The proof is a direct consequence of the following three facts. (i) For any $t>0$ and any integer $k\ge 1$
\begin{equation}
\label{5.0.1}
\lim_{N\to \infty} \max_{ \eta \in \{0,1\}^{\La_N}}\max_{(x_1,..,x_k)\in \La_N^{k,\ne}}
\Big| \mathbb E_{\eta}\big(\prod_{i=1}^k \eta(x_i,t)\big)-\prod_{i=1}^k \mathbb E_{\eta}\big(\eta(x_i,t)\big)\Big|=0
\end{equation}
(ii) For any $t>0$
\begin{equation}
\label{5.0.2}
\lim_{N\to \infty}
\max_{ \eta \in \{0,1\}^{\La_N}}\max_{x\in \La_N}
\Big| \mathbb E_{\eta}\big( \eta(x,t)\big)-\rho(x/N,t| \eta)\Big|=0
\end{equation}
(iii)
\begin{equation}
\label{5.0.3}
\lim_{t\to \infty}
\sup_{ \rho_0\in L^\infty([-1,1];[0,1])}
\| \rho(\cdot,t| \rho_0)-\rho^*(\cdot)\|_\infty=0
\end{equation}
(i) and (ii) are proved in \cite{DPTVpro}--\cite{DPTVjsp}, \eqref{5.0.3} is proved in Theorem \ref{thm3.3}.
\vskip.5cm
{\bf Acknowledgments.}
The work is partially supported by the PRIN project, n.2009TA2595. M.E.V. thanks the Universities of Rome and L'Aquila for kind hospitality. M.E.V. is
partially supported by CNPq grant 302796/2002-9. The research of D.T. has been partially supported by a
Marie Curie Intra European Fellowship within the 7th European
Community Framework Program.
\vskip1cm
\bibliographystyle{amsalpha}
|
1,941,325,220,565 | arxiv | \section{Introduction}\label{s:intro}
Galaxy groups form the environment of the majority of galaxies in the local Universe. Many groups also host an extended hot intra-group medium (IGM) whose radiative cooling appears to fuel, and be stabilised by, AGN feedback from group-central galaxies. The group environment is conducive to galaxy evolution; low relative velocities and small galaxy separations drive mergers and tidal interactions, while motion through the IGM can produce gas compression or stripping. As such, groups are a critical environment in which to study the interplay between galaxy evolution, the development of the hot IGM, and AGN feedback. However, until recently, studies of groups have been hampered by the lack of reliable, statistically representative samples. Optically selected samples tend to become unreliable for low mass systems owing to the small number of detected group member galaxies. X-ray selection in the nearby Universe is dependent on the ROSAT All-Sky Survey, which is known to be biased toward centrally concentrated, cool core systems at low luminosities \citep{Eckertetal11}. This bias can be illustrated by comparing the cool-core fractions of clusters measured from statistical samples \citep[e.g., $\sim$50\%, ][]{Sandersonetal06} with the best available estimates from non-statistical samples of groups \citep[$\sim$85\%,][]{Dongetal10}. Without a reliable, statistical sample of nearby groups we cannot know whether this difference indicates a change in the IGM heating balance between groups and clusters, or whether it is merely a product of bias.
To resolve this problem, we have created a Complete Local-Volume Groups Sample consisting of 53 groups within 80~Mpc. We initially draw groups from the Lyon Galaxy Group sample \citep{Garcia93}, selecting systems with at least 4 members and at least one luminous early-type galaxy (L$_B>$3$\times$10$^{10}$L$_\odot$). Groups must also have Declination $>$-30$^\circ$, to ensure visibility from GMRT and VLA. We then refine and expand group membership, and exclude the richest and poorest groups, which either correspond to known clusters, or lack the numbers of galaxies needed for accurate estimation of properties such as velocity dispersion and spiral fraction. Further details of the selection can be found at \texttt{http://www.sr.bham.ac.uk/$\sim$ejos/CLoGS.html}.
Our goal is then to observe these optically selected groups in the X-ray, since the presence of a hot X-ray emitting IGM would confirm them as gravitationally bound systems, and in the radio, to allow examination of the AGN and star formation in the member galaxies. At present, we have observed the full 53-group sample with GMRT, using dual-frequency 235/610~MHz observations with a typical on-source time of 3-4 hours. In the X-ray we have observed a high-richness subsample of 26 groups, itself statistically complete, with a limiting sensitivity of L$_{0.5-7~keV}>$1.2$\times$10$^{42}$~erg~s$^{-1}$.
\section{Radio and X-ray detection rates}
In the 26-group high richness subsample, we detect 14 systems with diffuse X-ray emission extending $>$65~kpc and temperatures typical of galaxy groups (0.5-1.5~keV). A further 7 systems have smaller scale gaseous haloes associated with the dominant early-type galaxy. We therefore consider $\sim$55\% of this subsample to be confirmed as gravitationally bound groups. The sample is dynamically active, with two group-group mergers and two recently disturbed ``sloshing'' systems \citep[NGC~5044 and NGC~5846,][]{Gastaldelloetal13}.
We detect $\sim$45\% of the group member galaxies in the radio, with a higher detection rate ($\sim$70\%) for spiral galaxies than other morphologies ($\sim$25\%). This excludes the dominant early-type galaxies, whose location at the group centre makes them more likely to host AGN. We detect 25 of these dominant early-types at 235 and/or 610~MHz, and 7 host jet sources. The jet sources have a variety of morphologies, from small-scale jets $<$10~kpc long to large-scale FR-I radio galaxies such as 3C~270 and 4C+03.01 (see Fig.~\ref{f.ims}). In most cases we observe IGM cavities associated with the jets, and we estimate that the age and total energy outputs of the jets to be in the range 1-100~Myr and 10$^{55}$-10$^{59}$~erg. Further details of the radio properties of some of our groups are presented in \citet{Kolokythasetal14}.
\begin{figure}
\centerline{
\includegraphics[width=7cm]{ugc408_0.3_2_610con.eps}
\includegraphics[width=7cm]{n5846_0.5_3_1400con.eps}
}
\caption{\textit{Chandra} X-ray images of NGC 193 (\textit{left}) and NGC~5846 (\textit{right}) with radio contours overlaid. GMRT 610~MHz contours show the jets and cocoon of the FR-I radio galaxy 4C+03.01 in NGC~193, while VLA 1.4~GHz contours reveal a much smaller jet source in NGC~5846. In both cases the jets have opened cavities in the IGM, see \citet{Giacintuccietal11} for more details.\label{f.ims}}
\vspace{-0.1cm}
\end{figure}
\section{Cool core fraction and relation to the central AGN}
We define systems with a central decline in X-ray temperature profile of $>$3$\sigma$ significance as having cool cores. Previous studies have shown that cool cores are linked to the presence of H$\alpha$ emission and nuclear activity in cluster central galaxies \citep[e.g.,][]{Cavagnoloetal08} indicating that gas can cool from the X-ray phase to fuel the central AGN. We find that 9 of the 14 confirmed groups in our high richness subsample ($\sim$65\%) have cool cores, a similar fraction to that measured in clusters. Central galaxies with AGN jets are only found in cool core groups within our sample, in line with previous findings from more massive systems \citep{Sun09}.
\begin{figure}
\centerline{
\includegraphics[width=7cm,bb=20 210 560 740]{tcool_plot.eps}
\includegraphics[width=7cm,bb=20 210 560 740]{entropy_plot.eps}
}
\caption{Profiles of IGM cooling time (\textit{left}) and entropy (\textit{right}) for the X-ray bright CLoGS groups. The grey bar indicates the range of radii corresponding to 0.05 R$_{500}$ for these 0.5-1.5~keV systems. \label{f.profs}}
\end{figure}
Figure~\ref{f.profs} shows profiles of cooling time and entropy for our confirmed groups. The entropy profiles follow the canonical r$^{1.1}$ gradient in their outer parts, and many of the groups have central entropies below the 30~keV~cm$^2$ limit found to be associated with nuclear activity and H$\alpha$ nebulae. The groups also typically have short central cooling times of order 0.5-3~Gyr. \citet{Hudsonetal10} suggest a cool core classification scheme for clusters based on cooling time at 0.05 R$_{500}$, with strong cool cores (SCCs) having t$_{cool}<$1~Gyr and weak cool cores (WCCs) having t$_{cool}<$7.7~Gyr. Applying this to our groups, we would find no SCCs and only four WCCs, two of which have rising central temperature profiles. We therefore conclude that current nuclear jet activity is more closely linked to a declining central temperature profile than to the central entropy or cooling time in our groups.
|
1,941,325,220,566 | arxiv | \section{Introduction}
\label{sec:introduction}
The suggestion that electrons can be supplied one by one by a mesoscopic
circuit has been proposed over two decades ago \cite{thouless}.
Instead of applying a constant bias voltage to the system, it is possible
to supply a.c. gate voltages which perturb the system periodically.
Under certain conditions, the charge transferred from one lead to the other,
during one period, can be almost quantized.
Adiabatic pumping of electrons could in principle be used in
future nanoelectronics schemes based on single electron transfer,
and it also has applications to
quantum information physics. Over the years,
theoretical approaches to this adiabatic pumping based
on scattering theory have become available
\cite{buttiker_thomas_pretre,brouwer,MB02}.
These situations typically describe mesoscopic systems which are large enough,
or sufficiently well connected to leads that electronic interactions (charging effects
for instance) can be discarded.
Scattering theory has been
applied\cite{MB02} to calculate both the charge and noise in systems
in the absence of electron-electron interactions.
On the experimental scene, Coulomb blockade effects have
been successfully exploited to achieve pumping with isolated quantum dots \cite{kouvenhoven}.
To our knowledge, pumping experiments which are not entirely based on Coulomb blockade,
where the shape of the electron wave functions is modified in an adiabatic
drive are rather scarce. A recent study \cite{marcus} has dealt with
the transport through an open quantum dot where such interactions are minimized.
Besides Coulomb blockade physics, the effect of electron-electron
interactions in conductors with reduced dimensionality have been
discussed by several authors. The case of strong interactions
in a one dimensional
quantum wire was presented in Ref. \onlinecite{sharma_chamon},
using Luttinger liquid theory. Alternatively, Ref. \onlinecite{rao}
discussed the opposite limit, where the effect of weak interactions
can be included in a scattering formulation of pumping using renormalized
transmission/reflection amplitudes\cite{DGM94}.
However, the results for the pumped charge remain mostly numerical
in this work.
The conditions under which pumping amplitude and interactions
must be tuned to achieve quantized pumping are not obvious. Many
physical parameters enter this problem, such as
the amplitude of the pumping potentials, the phase difference between these,
the possibility of a constant offset on these potentials, the overall
conductance of the unperturbed structure, and to what extent
the strength of electron-electron interactions play a role.
Analytical results on this issues are highly desirable, as well as
information about the noise.
With regard to the experiment of Ref. \onlinecite{marcus}, there is clearly a need for
further understanding the role of weak interactions in such mesoscopic
systems in the presence of pumping.
The purpose of the present work goes in this direction, in the sense that
we provide analytical expressions for the pumped charge and the noise for
a one dimensional wire in the presence of interactions.
This allows us to explore all pumping regimes\cite{GAO05} (weak to strong pumping)
and to determine in which manner and to what extent the pumped charge can help
to achieve single electron transfer.
Besides addressing the question of the ideal
conditions for good charge quantization, we shall establish relationships
between charge and noise in different regimes.
For concreteness, a two-delta potentials model will be used
and interactions will be added on top of it.
\section{Pumped charge and noise}
\subsection{Adiabatic pumping in non interacting systems}
Here, we recall the formula which was established for the
charge transferred during the single period of an adiabatic pumping cycle
through a quasi one-dimensional system. The system is in general described by
a potential $V(x)$ containing two internal parameters which
are modulated periodically. The time dependence is assumed to be sufficiently
slow so that, although the scattering matrix depends on time, its variations
are minute when an electron is scattered in the mesoscopic wire.
At finite temperature, the pumped current reads\cite{Polianski02}
\begin{eqnarray}
Q \, =\, e \int_0^{{2 \pi \over \omega}}
dt \int f(E) \, Tr \bigl\lbrace S^{\dagger}(E,t) \sigma_z S(E,t) - I \bigr\rbrace
{dE \over 2 \pi}.
\end{eqnarray}
where $S(E,t)$ is the Wigner transform of the scattering matrix $S(t,t^{\prime})$,
and to, a good approximation the scattering matrix for the problem with
the potential frozen. $\sigma_z$ is the ususal Pauli matrix, $I$ the identity matrix and
$f$ the Fermi-Dirac function.
\begin{eqnarray}
S(E,t) \, =\, \int_{- \infty}^{\infty} e^{-i E (t-t^{\prime})} S(t,t^{\prime}) dt^{\prime}.
\end{eqnarray}
The pumping potential will generate sidebands at $E \pm \hbar \omega$
and we assume that temperature is much smaller than $\omega$,
i.e. $k_B T \ll \hbar \omega$, so that we can approximate
the Fermi function by a step function. In fact temperature dependence
occurs in two places in this problem. First in the Fermi function
and second, the scattering matrices elements depend on the temperature because
of the renormalization due to the interactions (see next chapter). Formulas
for averaged current and zero frequency noise can be carried out using results
of the literature in the ``zero temperature'' formalism, except that the
scattering matrix elements are in fact temperature dependent.
The pumped charge reduces to a time integral over a pumping cycle\cite{
buttiker_thomas_pretre,rao}
\begin{eqnarray}
Q \, =\, {e \over 2 \pi} \int_0^{2 \pi}
Im \Biggl\lbrack \Biggl( {\partial s_{11}\over \partial X} s_{11}^*
+ {\partial s_{12} \over \partial X} s_{12}^* \Biggr) {dX \over dt} +
(X \leftrightarrow Y) \Biggr\rbrack \, dt,
\label{Qequation}
\end{eqnarray}
where $s_{1i}$ ($i=1,2$) are the elements of the scattering matrix $s(E)$:
\begin{equation}
{s(E)}=e^{i\phi}\left(\begin{array}{cc}
-i \sqrt R e ^{i\theta} &\sqrt T \\
\sqrt T& -i \sqrt R e^{-i\theta}
\end{array} \label{sm}\right),
\end{equation}
where $\phi$ is the phase accumulated in a transmission event and $\theta$ is the phase
characterizing the asymetry between the reflection from the left-hand-side and from
the right-hand-side of the potential. Conservation of probabilities
imposes $R+T=1$. We assume the quantities $\sqrt{R}$, $\sqrt{T}$,
$\theta$ and $\phi$ to be functions
of the Fermi energy $E_F$ and of
the external time-varying parameters $X(t)$ and $Y(t)$.
\subsection{Inclusion of weak interactions}
In the case of weak interactions, the transmission and
the reflection amplitudes $s_{12}$ and $s_{11}$ can be calculated
in the presence of Coulomb interaction via
a renormalization procedure\cite{DGM94}.
High energy scales above a given cutoff
are eliminated. The high energy cutoff is lowered progressively.
The renormalization
has to be stopped when the temperature becomes comparable to this cutoff.
Finally,
if $s_{12}^{(0)}$ and
$s_{11}^{(0)}$ denote respectively,
the transmission and reflexion coefficient without interactions, $s_{12}$ and $s_{11}$
can be expressed in the form
\cite{DGM94}:
\begin{eqnarray}
s_{12}&=&{s_{12}^{(0)} l^{\alpha}\over \sqrt{1+T_0(l^{2\alpha}-1)}}, \label{t1}\\
s_{11}&=&{s_{11}^{(0)}\over \sqrt{1+T_0(l^{2\alpha}-1)}}\label{r1},
\end{eqnarray}
where $l= k_B \Theta/W$; $\Theta$ is the temperature, $k_B$ the Boltzmann
constant and $W$ the original bandwidth.
$\alpha$ is a negative exponent related to
the strength of the screened Coulomb interaction potential\cite{DGM94}. Specifically:
\begin{equation}
\alpha={V_c(2k_F)-V_c(0)\over 2\pi v_F},
\end{equation}
with $V_c(q)$ the Fourier transform of the screened Coulomb potential at $q=2k_F$ and $q=0$,
respectively, $V_c(0)$ is finite due to screening.
$\alpha=0$ corresponds to the absence of electron-electron interactions.
$T_0=|s_{12}^{(0)}|^2$
represents the conductance of the wire in units of $e^2/h$.
>From now on, $Q$ will denote the pumped charge with
interactions and $Q_0$ without interactions.
The integrand of Eq. (\ref{Qequation}) is therefore
modified by the presence of the interactions.
Thus, temperature dependence occurs through the renormalization of the $S$-matrix.
We shall be interested in the regime where temperature is much lower than pumping frequency,
$k_B \Theta \ll \omega$, as said before,
but renormalization of $S$-matrix should not be too
severe so that the renormalization of $S$-matrix
still makes sense. At very low temperature, all barriers become almost opaque and
bosonization is required \cite{sharma_chamon}, so typically $l^{\alpha} > 10^{-1}$,
that is $k_B \Theta \gg W \, 10^{1 / \alpha}$.
For example, for nanotubes having $\alpha$ around $-0.3$,
this gives $\hbar \omega \gg\ k_B \Theta \gg \, 10^{-3} W$.
\subsection{Two-delta potentials model and pumped charge}
Consider the pumping charge $Q$ transferred during a single period through
a 1D chain of an arbitrary potential shape. Let two parameters of the system be modulated
periodically. The single particle Hamiltonian reads
\begin{eqnarray}
H \, =\, {\hbar^2 k^2 \over 2 m} + V(x) + V_p(x,t),
\label{Hsingle}
\end{eqnarray}
where $V_p(x,t)$ is the time-dependent perturbation part of the arbitrary potential and
has $\delta-$like potentials form
\begin{eqnarray}
V_p(x,t) \,= \, 2k_F X(t) \delta(x-x_i) + 2 k_F Y(t) \delta(x-x_f),
\label{definitionofVp}
\end{eqnarray}
with the amplitudes $X(t) = V_i(t)/2k_F$ , $Y(t) = V_i(t)/2k_F$,
where $V_i(t)$ and $V_f(t)$ have periodic time evolution with
the same period $t_0 = 2 \pi/\omega$ and $k_F$ denotes the Fermi wave-vector,. $V(x)$
is the static potential
in between two $\delta$ potentials. Below, superscript indexes
$(0)$ indicate non-interacting systems.
Using the known relation \cite{Fisherlee81},
\begin{eqnarray}
s_{\alpha \beta}^{(0)} \, =\, - \delta_{\alpha \beta} + 2 i k_F G^0(x_{\alpha},x_{\beta}),
\end{eqnarray}
where $G^0(x_{\alpha},x_{\beta})$ are the usual real space retarded Green's functions
and the fact that the functional derivative of the Green's function
$\delta G^0/\delta V(x)$ can be written as the product of
two Green's functions\cite{GCB96}
\begin{eqnarray}
{\delta G^0 (x_{\alpha},x_{\beta})\over \delta V(x_j)} \, =\, G^0(x_{\alpha},x_j)
G^0(x_j,x_{\beta}).
\end{eqnarray}
For the first bracket of Eq. (\ref{Qequation}), we get
\begin{eqnarray}
{\partial s_{11}^{(0)} \over \partial X} s^{(0) \, *}_{11} +
{\partial s_{12}^{(0)} \over \partial X} s^{(0) \, *}_{12}
\, =\, {\bigl(1+s_{11}^{(0)}\bigr) \bigl(1+s_{11}^{(0) \,*}\bigr) \over 2 i k_F},
\label{firstbracket}
\end{eqnarray}
where we used the condition $s^{(0)}_{11} s^{(0) \, *}_{11} +
s_{12}^{(0)} s_{12}^{(0) \,*} = 1$, $s_{12}^{(0)}$ and $s_{11}^{(0)}$ are
the bare transmission and reflection amplitudes from
the disordered system, without taking account the electron-electron interactions.
For the second bracket of Eq. (\ref{Qequation}), we get
\begin{eqnarray}
{\partial s_{11}^{(0)} \over \partial Y} s_{11}^{(0) \, *} +
{\partial s_{12}^{(0)} \over \partial Y} s_{12}^{(0) \, *} \, =\,
{s_{12}^{(0)} \over 2 i k_F} \, \Bigl(s_{12}^{(0)} s_{11}^{(0) \,*} +
s_{12}^{(0) \,*} (1+s_{22}^{(0)})\Bigr) \, =\,
{\vert T_0\vert^2 \over 2 i k_F}.
\label{secondbracket}
\end{eqnarray}
Here, we used the
current conservation requirement $s_{21}^{(0)} s_{11}^{(0) \,*}
+ s_{12}^{(0) \,*} s_{22}^{(0)} =0$.
$s_{22}^{(0)}$ is the reflection
amplitude from the right of the scaterrer and can be presented as
\begin{eqnarray}
s_{22}^{(0)} \, =\, 2 i k_F {\partial ln s_{12}^{(0)} \over \partial Y} -1.
\label{rprimeidentity}
\end{eqnarray}
Substituting Eq. (\ref{firstbracket}) and (\ref{secondbracket}) into Eq. (\ref{Qequation}),
using the identity (\ref{rprimeidentity}), we finally arrive at
\begin{eqnarray}
Q_0 \, =\, -{e \over 2 \pi} \int_0^{{2 \pi \over \omega}} \Biggl(
{\partial ln s_{12}^{(0)} \over \partial X} {\partial ln s_{12}^{(0) \,*} \over \partial X}
{dX \over dt} + T_0 {dY \over dt} \Biggr) \, dt.
\label{Q0sint}
\end{eqnarray}
Similar expression for $Q_0$, can be found in Ref. \onlinecite{avron}.
Using the same method for $Q$, we obtain, in the presence of interactions
\begin{eqnarray}
&Q& \, = \, Q_0 - {e (l^{2 \alpha} -1) \over 2 \pi} \nonumber \\
&\times& \! \int_0^{{2 \pi \over \omega}}
\Biggl\lbrack
\biggl(
Im \Bigl\lbrace {\partial \ln \, s_{12}^{(0)} \over \partial X} \Bigr\rbrace
\, -
\Bigl\vert{\partial \ln \,s_{12}^{(0)} \over \partial {X}}\Bigr\vert^2
\biggr) \, {d {X} \over dt}
\, + \,
\biggl( Im \Bigl\lbrace{\partial \ln \, s_{12}^{(0)}
\over \partial {Y}}\Bigr\rbrace
- T_0\biggr) \, {dY \over dt}
\Biggr\rbrack
\, {T_0 \over 1 + T_0(l^{2 \alpha}-1)}\, dt.
\label{Qsint}
\end{eqnarray}
For numerical simulations, we specialize to the case where the static potential
$V(x)=0$ in Eq. (\ref{Hsingle}) and the time-dependent perturbations are two
$\delta$-like potentials separated by a distance $2a$.
The expressions of $s_{12}^{(0)}$ and $s_{11}^{(0)}$
are needed, when the single particle Hamiltonian reads
\begin{eqnarray}
H \, = \, -\hbar^2 k^2/2m + V_p(x,t).
\end{eqnarray}
The elements of the $S$-matrix in the absence of electron-electron interactions, are given by
\begin{eqnarray}
s^{(0)}_{11} &=& {\Bigl\lbrack ({\overline Y} - {\overline X}) sin(2 k_F a)
- i
\bigl\lbrace 2 {\overline X} \, {\overline Y} sin(2 k_F a) + ({\overline X}
+ {\overline Y}) cos(2k_F a) \bigl\rbrace \Bigr\rbrack \over D}, \label{equationsforsmatrix1} \\
s^{(0)}_{12} \, &=& \, {1 \over D},
\label{s12}
\end{eqnarray}
with
\begin{eqnarray}
D \, &=&\, \biggl(1 - 2 {\overline X} \, {\overline Y} sin^2(2 k_F a) + i \bigl\lbrack
{\overline X} + {\overline Y} + {\overline X} \,{\overline Y} sin(4 k_F a) \bigr\rbrack
\biggr),
\label{equationsforsmatrixlast}
\end{eqnarray}
and ${\overline X} = X/2$, same for ${\overline Y}$.
$s^{(0)}_{21} = s^{(0)}_{12}$ and $s^{(0)}_{22}$ is obtained by replacing ${\overline Y}$
by ${\overline X}$ in $s^{(0)}_{11}$.
\subsection{Noise}
Ref. \onlinecite{avron} studied adiabatic quantum pumping in the context
of scattering theory.
Their goal was to derive under what conditions pumping could be achieved optimally,
in a noiseless manner, with
the assumption that the pumping frequency is small compared to the temperature.
This enabled the authors to derive
expressions not only for the pumped charge per cycle, but
also for the pumped noise, the current-current correlation function, averaged
over a time which is long compared to the period of the adiabatic drive, at zero
frequency. Specifically, the noise is defined
from the current-current time correlator:
\begin{eqnarray}
S(t,t^{\prime}) \, = \, {1 \over 2} \langle \delta I(t) \delta I(t^{\prime})
+ \delta I(t^{\prime}) \delta I(t) \rangle,
\end{eqnarray}
with $\delta I \, = \, I - \langle I \rangle$. This correlator is then averaged over
$n_0$ periods of the pumping drive with $n_0$ large, and it is taken at zero frequency by performing
an integral over the remaining time argument. Setting $\tau_0 = n_0 \, 2 \pi/\omega$,
\begin{eqnarray}
S(\Omega=0) \, = \, {\omega \over 2 \pi \,n_0} \int_0^{\tau_0} dt
\int_{- \infty}^{\infty} dt^{\prime} S(t,t^{\prime}).
\end{eqnarray}
Ref. \onlinecite{MB02} extended the results of Ref. \onlinecite{avron} to the case where this limiting assumption
is relaxed, yielding a complete description of the quantum statistical properties of an
adiabatic quantum pump, albeit restricted to small pumping amplitudes. Results
made use of the
generalized emissivity matrix. These results were generalized to arbitrary
pumping amplitudes by Ref. \onlinecite{Polianski02}. Here, our goal is to address the
question whether electron-electron interactions affect the pumping noise and how.
Following formula (14) of Ref. \onlinecite{Polianski02}
and applying their Eq. (15) without assuming that the
pumping amplitudes $X(t)$ are small,
it is possible to put the zero frequency
noise $S$ for arbitrary pumping amplitudes into the form
\begin{eqnarray}
S(\Omega=0) \, &=&\,
{1 \over (2 \pi \hbar)^2}
{e^2 \over \tau_0} \int_0^{\tau_0}dt \int_{-\infty}^{\infty} d(t^{\prime}-t) \,
\int_{- \infty}^{\infty} f(- \epsilon_1)
\int_{- \infty}^{\infty} f(\epsilon_2) \nonumber \\
& & \, \times \, Tr\Bigl\lbrack
s(\epsilon_1,t)^{\dagger} \sigma_z s(\epsilon_2,t)
s(\epsilon_2, t^{\prime})^{\dagger} \sigma_z
s(\epsilon_1,t^{\prime})- I \Bigr\rbrack
e^{i{(t - t^{\prime})(\epsilon_1 - \epsilon_2)\over \hbar}} \,
d \epsilon_1 d \epsilon_2,
\label{Sbasic}
\end{eqnarray}
where $s(\epsilon_1,t)$ is the $2 \times 2$ $S$-matrix, for an incoming wave at
energy $\epsilon_1 + \epsilon_F$ and value of the
pumping parameter $X(t)$.
$\tau_0$ is a time which is much larger than the period $\tau$.
$\epsilon_1 = \hbar^2 k_1^2 / 2 m - \epsilon_F$,
where $k_1$ is a wave vector and $\epsilon_F$ is
the Fermi energy $\epsilon_F = \hbar^2 k_F^2 / 2m$.
$f$ is the Fermi-Dirac function. Here, $s(\epsilon_1,t)$
is taken for fixed $t$\cite{Polianski02}.
Since we work at temperature much smaller than $\hbar \omega/k_B$, as explained before,
we can set
$f(- \epsilon_1) =1$ for $\epsilon_1\,> \,0$ and $0$ otherwise.
We set
$\epsilon^{\prime}_2 = -\epsilon_2$ so
that both $\epsilon_1$ and $\epsilon^{\prime}_2$ will be positive.
$M$ is defined as
\begin{eqnarray}
M(\epsilon_1,\epsilon^{\prime}_2,t) \, &=&\, s(\epsilon_1,t)^{\dagger}
\sigma_z s(-\epsilon^{\prime}_2,t).
\label{Mequation}
\end{eqnarray}
$S$ now reads
\begin{eqnarray}
S =
{e^2 \over (2 \pi \hbar)^2}
\, lim_{\tau_0 \rightarrow \infty}
\int_0^{\tau_0} \!\! {dt \over \tau_0} \!
\int_{- \infty}^{\infty}\!\!\!\! dt^{\prime} \!
\int_0^{\infty} \!\!\!\!\! d \epsilon_1 \! \int_0^{\infty}
\!\!\!\!\! d \epsilon^{\prime}_2 \,
Tr \Bigl\lbrace
M(\epsilon_1,\epsilon^{\prime}_2,t)
M(\epsilon_1,\epsilon^{\prime}_2,t^{\prime})^{\dagger} \! - I \Bigr\rbrace \,
e^{-i{(t-t^{\prime}) (\epsilon_1 + \epsilon^{\prime}_2) \over \hbar}}.
\end{eqnarray}
Now, for large pumping amplitudes, the above formula needs to be rearranged,
using the fact that $X(t)$ is a periodic function of
period $2 \pi / \omega$.
Note that the dependence of $M$ on $\epsilon_1$ and $\epsilon^{\prime}_2$ prevents
the direct use of fast Fourier transform. Nevertheless, we can use the fact that,
for given values of $\epsilon_1$ and $\epsilon^{\prime}_2$, $M(t)$
and $M(t^{\prime})$ are periodic functions of $t$.
Switching to Fourier transform
\begin{eqnarray}
{\hat M}_n(\epsilon_1, \epsilon^{\prime}_2) \, &=& \,
{\omega \over 2 \pi} \,\int_{0}^{{2 \pi \over \omega}}
M(\epsilon_1, \epsilon^{\prime}_2,t) \, e^{-in \omega t} \, dt, \\
M(\epsilon_1,\epsilon^{\prime}_2, \, t) \, &=& \,
\sum_{n = - \infty}^{+ \infty} {\hat M}_n(\epsilon_1,\epsilon^{\prime}_2) \,
e^{i n \omega t}.
\end{eqnarray}
Performing the trace, we arrive at
\begin{eqnarray}
S = {e^2 \over 2 \pi \hbar^2}
\int_0^{\infty} \!\!\! d \epsilon_1 \!\int_0 ^{\infty} \!\!\! d \epsilon^{\prime}_2
\!\sum_{n=- \infty}^{\infty}
\!\Bigl(\vert {\hat M}_{1,1 \, n}\vert^2 +
\vert {\hat M}_{1,2 \, n}\vert^2 +
\vert {\hat M}_{2,1 \, n}\vert^2 +
\vert {\hat M}_{2,2 \, n}\vert^2 - 2 \delta_{n,0}\Bigr) \,
\delta \Bigl( {\epsilon_1 + \epsilon^{\prime}_2 \over \hbar} - n \omega\Bigr),
\label{fullformulaforS}
\end{eqnarray}
where
$\delta_{n,0}$ is $1$ if $n=0$ and zero otherwise and
${\hat M}_{i,j \, n}$ is the $(i,j)$ element of matrix of
${\hat M}_n(\epsilon_1, \epsilon^{\prime}_2)$, where energy dependences
have been omitted to ease the notations.
When $\hbar \omega$ is much smaller than $\epsilon_F$, formula (\ref{fullformulaforS}) can be
simplified.
In this case, $M(\epsilon_1,\epsilon^{\prime} _2,t)$ will be different from
$M(0,0,t)$ only when $\epsilon_1$ or $\epsilon^{\prime}_2$ are a non negligible
fraction of $\epsilon_F$. This occurs because
$M(0,0,t)$ corresponds to matrix $M$ for incident wave and outgoing wave
at energy $\epsilon_F$. We denote by $\epsilon_{1F}$ typical energies of the order
$\epsilon_F$. $\epsilon_{1F}$ will correspond to
$n$ of the order $\bigl(\epsilon_{1F}/\hbar \omega\bigr)$, which is very large.
The Fourier transform
${\hat M}_n(\epsilon_1,\epsilon^{\prime} _2)$ will decrease exponentially
with $n$ for large $n$. Thus, we can neglect the dependence on $\epsilon_1$
and $\epsilon^{\prime}_2$ and replace them by zero, which amounts to replacing
the energies by $\epsilon_F$, except in the argument of the $\delta$ function.
Under these conditions, we have
\begin{eqnarray}
S \, \simeq \, e^2 {\omega \over \ 2 \pi}
\Biggl\lbrack
\sum_{n \geq 1} \, n\,
\biggl(
\vert {\hat M}_{1,1 \,n} \vert^2 +
\vert {\hat M}_{1,2 \,n} \vert^2 +
\vert {\hat M}_{2,1 \,n} \vert^2 +
\vert {\hat M}_{2,2 \,n} \vert^2
\biggr)
\Biggr\rbrack.
\label{noiseMn}
\end{eqnarray}
With our form of the $S$-matrix, this formula is equivalent to
Eqs. (24a), (24b) and (24c) of Ref. \onlinecite{MB04}, apart from am overall factor 2. See
Appendix C for details.
For numerical simulations however, we did not make this simplification
and kept the dependence on $\epsilon_1$ and $\epsilon^{\prime}_2$
of Eq. (\ref{fullformulaforS}).
\vspace{2. mm}
\section{Discussion of physical results}
We now illustrate these formulas by computing the charge and noise, in the case of two-delta potentials model.
The two parameters
of the drive $X$ and $Y$ (Eq.(9)) are chosen to vary periodically according to:
\begin{eqnarray}
X \, &=& \, X_0 + \eta \, cos(\omega t), \\
Y \, &=& \, Y_0 + \eta \, cos(\omega t - \varphi),
\end{eqnarray}where $X_0$ is a constant offset potential and $\varphi$ a phase difference.
Note that $X$, $X_0$ and $\eta$ are all dimensionless,
see Eq. (\ref{definitionofVp}). To
ensure maximal pumping, we shall specialize
\cite{GAO05} to $\varphi = \pi / 2$.
\subsection{Zero offset}
\begin{figure}[h]
\epsfxsize 8. cm
\centerline{\epsffile{fig1.eps}}
\caption{$Q$, charge with interactions, (solid line),
$Q_0$, charge without interactions, (dashed line), both in units
of $e$. $S$, noise with interactions, (dotted line) and
$S_0$ without, (dashed dotted line), multiplied by
$2 \pi / \omega$, in units of
$e^2$, vs. $\eta$. Essential
parameters are $X_0=0$, no offset, $l^{2 \alpha} = 0.3$, $k_Fa \, = \, 0.5$ and
$\hbar \omega = 10^{-2} \epsilon_F$.}
\end{figure}
\begin{figure}[h]
\epsfxsize 7. cm
\centerline{\epsffile{fig2.eps}}
\caption{$Q$, charge with interactions, (pluses),
and $Q_0 \,l^{-2 \alpha}$, (upper dashed line), both in units
of $e$ vs. $l^{2 \alpha}$, for
$\eta=0.3$, $X_0=0$, $k_F a \, = \, 0.5$ and $\hbar \omega = 10^{-2} \epsilon_F$.}
\end{figure}
First, the case without offset $X_0=Y_0=0$ is studied.
To look at the influence of interactions, we plot in Fig. 1
the pumped charge in units of $e$, with interactions and without,
versus the amplitude of the drive $\eta$,
for an interaction parameter $l^{2 \alpha}= 0.3$
(moderate electron-electron interactions). There are three regimes: weak pumping,
$\eta \ll 1$, intermediate pumping, $\eta$ of order $1$ and large pumping amplitudes,
$\eta \gg 1$. The current noise times
$2 \pi / \omega$, in the limit of small $\omega$,
is plotted together on the same figure in units of $e^2$.
Analytically, for $\eta \ll 1$, $Q$ reads
\begin{eqnarray}
Q \, =\,{e \over 4} sin(4 k_F a)\, l^{-2 \alpha} \eta^2.
\end{eqnarray}
As noted in Ref. \onlinecite{rao}, in the weak pumping regime,
charge $Q$ is larger with interactions by a factor $l^{-2 \alpha}$, see Fig. 2.
Results of Ref. \onlinecite{MB02} for the noise,
valid for weak pumping and no interactions can be adapted in a straightforward fashion to
the case with interactions. We find the following formula for the noise for weak pumping.
\begin{eqnarray}
S \, =\, e^2 l^{-2 \alpha} \eta^2 {\omega \over 2 \pi}.
\end{eqnarray}
\begin{figure}[h]
\epsfxsize 7. cm
\centerline{\epsffile{fig3.eps}}
\caption{$Q$, charge with interactions, (pluses),
and $Q_0 \,l^{2 \alpha}$, (lower dashed line), both in units
of $e$, vs. $l^{2 \alpha}$, for $\eta=15$, $X_0=0$, $k_F a \, = \, 0.5$ and
$\hbar \omega = 10^{-2} \epsilon_F$. In this regime, $Q$ is approximately larger
than $Q_0$ by a factor $l^{- 2 \alpha}$.}
\end{figure}
\begin{figure}[h]
\epsfxsize 7. cm
\centerline{\epsffile{fig4.eps}}
\caption{$(2 \pi / \omega) \, \eta^2 S$, noise with interactions scaled by $\eta^2$ and by
the period, $(+)$
and $(2 \pi / \omega) \, \eta^2 S_0 l^{2 \alpha}$, lower solid line, for moderate $\eta =15$.
Also illustrated by the top two curves, is
the very large pumping amplitude regime,
$(2 \pi / \omega) \,\eta^2 S$ ($\times$) and
$(2 \pi / \omega) \, \eta^2 S_0$ ($*$), for very large $\eta$; $\eta=100$.
The top two curves are close to each other.
An attempt to fit $\eta^2 S$ by $l^{2 \alpha} \eta^2 S_0$ (dashed line) for $\eta=100$,
clearly fails for this regime of pumping amplitudes. $k_F a =0.5$ and
$\hbar \omega = 10^{-2} \epsilon_F$ for all cases.}
\end{figure}
The noise is thus increased by the same factor as the
current. The Fano factor, defined as the ratio $S/e \langle I \rangle$,
is $4 / sin(4 k_F a)$ and
remains independent of the interactions, as long as we remain in the
weak pumping regime. This corresponds to the very left part of Fig. 1, for
$\eta$ smaller than $0.25$, typically.
At intermediate pumping amplitudes, $Q$ reaches a maximum value $Q_{max}$ which is
again larger than its non-interacting analog $Q_{0 \, max}$. This maximum is of
the order of the single electron charge, but less than it.
Meanwhile, the noise decreases. This is a reminder
of the reduction of the noise by a factor $T (1-T)$,
where $T$ is the electrons transmission coefficient in quantum wires.
This explains why the noise exhibits a first maximum around
$\eta$ close to $1$, since $Q$ gets closer to one electron charge, noise will decrease.
Then, for moderate amplitudes, $\eta$ around $6$, charge decreases and passes through
the value $0.5 e$, this corresponds then to the second maximum of the noise.
For large, but not very large pumping amplitudes,
typically $\eta =10$, $Q$ remains smaller than $Q_0$ but behaves in the same way,
namely as $\eta^{-3}$, as noted in Ref. \onlinecite{GAO05}.
As a function of the interaction parameter,
$Q$ behaves as $Q_0 l^{2 \alpha}$. See Fig. 3.
For very large pumping amplitudes, (typically order $100$ or more),
$Q$ becomes practically equal to $Q_0$,
$Q-Q_0$ behaves as $\eta^{-4}$, see appendix B for details.
For the noise, we found numerically that $S$ and its analog without interactions, $S_0$,
both decrease as $\eta^{-2}$, much slower than the charge. See below for analytical derivations.
As concerns now the interaction dependence of the noise,
$S$ is always smaller than $S_0$, but for very large $\eta$,
$S$ tends towards $S_0$. More precisely, for large but still reasonable
$\eta$, of the order $10$ typically,
$S$ is almost equal to $S_0 l^{2 \alpha}$, whereas for very large $\eta$,
of the order $100$ or more, $S$ and $S_0$ are practically the same. This is not surprising
since $Q$ and $Q_0$ are then also practically equal in the end.
This dependence on
the interaction parameter is shown in fig. 4. We have to plot
$(2 \pi / \omega) \, S \eta^2$, vs. $l^{2 \alpha}$, but the overall factor
$2 \pi / \omega$ is unimportant; the product $S \eta^2$
can be compared to both $l^{2 \alpha} S_0 \eta^2$ and to $S_0 \eta^2$
for $\eta=15$ and $\eta=200$. For $\eta=15$, we see that $\eta^2 S$ is fairly well approximated
by $\eta^2 S_0 l^{2 \alpha}$. On the contrary, for $\eta=200$, such a fit fails and instead,
$\eta^2 S$ is almost equal to $\eta^2 S_0$.
The results at large $\eta$
can be derived from analytical formula for the charge and noise. An expansion
for large $\eta$ is performed, $X$ and $Y$ behave
as $\eta$, except at particular points where $X$ or $Y$ are zero. See Appendix A for details.
\subsection{Non-zero offset}
We now turn to the case where $X_0$ is non-zero, which enables to have regions
where $Q$ is almost quantized. There are basically three cases, according
to the value of $k_F a$.
The first case corresponds to $k_F a = n \pi / 2$ (rigorously), where
$n$ is an integer.
In this case, it is impossible to pump anything. The reason is given below.
The second case corresponds to the case where $k_F a$ is small but non-zero,
$0.1$ typically. We first describe the behavior, then give numerical illustrations and last
provide analytical justifications. In this case, the charge is almost zero up to $\eta=X_0$.
It rises quickly around $\eta = \sqrt{2} X_0$ and reaches a value close
to quantized $e$ for a wide range of values of $\eta$. This is the quantized
region of $\eta$. The width of this region can be shown
to scale approximately as $(k_F a)^{-1}$. After the end of this region,
$Q$ and $Q_0$ first decrease abruptly and for even larger values
of $\eta$, decrease slower, as $\eta^{-3}$. The noise in the quantized
region and around it seems to be well approximated by $Q (e-Q)$, reminiscent
of the noise for fermions in narrow quantum wires.
However, this does not last when $\eta$ becomes noticeably
out of the region of almost quantized charge,
since $Q$ and $Q_0$ behave as $\eta^{-3}$, whereas
$S$ and $S_0$ decay only as $\eta^{-2}$.
Fig. 5 shows noise and charge with and without interactions, versus $\eta$ for $k_Fa =0.1$.
Fig. 6 shows a comparison between $S$ and a least square fit of the form
$C \, Q(e-Q)$, in the quantized charge regime, where $C$ is the only adjustable parameter.
See captions for details.
\begin{figure}[h]
\epsfxsize 8. cm
\centerline{\epsffile{fig5.eps}}
\caption{$Q$, (solid line), curve with a plateau for $\eta$ between $28$ and $43$, (the
quantized region), $Q_0$, (dashed line). The two curves with
two spikes located at $\eta$ around $28$ and at $\eta$ around $43$ represent
$(2 \pi / \omega) \, S$, (dotted line) and
$(2 \pi / \omega) \, S_0$, (dashed dotted line), vs. $\eta$. Essential
parameters are $X_0=20$, $l^{2 \alpha} = 0.3$, $k_Fa \, = \, 0.1$ and
$\hbar \omega = 10^{-2} \epsilon_F$. Electron charge $e$ is set to $1$.
The solid horizontal line of ordinate $1$ and the
thick vertical line at $\eta = X_0 \sqrt{2} \simeq 28.28$
are guides to the eye.}
\end{figure}
\begin{figure}[h]
\epsfxsize 6. cm
\centerline{\epsffile{fig6.eps}}
\caption{$S$ (solid line), in units of $e^2$, and best fit of the form $C\, Q(e-Q)$
(dashed line) in the interval $20 \leq\eta\leq 45$, for
the same situation as Fig. 5. The fit no longer works in the large pumping regime,
for $\eta$ between $45$ and $80$ here.}
\end{figure}
\vspace{ 4. mm}
We now turn to analytical justifications of the previous assertions.
We now explain
why pumping is impossible for $k_Fa = n \pi / 2$.
Since $sin(2 k_F a)=0$,
the scattering matrix now depends only on a single parameter,
the combination $({\overline X}+ {\overline Y})$,
see eqs. (\ref{equationsforsmatrix1}-\ref{equationsforsmatrixlast}), so we denote
by $s_{ij}^{(0) \, \prime}$ its derivative with respect to ${\overline X} + {\overline Y}$.
Thus,
\begin{eqnarray}
Q_0 \, =\, {e \over \pi} \int_0^{{2 \pi \over \omega}}
Im \biggl\lbrack \Bigl(\sum_{j=1}^2 s_{1j}^{(0) \, \prime}({\overline X} + {\overline Y})
\, s_{1j}^{(0) \,*}({\overline X} + {\overline Y}) \Bigr)
\biggr\rbrack
\, {d \over dt}({\overline X} + {\overline Y}) \, \, dt \, = \, 0,
\end{eqnarray}
because the bracket is just
$(1/2) \, d/d({\overline X} + {\overline Y})\,
\bigl(\vert s_{11}^{(0)}\vert^2 + \vert s_{12}^{(0)}\vert^2 \bigr)$.
Then, we look at the case when $k_F a$ is close to $n \pi / 2$, but different from it.
Clearly, when $sin(2k_F a)$ is small, for
$\eta \vert sin(k_F a) \vert < 1$, we will be back to the former case, so
$Q_0$ can start to level noticeably from $0$ only for $\eta$ values larger than a critical value
$\eta_{c1}$, which is proportionnal to $1/\vert sin(2 k_F a)\vert$, independently of $X_0$.
There is at least another scale, namely $X_0$. For $X_0$ large,
(typically larger than $10$)
and $\eta$ smaller than $X_0$, the pumping contour is
a circle which does not enclose the origin and both transmission
$s_{12}^{(0)}$ and derivatives of the transmission coefficients,
$\partial s_{12}^{(0)} /\partial X$ are small. $Q_0$ will remain very small.
Thus, to have
a significant $Q_0$, one needs $\eta > max \bigl(X_0, 1/\vert sin(2 k_Fa)\vert\bigr)$,
where $max(x,y)$ is the larger of $x$ and $y$. For even larger $\eta$,
when terms like $sin(2 k_F a) {\overline X} \, {\overline Y}$
dominate over terms linear in ${\overline X}$ or ${\overline Y}$, i.e.
$\bigl(\vert sin(2 k_F a)\vert \, \eta\bigr) \, \gg 1$,
it is possible to expand in $\eta^{-1}$ and
we are back to the large pumping regime where $Q$ decays as $\eta^{-3}$.
So, for $\vert sin(2 k_Fa) \vert$ smaller than $X_0$, there will be a region between
$max(X_0, C_1/sin(2 k_F a))$ and $C_2 /sin(2 k_F a)$, where
$C_1$ and $C_2$ are constants, where $Q_0$ is appreciable. These
very qualitative arguments however do not
prove that the charge is almost quantized in this interval,
whose width is of the order $\vert sin(2 k_Fa)\vert^{-1}$.
We now turn to the analytical explanation of the $Q (e-Q)$ behaviour in the quantized charge
regime. It is useful to disentangle the effects of the fluctuations of $T$ and
those of the phase $\theta$.
We start from (\ref{noiseMn}) and use the fact that for any periodic function
$f(x)$, if ${\hat f}_n$ denotes the Fourier component at frequency $n \omega$,
denoting $x = \omega t$,
\begin{eqnarray}
\sum_{n \geq 1} n \vert {\hat f}_n \vert^2 \, =\,
\int_0^{2 \pi} \!\int_0^{2 \pi}
{\vert f(x) - f(x^{\prime}) \vert \over (x-x^{\prime})^2} {dx \, dx^{\prime} \over 4 \pi^2}.
\end{eqnarray}
Applying the former equality to ${\hat M}_{1,1 \, n}$,
${\hat M}_{1,2 \, n}$,
${\hat M}_{2,1 \, n}$
and ${\hat M}_{2,2 \, n}$, and denoting
by $M_{11}(x)$ the inverse Fourier transform of ${\hat M}_{1,1 \, n}$,
we get
\begin{eqnarray}
S \, &=& \, e^2 {\omega \over 2 \pi} \! \int_0^{2 \pi} \!\! \int_0^{2 \pi}
\! \biggl(
\vert M_{11}(x) - M_{11}(x^{\prime}) \vert^2 +
\vert M_{12}(x) - M_{12}(x^{\prime}) \vert^2 +
\vert M_{21}(x) - M_{12}(x^{\prime}) \vert^2 +
\vert M_{22}(x) - M_{22}(x^{\prime})\vert^2 \biggr)\nonumber \\
&\,\,&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
(x-x^{\prime})^{-2} \,\,{dx \, dx^{\prime} \over 4 \pi^2}.
\end{eqnarray}
In terms of $\theta$ and $T$, using (\ref{sm}) and (\ref{Mequation}), this reads
\begin{eqnarray}
S \, = \, 8 e^2 \Bigl({\omega \over 2 \pi}\Bigr) \Bigl(I_1 + I_2\Bigr),
\end{eqnarray}
with
\begin{eqnarray}
I_1 \, &=& \, \int_0^{2 \pi} \! \int_0^{2 \pi}
{\vert T(x) - T(x^{\prime})\vert^2 \over
(x-x^{\prime} )^2} \, {dx \, dx^{\prime} \over 4 \pi^2},
\label{I1}\\
I_2 \, &=& \, \int_0^{2 \pi} \! \int_0^{2 \pi}
{\vert g(x) - g(x^{\prime})\vert^2 \over
(x-x^{\prime})^2} \, {dx \, dx^{\prime} \over 4 \pi^2},
\label{I2}
\end{eqnarray}
with $g(x) = \sqrt{T(1-T)} \, e^{i \theta}$.
We expand $\vert g(x) -g(x^{\prime})\vert^2$ and rewrite $I_2$ in the form
\begin{eqnarray}
I_2 \, =\, J_1 + J_2,
\end{eqnarray}
with
\begin{eqnarray}
J_1 \, &=& \, \int_0^{2 \pi} \! \int_0^{2 \pi}
\Biggl\lbrack
{ \sqrt{T(x) \bigl(1-T(x)\bigr)} - \sqrt{T(x^{\prime}) \bigl(1-T(x^{\prime})\bigr)}
\over (x-x^{\prime})}\Biggr\rbrack^2 {dx \, dx^{\prime} \over 4 \pi^2}, \\
J_2 \, &=& \,
\int_0^{2 \pi} \! \int_0^{2 \pi}
\sqrt{T(x) \bigl(1-T(x)\bigr)}
\sqrt{T(x^{\prime}) \bigl(1-T(x^{\prime})\bigr)}
{\vert e^{- i \theta(x)} - e^{-i \theta(x^{\prime})}\vert^2 \over (x-x^{\prime})^2} \,
{dx \, dx^{\prime} \over 4 \pi^2}.
\end{eqnarray}
Thus, noise breaks into three parts, one related to the fluctuations of $T$,
another to the fluctuations of $\sqrt{T (1-T)}$, and the last
to the fluctuations of $\theta$, in fact, to the variations of
the slope $d\theta/dx$, since $\theta$ has to vary by $2 \pi$ in one cycle
to get appreciable pumped charge. The physical message is that if the fluctuations
of $T$ are much smaller than the fluctuations of the phase, then noise
shows a $Q (e-Q)$ behaviour. Otherwise, fluctuations of $T$ bring an extra noise that
does not contribute to the pumped charge and overall noise is thus larger
than $Q (e-Q)$. We now try to establish this more firmly.
\vspace{3. mm}
In this paragraph, we now show that, for $Q_0 \, \leq \, e/2$,
\begin{eqnarray}
S_0 \, \geq \, 8 \Bigl({\omega \over 2 \pi}\Bigr) \, Q_0 \, (e-Q_0).
\label{S0bound}
\end{eqnarray}
The first integral $J_1$ involves solely the fluctuations of $T$ and can be rewritten as
\begin{eqnarray}
J_1 \, =\,
\sum_{n=1}^{\infty} n \vert \widehat{ \sqrt{T (1-T)}}_{n} \vert^2.
\end{eqnarray}
We have a lower bound for $J_1$ by replacing $n$ by $1$ in all terms of the sum except the term for $n=0$,
\begin{eqnarray}
J_1 \, \geq \, \sum_{n=0}^{\infty} \vert \widehat{ \sqrt{T (1-T)}}_{n} \vert^2 -
\vert \widehat{ \sqrt{T (1-T)}}_{0} \vert^2.
\end{eqnarray}
Using then the Parseval identity for the first term and using the notation
$\langle f \rangle \, \equiv \, \int_0^{2 \pi} f(x) \, dx/2 \pi$ for the second,
\begin{eqnarray}
J_1 \geq
\Bigl\langle \bigl\vert \sqrt{T(1-T)} \bigr\vert^2 \Bigr\rangle -
\Bigl\langle \sqrt{T(1-T)} \Bigr\rangle^2 \, = \,
\langle T \rangle - \bigl\langle T^2 \bigr\rangle -
\Bigl\langle \sqrt{T(1-T)} \Bigr\rangle^2.
\end{eqnarray}
Now for $J_2$, applying twice the H{\"o}lder inequality,
\begin{eqnarray}
J_2 \, &=& \,
\int_0^{2 \pi} {dx^{\prime} \over 2 \pi}
\sqrt{T(x^{\prime}) \bigl(1-T(x^{\prime})\bigr)}
\int_0^{2 \pi}
\sqrt{ T(x) \bigl(1-T(x)\bigr)}
{\vert e^{- i \theta(x)} - e^{- i \theta(x^{\prime})} \vert^2 \over (x-x^{\prime})^2}
\, {d x \over 2 \pi} \nonumber \\
& \geq & \,\int_0^{2 \pi} {dx^{\prime} \over 2 \pi} \,
\sqrt{T(x^{\prime}) \bigl(1-T(x^{\prime})\bigr)}
\, \Bigl\langle \sqrt{T(1-T)} \Bigr\rangle
\int_0^{2 \pi}
{\vert e^{- i \theta(x)} - e^{- i \theta(x^{\prime})} \vert^2 \over (x-x^{\prime})^2}
{d x \over 2 \pi} \, \nonumber \\
& \geq &
\Bigl\langle \sqrt{T(1-T)} \Bigr\rangle^2
\int_0^{2 \pi} {dx \over 2 \pi} \int_0^{2 \pi} {dx^{\prime} \over 2 \pi}
{\vert e^{- i \theta(x)} - e^{- i \theta(x^{\prime})} \vert^2 \over (x-x^{\prime})^2} .
\end{eqnarray}
For the last double integral, we proceed as before, going again to Fourier transform,
isolating the component of order $0$
and using the Parseval identity, it is larger than
$\bigl\langle \vert e^{i \theta} \vert^2 \bigr\rangle
- \vert \langle e^{i \theta} \rangle\vert^2$.
Always, $\vert e^{i \theta} \vert=1$ and, for
reasonable $\theta(x)$, we can assume symmetry $x$ into $-x$ which implies
$\langle sin \theta \rangle =0$. We can also assume symmetry when
$x$ is changed into $\pi -x$, which implies $\langle cos \theta \rangle =0$.
We have
\begin{eqnarray}
J_2 \geq
\Bigl\langle \sqrt{T(1-T)} \Bigr\rangle^2.
\label{J2}
\end{eqnarray}
Now, for $I_1$, using the same technique, (going to Fourier
transform and isolating the $n=0$ component)
\begin{eqnarray}
I_1 \, \geq \, \langle T^2 \rangle - \langle T \rangle^2.
\end{eqnarray}
Putting everything together
\begin{eqnarray}
I_1 + I_2 \geq
\langle T^2 \rangle - \langle T \rangle^2 +
\langle T \rangle - \langle T^2 \rangle - \Bigl\langle \sqrt{T(1-T)} \Bigr\rangle^2
+ \Bigl\langle \sqrt{T(1-T)} \Bigr\rangle^2 \, = \,
\langle T \rangle - \langle T \rangle^2.
\end{eqnarray}
Now, assuming that $\int_0^{2 \pi} {d \phi \over dx} \, {dx \over 2 \pi} =0$,
(the circulation of $\phi$ is zero in one cycle),
\begin{eqnarray}
Q \, =\, e \, \Bigl(1 - \Bigl\langle T {d\theta \over dx} \Bigr\rangle \Bigr),
\end{eqnarray}
which implies
\begin{eqnarray}
Q (e-Q) \, =\, e^2 \, \Bigl( \Bigl\langle T {d \theta \over dx} \Bigr\rangle -
\Bigl\langle T {d \theta \over dx} \Bigr\rangle^2 \Bigr).
\end{eqnarray}
Moreover, by the H\"older inequality
\begin{eqnarray}
\Bigl\langle T {d \theta \over dx} \Bigr\rangle \,
\geq \, \Bigl\langle T \Bigr\rangle \,
\Bigl\langle {d \theta \over dx} \Bigr\rangle \, =\, \langle T \rangle.
\end{eqnarray}
Now, for any $y \geq 1/2$, $y(1-y)$ is a decreasing function of $y$.
Thus, if $\Bigl\langle T {d \theta \over dx} \Bigr\rangle \, \geq \, 1/2$ , i.e.
$Q \, \leq \, e/2$, then
\begin{eqnarray}
\langle T \rangle - \langle T \rangle^2 \, \geq
\Bigl\langle T {d\theta \over dx} \Bigr\rangle -
\Bigl\langle T {d \theta \over dx} \Bigr\rangle^2.
\end{eqnarray}
Thus $S \, \geq 8 \bigl({\omega \over 2 \pi}\bigr) Q_0 (e-Q_0)$, which is (\ref{S0bound}).
\vspace{4. mm}
Now we turn to the case of interest, when $Q_0 \, \geq \, e/2$, for example in
the quantized region. We were not able to provide a general
analytical proof of $Q_0 (e-Q_0)$ behaviour. We first look at simple limiting cases.
Then, we examine the particular case of the two delta-potentials model.
First, in a situtation where $T(x)$ is a constant $T$, (with $T$ small to have almost
charge quantization), then $Q_0 = e (1-T)$ and
$S_0 \,=\, 8 C_0 \bigl({\omega \over 2 \pi} \bigr) Q_0 \, (e-Q_0)$ with
$C_0$ depending on the shape of $\theta(x)$, but always $C_0$ is greater than $1$.
In the case of constant slope $d \theta/dx \, = \, 1$, $C_0=1$.
Second, in the situation where $d\theta/dx$ is constant but $T(x)$ arbitrary,
we have
$\langle T \, ( d \theta/dx ) \rangle \, =\, \langle T \rangle$
and thus
again (\ref{S0bound}) holds for any $Q_0$, not just for $Q_0$ smaller than $e/2$.
\vspace{3. mm}
However, in practise, $\theta$ varies abruptly by $\pi$ in the vicinity of resonances. This is
different from optimal pumping strategies which have been studied before\cite{AndreevKamenev,avron}.
We first give qualitative arguments and then give precise calculations for
the model studied here. Let us look at
the contribution of $J_2$, Eq. (\ref{J2}), to the noise.
When $x$ and $x^{\prime}$ are both close to resonances
$\vert e^{- i \theta(x)} - e^{-i \theta(x^{\prime})} \vert^2 /(x-x^{\prime})^2$
behaves as $(d \theta / dx)^2$ for $(x-x^{\prime}) (d\theta/dx) <1 $.
If meanwhile, $T(x)$ does not vary too much and assumes the value $T_i$,
the contribution of this region in the plane $(x,x^{\prime})$ to $J_2$ will be $T_i (1-T_i)/4$.
The $1/4$ comes from the fact that $\theta$ varies suddenly only by $\pi$ and not by $2 \pi$
at each resonance.
Another contribution will come from the regions where $x$
is within the resonance but $x^{\prime}$ just outside or vice versa.
Let us take $x^{\prime}$ outside to be precise.
Then $\vert e^{-i \theta (x)} - e^{- i \theta(x^{\prime})}\vert = 2$ and integration
on $x$ and $x^{\prime}$ will give a contribution mainly from
$x^{\prime}$ just outside, due to the rapidly decreasing
factor $(x-x^{\prime})^{-2}$. This will eventually
give another factor $T_i (1-T_i)/4$. Despite the non-local character of the integrand in
$J_2$, regions where $x$ and $x^{\prime}$ are both
far from a resonance make very little
contribution to $J_2$. For $Q$, one gets $Q= e(1 - \sum_i T_i/2)$.
In the simple case where there are only two resonances and the $T_i$'s are equal,
then $Q = e (1 - T_1)$ and $J_2 = T_1 (1-T_1)$. If $I_1$ and $J_1$, which are
related to the fluctuations of $T(x)$ are much smaller than $J_2$, then, this
leads to $S_0 = 8 \, (\omega / 2 \pi) \, Q_0 (e-Q_0)$.
\vspace{3. mm}
We now test the former very qualitative ideas by analytical calculations
on our particular model. A first thing to be noted is that
the phase $\phi$ does not intervene in the noise, which is normal since
noise is related to the two-particle scattering matrix.
On the contrary, it does formally enter the equation for the pumped charge,
see Eqs. (\ref{sm}),(\ref{Q0sint}) and (\ref{Qsint}). Nevertheless, the variation of $\phi$ when
$\omega t$ varies in one period, is always zero so that $\phi$ does not
play any role.
This is due to the fact that $\phi$ is the phase of $D$,
see Eqs. (\ref{s12}) and (\ref{equationsforsmatrixlast}).
If the real part of $D$ becomes negative, then, its imaginary part cannot be zero
and thus, $\phi$ can never be equal to $-\pi$ or $\pi$. Moerover, $\phi$ varies
by strictly less than $2 \pi$ during one period and thus, its circulation is zero, giving no
contribution to $Q$ and $Q_0$.
We now turn to the variations of $\theta$.
Charge quantization necessitates $\eta$ larger than $\sqrt{2} \, X_0$ and
$k_Fa$ small. We thus approximate $cos(2 k_Fa)$ by $1$ and set
$u = sin(2 k_F a) \, \ll \, 1$.
$\theta$ can be written as
\begin{eqnarray}
\theta \, =\, arg(n),
\end{eqnarray}
with
\begin{eqnarray}
n \, =\, {\overline X}\, {\overline Y}\, u \, + {\overline X} + {\overline Y} +
i u ({\overline Y} - {\overline X}).
\end{eqnarray}
There exist two values of $\eta$, $\eta_1$ and $\eta_2$, which play a particular
role.
For $\eta_1 \leq \eta\leq \eta_2$, when $\omega t$ varies by $2 \pi$, $Re(n)$ changes
twice its sign and $\theta$ varies by $2 \pi$. In the model studied here,
$\eta_1= X_0 \sqrt{2}$ and $\eta_2 = X_0 \sqrt{2} + 2 \sqrt{2} u^{-1}$.
Outside this interval of $\eta$, the increase of $\theta$ when $\omega t$ varies
by $2 \pi$ is zero, not $2 \pi$.
The reason is the following.
For $\eta \leq \eta_1$, $Re(n)$ is always positive and the phase
$\theta$ remains confined in an interval contained in
$\lbrack - {\pi \over 2} \, , \, {\pi \over 2} \rbrack$.
For $\eta > \eta_2$, $Re(n)$ changes four times its sign but the contour described by
$n$, in the complex plane, as $\omega t$ is varies by $ 2 \pi$,
does not enclose the origin. It can be seen directly, for it is impossible to have
$Im(n)=0$ and $Re(n) \leq 0$ at the same time. Thus, it is not surprising
that the quantized region corresponds approximately to the interval
$\lbrack \eta_1, \, , \, \eta_2 \rbrack$.
We now examine more precisely the variations of $\theta$ and $T$.
Apart from a small variation around $x \equiv \omega t = - {3 \pi/4}$,
$\theta$, as a function of $x$, is essentially flat, except around
two points $x_1$ and $x_2$. In practise, $x_1$ is close to
$- {\pi \over 4}$ and $x_2$ close to $3 \pi /4$.
Around those two points, $\theta$ increases fastly by almost $\pi$ each time.
We can thus model the function $\theta$ by
\begin{eqnarray}
\theta \, &=& \, - \pi, \,\,\,\, {\rm for} \,\,\,\,\,\,\,\,\,\, x \leq x_1-l, \nonumber \\
\theta \, &=& \, {\pi \over 2} \Bigl({x - x_1 \over l} -1\Bigr)
\,\,\,\, {\rm for} \,\,\, x_1- l \leq x \leq x_1+l, \nonumber \\
\theta \, &=& \, 0, \,\,\,\,\,\,\,\,\,\,
\,\,\,\, {\rm for} \,\,\, x_1+l \leq x \leq x_2-l, \nonumber \\
\theta \, &=& \, {\pi \over 2} \Bigl(1 + {x-x_2 \over l}\Bigr)
\,\,\,\, {\rm for} \, \leq x_2-l \leq x \leq x_2 +l, \nonumber \\
\theta \, &=& \, \pi, \,\,\,\,\,\,\,\,\,\,\,{\rm for} \,\,\, x \geq x_2+l,
\label{thetaparticular}
\end{eqnarray}
$l$ being a small distance.
As for $T$, $\,T(x)$ shows a large peak around $x=x_s = - 3 \pi / 4$, and
two smaller peaks, practically identical, centered around $x_1$ and $x_2$.
Away from these values of $x$, $T(x)$ is
almost zero.
Now, we look at the implications for the charge and noise. When calculating
the pumped charge without interaction via the formula
${1 \over 2 \pi} \int_0^{2 \pi} {d \theta \over dx} \, \bigl(1-T(x)\bigr) \, dx$,
the region around $x=x_s$ does not bring much contribution because variations of
$\theta$ are small here.
For the calculation of $Q_0$ and $S_0$,
we can ignore the large peak in $T$ and thus model $T(x)$ by
\begin{eqnarray}
T(x) \, &=& \, T_1 \,
\exp \biggl({ (x-x_i)^2 \over (x-x_i)^2-l^2}\biggr)
\,\,{\rm for} \,\,\, \vert x - x_i \vert \leq \, l, \nonumber \\
T(x) \, &=& \, 0 \,\,\, {\rm otherwise}.
\label{Tparticular}
\end{eqnarray}
$T$ has to be derivable in order to avoid
logarithmic divergencies due to the factor
$1/(x-x^{\prime})^2$ in Eqs. (\ref{I1}) and (\ref{I2}).
Then, it is possible to perform analytical calculations which give
\begin{eqnarray}
Q_0 \,&=& \, e \bigl(1 - C_1 T_1 \bigr), \\
S_0 \, &=& \, C_2 \Bigl({\omega \over 2 \pi} \Bigr)
\, (e-Q_0) \, \bigl\lbrack e (1- C_3) + C_3 Q_0) \bigr\rbrack.
\end{eqnarray}
Detailed calculations, involving the integrals $I_1$ and $I_2$ and the constants
$C_1$, $C_2$ and $C_3$ are given in the appendix.
$C_3$ is smaller than $1$, (approximately $0.58$). Even if it is not exactly of the form
$Q_0 (e-Q_0)$, when there is good charge quantization,
i.e. when $Q_0$ is not far from $1$, $S_0$ goes as $(e-Q_0)$.
The essential thing is that $T$ and $\theta$ vary rapidly around certain
values of $\omega t$. We do not get exactly $Q_0(e-Q_0)$ because
$T$ varies substantially when $\theta$ jumps.
One might wonder if the results here are particular to our model. In fact,
brisk variations of the phase are widely shared by many types of models\cite{EntinWohlman,Levinson}.
\vspace{2. mm}
The third case corresponds to $k_F a$ not close to $n \pi / 2$.
In this case, for large $X_0$, $Q$ is almost zero except in the vicinity
of a value $\eta_c$, which is very near $\sqrt{2} X_0$; numerically it seems
that $\eta_c$ is always a little less than this value.
The maximum pumped charge is
of order $e$ but no longer close to one electron charge. Noise has a
double peak structure around $\eta_c$.
A rough qualitative picture of this can be seen in Eq. (\ref{s12}),
because, as soon as the integration contour does not get close
to the point $X(t) =Y(t) =0$, for any $t$, the integrands in Eqs. (\ref{Q0sint}) and
(\ref{Qsint}) are very small. An example is shown in Fig 7.
\begin{figure}[h]
\epsfxsize 7. cm
\centerline{\epsffile{fig7.eps}}
\caption{Same as Fig. 5, except that $k_Fa \, = \, 0.4$. The solid
horizontal line
of ordinate $1$ and the thick vertical line
at abscissa $X_0 \sqrt{2}$ are just
guides to the eye.}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
We have studied the influence of weak electron-electron interactions on pumped charge and noise
in the adiabatic regime, in a mesoscopic one-dimensional disordered wire. Within
the two-delta potentials barrier model, analytical results were
obtained for the charge and noise. Results were analyzed numerically for local pumping fields with
a harmonic dependence. Without any voltage offset, at weak pumping amplitudes, interactions tend to
enhance the pumped charge, as $l^{-2 \alpha}$, where $l$ is the interaction parameter.
For fairly large pumping amplitudes, it is exactly the reverse,
$Q$ and $Q_0$ both decrease as $\eta^{-3}$,
but $Q$ remains smaller than $Q_0$ by a factor $l^{2 \alpha}$. At very
large pumping amplitudes, $Q$ and $Q_0$ are practically the same. As to the pumping noise,
at weak amplitudes, it increases with interactions, but in the same way as the charge, so that,
the Fano factor remains constant, independent of the interactions. For moderate pumping amplitudes,
noise has a double peak structure around the maximum of pumped charge. For large amplitudes, the noise
decreases slower than the charge, as $\eta^{-2}$, and for very large amplitudes, noise with and without
interactions become approximately the same.
As emphasised in Ref. \onlinecite{rao}, interactions
tend to make resonances sharper, which is conducive to
obtaining an almost quantized pumped charge. However, it is not sufficient to enclose a resonance, it is
also necessary that the pumping contour does not go too far from the resonance. Otherwise,
the noise is appreciable and the signal $Q$ can even be very small.
In the case of constant offset $X_0$, the behavior depends if we are close to a resonance,
$k_F a = n \pi /2$ in the two-delta potentials model.
Close to a resonance, there is a region of almost quantized pumped charge where the noise
seems to follow a $Q(e-Q)$ behavior, reminiscent of the noise reduction
in quantum wires for good transmission by a $T(1-T)$ factor, where $T$ is the modulus
of the energy transmision coefficient. Quite generally, noise breaks up
into pieces due to fluctuations of $T$ and those due to fluctuations of $\theta$.
We believe that in the quantized region, the fluctuations of $\theta$ are predominant
and give rise to the $Q(e-Q)$ behaviour.
Interactions do help in having
a charge closer to $e$ and to reduce the noise. However, it does
not change the range of pumping amplitudes, where quantized charge is observed,
i.e. the width of the quantized region practically
does not depend on the interactions. Qualitative arguments seem to indicate that this
width scales as $sin(2k_F a)^{-1}$ close to $k_F a = n \pi / 2$.
Further from the resonance, the maximum charge which
can be pumped becomes of order, but less than $e$. Moreover, the region of quantized
charge shrinks to a narrow window of pumping amplitudes around a value close to $X_0 \sqrt{2}$.
In summary, our study of noise shows that interactions tend to increase the
quality of pumping. However, two
conditions need to be met; first, to operate at certain wavevectors favouring
sharp resonances and second to have a
pumping contour which encircles the resonant point, passing not very far from it.
Otherwise, only noise is produced and the quantization of the charge is not achieved. These
restrictions were not pointed out in previous works. In addition, in the quantized
charge region, noise vanishes as $e-Q$.
\acknowledgments
One of us (T.M) acknowledges the support from ANR grant ``molspintronics'' of the
French ministry of research. V.G. acknowledges the kind hospitality of the
Centre de Physique Th\'eorique de Marseille, where inital parts
of this work were done.
|
1,941,325,220,567 | arxiv | \section{Introduction}
One of the most basic operations of linear optics is the interference between two optical modes in a two-path interferometer. In quantum optics, this scenario has been widely studied in the context of quantum metrology, where it serves to illustrate the role of non-classical correlations in the phase sensitivity of multi-photon interference \cite{Hol93,Fiu02,Ste04,Uys07,Hof07,Dow08,Pez08,Hof09,Bir12,Roz14,Sah15,Pez15}. Unfortunately, these arguments about phase sensitivity often leave out the details of the physics, focusing merely on the perceived figures of merit and not on the mechanisms by which multi-photon interference patterns emerge. Recent experiments on multi-photon interference are providing a much more detailed picture of non-classical effects in two-mode interferences \cite{Mit04,Nag07,Sun08,Afe10,Isr12,Coo13,Xia13}, and it might be time to ask what these experimental observation of multi-photon fringes can tell us about the nature and origin of quantum interference.
To properly address this question, it is necessary to distinguish carefully between classical wave interference and quantum interference. In typical experiments, multi-photon fringes are detected by measuring the phase dependence of count rates for a fixed combination of input and output conditions. Quantum interference thus appears as a modulation in the phase dependence of a single multi-photon probability. This is quite different from classical interference, where the output intensity is a deterministic function of the phase shift and probabilities appear only as a technical noise background. In quantum optics, classical interference is usually associated with the Poissonian photon counting statistics of coherent states, which describes a highly localized increase of detection probability for the classically expected intensity distribution at a specific phase shift in the interferometer. In the limit of high photon number, we can distinguish classical interference and multi-photon quantum interference qualitatively, since only the latter effect is associated with multiple interference fringes in the phase dependence of the output probabilities, and these interference fringes have distinct periodicities that are much shorter than $2 \pi$. It is therefore a non-trivial challenge to explain the relation between the modulations of probability seen in multi-photon interference and the statistics of classical interference effects observed in the phase dependence of output intensities.
In the present paper, we address this problem by examining the role of the operators representing the coherence of the two field modes. These operators have a clear classical meaning, describing the intensity differences between any two modes as a component of a three dimensional vector in close analogy to the algebra of quantized spins. We point out that the phase dependence of the quantum state components representing a specific measurement outcome can be described in terms of weak values of the intensity difference between the two arms of the interferometer. These weak values can be written as a function of the eigenvalues of the initial and the final state using relations that correspond to the classical causality of two-path interference. For the case of photon number states in both the input and the output, we can then derive a differential equation describing the phase dependence of a single interference fringe, where the main contribution to the phase dependence originates from the rapid oscillation of the fringes, while the slowly varying envelope describes the statistics expected for a classical interference of two fields when the optical phases of the input fields are completely random. It is possible to identify the periodicity of the multi-photon fringes that describe the quantum mechanical modulation of output probability with the intensity difference between the paths obtained from the weak value of the squared operator representing this intensity difference. The periodicity of multi-photon interference fringes can thus be explained in terms of a classical estimate of the intensity difference between the paths obtained from the experimentally controlled input and output conditions.
Our results show that multi-photon interference can be understood in terms of the classical relations between the intensities and coherences of the two modes. Specifically, the quantum mechanical phase of the multi-photon interference fringes is given by a classical action that relates the optical phase shift to its generator, the intensity difference between the paths. The non-classical effects of multi-photon interference can thus be traced to the fundamental role of the action in quantum physics \cite{Hof14,Hof16}. It may be worth noting that the direct identification of the action of phase shifts also provides a more efficient approach to the quantization of fields that starts from the macroscopically observable physics and hence avoids many of the ambiguities associated with the mathematical concepts of superpositions and state vectors. Multi-photon interference may thus help us bridge the gap between quantum mechanical concepts and classical intuition in a new and unexpected manner.
The paper is organized as follows: In Sec. \ref{sec:phase}, we review the operator algebra of two-path interference and its relation to the interference fringes observed in multi-photon experiments. In Sec. \ref{sec:wv}, it is shown that weak values can be used to express the phase dependence of a single multi-photon interference fringe. Based on this observation, we derive a differential equation for the interference fringes obtained with well-defined photon numbers in the input and the output. In Sec. \ref{sec:separate}, the differential equation derived in Sec. \ref{sec:wv} is solved approximately by separating the interference fringe into an interference term described by a phase dependent action $S$ and an envelope function $A$. It is shown that the action evolves according to a classical Hamilton-Jacobi equation, while the envelope describes the statistics of classical random phase interference. In Sec. \ref{sec:examples} we apply the theory to a number of characteristic cases, comparing the approximate results to exact results obtained for eight and sixteen photons. The results show that the approximation correctly describes the main features of multi-photon interference, especially with regard to the separation of quantum interference effects and classical random phase field interference. In Sec. \ref{sec:action} we identify the necessary conditions for the identification of multi-photon interference fringes with a classical action function. It is shown that weak values can be used to identify the action of multi-photon phase interferences for a wide variety of possible input states. In Sec. \ref{sec:NOON} we discuss the relation between the general analysis of multi-photon interference using the action and the definition of multi-photon interference based on superpositions of photon number eigenstates in the interferometer paths associated with the well known NOON states. It is pointed out that the NOON states represent a special case of the general action-based theory. Sec. \ref{sec:concl} summarizes the results and concludes the paper.
\section{Phase shifts in two-path interferometers}
\label{sec:phase}
The physics of two optical modes can be described in terms of the field operators $\hat{a}$ and $\hat{b}$ that describe the complex field amplitudes of the two modes. Due to their mathematical effects on photon number states, these field operators are commonly known as annihilation operators, although it should be kept in mind that this mathematical effect is not related to the physical properties described by the operators in any obvious or intuitive manner. The proper connection between two mode coherences and photon number is obtained by considering the second order products of field amplitudes,
\begin{eqnarray}
\hat{J}_1 &=&\frac{\hbar}{2}\left(\hat{a}^\dagger \hat{b} + \hat{b}^\dagger\hat{a}\right)
\nonumber \\
\hat{J}_2 &=& -i \frac{\hbar}{2}\left(\hat{a}^\dagger \hat{b} - \hat{b}^\dagger\hat{a}\right)
\nonumber \\
\hat{J}_3 &=&\frac{\hbar}{2}\left(\hat{a}^\dagger \hat{a} - \hat{b}^\dagger\hat{b}\right).
\end{eqnarray}
Each component of this three dimensional vector represents an intensity difference between two orthogonal modes in units of $\hbar/2$ per photon. The motivation for this choice of units is the representation of phase shifts between the modes $\hat{a}$ and $\hat{b}$, which is generated by the component $\hat{J}_3$ in the same way that a Hamiltonian generates the time evolution. Specifically, the unitary transformation of a phase shift is given by
\begin{equation}
\label{eq:unitary}
\hat{U}(\phi) = \exp\left(-i\frac{1}{\hbar} \hat{J}_3 \phi \right)
\end{equation}
and the effect of a phase shift on an arbitrary state $\mid \psi \rangle$ can be described by
\begin{equation}
\label{eq:Schroed}
\frac{\partial}{\partial \phi} \mid \psi \rangle = -i \frac{1}{\hbar} \hat{J}_3 \mid \psi \rangle.
\end{equation}
Here, the product $\hat{J}_3 \phi$ in Eq. (\ref{eq:unitary}) represents the action of a phase shift, just as the action of a time evolution is given by the energy-time product $\hat{H} t$.
In a two path interferometer, $\hat{J}_3$ describes the intensity difference between the two paths in the interferometer. The intensity difference between the input modes is usually given by $\hat{J}_1$. The remaining component $\hat{J}_2$ describes the phase coherence between the input fields that results in interferences when a phase shift of $\phi$ is applied. For a phase shift of $\phi$, the intensity difference observed in the output can be given by
\begin{eqnarray}
\label{eq:Heis}
\hat{J}_\phi &=& \hat{U}^\dagger(\phi) \hat{J}_1 \hat{U}(\phi)
\nonumber \\
&=& \cos(\phi) \hat{J}_1 - \sin(\phi) \hat{J}_2.
\end{eqnarray}
In the following, we will consider input states $\mid \psi (\phi=0) \rangle$ that are eigenstates of the input intensity difference $\hat{J}_1$, and measurements of the output intensity difference $\hat{J}_\phi$. Specific measurement results can be given by an integer or half-integer value of $m$ representing one half of the photon number difference between the output ports. Since the total photon number $N$ is conserved, $m$ represents a photon number state of the two output ports with output photon numbers of $N/2+m$ and $N/2-m$. The quantum states $\{\mid m \rangle \}$ representing these measurement results are eigenstates of $\hat{J}_\phi$ with eigenvalues of $\hbar m$. We are therefore interested in the phase dependent values of the output probabilities given by
\begin{equation}
P(m;\phi) = |\langle m \mid \hat{U}(\phi) \mid \psi (0) \rangle|^2,
\end{equation}
where the unitary transformation is used to relate the state $\mid \psi(\phi)\rangle$ to the input state $\mid \psi(0)\rangle$. Experimentally, these probabilities are obtained as multi-photon coincidence rates for the detection of $N/2-m$ photons in one output port and $N/2+m$ photons in the other. Quantum interference is observed as a phase dependent modulation of the probability, with a periodicity that can be as short as $2 \pi/N$ for $N$ photons. We will now analyze the phase dependence of $\langle m \mid \psi (\phi) \rangle$ to identify the origin of these quantum interference fringes in multi-photon interference.
\section{Derivation of multi-photon interference fringes using weak values}
\label{sec:wv}
We can apply the transformation of quantum states given by Eq.(\ref{eq:Schroed}) to describe the phase evolution of the probability amplitudes $\langle m \mid \psi (\phi) \rangle$ associated with specific measurement outcomes $m$. In its most conventional form, the resulting differential equation is given by
\begin{equation}
\frac{\partial}{\partial \phi} \langle m \mid \psi(\phi) \rangle = -i \frac{1}{\hbar} \langle m \mid \hat{J}_3 \mid \psi(\phi) \rangle.
\end{equation}
In the textbook approach to quantum dynamics, the operator $\hat{J}_3$ is usually expanded into its matrix representation in the measurement basis $\{\mid m \rangle \}$. However, this may not be the most meaningful analysis of the role of the generator $\hat{J}_3$ in the evolution of the input-output relation $\langle m \mid \psi \rangle $. A closer correspondence to classical dynamics can be maintained by identifying the contribution of $\hat{J}_3$ with its weak value,
\begin{equation}
\label{eq:action}
\frac{\partial}{\partial \phi} \langle m \mid \psi(\phi) \rangle = -i \frac{1}{\hbar} \frac{\langle m \mid \hat{J}_3 \mid \psi(\phi) \rangle}{\langle m \mid \psi(\phi) \rangle} \langle m \mid \psi(\phi) \rangle.
\end{equation}
It is interesting to note that the weak value can be used to express the effects of the operator $\hat{J}_3$ on the phase evolution of the state component $\langle m \mid \psi(\phi) \rangle$. The advantage of this approach is that it is often possible to find the weak value without having to solve the complete dynamics in the Schr\"odinger picture. Specifically, weak values can be determined by expressing the operator as a function of two operators, where the initial state is an eigenstate of the first and the final state is an eigenstate of the second operator.
In the present case, $\mid \psi (\phi=0) \rangle$ is an eigenstate of $\hat{J}_1$ and $\mid m \rangle$ is an eigenstate of $\hat{J}_\phi$, which can be related to the operators at $\phi=0$ using Eq.(\ref{eq:Heis}). Since the algebra of the $\hat{J}_i$ operators is the familiar algebra of spin operators for a total spin quantum number of $l=N/2$, it is possible to derive a relation between the phase shift generator $\hat{J}_3$ and the components $\hat{J}_1$ and $\hat{J}_\phi$ in the $\hat{J}_1$-$\hat{J}_2$ plane orthogonal to $\hat{J}_3$ by using the total length of the $J$-vector,
\begin{eqnarray}
\label{eq:Jquad}
\hat{J}_3^2 &=& \frac{\hbar^2}{4} N(N+2) - \hat{J}_1^2 - \hat{J}_2^2
\nonumber \\
&=& \frac{\hbar^2}{4} N(N+2) - \frac{1}{(\sin(\phi))^2}\left(\hat{J}_1^2 - \cos(\phi) (\hat{J}_1 \hat{J}_\phi + \hat{J}_\phi \hat{J}_1) + \hat{J}_\phi^2\right).
\end{eqnarray}
Since the initial and the final state are eigenstates of $J$-vector components orthogonal to $\hat{J}_3$, it is not possible to distinguish negative values of $\hat{J}_3$ from positive values of $\hat{J}_3$. As a result of this symmetry, the real part of the weak value of $\hat{J}_3$ is zero at all phases $\phi$, and the ratio between the amplitude $\langle m \mid \psi (\phi) \rangle$ and its phase derivative in Eq.(\ref{eq:action}) is always real. It is therefore possible to express $\langle m \mid \psi (\phi) \rangle$ by real numbers for all values of $\phi$.
To make optimal use of the relation in Eq.(\ref{eq:Jquad}), we now consider the second derivative of the phase dependence,
\begin{eqnarray}
\label{eq:quadact}
\frac{\partial^2}{\partial\phi^2} \langle m \mid \psi(\phi) \rangle &=& - \frac{1}{\hbar^2} \langle m \mid \hat{J}^2_3 \mid \psi(\phi) \rangle
\nonumber \\
&=& - \frac{1}{\hbar^2} \frac{\langle m \mid \hat{J}^2_3 \mid \psi(\phi) \rangle}{\langle m \mid \psi(\phi) \rangle} \langle m \mid \psi(\phi) \rangle.
\end{eqnarray}
This derivative is described by the weak value of $\hat{J}_3^2$, and this weak value can be determined using the eigenvalues $m_\psi$ and $m$ for the input state and the measurement outcome, respectively. It is important to arrange the order of the operators so that the operator $\hat{J}_1$ is always to the right of the operator $\hat{J}_\phi$, and the necessary application of the commutation relations results in a contribution from the imaginary weak value of $\hat{J}_3$. The final relation between the weak values and the eigenvalues therefore reads
\begin{equation}
\frac{\langle m \mid \hat{J}^2_3 \mid \psi(\phi) \rangle}{\langle m \mid \psi(\phi) \rangle}
+i \hbar \frac{\cos(\phi)}{\sin(\phi)} \frac{\langle m \mid \hat{J}_3 \mid \psi(\phi) \rangle}{\langle m \mid \psi(\phi) \rangle}
= \frac{\hbar^2}{4} N(N+2) - \frac{\hbar^2}{(\sin(\phi))^2}\left(m_\psi^2 - 2 \cos(\phi) m_\psi m + m^2\right).
\end{equation}
Since both the weak value of $\hat{J}_3^2$ and the weak value of $\hat{J}_3$ appear in the phase derivatives of $\langle m \mid \psi(\phi) \rangle$, we can use this relation between the weak values and the eigenvalues to find a differential equation for the phase dependence of the probability amplitude $\langle m \mid \psi(\phi) \rangle$ that does not depend on the probability amplitudes of any other measurement outcomes $m$. According to Eqs.(\ref{eq:action}) and (\ref{eq:quadact}), this differential equation can be written as
\begin{equation}
\label{eq:fringes}
\frac{\partial^2}{\partial\phi^2} \langle m \mid \psi(\phi) \rangle
+ \frac{\cos(\phi)}{\sin(\phi)} \frac{\partial}{\partial\phi} \langle m \mid \psi(\phi) \rangle
= - \left(\frac{1}{4} N(N+2) - \frac{1}{(\sin(\phi))^2}\left(m_\psi^2 - 2 \cos(\phi) m_\psi m + m^2\right)\right) \langle m \mid \psi(\phi) \rangle.
\end{equation}
We have thus derived a general mathematical description of the multi-photon interference fringes observed with any photon number input. In the following, we will consider the physics described by this differential equation and identify the characteristic features of its solutions.
\section{Separation of quantum effects and field statistics}
\label{sec:separate}
The description of the effects of phase shifts on multi-photon states given by Eq.(\ref{eq:fringes}) makes it possible to separate quantum effects from the classical limit by considering how the relation changes with total photon number. Classical effects should all scale with the total intensity, whereas the magnitude of quantum effects will always depend on absolute photon numbers. In the present case, such a separation of scales can be achieved by expressing the probability amplitudes $\langle m \mid \psi(\phi) \rangle$ by a product of a slowly varying envelope function $A(\phi)$ and a quantum interference effect given by an action $S(\phi)$ that describes the rapid modulation of probability associated with multi-photon interference,
\begin{equation}
\label{eq:separate}
\langle m \mid \psi(\phi) \rangle = 2 A(\phi) \cos\left(\frac{1}{\hbar} S(\phi)\right).
\end{equation}
The factor of two represents the interference between two classical solutions, as will be seen more clearly in the following discussion. Note that there is no approximation involved at this point, and the separation in Eq.(\ref{eq:separate}) can be used to obtain an exact solution of Eq.(\ref{eq:fringes}) for a specific photon number. However, our main concern is the comparison between the quantum effects that are expressed by the rapid oscillation of $\cos(S/\hbar)$ and the much slower variation of $A(\phi)$ that corresponds more closely to the classical statistics of random phase interference. In the following, we will therefore focus on approximate solutions, where the different scales of the phase dependence allow a complete separation between the dynamics of $S(\phi)$ and the dynamics of $A(\phi)$.
Specifically, the use of $\hbar$ in Eq.(\ref{eq:separate}) allows us to expand Eq.(\ref{eq:fringes}) in $\hbar$, where the classical limit emerges when action differences of $\hbar$ are not resolved. In the limit of sufficiently large photon numbers, we can therefore identify quantum effects as effects that depend on the precise ratio between the macroscopic action $S(\phi)$ and the fundamental constant $\hbar$.
In Eq.(\ref{eq:fringes}), the derivatives result in contributions that depend explicitly on the fundamental constant $\hbar$. Due to the smallness of $\hbar$, we can expand the equation and neglect higher order contributions of $\hbar$ in favor of the lower order contributions. The leading terms of the expansion are proportional to $1/\hbar^2$, which means that the left hand side of Eq. (\ref{eq:fringes}) can be represented by the square of the derivative of $S(\phi)$,
\begin{equation}
\label{eq:second}
\frac{\partial^2}{\partial\phi^2} \langle m \mid \psi(\phi) \rangle \approx - \left(\frac{1}{\hbar} \frac{\partial}{\partial \phi} S(\phi) \right)^2 \langle m \mid \psi(\phi) \rangle
\end{equation}
Note that the largest contribution of the first derivative of $\langle m \mid \psi(\phi) \rangle$ in Eq.(\ref{eq:fringes}) is proportional to $1/\hbar$, so that the contribution of the first derivative to the left hand side of Eq.(\ref{eq:fringes}) can be neglected except at phases where the leading contribution of $\partial S/\partial \phi$ becomes very small. For non-vanishing values of $\partial S/\partial \phi$, the approximate solution of the equation is given by
\begin{equation}
\label{eq:Sapprox}
\frac{\partial}{\partial \phi} S(\phi) \approx - \sqrt{\frac{\hbar^2}{4} N(N+2) - \frac{\hbar^2}{(\sin(\phi))^2}\left(m_\psi^2 - 2 \cos(\phi) m_\psi m + m^2\right)},
\end{equation}
where the choice of sign is arbitrary since the action $S$ is defined by the phase of a cosine in Eq.(\ref{eq:separate}). We choose the negative sign because it corresponds to the conventional definition of action in classical Hamilton-Jacobi theory, as will be seen below. Importantly, the right hand side of this equation has a very intuitive physical meaning: it is the classical value of $\hat{J}_3$ obtained from the length of the $J$-vector and the values of $\hat{J}_1$ and $\hat{J}_\phi$ when the Heisenberg relation between the operators in Eq.(\ref{eq:Heis}) is converted to a classical relation between the eigenvalues by neglecting the non-commutativity of $\hat{J}_\phi$ and $\hat{J}_1$. We can define this classical approximation of $J_3$ as
\begin{equation}
\label{eq:J3}
J_3(\hbar m, \hbar m_\psi, \phi) = \sqrt{\hbar \frac{N}{2} \hbar\left(\frac{N}{2}+1\right) - \frac{1}{(\sin(\phi))^2}\left((\hbar m_\psi)^2 - 2 \cos(\phi) (\hbar m_\psi)(\hbar m) + (\hbar m)^2\right)},
\end{equation}
which describes the intensity difference between the arms of a classical two-path interferometer when an input intensity difference of $\hbar m_\psi$ and a phase shift of $\phi$ result in an output intensity difference of $\hbar m$. The approximate quantum mechanical solution to Eq. (\ref{eq:fringes}) given by Eq.(\ref{eq:Sapprox}) then corresponds to the classical relation between the generator $J_3$ and the action $S$ given by a version of the Hamilton-Jacobi equation,
\begin{equation}
\label{eq:HJE}
\frac{\partial}{\partial \phi} S(\hbar m, \hbar m_\psi, \phi)
= - J_3(\hbar m, \hbar m_\psi, \phi).
\end{equation}
The reason why this action describes a quantum effect is that it appears as the quantum phase of the multi-photon interference fringes described by Eq.(\ref{eq:separate}), where the classical action $S$ is converted into a quantum phase by dividing it by $\hbar$. Thus, the larger the classical action $S$ becomes, the faster the output probability oscillates, resulting in microscopic (and therefore highly phase sensitive) modulations of probability in the limit of macroscopic action. The approximations used above apply whenever the modulation dominates the phase dependence, which is the case whenever $J_3$ is sufficiently larger than $\hbar$.
We can now turn to the approximate solution for the slowly varying envelope function $A(\phi)$. Since the quantum effects are for the most part described by the interference effects associated with the action in Eq.(\ref{eq:separate}), we expect the square of the envelope function $A(\phi)$ to represent the classical probability density of the multi-photon interference scenario. Specifically, the phase dependence of $A(\phi)$ can be determined by considering the second largest term in the expansion of Eq.(\ref{eq:fringes}) in $\hbar$, which is proportional to $1/\hbar$. The result can be written as
\begin{equation}
-\frac{2}{\hbar}\left(A(\phi) \frac{\partial^2}{\partial\phi^2} S(\phi) + 2 \left(\frac{\partial}{\partial\phi} A(\phi) \right)\left( \frac{\partial}{\partial\phi} S(\phi) \right) + \frac{\cos(\phi)}{\sin(\phi)} A(\phi) \frac{\partial}{\partial\phi} S(\phi)\right)\sin\left(\frac{S(\phi)}{\hbar}\right) = 0.
\end{equation}
The general solution of this equation does not depend on the specific value of $J_3$ and can be expressed by a relation between $A(\phi)$ and $S(\phi)$ given in more compact form as
\begin{equation}
\frac{\partial}{\partial\phi}\left(\sin(\phi) (A(\phi))^2 \frac{\partial}{\partial\phi} S(\phi) \right)=0.
\end{equation}
Since the phase derivative of the action is given by (\ref{eq:HJE}), it is possible to express the squared envelope function $A^2$ as a function of $J_3$ and a normalization factor of $\rho_0$,
\begin{equation}
\label{eq:ergo}
A^2 (\hbar m, \hbar m_\psi, \phi) = \frac{\hbar \; \rho_0}{|\sin(\phi) J_3(\hbar m, \hbar m_\psi, \phi)|}.
\end{equation}
This relation corresponds to a classical density of output intensities, where the factor of $\hbar$ originates from the quantized distance between two eigenvalues of $\hat{J}_1$. Once the factor of $\hbar$ is removed, the probability density scales with total intensity, indicating that the envelope function $A$ represents statistics that are also observable in the classical limit.
It is in fact possible to derive an expression for $\rho_0$ based on classical considerations. If we assume that the probability distribution originates from random phase interference between the two input intensities, we can explain the probability densities in terms of the phase difference $\theta$ between the input modes. Since the output value of $\hat{J}_\phi$ is a deterministic function of the fixed input value of $\hat{J}_1$ and the random phase $\theta$, the ratio of probability densities for intervals of $d \theta$ and intervals of $d J_\phi$ is given by
\begin{equation}
\left| \frac{d J_\phi}{d \theta} \right| = \left|\sin(\phi) J_3 \right|.
\end{equation}
Thus the factor in the denominator of Eq.(\ref{eq:ergo}) effectively converts probability densities in the random phase $\theta$ into probability densities in the output observable $J_\phi$. For homogeneous phase distributions, the factor of $\rho_0$ in Eq.(\ref{eq:ergo}) is given by
\begin{equation}
\rho_0=\frac{d\rho}{d\theta} = \frac{1}{2 \pi}.
\end{equation}
Using these relations, it is possible to find a properly normalized solution for the envelope function $A(\phi)$.
In the light of this result, we can conclude that the slowly varying envelope function describes the classical statistics originating from the random optical phase of the photon number states in the input. Specifically, the phase dependence of $A^2$ given by Eq.(\ref{eq:ergo}) is independent of the absolute photon number if the photon number differences of $2m$ and $2 m_\psi$ are given by constant fractions of total photon number. In the limit of high $N$, this means that the shape of the envelope $A(\phi)$ does not change much if the photon number changes by about $\sqrt{N}$, corresponding to the Poissonian photon number fluctuations of coherent states. It is therefore also possible to observe $A(\phi)$ using random phase interferences between two coherent states of light, where the relative error in $m/N$ caused by photon number fluctuations drops to zero with $1/\sqrt{N}$. On the other hand, the multi-photon interference patterns described by $S(\phi)$ disappear in the statistics of coherent states because their periodicities are much shorter than the absolute shot noise value of $\sqrt{N}$ in the limit of high photon number. We can therefore confirm that $A(\phi)$ is classical in the sense that it can also be observed using input states that can be represented by mixtures of coherent states as described by a positive P-function, while the multi-photon fringes described by $S(\phi)$ are non-classical since they disappear when the input can be described as a mixture of coherent states.
In combination with the solution for the action $S$, we now have an approximate solution for $\langle m \mid \psi(\phi) \rangle$ that is valid whenever the values of $J_3$ are sufficiently large. For multi-photon states, this condition is usually satisfied if there is a classical solution that relates the input intensity to the output intensity for a phase shift of $\phi$. The condition only breaks down when the argument in the square root of Eq.(\ref{eq:J3}) is very small or negative. Note that Eq.(\ref{eq:fringes}) can also be solved in the regime of negative $J_3^2$ values, where the differential equation describes evanescent solutions characterized by an exponential suppression of the amplitude $\langle m \mid \psi(\phi) \rangle$ with increasing distance from the region of positive $J_3^2$. Mathematically, this situation is equivalent to the evanescent wavefunctions known from tunneling. In fact, a comparison of Eq.(\ref{eq:fringes}) with the time independent Schr\"odinger equation shows that the mathematics are quite similar, and that the method of approximation used above corresponds to the WKB-approximation in the region where the energy is larger than the potential energy. The regions where $J_3$ drops to zero and becomes imaginary can therefore be treated in close analogy to the WKB approximation for the tunneling problem, by allowing evanescent solutions where the cosine function in Eq.(\ref{eq:separate}) is replaced by a corresponding exponential function. The intermediate regime close to $J_3=0$ corresponds to the turning points of the WKB solution and can be approximated by Airy functions. In this manner, it is possible to obtain an approximate solution of Eq.(\ref{eq:fringes}) at all photon numbers. However, we are mostly concerned with the multi-photon interference fringes observed for $J_3 \gg 1$, so a detailed analysis of the evanescent parts of the phase dependence is not necessary in the present context.
The main merit of the present analysis is that it allows us to separate the quantum interference effects described by the action $S$ from the effects of classical field statistics originating from the randomness of the optical phases in the input, which are represented by the envelope function $A$. In the following, we will apply this analysis to several representative examples of multi-photon interference fringes in order to illustrate how the approximations used here reproduce the characteristic features of multi-photon interference for different input-output relations.
\section{Multi-photon fringes for different photon number distributions}
\label{sec:examples}
In the measurement scenarios we are considering, the input photon number distribution and the output photon number distribution are both fixed, and the phase dependence of the probability for this specific combination is investigated. For a given total photon number $N$, different measurement scenarios are defined by the combination of input photon number difference $2 m_\psi$ and output photon number difference $2 m$. To illustrate our method of analysis, it is best to start with a particularly simple class of scenarios, where the photon numbers in both input and output ports are equal, so that $m_\psi=m=0$. In this case, we find that the intensity difference between the two arms of the interferometer does not depend on phase and is simply given by the total length of the $J$-vector,
\begin{equation}
\label{eq:HB}
|J_3(0,0,\phi)| = \frac{\hbar}{2} \sqrt{N(N+2)} \approx \frac{\hbar}{2}(N+1).
\end{equation}
In terms of classical wave interference, it is easy to understand why equal intensities in both the input and the output require a maximal intensity difference between the paths. Since both $\hat{J}_1$ and $\hat{J}_\phi$ are zero, we can immediately conclude that $\hat{J}_2$ should also be zero, so that the $J$-vector must point in the direction of $\hat{J}_3$. It is also possible to argue that we do not observe any interferences between the two paths, and this is only possible when all of the light travels along only one of the two paths. As mentioned above, $J_3$ is actually derived from the weak value of $\hat{J}_3^2$, so the choice of sign in Eqs.(\ref{eq:J3}) and (\ref{eq:HJE}) is merely an arbitrary convention. In terms of the actual physics, the scenario is complete symmetric in the two paths, so it is impossible to distinguish positive and negative values of $J_3$. The quantum interferences described by $\cos(S/\hbar)$ in Eq.(\ref{eq:separate}) are a consequence of this path symmetry \cite{Hof09}. According to Eq.(\ref{eq:HJE}), the distance $\Delta \phi$ between two minima of the interference fringes ($\Delta S = \hbar \pi$) is given by
\begin{equation}
\Delta \phi = \frac{\hbar \pi}{|J_3|} = \frac{2 \pi}{(N+1)}.
\end{equation}
It is worth noting that the value of $|J_3|$ is larger than the maximal eigenvalue of $\hat{J}_3$, resulting in fringes that have a shorter period than fringes observed for superpositions of $m_3=N/2$ and $m_3=-N/2$, the states known as NOON states \cite{Dow08}. The observation of shorter fringes is somewhat surprising because firstly, the NOON states achieve the maximal possible phase sensitivity of $N$-photon states and secondly the fringes of photon number states can be written as a sum of fringes from superpositions of $m_3$ and $-m_3$, where the NOON state contribution is the contribution with the shortest fringe periodicity. However, the weak value of $\hat{J}_3^2$ is not limited by the available range of eigenvalues, resulting in a fringe periodicity that corresponds to an additional one photon intensity difference between the paths. Specifically, the classical relation between $J_3$ and the eigenvalues $\hbar m_\psi$ and $\hbar m$ does not include any uncertainties, and the total length of the $J$-vector is given by $N+1$ because the ``$+1$'' contribution represents the necessary quantum fluctuations of an eigenstate with maximal $\hat{J}_3$ eigenvalue. The shorter fringes therefore represent the absence of quantum noise in the deterministic relation between input photon numbers and output photon numbers.
Importantly, the shorter fringes are only obtained for this specific combination of input and output,
which means that this effect is compensated by measurement results with less phase sensitivity and does not result in an enhanced overall phase sensitivity \cite{Xia13,Hof12}. In fact, the experimental results reported in \cite{Xia13} confirm the observation of fringes shorter than the NOON state limit of $2 \pi/N$. It is interesting that the significance of this experimental result seems to have escaped notice, probably because there was no proper theoretical explanation available at the time.
\begin{figure}[th]
\vspace{-1.5cm}
\begin{picture}(500,480)
\put(0,0){\makebox(480,480){
\scalebox{0.8}[0.8]{
\includegraphics{fig1.ps}}}}
\end{picture}
\vspace{-2.5cm}
\caption{\label{pairs}
Comparison of exact solution and approximate separation of the action $S$ and the envelope function $A$ for (a) eight photons and (b) sixteen photons. The dotted lines show the results of the approximation, while the full line is the precise result obtained from the complete state vector in the nine and seventeen dimensional Hilbert spaces, respectively. The approximate amplitude $\langle m \mid m_\psi \rangle$ only diverges from the exact solution close to $\phi=0$ and $\phi=\pi$, where the quantum mechanical solution is limited to a probability of one, while the classical amplitude describes a probability density that diverges to infinity.
}
\end{figure}
We can now find an approximate expression for the amplitude of the $\langle m \mid \psi \rangle$ in the case of equal photon numbers in the input and output. There is only the problem of the integration constant for the action $S$. We can solve this problem by noting that $\phi=\pi/2$ corresponds to the action of a 50:50 beam splitter, where it is known that the probabilities of finding an odd number of photons in the output ports is zero for equal photon numbers in the input. This means that, at $\phi=\pi/2$, $\cos(S/\hbar)=0$ for odd $N/2$ and $\cos(S/\hbar)=\pm 1$ for even $N/2$. The phase dependence of $S$ is therefore given by
\begin{equation}
S(0,0,\phi) = - \frac{\hbar}{2}\left((N+1) \phi - \frac{\pi}{2}\right).
\end{equation}
Note that this solution is obtained for phases between zero and $\pi$, since the approximation that $S$ changes more rapidly than $A$ breaks down near $\phi=0$ and near $\phi=\pi$. To include the range between $\phi=-\pi$ and $\phi=0$, it is convenient to use the absolute value of $\phi$, for an approximate solution of
\begin{equation}
\langle m=0 \mid \psi(m_\psi=0) \rangle \approx
2 \sqrt{\frac{1}{\pi (N+1) \sin(|\phi|)}} \cos\left(\frac{N+1}{2}|\phi| - \frac{\pi}{4}\right).
\end{equation}
Fig. \ref{pairs} shows the comparison between the exact solution and the approximation for eight photons and for sixteen photons. As the figure shows, the approximation correctly describes the amplitudes $\langle m \mid \psi \rangle$ except in the immediate vicinity of $\phi=0$ and $\phi=\pm \pi$, where the approximation diverges to values higher than one. This discrepancy is easy to understand, since the classical approximation describes a probability density instead of a discrete probability, so it can exceed a value of one if the probability distribution in $J_\phi$ is narrower than $\hbar$. Significantly, the discreteness of the outcomes has no effect on the results outside of these two very narrow regions. Even for eight photons, the effects of higher order terms in $\hbar$ is mostly negligible.
\begin{figure}[th]
\vspace{-1.5cm}
\begin{picture}(500,480)
\put(0,0){\makebox(480,480){
\scalebox{0.8}[0.8]{
\includegraphics{fig2.ps}}}}
\end{picture}
\vspace{-2.5cm}
\caption{\label{rates}
Experimentally observable multi-photon interference fringes for equal photon numbers in the input and the output. (a) shows the phase dependent count rates for eight photons and (b) shows the phase dependent count rate for sixteen photons. The dotted line shows the envelope function $4 A^2$ to indicate that the amplitude of the fringes follows the classical statistics of random phase interference.
}
\end{figure}
Since the experimental count rates of a multi-photon interference experiment are given by the probabilities $|~\langle~m~\mid~\psi~\rangle~|^2$, these are shown in Fig. \ref{rates}. Note that the envelope values indicated in the figure correspond to twice the classical probability densities expected for random phase interference with the same input and output conditions. The coincidence of this envelop with the peaks of the interference fringes in Fig. \ref{rates} thus illustrates how classical statistics emerge when the phase resolution fails to resolve the quantum interferences associated with the action $S$. As mentioned above experimental data of this type has already been reported, e.g. in \cite{Xia13} for a six photon state, where the minima are separated by phases of $\Delta \phi = 2 \pi/7$. In general, the positions of the minima are found at $\cos(S)=0$, which means that the minima for $0<\phi<\pi$ are located at
\begin{equation}
\phi_{\mbox{min.}}=\frac{3 \pi}{2 (N+1)}, \frac{7 \pi}{2 (N+1)}, \frac{11 \pi}{2 (N+1)}, \ldots
\end{equation}
For $-\pi < \phi < \pi$, there will be a total of $N$ minima, equal to the total number of photons. However, the width of the peaks at $\phi=0$ and at $\phi=\pm \pi$ is 1.5 times the width of the other peaks, resulting in $N-2$ interference fringes with a width of $\Delta \phi=2 \pi/(N+1)$ and two fringes with a width of $\Delta \phi^\prime=3 \pi/(N+1)$.
\begin{figure}[th]
\vspace{-1cm}
\begin{picture}(500,240)
\put(0,0){\makebox(480,240){
\scalebox{0.8}[0.8]{
\includegraphics{fig3.ps}}}}
\end{picture}
\vspace{-1.2cm}
\caption{\label{J3cross}
Dependence of the intensity difference between the paths in the interferometer $|J_3|$ on phase $\phi$ for an input photon number difference of $2m_\psi=0$ and an output photon number difference of $2m = N/2$. The value is given relative to the total length of the $J$-vector, which is $\hbar (N+1)/2$.
}
\end{figure}
\begin{figure}[th]
\vspace{-1.5cm}
\begin{picture}(500,480)
\put(0,0){\makebox(480,480){
\scalebox{0.8}[0.8]{
\includegraphics{fig4.ps}}}}
\end{picture}
\vspace{-2.5cm}
\caption{\label{Pcross}
Experimentally observable multi-photon interference fringes for equal photon numbers in the input and a photon number difference of $2m=N/2$ in the output. (a) shows the phase dependent count rates for eight photons and (b) shows the phase dependent count rate for sixteen photons. The dotted line shows the envelope function $4 A^2$ to indicate that the amplitude of the fringes follows the classical statistics of random phase interference.
}
\end{figure}
Next we consider cases where the photon number difference is non-zero in either the input or the output. Experimentally, it is relatively easy to obtain this kind of data from an $m_\psi=0$ input state by simply arranging the detectors to detect a non-zero photon number difference of $2 m$ between the output ports. From the output photon number difference $2 m$, we can then estimate the intensity difference between the paths in the interferometer as
\begin{equation}
\label{eq:m0}
|J_3(\hbar m,0,\phi)| = \hbar \sqrt{\frac{N(N+2)}{4} - \frac{m^2}{(\sin(\phi))^2}}
\end{equation}
Since a minimal phase shift of about $\phi = \arcsin(m/(N+1))$ is necessary to achieve an output photon number difference of $m$ by optical interference, this equation only has valid solutions for a limited range of phase shifts centered around $\phi=\pi/2$. In all cases, the magnitude of $m$ limits the maximal value of $J_3$, and hence the periodicity of the multi-photon interference fringes. Fig. \ref{J3cross} shows the dependence of $|J_3|$ on phase for $m \approx (N+1)/4$. In this case, the maximal value at $\phi=\pi/2$ is equal to $\sqrt{3/4}$ times the maximal value of $\hbar (N+1)/2$.
Although it is possible to find an analytical solution for the action, the merit of the approximation is that it provides an insight into the physics that determines the periodicity of multi-photon interferences, and into the relation between the action and the envelope function that describes the classical statistics of random phase interference. We will therefore analyze both aspects using the exact solutions of the interference fringes obtained for specific photon numbers. Fig. \ref{Pcross} (a) shows the interference fringes for eight photons and $m=N/4=2$ in the output, and Fig. \ref{Pcross} (b) shows the fringes for sixteen photons and $m=N/4=4$ in the output. The approximate envelop function given by Eq.(\ref{eq:ergo}) is also indicated, showing that the approximation works well within the range of classical solutions. Outside this range, there is a non-zero evanescent solution that quickly drops to zero. Within the range of classical solutions, we observe multi-photon interference effects in the form of fringes of width $\Delta \phi_{\mathrm{exp.}}$. This width is easy to observe experimentally, since it can be defined as the phase difference between two consecutive minima of the output count rate. We can use this definition to obtain an experimental value of $|J_3|$ for a specific fringe,
\begin{equation}
\label{eq:evaluate}
|J_3|_{\mathrm{exp.}} = \frac{\pi}{\Delta \phi_{\mathrm{exp.}}}.
\end{equation}
We now apply this method of evaluation to the exact solutions of $|\langle m \mid \psi \rangle|^2$ in order to find out how well the approximation given by Eq.(\ref{eq:J3}) corresponds to the actual periodicities of multi-photon interference. For the eight photon case shown in Fig. \ref{Pcross} (a), probabilities of zero occur at $\phi=1.183$ and at $\phi=1.958$, resulting in a single interference fringe with $\Delta \phi_{\mathrm{exp.}}=0.775$. The corresponding value of $|J_3|$ is $4.05 \hbar$. According to Eq. (\ref{eq:m0}), the maximal possible value of $|J_3|$ for $N=8$ and $m=2$ should be $4.00$, indicating that the approximation slightly underestimates the value of $|J_3|$.
For the sixteen photon case shown in Fig. \ref{Pcross} (b), probabilities of zero occur at $\phi=0.931$, $\phi=1.362$, $\phi=1.780$, and $\phi=2.211$, resulting in three interference fringes with $\Delta \phi_{\mathrm{exp.}}=0.431$, $\Delta \phi_{\mathrm{exp.}}=0.418$, and $\Delta \phi_{\mathrm{exp.}}=0.431$. The corresponding values of $|J_3|$ are $7.29 \hbar$, $7.52 \hbar$, and $7.29 \hbar$. As expected from the theory, $|J_3|$ is maximal around $\phi=\pi/2$, although the theory again underestimates the value, with $|J_3|=7.48 \hbar$ at $\phi=\pi/2$ for $N=16$ and $m=4$. As shown in Fig.\ref{J3cross}, the value of $|J_3|$ drops off symmetrically as the phase is shifted from $\phi=\pi/2$ to higher or lower phases. We can use Eq.(\ref{eq:m0}) to find the phase at which the expected value of $|J_3|$ is $7.29 \hbar$. We obtain phases of $\phi=1.263$ and $\phi=1.879$. Both of these phases are located near the center of the interference fringes from which the $|J_3|$ values are derived.
In summary, the comparison between the approximate separation of quantum interference and (classical) envelope function can also be confirmed for different photon numbers in either the input or the output. Specifically, we can explain both the confinement of output probabilities to a finite phase interval around $\phi=\pi/2$ and the width of quantum interference fringes using the input-output relations given by Eqs.(\ref{eq:J3}) and Eq.(\ref{eq:ergo}). Remarkably, both of these relations also apply to classical random phase interference, although classical statistics would result in an output probability given by $2 A^2$, without the quantum interference effects associated with the value of $|J_3|$.
\begin{figure}[th]
\vspace{-1cm}
\begin{picture}(500,240)
\put(0,0){\makebox(480,240){
\scalebox{0.8}[0.8]{
\includegraphics{fig5.ps}}}}
\end{picture}
\vspace{-1.2cm}
\caption{\label{J3self}
Dependence of the intensity difference between the paths in the interferometer $|J_3|$ on phase $\phi$ for an input photon number difference of $2m_\psi=N/2$ and an output photon number difference of $2m = N/2$. The value is given relative to the total length of the $J$-vector, which is $\hbar (N+1)/2$.
}
\end{figure}
\begin{figure}[th]
\vspace{-1.5cm}
\begin{picture}(500,480)
\put(0,0){\makebox(480,480){
\scalebox{0.8}[0.8]{
\includegraphics{fig6.ps}}}}
\end{picture}
\vspace{-2.5cm}
\caption{\label{Pself}
Experimentally observable multi-photon interference fringes for photon number differences of $2m_\psi=2m=N/2$ in both the input and the output. (a) shows the phase dependent count rates for eight photons and (b) shows the phase dependent count rate for sixteen photons. The dotted line shows the envelope function $4 A^2$ to indicate that the amplitude of the fringes follows the classical statistics of random phase interference.
}
\end{figure}
Finally, it may also be good to consider the case where both the input and the output have a non-vanishing photon number difference between the ports. To keep the mathematics simple, we choose the case of $m=m_\psi$, where the output photon number distribution happens to be identical to the input photon number distribution. The intensity difference $|J_3|$ between the paths is then given by
\begin{equation}
\label{eq:mm}
|J_3(\hbar m,m,\phi)| = \hbar \sqrt{\frac{N(N+2)}{4} - \frac{2 m^2}{1+\cos(\phi)}}.
\end{equation}
Note that the maximal value of $|J_3|$ is the same as the maximal value in Eq.(\ref{eq:m0}), but it occurs at $\phi=0$ instead of at $\phi=\pi/2$. The value of $|J_3|$ drops continuously as $|\phi|$ increases, until it reaches zero at $1+\cos(\phi) \approx 8 m^2/(N+1)^2$. Fig. \ref{J3self} shows the dependence of $|J_3|$ on phase for $m\approx(N+1)/4$, where the value of $|J_3|$ reaches zero at $|\phi|=2 \pi/3$.
Fig. \ref{Pself} (a) shows the interference fringes for eight photons and $m=N/4=2$ in both input and output, and \ref{Pself} (b) shows the fringes for sixteen photons and $m=N/4=4$ in input and output. The approximate envelope function is also shown, indicating that Eq.(\ref{eq:ergo}) is a good approximation of the precise results up to the limit of validity near $|\phi|=2 \pi/3$. We can now evaluate the width of the fringes and obtain the corresponding values of $|J_3|$ according to Eq.(\ref{eq:evaluate}). For the eight photon case shown in Fig. \ref{Pself} (a), probabilities of zero occur at $\phi=0.597$ and at $\phi=1.397$, resulting in a single interference fringe with $\Delta \phi_{\mathrm{exp.}}=0.800$. The corresponding value of $|J_3|$ is $3.93\hbar$. According to Eq.(\ref{eq:mm}), this value of $|J_3|$ is obtained at a phase of $|\phi|=0.713$. Although this phase is found within the fringe, it is much closer to the low phase minimum than to the high phase minimum, indicating that the approximation slightly underestimates the $|J_3|$ value. In the middle of the fringe, at $\phi=0.997$, Eq.(\ref{eq:mm}) assigns a $|J_3|$ value of only $3.85 \hbar$. This is consistent with the result from Fig.(\ref{Pcross}) (a) above, suggesting an error margin of about two percent for eight photon fringe widths obtained from Eq.(\ref{eq:J3}).
For the sixteen photon case shown in Fig. \ref{Pself} (b), probabilities of zero occur at $\phi=0.321$, $\phi=0.740$, $\phi=1.175$, and $\phi=1.644$, resulting in three interference fringes with $\Delta \phi_{\mathrm{exp.}}=0.419$, $\Delta \phi_{\mathrm{exp.}}=0.435$, and $\Delta \phi_{\mathrm{exp.}}=0.469$. The corresponding values of $|J_3|$ are $7.50\hbar$, $7.22\hbar$, and $6.70\hbar$. As expected, the values of $|J_3|$ decrease as the phase shift increases. The $|J_3|$ value of the first fringe exceeds the maximal value of $|J_3|=7.48 \hbar$ obtained from Eq.(\ref{eq:mm}), indicating once more that the theory underestimates the values of $|J_3|$. For the other two fringes, the values of $|J_3|$ obtained for their respective widths are found at $|\phi|=0.914$ for $|J_3|=7.22\hbar$, and at $|\phi|=1.389$ for $|J_3|=6.70\hbar$. Both results are very close to the centers of the fringes, confirming that Eq.(\ref{eq:HJE}) provides a good approximate description of multi-photon interferences.
The results obtained for all examples show that the physics of multi-photon interference can be understood in terms of the separation between the action $S$ that describes quantum interferences and the amplitude $A$ that describes the classical statistics of random phase interference. In particular, it is possible to explain the periodicity of multi-photon interference in terms of the intensity difference between the paths expected from the {\it combination} of input and output conditions. It is therefore important to recognize that multi-photon interference is not just the result of superpositions in the input state, but also involves quantum coherences in the multi-photon statistics of the measurement outcome. In the case of photon number states in the input, the relation is completely symmetric, and the same phase dependent interference fringes will be obtained when input and output are exchanged.
\section{The role of the action in multi-photon interference}
\label{sec:action}
In the analysis presented above, we have studied the multi-photon interference fringes of photon number state inputs in the two input ports of a two-path interferometer. However, the approach developed at the start of Sec. \ref{sec:wv} is not limited to these specific states. Specifically, Eqs.(\ref{eq:action}) and (\ref{eq:quadact}) are generally valid expressions of the phase evolution of interference fringes in terms of the weak values of $\hat{J}_3$ and $\hat{J}_3^2$. The separation of amplitude $A$ and action $S$ introduced in Eq.(\ref{eq:separate}) is possible whenever the amplitudes have only real values. This means that the real part of the weak values of $\hat{J}_3$ must be zero for all phases $\phi$,
\begin{equation}
\label{eq:cond1}
\mbox{Re}\left(\frac{\langle m \mid \hat{J}_3 \mid \psi \rangle}{\langle m \mid \psi \rangle}\right)=0.
\end{equation}
Since the phases of different eigenstate components of $\hat{J}_3$ evolve differently, this criterion cannot be satisfied unless each pair of components with opposite eigenvalues have the same amplitude and opposite phases. If the eigenstates of $\hat{J}_3$ with eigenvalues of $\hbar m_3$ are written as $\mid m_3 \rangle$, this criterion can be expressed by the condition
\begin{equation}
\langle m \mid m_3 \rangle \langle m_3 \mid \psi \rangle = \langle \psi \mid -m_3 \rangle \langle -m_3 \mid m \rangle.
\end{equation}
Usually, this condition is satisfied separately by both the input state and the output state, so that both states are symmetric in the path basis. Specifically, the eigenstates of $\hat{J}_\phi$ that represent the results of photon number measurements in the output ports all satisfy the condition
\begin{equation}
\langle m \mid m_3 \rangle = \langle -m_3 \mid m \rangle.
\end{equation}
Therefore, the amplitudes $\langle m \mid \psi \rangle$ will be real at all phases $\phi$ if the $\hat{J}_3$ components of $\mid \psi \rangle$ satisfy
\begin{equation}
\label{eq:pathsym}
\langle m_3 \mid \psi \rangle = \langle \psi \mid -m_3 \rangle.
\end{equation}
Note that it has been shown elsewhere that input states satisfying this path symmetry condition achieve their maximal phase sensitivity in photon number measurements \cite{Hof09}. The present analysis investigates the more detailed mechanism by which this is achieved.
In general, the approximate solution of Eq.(\ref{eq:quadact}) is given by Eq.(\ref{eq:second}), so that the multi-photon interference fringes can be described by the phase dependent action $S$ with
\begin{equation}
\label{eq:act2}
\frac{\partial}{\partial \phi} S(\phi) = -\sqrt{\frac{\langle m \mid \hat{J}_3^2 \mid \psi \rangle}{\langle m \mid \psi \rangle}}
\end{equation}
It is therefore possible to derive the periodicity of the interference fringes from the weak value of the squared phase shift generator $\hat{J}_3^2$ for any path symmetric state. Note that Eq.(\ref{eq:quadact}) indicates that the weak value of $\hat{J}_3^2$ is always real if the states and measurements satisfy Eq.(\ref{eq:cond1}) at all phases. In the limit of high photon number $N$, it is usually possible to find a classical relation that defines the value of $J_3$ as a function of initial conditions $\psi$, measurement outcome $m$, and phase shift $\phi$. It is then possible to relate the multi-photon interference effects to the deterministic relations between initial and final conditions known from classical wave interference.
In the present case, the action $S$ is directly observable as an interference fringe because the amplitudes $\langle m \mid \psi \rangle$ are all real. This is a result of the symmetry between negative and positive intensity differences between the paths. If there is only a single classical solution for $J_3$, the action given by Eq.(\ref{eq:HJE}) will merely describe an unobservable phase evolution of $\langle m \mid \psi \rangle$. In this sense, one can explain quantum interference as an interference between the two classical solutions $+J_3$ and $-J_3$, both of which connect $\psi$ to $m$ under phase shifts of $\phi$. Importantly, this interference can be described without any reference to Hilbert space vectors, since the physical properties $J_i$ can all be defined within classical wave theory. It may therefore be useful to take a closer look at the relation between classical wave interference and the quantum interference of multi-photon states.
\section{Classical interference versus quantum interference}
\label{sec:NOON}
As shown above, multi-photon interference can be understood in terms of the weak values of the generator $\hat{J}_3$ that describes the phase shift induced by a two-path interferometer. To explain the connection with more conventional ideas of quantum interference as an effect of superpositions in the initial state, it may be useful to consider a particularly simple solution of Eq.(\ref{eq:act2}), where the periodicity of the interference fringes is determined by the input state and does not depend on output photon number or phase. This situation only occurs if the input state is an eigenstate of $\hat{J}_3^2$, so that the weak value is given by the initial eigenvalue and does not depend on phase anymore. Except for arbitrary phase shifts, the complete set of states with constant fringe periodicity is therefore given by
\begin{equation}
\label{eq:super}
\mid \psi(m_3) \rangle = \frac{1}{\sqrt{2}} \left(\mid m_3 \rangle + \mid - m_3 \rangle\right),
\end{equation}
where $\mid m_3 \rangle$ are the $\hat{J}_3$ eigenstates with eigenvalues of $\hbar m_3$. Similar to the case of equal photon numbers in input and output, the derivative of the action is constant and a multiple of $\hbar$,
\begin{equation}
\frac{\partial}{\partial \phi} S(\phi) = - \hbar m_3.
\end{equation}
For an $N$-photon state, the maximal number of fringes is obtained for the so-called NOON state, where $m_3=N/2$. It is therefore tempting to think of multi-photon interference as an $N$-fold increase in the number of fringes, or an $N$-fold enhancement of phase sensitivity. However, even the simple superposition states given by Eq.(\ref{eq:super}) show that the number and the periodicity of fringes is not a function of photon number, but a function of photon number distribution between the paths. In general, the weak value of $\hat{J}_3^2$ provides an estimate of the intensity distribution between the paths for a specific combination of input state, measurement outcome, and phase shift. This estimate identifies the proper physics of multi-photon interference.
What is perhaps most remarkable about the results presented above is that the quantum effects of multi-photon interference can be explained by using the deterministic relations between field coherences already known from classical two-path interference. The classical interference effect is actually described by the operator relation in Eq.(\ref{eq:Heis}), where the output intensities are expressed as a function of input intensities, coherence, and phase shift. In classical theory, probabilities enter the picture only if there is some randomness in the initial conditions. In the case of photon number inputs discussed above, this randomness can be identified as a random phase difference between the two input fields. As the approximation shows, the quantum mechanical result reproduces this classical probability distribution over random phases in the form of the envelope function that describes the amplitude of the quantum interference fringes.
Importantly, quantum interference is not an effect that emerges gradually from classical field interference as photon number increases, but should be considered as a fundamentally different effect associated with the non-classical relation between the phase shift $\phi$ and its generator $\hat{J}_3$. The only reason why this effect is related to photon number is that, as a quantum effect, its scale is determined by $\hbar$. All classical effects scale with intensity, only the quantum interference fringes decrease in width as photon number increases. The analysis in this paper shows that it is possible to separate the quantum scale and the classical scale in the quantum theory of multi-photon interference fringes, highlighting the different physics involved in the two distinct processes.
\section{Conclusions}
\label{sec:concl}
In conclusion, multi-photon interferences can be explained in terms of the weak values of the intensity difference between the paths of the interferometer. For photon number states, it is then possible to describe each multi-photon interference fringe by its own differential equation. The approximate solution of this differential equation identifies the periodicity of the fringes with the intensity difference between the paths inside the interferometer, which can be determined from the classical relation between input intensity difference, output intensity difference, and phase.
We have shown that it is possible to separate the quantum effects of multi-photon interference from the classical statistics of random phase interference. Specifically, we can trace quantum interference back to the action of the phase shift, which is given by the product of the generator $\hat{J}_3$ and the phase shift $\phi$. The photon number appears in this relation only because $\hat{J}_3$ is quantized in units of $\hbar$. The classical units of intensity difference are action units, where $\hbar$ is the ratio between the action and the quantum phase. Our analysis thus shows that an detailed analysis of multi-photon interference can be achieved by focusing on the classical field properties if both the input state and the measurement outcomes are treated on an equal footing.
Our work provides a general characterization of multi-photon interference that may serve as a road map to future experiments involving larger photon numbers. We have identified the relevant features of different two-path interference scenarios and explained the origin and the physical meaning of the experimentally observable fringe width. We believe that these observations will be extremely helpful in the characterization of multi-photon quantum devices and in the development of new approaches to multi-particle quantum physics.
\section*{Acknowledgment}
This work was supported by
JSPS KAKENHI Grant Number 26220712.
\vspace{0.5cm}
|
1,941,325,220,568 | arxiv | \section{Introduction}
\label{sec:Intro}
The spectrum of excited states has historically given essential information on the nature of any
composite quantum system. The careful mapping of the excited states of baryons shines light on the
nature of the nonperturbative regime of quantum chromodynamics (QCD). This spectrum specifically
depends on the effective degrees of freedom and the forces confining the quarks. Symmetric quark models,
which attempt to describe the baryon system, predict the pattern of low-mass baryons reasonably well.
However, the predicted baryon states for masses above 1.8~GeV/$c^2$ greatly outnumber those which have
been found experimentally. Most known light-flavor baryon resonances lie below 2 GeV/$c^2$ and were
discovered in elastic $\pi N$ scattering experiments. Quark model calculations have shown that many of
these so-called ``missing'' baryons have weak $\pi N$ couplings; and moreover, they could strongly couple
to $\eta N$ and $\omega N$ without a small coupling to $\gamma N$. In recent years, many laboratories
around the world (ELSA, GRAAL, Jefferson Laboratory, MAMI, SPring-8, etc.) have published differential cross
sections and polarization observables in photoproduced reactions. For a recent review on baryon resonances,
see \cite{Klempt:2009pi,Crede:2013kia}.
According to the predictions of the Constituent Quark Model, e.g.~\cite{Capstick:1993kb}, data on
$\omega$ photoproduction have a good chance of revealing some of the ``missing'' baryon resonances.
However, since the $\omega$~meson has the same quantum numbers as the incoming photon, meson
exchange ($t$-channel) processes are likely to contribute strongly. To disentangle the $t$-channel from
the resonant ($s$-channel) amplitude, data with full angular coverage are needed. Of particular importance
is the very forward direction where the $t$-channel amplitude has its maximum.
Moreover, the $\omega$ meson acts as an isospin filter for baryon resonances. Since the isospin of the
$\omega$ meson is zero, any baryon resonance decaying to $N\omega$ must have isospin $I = 1/2$, and
therefore contributions from $\Delta$ states are excluded.
In this paper, the differential cross sections and spin-density matrix elements for the reaction
\begin{equation} \gamma p \to p\omega\label{eqn:reaction}\end{equation}
are presented by reconstructing the $\omega$ from the neutral decay,
\begin{equation} \omega \to \pi^0 \gamma \to \gamma\gamma\gamma\:.\end{equation}
\section{Experimental Setup}
\label{sec:ExpSetup}
The CBELSA/TAPS experiment was conducted at the electron stretcher accelerator (ELSA) facility~\cite{ELSApast}
located at the University of Bonn in Germany. A 3.175~GeV electron beam from ELSA interacted with a radiator
target and produced bremsstrahlung photons. The radiator target was situated in a goniometer which contained
copper radiators of varying thickness along with a diamond radiator for linear polarization. The unpolarized data
utilized a copper radiator of thickness 3/1000\,$X_R$ (radiation length). The polarized data used a diamond
radiator. The bremsstrahlung electrons were deflected by a dipole magnet into the tagging detector system
(tagger). The tagger consisted of 480 scintillating fibers on top of 14~scintillating bar counters which partly
overlapped. Using the knowledge of the magnetic field strength and the hit position in the tagger, the energy
of each electron was determined and used to tag each bremstrahlung photon with energy and time information.
A fraction of the tagged bremsstrahlung photons continued down the beam line and interacted with the
protons in the liquid hydrogen target to produce mesons which decayed to final-state photons. The energy
and position of these photons were detected by the crystal modules in the two electromagnetic calorimeters,
Crystal Barrel and TAPS. The Crystal Barrel detector, in its configuration during the CBELSA/TAPS experiment of
2002/2003, consisted of 1290 CsI(Tl) crystals, which were read out by photodiodes. The TAPS detector consisted
of 528~BaF$_2$ crystals, which were read out by photomultiplier tubes. They formed a hexagonal wall that covered
the forward hole left open by the Crystal Barrel detector. Together the Crystal Barrel and TAPS detectors covered
more than 98\,\% of the full $4\pi$ solid angle. Protons or any other charged particles were identified by either
5~mm thick plastic scintillators placed in front of each TAPS crystal or by a three layer scintillating fiber detector
which closely surrounded the target. For more information on this setup, see~\cite{CREDEeta}.
\section{Data Analysis}
\label{sec:Danalysis}
The unpolarized data were recorded in October 2002 and November 2002. The linearly-polarized data
were recorded in March and May 2003. The polarized data used a diamond radiator optimized to have a
coherent polarization edge at 1350~MeV and 1600~MeV with a polarization maximum of 49\% and 39\%,
respectively. More information on the goniometer and the linear beam polarization can be found in
\cite{Elsner:2008sn}. The ELSA beam energy for the unpolarized runs was 3.175~GeV, but for this analysis,
only photons up to 2.55~GeV were used due to the lack of tagger scintillating fibers above this energy. The
fibers provided a fine energy resolution and additional timing information. The trigger for these datasets
relied on Leading-Edge Discriminator~(LED) outputs which signaled if the energy deposit in a group of TAPS
crystals was above either a low-energy threshold~(LED-low) or a higher-energy threshold~(LED-high). Each
TAPS crystal belonged to one of eight LED-low sectors and one of eight LED-high sectors. The trigger required
either 1) two LED-low sectors in the TAPS detector firing above a low energy threshold or 2) one LED-high sector
in TAPS above a higher energy threshold and at least one hit (two hits for the polarized data) in the Crystal Barrel.
For sector definitions and more information, see~\cite{CREDEeta,Sparks:2010vb}. The same data were used for
several previously published analyses on a variety of final
states~\cite{CREDEeta,Elsner:2008sn,Credepi0,NanovaKstarSigma,Ewald:2011gw,Castelijns:2007qt,Gutz:2009zh,
Sparks:2010vb,Klein:2008aa,Elsner:2007hm}.
In order to study Reaction~(\ref{eqn:reaction}), the $p \pi^0 \gamma$ final state was reconstructed first
using kinematic fitting. All events based on three distinct neutral hits and less than two charged hits were
subjected to the $\gamma p\to p_{\rm missing}\,\pi^0\gamma$ hypothesis. The $\omega$ yields were then
extracted from $\pi^0\gamma$ invariant masses by carefully subtracting the background contribution. The
proton was chosen as a missing particle in the fit due to the relatively large uncertainty in reconstructing the
proton energy and momentum from calorimeter output. The resulting confidence-level (CL) values from kinematically
fitting the data events and Monte Carlo simulated $p\omega$ events were used first to reduce background in the
analysis. A cut of CL$_{p_{\rm missing}\,\pi^0\gamma}\:>\:0.005 \label{eqn:CLcut}$ was applied. This very small CL
cut simply guaranteed convergence of the kinematic fit, and therefore energy-momentum conservation, but
had essentially no impact on the $p\omega$~yield. The remaining $\omega$~background events were removed
by applying a probabilistic method which is described below. More information on kinematic fitting used at the
CBELSA/TAPS experiment can be found in~\cite{CREDEeta}.
To isolate the incoming photon, a coincident timing cut between the tagger and TAPS was used to reduce
the number of initial photon candidates. The remaining photons were subjected to kinematic fitting, which
required energy and momentum conservation. The beam photon with the largest CL value was chosen as the
initial photon. An equivalent analysis of events with timing outside of this coincident timing cut was performed
and used to eliminate the effect of accidental background.
The contribution of $p\pi^0$ events which were poorly reconstructed as $p\pi^0\gamma$~events could
be effectively separated from good $p\omega$~events by studying the momentum-depen\-dence of the
opening angle between the $\pi^0$ and the final-state bachelor-photon in the center-of-mass frame,
$\theta_{c.m.}^{\pi^0,\gamma}$. In this two-dimensional distribution, events which satisfied
$|p^{\omega}_{c.m.}| < (-13.33\; \theta_{c.m.}^{\pi^0,\gamma} + 2400)\;\text{MeV}/c \label{eqn:OpAC}$ were
kept for further analysis. Simulations showed that this cut removed $p\pi^0$ events which caused
structures in the $\pi^0\gamma$ background mass and facilitated the modeling of the background.
When a proton interacted with TAPS, the energy deposit was typically smaller than for a photon of the same
energy. The TAPS crystal trigger thresholds, LED-low and LED-high, were calibrated for photon triggering and
responded much differently for protons. Due to this issue, triggering on low-momentum proton energy deposits
was difficult to model correctly and could have caused errors in the detector acceptance correction. Fortunately,
the kinematics of the reaction allowed this class of events to be excluded from the analysis without losing angular
coverage. All events which had an incident photon energy less than 1600~MeV, a proton reconstructed in TAPS,
and no additional photons triggering in TAPS above the LED-high threshold were cut out of both the data and
the Monte Carlo events. The initial photon energy requirement was chosen to ensure that only low-momentum
protons were discarded. The detector response to high-energy protons was similar to that of photons.
For more details on the $\omega$~event selection, please see~\cite{DissWilson}.
To further remove non-$p\omega$ events (background) from the data, a method was applied which assigned
a quality factor (Q-factor) to each event. This factor describes the probability for an event to be a $p\omega$ event.
The Q-factors were then used to weight each event in the analysis when an observable was formed. This method
is described in more detail in~\cite{Williams:2008sh} and its previous application to the CBELSA/TAPS experiment
in \cite{Sparks:2010vb}. Figure~\ref{fig:Total_inv_mass} shows the resulting separation of signal and background
in the total $\pi^0\gamma$ invariant mass distribution. The number of experimentally observed $p \omega$ data
events was approximately 128,000 in the unpolarized dataset and 60,000 in the polarized combined datasets.
\begin{figure}[bt]
\centering
\label{fig:Total_inv_mass_data}\includegraphics[width=0.49\textwidth]{Fig1}
\caption[Final $\pi^0\gamma$ Invariant Mass distribution]{(Color Online) The invariant $\pi^0\gamma$
mass distribution for unpolarized data events which were subjected to the Q-factor fitting (background
subtraction). These events survived all kinematic cuts. The solid blue area shows the background subtracted
using the Q-factor method. There were approximately 128,000 $\omega$'s in this dataset.}
\label{fig:Total_inv_mass}
\end{figure}
The response of the CBELSA/TAPS experiment was studied with a GEANT3-based Monte Carlo simulation.
The response of the Crystal Barrel and TAPS detectors to photons was well reproduced by the Monte Carlo
simulations. The acceptance for the $p\omega$ events was determined by simulating events which were
evenly distributed over the available kinematic phase space. The detector acceptance is defined as the ratio
of reconstructed Monte Carlo events to generated Monte Carlo events for each kinematic bin. The tagging
and timing of initial state photons were not simulated. The Monte Carlo events were subject to exactly the
same reconstruction as the data events from the experiment. The Q-factor method of background
subtraction was not applied to the Monte Carlo events which are background-free by construction.
The angular distributions of the decay products of the $\omega$~meson were analyzed to obtain more
information about the production mechanisms leading to this final state. For example, the spin-density
matrix elements (SDMEs) are a frame-dependent expression of the $\omega$~meson helicity.
The unpolarized SDMEs were extracted by combining the polarized and unpolarized datasets in a fit. They were
extracted from the data for each kinematic bin by performing an Extended Maximum Likelihood fit to \cite{Zhao:2005vh}
\begin{eqnarray} W^0(\theta_d,\phi_d,\rho^0)& =& \frac{3}{8\pi} \{ \sin^2\theta_d\;\rho^0_{00}\label{Eqn:SDME0} \\[1ex]
& & + \frac{1}{2}\left( 1+\cos^2\theta_d \right)(1-\rho^0_{00}) \nonumber\\[1ex]
& & + \sin^2\theta_d \cos 2\phi_d\,\rho^0_{1-1} \nonumber\\[0.6ex]
& & + \sqrt{2} \sin 2\theta_d \cos\phi_d\,\text{Re}\rho^0_{10} \} \;,\nonumber
\end{eqnarray}
where $\rho^0$ is the unpolarized spin-density matrix, and $\theta_d$ and $\phi_d$ are the polar and the
azimuthal angle of the $\omega$ bachelor-photon measured in the rest frame of the $\omega$ meson.
Acceptance corrections were taken into account in the likelihood function during the fitting. The statistical
uncertainties for these data were studied by using a large number of Monte Carlo datasets with varying numbers
of events and with the same SDME values. The uncertainty in the fit parameters as a function of the number of
events in the dataset was studied and used to define the overall statistical uncertainty for each reported SDME.
All SDMEs in this analysis were extracted in the helicity, Gottfried-Jackson, and Adair frames~\cite{Schilling:1969um}.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.95\textwidth,height=0.35\textheight]{Fig2_final}
\caption[Excitation functions]{(Color Online) Excitation functions for $\gamma p \to p \omega$ from
CBELSA/TAPS ($\bullet$) for selected angle bins. For comparison, the CLAS data~\cite{Williams:2009yj}
are represented by {\tiny \color{blue}$\square$}, the LEPS data \cite{LEPSeta} are represented by {\scriptsize
\color{lepsaw}$\bigstar$} (for the $-0.8$ - $-0.7$ panel), and the Gie\ss en Lagrangian fit in \cite{Shklyar:2004ba}
is represented by {\color{red}\rule[0.5ex]{0.1cm}{1pt} \hspace{0.02mm}\rule[0.5ex]{0.1cm}{1pt}
\hspace{0.02mm}\rule[0.5ex]{0.1cm}{1pt}} . The Bonn-Gatchina PWA solutions are represented by green lines:
{\color{green}\rule[0.5ex]{0.34cm}{1pt}} (full solution),
{\color{green}\rule[0.5ex]{0.1cm}{1pt}\,\hspace{0.02mm}\rule[0.5ex]{0.1cm}{1pt}\,\hspace{0.02mm}\rule[0.5ex]{0.1cm}{1pt}}
(solution without $3/2^+${} partial wave). Statistical uncertainties are reported as vertical bars on each
data point. The total systematic uncertainty for the CBELSA/TAPS data is shown as a black band at the bottom
of each plot. Each plot is labeled with its range in cosine of the polar angle in the center-of-mass frame of the
$\omega$ meson ($\cos \theta^{\,\omega}_{\rm c.m.}$). The horizontal axis is measured in the energy of the initial
photon. An additional horizontal axis at the top of the figure shows the center-of-mass energy.}
\label{fig:Dcross_EF}
\end{figure*}
Polarized SDMEs were extracted from data by combining the two polarized datasets in a fit where each
kinematic bin was subjected to a Maximum Likelihood fit to~\cite{Zhao:2005vh,Schilling:1969um}
\begin{eqnarray} W^L(\Omega_d,\Phi_{\rm pol},\rho) &=& W^0(\Omega_d,\rho)\label{Eqn:SDMEPL}\\
&&+\frac{3}{8\pi} P_\gamma\,[ \cos\Phi_{\rm pol}W_1(\Omega_d,\rho)
\nonumber\\&&+ \sin\Phi_{\rm pol}W_2(\Omega_d,\rho) ] \nonumber \\[2ex]
{\rm with}~~W_1(\Omega_d,\rho) &=& \sin^2\theta_d\,\rho^1_{00} \label{Eqn:SDMEP1} \\
&&+ (1+\cos^2\theta_d)\,\rho^1_{11} \nonumber\\&&+ \sin^2\theta_d\cos 2 \phi_d\,\rho^1_{1-1}\nonumber\\
&&+\sqrt{2}\sin 2 \theta_d\cos \phi_d\,\text{Re}\,\rho^1_{10}\nonumber\\[1ex]
W_2(\Omega_d,\rho) &=& \sin^2\theta_d\sin 2 \phi_d \,\text{Im}\,\rho^2_{1-1}\label{Eqn:SDMEP2}\\
&&+ \sqrt{2}\sin 2\theta_d\sin\phi_d\,\text{Im}\,\rho^2_{10}\nonumber \;,\end{eqnarray}
where $P_\gamma$ is the degree of polarization of the photon beam and $\Phi_{\rm pol} $ is the polarization
angle. Statistical uncertainties were estimated in the same manner as for the unpolarized SDMEs. The values
of the polarized SDMEs were limited in the fit by the restrictions listed in \cite{Schilling:1969um}.
For the differential cross sections, the statistical errors were determined from the number of events in each
$(E_\gamma,\,{\rm cos}\,\theta_{\rm c.m.})$ bin. The systematic uncertainties are given as error bands at the
bottom of each distribution in Figure~\ref{fig:Dcross_EF}; the statistical and systematic uncertainties for
each data point in Figures~\ref{fig:SDME_unpol} and \ref{fig:SDME_pol}~(SDMEs) were added in quadrature.
Sources of the systematic uncertainties include kinematic fitting and a possible target shift away from the
known position in the Monte Carlo simulation. The corresponding uncertainties were determined by applying
different CL cuts as well as varying the target position in the Monte Carlo ($\pm\,1.5$~mm~\cite{CREDEeta})
and evaluating changes in the re-extracted observables. The effects were observed to be $<$\,1.5\,\% and
$<$\,4\,\% on average, respectively. The errors of the decay branching fractions were negligible. Further
contributions include Q-factor fitting errors of $<$\,8\,\% (for details on determining the methodical errors,
see~\cite{Williams:2008sh}), photon flux uncertainties of $<$\,10\,\% for the differential cross sections, and
a 5\,\% degree of polarization uncertainty for the polarized SDMEs. All these errors were then added
quadratically to give the total systematic error.
\section{Experimental Results}
\label{sec:EResults}
The differential cross sections, $d\sigma/d\Omega$, for $\gamma p\to p\omega$ from this analysis are
shown in Figure~\ref{fig:Dcross_EF}~(black dots) for a few selected angle bins. These serve as representative
distributions of the entire dataset. The data are binned in 50~MeV-wide initial photon energy bins from 1.15~GeV
to 2.5~GeV. For $-1.0 < {\rm cos}\,\theta_{\rm c.m.}^{\,\omega} < 0.8$, the angular bin width is 0.1. To show the forward
region's ($0.8 < {\rm cos}\,\theta_{\rm c.m.}^{\,\omega} < 1.0$) strong rise, the bins are 0.033 wide. The vertical error
bars on each point are statistical only and the systematic uncertainties, on the order of 15\,\%, are shown as a black
band at the bottom of each distribution. The CLAS~\cite{Williams:2009yj} and LEPS cross sections \cite{LEPSeta} are
also shown for comparison. The dashed red line represents a calculation within a coupled-channel effective Lagrangian
approach in the energy region from the pion threshold up to 2~GeV (Gie\ss en model)~\cite{Shklyar:2004ba}, which
also includes data on $\pi$-induced reactions. The Gie\ss en group reported a reasonable description of the SAPHIR
data~\cite{SAPHIRw} in their analysis~(\cite{Shklyar:2004ba},~Figure~9 therein). Since the SAPHIR data were available
only as a function of the mandelstam variable~$t$, the Gie\ss en model is shown as a representation of that dataset.
The differential cross sections are reported over the full kinematic phase space, and they can be integrated over
to measure the total cross section without any extrapolation. The total cross section in Figure~\ref{fig:TotCross}
is reported for initial photon energies ranging from 1.15~GeV to 2.5~GeV.
The overall shape of the various experimental distributions is in fair agreement. However, some systematic
differences can be seen, in particular in the very backward direction and at higher energies. The earlier results
from SAPHIR, as represented by the Gie\ss en model, appear to be systematically lower for
${\rm cos}\,\theta_{\rm c.m.}^{\,\omega} < 0.0$ than all other results, whereas an almost linear energy-dependent
normalization discrepancy is observed between CLAS and CBELSA/TAPS. It is worth noting that older SLAC
results~\cite{Ballam:1972eq} at 2.8~GeV exhibit a similar normalization discrepancy with CLAS as do the new
data presented here.
It is important to resolve the normalization discrepancy between the results from CBELSA/TAPS and CLAS for a
reliable extraction of physics contributions to the cross section. A similar normalization discrepancy has been
observed recently between the two experiments for the reaction $\gamma p\to p\eta$~\cite{CREDEeta}. The very
similar behavior of the discrepancies in both reactions hints again at a normalization issue. Significant efforts have
been invested in both collaborations to understand this normalization issue, but the nature of the discrepancy has
remained unclear.
The shape of the $\gamma p\to p\omega$ differential cross sections suggests two dominant processes. The
low-energy cross sections are nearly flat and suggest $s$-channel resonance production. The effect of this
resonance production can be seen at low energies in the total cross section as a resonant peak. In the
higher-energy region, the increase in the cross sections toward ${\rm cos}\,\theta_{\rm c.m.}^{\,\omega}\sim1$
indicates that a $t$-channel or meson exchange process contributes strongly. For $W < 2$~GeV, any $u$-channel
(baryon exchange) contribution must be small due to the lack of any visible peak in the backwards cross section.
\begin{figure}[t]
\centering
\includegraphics[width=0.46\textwidth]{Fig3}
\caption[Spin Density Matrix Total Cross Section]{(Color Online) Total cross cection for $\gamma p\to p\omega$
from the CBELSA/TAPS experiment ($\bullet$) as a function of the initial photon ($E_{\gamma}$) and center-of-mass (W)
energy. The error bars represent the statistical uncertainty and the gray band represents the systematic uncertainty. The
Bonn-Gatchina fit is represented with a solid black line. The largest contributions to this fit are (listed from largest to
smallest at the highest energies) pomeron-exchange~(\mbox{\color{red}\rule[0.15ex]{0.25cm}{2pt}\hspace{1.5mm}\rule[0.15ex]{0.04cm}{2pt}\hspace{1.5mm}\rule[0.15ex]{0.25cm}{2pt}}), resonant production of the $J^p$ = $3/2^+$ partial
wave~(\mbox{\color{blue}\rule[0.15ex]{0.2cm}{2pt}\hspace{0.5mm}\rule[0.15ex]{0.2cm}{2pt}}), $3/2^-$ partial wave~(\mbox{\color{magentaaw}\rule[0.05ex]{0.09cm}{2pt}\hspace{0.65mm}\rule[0.05ex]{0.09cm}{2pt}\hspace{0.65mm}\rule[0.05ex]{0.09cm}{2pt}}), $5/2^+$ partial wave~(\mbox{\color{cyanaw}\rule[0.05ex]{0.12cm}{2pt}\hspace{0.5mm}\rule[0.05ex]{0.025cm}{2pt}\hspace{0.5mm}\rule[0.05ex]{0.025cm}{2pt}\hspace{0.5mm}\rule[0.05ex]{0.12cm}{2pt}}), and $1/2^-$ partial
wave~(\mbox{\color{green}\rule[0.05ex]{0.085cm}{2pt}\hspace{0.65mm}\rule[0.05ex]{0.03cm}{2pt}\hspace{0.65mm}\rule[0.05ex]{0.085cm}{2pt}}).}
\label{fig:TotCross}
\end{figure}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.89\textwidth]{Fig4a}
\includegraphics[width=0.89\textwidth]{Fig4b}
\includegraphics[width=0.89\textwidth]{Fig4c}
\caption[Spin Density Matrix Elements Comparison]{(Color Online) Unpolarized spin density matrix elements
in the Adair frame for selected energy bins. The $\bullet$ are the SDMEs extracted in this analysis with the total
uncertainty for each data point represented as a vertical bar. The {\tiny \color{blue} $\square$} are the SDMEs
published by the CLAS collaboration~\cite{Williams:2009yj}. The Bonn-Gatchina PWA solutions are represented
by green lines: {\color{green}\rule[0.5ex]{0.34cm}{1pt}} (full solution),
{\color{green}\rule[0.5ex]{0.1cm}{1pt}\,\hspace{0.02mm}\rule[0.5ex]{0.1cm}{1pt}\,\hspace{0.02mm}\rule[0.5ex]{0.1cm}{1pt}}
(solution without $3/2^+${}~partial wave). Each plot is labeled with its range in initial photon energy
measured in the lab frame~(center-of-mass (W) energy).}
\label{fig:SDME_unpol}
\end{figure*}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.86\textwidth]{Fig5}
\caption[Spin Density Matrix Elements Comparison]{(Color Online) Polarized spin density matrix elements
in the Adair frame for selected energy bins. The fitted Bonn-Gatchina PWA solution is represented by a
{\color{green}\rule[0.5ex]{0.34cm}{1pt}} . The statistical and systematic uncertainties for each data point
have been added in quadrature and are represented as a vertical bar. Each plot is labeled with its range in
initial photon energy in the lab frame~(center-of-mass (W) energy).}
\label{fig:SDME_pol}
\end{figure*}
Figure~\ref{fig:SDME_unpol} shows representative distributions in the Adair frame, unpolarized SDMEs from this
analysis ($\bullet$) extracted from the $\omega$ meson's radiative decay along with the data published by the
CLAS collaboration ({\tiny \color{blue}$\square$}) in \cite{Williams:2009yj} extracted from the $\omega$ meson's
$3\pi$ decay. These data have 11 $\cos\theta^{\,\omega}_{\rm c.m.}$ angular bins with an energy binning identical to
the differential cross sections already shown. The comparison of these datasets for the SDME $\rho^0_{00}$ shows
very good agreement in contrast to the normalization discrepancy in the differential cross sections. The extraction
of SDMEs does not depend on the absolute normalization. Some differences were found in the SDME
$\text{Re}\:\rho^0_{10}$ and to a smaller extent, mostly at higher energies, in $\rho^0_{1\:-1}$. Since the data agree
in $\rho^0_{00}$, these differences seem to be related to the kinematic variable $\phi_d$ in Equation~\ref{Eqn:SDME0}.
However, little impact on the results of the partial wave analysis (PWA) was observed since the $\text{Re}\:\rho^0_{10}$
and $\rho^0_{1\:-1}$ values are very small.
Four of the six polarized SDMEs measured in the Adair coordinate system are shown in Figure \ref{fig:SDME_pol} for
selected energy bins. These data have four $\cos \theta^{\,\omega}_{\rm c.m.}$ bins and the same $E_\gamma$ binning as
the above unpolarized data. With the exception of $\rho^1_{00}$, the values of the polarized SDMEs above 1.4~GeV in
photon energy extracted in the Adair frame are very small and seem to be consistent with zero within our statistical
uncertainties.
\section{Partial Wave Analysis}
\label{sec:PWA}
Owing to the broad and overlapping nature of baryon resonances, amplitude analyses or PWAs need to be
performed to extract $N^\ast$ parameters from the data. The situation is particularly complicated above the
$\Delta$ region. Many open channels need to be considered and any reliable extraction of resonance properties
must be based on a coupled-channel approach. While several groups have significantly contributed to our
understanding of baryon resonances, a comprehensive PWA based on a larger database of observables has
been performed only at very few institutions. The photoproduction data in most channels still suffer from
an insufficient number of observables with good precision. For this reason, different groups have made
different claims about which resonances are important and in which regions and for which processes.
However, an improved understanding of these different solutions has been clearly observed in recent
years due to an increasing database of high-quality measurements from current facilities. This section
describes the results of a PWA in the framework of the Bonn-Gatchina (BnGa) coupled-channel
approach~\cite{Anisovich:2004zz} that includes the data presented here.
The $\gamma p\to p \omega $ differential cross section has a strong peak in the forward direction; and therefore,
a large contribution from $t$-channel exchange amplitudes is expected at higher energies. An initial fit with
only two $t$-channel amplitudes defined by the reggezied pomeron- and $\pi$-exchange described the
forward peak fairly well but failed to reproduce the entire angular range. This fit returned very small and almost
flat angular distributions for all three unpolarized SDMEs, and it also failed to reproduce the polarization
observables. Contributions from nucleon resonances are a necessary additional ingredient to describe the data.
The new data were added to the BnGa database which already included a large set of data on pion- and photo-induced
meson production reactions, with up to two pseudoscalar mesons in the final state~\cite{Anisovich:2011fc}. In a first
attempt at extracting resonance contributions from this analysis, additional CBELSA/TAPS $\omega$ polarization
observables were included~\cite{Klein:2008aa,Eberhardt:2015lwa} in the PWA and all photo-nucleon and pseudoscalar
meson-nucleon couplings from the previous BnGa solution (BG2014-02)~\cite{Sokhoyan:2015fra} were fixed. Only the
$\omega N$~couplings of the nucleon states and $t$-channel exchange amplitudes were fitted. More than
100 fits were tested for different initial conditions.
The elements of the density matrix are connected to one another by the production amplitudes. The SDMEs, in
particular $\rho^0_{00}$, were therefore essential to describe the contributions from nucleon resonances. Moreover,
the inclusion of the polarized SDMEs allowed the study of the production process in more detail and helped separate
the natural and unnatural parity-exchange contributions. In this analysis, it was possible to distinguish between
pomeron- (natural) and $\pi$-exchange (unnatural), which is a clear advantage over previous approaches.
Figure~\ref{fig:Dcross_EF} shows the description of the differential cross sections; the solid line represents the
best solution. The description of the total cross section and the contribution of the five largest amplitudes for
the best solution are shown in Figure~\ref{fig:TotCross}. The pomeron-exchange amplitude provides the largest
contribution; it dominates the total cross section in the high-mass region. The $\pi$-exchange contributes much
less than the pomeron-exchange. Moreover, its contribution was found to be unstable; it varied in different solutions
from 5\,\% to 30\,\% depending on the behavior of the form factors. The description of the unpolarized SDMEs and
polarized SDMEs is shown in Figure~\ref{fig:SDME_unpol} and Figure~\ref{fig:SDME_pol}, respectively. A discussion
of the CBELSA/TAPS $\omega$ double-polarization observables included in this analysis and their description can
be found in~\cite{Eberhardt:2015lwa}.
At low energies, the $J^P=$ $3/2^+${} is the leading resonant partial wave and shows a strong peak with a
maximum around $W = 1.8$~GeV. Such a behavior is identified with a strong $\omega N$ coupling of the
$N(1720)$\,$3/2^+${} state which is situated just below the reaction threshold. This state decays into the
$\omega N$ channel with orbital angular momentum $L=1$, therefore a large contribution near threshold came as
a surprise. The contributions from the $1/2^-${} and $3/2^-${} partial waves are notably smaller, in
spite of the fact that the $\omega N$~channel can couple to these partial waves with \mbox{$L=0$}. The $3/2^+${}
wave has a more complex structure and hints for at least one more resonance around $W = 1.9$~GeV were found. Fitting
without the $3/2^+${} wave significantly deteriorated the description of the data, particularly of the beam
asymmetry (not shown here) and of the SDME $\rho^0_{00}$. The dashed (green) line in Figures~\ref{fig:Dcross_EF}
and \ref{fig:SDME_unpol} shows the PWA solution without the $3/2^+${} partial wave included.
The contribution from the $3/2^-${} partial wave has two maxima: one at $W = 1.87$~GeV and a second
around $2.1$~GeV. This structure is identified with the contributions from the two $3/2^-${} states,
$N(1875)$ and $N(2120)$. The $1/2^-${} partial wave has a maximum close to the threshold region which is
identified with the tail from the two lowest $1/2^-${} states and a minimum due to the destructive interference
with the $N(1895)$\,$1/2^-${} state. The PWA also found a notable contribution from the $5/2^+${} partial
wave. This wave has some structure close to the threshold and also around $W =2$~GeV; the latter is identified with
the $N(2000)$\,$5/2^+${} state. The contributions from the $5/2^-$, $7/2^+$ and $7/2^-$ partial waves appeared
to be smaller. In all fits, they were found to be less than 5\,\%. However, the $7/2$ partial waves play an important
role in the description of the density matrices at masses above 2.1~GeV. They produced particular structures in the
$\rho^0_{00}$ and $\rho^0_{1-1}$ angular distributions.
A recent CLAS PWA~\cite{Williams:2009aa} also observed a $5/2^+$ partial wave around 2~GeV and a resonance
contribution above 2.1~GeV, which was identified as $7/2^-$. The dominant $t$-channel exchange amplitude
was found to be $\pi$-exchange; $3/2^-$ and $5/2^+$ were reported as the dominant waves at threshold. The
authors also observed a significant $3/2^+$ amplitude but did not claim resonance contributions. This partial
wave appeared to be inconsistent with a single resonance. This is compatible with the BnGa solution. It must be
noted that the CLAS PWA was a single-channel analysis and did not include polarization observables. In the BnGa
PWA, the multichannel approach provided signifcant constraints. The $t$-channel amplitude was identified
by the polarized SDMEs and the additional polarization data~\cite{Klein:2008aa,Eberhardt:2015lwa} were important
to identify clear fit minima.
In utilizing linearly-polarized photons, vector-meson photoproduction is defined by six density matrix
elements $\rho^1_{00}$, $\rho^1_{11}$, $\text{Re}\,\rho^1_{10}$, $\rho^1_{1-1}$, $\text{Im}\,\rho^2_{10}$,
$\text{Im}\,\rho^2_{1-1}$, defined in Equations~(\ref{Eqn:SDMEPL})-(\ref{Eqn:SDMEP2}). For example, the $\omega$
photoproduction beam asymmetry is expressed through density matrices as~\cite{Schilling:1969um}
\begin{eqnarray}
\Sigma_\omega=\rho^1_{00}+2\rho^1_{11}\,.
\end{eqnarray}
Therefore, the $\Sigma$ observable alone only provides one angular distribution in contrast to the full dataset of six
observables mentioned above.
The final fit of the full datasets with free photo- and meson-nucleon couplings did not significantly change the
description of the $\gamma p\to p \omega $ reaction. Therefore, the present $\omega$~photoproduction data are
compatible with previous fits to pseudoscalar meson photoproduction data. However, a good description of the new
high-energy $p\omega$ data required a contribution from at least one new state. The available datasets do not allow to
unambiguously determine the quantum numbers; the hypothesis of a $1/2^-${}, $1/2^+$, $3/2^+$, or $5/2^+$
partial wave with a mass around 2.2~GeV leads to comparable data dscriptions. More polarization data are needed at these
higher energies to define this state.
In the $\cos \theta_{\rm c.m.}^{\,\omega} = [0.967-1.000]$ bin in Figure~\ref{fig:Dcross_EF}, a small discrepancy between
the data and the best solution can be seen. It is assumed that the deviation is caused by the $t$-channel exchange of a
higher-mass meson. It should not greatly affect the identification of contributing resonances.
\section{Summary}
\label{sec:Summary}
The $\gamma p \to p \omega$ differential cross sections and spin-density matrix elements detected in the $\omega$
meson's radiative decay measured at the CBELSA/TAPS experiment have been presented. The experimentally measured
events were obtained by irradiating a liquid hydrogen target with tagged photons ranging in energy from threshold up to
2.5 GeV. The measured cross sections presented here seem to be systematically higher than some previous measurements.
This indicates a normalization discrepancy in particular between CBELSA/TAPS and CLAS. The spin-density matrix element,
$\rho^0_{00}$, agrees well with the CLAS measurement and further suggests that the cross section discrepancy is related to
an unknown issue with the absolute normalization, either at CLAS or CBELSA/TAPS.
The BnGa PWA solution indicated that the dominant contributions to the cross section near threshold were the
sub-threshold $N(1720)$\,$3/2^+${} resonance as well as the $3/2^-${} and $5/2^+${} partial
waves. Toward higher energies, the $t$-channel contributions increased in strength. They were defined by a dominant
pomeron-exchange and a smaller $\pi$-exchange. In addition to the $t$-channel amplitude, further contributions from
nucleon resonances were required to describe the data. The $1/2^-${}, $3/2^-${}, and $5/2^+${} partial
waves showed significant contributions to the PWA solution. The $1/2^-${} wave was defined by two sub-threshold
resonances interfering destructively with the $N(1895)$\,$1/2^-${} resonance. The $N(1875)$ and $N(2120)$
resonances created the two-peak structure seen in the $1/2^-${} partial wave in Figure~\ref{fig:TotCross}. The
$5/2^+${} partial wave had sub-threshold contributions along with the $N(2000)$ resonance. In addition, at least
one previously unseen higher-mass resonance above 2~GeV was needed to describe the data.
\subsection*{Acknowledgements}
We thank the technical staff at ELSA and at all the participating institutions for their contributions to the success of this
experiment and this analysis. Financial support is acknowledged from the Deutsche Forschungsgemeinschaft (DFG) within
the SFB/TR16 and from the Schweizerischer Nationalfonds. The presented results are partly based upon work supported by
the U.S. National Science Foundation under contract 0456463 and 0754674.
|
1,941,325,220,569 | arxiv | \section{Introduction}
Observations have suggested that the radio-loud active galactic nuclei (AGNs) can play an important role in feedback, and thus have a significant impact on galaxy evolution, with AGN outflows being responsible for controlling or terminating star formation
\citep[SF; e.g.][]{2009Natur.460..213C,2012ARA&A..50..455F,2014MNRAS.445..955B,2014ARA&A..52..589H}. To understand the significance of such AGN feedback processes and their influence on the global SF history of the Universe \citep{2013MNRAS.436.1084M}, it is important to first understand the evolution of the radio luminosity function (RLF) to high redshift.
In the study of RLF, an increasingly acceptable viewpoint is that the powerful radio AGNs undergo very rapid evolution to a redshift of $z\sim 3$ , while the lower luminosity radio population only experiences much milder positive evolution to $z\sim 1$ \citep[e.g.][]{2001MNRAS.328..882W,2004MNRAS.352..909C,2011MNRAS.416.1900R,2013MNRAS.436.1084M, 2015A&A...581A..96R}. The different evolution for high- and low-luminosity radio-loud AGNs may reminds one to regard them as two essentially different populations. However, the simple division based on the radio luminosity may not reveal the physical essence. Over recent years, it has become clear that a more essential classification of radio-loud AGNs is based on the AGN accretion state. Radio sources powered by radiatively efficient accretion of cold gas onto a geometrically thin, optically thick accretion disk are referred to as high-excitation radio galaxies (HERGs), or `quasar-mode' accretors \citep[][]{2012MNRAS.421.1569B,2013MNRAS.430.3086G}. While those powered by the radiatively inefficient accretion of hot gas \citep{2007MNRAS.376.1849H,2014ARA&A..52..529Y} onto a geometrically thick accretion disk are referred to as low-excitation radio galaxies (LERGs), or `radio-model' accretors. Quasar-mode and radio-model AGNs dominate the radio-AGN population at higher and lower radio luminosities respectively \citep{2014MNRAS.445..955B}. However, this does not mean one can separate the two radio populations by a simple division in radio luminosity. As shown by \citet{2012MNRAS.421.1569B}, both quasar-mode and radio-mode radio AGNs are found across all radio luminosities.
In the past two decades, research on the RLF of radio-loud AGNs has been increasingly abundant\citep[e.g.][]{2001MNRAS.328..882W,2004MNRAS.352..909C,2009ApJ...696...24S,2009MNRAS.392..617D,2013MNRAS.436.1084M,2015A&A...581A..96R,
2015MNRAS.452.2692F}. However, most of these works are non-parametric RLFs derived by the classical binned $1/V_{max}$ method\citep[see][]{1968ApJ...151..393S,2013Ap&SS.345..305Y}. The two most representative papers \citep{1990MNRAS.247...19D,2001MNRAS.322..536W} on the parametric analysis of RLFs were published more than a decade ago. The above authors artificially divided the RLFs to be the sum of two components, i.e., a high-luminosity evolving component $\rho_h$, and a low-luminosity non-evolving/mild-evolving component $\rho_l$. Then they considered $\rho=\rho_l+\rho_h$ and modeled $\rho$ using pure density evolution (PDE) or pure luminosity evolution (PLE) models. In this work, we do not artificially separate the low-luminosity and high-luminosity populations into two evolving components. We use mixture evolution models, which consider a combination of density and luminosity evolution, to parameterize the steep-spectrum AGN RLFs. For the parameterizing procedure, a Bayesian method is used.
Throughout the paper, we adopt a Lambda Cold Dark Matter cosmology with the parameters $\Omega_{m}$ = 0.27, $\Omega_{\Lambda}$ = 0.73, and $H_{0}$ = 71 km s$^{-1}$ Mpc$^{-1}$.
\section{The sample}
The data we used in this work is the combined sample established by \citet{2012ApJ...744...84Y}. It consists of four sub-samples: the MRC1 \citep{1996ApJS..107...19M}, the MS4 \citep{2006AJ....131..100B}, the BRL \citep{1999MNRAS.310..223B} and the 3CRR \citep{1983MNRAS.204..151L} samples. The sources (totaling 1063) in our sample are all steep-spectrum ones, which mainly consist of radio galaxies (RGs) and steep-spectrum quasars. All of the sources have flux density data observed at 408 MHz. At this relatively lower frequency, the radio emission is mainly from the extended lobes of the radio-loud AGNs, minimizing the effect of orientation bias. For more details about the combined sample, please refer to \citet{2012ApJ...744...84Y}.
\section{Methods}
\subsection{The Likelihood function for the RLF}
The RLF $\rho(z,L)$ is defined as the number of sources per comoving volume $V(z)$ with radio luminosities in the range of $L,L+dL$. It is related to the probability distribution of $(z,L)$ by
\begin{eqnarray}
\label{eqpc}
p(z,L)=\frac{1}{N}\rho(z,L)\frac{dV}{dz}.
\end{eqnarray}
where $N$ is the total number of sources in the universe, and is given by the integral of $\rho$ over $L$ and $V(z)$ \citep{2008ApJ...682..874K}. The likelihood function for the observed data $p(L_{obs},z_{obs}|\theta)$can be derived, once we assume a parametric form for $\rho(z,L)$, with parameters $\theta$. \citet{1983ApJ...269...35M} give a likelihood function based on the Poisson distribution and define $S=-2ln(p(L_{obs},z_{obs}|\theta))$. Dropping terms independent of the model parameters, one finds
\begin{eqnarray}
\label{likelihood1}
\begin{aligned}
S=-2\sum_{i}^{N_{obs}}&ln[\rho(z_{i},L_{i})]+\\
&2\int\int\rho(z,L)\Omega(z,L)\frac{dV}{dz}dzdL.
\end{aligned}
\end{eqnarray}
Usually, best estimates for the model parameters are obtained by minimizing $S$. In this work, we use a Bayesian method as described in section \ref{Bayes} to obtain not only the best estimates for the model parameters, but also their probability distribution.
The limits of the integral in $S$ should consider our combined sample, which consists of four sub-samples. Then we have
\begin{eqnarray}
\label{likelihood2}
\begin{aligned}
S=-2\sum_{i}^{N_{obs}}&ln[\rho(z_{i},L_{i})]+\\
&\sum_{j}^{4}\Omega^j\int^{z_2^j}_{z_1^j}dz\frac{dV}{dz}\int_{max[L_1^j,L_{min}^j(z)]}^{L_2^j}\rho(z,L)dL,
\end{aligned}
\end{eqnarray}
where $(z_1^j,z_2^j)$ and $(L_1^j,L_2^j)$ are the redshift and luminosity limits of the $j$th sub-sample, respectively, $L_{min}^j(z)$ is the luminosity limit corresponding to the flux density limit, and $\Omega^j$ is the solid angle subtended by the sub-sample j.
Thus, we have given the general form of the RLF likelihood function. It can be used to perform Bayesian inference by combining with a prior distribution.
\begin{figure*}[!htb]
\centering
\includegraphics[width=1.02\columnwidth]{f1a.eps}
\includegraphics[width=1.02\columnwidth]{f1b.eps}
\includegraphics[width=1.02\columnwidth]{f1c.eps}
\includegraphics[width=1.02\columnwidth]{f1d.eps}
\includegraphics[width=1.02\columnwidth]{f1e.eps}
\includegraphics[width=1.02\columnwidth]{f1f.eps}
\caption{Left: 1-D probability distribution of the parameters in the models \citep[the red dash dot curves are the mean likelihoods of MCMC samples and the black solid curves are the marginalized probabilities, see][]{2002PhRvD..66j3511L}; Right: 2-D confidence contours of the parameters. The contours are for 1 and 2 $\sigma$ levels. For the 2D confidence regions of the parameters, we only show combinations with relatively large correlation. The upper, middle and lower panels are for model A, C and D respectively.}
\label{fig:mcmc}
\end{figure*}
\begin{figure}
\centerline{
\includegraphics[scale=0.40,angle=0]{f2.eps}}
\caption[R-z relation]{\label{f3} The RLFs derived for model A (top panel), B (middle panel) and C (bottom panel) at z=0.1,0.55,1,2,3 and 4 (red, blue, green, violet, black solid lines and red dashed lines respectively). The RLFs at z=4 are marked as red dashed lines to clarify whether a redshift cut-off occurs or not at z$>$3. The red and gray lines with error bars show the local RLFs of \citet{2012MNRAS.421.1569B} and \citet{2007MNRAS.375..931M} respectively. The blue lines with error bars show the RLF at $\langle z \rangle$ =0.55 derived by \citet{2009MNRAS.392..617D}. These RLFs have been converted to the same frequency and cosmology used by our work.}
\label{RLF}
\end{figure}
\subsection{Bayesian inference for the RLF parameters}
\label{Bayes}
The purpose of Bayesian analysis is to estimate the probability distribution of the model parameters (called as the posterior distribution), given by the observed data. It is
\begin{eqnarray}
\begin{aligned}
p(\theta|L_{obs},z_{obs})\varpropto p(\theta)p(L_{obs},z_{obs}|\theta),\\
&
\end{aligned}
\end{eqnarray}
where $p(\theta)$ is the prior distribution of $(\theta)$, which should convey information known prior to the analysis, and $p(L_{t,obs},z_{obs},I|\theta)$ is the observed data likelihood function defined in Equations \ref{likelihood1} and \ref{likelihood2}.
In this work, we assume a uniform prior on $\theta$. This is equivalent to setting a value range for it.
The posterior distribution of $\theta$ given above can be used to perform Bayesian inference. This can be achieved by drawing values of $\theta$ from its posterior distribution using a Markov chain Monte Carlo (MCMC) sampling algorithm. The MCMC sampler here used is the public code ``COSMOMC", which was presented by \citet{2002PhRvD..66j3511L}. See \citet{2003itil.book.....M} for more details about the MCMC method.
\subsection{The form of the RLF}
For the forms of the RLFs, pure density evolution (PDE) models \citep[e.g.,][]{2001MNRAS.322..536W} and pure luminosity evolution (PLE) models \citep[][]{1990MNRAS.247...19D} have been adopted in the literature. To achieve a good fitting, the above authors have to consider the RLFs to be the sum of two components, i.e., a high-power evolving component $\rho_h$, and a low-power non-evolving/mild-evolving component $\rho_l$. We use the mixture evolution models first introduced by \citet{1988ApJ...325..103H} to model the RLF. The mixture evolution models consider a combination of density and luminosity evolution to parameterize the RLF. Mathematically, suppose the evolution of RLF is a vector $\vec{E}$, then it can be written as
\begin{eqnarray}
\label{vector}
\vec{E}=e_1\vec{E}_d+e_2\vec{E}_l,
\end{eqnarray}
where $\vec{E}_d$ and $\vec{E}_l$ are the base vectors of density and luminosity evolution respectively. $e_1$ and $e_2$ are the coefficients, which are usually the functions of redshift. The physical meaning of density evolution is whether the sources are more/less numerous than that of today, while the luminosity evolution represents whether the sources are more/less luminous than that of today. Thus the advantage of mixture evolution models is that they can discriminate the above two evolutions with different physical meaning.
Our mixture evolution model RLF can be written as the following general form.
\begin{eqnarray}
\label{eqn:rhoform}
\rho(z,L_{t})=e_1(z)\rho(z=0,L/e_2(z)),
\end{eqnarray}
where $e_1(z)$ and $e_2(z)$ are the functions describing the density and the luminosity evolution respectively.
The density evolution function $e_1(z)$ we used is very similar to the one adopted by \citet{2008ApJ...674..111C},
\begin{eqnarray}\label{eqn:LFPLEnoev}
e_1(z) =
\begin{cases}
\displaystyle z^m\exp\left[-\frac{1}{2}\left(\frac{z-z_0}{z_{\sigma}}\right)^2\right], & 0<z \leqslant z_0 \\
z^m, & z>z_0
\end{cases}
\end{eqnarray}
where $m$, $z_0$, and $z_{\sigma}$ are free parameters of the model.
The local luminosity function $\rho(z=0,L/e_2(z=0))$ is a double power-law form described as
\begin{eqnarray}
\label{eqn:rhot}
\begin{aligned}
\rho(z=0,L/e_2(z&=0))=\frac{dN}{d\log L} \\
&=\phi_0\left[\left(\frac{L}{L_*}\right)^{\alpha} + \left(\frac{L}{L_*}\right)^{\beta}\right]^{-1},
\end{aligned}
\end{eqnarray}
where $\phi_0$, $\alpha$, $L_*$ and $\beta$ are free parameters of the model. The luminosity evolution function $e_2(z)$ has three different forms depending upon the model: the traditional power-law form
\begin{eqnarray}
e_2(z)=(1+z)^{k_1}
\end{eqnarray}
for model A, the modified power-law form used by \citet{2012ApJ...751..108A}
\begin{eqnarray}
e_2(z)=(1+z)^{k_1}\mathrm{exp}(z/k_2)
\end{eqnarray}
for model B, and the polynomial form used by \citet{1990MNRAS.247...19D} and \citet{2000MNRAS.317.1014B}
\begin{eqnarray}
\label{eqn:flc}
e_2(z)=10^{k_1z+k_2z^2}
\end{eqnarray}
for model C.
\section{results}
\subsection[]{The modeled RLFs}
In Figure \ref{fig:mcmc}, we show the one-dimensional (1D) probability distributions \citep[the dashed curves are the mean likelihoods of MCMC samples and the solid curves are the marginalized probabilities, see][]{2002PhRvD..66j3511L,2015MNRAS.454.1310Y} and two-dimensional (2D) confidence regions (at 1$\sigma$ and 2$\sigma$ confidence levels) of the model parameters. The upper, middle, and lower panels are for models A, B, and C respectively. The best-fit model parameters correspond to the peak of the 1D probability distributions (the marginalized probabilities, the solid curves in Figure \ref{fig:mcmc}). For the 2D confidence regions of the parameters, we only show the combinations with relatively large correlations \citep[e.g.,][]{2011ApJ...735..120Y}. The values of the best-fitting parameters and their 1 $\sigma$ error are reported in Table \ref{tab:fit}.
\begin{table*}
\tablewidth{0pt}
\renewcommand{\arraystretch}{1.8}
\caption[]{Best-fit Parameters}
\begin{center}
\begin{tabular}{lccccccccc}
\hline\hline
\colhead{Model} & \colhead{$\log_{10}\phi_0$} &
\colhead{$\log_{10}L_*$} & \colhead{$\alpha$} &
\colhead{$\beta$} & \colhead{m} &
\colhead{z$_{\rm 0}$} & \colhead{z$_{\rm \sigma}$} &
\colhead{k$_1$} & \colhead{k$_2$} \\
\hline
A & -5.278$_{-0.280}^{+0.151}$ & 25.82$_{-0.09}^{+0.18}$ & 1.76$_{-0.03}^{+0.03}$ & 0.54$_{-0.07}^{+ 0.12}$ & -0.67$_{-0.11}^{+0.13}$ & 0.974$_{-0.041}^{+0.080}$ & 0.351$_{-0.016}^{+0.038}$ & 2.434$_{-0.185}^{+0.148}$ & - \\
B & $-5.836_{-0.191}^{+0.367}$ & $25.87_{-0.13}^{+0.15}$ & $1.79_{-0.03}^{+0.04}$ & $0.56_{-0.09}^{+0.10}$ & $-0.72_{-0.09}^{+0.15}$ & $0.876_{-0.041}^{+0.093}$ & $0.365_{-0.027}^{+0.037}$ & $4.321_{-1.068}^{+0.361}$ & $-1.496_{-1.953}^{+0.259}$\\
C & $-5.308_{-0.182}^{+0.286}$ & $25.86_{-0.13}^{+0.13}$ & $1.76_{-0.03}^{+0.03}$ & $0.53_{-0.08}^{+ 0.11}$ & $-0.73_{-0.07}^{+0.17}$ & $0.915_{-0.041}^{+0.076}$ & $0.320_{-0.015}^{+0.032}$ & $-0.107_{-0.021}^{+0.025}$ & $0.796_{-0.119}^{+0.072}$\\
\hline
\end{tabular}
\end{center}
~~Units -- $\phi_0$: [${\rm Mpc^{-3}}$],\,\, $L_*$: [${\rm W Hz^{-1}}$]. The best-fitting parameters as well as their 1 $\sigma$ errors for model A, B and C.
\label{tab:fit}
\end{table*}
\begin{table}
\renewcommand{\arraystretch}{1.0}
\caption[]{Goodness-of-fit}
\begin{center}
\begin{tabular}{lccc}
\hline\hline
\colhead{Model} & \colhead{1D-$P_{\mathrm{KS}}$-$L$} & \colhead{1D-$P_{\mathrm{KS}}$-$z$} & \colhead{2D-$P_{\mathrm{KS}}$} \\
\hline
A &0.58 & 0.99 & 0.34 \\
B &0.67 & 0.99 & 0.39 \\
C &0.56 & 0.99 & 0.33 \\
\hline
\end{tabular}
\end{center}
\label{tab:ks}
The goodness-of-fit of the models. These are the probabilities from the 1D (for $z$ and $L$) and 2D KS tests to the $L_{408}-z$ distribution.
\end{table}
In Figure \ref{RLF}, we show the best-fit RLFs of models A, B, and C at $z=0.1, 0.55, 1, 2, 3$ and $4$. All three models show luminosity-dependent evolution where the low-luminosity radio sources experience weaker evolution than their bright counterparts. The faint sources experience mild positive evolution out to $z \thicksim$ 1.0. Beyond this redshift, this evolution slows down. However, the bright sources evolve strongly to $z > 3.0$. The above pictures are in good agreement with previous studies, showing that our mixture evolution models are effective.
Figure \ref{RLF} shows that our three models present very similar results for the bright end RLFs, while they give slightly different results for the faint end RLFs. Model B gives a milder evolution for the faint sources than that given by Models A and C. We then perform the 1D and 2D Kolmogorov-Smirnov (KS) tests \citep{1983MNRAS.202..615P,1987MNRAS.225..155F,1992nrfa.book.....P} to estimate the goodness-of-fit of models. The goodness-of-fit (KS test probabilities) of the models are reported in Table \ref{tab:ks}. All the three models are acceptable because they satisfy the condition of $P_{\mathrm{KS}}\geqslant0.2$ \citep{1983MNRAS.202..615P}. The KS tests show that model B is marginally favored over models A and C.
\subsection[]{Comparing with local RLFs}
Up to now, multi-wavelength observed data of the radio-loud AGNs has become abundant. By combining different surveys, such as the Sloan Digital Sky Survey (SDSS), the NRAO (National Radio Astronomy Observatory) VLA (Very Large Array) Sky Survey (NVSS) and the Faint Images of the Radio Sky at Twenty Centimetres (FIRST) survey, one is able to construct large samples of radio-loud AGNs down to a flux limit of few mJy. Based on these large and deep samples, the local \citep{2007MNRAS.375..931M,2012MNRAS.421.1569B} and low ($z\thicksim0.55$) redshift \citep{2009MNRAS.392..617D} RLFs are well determined. Figure \ref{RLF} shows the comparison among these results with our model RLFs. Notably, all of these previous RLFs are calculated at 1.4 GHz. We have shifted luminosities from 1.4 GHz to 408 MHz according to a power law ($L/\nu \propto \nu^{-\alpha}$, with $\alpha=0.70$). Besides, we also need to convert their results to the one in the same cosmology used by our work. This can be achieved by using the following relation from \citet{1985MNRAS.217..601P},
\begin{eqnarray}
\label{convert}
\rho_1(z,L_1)\frac{dV_1}{dz}=\rho_2(z,L_2)\frac{dV_2}{dz}.
\end{eqnarray}
Figure \ref{RLF} shows both the local and $z=0.55$ RLFs are in good agreement with previous determinations, indicating that our mixture evolution model is successful.
\begin{figure}[!htb]
\centering
\includegraphics[width=1.02\columnwidth]{f3a.eps}
\includegraphics[width=1.02\columnwidth]{f3b.eps}
\caption{Model B and Model C RLFs as function of redshift, having been converted to the same frequency used by \citet{2015A&A...581A..96R}. The black, red, green, blue and violet dashed lines show the RLFs at $\log_{10}L_{1.4\mathrm{GHz}}$=24.5, 25.5, 26.5, 27.5 and 28.5 respectively. The light shaded areas take into account the 1 $\sigma$ error bands. The black, red, green and blue solid lines show the result of \citet{2015A&A...581A..96R}.}
\label{luminosity-dependence}
\end{figure}
\begin{figure}
\centerline{
\includegraphics[scale=0.45,angle=0]{f4.eps}}
\caption[A example LF]{\label{f2} Density and luminosity evolution functions for Models A (black solid lines), B (red solid lines) and C (green solid lines).}
\label{e1e2}
\end{figure}
\subsection[]{Luminosity-dependence of the high-redshift turnover}
In recent years, evidence is growing that the position of the steep-spectrum RLF peak is luminosity dependent, being interpreted as a sign of cosmic downsizing. Recent evidence is given by \citet{2015A&A...581A..96R}, showing that the space density of the most powerful sources peaks at higher redshift than that of their weaker counterparts. Figure \ref{luminosity-dependence} shows the space density changing with redshift for our model B and C at various luminosities. For the convenience in comparing with the result of \citet{2015A&A...581A..96R} (their Figure 4), we have shifted the luminosities to 1.4 GHz by assuming a spectral index of 0.70. It shows that our models are in broad agreement with their result. We also notice that a space density peak is more clearly shown at high radio luminosities than that at low radio luminosities. Similar phenomenon is also noticed by \citet{2015A&A...581A..96R}, who find that at $P_{1.4 \mathrm{GHz}} > 10^{26}$ W/Hz the redshift of the peak space density increases with luminosity, whilst at lower radio luminosities the position of the peak remains constant within the uncertainties.
\subsection[]{The Redshift Cut-off}
The redshift cut-off, involving whether the comoving number density of radio AGNs declines dramatically at the redshift beyond z $\sim$ 2.5, is a unsolved problem in the research of RLF. The RLFs of flat-spectrum radio sources support the existence of the redshift cut-off \citep[e.g., ][]{1990MNRAS.247...19D,1996Natur.384..439S,2000MNRAS.319..121J,2005A&A...434..133W}, while the redshift cut-off in the steep-spectrum radio sources has much controversy. \citet{1990MNRAS.247...19D} first proposed a red-shift cutoff for steep-spectrum radio sources, but several authors \citep[e.g., ][]{2001MNRAS.327..907J,2007MNRAS.375.1349C} subsequently argued that there is no compelling evidence of a high-redshift decline in steep spectrum radio sources.
Considering the existence of peaks for density and luminosity evolution (see Figure \ref{e1e2}), both model B and C in principle enable the occurrence of redshift cut-off. However, the evidence for a redshift cut-off does not appear to be significant given the uncertainties on the space density evolution shown in Figure \ref{luminosity-dependence} for Model B. In comparison, Model C support a more significant evidence for a redshift cut-off. Nevertheless, we can notice that the turnover redshift is at least greater than 4, which is remarkably bigger than previous determinations \citep[$z \sim$ 2.5, e.g., ][]{1990MNRAS.247...19D}. To sum up, we believe that the existence of redshift cut-off still depends on models. Given that the only model (C) which supports a significant existence of redshift cut-off is not overwhelmingly favoured over the others, we conclude that based on the current sample we do not find compelling evidence for a redshift cut-off for steep-spectrum radio sources. In the future, samples based on deeper and fainter radio surveys are expected to provide more significant constraint on the density evolution, especially the luminosity evolution of RLFs.
\begin{figure}
\centerline{
\includegraphics[scale=0.40,angle=0]{f5.eps}}
\caption[A example LF]{\label{f2} Example of LF shows that the flatter an LF is, the more sensitive is it to the density evolution, whereas the steeper an LF is, the more sensitive is it to the luminosity evolution.}
\label{example_RLF}
\end{figure}
\section[]{Discussion}
The main point of this work is that we use mixture evolution to model the RLFs. In this scenario, the shape of RLFs is determined by both the density and luminosity evolution. Our result (see Figure \ref{e1e2}) indicates that the positive density evolution continues to
a redshift of $\thicksim 0.9$ and then changes its sign (positive to negative), while the positive luminosity evolution continues
to a higher redshift ($\thicksim 5$ for model B and $\thicksim 3.5$ for model C) and then changes its sign. Usually, the flatter a LF is,
the more sensitive is it to the density evolution, whereas the steeper a LF is, the more sensitive is it to the luminosity evolution (see Figure \ref{example_RLF}). For a double power-law RLF, it is generally flatter at low luminosity and steeper at high luminosity, hence the turnover redshift of evolution for low-luminosity sources must be lower than that of high-luminosity sources,
displaying a luminosity-dependent evolution.
Intuitively understanding, the physical meaning of density evolution is whether the sources are more/less numerous than that of today, while the luminosity evolution represents whether the sources are more/less luminous than that of today. But the more profound physical significance behind them is still unknown. We speculate that the density evolution is associated with the density distribution of accreting black holes, while the luminosity evolution is related with the changing of accretion state. This will be the subject of a future work.
\section[]{Conclusions}
The main results of this work are as follows.
\begin{enumerate}
\item We propose a mixture evolution scenario to model the steep-spectrum AGN RLF based on a Bayesian method. In this scenario, the shape of the RLF is determined together by the density and luminosity evolution. Our models indicate that the positive density evolution continues to a redshift of $\thicksim 0.9$ and then changes its sign (positive to negative), while the positive luminosity evolution continues to a higher redshift ($\thicksim 5$ for model B and $\thicksim 3.5$ for model C) and then changes its sign.
\item The mixture evolution scenario can naturally explain the luminosity-dependent evolution of the RLF. In agreement with previous results, we find that a space density peak is more obviously shown at high radio luminosities than that at low radio luminosities.
\item The existence of redshift cut-off still depends on models. Given that the only model (C) that supports a significant existence of redshift cut-off is not overwhelmingly favored over the others, we conclude that based on the current sample we do not find compelling evidence for a redshift cut-off for steep-spectrum radio sources. Even if the redshift cut-off indeed exists, we believe the turnover redshift should be greater than previous determinations.
\end{enumerate}
\section*{Acknowledgments}
We are grateful to the referee for useful comments that improved this paper. We acknowledge the financial supports from the National Natural Science Foundation of China 11133006, 11163006, 11173054,11573060, the Policy Research Program of Chinese Academy of Sciences (KJCX2-YW-T24) and CAS "Light of West China" Program. J.M. is supported by the Hundred-Talent Program of Chinese Academy of Sciences and the Key Research Program of Chinese Academy of Sciences (grant No. KJZD-EW-M06). The authors gratefully acknowledge the computing time granted by the Yunnan Observatories, and provided on the facilities at the Yunnan Observatories Supercomputing Platform.
|
1,941,325,220,570 | arxiv | \section{Introduction}\label{introduction}
Situations where observation data are generated by different mechanisms in different contexts appear frequently in various scientific phenomena. Because of this reason, supervised machine learning models for predicting physical properties of materials still get frequently outmatched by empirical models created by human researchers. Finding a model that qualitatively determines the mixture effect is a constantly demanded task in both theoretical and experimental model construction. Use of unsupervised learning techniques, with the ability to screen predefined correlations at different data scales, can be a promising approach \cite{LeCun15}. Conventional unsupervised learning techniques for unveiling of mixture models in descriptive space are implemented using clustering methods \cite{Xu2015} such as the Gaussian mixture model, hierarchical clustering, K-means clustering, etc. Besides, the unveiling of mixture models using supervised information, which considers functions connecting the descriptive space and target space as center objects, has not gained considerable attention from the machine-learning community. One of the well-known methods in this research direction is the mixture of experts model \cite{Jacobs91, Yuksel12}, which learns the gating functions to appropriately partition the descriptive space for identifying the components of mixture models. Further, linear regression-based clustering was recently developed \cite{Eto2014FullyAutomaticBP, NIPS2013_5171, Nguyen18} without partitioning the descriptive space. However, these models are sensitive to parameters setting, including the number of clusters, complexity of the learners (linear model), etc.
In this study, we propose a data-driven method to unveil the mixture of information on the mechanism of physical properties of materials by using nonlinear supervised learning techniques. We pay attention to extracting dissimilarity between materials, with respect to a given target physical property. The method is based on an ensemble method with Kernel ridge regression as the predicting model. We apply a bagging algorithm to carry out random subset samplings of the materials for generating multiple prediction models. The distribution of the predicted values for each material is then approximated by a Gaussian mixture model. Further, the contributions of the reference training materials to each of the corresponding models are investigated in detail. Reference training materials that are avoided and do not contribute to a predictive model, which accurately predicts the physical properties of a particular material, are considered dissimilar to that material.
This paper is organized as follows: The components of the algorithm are described in Section \ref{Methodology}, including the base learning model in Section \ref{KRR}, the bagging algorithm in Section \ref{EnsembleLearning}, and the dissimilarity voting machine in Section \ref{VotingMachine}. Data preprocessing is elucidated in Section \ref{DataPreprocess}, and the results and discussions are presented in Section \ref{ResultDiscuss}.
\section{Methodology}\label{Methodology}
We consider a dataset $\mathcal{D}$ of $p$ materials. Assume that a material with index $i$ is described by an $m$-dimensional predictor variable vector, ${\bm{x}}_{i} = {\left( {x_i^1}, {x_i^2}, \dots , {x_i^m} \right)} \in {\mathbb{R}}^m$. The dataset $\mathcal{D} = \left\{ ({\bm x}_1, y_1), ({\bm x}_2, y_2) \dots ({\bm x}_p, y_p)\right\} $ is then represented using a $\left( p \times \left( m+1 \right) \right)$ matrix. The target physical property values of all materials in the dataset are stored as a $p$-dimensional target vector $ \bm{y} = \left( y_1, y_2 \dots y_p \right) \in \mathbb{R}^p$.
\subsection{Kernel ridge regression}\label{KRR}
To learn a regression function $\hat{f}$ for predicting the target variable, we utilize the kernel ridge regression (KRR) technique \cite{ML}, which has been recently applied successfully in several materials science studies \cite{Rupp_tutorial, Botu, Pilania}. We use the KRR with Laplacian kernel function as follows:
\begin{equation}
k(\bm{x}_{i}, \bm{x}_{j}) = \exp (\frac{-|\bm{x}_{i}-\bm{x}_{j}|}{\sigma})
\end{equation}
where $|\bm{x}_{i} - \bm{x}_{j}| = \sum_{a=1}^{m}|x_{i}^{a}-x_{j}^{a}|$ and $\sigma$ is the tuning variance parameter of the Laplacian kernel.
For a given new material $\bm{x}_*$, the predicted property $\hat{f}(\bm{x}_*)$ is expressed as the weighted sum of the kernel functions:
\begin{equation}
\hat{f}(\bm{x}_*) = \sum_{i=1}^N c_i k(\bm{x}_*, \bm{x}_{i})
\label{eq.fx}
\end{equation}
where $N$ is the number of training materials. The weighting coefficients $c_i$ for the corresponding materials $\bm{x}_{i}$ are determined by minimizing
\begin{equation}
\sum_{i=1}^N [\hat{f}(\bm{x}_i) - {\bm y}_i]^2 + \lambda\sum_{i=1}^{N} ||c_i||_2^2.
\label{eq.EL}
\end{equation}
The regularization parameter $\lambda$ and the hyper parameter $\sigma$ are selected by cross-validation\cite{Stone74, Picard84}, i.e., by excluding some of the materials during the training process, and maximizing the prediction accuracy for those excluded materials. We consider the component $c_i k(\bm{x}_*, \bm{x}_{i})$ in Eq. \ref{eq.fx} as the contribution of the training material $\bm{x}_{i}$ to the prediction model $\hat{f}$ and if it is not refered in the training set, $c_i$ is set to zero.
\subsection{Testing for homogeneity of dataset with ensemble learning}\label{EnsembleLearning}
Ensemble learning \cite{Hastie09, Dietterich2000} is a method in machine learning where multiple learners are trained to solve the same problem, which was initially developed to avoid over fitting for our designed base learner \cite{zhou2012}. There are various strategies, such as bagging, boosting, and stacking, for different learning purposes.
In this study, we intend to unveil the mixture of information in prediction model space, which is a linear combination of kernel functions constructed by training materials. Applying the bagging algorithm\cite{BALDI201478}, we carry out random subset samplings of the materials dataset to generate multiple prediction models. For each sampling, we prepare two datasets: bagging dataset, $\mathcal{D}_{bagg}$, and testing dataset, $\mathcal{D}_{test}$. These two datasets satisfy $\mathcal{D}_{bagg} \cap \mathcal{D}_{test} = \emptyset$ and $\mathcal{D}_{bagg} \cup \mathcal{D}_{test} = \mathcal{D}$. With each of the two datasets ${\mathcal{D}_{bagg}, \mathcal{D}_{test}}$, we generate a prediction model by regressing the bagging datasets $\mathcal{D}_{bagg}$ using a cross-validation technique \cite{Stone74, Picard84}. For each obtained prediction model, we collect the predicted values of the target property for all the materials in the corresponding testing dataset $\mathcal{D}_{test}$. The canonical size of $\mathcal{D}_{bagg}$ is selected as 66\% of the total number of data instances. By repeating the bagging process, each material $\bm{x}_i$ has an equal chance to appear in the test set $\mathcal{D}_{test}$. Finally, we can obtain a distribution of the predicted values of target property $p(\hat{y}(\bm{x}_i))$ for all materials.
Here, the null hypothesis stands for an assumption of the homogeneity of the dataset in the kernel space or the existence of a single regression function. If the null hypothesis is true, the distribution $p(\hat{y}(\bm{x}_i))$ should be Gaussian for every material $\bm{x}_i$; else, we can significantly approximate the distribution $p(\hat{y}(\bm{x}_i))$ for a particular material $\bm{x}_i$ in the form of a mixture of Gaussian distributions. By examining the distribution $p(\hat{y}(\bm{x}_i))$, we can test the hypothesis on the homogeneity of our dataset.
The approximation of the distribution $p(\hat{y}(\bm{x}_i))$ by a mixture model of $K$ Gaussian distributions \cite{GMM} is as follows:
\begin{equation}
p(\hat{y}(\bm{x}_i)|\theta) = \sum_{k=1}^{K} {\pi}_{i}^{k} \mathcal{N}({\mu}_{i}^{k}, {\sigma}_{i}^{k}),
\label{eq.GMM}
\end{equation}
where ${\pi}_{i}^{k}$, ${\mu}_{i}^{k}$, and ${\sigma}_{i}^{k}$ are the weights, centers, and coefficient matrices of the constituent Gaussians components. For a given number of mixture components, the parameters are estimated using an expectation-maximization algorithm, which is explained in detail in \cite{GMM}. To determine the number of mixture components, a maximizing Bayesian information criterion \cite{SchwarzBIC} process is utilized by applying several different trials to randomize the initial states.
\subsection{Dissimilarity voting machine}\label{VotingMachine}
In this section, we utilize the information from the bagging experiment to vote for the dissimilarity among materials. To perform the dissimilarity voting procedure, first, under a predefined tolerance in prediction error $\delta_{thres}$, all prediction models $\hat{f}$ learned from a dataset $\mathcal{D}_{bagg}$ (Eq. \ref{eq.fx}), which satisfies $|\hat{f}(\bm{x}_{i})-y_i|< \delta_{thres}$, are collected. Then, under a predefined neighborhood condition in description space $k_{thres}$, for a given $\bm{x}_{j}$ in the $\mathcal{D}_{test}$ and all $\bm{x}_{i}$ in {$\mathcal{D} - \{x_j\}$, a vote $ds(\bm{x}_{i}, \bm{x}_{j})$ for the dissimilarity between $\bm{x}_{i}$ and $\bm{x}_{j}$ is defined:
\begin{equation}
ds(\bm{x}_{i}, \bm{x}_{j}) =
\begin{cases}
1, & \text{if}\ c_{i} = 0 \ and \ k(\bm{x}_{i}, \bm{x}_{j}) < k_{thres}\\
0, & otherwise
\end{cases}.
\label{eqn.voting}
\end{equation}
In this voting machine, for each material, we pay more attention on its relationship with the neighborhood materials (in the data set) in the description space. If the neighborhood materials are avoided and make no contribution to the predictive models that accurately predict the physical property value of the concerning material, those neighborhood materials should be considered dissimilar to that material. Finally, the bagging-based dissimilarity voting algorithm is summarized as follows:
\begin{algorithm}
\caption{Bagging-based dissimilarity voting algorithm}\label{algorithm}
\KwData{
\begin{description}
\item \makebox[4.2cm][l]{Dataset:} $\mathcal{D} = \left\{ ({\bm x}_1, y_1), ({\bm x}_2, y_2) \dots ({\bm x}_p, y_p)\right\} $
\item \makebox[4.2cm][l]{Base learning:} $\hat{f}$
\item \makebox[4.2cm][l]{Number of base learners:} $T$
\end{description}
}
\Parameter{ $k_{thres}, \delta_{thres}$}
\KwResult{Dissimilarity matrix, $\bm{dS}$}
\SetKwBlock{Begin}{begin}{End}
\Begin{
\For{$t\gets1$ \KwTo $T$ \KwBy $1$}{
$h_t = \hat{f}(\mathcal{D}, \mathcal{D}_{bagg})$ \\
}
$H({\bm x}) = \sum_{t=1}^{T} \mathbb{I}\left(h_{t}({\bm x}_{*}) \leq \delta_{thres}\right)$ \\
${\bm{dS}} = 0$ \text{with} ${\bm{dS}} = {\left[ ds_{ij} \right]}_{p \times p}$ \\
\ForEach{$h_t \in H$}{
\ForAll{ $k({\bm x}_{i}, {\bm x}_{*t}) \leq k_{thres}$ }{
\uIf{ $c_{i} = 0$}{
${ds_{ij}} \mathrel{+}= 1\ \forall {\bm x}_{j} \in h_t $
}
}
}
}
\Return ${\bm{dS}}$.
\end{algorithm}
\section{Data collection and representation}\label{DataPreprocess}
\subsection{Prototype model} \label{DataToymodel}
We simulate a dataset containing 70 data instances with a one-dimensional descriptive variable, $x$, and target variable, $y$, as shown in Figure \ref{fig.ToyModel}a. To illustrate the capability of the bagging prediction model in unveiling the mixture of nonlinear models, the dataset is designed as a mixture of three main functions. In the range of $x$ lesser than -0.4, the function $y = f(x)$ is monotonic and centered at 0.1. In the range of $x$ greater than -0.4, the function $f$ is bifurcated with a branch fluctuation increasing from 0.1 to 0.3 and the other variation decreasing from 0.1 to -0.15.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\textwidth]{Fig1.pdf}
\caption{ a) Visualization of the prototype data with the one-dimensional predictor variable, $x$, and target variable, $y$. b) Dissimilarity voting matrix with colored cells show the dissimilarity pairs of materials. Dark blue cells indicate pairs of data instances associated with high dissimilarity voting information and vice versa. c) Hierarchical clustering graph is constructed by embedding information of dissimilarity voting matrix. d) Distribution of the predicted values $y$ along the $x$ axis, applying the bagging model (orange lines) and observed data (green and red points), which are clustered using hierarchical clustering technique and information from dissimilarity voting results.} \label{fig.ToyModel}
\end{figure}
\subsection{Curie temperature data} \label{DataTc}
We collected experimental data on 101 binary materials, consisting of transition metals and rare-earth metals, from the Atomwork database of NIMS \cite{paulingfile, atomwork}, including the crystal structure of the materials and their observed {$T_{\rm C}$} values. Our task was to develop a model for estimating the {$T_{\rm C}$} of a new material based on the training data of known materials. For this, one of the crucial steps is the selection of an appropriate data representation that reflects the application domain, i.e., a model of the underlying physics. On the other hand, data representation that derives a good estimation model may imply the discovery of the underlying physics. To represent the structural and physical properties of each binary material, we designed 21 descriptive variables. We divided the 21 variables into three groups; the first and second categories contained descriptive variables that describe the atomic properties of the transition-metal ($T$) and rare-earth ($R$) constituents. The properties were as follows: (1, 2) atomic number ($Z_R$, $Z_T$), (3, 4) covalent radius ($r_{covR}$, $r_{covT}$), (5, 6) first ionization (${IP}_R$, ${IP}_T$), and (7, 8) electronegativity (${\chi}_R$, ${\chi}_T$). In addition, descriptive variables related to the magnetic properties were included, as follows: (9, 10) total spin quantum number ($S_{3d}$, $S_{4f}$), (11, 12) total orbital angular momentum quantum number ($L_{3d}$, $L_{4f}$), and (13, 14) total angular momentum ($J_{3d}$, $J_{4f}$).
The selection of these variables originates from the physical consideration that the intrinsic electronic and magnetic properties determine the 3$d$ orbital splitting at the transition-metal sites. To better capture the effect of the $4f$ electrons, the strong spin-orbit coupling effect in particular, we included three additional variables for describing the properties of the constituent rare-earth metal ions: the (15) projection of the total magnetic moment onto the total angular moment ($J_{4f}g_{j}$), and (16) projection of the spin magnetic moment onto the total angular moment ($J_{4f}\left(1-g_{j}\right)$) of the 4$f$ electrons. The selection of these variables was based on the physical consideration that the magnitude of the magnetic moment determines {$T_{\rm C}$}. It has been well established that information related to the crystal structure is highly important for understanding the physics of binary materials with transition metals and rare-earth metals. Therefore, we designed a third group with structural variables that approximately represent the structural information at the transition metal and rare-earth metal sites, including the (17) concentration of the transition metal ($C_T$), and (18) concentration of the rare-earth metal ($C_R$). Note that, if we use the atomic percentage for the concentration, the two quantities are not independent. Therefore, in this study, we measured the concentrations in units of atoms/\AA$^3$. This unit is more informative than the atomic percentage because it contains information on the constituent atomic size. As a consequence, $(C_T)$ and $(C_R)$ are not completely dependent on each other. Other structure variables were also included: the mean radius of the unit cell (19) between two rare-earth elements, $r_{RR}$, (20) between two transition metal elements, $r_{TT}$, and (21) between transition and rare-earth elements, $r_{TR}$. We set the experimentally observed {$T_{\rm C}$} as the target variable.
\section{Results and discussion}\label{ResultDiscuss}
\subsection{Prototype model} \label{ResultToymodel}
To demonstrate the effect of applying the bagging prediction model in investigating the structural insight dataset, we present the results of applying the model to two-dimensional simulation data. The dataset contains 70 instances with a one-dimensional descriptive variable, $x$, and target variable, $y$, as depicted in Figure \ref{fig.ToyModel}b and Section \ref{DataToymodel}. The bagging prediction model includes one million random samplings, with the sampling size is 35\% of the total number of data instances. Details about setting parameters are described in Table 1 in Supplemental Materials.
Figure \ref{fig.ToyModel}a shows the distributions of the predicted values, ${\hat y}$, obtained using the bagging model. It is obvious that, for $x$ values lesser than -0.4, the ${\hat y}$ distributions include a single distribution centered around 0.1. On the other hand, for $x$ values greater than -0.4, almost all the ${\hat y}$ distributions can be considered to be a mixture of two main Gaussians, whose mean are approximately 0.2 and -0.1, respectively. These distribution components of the predicted value, ${\hat y}$ reflect the actual shape of the designed data, shown by colored points.
Figure \ref{fig.ToyModel}b shows the dissimilarity $(70 \times 70)$ matrix obtained by our developed dissimilarity voting machine. In the matrix, dark blue cells represent dissimilarity pairs of data instances. Zero value cells show no dissimilarity information between corresponding pairs. For convenience, the ordered data instances shown in the matrix are sorted by the $x$ values. Details about how the voting machine works to detect dissimilarity effect are shown by zero contribution example profiles in Supplemental Materials.
There are two noticeable points extracted from this figure. First, the upper left of the matrix represents a large bright region or the region of non-dissimilarity among instances. It is consistent with the monotone and smoothly changes for $x$ values lesser than -0.4. Second, for $x$ values larger than -0.4, one can notice that any data instances in this region are dissimilar with their two closest neighbors and similar to the next ones. Once again, this extracted information shows consistency with the actual distribution of the designed dataset.
By transferring information from the dissimilarity matrix to hierarchical clustering model \cite{hac11}, we obtain clustering results as shown in figure \ref{fig.ToyModel}c. The dissimilarity information helps us to identify a mixture of two main groups, which are labeled by green and red. These two groups successfully reconstruct the shape of two branches in the bifurcated region of the dataset. To conclude, through the prototype model result, the dissimilarity voting machine based on the bagging algorithm is shown to possess the ability to identify the mixture phenomena regarding a specific target property.
\begin{figure*}[t]
\centering
\subfloat[][]{\includegraphics[width=0.33\textwidth]{pred_ensemble_Co5La-838_2.pdf}}
\subfloat[][]{\includegraphics[width=0.33\textwidth]{Co5La-838_65.pdf}}
\subfloat[][]{\includegraphics[width=0.33\textwidth]{Co5La_ciki_edited.pdf}}
\caption{a) {$T_{\rm C}$} predicted-value distribution of $\rm Co_{5}La$ for different bagging sizes. The constant plane shows the position of the observed value. b) {$T_{\rm C}$} predicted-value distribution of $\rm Co_{5}La$ with a bagging size of 65\% of the total data instances in the dataset. The distribution is a mixture of seven Gaussian components. The red dashed lines indicate the positions of the observed values. c) Heat map matrix depicting the contribution of the training materials to the target material for the different learners in Figure b). The horizontal axis represents training materials sorted by the L1 distance to the target material on the description space. The vertical axis represents those sorted by the predicted {$T_{\rm C}$} value, i.e., the summation of all the contributions.}
\label{fig.PredictedDistTest}
\end{figure*}
\subsection{Curie temperature analysis} \label{ResultTc}
For the designed descriptive variables, the maximum prediction ability with an $R^2$ score of $0.967 \pm 0.004$ was achieved by a model derived from the variable combinations, $\left\{ {\chi}_R, {\chi}_T, J_{4f}\left(1-g_{j}\right), Z_T, r_{covT}, {IP}_T, S_{3d}, L_{3d}, J_{3d}, C_R\right\}$. Details about setting parameters are described in Table 1 in Supplemental Materials. The model selection and relation among variables are discussed in \cite{Dam2018, Nguyen2018}. The high prediction accuracy level of this model shows that it is possible to accurately predict the {$T_{\rm C}$} values of rare-earth transition bimetal materials with the designed variables. With the bagging size at level of 90\% total data instances in the data set, the mean absolute value of cross-validation is approximate 40K. The summation about the bagging size dependence of mean absolute error in cross-validation process is shown in Figure 2 in Supplemental Materials. Details about training-testing prediction errors for all materials is shown in Figure 1 in Supplemental Materials. From these evidences, under the description of this variable combination, the regression function to predict {$T_{\rm C}$} is approximately a single function associated with a number of probable unknown anomalies. Our designed dissimilarity voting machine could help to address this problem.
Next, we discuss the distribution of the predicted {$T_{\rm C}$}. Almost all materials obtained its own predicted values follow a single Gaussian function distribution, whereas some materials were associated with distributions that were a mixture of Gaussians. For example, figure \ref{fig.PredictedDistTest}a shows the {$T_{\rm C}$} predicted-value distribution of $\rm Co_{5}La$ (with observed {$T_{\rm C}$} of 838~K) for bagging sizes varying from 50--95 percent of the total dataset. It is clear that for all values of the size in the subset sampling selection, there is a consistent form of distribution. On changing the size, the corresponding peaks at 686~K, 739~K, 925~K, and 980K remained. Figure \ref{fig.PredictedDistTest}b displays an enlarged view of the distribution for a bagging size of 65 percent of the total dataset. The detailed result, on applying the Gaussian mixture model (Sect. \ref{EnsembleLearning}), shows that this distribution is a mixture of seven Gaussian distributions, whose means are at 580.34~K, 686.81~K, 739.52~K, 881.59~K, 925.18~K, 980.15~K, and 1101.72~K, respectively. This indicates the presence of a mixture of nonlinear functions in the structural insight dataset. Further investigation will reveal the significance of appearance of these functions.
Figure \ref{fig.PredictedDistTest}c shows the contribution of each training material in the dataset to the target material, $\rm Co_{5}La$, in detail. The color bar shows the color encoding of the contributions, in Eqn. \ref{eq.EL} for all materials in the dataset to $\rm Co_{5}La$. It is zero-center symmetrical color encoding with the white color corresponds to non referring materials in training set. The vertical axis represents those sorted by the predicted {$T_{\rm C}$} value, i.e., the summation of all the contribution values as in Eqn. \ref{eq.EL}. The horizontal axis represents materials with an ascending order of the $L_{1}$ distance to the target material. The top five closest to $\rm Co_{5}La$ are: $\rm Co_{5}Ce~(662~K)$, $\rm Co_{13}La~(1298~K)$, $\rm Co_{17}Ce_{2}~(1090~K)$, $\rm Co_{5}Pr~(931~K)$,and $\rm Co_{7}Ce_{2}~(50~K)$. This figure also shows that models with the closest predicted value of {$T_{\rm C}$} (purple distribution with a mean of 881.59~K) are constructed from a combination of instances, $\mathcal{D}_{bagg}$, with no contribution from $\rm Co_{5}Ce$, $\rm Co_{13}La$, $\rm Co_{17}Ce_{2}$, and $\rm Co_{7}Ce_{2}$. Only $\rm Co_{5}Pr$ shows significant contribution to $\rm Co_{5}La$. Details about zero-contribution counting profiles are shown in Figure 4 in Supplemental Materials section.
For comparison, the two nearest neighbor models of the purple model, namely, the green model with mean 739~K, and blue model with mean 925.18~K were considered. For the blue model, all the five nearest neighbors were considered as contributors, and in the green model, the contribution of $\rm Co_{13}La$ was removed. The results from the training data contributions, shown above, provide certain conclusions on the actual physical meaning. The three materials, $\rm Co_{5}Ce$, $\rm Co_{13}La$, and $\rm Co_{17}Ce_{2}$, should not be considered highly "similar" to $\rm Co_{5}La$ with respect to $T_C$, even though they have the same constituent $T$-metal and the same constituent $R$-metals or are positioned next to each other on the periodic table ($Z_{La}$=57 and $Z_{Ce}$=58). In other words, the distance between these materials with respect to the rare-earth element difference should not be close, as measured by the three $R$ predictive variables, $\chi{R}$, $J_{4f}(1 - g_{j})$, and $C_{R}$. As the concentration of the $R$-metal efficiently indicates the change in {$T_{\rm C}$} values among those sharing the same $R$ and $T$ materials, e.g., $\rm Co_{5}La$ vs $\rm Co_{13}La$, these dissimilarity results show that the other two $R$ variables do not capture the real mechanism of {$T_{\rm C}$}.
\begin{figure}[t]
\centering
\subfloat[][]{\includegraphics[width=0.5\textwidth]{Tc_tree_bold.pdf}\label{fig.TcHac}}
\subfloat[][]{\includegraphics[width=0.5\textwidth]{Tc_plt.pdf}\label{fig.Tc_CR}}
\caption{a) Hierarchical clustering model by utilizing information from dissimilarity matrix. b) Relationship between {$T_{\rm C}$} and the concentration of rare-earth element, $C_R$. Red and blue lines show anomalies detection which is discussed in detail in Section \ref{ResultTc} }
\end{figure}
The entire dataset can be divided into four main groups based on transition metals: cobalt-based, iron-based, manganese-based, and nickel-based materials. In our defined $k_{thres}$ neighbor regions, the number of dissimilarity values was not identical for all the groups of materials. Here, we analyze the cobalt-based and iron-based material groups. In the cobalt-based group, we can notice that $\rm Co_{5}Ce$ does not receive contributions from the other materials of the group $\rm Co_{5}R$. The dissimilarity between the $\rm Co_{5}Ce$ and the $\rm Co_{5}La$ material was shown in the previous analysis. Here, the dissimilarity can be observed more distinctly. Compared to the other $\rm Co_{5}R$ materials, $\rm Co_{5}Ce$ has a {$T_{\rm C}$} of 662~K, which is considerably lower than those of $\rm Co_{5}La$ at 838~K, $\rm Co_{5}Pr$ at 931~K, $\rm Co_{5}Nd$ at 910~K, and $\rm Co_{5}Sm$ at 1016~K. In this family, except for $\rm Co_{5}Ce$, an increase in the atomic number of the rare-earth element correlates to an increasing {$T_{\rm C}$} value. Figure \ref{fig.TcHac} shows the hierarchical clustering result obtained by utilizing the information from the dissimilarity voting machine. It is obvious that $\rm Co_{5}Ce$ is isolated from other Co-based materials. We can also confirm the anomalousness of $\rm Co_{5}Ce$ by comparing $\rm Co_{5}Ce$ with other Co-based materials (Fig. \ref{fig.Tc_CR}, the compounds surrounded by blue line). This result confirms the significance of our method of dissimilarity measurement.
In the group of iron-based materials, Figure. \ref{fig.TcHac} shows that $\rm Fe_{5}Gd$ appears with a large number of dissimilarity values compared to other Fe-based materials -- especially the $\rm Fe_{x}Gd_{y}$ group, indicating that $\rm Fe_{5}Gd$ is out of trend with its nearest neighbors. From Figure \ref{fig.Tc_CR} (the compounds surrounded by blue line), it can be shown that, for an increasing concentration of rare-earth elements {$C_R$}, $\rm Fe_{17}Gd_{2}$, $\rm Fe_{5}Gd$, $\rm Fe_{23}Gd_{6}$, $\rm Fe_{3}Gd$, and $\rm Fe_{2}Gd$, the {$T_{\rm C}$} values are 479~K, 465~K, 659~K, 725~K, and 814~K, respectively. It is clear that, $\rm Fe_{5}Gd$ does not follow the general trend of $\rm Fe_{x}Gd_{y}$ groups. Thus, it is again demonstrated that the information extracted by dissimilarity voting using the bagging algorithm could be used as a useful method to detect anomalies.
\section{Conclusion}
In this study, we have proposed a method to extract dissimilarity between materials, with respect to a given target property. This technique is based on an ensemble method with Kernel ridge regression as the predicting model; multiple random subset sampling of the materials is performed to generate prediction models and corresponding contributions of the reference training materials in detail. The mixture distribution of predicted values was unveiled using a Gaussian mixture models. The reference training materials contributed to the prediction model that accurately predicts the physical property value of a specific material, are considered to be similar to that material, or vice versa. Evaluations using synthesized data demonstrate that the proposed method can effectively measure the dissimilarity between data instances. Next, the algorithm was applied for analyzing the Curie temperature ({$T_{\rm C}$}) prediction of the binary 3$d$ transition metal - 4$f$ rare-earth binary alloy problem and exhibited satisfactory results. The proposed method can be considered as a potential tool for obtaining a deeper understanding of the structure of data, with respect to a target property, in particular.
\section*{Acknowledgments}
This work was partly supported by PRESTO and the ``Materials Research by Information Integration'' initiative (MI$^2$I) project of the Support Program for Starting Up Innovation Hub, by the Japan Science and Technology Agency (JST) and the Elements Strategy Initiative Project under the auspices of MEXT, and also by MEXT as a social and scientific priority issue (Creation of New Functional Devices and High-Performance Materials to Support Next-Generation Industries; CDMSI) to be tackled using a post-K computer.
\section*{References}
|
1,941,325,220,571 | arxiv | \section{Introduction}
Suppose that we are given a finite simple graph $\Gamma = (V,E)$ and we are asked to count its number of \emph{independent sets}. An independent set is a subset $I \subseteq V$ such that $(v,v') \notin E$ (i.e., $(v,v')$ is not an edge) for all $v,v' \in I$. For example, if $\Gamma$ is the $4$-cycle $C_4$ with $V = \{v_1,v_2,v_3,v_4\}$ and $E = \{(v_1,v_2),(v_2,v_3),(v_3,v_4),(v_4,v_1)\}$, it can be checked that there are exactly $7$ different independent sets, namely $\emptyset$, $\{v_1\}$, $\{v_2\}$, $\{v_3\}$, $\{v_4\}$, $\{v_1,v_3\}$, and $\{v_2,v_4\}$. A common generalization of this question is to ask for the ``number'' of \emph{weighted} independent sets in $\Gamma$: given a parameter $\lambda > 0$ ---usually called \emph{activity} or \emph{fugacity}---, we ask for the value of the summation
$$
Z_\Gamma(\lambda) := \sum_{I \in X(\Gamma)} \lambda^{|I|},
$$
where $X(\Gamma)$ denotes the collection of all independent sets in $\Gamma$ and $|I|$, the cardinality of a given independent set $I$. Notice that we recover the original problem ---i.e., to compute $|X(\Gamma)|$--- if we set $\lambda = 1$ and that $Z_\Gamma(\lambda) \geq \lambda|V|$, by considering all the singletons $\{\{v\}\}_{v \in V}$. The sum $Z_\Gamma(\lambda)$ corresponds to the normalization factor of the probability distribution $\mathbb{P}_{\Gamma}$ on $X(\Gamma)$ that assigns to each $I \in X(\Gamma)$ a probability proportional to $\lambda^{|I|}$, i.e., the so-called \emph{partition function} (also known as the \emph{independence polynomial}) of the well-known \emph{hardcore model} from statistical physics.
In general, it is not possible to compute exactly $Z_\Gamma(\lambda)$ efficiently \cite{1-luby}, even for the case $\lambda = 1$ \cite{1-xia} (technically, to compute $Z_\Gamma(\lambda)$ is an \emph{$\mathrm{NP}$-hard} problem and to compute $|X(\Gamma)|$ is a \emph{$\#\mathrm{P}$-complete} problem). Therefore, one may attempt to at least finding ways to approximate $Z_\Gamma(\lambda)$ efficiently.
In recent years, there has been a great deal of attention to the complexity of approximating partition functions of spin systems (e.g., see \cite{1-barvinok}), and the hardcore model, possibly together with the Ising model, occupies the most important place. One of the most noticeable results, due to Weitz \cite{1-weitz}, and then Sly \cite{1-sly} and Sly-Sun \cite{2-sly}, is the existence of a \emph{computational phase transition} for having a \emph{fully polynomial-time approximation scheme (FPTAS)} for the approximation of $Z_\Gamma(\lambda)$. In simple terms, Weitz developed an FPTAS, a particular kind of efficient deterministic approximation algorithm on the family of finite graphs with bounded degree $\Delta$, provided $\lambda < \lambda_c(\Delta)$, where $\lambda_c(\Delta) := \frac{(\Delta-1)^{(\Delta-1)}}{(\Delta-2)^{\Delta}}$ denotes the critical activity for the hardcore model on the $\Delta$-regular tree $\mathbb{T}_\Delta$. Conversely, a couple of years later, Sly and Sun managed to prove that the existence of such an algorithm for $\lambda > \lambda_c(\Delta)$ would imply that $\mathrm{NP} = \mathrm{RP}$, the equivalence of two well-known computational complexity classes which are widely believed to be different \cite{1-arora}.
The work of Weitz exploited a technique based on trees of self-avoiding walks and a special notion of correlation decay known as \emph{strong spatial mixing}. Later, Sinclair et al. \cite{1-sinclair} studied refinements of Weitz's result by considering families of finite graphs parameterized by their \emph{connective constant} instead of their maximum degree, and established that there exists an FTPAS for $Z_\Gamma(\lambda)$ for families of graphs with connective constant bounded by $\mu$, whenever $\lambda < \frac{\mu^\mu}{(\mu-1)^{(\mu+1)}}$.
Now, if $\Gamma$ is an infinite graph, most of these concepts stop making sense. One way to deal with this issue is by choosing an appropriate normalization and by using the \emph{DLR formalism}. The idea is roughly the following: suppose that we have a sequence $\{\Gamma_n\}_n$ of finite subgraphs that ``exhausts'' $\Gamma$ in some way. This sequence induces two other sequences: a sequence $\{Z_{\Gamma_n}(\lambda)\}_n$ of partition functions and a sequence $\{\mathbb{P}_{\Gamma_n}\}_n$ of probability distributions. A way to extend the idea of ``number of weighted independent sets (per site)'' in $\Gamma$ is by considering the sequence $\{Z_{\Gamma_n}(\lambda)^{1/|\Gamma_n|}\}_n$ and hoping that it converges. Under the right assumptions on $\Gamma$ and $\{\Gamma_n\}_n$, this is exactly the case and something similar happens to $\{\mathbb{P}_{\Gamma_n}\}_n$. Moreover, there is an intimate connection between the properties of the limit measures and our ability to estimate the value of $\lim_n |\Gamma_n|^{-1} \log Z_{\Gamma_n}(\lambda)$, i.e., to ``approximately count'' it. We denote such limit by $f(\Gamma,\lambda)$ and call it the \emph{free energy} of the hardcore model $(\Gamma,\lambda)$, one of the most crucial quantities in statistical physics \cite{1-baxter,1-georgii,1-simon}.
It can be checked that if $\Gamma$ is finite, to approximate the partition function $Z_\Gamma(\lambda)$ with a multiplicative error (in polynomial time) is equivalent to approximate the free energy $f(\Gamma,\lambda)$ with an additive error \cite{2-sinclair} (in polynomial time), so the problem of approximating $f(\Gamma,\lambda)$ recovers the problem of approximating the partition function in the finite case and, at the same time, extends the problem to the infinite setting.
The main goal of this paper is to establish a computational phase transition for the free energy on ensembles of ---possibly infinite--- hardcore models, i.e., we aim to prove the existence of an efficient additive approximation algorithm for the free energy when the activity is low and to establish that there is no efficient approximation algorithm for the free energy when the activity is high.
There have been many recent works related to the study of correlation decay properties and its relation to approximation algorithms for the free energy (and related quantities such as pressure, capacity, and entropy) in the infinite setting \cite{1-gamarnik,1-marcus,1-briceno,1-wang,1-pavlov,3-marcus,2-marcus}. In this work we put all these results in a single framework, which also encompasses the results from Weitz, Sly and Sun, and Sinclair et al., and at the same time generalizes them.
In 2009, Gamarnik and Katz \cite{1-gamarnik} introduced what they called the \emph{sequential cavity method}, which can be regarded as a sort of infinitary \emph{self-reducibility} property \cite{1-jerrum}. Combining such method with Weitz's results, they managed to prove that the free energy of the hardcore model in the Cayley graph of $\mathbb{Z}^d$ with canonical generators has an $\epsilon$-additive approximation algorithm that runs in time polynomial in $\epsilon^{-1}$ whenever $\lambda < \lambda_c(2d)$, where $2d$ is the maximum degree of the graph. Related results were also proven by Pavlov in \cite{1-pavlov}, who developed an approximation algorithm for the \emph{hard square entropy}, i.e., the free energy of the hardcore model in the Cayley graph of $\mathbb{Z}^2$ with canonical generators and activity $\lambda = 1$. Later, there were also some explorations due to Wang et al. \cite{1-wang} in Cayley graphs of $\mathbb{Z}^2$ with respect to other generators (e.g., the \emph{non-attacking kings} system) in the context of information theory and algorithms for approximating capacities.
In this paper we prove that all these results fit and can be generalized to hardcore models $(\Gamma,\lambda)$ such that $\Gamma$ is a locally finite graph, $G \curvearrowright \Gamma$ is free and almost transitive for some countable amenable subgroup $G \leq \mathrm{Aut}(\Gamma)$, and $\lambda: V \to \mathbb{R}_{>0}$ is a ---not necessarily constant--- $G$-invariant activity function.
We denote the free energy by $f_G(\Gamma,\lambda)$ to emphasize the group $G$. First, based on results from \cite{1-gurevich,2-briceno}, we prove in Theorem \ref{thm:smb} that $f_G(\Gamma,\lambda)$ can be obtained as the pointwise limit of a Shannon-McMillan-Breiman type ratio with regards to any Gibbs measure on $X(\Gamma)$. Next, in Theorem \ref{thm:rep1}, we prove that if $\lambda$ is such that $(\Gamma,\lambda)$ satisfies strong spatial mixing, then $f_G(\Gamma,\lambda)$ corresponds to the evaluation of a random information function, based on ideas about random invariant orders and the \emph{Kieffer-Pinsker formula} for measure-theoretical entropy introduced in \cite{1-alpeev}. Then, in Theorem \ref{thm:rep2}, using the previous representation theorem and the techniques from \cite{1-weitz}, we provide a formula for $f_G(\Gamma,\lambda)$ in terms of trees of self-avoiding walks in $\Gamma$. These first three theorems can be regarded as a preprocessing treatment of $f_G(\Gamma,\lambda)$ in order to obtain an \emph{arboreal representation of free energy} to develop approximation techniques, but we believe that they are of independent interest.
Later, in Theorem \ref{thm:phase}, given a finitely generated amenable orderable group $G$ with a prescribed set of generators $S$ such that some algebraic past is decidable in exponential time (for example, $\mathbb{Z}^d$ with the \emph{lexicographic order} or other groups such as the \emph{Heisenberg group} $H_3(\mathbb{Z})$ with its usual order), a positive integer $\Delta$, and $\lambda_0 > 0$, we denote by $\mathcal{H}_G^\Delta(\lambda_0)$ the ensemble of hardcore models $(\Gamma,\lambda)$ such that $G \curvearrowright \Gamma$ is free and almost transitive, the maximum degree of $\Gamma$ is bounded by $\Delta$, the values of $\lambda$ are bounded from above by $\lambda_0$, and establish the following algorithmic implications: if $\lambda_0 < \lambda_c(\Delta)$, there exists an $\epsilon$-additive approximation algorithm on $\mathcal{H}_G^{\Delta}(\lambda_0)$ for $f_G(\Gamma,\lambda)$ that runs in time $\mathrm{poly}((1+\epsilon^{-1})|\Gamma/G|)$ and, if $\lambda_0 > \lambda_c(\Delta)$, there is no such algorithm, unless $\mathrm{NP} = \mathrm{RP}$, where $\lambda_c(\Delta)$ denotes the critical activity on the $\Delta$-regular tree $\mathbb{T}_\Delta$.
Here, by an additive approximation algorithm, we mean a procedure that given $(\Gamma,\lambda)$ and $\epsilon > 0$, outputs a number $\hat{f}$ such that $|f_G(\Gamma,\lambda) - \hat{f}| < \epsilon$ and by polynomial time, we mean that the procedure takes time polynomial in $|\Gamma / G|$ and $\epsilon^{-1}$, where $|\Gamma / G|$ denotes the size of a (or any) fundamental domain of the action $G \curvearrowright \Gamma$, and therefore, all the information we need in order to construct $\Gamma$. In particular, we show that if $G$ is the trivial group, Theorem \ref{thm:phase} directly recovers the results from Weitz, Sly, and Sun.
Finally, as an application in symbolic dynamics, we show how to use these results to establish representation formulas and efficient approximation algorithms for the topological entropy of nearest-neighbor subshifts of finite type with enough safe symbols and the pressure of single-site potentials with a vacuum state, which includes systems such as the \emph{Widom-Rowlinson model}, some other weighted graph homomorphisms from $\Gamma$ to any finite graph, among others. These results can also be regarded as an extension of the works by Marcus and Pavlov in $\mathbb{Z}^d$ (see \cite{1-marcus,2-marcus,3-marcus}), who developed additive approximation algorithms for the entropy and free energy (or pressure) of general $\mathbb{Z}^d$-subshifts of finite type, with special emphasis in the $d=2$ case. We believe that these implications are relevant, especially in the light of results like the one from Hochman and Meyerovitch \cite{1-hochman}, who proved that the set of topological entropies that a nearest-neighbor $\mathbb{Z}^2$-subshift of finite type can achieve coincides with the set of \emph{non-negative right-recursively enumerable real numbers}, which includes numbers that are poorly computable or even not computable. In addition, we discuss the case of the monomer-dimer model and counting independent sets of line graphs, which is a special case that does not exhibit a phase transition. As a byproduct of our results, we also give sufficient conditions for the existence of a unique measure of maximal entropy for subshifts on arbitrary amenable groups.
The paper organized as follows: in Section \ref{sec2}, we introduce the basic concepts regarding graphs, homomorphisms, independent sets, group actions, Cayley graphs, and partition functions; in Section \ref{sec3}, we rigorously define free energy based on the notion of amenability and provide a way to reduce every almost free action $G \curvearrowright \Gamma$ to a free one; in Section \ref{sec4}, we define Gibbs measures and relevant spatial mixing properties; in Section \ref{sec5}, we develop the formalism based on tree of self-avoiding walks and discuss some of its properties; in Section \ref{sec6}, we present the formalism of invariant (deterministic and random) orders of a group; in Section \ref{sec7}, we prove Theorem \ref{thm:smb}, Theorem \ref{thm:rep1}, and Theorem \ref{thm:rep2}, which provide a randomized sequential cavity method that allows us to obtain an arboreal representation of free energy; in Section \ref{sec8}, we prove Theorem \ref{thm:phase} and establish the algorithmic implications of our results; in Section \ref{sec9}, we provide reductions that allow us to translate the problem of approximating pressure of a single-site potential and the entropy of a subshift into the problem of counting independent sets and discuss other consequences that are implicit in our results.
\section{Preliminaries}
\label{sec2}
\subsection{Graphs}
A {\bf graph} will be a pair $\Gamma = (V,E)$ such that $V$ is a countable set ---the {\bf vertices}--- and $E \subseteq V \times V$ is a symmetric relation ---the {\bf edges}---.
Let $\leftrightarrow$ be the equivalence relation generated by $E$, i.e., $v \leftrightarrow v'$ if and only if there exist $n \in \mathbb{N}_0$ and $\{v_i\}_{0 \leq i \leq \ell}$ such that $v = v_0$, $v' = v_n$, and $(v_i,v_{i+1}) \in E$ for every $0 \leq i < \ell$. Denote by $n(v,v')$ the smallest such $n$. This induces a notion of distance in $\Gamma$ given by
$$
\mathrm{dist}_\Gamma(v,v') =
\begin{cases}
n(v,v') & \text{if } v \leftrightarrow v', \\
+\infty & \text{otherwise.}
\end{cases}
$$
Given a set $U \subseteq V$, we define its {\bf boundary} $\partial U$ as the set $\{v \in V: \mathrm{dist}_\Gamma(v,U) = 1\}$, where $\mathrm{dist}_\Gamma(U,U') = \inf_{v \in U, v' \in U'} \mathrm{dist}_\Gamma(v,v')$. In addition, given $\ell \geq 0$ and $v \in V$, we define the {\bf ball centered at $v$ with radius $\ell$} as $B_\Gamma(v,\ell) := \{v' \in V: \mathrm{dist}_\Gamma(v,v') \leq \ell\}$.
A graph $\Gamma$ is
\begin{itemize}
\item {\bf loopless}, if $E$ is anti-reflexive (i.e., there is no vertex related to itself);
\item {\bf connected}, if $v \leftrightarrow v'$ for every $v,v' \in V$; and
\item {\bf locally finite}, if every vertex is related to only finitely many vertices.
\end{itemize}
Sometimes we will write $V(\Gamma)$ and $E(\Gamma)$ ---instead of just $V$ and $E$--- to emphasize $\Gamma$.
\subsection{Homomorphisms}
Consider graphs $\Gamma_1$ and $\Gamma_2$. A {\bf graph homomorphism} is a map $g: V(\Gamma_1) \to V(\Gamma_2)$ such that
$$
(v,v') \in E(\Gamma_1) \implies (g(v), g(v')) \in E(\Gamma_2).
$$
We denote by $\mathrm{Hom}(\Gamma_1,\Gamma_2)$ the set of graph homomorphisms from $\Gamma_1$ to $\Gamma_2$.
A {\bf graph isomorphism} is a bijective map $g: V(\Gamma_1) \to V(\Gamma_2)$ such that
$$
(v,v') \in E(\Gamma_1) \iff (g(v), g(v')) \in E(\Gamma_2).
$$
If such a map exists, we say that $\Gamma_1$ and $\Gamma_2$ are \emph{isomorphic}, denoted by $\Gamma_1 \cong \Gamma_2$.
A {\bf graph automorphism} is a graph isomorphism from a graph $\Gamma$ to itself. We denote by $\mathrm{Aut}(\Gamma)$ the set of graph automorphisms of $\Gamma$. This set is a group when considering composition $\circ$ as the group operation and the identity map $\mathrm{id}_\Gamma: V \to V$ as the identity group element $1_{\mathrm{Aut}(\Gamma)}$. In this case, instead of writing $g_1 \circ g_2$, we will simply write $g_1g_2$ to emphasize the group structure.
\subsection{Independent sets}
Given a subset $U \subseteq V$, the {\bf subgraph} \emph{induced} by $U$, denoted by $\Gamma[U]$, is the graph with set of vertices $U$ and set of edges $E \cap (U \times U)$. A subset $I \subseteq V$ is called an {\bf independent set} if $\Gamma[I]$ has no edges. We can also represent an independent set by its indicator function, i.e., by a map $x: V \to \{0,1\}$ so that
$$
\left[x(v) = 1 \text{ and } (v,v') \in E \right] \implies x(v') = 0.
$$
In addition, if we consider the finite graph $H_0 := (\{0,1\},\{(0,0),(0,1),(1,0)\})$, then $x$ can be also understood as a graph homomorphism from $\Gamma$ to $H_0$ (see Figure \ref{fig:hard}).
\begin{figure}[ht]
\centering
\includegraphics[scale = 0.6]{hard.pdf}
\caption{The graph $H_0$.}
\label{fig:hard}
\end{figure}
We denote by $X(\Gamma)$ the {\bf set of independent sets} of $\Gamma$. Notice that $X(\Gamma) \subseteq \{0,1\}^{V}$ can be identified with the set $\mathrm{Hom}(\Gamma,H_0)$ and that the \emph{empty independent set} $0^{V}$ always belongs to $X(\Gamma)$. Sometimes we will denote such independent set by $0^\Gamma$.
\subsection{Group actions}
Let $G$ be a subgroup of $\mathrm{Aut}(\Gamma)$. Given $g \in G$ and $v \in V$, the map $(g,v) \mapsto g \cdot v := g(v)$ is a {\bf (left) group action}, this is to say, $1_G \cdot v = v$ and $(gg') \cdot v = g \cdot (g' \cdot v)$ for all $g' \in G$, where $1_G = 1_{\mathrm{Aut}(\Gamma)}$. In this case, we say that \emph{$G$ acts on $\Gamma$} and denote this fact by $G \curvearrowright \Gamma$.
The group $G$ also acts on $\{0,1\}^{V}$ by precomposition. Given $g \in G$ and $x \in \{0,1\}^{V}$, consider the map $(g,x) \mapsto g \cdot x := x \circ g^{-1}$. A subset $X \subseteq \{0,1\}^{V}$ is called {\bf $G$-invariant} if $g \cdot X = X$ for all $g \in G$, where $g \cdot X := \{g \cdot x: x \in X\}$. Clearly, if $x \in X(\Gamma)$, then $g \cdot x$ and $g^{-1} \cdot x$ also belong to $X(\Gamma)$, since $g \in \mathrm{Aut}(\Gamma)$ and $x \in \mathrm{Hom}(\Gamma,H_0)$. Therefore, $X(\Gamma)$ is $G$-invariant and the action $G \curvearrowright X(\Gamma)$ is well defined.
We will usually use the letter $v$ to denote vertices in $V$, the letter $g$ to denote graph automorphisms in $G$, and the letter $x$ to denote independent sets in $X(\Gamma)$. In order to distinguish the action of $G$ on $V$ from the action of $G$ on $X(\Gamma)$, we will write $g v$ instead of $g \cdot v$, without risk of ambiguity.
The action $G \curvearrowright \Gamma$ is always {\bf faithful}, i.e., for all $g \in G \setminus \{1_G\}$, there exists $v \in V$ such that $g v \neq v$. The {\bf $G$-orbit} of a vertex $v \in V$ is the set $Gv := \{g v: g \in G\}$. The set of all $G$-orbits of $\Gamma$, denoted by $\Gamma/G$, is a partition of $V$ and it is called the {\bf quotient} of the action.
We say that a subset $\emptyset \neq U \subseteq V$ is {\bf dynamically generating} if $GU = V$, where $GF := \{g v: g \in F, v \in U\}$ for any $F \subseteq G$, and a {\bf fundamental domain} if it is also minimal, i.e., if $U' \subsetneq U$, then $GU' \subsetneq V$. The action $G \curvearrowright \Gamma$ is {\bf almost transitive} if $|\Gamma/G| < +\infty$ and {\bf transitive} if $|\Gamma/G| = 1$. A graph $\Gamma$ is called \emph{almost transitive} (resp. \emph{transitive}) if $\mathrm{Aut}(\Gamma) \curvearrowright \Gamma$ is almost transitive (resp. transitive).
The {\bf index} of a subgroup $H \leq G$, denoted by $[G:H]$, is the cardinality of the set of cosets $\{Hg: g \in G\}$. We will usually consider subgroups of finite index. In this case, we have that $|\Gamma/H| = |\Gamma/G|[G:H]$.
The {\bf $G$-stabilizer} of a vertex $v \in V$ is the subgroup $\mathrm{Stab}_G(v) := \{g \in G: g v = v\}$. Notice that $|\mathrm{Stab}_G(v)| = |\mathrm{Stab}_G(v')|$ for all $v' \in Gv$. If $|\mathrm{Stab}_G(v)| < \infty$ for all $v$, we say that the action is {\bf almost free}, and if $|\mathrm{Stab}_G(v)| = 1$ (i.e., if $\mathrm{Stab}_G(v) = \{1_G\}$) for all $v,$ we say that the action is {\bf free}.
A relevant observation is that if we assume that $\Gamma$ is countable and $G \curvearrowright \Gamma$ is almost transitive and almost free, then $G$ must be a countable group. In this work, we will only consider almost free and almost transitive actions. In this case, there exists a finite fundamental domain $U_0 \subseteq V$ such that $|U_0| = |\Gamma/G|$ and, if $\Gamma$ is locally finite, then $\Gamma$ must have {\bf bounded degree}, i.e., there is a uniform bound on the number of vertices that each vertex is related to. In such case, we denote by $\Delta(\Gamma)$ the maximum degree among all vertices of the graph $\Gamma$.
\subsection{Transitive case: Cayley graphs}
Consider a subset $S \subseteq G$ that we assume to be \emph{symmetric}, i.e., $S = S^{-1}$, where $S^{-1} = \{s^{-1} \in S: s \in S\}$. We define the {\bf (right) Cayley graph} as $\mathrm{Cay}(G,S) = (V,E)$, where
$$
V = G \quad \text{ and } \quad E = \{(g,sg): g \in G, s \in S\}.
$$
Cayley graphs are a natural construction used to represent groups in a geometric fashion. In this context, it is common to ask that $1_G \notin S$, $S$ to be finite, and $S$ to be \emph{generating}, i.e., $G = \left<S\right>$, where
$$
\left<S\right> := \left\{s_1 \cdots s_k: s_i \in S \text{ for all } 1 \leq i \leq k \text{ and } k \in \mathbb{N}\right\}.
$$
Notice that if $1_G \notin S$, then $\mathrm{Cay}(G,S)$ is loopless, if $S$ is finite, then $\mathrm{Cay}(G,S)$ has bounded degree and, if $S$ is generating, then $\mathrm{Cay}(G,S)$ is connected.
Now, suppose that $G \curvearrowright \Gamma$ is transitive (and free). Then, there exists a symmetric set $S \subseteq G$ such that
$$
\Gamma \cong \mathrm{Cay}(G,S).
$$
Indeed, it suffices to take $S = \{g \in G: (v,g v) \in E\}$, where $v \in V$ is arbitrary (see \cite{1-sabidussi}).
We will be interested in Cayley graphs $\Gamma = \mathrm{Cay}(G,S)$ and their subgroup of automorphisms induced by group multiplication as a special and relevant case: given $g \in G$, we can define $f_g: \Gamma \to \Gamma$ as $f_g(g') = g'g$ and it is easy to check that $f_g \in \mathrm{Aut}(\Gamma)$. Then $G$ acts (as a group, from the left) on $\Gamma$ so that $g \cdot g' = f_g(g') = g'g$ for all $g' \in G$ and $G \hookrightarrow \mathrm{Aut}(\Gamma)$ by identifying $g$ with $f_{g^{-1}}$. In addition, via this identification, $G$ acts transitively on $\Gamma$ as a subgroup of graph automorphisms. In particular, every Cayley graph is transitive.
\subsection{Partition functions}
Given a graph $\Gamma = (V,E)$, let's consider $\lambda: V \to \mathbb{R}_{>0}$, an {\bf activity function}. We call the pair $(\Gamma,\lambda)$ a {\bf hardcore model}. We will say that a hardcore model $(\Gamma,\lambda)$ is \emph{finite} if $\Gamma$ is finite. If $U \subseteq V$ is a finite subset, a fact that we denote by $U \Subset V$, and $x \in X(\Gamma)$ is an independent set, we define the {\bf $\lambda$-weight} of $x$ on $U$ as
$$
\mathrm{w}_\lambda(x,U) := \prod_{v \in U} \lambda(v)^{x(v)}
$$
and the {\bf $(\Gamma,U,\lambda)$-partition function} as
$$
Z_\Gamma(U,\lambda) := \sum_{x \in X(\Gamma,U)} \mathrm{w}_\lambda(x,U) = \sum_{x \in X(\Gamma,U)} \prod_{v \in U} \lambda(v)^{x(v)},
$$
where $X(\Gamma,U) := \{x \in X(\Gamma): x(v) = 0 \text{ for all } v \notin U\}$ is the finite set corresponding to the subset of independent sets of $\Gamma$ \emph{supported} on $U$. It is easy to check that there is an identification between $X(\Gamma,U)$ and $X(\Gamma[U])$. Then, the quantity $Z_\Gamma(U,\lambda)$ corresponds to the summation of independent sets of $\Gamma[U]$ weighted by $\lambda$. In the special case $\lambda \equiv 1$, we have that $Z_\Gamma(U,1) = |X(\Gamma,U)| = |X(\Gamma[U])|$, i.e., the partition function is exactly the number of independent sets supported on $U$. If $(\Gamma,\lambda)$ is finite, we will simply write $Z_\Gamma(\lambda)$ instead of $Z_\Gamma(V,\lambda)$.
\begin{remark}
Notice that if $(v,v) \in E$ or $\lambda(v) = 0$, then $Z_\Gamma(U,\lambda) = Z_{\Gamma}(U \setminus \{v\},\lambda)$; due to this fact, we usually ask $\lambda$ to be strictly positive and that $\Gamma$ is loopless.
\end{remark}
\section{Free energy}
\label{sec3}
Now, suppose that we have an increasing sequence $\{U_n\}_n$ of finite subsets of vertices exhausting $\Gamma$, i.e., $U_n \subseteq U_{n+1}$ and $\bigcup_n U_n = V$. Tentatively, we would like to define the exponential growth rate of $Z_\Gamma(V_n,\lambda)$ as
$$
\lim_n \frac{\log Z_\Gamma(U_n, \lambda)}{|U_n|}.
$$
In order to guarantee the existence of such limit, we will provide a self-contained argument based on the particular properties of the hardcore model and \emph{amenability}. The reader that is familiar with this kind of arguments may skip the next part and go straight to Section \ref{subsec:3-3}.
\subsection{Amenability}
Let
$$
\mathcal{F}(G) := \{F \subseteq G: 0 < |F| < \infty\}
$$
be the set of finite nonempty subsets of $G$. Given $g \in G$ and $K,F \subseteq G$, we denote $Fg = \{hg: h \in F\}$, $gF = \{g h: h \in F\}$, $F^{-1} := \{g^{-1}: g \in F\}$, and $KF = \{hg: h \in K, g \in F\}$.
We say that $\{F_n\}_n \subseteq \mathcal{F}(G)$ is a {\bf right F{\o}lner sequence} if
$$
\lim_n \frac{|F_ng \triangle F_n|}{|F_n|} = 0 \quad \text{for all } g \in G,
$$
where $\triangle$ denotes the symmetric difference. Similarly, $\{F_n\}_n$ is a {\bf left F{\o}lner sequence} if
$$
\lim_n \frac{|g F_n \triangle F_n|}{|F_n|} = 0 \quad \text{for all } g \in G,
$$
and $\{F_n\}_n$ is a {\bf two-sided F{\o}lner} if it is both a left and a right F{\o}lner sequence. The group $G$ is said to be {\bf amenable} if it has a (\emph{right} or \emph{left}) \emph{F{\o}lner sequence}. Notice that $\{F_n\}_n$ is left F{\o}lner if and only if $\{F_n^{-1}\}_n$ is right F{\o}lner. A F{\o}lner sequence $\{F_n\}_n$ is a {\bf F{\o}lner exhaustion} if in addition $F_n \subseteq F_{n+1}$ and $\bigcup_n F_n = G$. Every countable amenable group has a two-sided F{\o}lner exhaustion (see \cite[Theorem 4.10]{1-kerr}).
\subsection{Growth rate of independent sets}
Given $\emptyset \neq U \Subset V$, define $\varphi_U: \mathcal{F}(G) \to \mathbb{R}$ as
$$
\varphi_{U}(F) := \log Z_\Gamma(F U, \lambda).
$$
From now on, we will assume that $\lambda: V \to \mathbb{R}_{>0}$ is $G$-invariant, this is to say,
$$
\lambda(g v) = \lambda(v) \quad \text{for all } g \in G.
$$
In other words, $\lambda$ is constant along the $G$-orbits, so it achieves at most $|\Gamma/G|$ different values. We denote by $\lambda_+$ and $\lambda_-$ the maximum and minimum among such values, respectively.
Now, let $W$ be an abstract set, $M$ a finite subset of $W$, and $k \in \mathbb{N}$. We will say that a finite collection $\mathcal{K}$ of nonempty finite subsets of $M$, with possible repetitions, is a {\bf $k$-cover} of $W$ if $\sum_{K \in \mathcal{K}} \mathbbm{1}_K \geq k \mathbbm{1}_M$, where $\mathbbm{1}_A: W \to \{0,1\}$ denotes the indicator function of a set $A \subseteq W$. The following lemma is due to Downarowicz, Frej, and Romagnoli.
\begin{lemma}[{\cite{1-downarowicz}}]
\label{lem:downarowicz}
Let $Y$ be a subset of $A^{n}$, where $A$ is a finite set and $n \in \mathbb{N}$. Let $\mathcal{K}$ be a $k$-cover of the set of coordinates $M = \{1,\dots, n\}$. For $K \in \mathcal{K}$, let $Y_{K} = \{y_K: y \in Y\}$, where $y_K$ is the restriction of $y$ to $K$. Then,
$$
|Y| \leq \prod_{K \in \mathcal{K}}\left|Y_{K}\right|^{\frac{1}{k}}.
$$
\end{lemma}
Given $\varphi: \mathcal{F}(G) \to \mathbb{R}$, we will say that $\varphi$ satisfies {\bf Shearer's inequality} if
$$
\varphi(F) \leq \frac{1}{k} \sum_{K \in \mathcal{K}} \varphi(K)
$$
for all $F \in \mathcal{F}(G)$ and for all $k$-cover $\mathcal{K}$ of $F$. We have the following theorem.
\begin{theorem}[{\cite[Theorem 4.48]{1-kerr}}]
\label{kerr-li}
Given a countable amenable group $G$, suppose that $\varphi: \mathcal{F}(G) \to \mathbb{R}$ satisfies Shearer's inequality and $\varphi(Fg) = \varphi(F)$ for all $F \in \mathcal{F}(G)$ and $g \in G$. Then,
$$
\lim_n \frac{\varphi(F_n)}{|F_n|} = \inf_{F \in \mathcal{F}(G)} \frac{\varphi(F)}{|F|}
$$
for any F{\o}lner sequence $\{F_n\}_n$.
\end{theorem}
Considering the two previous results, we obtain the next lemma.
\begin{lemma}
\label{lem:rational}
Given a fundamental domain $U_0$ of $G \curvearrowright \Gamma$ and $\lambda: V \to \mathbb{Q}_{> 0}$ such that $\lambda(v) = \frac{p_v}{q_v}$ with $p_v, q_v \in \mathbb{N}$ for all $v \in V$, we have that, for any F{\o}lner sequence $\{F_n\}_n$,
$$
\lim_n \frac{\varphi(F_n)}{|F_n|} = \inf_{F \in \mathcal{F}(G)} \frac{\varphi(F)}{|F|},
$$
where $\varphi: \mathcal{F}(G) \to \mathbb{R}$ is given by $\varphi(F) = \log Z_\Gamma(FU_0, \lambda) + |F| \sum_{v \in U_0} q_v$.
\end{lemma}
\begin{proof}
Given $F \in \mathcal{F}(G)$ and $k \in \mathbb{N}$, let $\mathcal{K}$ be a $k$-cover of $F$. Notice that
$$
Z_\Gamma(FU_0,\lambda) = \sum_{x \in X(\Gamma, FU_0)} \prod_{v \in FU_0} \left(\frac{p_v}{q_v}\right)^{x(v)} = \frac{1}{\prod_{v \in FU_0} q_v} \sum_{x \in X(\Gamma, FU_0)} \prod_{v \in FU_0} p_v^{x(v)} q_v^{1-x(v)}.
$$
Consider $q := \max_v q_v$, $p := \max_v p_v$, $A := \{-q,\dots,-1\} \cup \{1,\dots,p\}$, and
$$
Y := \left\{y \in A^{FU_0}: \begin{array}{cc}-q_v \leq y(v) \leq p_v \text{ and, for all } (v,v') \in E(\Gamma), \\ y(v) \in \{1,\dots,p_v\} \implies y(v') \in \{-q_v,\dots,-1\}\end{array}\right\}.
$$
Notice that
$$
|Y| = \sum_{x \in X(\Gamma,FU_0)} \prod_{v \in FU_0} p_v^{x(v)} q_v^{1-x(v)}.
$$
Therefore, by Lemma \ref{lem:downarowicz}, and noticing that $\left|Y_{KU_0}\right| \leq Z_\Gamma(KU_0,\lambda)$, we have that
$$
\prod_{v \in FU_0} q_v \cdot Z_\Gamma(FU_0,\lambda) = |Y| \leq \prod_{K \in \mathcal{K}}\left|Y_{KU_0}\right|^{\frac{1}{k}} \leq \prod_{K \in \mathcal{K}} \left(\prod_{v \in KU_0} q_v \cdot Z_\Gamma(KU_0,\lambda)\right)^{\frac{1}{k}},
$$
where we use that $\{KU_0: K \in \mathcal{K}\}$ is a $k$-cover of $FU_0$. Therefore, by $G$-invariance of $\lambda$,
\begin{align*}
\varphi(F) & = \log Z_\Gamma(FU_0, \lambda) + |F| \sum_{v \in U_0} q_v \\
& \leq \frac{1}{k} \sum_{K \in \mathcal{K}}\left( \log Z_\Gamma(KU_0,\lambda) + |K|\sum_{v \in U_0} q_v \right) \\
& = \frac{1}{k} \sum_{K \in \mathcal{K}} \varphi(K),
\end{align*}
so $\varphi$ satisfies Shearer's inequality. On the other hand, by $G$-invariance of $X(\Gamma)$ and $\lambda$, it follows that $\varphi(Fg) = \varphi(F)$ for all $F \in \mathcal{F}(G)$ and $g \in G$. Therefore, by Theorem \ref{kerr-li}, we conclude.
\end{proof}
\begin{proposition}
\label{prop:energy}
Given a fundamental domain $U_0$ of $G \curvearrowright \Gamma$, we have that
$$
\lim_n \frac{\log Z_\Gamma(F_nU_0, \lambda)}{|F_n|} = \inf_{F \in \mathcal{F}(G)} \frac{\log Z_\Gamma(FU_0, \lambda)}{|F|},
$$
for any F{\o}lner sequence $\{F_n\}_n$.
\end{proposition}
\begin{proof}
First, suppose that $\lambda$ only takes rational values, i.e., $\lambda: V \to \mathbb{Q}_{>0}$ so that $\lambda(v) = \frac{p_v}{q_v}$ for all $v \in V$. By Lemma \ref{lem:rational}, for $\varphi(F) = \log Z_\Gamma(FU_0, \lambda) + |F| \sum_{v \in U_0} q_v$, we have that
\begin{align*}
\lim_{n} \frac{\log Z_\Gamma(F_nU_0, \lambda)}{|F_n|} + \sum_{v \in U_0} q_v & = \lim_{n} \frac{\varphi(F_n)}{|F_n|} \\
& = \inf_{F \in \mathcal{F}(G)} \frac{\varphi(F)}{|F|} \\
& = \inf_{F \in \mathcal{F}(G)} \frac{\log Z_\Gamma(FU_0, \lambda)}{|F|} + \sum_{v \in U_0} q_v,
\end{align*}
and, after cancelling out $\sum_{v \in U_0} q_v$, we obtain that
$$
\lim_{n} \frac{\log Z_\Gamma(F_nU_0, \lambda)}{|F_n|} = \inf_{F \in \mathcal{F}(G)} \frac{\log Z_\Gamma(FU_0, \lambda)}{|F|}.
$$
Now, given a general $\lambda$, we can always approximate it by some $G$-invariant $\tilde{\lambda}: V \to \mathbb{Q}_{>0}$ arbitrarily close in the supremum norm. Given $\epsilon > 0$, pick such $\tilde{\lambda}$ so that $\tilde{\lambda}(v) \leq \lambda(v)\ \leq (1+\epsilon)\tilde{\lambda}(v)$ for every $v$. Then,
\begin{align*}
\log Z_\Gamma(FU_0,\tilde{\lambda}) & \leq \log Z_\Gamma(FU_0,\lambda) \\
& \leq \log Z_\Gamma(FU_0,(1+\epsilon)\tilde{\lambda}) \\
& \leq |FU_0|\log(1+\epsilon) + \log Z_\Gamma(FU_0,\tilde{\lambda}),
\end{align*}
so,
$$
\frac{\log Z_\Gamma(FU_0,\tilde{\lambda})}{|F|} \leq \frac{\log Z_\Gamma(FU_0,\lambda)}{|F|} \leq |U_0|\log(1+\epsilon) + \frac{\log Z_\Gamma(FU_0,\tilde{\lambda})}{|F|}.
$$
Therefore,
\begin{align*}
\liminf_n \frac{\log Z_\Gamma(F_nU_0,\lambda)}{|F_n|} & \geq \inf_{F \in \mathcal{F}(G)} \frac{\log Z_\Gamma(FU_0,\lambda)}{|F|} \\
& \geq \inf_{F \in \mathcal{F}(G)} \frac{\log Z_\Gamma(FU_0,\tilde{\lambda})}{|F|} \\
& = \lim_n \frac{\log Z_\Gamma(F_nU_0,\tilde{\lambda})}{|F_n|} \\
& \geq \limsup_n \frac{\log Z_\Gamma(F_nU_0,\lambda)}{|F_n|} - |U_0|\log(1+\epsilon),
\end{align*}
and since $\epsilon$ was arbitrary, we conclude.
\end{proof}
In order to fully characterize $\lim_n \frac{\log Z_\Gamma(U_n, \lambda)}{|U_n|}$, we have the following lemma.
\begin{lemma}
\label{lem:stab}
Let $\{F_n\}_n$ be F{\o}lner sequence and $U_0$ a fundamental domain. Then, for any F{\o}lner sequence $\{F_n\}_n$,
$$
\lim_n \frac{|F_n U_0|}{|F_n|} = \sum_{v \in U_0} |\mathrm{Stab}(v)|^{-1}.
$$
\end{lemma}
\begin{proof}
First, pick $v \in U_0$. Since $\mathrm{Stab}(v)$ is finite, we have that $\lim_n \frac{|F_n \mathrm{Stab}(v)|}{|F_n|} = 1$ due to amenability. On the other hand, $F_n \mathrm{Stab}(v) v = F_n v$ and for each $v' \in F_n v$, there exist exactly $|\mathrm{Stab}(v)|$ different elements $g \in F_n \mathrm{Stab}(v)$ such that $g v = v'$. In other words,
$$
|F_n \mathrm{Stab}(v)| = |F_n v| |\mathrm{Stab}(v)|,
$$
so,
$$
\lim_n \frac{|F_n v|}{|F_n|} = \lim_n \frac{|F_n v|}{|F_n||\mathrm{Stab}(v)|} = |\mathrm{Stab}(v)|^{-1}.
$$
Therefore,
$$
\lim_n \frac{|F_n U_0|}{|F_n|} = \sum_{v \in U_0} \lim_n \frac{|F_n v|}{|F_n|} = \sum_{v \in U_0} |\mathrm{Stab}(v)|^{-1}.
$$
\end{proof}
Now, given a fundamental domain $U_0$, define
$$
f_{G}(\Gamma,U_0,\lambda) := \inf_{F \in \mathcal{F}(G)}\frac{\log Z_\Gamma(FU_0, \lambda)}{|FU_0|}.
$$
which, by Proposition \ref{prop:energy} and Lemma \ref{lem:stab}, is equal to
$$
\sum_{v \in U_0} |\mathrm{Stab}(v)|^{-1} \lim_n \frac{\log Z_\Gamma(F_nU_0, \lambda)}{|F_n|}
$$
for any F{\o}lner sequence $\{F_n\}_n$ and, in particular, for any F{\o}lner exhaustion. Notice that, since $GU_0 = V$, the sequence $\{U_n\}_n$ defined as $U_n = F_nU_0$ is an exhaustion of $V$ in the sense that we were looking for. Now we will see that $f_{G}(\Gamma,U_0,\lambda)$ is independent of $U_0$.
\begin{proposition}
Given two fundamental domains $U_0$ and $U_0'$ of $G \curvearrowright \Gamma$, we have that
\begin{equation*}
f_{G}(\Gamma,U_0,\lambda) = f_{G}(\Gamma,U_0',\lambda).
\end{equation*}
\end{proposition}
\begin{proof}
Since $V = GU_0 = GU_0'$, there must exist $K,K' \in \mathcal{F}(G)$ such that $U_0' \subseteq KU_0$ and $U_0 \subseteq K'U_0'$. Then, for every $F \in \mathcal{F}(G)$,
\begin{align*}
FU_0 \triangle FU_0' & = (FU_0 \setminus FU_0') \cup (FU_0' \setminus FU_0) \\
& \subseteq (FK'U_0' \setminus FU_0') \cup (FKU_0 \setminus FU_0) \\
& = (FK' \setminus F)U_0' \cup (FK \setminus F)U_0.
\end{align*}
Therefore, $|FU_0 \triangle FU_0| \leq |FK' \setminus F||U_0'| + |FK \setminus F||U_0|$. Now, notice that for $U, U' \Subset V$, we always have that
\begin{enumerate}
\item $Z_\Gamma(U \cup U',\lambda) \leq Z_\Gamma(U,\lambda) \cdot Z_\Gamma(U',\lambda)$, provided $U \cap U' = \emptyset$;
\item $Z_\Gamma(U,\lambda) \leq Z_\Gamma(U',\lambda)$, provided $U \subseteq U'$;
\item $Z_\Gamma(U,\lambda) \leq (2\max\{1,\lambda_+\})^{|U|}$.
\end{enumerate}
Therefore,
\begin{align*}
\log Z_\Gamma(FU_0,\lambda) & \leq \log Z_\Gamma(FU_0 \cap FU_0',\lambda) + \log Z_\Gamma(FU_0 \setminus FU_0',\lambda) \\
& \leq \log Z_\Gamma(FU_0',\lambda) + \log Z_\Gamma(FU_0 \triangle FU_0',\lambda) \\
& \leq \log Z_\Gamma(FU_0',\lambda) + |FU_0 \triangle FU_0'|\log(2\max\{1,\lambda_+\}) \\
& \leq \log Z_\Gamma(FU_0',\lambda) + (|FK' \setminus F||U_0'| + |FK \setminus F||U_0|)\log(2\max\{1,\lambda_+\}).
\end{align*}
Finally, since $|U_0| = |U_0'|$ and $|FU_0| = |F||U_0|$, it follows by amenability that
\begin{align*}
f_G(\Gamma,U_0,\lambda) & = \lim_n \frac{\log Z_\Gamma(F_nU_0,\lambda)}{|F_nU_0|} \\
& \leq \lim_n \frac{\log Z_\Gamma(F_nU_0',\lambda)}{|F_nU_0'|} \\
& \qquad + \lim_n\left( \frac{|F_nK' \setminus F_n|}{|F_n|} + \frac{|F_nK \setminus F_n|}{|F_n|}\right)\log(2\max\{1,\lambda_+\}) \\
& = f_{G}(\Gamma,U_0',\lambda),
\end{align*}
and by symmetry of the argument, we conclude.
\end{proof}
Then, we can consistently define the {\bf Gibbs $(\Gamma,\lambda)$-free energy according to $G$} as
\begin{equation*}
f_G(\Gamma,\lambda) := f_G(\Gamma,U_0,\lambda),
\end{equation*}
where $U_0$ is an arbitrary fundamental domain of $G \curvearrowright \Gamma$. In addition, it is easy to see that if $G_1$ and $G_2$ act almost transitively on $\Gamma$ and the $G_1$-orbits and $G_2$-orbits coincide, i.e., $G_1 v = G_2 v$ for all $v \in V$, then
\begin{equation*}
f_{G_1}(\Gamma,\lambda) = f_{G_2}(\Gamma,\lambda).
\end{equation*}
In particular, we have that $f_G(\Gamma,\lambda)$ is equal for all $G$ acting transitively on $\Gamma$. Then, if there exists $G \leq \mathrm{Aut}(\Gamma)$ acting transitively on $\Gamma$, it makes sense to define the {\bf Gibbs $(\Gamma,\lambda)$-free energy} as
\begin{equation}
\label{infimumrule2}
f(\Gamma,\lambda) := \inf_{\emptyset \neq U \Subset V} \frac{\log Z_\Gamma(U, \lambda)}{|U|},
\end{equation}
which is a quantity that only depends on the graph $\Gamma$ and the activity function $\lambda$.
\begin{remark}
In the almost transitive case, Equation (\ref{infimumrule2}) does not necessarily hold: consider the graph $\Gamma$ obtained by taking the disjoint union of $\Gamma_1 = \mathrm{Cay}(\mathbb{Z},\emptyset)$ and $\Gamma_2 = \mathrm{Cay}(\mathbb{Z},\{1,-1\})$ and the constant activity function $\lambda \equiv 1$. Then, $f_\mathbb{Z}(\Gamma_1,1) = \log 2$ and $f_\mathbb{Z}(\Gamma_2,1) = \log\left(\frac{1+\sqrt{5}}{2}\right)$, so
$$
f_\mathbb{Z}(\Gamma,1) = \frac{1}{2}(f_\mathbb{Z}(\Gamma_1,1) + f_\mathbb{Z}(\Gamma_2,1)) > f_\mathbb{Z}(\Gamma_2,1) \geq \inf_{\emptyset \neq U \Subset V} \frac{\log Z_\Gamma(U, 1)}{|U|}.
$$
The value of $f_\mathbb{Z}(\Gamma_2,1)$ corresponds to the \emph{topological entropy} of the \emph{golden mean shift} (see \cite[Example 4.1.4]{1-lind} and Section \ref{sec9}).
\end{remark}
The main theme of this paper will be to explore our ability to compute $f_G(\Gamma,\lambda)$. The next subsection establishes that, without loss of generality for our purposes, we can assume that $G \curvearrowright \Gamma$ is free.
\subsection{From almost free to free}
\label{subsec:3-3}
We proceed to show how to represent $f_G(\Gamma,\lambda)$ for an almost free action $G \curvearrowright \Gamma$ as $f_G(\Gamma_b,\lambda_b)$ for a free action $G \curvearrowright \Gamma_b$, where $\Gamma_b$ and $\lambda_b$ are some suitable auxiliary graph and activity function, respectively.
Given a graph $\Gamma$ and $b:V(\Gamma) \to \mathbb{N}$, let $\Gamma_b$ be the new graph obtained by setting
$$
V(\Gamma_b) = \{(v,i): v \in V(\Gamma), 1 \leq i \leq b(v)\},
$$
and
$$
E(\Gamma_b) = \bigcup_{\substack{v \in V(\Gamma)\\1 \leq i,j \leq b(v)\\i \neq j}} \{((v,i),(v,j))\} \cup \bigcup_{\substack{(v,v') \in E(\Gamma)\\1 \leq i \leq b(v)\\1 \leq j \leq b(v')}} \{((v,i),(v',j))\}.
$$
In simple words, for each $v$ in $V(\Gamma)$ there are $b(v)$ copies in $V(\Gamma_b)$ such that
\begin{enumerate}
\item the $b(v)$ copies of $v$ form a clique in $\Gamma_b$;
\item for $(v,v') \in E(\Gamma)$, each copy of $v$ is connected to all copies of $v'$ in $\Gamma_b$.
\end{enumerate}
The idea is to define $\lambda_b: V(\Gamma_b) \to \mathbb{N}$ so that for every $U \Subset V(\Gamma)$ we have that
$$
Z_\Gamma(U,\lambda) = Z_{\Gamma_b}(U_b,\lambda_b),
$$
where $U_b$ is the set of all copies of vertices in $U$. Notice that each independent set in $\Gamma_b[U_b]$ can be naturally identified with a unique independent set in $\Gamma[U]$: for $x' \in X(\Gamma_b)$, we can define $x \in X(\Gamma)$ as $x(v) = 1$ if and only if there exists $1 \leq i \leq b(v)$ so that $x'((v,i)) = 1$. Conversely, if $\Gamma$ is finite, each independent set $x \in X(\Gamma)$ can be identified with $\prod_{v \in \Gamma} b(v)^{x(v)}$ copies in $X(\Gamma_b)$. Therefore,
\begin{align*}
Z_{\Gamma_b}(U_b,\lambda_b) & = \sum_{x' \in X(\Gamma_b[U_b])} \prod_{(v,i) \in U_b} \lambda_b((v,i))^{x'((v,i))} \\
& = \sum_{x \in X(\Gamma[U])} \prod_{v \in U} b(v)^{x(v)} \prod_{v \in U} \lambda_b(v)^{x(v)} \\
& = \sum_{x \in X(\Gamma[U])} \prod_{v \in U} (b(v)\lambda_b(v))^{x(v)} \\
& = Z_{\Gamma}(U,\lambda),
\end{align*}
where the last equality holds if we take $\lambda_b$ so that $(v,i) \mapsto \lambda_b(v,i) = \lambda(v)/b(v)$. Now, given an almost free action $G \curvearrowright \Gamma$, we can define a free action $G \curvearrowright \Gamma_b$ as follows. Take $b(v) := |\mathrm{Stab}_G(v)|$ and, for $v \in V(\Gamma)$, label the elements of $\mathrm{Stab}_G(v)$ as $\{g_{v,1},\dots,g_{v,b(v)}\}$. Then, set $g (v,i) := (g v,g i)$, where
$$
g i :=
\begin{cases}
i & \text{if } g \notin \mathrm{Stab}_G(v), \\
j & \text{if } g \in \mathrm{Stab}_G(v), g = g_{v,k}, \text{ and } g_{v,k}g_{v,i} = g_{v,j}.
\end{cases}
$$
Clearly, $G \curvearrowright \Gamma_b$ is free: if $g (v,i) = (v,i)$, then $g v = v$ and $g i = i$. Since $g v = v$, then $g \in \mathrm{Stab}_G(v)$, so $g = g_{v,k}$ for some $1 \leq k \leq b(v)$. Since $g i = i$, then $g_{v,k}g_{v,i} = g_{v,i}$, so $g_{v,k} = 1_G$.
Now, if $G \curvearrowright \Gamma$ is almost transitive, then $G \curvearrowright \Gamma_b$ is almost transitive, too, with $|\Gamma_b/G| = |\Gamma/G|$. Given $v' \in Gv$, we claim that $(v',j) \in G(v,i)$ for any $1 \leq i,j \leq b(v) = b(v')$. First, since $G \curvearrowright \Gamma$ is almost transitive, we can always find $g_1 \in G$ such that $g_1(v,i) = (v',i)$. In addition, we can find $g_2 \in G$ such that $g_2 (v',i) = (v',j)$. In fact, it suffices to take $g_2 = g_{v',j} g_{v',2}^{-1}$. Then, we have that $g_2g_1 (v,i) = (v',j)$, so $G(v,i) = Gv \times \{1,\dots,b(v)\}$. Moreover, if $U_0$ is a fundamental domain of $G \curvearrowright \Gamma$, then $U_0 \times \{1\}$ is a fundamental domain of $G \curvearrowright \Gamma_b$.
\section{Gibbs measures}
\label{sec4}
Given a graph $\Gamma = (V,E)$, consider the set $\{0,1\}^{V}$ endowed with the product topology and the set $X(\Gamma)$, with the subspace topology. The set of independent sets $X(\Gamma)$ is a compact and metrizable space. A base for the topology is given by the {\bf cylinder sets}
$$
[x_U] := \{x' \in X(\Gamma): x'_U = x_U\}
$$
for $U \Subset V$ and $x \in X(\Gamma)$, where $x_U$ denotes the \emph{restriction} of $x$ from $V$ to $U$. If $U$ is a singleton $\{v\}$, we will omit the brackets and simply write $x_v$ and the same convention will hold in analogue instances. Given $W \subseteq V$, we denote by $\mathcal{B}_W$ the smallest $\sigma$-algebra generated by
$$
\{[x_U]: U \Subset W, x \in X(\Gamma)\},
$$
and by $\mathcal{B}_\Gamma$ the {\bf Borel $\sigma$-algebra}, which corresponds to $\mathcal{B}_{V}$.
Let $\mathcal{M}(X(\Gamma))$ be the set of Borel probability measures $\mathbb{P}$ on $X(\Gamma)$. We say that $\mathbb{P}$ is {\bf $G$-invariant} if $\mathbb{P}(A) = \mathbb{P}(g \cdot A)$ for all $A \in \mathcal{B}_\Gamma$ and $g \in G$, and {\bf $G$-ergodic} if $g \cdot A = A$ for all $g \in G$ implies that $\mathbb{P}(A) \in \{0,1\}$. We will denote by $\mathcal{M}_G(X(\Gamma))$ and $\mathcal{M}_G^{\mathrm{erg}}(X(\Gamma))$ the set of $G$-invariant and the subset of $G$-ergodic measures that are $G$-invariant, respectively.
For $\mathbb{P} \in \mathcal{M}(X(\Gamma))$, define the {\bf support} of $\mathbb{P}$ as
$$
\mathrm{supp}(\mathbb{P}) := \left\{x \in X(\Gamma): \mathbb{P}([x_U]) > 0 \text{ for all } U \Subset V\right\}.
$$
Given $\emptyset \neq U \Subset V$ and $y \in X(\Gamma)$, we define $\pi^y_U$ to be the probability distribution on $X(\Gamma,U)$ given by
$$
\pi_U^y(x) := \mathrm{w}^y_\lambda(x,U) Z^y_\Gamma(U,\lambda)^{-1},
$$
where $\mathrm{w}^y_\lambda(x,U) = \mathrm{w}_\lambda(x,U) \mathbbm{1}_{[y_{U^{\rm c}}]}(x)$ and $Z^y_\Gamma(U,\lambda) = \sum_x \mathrm{w}^y_\lambda(x,U)$. In other words, to each independent set $x$ supported on $U$, we associate a probability proportional to its $\lambda$-weight over $U$, $\prod_{v \in U} \lambda(v)^{x(v)}$, provided $x_U$ is \emph{compatible} with $y_{U^{\rm c}}$, in the sense that the element $z \in \{0,1\}^{V}$ such that $z_U = x_U$ and $z_{U^{\rm c}} = y_{U^{\rm c}}$ is an independent set.
Now, given an activity function $\lambda: V \to \mathbb{R}_{>0}$, consider the hardcore model $(\Gamma, \lambda)$ and the collection $\pi_{\Gamma,\lambda} = \{\pi^y_U: U \Subset V, y \in X(\Gamma)\}$. We call $\pi_{\Gamma,\lambda}$ the {\bf Gibbs $(\Gamma,\lambda)$-spe\-ci\-fi\-ca\-tion}. A measure $\mathbb{P} \in \mathcal{M}(X(\Gamma))$ is called a {\bf Gibbs measure (for $(\Gamma,\lambda)$)} if for all $U \Subset V$, $U' \subseteq U$, and $x \in X(\Gamma)$,
$$
\mathbb{P}([x_{U'}] \vert \mathcal{B}_{U^{\rm c}})(y) = \pi^y_U([x_{U'}]) \quad \mathbb{P}\text{-a.s. in } y,
$$
where $\pi^y_U([x_U'])$ denotes the marginalization
$$
\pi^y_U([x_U']) = \sum_{x' \in X(\Gamma[U]): x'_{U'} = x_{U'}} \pi^y_U(x')
$$
and $\mathbb{P}(A \vert \mathcal{B}_U) = \mathbb{E}_{\mathbb{P}}(\mathbbm{1}_{A} \vert \mathcal{B}_U)$ for $A \in \mathcal{B}_\Gamma$. We denote by $\mathcal{M}_{\mathrm{Gibbs}}(\Gamma,\lambda)$ the set of Gibbs measures for $(\Gamma,\lambda)$.
An important question in statistical physics is whether the set of Gibbs measures is empty or not, and if not, whether there is a unique or multiple Gibbs measures \cite{1-georgii}.
\subsection{The locally finite case}
The model described in \cite[Example 4.16]{1-georgii} can be understood as an attempt to formalize the idea of a system where there is a single particle $1$ (uniformly distributed) or none, i.e., $0$ everywhere. There, it is proven that such model cannot be represented as a Gibbs measure. This example can be also viewed as a hardcore model in a countable graph that is \emph{complete} (i.e., there is an edge between any pair of different vertices) and, in particular, in a non-locally finite graph. In other words, there exist examples of non-locally finite graphs $\Gamma$ such that the $(\Gamma,\lambda)$-specification $\pi_{\Gamma,\lambda}$ has no Gibbs measure.
From now on, we will always assume that $\Gamma$ is locally finite. In this case, the existence of Gibbs measures is guaranteed (see \cite{2-brightwell,2-dobrushin}) and, moreover, Gibbs measures must be a \emph{Markov random field} that is \emph{fully supported}.
Indeed, it can be checked that $\pi_{\Gamma,\lambda}$ is an example of a \emph{Markovian specification} (see \cite[Example 8.24]{1-georgii}).
In this case, any Gibbs measure $\mathbb{P} \in \mathcal{M}_{\mathrm{Gibbs}}(\Gamma,\lambda)$ satisfies the following \emph{local Markov property}:
$$
\mathbb{P}([x_U] \vert \mathcal{B}_{U^{\rm c}})(y) = \mathbb{P}([x_U] \vert \mathcal{B}_{\partial U})(y) \quad \mathbb{P}\text{-a.s. in } y,
$$
for any $U \Subset V$ and $x \in X(\Gamma)$. In other words, $\mathbb{P}$ is a {\bf Markov random field}, so any event supported on a finite set conditioned to a specific value on its boundary is independent of events supported on the complement.
In addition, it can be checked that any such $\mathbb{P}$ must be {\bf fully supported}, i.e., $\mathrm{supp}(\mathbb{P}) = X(\Gamma)$. Indeed, it suffices to check that $X(\Gamma) \subseteq \mathrm{supp}(\mathbb{P})$; the other direction follows directly from the definition of $\pi_{\Gamma,\lambda}$ and Gibbs measures. Now, given $x \in X(\Gamma)$ and $U \Subset V$, we would like to check that $\mathbb{P}([x_U]) > 0$. To prove this, observe that given $x \in X(\Gamma)$, we have that $z \in \{0,1\}^{V}$ defined as $z_U = x_U$, $z_{\partial U} \equiv 0$, and $z_{W^c} = y_{W^c}$, always belongs to $X(\Gamma)$ for any $y \in X(\Gamma)$, where $W = U \cup \partial U$. In particular, $\pi^y_{W}(z) > 0$ for any $y \in X(\Gamma)$. Then, considering that $\partial (W^c)$ is finite,
\begin{align*}
\mathbb{P}([x_U]) & \geq \mathbb{P}([z_{W}]) \\
& = \sum_{y \in X(\Gamma,\partial W): \mathbb{P}([y_{\partial W}]) > 0} \mathbb{P}([z_{W}] \vert [y_{\partial W}])\mathbb{P}([y_{\partial W}]) \\
& = \sum_{y \in X(\Gamma,\partial W): \mathbb{P}([y_{\partial W}]) > 0} \pi^y_{W}(z)\mathbb{P}([y_{\partial W}]) \\
& \geq \pi^{y^*}_{W}(z)\mathbb{P}([y^*_{\partial W}]) > 0,
\end{align*}
since $\mathbb{P}$ is a probability measure and there must exist $y^* \in X(\Gamma)$ such that $\mathbb{P}([y^*_{\partial W}]) > 0$. In other words, $X(\Gamma)$ satisfies the property (D*) introduced in \cite[1.14 Remark]{1-ruelle}, which guarantees full support.
\subsection{Spatial mixing and uniqueness}
Given a Gibbs $(\Gamma,\lambda)$-specification $\pi_{\Gamma,\lambda}$, we define two \emph{spatial mixing} properties fundamental to this work.
\begin{definition}
We say that a hardcore model $(\Gamma,\lambda)$ exhibits {\bf strong spatial mixing (SSM)} if there exists a \emph{decay rate} function $\delta: \mathbb{N} \to \mathbb{R}_{\geq 0}$ such that $\lim_{\ell \to \infty} \delta(\ell) = 0$ and for all $U \Subset V$, $v \in U$, and $y,z \in X(\Gamma)$,
$$
\left| \pi^y_U([0^v]) - \pi^z_U([0^v]) \right| \leq \delta(\mathrm{dist}_\Gamma(v,D_U(y,z))),
$$
where $[0^v]$ denotes the event that the vertex $v$ takes the value $0$ and
$$
D_U(y,z) := \{v' \in U^c: y(v') \neq z(v')\}.
$$
\end{definition}
This definition is equivalent (see \cite[Lemma 2.3]{2-marcus}) to the ---a priori--- stronger following property: for all $U' \subseteq U \Subset V$ and $x,y,z \in X(\Gamma)$,
$$
\left| \pi^y_U([x_{U'}]) - \pi^z_U([x_{U'}]) \right| \leq |U'|\delta(\mathrm{dist}_\Gamma(U',D_U(y,z))).
$$
Similarly, we say that $(\Gamma,\lambda)$ exhibits {\bf weak spatial mixing (WSM)} if for all $U' \subseteq U \Subset V$ and $x,y,z \in X(\Gamma)$,
$$
\left| \pi^y_U([x_{U'}]) - \pi^z_U([x_{U'}]) \right| \leq |U'| \cdot \delta(\mathrm{dist}_\Gamma(U',U^c)).
$$
Clearly, SSM implies WSM. Moreover, it is well-known that, in this context, WSM (and therefore, SSM) implies uniqueness of Gibbs measures \cite{2-weitz}. In other words, $\mathcal{M}_{\mathrm{Gibbs}}(\Gamma,\lambda) = \{\mathbb{P}_{\Gamma,\lambda}\}$, where $\mathbb{P}_{\Gamma,\lambda}$ denotes the unique Gibbs measure for $(\Gamma,\lambda)$. In this case, $\mathbb{P}_{\Gamma,\lambda}$ is always $\mathrm{Aut}(\Gamma)$-invariant.
We say that $(\Gamma,\lambda)$ exhibits {\bf exponential SSM} (resp. {\bf exponential WSM}) if there exist constants $C,\alpha > 0$ such that $\pi_{\Gamma,\lambda}$ exhibits SSM (resp. WSM) with decay rate function $\delta(n) = C \cdot \exp(-\alpha \cdot n)$.
Given $U \subseteq V$, we denote by $\Gamma \setminus U$ the subgraph induced by $V \setminus U$, i.e., $\Gamma[V \setminus U]$. We have the following result due to Gamarnik and Katz.
\begin{proposition}[{\cite[Proposition 1]{1-gamarnik}}]
\label{prop:GKssm}
If a hardcore model $(\Gamma,\lambda)$ satisfies SSM, then so does the hardcore model $(\Gamma',\lambda)$ for any subgraph $\Gamma'$ of $\Gamma$. The same assertion applies to exponential SSM. Moreover, for every $U \subseteq V$ and $v \in V \setminus U$, the following identity holds:
$$
\mathbb{P}_{\Gamma,\lambda}([0^v] \vert [0^U]) = \mathbb{P}_{\Gamma \setminus U,\lambda}([0^v]),
$$
where $\mathbb{P}_{\Gamma,\lambda}$ and $\mathbb{P}_{\Gamma \setminus U,\lambda}$ are the unique Gibbs measures for $(\Gamma,\lambda)$ and $(\Gamma \setminus U,\lambda)$, respectively, and $[0^U]$ denotes the event that all the vertices in $U$ take the value $0$. In particular, $\mathbb{P}_{\Gamma,\lambda}([0^v] \vert [0^U])$ is always well-defined, even if $U$ is infinite.
\end{proposition}
\begin{remark}
\label{ssm}
Notice that any event of the form $[x_U]$ can be translated into an event of the form $[0^{U' }]$ for a suitable set $U'$: it suffices to define $U' = U \cup \partial\{v \in U: x_U(v) = 1\}$ since, deterministically, every neighbor of a vertex colored $1$ must be $0$, so Proposition \ref{prop:GKssm} still holds for more general events. We also remark that in \cite{1-gamarnik} it is assumed that $\lambda$ is a constant function. Here we drop this assumption, but it is direct to check that the same proof of \cite[Proposition 1]{1-gamarnik} also applies to the more general non-constant case.
\end{remark}
\subsection{Families of hardcore models}
We introduce some notation regarding relevant families of hardcore models.
First, we will denote by $\mathcal{H}$ the family of hardcore models $(\Gamma, \lambda)$ such that $\Gamma$ is a countable locally finite graph and $\lambda$ is any activity function $\lambda: V(\Gamma) \to \mathbb{R}_{>0}$.
Given a countable group $G$, we will denote by $\mathcal{H}_G$ the set of hardcore models $(\Gamma,\lambda)$ in $\mathcal{H}$ for which $G$ is isomorphic to some subgroup of $\mathrm{Aut}(\Gamma)$ such that $G \curvearrowright \Gamma$ is free and almost transitive and $\lambda: V(\Gamma) \to \mathbb{R}_{>0}$ is a $G$-invariant activity function.
Given a positive integer $\Delta$, we will denote by $\mathcal{H}^\Delta$ the set of hardcore models $(\Gamma,\lambda)$ in $\mathcal{H}^\Delta$ such that $\Delta(\Gamma) \leq \Delta$. Notice that any hardcore model defined on the $\Delta$-regular (infinite) tree $\mathbb{T}_\Delta$ belongs to $\mathcal{H}^\Delta$.
Given $\lambda_0 > 0$, we will denote by $\mathcal{H}(\lambda_0)$ the family of hardcore models $(\Gamma,\lambda)$ in $\mathcal{H}$ such that $\lambda_+ \leq \lambda_0$.
We will also combine the notation for these families in the natural way; for example, $\mathcal{H}_G^\Delta(\lambda_0)$ will denote the set of hardcore models $(\Gamma,\lambda)$ in $\mathcal{H}$ such that $G \curvearrowright \Gamma$ is free and almost transitive, $\lambda$ is $G$-invariant, $\Delta(\Gamma) \leq \Delta$, and $\lambda_+ \leq \lambda_0$.
\section{Trees}
\label{sec5}
Given a graph $\Gamma$, a {\bf trail} $w$ in $\Gamma$ is a finite sequence $w = (v_1,\dots,v_n)$ of vertices such that consecutive vertices are adjacent in $\Gamma$ and the edges $(v_i,v_{i+1})$ involved are not repeated. For a fixed vertex $v \in V(\Gamma)$, the {\bf tree of self-avoiding walks starting from $v$}, denoted by $T_{\mathrm{SAW}}(\Gamma,v)$, is defined as follows:
\begin{enumerate}
\item Consider the set $W_0$ of trails starting from $v$ that repeat no vertex and the set $W_1$ of trials that repeat a single vertex exactly once and then stop (i.e., the set of non-backtracking walks that end immediately after performing a cycle). We define $T_{\mathrm{SAW}}(\Gamma,v)$ to be a rooted tree with root $\rho = (v)$ such that the set of vertices $V(T_{\mathrm{SAW}}(\Gamma,v))$ is $W_0 \cup W_1$ and the set of (undirected) edges $E(T_{\mathrm{SAW}}(\Gamma,v))$ corresponds to all the pairs $(w,w')$ such that $w'$ is a one vertex extension of $w$ or vice versa. In simple words, $T_{\mathrm{SAW}}(\Gamma,v)$ is a rooted tree that represents all self-avoiding walks in $\Gamma$ that start from $v$. It is easy to check that the set of leaves of $T_{\mathrm{SAW}}(\Gamma,v)$ contains $W_1$, but they are not necessarily equal (e.g., see vertex $b$ in Figure \ref{fig:diagram1}).
\item For $u \in V(\Gamma)$, consider an arbitrary ordering of its neighbors $\partial\{u\} = \{u_1, \dots, u_d\}$. Given $w \in W_1$, we can represent such walk as a sequence $w = (v,\dots, u,u_i,\dots,u_j,u)$, with $u_i,u_j \in \partial\{u\}$. Notice that $i \neq j$, since we are not repeating edges. Considering this, we ``tag'' $w$ with a $1$ (\emph{occupied}) if $i < j$ and with a $0$ (\emph{unoccupied}) if $i > j$.
\end{enumerate}
Given a hardcore model $(\Gamma,\lambda)$, a vertex $v \in V(\Gamma)$, a subset $U \subseteq V(\Gamma)$, and an independent set $x \in X(\Gamma)$, we are interested in computing the marginal probability that $v$ is unoccupied in $\Gamma$ given the partial configuration $x_U$, i.e., $\mathbb{P}_{\Gamma,\lambda}([0^v] \vert [x_U])$. Notice that if $(\Gamma,\lambda)$ satisfies SSM (which includes the particular but relevant case of $\Gamma$ being finite), then such probability is always well defined due to Proposition \ref{prop:GKssm}, even if $U$ is infinite.
\begin{figure}[ht]
\centering
\includegraphics[scale = 0.8]{saw-1.pdf}
\caption{A graph $\Gamma$ and its corresponding (tagged) tree of self-avoiding walks $T_{\mathrm{SAW}}(\Gamma,v)$, where every trial/vertex is represented by the final vertex of the trial in $\Gamma$ starting from $v = a$.}
\label{fig:diagram1}
\end{figure}
To understand better $\mathbb{P}_{\Gamma,\lambda}([0^v] \vert [x_U])$, we consider the hardcore model $(T_{\mathrm{SAW}}(\Gamma,v),\overline{\lambda})$, where $\overline{\lambda}(w) = \lambda(u)$ for every trial $w$ ending in $u$. In this context, a condition $x_U$ in $(\Gamma,\lambda)$ is translated into the condition $\overline{x_U}$ in $T_{\mathrm{SAW}}(\Gamma,v)$, whose support is the set $W(U)$ of trials $w$ that end in $u$ for some $u \in U$, and $\overline{x}(w) = x(u)$ for all such $w$. We have the following result from \cite{1-weitz}, that we adapt to the more general non-constant $\lambda$ case and we include its proof for completeness.
\begin{theorem}[{\cite[Theorem 3.1]{1-weitz}}]
\label{thm21}
For every finite hardcore model $(\Gamma,\lambda)$, every $v \in V(\Gamma)$, and $U \subseteq V(\Gamma)$,
$$
\mathbb{P}_{\Gamma,\lambda}([0^v] \vert [x_U]) = \mathbb{P}_{T_{\mathrm{SAW}}(\Gamma,v),\overline{\lambda}}([0^\rho] \vert [\overline{x_U}]).
$$
\end{theorem}
\begin{proof}
Instead of probabilities, we work with the ratios
$$
R_{\Gamma,\lambda}(v,x_U) := \frac{\mathbb{P}_{\Gamma,\lambda}([1^v] \vert [x_U])}{\mathbb{P}_{\Gamma,\lambda}([0^v] \vert [x_U])},
$$
where if $v \in U$ and $x_U(v)$ is equal to $1$ or $0$, we let $R_{\Gamma,\lambda}(v,x_U)$ to be $\infty$ or $0$, respectively. Notice that
$$
\mathbb{P}_{\Gamma,\lambda}([0^v] \vert [x_U]) = \frac{1}{1+R_{\Gamma,\lambda}(v,x_U)} \quad \mbox{ and } \quad \mathbb{P}_{\Gamma,\lambda}([1^v] \vert [x_U]) = \frac{R_{\Gamma,\lambda}(v,x_U)}{1+R_{\Gamma,\lambda}(v,x_U)}.
$$
Given a finite tree $T$ rooted at $\rho$, let's denote by $\left\{\rho_1,\dots,\rho_d\right\}$ the set of neighbors $\partial \{\rho\}$ of $\rho$ and by $T_i$, for $i=1,\dots,d$, the corresponding subtrees starting from $\rho_i$, i.e., $V(T) = \{\rho\} \cup V(T_1) \cup \cdots \cup V(T_d)$. If we have a condition $x_U$ on $U$, we define $U_i = U \cap V(T_i)$ and $x_{U_i} = (x_{U})\vert_{U_i}$. Considering this, we have that
\begin{align*}
R_{T,\lambda}(\rho, x_U) & = \frac{\mathbb{P}_{T,\lambda}([1^\rho] \vert [x_U])}{\mathbb{P}_{T,\lambda}([0^\rho] \vert [x_U])} \\
& = \frac{\lambda(\rho) \cdot Z^{x_U}_{T \backslash \{\rho \cup \partial \{\rho\}\}}(\lambda)}{Z^{x_U}_T(\lambda)} \cdot \frac{Z^{x_U}_T(\lambda)}{Z^{x_U}_{T \backslash \{\rho\}}(\lambda)} \\
& = \lambda(\rho) \cdot \prod_{i=1}^{d}\frac{Z^{x_{U_i}}_{T_i \backslash \{\rho_i\}}(\lambda_i)}{Z^{x_{U_i}}_{T_i}(\lambda)} \\
& = \lambda(\rho) \cdot \prod_{i=1}^{d} \mathbb{P}_{T_i,\lambda}([0^{\rho_i}] \vert [x_{U_i}]) \\
& = \lambda(\rho) \cdot \prod_{i=1}^{d} \frac{1}{1+R_{T_i,\lambda}(\rho_i, x_{U_i})},
\end{align*}
where
$$
Z^{x_U}_\Gamma(\lambda) := \sum_{y \in X(\Gamma): y_U = x_U} \prod_{v \in V(\Gamma)} \lambda(v)^{y(v)}.
$$
Notice that this gives us a linear recursive procedure for computing $R_{T,\lambda}(\rho, x_U)$, and therefore $\mathbb{P}_{T,\lambda}([0^\rho] \vert [x_U])$, with base cases: $R_{T,\lambda}(\rho, x_U) = 0 \text{ or } +\infty$ if $\rho$ is fixed, and $R_{T,\lambda}(\rho, x_U) = \lambda(\rho)$ if $\rho$ is free and isolated.
Now, consider an arbitrary hardcore model $(\Gamma,\lambda)$ and $v \in V(\Gamma)$ with neighbors $\partial\{v\} = \{u_1,\dots,u_d\}$. We consider the auxiliary hardcore model $(\Gamma',\lambda')$, where
\begin{itemize}
\item $V(\Gamma') = V(\Gamma) \backslash \{v\} \cup \{v_1,\dots,v_d\}$,
\item $E(\Gamma') = E(\Gamma) \backslash \{(v,u_i)\}_{i=1,\dots,d} \cup \{(v_i,u_i)\}_{i=1,\dots,d}$,
\item $\lambda'(v_i) = \lambda(v)^{1/d}$ for $i=1,\dots,d$, and $\lambda'(u) = \lambda(u)$, otherwise.
\end{itemize}
Notice that
\begin{align*}
R_{\Gamma,\lambda}(v, x_U) & = \frac{\mathbb{P}_{\Gamma,\lambda}\left([1^v] \middle\vert [x_U]\right)}{\mathbb{P}_{\Gamma,\lambda}\left([0^v] \middle\vert [x_U]\right)} \\
& = \frac{\mathbb{P}_{\Gamma',\lambda'}\left([1^{\{v_1, \dots, v_d\}}] \middle\vert [x_U]\right)}{\mathbb{P}_{\Gamma',\lambda'}\left([0^{\{v_1, \dots, v_d\}}] \middle\vert [x_U]\right)} \\
& = \prod_{i=1}^{d}\frac{\mathbb{P}_{\Gamma',\lambda'}\left([0^{\{v_1,\dots,v_{i-1}\}}1^{\{v_i, \dots, v_d\}}] \middle\vert [x_U]\right)}{\mathbb{P}_{\Gamma',\lambda'}\left([0^{\{v_1,\dots,v_i\}}1^{\{v_{i+1}, \dots, v_d\}}] \middle\vert [x_U]\right)} \\
& = \prod_{i=1}^{d}\frac{\mathbb{P}_{\Gamma',\lambda'}\left([1^{v_i}] \middle\vert [x_U\tau_i]\right)}{\mathbb{P}_{\Gamma',\lambda'}\left([0^{v_i}] \middle\vert [x_U\tau_i]\right)} \\
& = \prod_{i=1}^{d}R_{\Gamma',\lambda'}(v_i,x_Uz_i),
\end{align*}
where $z_i = 0^{\{v_1,\dots,v_{i-1}\}}1^{\{v_{i+1}, \dots, v_d\}}$ and $x_Uz_i$ is the concatenation of $x_U$ and $z_i$. Now, since $v_i$ is connected only to $u_i$, notice that
$$
R_{\Gamma',\lambda'}(v_i,x_Uz_i) = \frac{\lambda'(v_i) \cdot Z^{x_Uz_i}_{\Gamma' \backslash \{v_i,u_i\}}(\lambda')}{Z^{x_U z_i}_{\Gamma' \backslash \{v_i\}}(\lambda')}
= \frac{\lambda^{1/d}(v)}{1+R_{\Gamma' \backslash \{v_i\},\lambda'}(u_i,x_Uz_i)}.
$$
Therefore,
$$
R_{\Gamma,\lambda}(v, x_U) = \prod_{i=1}^{d}\frac{\lambda^{1/d}(v)}{1+R_{\Gamma' \backslash \{v_i\},\lambda'}(u_i,x_Uz_i)} = \lambda(v) \cdot \prod_{i=1}^{d}\frac{1}{1+ R_{\Gamma' \backslash \{v_i\},\lambda'}(u_i,x_Uz_i)}.
$$
Notice that the previous recursion can increase the original number of vertices, but the number of free vertices always decrease, so the recursion ends. Then, we have that
\begin{enumerate}
\item $R_{T,\lambda}(v, x_U) = \lambda(\rho) \cdot f\left(R_{T_1,\lambda}(\rho_1, x_{U_1}),\dots,R_{T_d,\lambda}(\rho_d, x_{U_d})\right)$ and
\item $R_{\Gamma,\lambda}(v, x_U) = \lambda(v) \cdot f\left(R_{\Gamma' \backslash \{v_1\},\lambda'}(u_1,x_U\tau_1),\dots,R_{\Gamma' \backslash \{v_d\},\lambda'}(u_d,x_U\tau_d)\right)$,
\end{enumerate}
where $f(r_1,\dots,r_d) = \prod_{i=1}^{d}\frac{1}{1+r_i}$. Now we proceed by induction in the number of free vertices. We can consider the base case where there are no free vertices (besides $v$) and the theorem is trivial. Then, if we know that the theorem is true when we have $n$ free vertices, we prove it for $n+1$. Notice that if $R_{\Gamma',\lambda'}(v,x_U)$ involves $n+1$ free vertices, then $R_{\Gamma' \setminus \{v\},\lambda'}(v_i,x_Uz_i)$ involves $n$ free vertices, so by the induction hypothesis,
$$
R_{\Gamma' \backslash \{v_i\},\lambda}(u_i,x_Uz_i) = R_{T_{\mathrm{SAW}}(\Gamma,v),\overline{\lambda}}(\rho_i,\overline{x_{U}z_i}).
$$
Then, noticing that the rooted subtree $(T_i,\rho_i)$ and the condition $\overline{x_{U}z_i}$ gives exactly the tree of self-avoiding walks of $\Gamma' \backslash \{v_i\}$ starting from $u_i$ under the condition $x_Uz_i$, we are done.
\end{proof}
\begin{figure}[ht]
\centering
\includegraphics[scale = 0.8]{saw-2.pdf}
\caption{Condition on $\Gamma$ and its representation on the tree of self-avoiding walks $T_{\mathrm{SAW}}(\Gamma,v)$ for $v = a$.}
\label{fig:diagram2}
\end{figure}
\begin{remark}
The recursions presented in the proof of Theorem \ref{thm21} give us a recursive procedure to compute the marginal probability of the root $\rho$ of a tree $T$ being occupied which requires linear time with respect to the size of the tree. On the other hand, if $\Gamma$ is such that $\Delta(\Gamma) \leq \Delta$, then $T_{\mathrm{SAW}}(\Gamma,v)$ is a subtree of $\mathbb{T}_\Delta$ and its size of $T_{\mathrm{SAW}}(\Gamma,v)$ can be (at most) exponential in the size of $\Gamma$. Since hardcore models are Markov random fields and we are interested in the sensitivity of the root $\rho$ associated to $v$, we only need to consider the graph obtained after pruning all the subtrees below $W(U)$ (see Figure \ref{fig:diagram2}).
\end{remark}
Before stating the main results concerning hardcore models and strong spatial mixing, we will establish the following bounds.
\begin{lemma}
\label{lem:probbound}
Given a finite hardcore model $(\Gamma,V) \in \mathcal{H}^\Delta$ and $v \in V$, we have that
$$
0 < \frac{1}{1+\lambda_+} \leq \mathbb{P}_{\Gamma,\lambda}([0^v]) \leq \frac{(1+\lambda_+)^\Delta}{\lambda_- + (1+\lambda_+)^\Delta} < 1
$$
and
$$
0 < \frac{\lambda_-}{\lambda_- + (1+\lambda_+)^\Delta} \leq \mathbb{P}_{\Gamma,\lambda}([1^v]) \leq \frac{\lambda_+}{1+\lambda_+} < 1.
$$
\end{lemma}
\begin{proof}
Notice that, since $\mathbb{P}_{\Gamma,\lambda}$ is a Markov random field and a $1$ at $v$ forces $0$s in $\partial\{v\}$,
$$
\mathbb{P}_{\Gamma,\lambda}([1^v]) = \mathbb{P}_{\Gamma,\lambda}([1^v] \vert [0^{\partial \{v\}}]) \mathbb{P}_{\Gamma,\lambda}([0^{\partial \{v\}}]) \leq \mathbb{P}_{\Gamma,\lambda}([1^v] \vert [0^{\partial \{v\}}]),
$$
so, considering that $\frac{\lambda}{1+\lambda}$ is increasing in $\lambda > 0$, we obtain that
$$
\mathbb{P}_{\Gamma,\lambda}([1^v]) \leq \mathbb{P}_{\Gamma,\lambda}([1^v] \vert [0^{\partial \{v\}}]) = \frac{\lambda(v)}{1+\lambda(v)} \leq \frac{\lambda_+}{1+\lambda_+} < 1,
$$
and
$$
\mathbb{P}_{\Gamma,\lambda}([0^v]) = 1 - \mathbb{P}_{\Gamma,\lambda}([1^v]) \geq 1 -\frac{\lambda_+}{1+\lambda_+} = \frac{1}{1+\lambda_+} > 0.
$$
On the other hand, by Theorem \ref{thm21}, without loss of generality, we can suppose that $\Gamma$ is a tree rooted at $v$. Then, if $\Gamma_i$ denotes the $i$th subtree of $\Gamma$ rooted at $v_i \in \partial \{v\}$,
$$
\frac{\mathbb{P}_{\Gamma,\lambda}([1^v])}{\mathbb{P}_{\Gamma,\lambda}([0^v])} = \lambda(v) \cdot \prod_{i=1}^{d} \frac{1}{1+\frac{\mathbb{P}_{(\Gamma_i,\lambda)}([1^{v_i}])}{\mathbb{P}_{(\Gamma_i,\lambda)}([0^{v_i}])}} \geq \lambda_- \cdot \prod_{i=1}^{d} \frac{1}{1+\frac{\frac{\lambda_+}{1+\lambda_+}}{\frac{1}{1+\lambda_+}}} \geq \frac{\lambda_-}{(1+\lambda_+)^{\Delta}}.
$$
Therefore, since $\mathbb{P}_{\Gamma,\lambda}([0^v]) = 1 - \mathbb{P}_{\Gamma,\lambda}([1^v])$, we have that
$$
\mathbb{P}_{\Gamma,\lambda}([1^v]) \geq \frac{\lambda_-}{\lambda_- + (1+\lambda_+)^\Delta} > 0 \quad \text{ and } \quad \mathbb{P}_{\Gamma,\lambda}([0^v]) \leq \frac{(1+\lambda_+)^\Delta}{\lambda_- + (1+\lambda_+)^\Delta} < 1.
$$
\end{proof}
We define the {\bf critical activity function} $\lambda_c: [2,+\infty) \to (0,+\infty]$ as
$$
\lambda_c(t) := \frac{(t-1)^{(t-1)}}{(t-2)^{t}}.
$$
From the works of Kelly \cite{1-kelly} and Spitzer \cite{1-spitzer}, we have the following result.
\begin{proposition}
\label{prop:kelly}
For every $\Delta \in \mathbb{N}$, the hardcore model $(\mathbb{T}_{\Delta}, \lambda_0)$ exhibits WSM if and only if $\lambda_0 \leq \lambda_c(\Delta)$. If the inequality is strict, then $(\mathbb{T}_{\Delta}, \lambda_0)$ exhibits exponential WSM with a decay rate $\delta$ involving constants that depend on $\Delta$ and $\lambda_0$.
\end{proposition}
We summarize in the following theorem the main results from \cite{1-weitz}, that relate the correlation decay in $(\mathbb{T}_{\Delta}, \lambda_0)$ with the correlation decay in $\mathcal{H}^\Delta(\lambda_0)$. Here again, as in Theorem \ref{thm21}, the results in \cite{1-weitz} are focused on the constant activity case. However, we can also adapt the results to the non-constant case by considering that the main tool used in \cite{1-weitz} to prove them is \cite[Theorem 4.1]{1-weitz}, which is based on hardcore models with non-constant activity functions.
\begin{theorem}[{\cite[Theorem 2.3 and Theorem 2.4]{1-weitz}}]
\label{thm:wsmssm}
Fix $\Delta \in \mathbb{N}$ and $\lambda_0 > 0$. Then,
\begin{enumerate}
\item If $(\mathbb{T}_\Delta,\lambda_0)$ exhibits WSM with decay rate $\delta$, then $(\mathbb{T}_\Delta,\lambda_0)$ exhibits SSM with rate
$$\frac{(1+\lambda_0)(\lambda_0 + (1+\lambda_0)^{\Delta})}{\lambda_0}\delta;$$
\item If $(\mathbb{T}_\Delta,\lambda_0)$ exhibits SSM with decay rate $\delta$, then $(\Gamma,\lambda)$ exhibits SSM with rate $\delta$ for every $(\Gamma,\lambda) \in \mathcal{H}^\Delta(\lambda_0)$.
\end{enumerate}
\end{theorem}
Then, combining Proposition \ref{prop:kelly} and Theorem \ref{thm:wsmssm}, we have that if $\lambda_0 \leq \lambda_{c}(\Delta)$, then every hardcore model $(\Gamma,\lambda) \in \mathcal{H}^\Delta(\lambda_0)$ exhibits SSM with the same decay rate $\delta$, that would be exponential if the inequality is strict. In addition, observe that if $(\Gamma,\lambda)$ is a hardcore model such that $(T_{\mathrm{SAW}}(\Gamma,v),\overline{\lambda})$ exhibits SSM with decay rate $\delta$ for every $v \in V(\Gamma)$, then $(\Gamma,\lambda)$ exhibits SSM with decay rate $\delta$ as well. This follows from Theorem \ref{thm21}, since SSM is a property that depends on finitely supported events and the probabilities involved can be translated into probabilities defined on finite hardcore models which at the same time can be translated into events on finite subtrees of $T_{\mathrm{SAW}}(\Gamma,v)$. Considering this, we have the following theorem, which can be understood as a generalization of Theorem \ref{thm21} to the infinite setting.
\begin{theorem}
\label{thm:tsaw}
Given a hardcore model $(\Gamma,\lambda)$ and $v \in V(\Gamma)$ such that $(T_{\mathrm{SAW}}(\Gamma,v),\overline{\lambda})$ exhibits SSM, then for every $x \in X(\Gamma)$ and $U \subseteq V(\Gamma)$,
$$
\mathbb{P}_{\Gamma,\lambda}([0^v] \vert [x_U]) = \mathbb{P}_{T_{\mathrm{SAW}}(\Gamma,v),\overline{\lambda}}([0^\rho] \vert [\overline{x_U}]).
$$
\end{theorem}
\begin{proof}
Assume that $(T_{\mathrm{SAW}}(\Gamma,v),\overline{\lambda})$ exhibits SSM with decay rate $\delta$. Then, for every $\ell \in \mathbb{N}$,
\begin{align*}
\mathbb{P}_{\Gamma,\lambda}([0^v] \vert [x_U]) & = \sum_{w \in \{0,1\}^{\partial B_\Gamma(v,\ell) \setminus U}} \mathbb{P}_{\Gamma,\lambda}([0^v] \vert [x_Uw])\mathbb{P}_{\Gamma,\lambda}([w] \vert [x_U]) \\
& \leq \sum_{w \in \{0,1\}^{\partial B_\Gamma(v,\ell) \setminus U}} (\mathbb{P}_{\Gamma,\lambda}([0^v] \vert [x_U0^{\partial B_\Gamma(v,\ell) \setminus U}]) + \delta(\ell))\mathbb{P}_{\Gamma,\lambda}([w] \vert [x_U]) \\
& = \mathbb{P}_{\Gamma,\lambda}([0^v] \vert [x_U0^{\partial B(v,\ell) \setminus U}]) + \delta(\ell)
\end{align*}
and, similarly,
\begin{align*}
\mathbb{P}_{\Gamma,\lambda}([0^v] \vert [x_U]) & \geq \mathbb{P}_{\Gamma,\lambda}([0^v] \vert [x_U0^{\partial B_\Gamma(v,\ell) \setminus U}]) - \delta(\ell).
\end{align*}
Therefore, since $\lim_{\ell \to \infty} \delta(\ell) = 0$,
$$
\mathbb{P}_{\Gamma,\lambda}([0^v] \vert [x_U]) = \lim_{\ell \to \infty} \mathbb{P}_{\Gamma,\lambda}([0^v] \vert [x_U0^{\partial B_\Gamma(v,\ell) \setminus U}]),
$$
and, by the same argument,
$$
\mathbb{P}_{(T_{\mathrm{SAW}}(\Gamma,v),\overline{\lambda})}([0^\rho] \vert [\overline{x_U}]) = \lim_{\ell \to \infty} \mathbb{P}_{(T_{\mathrm{SAW}}(\Gamma,v),\overline{\lambda})}([0^\rho] \vert [\overline{x_U}0^{\partial B_{T_{\mathrm{SAW}}(\Gamma,v)}(\rho,\ell) \setminus W(U)}]).
$$
Considering this, the Markov random field property, and Proposition \ref{prop:GKssm}, we have that
\begin{align*}
\mathbb{P}_{\Gamma,\lambda}([0^v] \vert [x_U]) & = \lim_{\ell \to \infty} \mathbb{P}_{\Gamma,\lambda}([0^v] \vert [x_U0^{\partial B_\Gamma(v,\ell) \setminus U}]) \\
& = \lim_{\ell \to \infty} \mathbb{P}_{\Gamma \cap B(v,\ell),\lambda}([0^v] \vert [x_{U \cap B_\Gamma(v,\ell)}]) \\
& = \lim_{\ell \to \infty} \mathbb{P}_{T_{\mathrm{SAW}}(\Gamma \cap B_{T_{\mathrm{SAW}}(\Gamma,v)}(v,\ell),v),\overline{\lambda}}([0^\rho] \vert [\overline{x_{U \cap B_\Gamma(v,\ell)}}]) \\
& = \lim_{\ell \to \infty} \mathbb{P}_{T_{\mathrm{SAW}}(\Gamma,v),\overline{\lambda}}([0^\rho] \vert [\overline{x_U}0^{\partial B_{T_{\mathrm{SAW}}(\Gamma,v)}(\rho,\ell) \setminus W(U)}]) \\
& = \mathbb{P}_{T_{\mathrm{SAW}}(\Gamma,v),\overline{\lambda}}([0^\rho] \vert [\overline{x_U}]).
\end{align*}
\end{proof}
Notice that Theorem \ref{thm:tsaw} requires that $(T_{\mathrm{SAW}}(\Gamma,v),\overline{\lambda})$ exhibits SSM rather than the graph $(\Gamma,\lambda)$, since SSM on $T_{\mathrm{SAW}}(\Gamma,v)$ may be a stronger condition than SSM on $\Gamma$. A key fact is that if $(\mathbb{T}_\Delta, \lambda_0)$ exhibits SSM, then $(T, \lambda_0)$ exhibits SSM for every subtree $T$ of $\mathbb{T}_\Delta$. Then, since for every $\Gamma$ with $\Delta(\Gamma) \leq \Delta$, we have that $T_{\mathrm{SAW}}(\Gamma,v)$ is a subtree of $\mathbb{T}_\Delta$, it follows that $(T_{\mathrm{SAW}}(\Gamma,v),\overline{\lambda})$, and therefore $(\Gamma,\lambda_0)$, exhibit SSM. Considering this, we have the following corollary.
\begin{corollary}
Fix $\Delta \in \mathbb{N}$. Then, every $(\Gamma,\lambda) \in \mathcal{H}^\Delta(\lambda_c(\Delta))$ exhibits SSM and for every $v \in V(\Gamma)$, $x \in X(\Gamma$), and $U \subseteq V(\Gamma)$,
$$
\mathbb{P}_{\Gamma,\lambda}([0^v] \vert [x_U]) = \mathbb{P}_{T_{\mathrm{SAW}}(\Gamma,v),\overline{\lambda}}([0^\rho] \vert [\overline{x_U}]).
$$
\end{corollary}
Since we are ultimately interested in studying the interplay between the SSM property on $T_{\mathrm{SAW}}(\Gamma,v)$ and $\Gamma$, we may wonder whether is really necessary to have control over the full $\Delta$-regular tree $\mathbb{T}_\Delta$. In \cite{1-sinclair}, it was proven a refinement of this fact by considering the \emph{connective constant} of the graphs involved.
\subsection{Connective constant}
Following \cite{1-sinclair}, given a family of finite graphs $\mathcal{G}$, we define the {\bf connective constant} of $\mathcal{G}$ as the infimum over all $\mu > 0$ for which there exist $a,c > 0$ such that for any $\Gamma \in \mathcal{G}$ and any $v \in V(\Gamma)$, it holds that $\sum^\ell_{k=1} N_\Gamma(v,k) \leq c\mu^\ell$ for all $\ell \geq a\log|V(\Gamma)|$, where $N_\Gamma(v,k)$ denotes the \emph{number of self-avoiding walks} in $\Gamma$ of length $k$ starting from $v$. We denote by $\mu(\mathcal{G})$ the connective constant of $\mathcal{G}$. This definition extends the more usual definition of connective constant for a single infinite almost transitive graph $\Gamma$, which is given by
$$
\mu(\Gamma) := \max_{v \in V(\Gamma)} \lim_{\ell \to \infty} N_\Gamma(v,\ell)^{1/\ell}.
$$
Indeed, if $\Gamma$ is almost transitive, then $\mu(\Gamma) = \mu(\mathcal{G}(\Gamma))$, where $\mathcal{G}(\Gamma)$ denotes the family of finite subgraphs of $\Gamma$. Notice that $\mu(\Gamma)$ exists due to Fekete's lemma and that, if $\Gamma$ is connected, then $\mu(\Gamma) = \lim_{\ell \to \infty} N_\Gamma(v,\ell)^{1/\ell}$ for arbitrary $v$. Roughly, the connective constant measures the growth rate of the number of self-avoiding walks according to their length or, equivalently, the \emph{branching} of $T_{\mathrm{SAW}}(\Gamma,v)$. In general, it is not an easy task to compute $\mu(\Gamma)$ (e.g., see \cite{1-duminil}).
Considering this, we extend the definition of strong spatial mixing to families of graphs as follows: Given a family of graphs $\mathcal{G}$ and a family of activity functions $\Lambda = \{\lambda^\Gamma\}_{\Gamma \in \mathcal{G}}$ with $\lambda^\Gamma: V(\Gamma) \to \mathbb{R}_{>0}$, we say that $(\mathcal{G}, \Lambda)$ satisfies strong spatial mixing if there exists a decay rate function $\delta: \mathbb{N} \to \mathbb{R}_{\geq 0}$ such that $\lim_{\ell \to \infty} \delta(\ell) = 0$ and for all $\Gamma \in \mathcal{G}$, for all $U \Subset V(\Gamma)$, $v \in U$, and $y,z \in X(\Gamma)$,
$$
\left| \pi^y_{\Gamma, U}([0^v]) - \pi^z_{\Gamma, U}([0^v]) \right| \leq \delta(\mathrm{dist}_\Gamma(v,D_U(y,z))),
$$
where $\pi^y_{\Gamma, U}$ denotes the specification element corresponding to the hardcore model $(\Gamma, \lambda^\Gamma)$. We translate into this language the following result from \cite{1-sinclair}.
\begin{theorem}[\cite{1-sinclair}]
\label{thm:sinclair}
Let $\mathcal{G}$ be a family of almost transitive locally finite graphs and $\Lambda = \{\lambda^{\Gamma}\}_{\Gamma \in \mathcal{G}}$ a set of activity functions such that
$$
\sup_{\Gamma \in \mathcal{G}} \lambda^\Gamma_+ < \lambda_{c}(\mu(\mathcal{G})+1).
$$
Then, $(\mathcal{G}, \Lambda)$ exhibits exponential SSM.
\end{theorem}
Notice that if a graph has maximum degree $\Delta$, then $\mu(\Gamma) \leq \Delta-1$. In addition, observe that $N_\Gamma(v,\ell) = N_{T_{\mathrm{SAW}}(\Gamma,v)}(\rho,\ell)$. We have the following corollary.
\begin{corollary}
If $(\Gamma,\lambda)$ is a hardcore model such that
$$
\lambda_+ < \lambda_c(\mu(\Gamma)+1),
$$
then $(T_{\mathrm{SAW}}(\Gamma,v),\overline{\lambda})$ exhibits (exponential) SSM for every $v \in V(\Gamma)$. In particular, $(\Gamma,\lambda)$ exhibits (exponential) SSM and for every $v \in V(\Gamma)$, $x \in X(\Gamma$), and $U \subseteq V(\Gamma)$,
$$
\mathbb{P}_{\Gamma,\lambda}([0^v] \vert [x_U]) = \mathbb{P}_{T_{\mathrm{SAW}}(\Gamma,v),\overline{\lambda}}([0^\rho] \vert [\overline{x_U}]).
$$
\end{corollary}
\section{Orders}
\label{sec6}
We have already explored the main combinatorial and measure-theoretical tools that we require to establish the main results. In this section, we present some concepts of a more group-theoretical nature, namely, our ability to order a given group.
\subsection{Orderable groups}
Let $\prec$ be a strict total order on $G$. We say that $\prec$ is an {\bf invariant (right) order} if, for all $h_1,h_2,g \in G$,
$$
h_1 \prec h_2 \implies h_1g \prec h_2g.
$$
If $\prec$ is an invariant order, then the associated {\bf algebraic past} $\Phi_\prec := \{g \in G: g \prec 1_G\}$ is a semigroup such that
$$
G = \Phi_\prec \sqcup \{1_G\} \sqcup \Phi_\prec^{-1}.
$$
Notice that $h \prec g \iff hg^{-1} \in \Phi_\prec$, so $\Phi_\prec$ fully determines $\prec$ and vice versa.
A group $G$ will be called {\bf orderable} if it admits an invariant order and {\bf almost orderable} if there exists an orderable subgroup $H \leq G$ such that $[G:H] < \infty$. Notice that if $G$ is almost orderable and $G \curvearrowright \Gamma$ is almost transitive and free with fundamental domain $U_0$, then $H \curvearrowright \Gamma$ is also almost transitive and free with fundamental domain $KU_0$, where $K \in \mathcal{F}(G)$ is any finite set of representatives. In particular, $|\Gamma/H| = |\Gamma/G|[G:H]$. For this reason, since we are interested in almost transitive actions, there is no loss of generality if, given an almost orderable group, we assume that is just orderable.
Orderability is a local property, that is, if every finitely-generated subgroup of a given group is orderable, then the whole group is orderable. Given a finitely generated group $G$, a generating set $S$, and its corresponding Cayley graph $\Gamma = \mathrm{Cay}(G,S)$, we define the volume growth function as $g_\Gamma(n) = |B_\Gamma(1_G,n)|$. We say that $G$ has {\bf polynomial growth} if $g_\Gamma(n) \leq p(n)$ for some polynomial $p$. It is well-known that groups with polynomial growth are amenable and a classic result due to Gromov asserts that they are virtually nilpotent \cite{1-gromov}. Without further detail, from Schreier's Lemma, it is also well-known that finite index subgroups of finitely generated groups are also finitely generated \cite[Proposition 4.2]{1-lyndon} and finitely generated nilpotent groups have a torsion-free nilpotent subgroup with finite index \cite[Proposition 2]{1-segal}. From this, and since torsion-free nilpotent groups are orderable \cite[p.37]{1-mura}, it follows that any finitely generated group $G$ with polynomial growth is amenable and almost orderable. In particular, all our results that apply to amenable and almost orderable groups will hold for groups of polynomial growth, but they will also hold in groups of super-polynomial ---namely, exponential--- growth, like solvable groups that are not virtually nilpotent \cite{1-milnor} and, more concretely, cases like the Baumslag-Solitar groups $\mathrm{BS}(1,n)$, that can also be ordered. On the other hand, not every amenable groups is almost orderable. In order to address this issue, we introduce a randomized generalization of invariant orders.
\subsection{Random orders}
Consider now the set of relations $\{0,1\}^{G \times G}$ endowed with the product topology and the closed subset $\mathrm{Ord}(G)$ of strict total orders $\prec$ on $G$. We will consider the action $G \curvearrowright \mathrm{Ord}(G)$ given by
$$
h_1(g ~\cdot \prec)h_2 \iff (h_1g)\prec(h_2g)
$$
for $h_1,h_2,g \in G$ and $\prec \in \mathrm{Ord}(G)$. An {\bf invariant random order} on $G$ is a $G$-invariant Borel probability measure on $\mathrm{Ord}(G)$. Notice that a fixed point for the action $G \curvearrowright \mathrm{Ord}(G)$ corresponds to a (deterministic) invariant order on $G$. The space of invariant random orders will be denoted by $\mathcal{M}_G(\mathrm{Ord}(G))$.
Invariant random orders were introduced in \cite{1-alpeev} in order to answer problems about predictability in topological dynamics through what they called the \emph{Kieffer-Pinsker formula} for the Kolmogorov-Sinai entropy of a group action.
Now, as in the deterministic case, we can also define a notion of past for the group. An {\bf invariant random past} on $G$ is a random function $\tilde{\Phi}: G \to \{0,1\}^G$ or, equivalently, a Borel probability measure on $(\{0,1\}^G)^G$ that satisfies, for almost every instance of $\tilde{\Phi}$, the following properties:
\begin{enumerate}
\item for all $g \in G$, the condition $g \notin \tilde{\Phi}(g)$ holds;
\item for all $g,h \in G$, if $g \in \tilde{\Phi}(h)$, then $\tilde{\Phi}(g) \subseteq \tilde{\Phi}(h)$;
\item if $g \neq h$, then either $g \in \tilde{\Phi}(h)$ or $h \in \tilde{\Phi}(g)$; and
\item for all $g \in G$, the random subsets $\tilde{\Phi}(g)$ and $\tilde{\Phi}(1_G)g$ have the same distribution.
\end{enumerate}
Notice that if $\prec$ is an invariant random total order, then the random function $g \mapsto \{h \in G: h \prec g\}$ defines an invariant random past.
In contrast to deterministic invariant orders, every countable group $G$ admits at least one invariant random total order. Namely, consider the random process $(\chi_g)_{g \in G}$ of independent random variables such that each $\chi_g$ has uniform distribution on $[0,1]$. This process is invariant and each realization of it induces an order on $G$ almost surely.
\section{Counting}
\label{sec7}
From now on, given $(\Gamma,\lambda) \in \mathcal{H}_G$, we always assume that there is some (or any) fixed fundamental domain $U_0$ for $G \curvearrowright \Gamma$ and we introduce the auxiliary function $\phi_\lambda: X(\Gamma) \to \mathbb{R}$ given by
$$
\phi_\lambda(x) = \frac{1}{|\Gamma / G|}\sum_{v \in U_0} x(v)\log\lambda(v).
$$
\subsection{A pointwise Shannon-McMillan-Breiman type theorem}
The next theorem establishes a pointwise Shannon-McMillan-Breiman type theorem for Gibbs measures (related results can be found in \cite{1-gurevich} and \cite{2-briceno}). In order to prove it we use the Pointiwise Ergodic Theorem \cite{1-lindenstrauss}, which requires F{\o}lner sequence $\{F_n\}_n$ to be \emph{tempered}, a technical condition that is satisfied by every F{\o}lner sequence up to a subsequence and that we will assume without further detail.
\begin{theorem}
\label{thm:smb}
Let $G$ be a countable amenable group. For every $(\Gamma,\lambda) \in \mathcal{H}_G$ and every $\mathbb{P} \in \mathcal{M}_{\mathrm{Gibbs}}(\Gamma,\lambda)$,
$$
\lim_n \left[ - \frac{1}{|F_nU_0|} \log \mathbb{P}([x_{F_nU_0}]) \right] = -\int{\phi_\lambda} d\mathbb{Q} + f_G(\Gamma,\lambda) \quad \mathbb{Q}(x)\text{-a.s. in } x,
$$
for any tempered F{\o}lner sequence $\{F_n\}_n$ and any $\mathbb{Q} \in \mathcal{M}_G^{\mathrm{erg}}(X(\Gamma))$.
\end{theorem}
\begin{proof}
Consider the sets $U_n = F_nU_0$ and $M_n = U_n \cup \partial U_n$. Notice that, by amenability, $\lim_{n \to \infty} \frac{|M_n|}{|U_n|} = 1$. Indeed, define $K = \{g \in G: \mathrm{dist}_\Gamma(U_0,gU_0) \leq 1\}$. Then, $1_G \in K$ and $U_0 \cup \partial U_0 \subseteq KU_0$. Since $\Gamma$ is locally finite and the action is free, $K$ is finite. In addition, $U_n \cup \partial U_n \subseteq F_nKU_0$. Therefore, by amenability,
$$
1 \leq \lim_n \frac{|U_n \cup \partial U_n|}{|U_n|} \leq \lim_n \frac{|F_nKU_0|}{|F_nU_0|} = 1,
$$
so $\lim_n \frac{|\partial U_n|}{|U_n|} = 0$. Fix independent sets $x \in X(\Gamma[U_n])$, $z_1,z_2 \in X(\Gamma[M_n \setminus U_n])$, and $y \in X(\Gamma)$ such that $xz_iy \in X(\Gamma)$ for $i = 1,2$. Then,
\begin{align*}
\frac{\pi^y_{M_n}(xz_1)}{\pi^y_{M_n}(xz_2)} & = \frac{\mathrm{w}^y_\lambda(xz_1,M_n) Z^y_\Gamma(M_n,\lambda)^{-1}}{\mathrm{w}^y_\lambda(xz_2,M_n) Z^y_\Gamma(M_n,\lambda)^{-1}} \\
& = \frac{\mathrm{w}_\lambda(xz_1,M_n)1_{[y_{M_n^c}]}(xz_1)}{\mathrm{w}_\lambda(xz_2,M_n) 1_{[y_{M_n^c}]}(xz_2)} \\
& = \frac{\prod_{v \in M_n \setminus U_n} \lambda(v)^{z_1(v)}}{\prod_{v \in M_n \setminus U_n} \lambda(v)^{z_2(v)}} \\
& = \prod_{v \in M_n \setminus U_n} \lambda(v)^{z_1(v) - z_2(v)}.
\end{align*}
Therefore, $\frac{\pi^y_{M_n}(xz_1)}{\pi^y_{M_n}(xz_2)} \leq \max\{1,\lambda_+\}^{|M_n \setminus U_n|}$. Taking $z_2 = 0^\Gamma$ and adding over all possible $z_1$, we obtain that
$$
1 \leq \frac{\pi^y_{M_n}([x_{U_n}])}{\pi^y_{M_n}(x_{U_n}0^{M_n \setminus U_n})} = \sum_{\substack{z \in \{0,1\}^{M_n \setminus U_n}:\\ x_{U_n}z \in X(\Gamma[M_n])}}\frac{\pi^y_{M_n}(x_{U_n}z)}{\pi^y_{M_n}(x_{U_n}0^{M_n \setminus U_n})} \leq (2\max\{1,\lambda_+\})^{|M_n \setminus U_n|}.
$$
On the other hand, we have that
$$
\frac{\pi^y_{M_n}([x_{U_n}0^{M_n \setminus U_n}])}{\pi^{0^\Gamma}_{M_n}([x_{U_n}0^{M_n \setminus U_n}])} = \frac{\mathrm{w}^y_\lambda(x_{U_n}0^{M_n \setminus U_n},M_n) Z^y_\Gamma(M_n,\lambda)^{-1}}{\mathrm{w}^{0^\Gamma}_\lambda(x_{U_n}0^{M_n \setminus U_n},M_n) Z^{0^\Gamma}_\Gamma(M_n,\lambda)^{-1}} = \frac{Z^y_\Gamma(M_n,\lambda)}{Z_\Gamma(M_n,\lambda)},
$$
since
$$
\mathrm{w}^y_\lambda(x_{U_n}0^{M_n \setminus U_n},M_n) = \mathrm{w}^{0^\Gamma}_\lambda(x_{U_n}0^{M_n \setminus U_n},M_n) = \mathrm{w}_\lambda(x_{U_n},U_n)
$$
and $Z^{0^\Gamma}_\Gamma(M_n,\lambda) = Z_\Gamma(M_n,\lambda)$. In addition,
$$
1 \leq \frac{Z^y_\Gamma(M_n,\lambda)}{Z_\Gamma(M_n,\lambda)} \leq (2\max\{1,\lambda_+\})^{|M_n \setminus U_n|},
$$
since
\begin{align*}
Z^y_\Gamma(M_n,\lambda) & \leq Z_\Gamma(M_n,\lambda) \\
& = \sum_{x \in X(\Gamma[U_n])}\sum_{\substack{z \in X(\Gamma[M_n \setminus U_n]):\\ xz \in X(\Gamma[M_n])}} \prod_{v \in U_n} \lambda(v)^{x(v)}\prod_{v \in M_n \setminus U_n} \lambda(v)^{z(v)} \\
& \leq \sum_{x \in X(\Gamma[U_n])}\sum_{\substack{z \in X(\Gamma[M_n \setminus U_n]):\\ xz \in X(\Gamma[M_n])}} \prod_{v \in U} \lambda(v)^{x(v)} \max\{1,\lambda_+\}^{|M_n \setminus U_n|} \\
& \leq \max\{1,\lambda_+\}^{|M_n \setminus U_n|}\sum_{x \in X(\Gamma[U])} 2^{|M_n \setminus U_n|} \prod_{v \in U_n} \lambda(v)^{x(v)} \\
& = (2\max\{1,\lambda_+\})^{|M_n \setminus U_n|} Z_\Gamma(U_n,\lambda) \\
& \leq (2\max\{1,\lambda_+\})^{|M_n \setminus U_n|} Z^y_\Gamma(M_n,\lambda).
\end{align*}
Therefore,
\begin{align*}
1 & \leq \frac{\pi^y_{M_n}([x_{U_n}])}{\pi^{0^\Gamma}_{M_n}([x_{U_n}0^{M_n \setminus U_n}])} \\
& = \frac{\pi^y_{M_n}([x_{U_n}])}{\pi^y_{M_n}([x_{U_n}0^{M_n \setminus U_n}])}\frac{\pi^y_{M_n}([x_{U_n}0^{M_n \setminus U_n}])}{\pi^{0^\Gamma}_{M_n}([x_{U_n}0^{M_n \setminus U_n}])} \\
& \leq (2\max\{1,\lambda_+\})^{2|M_n \setminus U_n|}.
\end{align*}
In particular, since $\mathbb{P}([x_{F_nU_0}]) = \mathbb{E}_{\mathbb{P}}(\mathbb{P}([x_{U_n}] \vert \mathcal{B}_{M_n^c})(y)) = \mathbb{E}_{\mathbb{P}}(\pi^y_{M_n}([x_{U_n}]))$, we have that
$$
1 \leq \frac{\mathbb{P}([x_{F_nU_0}])}{\pi^{0^\Gamma}_{M_n}([x_{U_n}0^{M_n \setminus U_n}])} \leq (2\max\{1,\lambda_+\})^{2|M_n \setminus U_n|},
$$
so
$$
|\log\mathbb{P}([x_{F_nU_0}]) - \log\pi^{0^\Gamma}_{M_n}(x_{U_n}0^{M_n \setminus U_n})| \leq 2|M_n \setminus U_n|\log(2\max\{1,\lambda_+\}).
$$
Now, since $\mathrm{w}^{0^\Gamma}_\lambda(x_{M_n},M_n) = \mathrm{w}_\lambda(x_{M_n},M_n)$ for every $x$, we have that
$$
\pi^{0^\Gamma}_{M_n}(x_{U_n}0^{M_n \setminus U_n}) = \mathrm{w}_\lambda(x_{U_n}0^{M_n \setminus U_n},M_n) Z_\Gamma(M_n,\lambda)^{-1} = \mathrm{w}_\lambda(x_{U_n},U_n) Z_\Gamma(M_n,\lambda)^{-1}.
$$
Therefore,
$$
|\log\mathbb{P}([x_{F_nU_0}]) - (\log\mathrm{w}_\lambda(x_{U_n},U_n) - \log Z_\Gamma(M_n,\lambda))| \leq 2|M_n \setminus U_n|\log(2\max\{1,\lambda_+\}),
$$
so
\begin{align*}
& \lim_{n \to \infty} \left| -\frac{\log\mathbb{P}([x_{F_nU_0}])}{|U_n|} + \left(\frac{\log\mathrm{w}_\lambda(x_{U_n},U_n)}{|U_n|} - \frac{\log Z_\Gamma(M_n,\lambda)}{|U_n|}\right)\right| \\
\leq & \lim_n 2\frac{|M_n \setminus U_n|}{|U_n|}\log(2\max\{1,\lambda_+\}) = 0,
\end{align*}
and we conclude that
\begin{align*}
-\lim_n \frac{\log\mathbb{P}([x_{F_nU_0}])}{|F_nU_0|} & = \lim_n \frac{-\log\mathrm{w}_\lambda(x_{U_n},U_n)}{|U_n|} + \frac{\log Z_\Gamma(M_n,\lambda)}{|U_n|} \\
& = -\lim_n \left(\frac{1}{|F_n|} \sum_{g \in F_n} \frac{1}{|U_0|} \sum_{v \in gU_0} x(v)\log\lambda(v)\right) + f_G(\Gamma,\lambda),
\end{align*}
where we have used that $Z_\Gamma(U_n,\lambda) \leq Z_\Gamma(M_n,\lambda) \leq (2\max\{1,\lambda_+\})^{|M_n \setminus U_n|}Z_\Gamma(U_n,\lambda)$. Finally, notice that
$$
\frac{1}{|F_n|} \sum_{g \in F_n} \frac{1}{|U_0|} \sum_{v \in gU_0} x(v)\log\lambda(v) = \frac{1}{|F_n|} \sum_{g \in F_n} \phi_\lambda(g \cdot x)
$$
and by the Pointwise Ergodic Theorem, we obtain that
$$
\lim_n \frac{1}{|F_n|} \sum_{g \in F_n} \phi_\lambda(g \cdot x) = \int{\phi_\lambda}d\mathbb{Q} \quad \mathbb{Q}\text{-a.s.},
$$
so
$$
-\lim_n \frac{\log\mathbb{P}([x_{F_nU_0}])}{|F_nU_0|} = - \int{\phi_\lambda}d\mathbb{Q} + f_G(\Gamma,\lambda),
$$
and we conclude the proof.
\end{proof}
We have the following lemma.
\begin{lemma}
\label{lem:bound}
Given $\Delta \in \mathbb{N}$ and $(\Gamma,\lambda) \in \mathcal{H}^\Delta$, there exists a constant $C = C(\Delta,\lambda_0^-,\lambda_0^+) > 0$ such that for every $U \Subset V(\Gamma)$, $x \in X(\Gamma)$, and $v \in U$,
$$
\pi^x_U([x_v]) \geq C.
$$
\end{lemma}
\begin{proof}
Fix $(\Gamma,\lambda) \in \mathcal{H}_G^\Delta$, $U \Subset V(\Gamma)$, $x \in X(\Gamma)$, and $v \in U$. Notice that if $U^c \cap \partial\{v\} \neq \emptyset$ and $u \in U^c \cap \partial\{v\}$ is such that $x(u) = 1$, then necessarily $x_v = 0^v$ and $\pi^x_U([x_v]) = 1$. On the other hand, if $U \cap \partial\{v\} = \emptyset$, then $\pi^x_U([x_v]) = \mathbb{P}_{\Gamma[U'],\lambda}([x_v])$ for $U' = U \setminus \{u \in U: x(u') = 1 \text{ for some } u' \in U^c \cap \partial\{u\}\}$, so, by Lemma \ref{lem:probbound},
$$
\pi^x_U([x_v]) = \mathbb{P}_{\Gamma[U'],\lambda}([x_v]) \geq \min\left\{\frac{1}{1+\lambda_+}, \frac{\lambda_-}{\lambda_- + (1+\lambda_+)^\Delta}\right\} \geq \frac{\min\{1,\lambda_-\}}{\lambda_- + (1+\lambda_+)^\Delta} > 0,
$$
Therefore, by taking $C = \min\left\{1, \frac{\min\{1,\lambda_-\}}{\lambda_- + (1+\lambda_+)^\Delta}\right\} = \frac{\min\{1,\lambda_-\}}{\lambda_- + (1+\lambda_+)^\Delta}$, we conclude.
\end{proof}
\subsection{A randomized sequential cavity method}
Suppose now that $(\Gamma,\lambda) \in \mathcal{H}_G$ is such that the Gibbs $(\Gamma,\lambda)$-specification satisfies SSM and let $\mathbb{P}$ be the unique Gibbs measure. Considering this, we define the function $I_{\mathbb{P}}: X(\Gamma) \times (\{0,1\}^G)^G \to \mathbb{R}$ given by
$$
I_{\mathbb{P}}(x,\Phi) := \limsup_{n \to \infty} I_{\mathbb{P}, n}(x,\Phi),
$$
where
$$
I_{\mathbb{P},n}(x,\Phi) := -\log\mathbb{P}([x_{U_0}] \vert [x_{(\Phi(1_G) \cap F_n)U_0}])
$$
and $\{F_n\}_n$ is any exhaustion of $G$ (not necessarily F{\o}lner).
\begin{lemma}
If $(\Gamma,\lambda) \in \mathcal{H}_G$ is such that the Gibbs $(\Gamma,\lambda)$-specification satisfies SSM and $\mathbb{P}$ is the unique Gibbs measure, then the function $I_{\mathbb{P}}$ is measurable, non-negative, defined everywhere, and bounded.
\end{lemma}
\begin{proof}
Since $\mathbb{P}([x_{U_0}] \vert [x_{(\Phi(1_G) \cap F_n)U_0}])$ depends on finitely many coordinates in both $X(\Gamma)$ and $(\{0,1\}^G)^G$, $0 < \mathbb{P}([x_{U_0}] \vert [x_{(\Phi(1_G) \cap F_n)U_0}]) < 1$, and $-\log(\cdot)$ is a continuous function, $I_{\mathbb{P},n}$ is measurable and since $I_{\mathbb{P}}$ is a limit superior, it is measurable as well.
By SSM and Proposition \ref{prop:GKssm}, $\lim_{n \to \infty} \mathbb{P}([x_{U_0}] \vert [x_{(\Phi(1_G) \cap F_n)U_0}]) = \mathbb{P}([x_{U_0}] \vert [x_{\Phi(1_G)U_0}])$ is always a well-defined limit. By Lemma \ref{lem:bound}, and since $(\Gamma,\lambda) \in \mathcal{H}^\Delta_G$ for some $\Delta$, there exists a constant $C > 0$ such that for every $v \in V$, $U \Subset V \setminus \{v\}$, and $x \in X(\Gamma)$,
$$
\pi^x_U([x_v]) \geq C.
$$
Now, combined with the SSM property, this implies that for every $v \in V$, $U \subseteq V$, and $x \in X(\Gamma)$, we have that
$$
\mathbb{P}([x_v] \vert [x_U]) \geq C.
$$
Indeed, if $v \in U$, this is direct since $\mathbb{P}([x_v] \vert [x_U]) = 1$. On the other hand, if $v \notin U$, by SSM,
$$
\mathbb{P}([x_v] \vert [x_U]) = \lim_{\ell \to \infty} \mathbb{P}([x_v] \vert [x_{U \cup B_\Gamma(v,\ell)^c}]) = \lim_{\ell \to \infty} \pi_{U \cup B_\Gamma(v,\ell)^c}^x([x_v]) \geq \lim_{\ell \to \infty} C = C.
$$
Therefore, by conditioning and iterating, we obtain that
$$
1 \geq \mathbb{P}([x_{U_0}] \vert [x_{\Phi(1_G) U_0}]) = \prod_{i=1}^{|U_0|} \mathbb{P}([x_{v_i}] \vert [x_{\Phi U_0 \cup \{v_1,\dots,v_{i-1}\}}]) \geq \prod_{i=1}^{|U_0|} C = C^{|\Gamma / G|},
$$
so
$$
0 \leq I_{\mathbb{P}}(x,\Phi) = -\log\mathbb{P}([x_{U_0}] \vert [x_{\Phi(1_G)U_0}]) \leq -|\Gamma / G|\log C < +\infty,
$$
i.e., $I_{\mathbb{P}}(x,\Phi)$ is bounded (and, in particular, integrable).
\end{proof}
Following \cite{1-alpeev}, given an invariant random past $\tilde{\Phi}: G \to \{0,1\}^G$ on $G$ with law $\tilde{\nu} \in \mathcal{M}_G((\{0,1\}^G)^G)$, we denote
$$
\mathbb{E}_{\tilde{\Phi}}f(\tilde{\Phi}) = \int{f(\tilde{\Phi})}d\tilde{\nu}(\tilde{\Phi}),
$$
for $f \in L^1(\tilde{\nu})$. Now, since $I_{\mathbb{P}}$ is measurable, non-negative, and bounded, we have that for every $\mathbb{Q} \in \mathcal{M}_G(X(\Gamma))$, the function $I_{\mathbb{P}}$ is integrable with respect to $\mathbb{Q} \times \tilde{\nu}$ and by Tonelli's theorem, the function $\mathbb{E}_{\tilde{\Phi}}I_{\mathbb{P}}: X(\Gamma) \to \mathbb{R}$ is integrable, defined $\mathbb{Q}$-almost everywhere, and satisfies that
$$
\int{\mathbb{E}_{\tilde{\Phi}}I_{\mathbb{P}}(x,\tilde{\Phi}})d\mathbb{Q}(x) = \mathbb{E}_{\tilde{\Phi}}\int{I_{\mathbb{P}}(x,\tilde{\Phi}})d\mathbb{Q}(x).
$$
We call $\mathbb{E}_{\tilde{\Phi}}I_{\mathbb{P}}$ the {\bf random $\mathbb{P}$-information function} (with respect to $\tilde{\Phi}$).
\begin{lemma}
\label{lem:info}
For every $\mathbb{Q} \in \mathcal{M}_G(X(\Gamma))$ and for $\mathbb{Q}$-almost every $x$,
$$
\lim_n \left[ - \frac{1}{|F_nU_0|} \log \mathbb{P}([x_{F_nU_0}]) - \frac{1}{|F_n|}\sum_{g \in F_n} \mathbb{E}_{\tilde{\Phi}}I_{\mathbb{P}}(g \cdot x,\tilde{\Phi}) \right] = 0.
$$
\end{lemma}
\begin{proof}
Fix a (tempered) F{\o}lner sequence $\{F_n\}_n$. By the properties of $\tilde{\Phi}$, for $\tilde{\nu}$-almost every instance $\Phi$, every $F_n$ can be ordered as $g_1, \dots,g_{|F_n|}$ so that $\Phi(g_i) \cap F_n = \{g_1,\dots,g_{i-1}\}$. Then,
$$
\mathbb{P}([x_{F_nU_0}]) = \prod_{g \in F_n} \mathbb{P}([x_{gU_0}] \vert [x_{(\Phi(g) \cap F_n)U_0}]).
$$
Given $\ell > 0$, let $K_\ell = \{g \in G: \mathrm{dist}_\Gamma(U_0,gU_0) \leq \ell\}$. If $g \in \mathrm{Int}_{K_\ell}(F_n) = \{g \in G: K_\ell g \subseteq F_n\}$, then
$$
|\mathbb{P}([x_{gU_0}] \vert [x_{(\Phi(g) \cap F_n)U_0}]) - \mathbb{P}([x_{gU_0}] \vert [x_{(\Phi(g) \cap K_\ell g)U_0}])| \leq |U_0|\delta(\ell)
$$
and
$$
|\mathbb{P}([x_{gU_0}] \vert [x_{(\Phi(g) \cap K_\ell g)U_0}]) - \mathbb{P}([x_{gU_0}] \vert [x_{\Phi(g)U_0}])| \leq |U_0|\delta(\ell),
$$
so
$$
|\mathbb{P}([x_{gU_0}] \vert [x_{(\Phi(g) \cap F_n)U_0}]) - \mathbb{P}([x_{gU_0}] \vert [x_{\Phi(g)U_0}])| \leq 2|U_0|\delta(\ell).
$$
On the other hand, by Lemma \ref{lem:bound} and the discussion after it, for every $g \in G$ and $U \subseteq V$,
$$
\mathbb{P}([x_{gU_0}] \vert [x_U]) \geq C^{|\Gamma / G|} > 0.
$$
Therefore, by the Mean Value Theorem,
$$
|\log\mathbb{P}([x_{gU_0}] \vert [x_{(\Phi(g) \cap F_n)U_0}]) - \log\mathbb{P}([x_{gU_0}] \vert [x_{\Phi(g)U_0}])| \leq \frac{2|U_0|}{C^{|\Gamma / G|}}\delta(\ell).
$$
Notice that
\begin{align*}
\log\mathbb{P}([x_{F_nU_0}]) & = \sum_{g \in \mathrm{Int}_{K_\ell}(F_n)} \log\mathbb{P}([x_{gU_0}] \vert [x_{(\Phi(g) \cap F_n)U_0}]) \\
& \qquad + \sum_{g \in F_n \setminus \mathrm{Int}_{K_\ell}(F_n)} \log\mathbb{P}([x_{gU_0}] \vert [x_{(\Phi(g) \cap F_n)U_0}]),
\end{align*}
so
\begin{align*}
|\log\mathbb{P}([x_{FU_0}]) - \sum_{g \in F_n} \log\mathbb{P}([x_{gU_0}] \vert [x_{\Phi(g)U_0}])| & \leq |\mathrm{Int}_{K_\ell}(F_n)|\frac{2|U_0|}{C^{|\Gamma / G|}}\delta(\ell) \\
& \qquad + 2|F_n \setminus \mathrm{Int}_{K_\ell}(F_n)|\log(C^{-|\Gamma / G|}).
\end{align*}
Now, given $\epsilon > 0$, there exists $\ell$ and $n_0$ such that for every $n \geq n_0$,
$$
\left|\frac{\log\mathbb{P}([x_{F_nU_0}])}{|F_n|} - \frac{1}{|F_n|}\sum_{g \in F_n} \log\mathbb{P}([x_{gU_0}] \vert [x_{\Phi(g)U_0}])\right| \leq \epsilon.
$$
By $G$-invariance of $\mathbb{P}$,
\begin{align*}
\mathbb{P}([x_{gU_0}] \vert [x_{\Phi(g)U_0}]) & = \mathbb{P}(g^{-1} \cdot [(g \cdot x)_{U_0}] \vert [x_{\Phi(g)U_0}]) \\
& = \mathbb{P}([(g \cdot x)_{U_0}] \vert g \cdot [x_{\Phi(g)U_0}]) \\
& = \mathbb{P}([(g \cdot x)_{U_0}] \vert [(g \cdot x)_{\Phi(g)g^{-1}U_0}]),
\end{align*}
and combining this fact with the previous estimate, we obtain that
$$
\left|\frac{\log\mathbb{P}([x_{F_nU_0}])}{|F_n|} - \frac{1}{|F_n|}\sum_{g \in F_n} \log\mathbb{P}([(g \cdot x)_{U_0}] \vert [(g \cdot x)_{\Phi(g)g^{-1}U_0}])\right| \leq \epsilon
$$
Integrating against $\tilde{\nu}$, we obtain that, for $\mathbb{Q}$-almost every $x$,
$$
\left|\frac{\log\mathbb{P}([x_{F_nU_0}])}{|F_n|} - \frac{1}{|F_n|}\sum_{g \in F_n} \mathbb{E}_{\tilde{\Phi}}\log\mathbb{P}([(g \cdot x)_{U_0}] \vert [(g \cdot x)_{\tilde{\Phi}(g)g^{-1}U_0}])\right| \leq \epsilon
$$
and since $\tilde{\Phi}(g)$ has the same distribution as $\tilde{\Phi}(1_G)g$, we get that
\begin{align*}
\mathbb{E}_{\tilde{\Phi}}\log\mathbb{P}([(g \cdot x)_{U_0}] \vert [(g \cdot x)_{\tilde{\Phi}(g)g^{-1}U_0}]) & = \mathbb{E}_{\tilde{\Phi}}\log\mathbb{P}([(g \cdot x)_{U_0}] \vert [(g \cdot x)_{\tilde{\Phi}(1_G)gg^{-1}U_0}]) \\
& = \mathbb{E}_{\tilde{\Phi}}\log\mathbb{P}([(g \cdot x)_{U_0}] \vert [(g \cdot x)_{\tilde{\Phi}(1_G)U_0}]) \\
& = -\mathbb{E}_{\tilde{\Phi}} I_\mathbb{P}(g \cdot x, \tilde{\Phi}),
\end{align*}
so
$$
\left|-\frac{\log\mathbb{P}([x_{F_nU_0}])}{|F_n|} - \frac{1}{|F_n|}\sum_{g \in F_n} \mathbb{E}_{\tilde{\Phi}} I_\mathbb{P}(g \cdot x, \tilde{\Phi})\right| \leq \epsilon,
$$
and since $\epsilon$ is arbitrary and the limit exists $\mathbb{Q}$-almost surely, we conclude.
\end{proof}
We have the following representation theorem for free energy, which can be regarded as a randomized generalization of the results in \cite{1-gamarnik,1-marcus,2-briceno} tailored for the specific case of the hardcore model, but that holds in every amenable group (not necessarily $\mathbb{Z}^d$ or an almost orderable group).
\begin{theorem}
\label{thm:rep1}
Let $(\Gamma,\lambda) \in \mathcal{H}_G$ such that the Gibbs $(\Gamma,\lambda)$-specification satisfies SSM and $\mathbb{P}$ is the unique Gibbs measure. Then,
$$
f_G(\Gamma,\lambda) = \int{(\mathbb{E}_{\tilde{\Phi}}I_{\mathbb{P}} + \phi_\lambda)}d\mathbb{Q}.
$$
for any $\mathbb{Q} \in \mathcal{M}_G(X(\Gamma))$ and for any invariant random past $\tilde{\Phi}$ of $G$. In particular,
$$
f_G(\Gamma,\lambda) = I_{\mathbb{P}_{\Gamma,\lambda},\tilde{\Phi}}(0^{\Gamma}).
$$
\end{theorem}
\begin{proof}
First, notice that if the statement holds for every $\mathbb{Q} \in \mathcal{M}^{\mathrm{erg}}_G(X(\Gamma))$, then it holds for every $\mathbb{Q} \in \mathcal{M}_G(X(\Gamma))$ by the Ergodic Decomposition Theorem. Then, without loss of generality, we can assume that $\mathbb{Q}$ is $G$-ergodic. Considering this, by Theorem \ref{thm:smb}, for $\mathbb{Q}$-almost every $x$,
$$
\lim_n \left[ - \frac{1}{|F_n|} \log \mathbb{P}([x_{F_n}]) \right] = -\int{\phi_\lambda} d\mathbb{Q} + f_G(\Gamma,\lambda)
$$
By Lemma \ref{lem:info}, for $\mathbb{Q}$-almost every $x$,
$$
\lim_n \left[ - \frac{1}{|F_n|} \log \mathbb{P}([x_{F_n}]) \right] = \lim_n \frac{1}{|F_n|}\sum_{g \in F_n} \mathbb{E}_{\tilde{\Phi}} I_\mathbb{P}(g \cdot x, \tilde{\Phi}).
$$
Therefore, for $\mathbb{Q}$-almost every $x$,
$$
f_G(\Gamma,\lambda) - \int{\phi_\lambda} d\mathbb{Q} = \lim_n \frac{1}{|F_n|}\sum_{g \in F_n} \mathbb{E}_{\tilde{\Phi}} I_\mathbb{P}(g \cdot x, \tilde{\Phi}).
$$
Integrating against $\mathbb{Q}$, we obtain that
\begin{align*}
\int{ \lim_n \frac{1}{|F_n|}\sum_{g \in F_n} \mathbb{E}_{\tilde{\Phi}}I_{\mathbb{P}}(g \cdot x,\tilde{\Phi})}d\mathbb{Q} & = \lim_n\int{ \frac{1}{|F_n|}\sum_{g \in F_n} \mathbb{E}_{\tilde{\Phi}}I_{\mathbb{P}}(g \cdot x,\tilde{\Phi})}d\mathbb{Q} \\
& = \lim_n\frac{1}{|F_n|}\sum_{g \in F_n}\mathbb{E}_{\tilde{\Phi}} \int{I_{\mathbb{P}}(g \cdot x,\tilde{\Phi})}d\mathbb{Q} \\
& = \lim_n\frac{1}{|F_n|}\sum_{g \in F_n}\mathbb{E}_{\tilde{\Phi}} \int{I_{\mathbb{P}}(x,\tilde{\Phi})}d\mathbb{Q} \\
& = \mathbb{E}_{\tilde{\Phi}} \int{I_{\mathbb{P}}(x,\tilde{\Phi})}d\mathbb{Q} \\
& = \int{\mathbb{E}_{\tilde{\Phi}} I_{\mathbb{P}}(x,\tilde{\Phi})}d\mathbb{Q},
\end{align*}
where the first equality is due to the Dominated Convergence Theorem, the second and last equalities are due to Tonelli's Theorem, and the third equality is due to the $G$-invariance of $\mathbb{Q}$. We conclude that
$$
f_G(\Gamma,\lambda) = \int{(\mathbb{E}_{\tilde{\Phi}}I_{\mathbb{P}} + \phi_\lambda)}d\mathbb{Q}.
$$
In particular, if $\mathbb{Q} = \delta_{0^\Gamma} \in \mathcal{M}_G(X(\Gamma))$, the Dirac measure supported on $0^\Gamma$, then
$$
f_G(\Gamma,\lambda) = \mathbb{E}_{\tilde{\Phi}} I_{\mathbb{P}}(0^\Gamma,\tilde{\Phi}) + \phi_\lambda(0^\Gamma) = \mathbb{E}_{\tilde{\Phi}} I_{\mathbb{P}}(0^\Gamma,\tilde{\Phi}).
$$
\end{proof}
\begin{figure}[ht]
\centering
\includegraphics[scale = 1.5]{pastbw.pdf}
\caption{The graph $\Gamma \setminus \Phi_\prec U_0$ and the corresponding graphs $\Gamma_i(\Phi_\prec)$ for $G = \mathbb{Z}^2$ and the lexicographic order $\prec$.}
\label{fig:lex}
\end{figure}
\subsection{An arboreal representation of free energy}
The following theorem tell us that, under some special conditions, $f_G(\Gamma,\lambda)$ can be expressed using $|\Gamma / G|$ terms that depend on the probability that the roots of some particular trees are unoccupied.
\begin{theorem}
\label{thm:rep2}
Let $(\Gamma,\lambda) \in \mathcal{H}_G$ such that the Gibbs $(T_{\mathrm{SAW}}(\Gamma,v),\overline{\lambda})$-specification satisfies SSM for every $v \in V(\Gamma)$ and let $v_1,\dots,v_{|\Gamma / G|}$ be an arbitrary ordering of a fundamental domain $U_0$. Given an invariant random order $\tilde{\Phi}$ of $G$, denote by $\Gamma_i(\tilde{\Phi})$ the random graph given by $\Gamma \setminus (\tilde{\Phi}(1_G)U_0 \cup \{v_1,\dots,v_{i-1}\})$. Then,
$$
f_G(\Gamma,\lambda) =-\sum_{i=1}^{|\Gamma / G|} \mathbb{E}_{\tilde{\Phi}} \log \mathbb{P}_{T_{\mathrm{SAW}}(\Gamma_i(\tilde{\Phi}),v_i),\overline{\lambda}}([0^{\rho_i}]),
$$
where $\rho_i$ denotes the root of $T_{\mathrm{SAW}(\Gamma_i(\Phi),v_i)}$. In particular, if $\prec$ is a deterministic invariant order of $G$,
$$
f_G(\Gamma,\lambda) = -\sum_{i=1}^{|\Gamma / G|} \log \mathbb{P}_{T_{\mathrm{SAW}}(\Gamma_i(\Phi_\prec),v_i),\overline{\lambda}}([0^{\rho_i}]).
$$
\end{theorem}
\begin{proof}
By Theorem \ref{thm:rep1}, we know that
$$
f_G(\Gamma,\lambda) = \mathbb{E}_{\tilde{\Phi}}I_{\mathbb{P}_{\Gamma,\lambda}}(0^\Gamma,\tilde{\Phi}) = \mathbb{E}_{\tilde{\Phi}}(-\log\mathbb{P}_{\Gamma,\lambda}([0^{U_0}] \vert [0^{\tilde{\Phi}(1_G)U_0}])).
$$
By iterating conditional probabilities, linearity of expectation, and Proposition \ref{prop:GKssm} (see Figure \ref{fig:lex}),
\begin{align*}
-\mathbb{E}_{\tilde{\Phi}}\log\mathbb{P}_{\Gamma,\lambda}([0^{U_0}] \vert [0^{\tilde{\Phi}(1_G)U_0}]) & = -\sum_{i=1}^{|\Gamma / G|}\mathbb{E}_{\tilde{\Phi}}\log\mathbb{P}_{\Gamma,\lambda}([0^{v_i}] \vert [0^{\tilde{\Phi}(1_G)U_0 \cup \{v_1,\dots,v_{i-1}\}}]) \\
& = -\sum_{i=1}^{|\Gamma / G|}\mathbb{E}_{\tilde{\Phi}}\log\mathbb{P}_{\Gamma \setminus (\tilde{\Phi}(1_G)U_0 \cup \{v_1,\dots,v_{i-1}\}),\lambda}([0^{v_i}]) \\
& = -\sum_{i=1}^{|\Gamma / G|}\mathbb{E}_{\tilde{\Phi}}\log\mathbb{P}_{\Gamma \setminus \Gamma_i(\tilde{\Phi}),\lambda}([0^{v_i}]).
\end{align*}
Finally, by Theorem \ref{thm:tsaw}, we obtain
$$
-\sum_{i=1}^{|\Gamma / G|}\mathbb{E}_{\tilde{\Phi}}\log\mathbb{P}_{\Gamma \setminus \Gamma_i(\tilde{\Phi}),\lambda}([0^{v_i}]) = -\sum_{i=1}^{|\Gamma / G|} \mathbb{E}_{\tilde{\Phi}}\log \mathbb{P}_{T_{\mathrm{SAW}}(\Gamma_i(\tilde{\Phi}),v_i),\overline{\lambda}}([0^{\rho_i}]).
$$
In particular, if $\prec$ is a deterministic invariant order on $G$ (see Figure \ref{fig:trees}), we have that
$$
f_G(\Gamma,\lambda) = -\sum_{i=1}^{|\Gamma / G|} \log \mathbb{P}_{T_{\mathrm{SAW}}(\Gamma_i(\Phi_\prec),v_i),\overline{\lambda}}([0^{\rho_i}]).
$$
\end{proof}
\section{A computational phase transition in the thermodynamic limit}
\label{sec8}
Given an amenable countable group $G$, we are interested in having an algorithm to efficiently approximate $f_G(\Gamma,\lambda)$ in some uniform way over $\mathcal{H}_G$.
Let $\mathbb{M} \subseteq \mathcal{H}_G$ be a family of hardcore models. We will say that $\mathbb{M}$ admits a {\bf fully polynomial-time additive approximation (FPTAA)} for $f_G(\Gamma,\lambda)$ if there is an algorithm such that, given an input $(\Gamma,\lambda) \in \mathbb{M}$ and $\epsilon > 0$, outputs $\hat{f}$ with
$$
|f_G(\Gamma,\lambda) - \hat{f}| \leq \epsilon,
$$
in polynomial time in $|\left<\Gamma,\lambda\right>|$ and $\epsilon^{-1}$, where $|\left<\Gamma,\lambda\right>|$ denotes the length of any reasonable representation $\left<\Gamma,\lambda\right>$ of $(\Gamma,\lambda)$. A FPTAA will be what we will regard as an efficient and uniform approximation algorithm for $f_G(\Gamma,\lambda)$.
\begin{remark}
In order to not having to deal with numerical details about the representation of $\lambda$, we will always implicitly assume that the values taken by $\lambda$ have a bounded number of digits uniformly on $\mathbb{M}$.
\end{remark}
\begin{figure}[ht]
\centering
\includegraphics[scale = 1.5]{pasttreebw2.pdf}
\caption{A representation of $T_{\mathrm{SAW}(\Gamma_i(\Phi_\prec),v_i)}$ and a logarithmic depth truncation for $G = \mathbb{Z}^2$ and $\Gamma = \mathrm{Cay}(\mathbb{Z}^2,\{\pm(1,0),\pm(0,1)\})$.}
\label{fig:trees}
\end{figure}
\subsection{FPTAS, Weitz's algorithm, and a computational phase transition}
Notice that if $G$ is the trivial group $\{1\}$, then $\mathcal{H}_{\{1\}}$ is exactly the family of finite hardcore models. In such case, we have that
$$
f_G(\Gamma,\lambda) = \frac{\log Z_\Gamma(\lambda)}{|V(\Gamma)|},
$$
and we can translate an approximation of $Z_\Gamma(\lambda)$ into an approximation of $f_G(\Gamma,\lambda)$ and vice versa.
In this finite context, it is common to consider a \emph{fully polynomial-time approximation scheme}. Given a family $\mathbb{M} \subseteq \mathcal{H}_{\{1\}}$ of finite hardcore models, we will say that $\mathbb{M}$ admits a {\bf fully polynomial-time approximation scheme (FPTAS)} for $Z_\Gamma(\lambda)$ if there is an algorithm such that, given an input $(\Gamma,\lambda) \in \mathbb{M}$ and $\epsilon > 0$, outputs $\hat{Z}$ with
$$
Z_\Gamma(\lambda)e^{-\epsilon} \leq \hat{Z} \leq Z_\Gamma(\lambda)e^{\epsilon},
$$
in polynomial time in $|V(\Gamma)|$ and $\epsilon^{-1}$. An FPTAS is regarded as an efficient and uniform approximation algorithm for $Z_\Gamma(\lambda)$.
Notice that if we take logarithms and divide by $|V(\Gamma)|$ in the previous equation, we obtain that
$$
|f_G(\Gamma,\lambda) - \hat{f}| = \left|\frac{\log Z_\Gamma}{|V(\Gamma)|} - \frac{\log \hat{Z}}{|V(\Gamma)|}\right| \leq \frac{\epsilon}{|V(\Gamma)|},
$$
where $\hat{f} = \frac{\log \hat{Z}}{|V(\Gamma)|}$, so an FTPAS for $Z_\Gamma(\lambda)$ is equivalent to an FPTAA for $f_G(\Gamma,\lambda)$, since a polynomial in $|V(\Gamma)|$ and $\epsilon^{-1}$ is also a polynomial in $|\left<\Gamma,\lambda\right>|$ and $|V(\Gamma)|\epsilon^{-1}$ and vice versa.
We will fix a positive integer $\Delta$ and $\lambda_0 > 0$. Given such parameters, we aim to develop a fully polynomial-time additive approximation on $\mathbb{M} = \mathcal{H}_G^\Delta(\lambda_0)$.
The main theorem in \cite{1-weitz} was the development of an FPTAS for $Z_\Gamma(\lambda_0)$ on $\mathcal{H}_{\{1\}}^\Delta(\lambda_0)$ for $\lambda_0 < \lambda_c(\Delta)$. It is not difficult to see that the theorem extends to non-constant activity functions $\lambda$. Then, and also translated into the language of free energy, we have the following result.
\begin{theorem}[{\cite{1-weitz}}]
\label{thm:weitz}
For every $\Delta \in \mathbb{N}$ and $0 < \lambda_0 < \lambda_c(\Delta)$, there exists an FPTAS (resp., an FPTAA) on $\mathcal{H}_{\{1\}}^\Delta(\lambda_0)$ for $Z_\Gamma(\lambda)$ (resp., for $f_G(\Gamma,\lambda)$).
\end{theorem}
This theorem was subsequently refined in \cite{1-sinclair} by considering connective constants instead of maximum degree $\Delta$. A very interesting fact is that when classifying graphs according to their maximum degree, then Theorem \ref{thm:weitz} is in some sense optimal due to the following theorem.
\begin{theorem}[{\cite{1-sly,2-sly}}]
\label{thm:sly}
For every $\Delta \geq 3$ and $\lambda_0 > \lambda_c(\Delta)$, there does not exist an FPTAS (resp., an FPTAA) on $\mathcal{H}_{\{1\}}^\Delta(\lambda_0)$ for $Z_\Gamma(\lambda_0)$ (resp., for $f_G(\Gamma,\lambda_0)$), unless $\mathrm{NP} = \mathrm{RP}$.
\end{theorem}
\begin{remark}
In \cite{1-sly,2-sly}, it is considered a stronger version of Theorem \ref{thm:sly}, where instead of the lack of existence of an FPTAS, it is proven the lack of existence of an FPRAS, which is a randomized (and therefore, weaker) version of an FPTAS.
\end{remark}
The combination of Theorem \ref{thm:weitz} and Theorem \ref{thm:sly} is what is regarded as a \emph{computational phase transition}. We aim to extend these theorems to the infinite setting. A theoretical advantage about considering $f_G(\Gamma,\lambda)$ instead of $Z_\Gamma(\lambda)$ is that the free energy still makes sense in infinite graphs and at the same time recovers the theory for $Z_\Gamma(\lambda)$ in the finite case.
\subsection{A computational phase transition in the infinite setting}
For algorithmic purposes, in this section we only consider finitely generated orderable groups $G$ with fixed symmetric set of generators $S$ and deterministic invariant order $\prec$. Recall that we are not losing generality if we assume $G$ to be orderable instead of just almost orderable, so in particular our results apply to every group of polynomial growth ---including the very relevant case $G = \mathbb{Z}^d$--- but are not restricted to them, i.e., our results also apply to some groups of exponential growth (e.g., $BS(1,n)$).
Notice that if $(\Gamma,\lambda) \in \mathcal{H}_{G}^\Delta$, then it suffices to know $\Gamma[U_0]$ for some fundamental domain $U_0$ and $L = \{(v_1,s,v_2) \in U_0 \times S \times U_0: (v_1,sv_2) \in E(\Gamma)\}$ in order to fully reconstruct the graph $\Gamma$. In particular, the size of the necessary information to reconstruct $\Gamma$ is bounded by a polynomial in $\Delta|\Gamma / G|$. In addition, given a $G$-invariant activity function $\lambda: V(\Gamma) \to \mathbb{Q}_{>0}$, we only need to know $\left.\lambda\right\vert_{U_0}$ to recover $\lambda$, i.e., just $|\Gamma / G|$ many rational numbers. Therefore, in this context, the length $|\left<\Gamma,\lambda\right>|$ of the representation $\left<\Gamma,\lambda\right>$ of a hardcore model $(\Gamma, \lambda)$ will be polynomial in $\Delta|\Gamma / G|$.
We are interested in being able to generate in an effective way balls of arbitrary radius in $\mathrm{Cay}(G,S)$ and to recognize the set $\Phi_\prec U_0$. To do this, assume that \emph{the algebraic past of $G$ is decidable in exponential time}, i.e., given an input word $w \in S^*$, we can decide whether $\pi(w) \in \Phi_\prec$ or not in time $\exp(O(|w|))$, where $|w|$ denotes the length of $w$ and $\pi: S^* \to G$ is the usual evaluation map. Notice that this implies that the word problem is solvable in exponential time, since $\pi(w) = 1_G$ if and only if $\pi(w) \notin \Phi_\prec$ and $\pi(w^{-1}) \notin \Phi_\prec$. Now, given $\ell > 0$, if we can solve the word problem in exponential time, then $\mathrm{Cay}(G,S)$ is constructible in exponential time as well (see \cite[Theorem 5.10]{1-meier}), this is to say, we can generate $B_{\mathrm{Cay}(G,S)}(1_G,\ell)$ in time $\exp(O(\ell))$, where the $O$-notation regards $|S|$ and $\Delta$ as constants. Now, having $B_{\mathrm{Cay}(G,S)}(1_G,\ell)$ it is possible to construct $\Gamma[B_{\mathrm{Cay}(G,S)}(1_G,\ell)U_0]$ in time $O(\mathrm{poly}(|\Gamma / G|)\exp(O(\ell)))$ by identifying each $g \in B_{\mathrm{Cay}(G,S)}(1_G,\ell)$ with a copy $\Gamma[gU_0]$ of $\Gamma[U_0]$ and by connecting it to other adjacent copies according to $L$. Finally, we can remove all the copies $\Gamma[gU_0]$ with $g \in \Phi_\prec$, and this can be done in time $|B_{\mathrm{Cay}(G,S)}(1_G,\ell)|\exp(O(\ell)) = \exp(O(\ell))$, since $|B_{\mathrm{Cay}(G,S)}(1_G,\ell)| \leq |S|^\ell$. Therefore, if $\Phi_\prec$ is decidable in exponential time, then $\Gamma[(B_{\mathrm{Cay}(G,S)}(1_G,\ell) \setminus \Phi_\prec)U_0]$ can be constructed in time $O(\mathrm{poly}(|\Gamma / G|)\exp(O(\ell)))$ and from this we can obtain directly the graphs $T_{\mathrm{SAW}}(\Gamma_i(\Phi_\prec),v_i)$ truncated at depth $\ell$, which are the ones that we are ultimately interested in.
\begin{proposition}
\label{prop:approx}
Let $G$ be a finitely generated amenable orderable group such that its algebraic past is decidable in exponential time. Then, for every $\Delta \in \mathbb{N}$ and $0 < \lambda_0 < \lambda_c(\Delta)$, there exists an FPTAA on $\mathcal{H}_G^\Delta(\lambda_0)$ for $f_G(\Gamma,\lambda)$.
\end{proposition}
\begin{proof}
Pick $(\Gamma,\lambda)$ as in the statement and enumerate as $U_0 = \left\{v_1,\dots,v_n\right\}$ the fundamental domain of $G \curvearrowright \Gamma$. Denote $n = |\Gamma / G|$. Then, by Theorem \ref{thm:rep2},
$$f_G(\Gamma,\lambda) = -\sum_{i=1}^n \log p_i,$$
where $p_i = \mathbb{P}_{T_{\mathrm{SAW}}(\Gamma_i(\Phi_\prec),v_i),\overline{\lambda}}([0^{\rho_i}])$. Given $\epsilon > 0$, our goal is generate numbers $\hat{q}_i$ and $\hat{r}_i$ such that
$$
\hat{q}_i \leq p_i \leq \hat{r}_i \leq \hat{q}_i\left(1 + \frac{\epsilon}{(1+\epsilon)n}\right)
$$
for every $i = 1,\dots,n$. If we manage to compute such approximations, we have that
$$
\prod_{i=1}^{n}\hat{q}_i \leq \prod_{i=1}^{n}p_i \leq \prod_{i=1}^{n}\hat{r}_i \leq \prod_{i=1}^{n}\left(\hat{q}_i \left(1 + \frac{\epsilon}{(1+\epsilon)n}\right)\right) = \left(1 + \frac{\epsilon}{(1+\epsilon)n}\right)^n \prod_{i=1}^{n}\hat{q}_i.
$$
Therefore, it is possible to generate a pair of numbers $\hat{Z}_1 = \left(\prod_{i=1}^{n}\hat{r}_i\right)^{-1}$ and $\hat{Z}_2 = \left(\prod_{i=1}^{n}\hat{q}_i\right)^{-1}$ such that
$$
\hat{Z}_2 \geq Z \geq \hat{Z}_1 \geq \hat{Z}_2\left(1 + \frac{\epsilon}{(1+\epsilon)n}\right)^{-n} \geq \hat{Z}_2\left(1 - \frac{\epsilon}{(1+\epsilon)}\right) = \hat{Z}_2\frac{1}{(1+\epsilon)},
$$
so $\hat{Z}_1 \leq Z \leq \hat{Z}_2 \leq (1+\epsilon)\hat{Z}_1$, where $Z = \left(\prod_{i=1}^{n}p_i\right)^{-1}$. Therefore,
$$
\log \hat{Z}_1 \leq -\sum_{i=1}^n \log p_i = f_G(\Gamma,\lambda) \leq \log\hat{Z}_1 + \log(1+\epsilon) \geq \log\hat{Z}_1 + \epsilon,
$$
so $\hat{f} = \log\hat{Z}_1$ would be the required approximation. Now, in order to estimate $p_i$, i.e., the probability that $\rho_i$ is unoccupied in $T_i = T_{\mathrm{SAW}}(\Gamma_i(\Phi_\prec),v_i)$, we denote
$$
\hat{q}_i = \min\{\mathbb{P}_{T_i,\overline{\lambda}}([0^{\rho_i}] \vert [0^{\partial B_{T_i}(\rho_i,\ell)}]),\mathbb{P}_{T_i,\overline{\lambda}}([0^{\rho_i}] \vert [1^{\partial B_{T_i}(\rho_i,\ell)}])\}
$$
and
$$
\hat{r}_i = \max\{\mathbb{P}_{T_i,\overline{\lambda}}([0^{\rho_i}] \vert [0^{\partial B_{T_i}(\rho_i,\ell)}]),\mathbb{P}_{T_i,\overline{\lambda}}([0^{\rho_i}] \vert [1^{\partial B_{T_i}(\rho_i,\ell)}])\}
$$
for $\ell > 0$ to be defined. It is known that the ``all $0$'' and ``all $1$'' conditions are the conditions that maximize and minimize the probability that $\rho_i$ is unoccupied among all conditions at distance $\ell$ and whether the ``all $0$'' (resp. ``all $1$'') condition maximizes or minimizes such probability depends on the parity of $\ell$. In particular, $\hat{q}_i \leq p_i \leq \hat{r}_i $ and, by SSM,
$$
0 \leq \hat{r}_i - \hat{q}_i \leq C\exp(-\alpha\ell).
$$
We want that $\hat{q}_i \leq \hat{r}_i \leq \hat{q}_i\left(1+\frac{\epsilon}{(1+\epsilon)n}\right)$. This is equivalent to have
$$
0 \leq \hat{r}_i - \hat{q}_i \leq \hat{q}_i\frac{\epsilon}{(1+\epsilon)n}.
$$
Now, by Lemma \ref{lem:probbound}, $\hat{q}_i \geq \frac{1}{1+\lambda_+}$ for every $i$, so it is enough to have that $\hat{r}_i - \hat{q}_i \leq \frac{1}{1+\lambda_+}\frac{\epsilon}{(1+\epsilon)n}$. In consequence, it suffices to take $\ell > 0$ such that
$$
C\exp(-\alpha\ell) \leq \frac{1}{1+\lambda_+}\frac{\epsilon}{(1+\epsilon)n},
$$
so we can pick
$$
\ell^* = \left\lceil\frac{1}{\alpha}\log\left(C(1+\lambda_0)(1+\epsilon^{-1})n\right)\right\rceil.
$$
For every $i = 1,\dots,n$, the size of the ball $B_{T_i}(\rho_i,\ell)$ in $T_i$ is bounded by
$$
|B_{T_i}(\rho_i,\ell^*)| \leq \Delta^{\ell^*} \leq \left(C(1+\lambda_0)(1+\epsilon^{-1})n\right)^{\frac{\log\Delta}{\alpha}},
$$
which is also a bound for the order of time required for computing $\hat{r}_i$, because $T_i$ is a tree (see Figure \ref{fig:trees}). Notice that to construct each $B_{T_i}(\rho_i,\ell^*)$ it suffices to be able to construct $B_{\Gamma_i(\Phi_\prec)}(v_i,\ell^*)$, which takes time $\mathrm{poly}(n)\exp(O(\ell^*)) = \mathrm{poly}((1+\epsilon^{-1})n)$. Finally, since we require to do this procedure $n$ times (one for each $i$), we have that the total order of the algorithm is still $\mathrm{poly}((1+\epsilon^{-1})n)$, i.e., a polynomial in $n$ and $\epsilon^{-1}$ where the constants involved depend only on $\Delta$, $\lambda_0$, and $|S|$.
\end{proof}
\begin{remark}
Notice that Proposition \ref{prop:approx} holds for groups of exponential growth, despite it involves a polynomial time algorithm. In addition, in virtue of Theorem \ref{thm:sinclair}, the families of graphs in Proposition \ref{prop:approx} could be parameterized according to their connective constant instead of the maximum degree.
\end{remark}
Next, we reduce the problem of approximating the partition function of a finite hardcore model to the problem of approximating the free energy of a hardcore model in $\mathcal{H}_G$.
\begin{proposition}
\label{prop:phase}
Let $G$ be an amenable group. Then, for every $\Delta \geq 3$ and $\lambda_0 > \lambda_c(\Delta)$, there does not exist an FPTAA on $\mathcal{H}_G^\Delta(\lambda_0)$ for $f_G(\Gamma,\lambda_0)$, unless $\mathrm{NP} = \mathrm{RP}$.
\end{proposition}
\begin{proof}
Suppose that we have a FPTAA on $\mathcal{H}_G^\Delta(\lambda_0)$ for $f_G(\Gamma,\lambda)$ for some amenable group $G$. Then, we claim that we would have an FPTAS on $\mathcal{H}_{\{1\}}^\Delta(\lambda_0)$ for $Z_\Gamma(\lambda_0)$, contradicting Theorem \ref{thm:sly}. Indeed, given the input $(\Gamma,\lambda) \in \mathcal{H}_{\{1\}}^\Delta(\lambda_0)$, it suffices to consider the graph $\Gamma^G$ made out of copies $\{\Gamma_g\}_{g \in G}$ of $\Gamma$ indexed by $g \in G$, where $v_g$ denotes the copy in $\Gamma_g$ of the vertex $v$ in $\Gamma$. Then, there is a natural action $G \curvearrowright \Gamma^G$ consisting on just translating copies of vertices, i.e., $gv_h = v_{hg}$, and a fundamental domain of the action is $U_0 = V(\Gamma_{1_G})$. Therefore, since $|\Gamma^G / G| = |V(\Gamma)|$, if we could $\epsilon$-approximate in an additive way $f_G(\Gamma,\lambda_0)$ in polynomial time in $|\Gamma^G / G|$ and $\epsilon^{-1}$, then we would be able to $\epsilon$-approximate in a multiplicative way $Z_\Gamma(\lambda_0)$ in polynomial time in $|V(\Gamma)|$ and $\epsilon^{-1}$, because
\begin{align*}
f_G(\Gamma,\lambda_0) & = \inf_{F \in \mathcal{F}(G)} \frac{\log Z_{\Gamma^G}(FU_0,\lambda_0)}{|FU_0|} \\
& = \inf_{F \in \mathcal{F}(G)} \frac{\log Z_{\Gamma^G}(U_0,\lambda_0)^{|F|}}{|F||U_0|} \\
& = \frac{\log Z_{\Gamma^G}(U_0,\lambda_0)}{|U_0|} \\
& = \frac{\log Z_{\Gamma}(\lambda_0)}{|V(\Gamma)|},
\end{align*}
but this contradicts Theorem \ref{thm:sly}.
\end{proof}
Considering Proposition \ref{prop:approx} and Proposition \ref{prop:phase}, we have the following theorem.
\begin{theorem}
\label{thm:phase}
Let $G$ be a finitely generated amenable orderable group such that its algebraic past is decidable in exponential time. Then, for every $\Delta \geq 3$ and $\lambda_0 > 0$, if $\lambda_0 < \lambda_c(\Delta)$, there exists an FPTAA on $\mathcal{H}_G^\Delta(\lambda_0)$ for $f_G(\Gamma,\lambda)$ and, if $\lambda_0 > \lambda_c(\Delta)$, there is no such FPTAA, unless $\mathrm{NP} = \mathrm{RP}$.
\end{theorem}
\begin{remark}
Notice that Theorem \ref{thm:phase} still holds for $\Delta = 1$ and $\Delta = 2$. The first case is trivial and in the second case, there is no phase transition and the conditions for the existence of an FPTAA hold for every $\lambda_0$.
\end{remark}
\section{Reductions}
\label{sec9}
In this section we provide a set of reductions to relate the results already obtained for hardcore models with other systems.
\subsection{$G$-subshifts and conjugacies}
Given a countable group $G$ and a finite set $\Sigma$ endowed with the discrete topology, the {\bf full shift} is the set $\Sigma^G$ of maps $\omega: G \to \Sigma$ endowed with the product topology. We define the {\bf $G$-shift} as the group action $G \times \Sigma^G \to \Sigma^G$ given by $(g,\omega) \mapsto g \cdot \omega$, where $(g \cdot \omega)(h) = \omega(hg)$ for all $h \in G$. A {\bf $G$-subshift} $\Omega$ is a $G$-invariant closed subset of $\Sigma^G$.
Given two $G$-subshifts $\Omega_1$ and $\Omega_2$, we say that a map $\varphi: \Omega_1 \to \Omega_2$ is a {\bf conjugacy} if it is bijective, continuous, and $G$-equivariant, i.e., $g \cdot \varphi(x) = \varphi(g \cdot x)$ for every $\omega \in \Omega_1$ and $g \in G$. In this context, such maps are characterized as \emph{sliding block codes} (e.g., see \cite{1-lind,1-ceccherini}) and provide a notion of isomorphism between $G$-subshifts.
Any $G$-subshift $\Omega$ is characterized by the existence of a family of forbidden patterns $\mathfrak{F} \subseteq \bigcup_{F \in \mathcal{F}(G)} \Sigma^F$ such that $\Omega = X_{\mathfrak{F}}$, where
$$
X_\mathfrak{F} = \{\omega \in \Sigma^G: (g \cdot x)_{F} \notin \mathfrak{F} \text{ for all } g \in G\}.
$$
If the family $\mathfrak{F}$ can be chosen to be finite, we say that $\Omega$ is a {\bf $G$-subshift of finite type ($G$-SFT)}. Given a finite set $S \subseteq G$, we can consider a family of $|\Sigma| \times |\Sigma|$ binary matrices ${\bf M} = \{M_s\}_{s \in S}$ with rows and columns indexed by the elements of $\Sigma$, and define the set
$$
\Omega_{{\bf M}} = \{\omega \in \Sigma^G: M_s(\omega(g),\omega(sg)) = 1 \text{ for all } g \in G, s \in S\}.
$$
The set $\Omega_{{\bf M}}$ is a special kind of $G$-SFT known as {\bf nearest neighbor (n.n.) $G$-SFT}. It is known that for every $G$-SFT there exists a conjugacy to a n.n. $G$-SFT, so we are not losing much generality by considering n.n. $G$-SFTs instead of general $G$-SFTs.
We say that a n.n. $G$-SFT $\Omega_{{\bf M}}$ has a {\bf safe symbol} if there exists $a \in \Sigma$ such that $a$ can be adjacent to any other symbol $b \in \Sigma$. Formally, this means that, for all $s \in S$ and $b \in \Sigma$, $M_s(a,b) = M_s(b,a) = 1$.
\subsection{Entropy and potentials}
Given a $G$-subshift $\Omega$, we define its {\bf topological entropy} as
$$
h_G(\Omega) := \lim_n \frac{\log |\Omega_{F_n}|}{|F_n|},
$$
where $\{F_n\}_n$ is a F{\o}lner sequence and $\Omega_F = \{\omega_F: \omega \in \Omega\}$ is the set of restrictions of points in $\Omega$ to the set $F \subseteq G$. It is known that the definition of $h_G(\Omega)$ is independent of the choice of F{\o}lner sequence and is also a \emph{conjugacy invariant}, i.e., if $\varphi: \Omega_1 \to \Omega_2$ is a conjugacy, then $h_G(\Omega_1) = h_G(\Omega_2)$.
A {\bf potential} is any continuous function $\phi: \Omega \to \mathbb{R}$. Given a potential, we define the {\bf pressure} as
$$
p_G(\phi) := \lim_n \frac{\log |Z_{F_n}(\phi)|}{|F_n|},
$$
where $Z_{F_n}(\phi) = \sum_{w \in \Omega_F} \sup_{\omega \in [w]}\exp(\sum_{g \in F} \phi(g \cdot \omega))$. Notice that $p_G(0) = h_G(\Omega)$.
A {\bf single-site potential} is any potential that only depends on the value of $\omega$ at $1_G$, i.e., $\omega_{1_G}$. In other words, and without risk of ambiguity, we can think that a single-site potential is just a function $\phi: \Sigma \to \mathbb{R}$. In this case, $Z_{F_n}(\phi)$ has the following simpler expression:
$$
Z_{F_n}(\phi) = \sum_{w \in \Omega_F} \prod_{g \in F}\exp(\phi(w(g))).
$$
In this context, we will say that a symbol $a \in \Sigma$ is a {\bf vacuum state} if $a$ is a safe symbol and $\phi(a) = 0$.
\subsection{Reduction 1: from a hardcore model to a n.n. $G$-SFT with a vacuum state}
Let $(\Gamma,\lambda)$ be a hardcore model in $\mathcal{H}_G$. If $G \curvearrowright \Gamma$ is transitive, then $\Gamma = \mathrm{Cay}(G,S)$ for some finite symmetric set $S \subseteq G$. Then, it is easy to see that if $\Sigma = \{0,1\}$ and, for all $s \in S$,
$$
M_s = \left(\begin{array}{cc} 1 & 1\\1 & 0\end{array}\right),
$$
then $\Omega_{{\bf M}}$ coincides with the set $X(\Gamma)$ and $0$ is a safe symbol. In addition, there is a natural relationship between the activity function $\lambda$ and the single-site potential given by $\phi(0) = 0$ and $\phi(1) = \log\lambda(v)$, where $v$ is some (or any) vertex $v$. In other words, if $G \curvearrowright \Gamma$ is transitive, then $(\Gamma,\lambda)$ corresponds to a n.n. $G$-SFT with a vacuum state.
More generally, if $G \curvearrowright \Gamma$ is almost transitive, then $(\Gamma,\lambda)$ can also be interpreted as a n.n. $G$-SFT with a vacuum state. Indeed, consider the set $\Sigma_\Gamma = X(\Gamma[U_0])$, i.e., the set of independent sets of the subgraph $\Gamma[U_0]$ induced by some fundamental domain $U_0$. Since $\Gamma$ is locally finite and $G \curvearrowright \Gamma$ is free, there must exist a finite set $S \subseteq G \setminus \{1_G\}$ such that $SU_0$ contains all the vertices adjacent to $U_0$. Considering this, we define a collection of matrices ${\bf M}_\Gamma = \{M_s\}_{s \in S}$, where
$$
M_s(x,x') =
\begin{cases}
1 & \text{ if } xx' \in X(\Gamma[U_0 \cup sU_0]), \\
0 & \text{ otherwise,}
\end{cases}
$$
and $xx'$ denotes the concatenation of the independent set $x$ of $\Gamma[U_0]$ and the independent set $x'$ of $\Gamma[sU_0]$. In other words, $M_s(x,x') = 1$ if and only if the union of the independent set $x$ and the independent set $x'$ is also an independent set of $\Gamma[U_0 \cup sU_0]$.
Then, there is a natural identification between $\Omega_{{\bf M}_\Gamma} \subseteq \Sigma_\Gamma^G$ and $X(\Gamma)$. In particular, the symbol $0^{U_0} \in X(\Gamma[U_0])$ plays the role of a safe symbol in $\Omega_{{\bf M}_\Gamma}$. Moreover, we can define the single-site potential $\phi_\lambda: \Omega_{{\bf M}_\Gamma} \to \mathbb{R}$ given by $\phi_\lambda(\omega) = \sum_{v \in U_0} \omega_{1_G}(v) \log \lambda(v)$. Then, for every $F \in \mathcal{F}(G)$,
\begin{align*}
Z_{F}(\phi_\lambda) & = \sum_{w \in \Omega_F} \prod_{g \in F}\exp(\phi_\lambda(w(g))) \\
& = \sum_{w \in X(\Gamma[FU_0])} \prod_{g \in F}\exp\left(\sum_{v \in U_0} w(g)(v) \log \lambda(v)\right) \\
& = \sum_{w \in X(\Gamma[FU_0])} \prod_{g \in F} \prod_{v \in U_0} \exp(w(g)(v) \log \lambda(v)) \\
& = \sum_{w \in X(\Gamma[FU_0])} \prod_{v \in FU_0} \lambda(v)^{w(v)} \\
& = Z_{\Gamma}(FU_0,\lambda).
\end{align*}
Therefore, $p_G(\Omega_\Gamma, \phi_\lambda) = f_G(\Gamma,\lambda)$. In the language of dynamics, for every almost transitive and locally finite graph $\Gamma$, there exists a n.n. $G$-SFT with a safe symbol $\Omega_\Gamma$ such that $G \curvearrowright X(\Gamma)$ and $G \curvearrowright \Omega_\Gamma$ are conjugated. Moreover, this gives us a way to identify any hardcore model $(\Gamma,\lambda) \in \mathcal{H}_G$ with the corresponding $G$-SFT $\Omega_\Gamma$ and the single-site potential $\phi_\lambda$.
\subsection{Reduction 2: from a n.n. $G$-SFT with a vacuum state to a hardcore model}
Conversely, given a n.n. $G$-SFT $\Omega$ and a potential with a vacuum state, we can translate such scenario into a hardcore model. Indeed, consider the graph $\Gamma_\Omega$ defined as follows:
\begin{itemize}
\item for every $g \in G$, consider a finite graph $\Gamma_g$ isomorphic to $K_{|\Sigma|}$, the complete graph with $|\Sigma|$ vertices. In other words, for each $g \in G$ and for each $a \in \Sigma$ there will be a vertex $v_{g,a} \in V(\Gamma_g)$ and for every $a \neq b$, the edge $(v_{g,a},v_{g,b})$ will belong to $E(\Gamma_g)$;
\item the graph $\Gamma_\Omega$ will be the union of all the finite graphs $\Gamma_g$ plus some extra edges;
\item for every $s \in S$ and $a,b \in \Sigma$, we add the edge $(v_{1_G,a},v_{s,b})$ if and only if $M_s(a,b) = 0$;
\item we define $\lambda_\phi: \Gamma_\Omega \to \mathbb{R}_{>0}$ as $\lambda_\phi(v_{g,a}) = \exp(\phi(a))$ for every $g \in G$ and $a \in \Sigma$.
\end{itemize}
Then, $G$ acts on $\Gamma_\Omega$ in the natural way and $V(\Gamma_{1_G})$ corresponds to a fundamental domain of the action $G \curvearrowright \Gamma_\Omega$. In the language of dynamics, for every n.n. $G$-SFT with a safe symbol $\Omega$, there exists an almost transitive and locally finite graph $\Gamma_\Omega$ such that $G \curvearrowright \Omega$ and $G \curvearrowright X(\Gamma_\Omega)$ are conjugated. Moreover, it is clear that
$$
f_G(\Gamma_\Omega,\lambda_\phi) = p_G(\Omega,\phi),
$$
so in particular, all the representation and approximation theorems for free energy of hardcore models can be used to represent and approximate the pressure of n.n. $G$-SFTs $\Omega$ and potentials $\phi$ with a vacuum state, provided $(\Gamma_\Omega,\lambda_\phi)$ satisfies the corresponding hypotheses. Relevant cases like the \emph{Widom-Rowlinson model} \cite{2-georgii} and graph homomorphisms from $\Gamma$ to any finite graph with some vertex (which plays the role of a safe symbol) connected to every other vertex fall in this category.
\begin{figure}[ht]
\centering
\includegraphics[scale = 0.8]{3col.pdf}
\caption{On the left, a sample of a configuration in the n.n. SFT $\Omega$ corresponding to proper $3$-colorings of $\mathrm{Cay}(\mathbb{Z}^2, \{\pm(1,0), \pm(0,1)\})$ plus a safe symbol $0$, where each square corresponds to an element of $\mathbb{Z}^2$. On the right, the independent set in the graph $\Gamma_\Omega$ representing the configuration in $\Omega$.}
\label{fig:3cols}
\end{figure}
\subsection{Topological entropy and constraintedness of n.n. $G$-SFTs with safe symbols}
\label{sec9-4}
Let $\Omega \subseteq \Sigma^G$ be a n.n. $G$-SFT with $|\Sigma| = n_s + n_u$, where $n_s$ denotes the number of safe symbols in $\Sigma$ and $n_u$ denotes the number of symbols that are not safe symbols (\emph{unsafe}). Consider the n.n. $G$-SFT $\Omega_{n_u} \subseteq \Sigma_{n_u}^G$ obtained after collapsing all the safe symbols in $\Sigma$ into a single one, so that the $|\Sigma_{n_u}| = 1 + n_u$, and construct the graph $\Gamma_{\Omega_{n_u}}$. Then, given $F \in \mathcal{F}(G)$, we have that
\begin{align*}
|\Omega_F| & = \sum_{x \in X(\Gamma_{\Omega_{n_u}},FU_0)} \prod_{v \in FU_0} 1^{x(v)}n_s^{1-x(v)} \\
& = \sum_{x \in X(\Gamma_{\Omega_{n_u}},FU_0)} \prod_{v \in FU_0} \left(\frac{1}{n_s}\right)^{x(v)}n_s \\
& = n_s^{|FU_0|} \sum_{x \in X(\Gamma_{\Omega_{n_u}},FU_0)} \prod_{v \in FU_0} \left(\frac{1}{n_s}\right)^{x(v)} \\
& = n_s^{|FU_0|} Z_{\Gamma_{\Omega_{n_u}}}(FU_0, \frac{1}{n_s}),
\end{align*}
so, considering that $n_u = |U_0| = |\Gamma_{\Omega_{n_u}} / G|$,
\begin{align*}
h_G(\Omega) & = \lim_n \frac{\log |\Omega_{F_n}|}{|F_n|} \\
& = |U_0|\log n_s + \lim_n \frac{Z_{\Gamma_{\Omega_{n_u}}}(FU_0, 1/n_s)}{|F_n|} \\
& = |U_0|\log n_s + |U_0|f_G(\Gamma_{\Omega_{n_u}},1/n_s) \\
& = n_u\left(\log n_s + f_G(\Gamma_{\Omega_{n_u}},1/n_s)\right).
\end{align*}
Therefore, to understand and approximate $h_G(\Omega)$ reduces to study the hardcore model on $\Gamma_{\Omega,n_s}$ with constant activity $\frac{1}{n_s}$. In particular, if
$$
\frac{1}{n_s} < \lambda_c(\mu(\Gamma_{\Omega_{n_u}})),
$$
the hardcore model $(\Gamma_{\Omega,n_s},1/n_s)$ satisfies exponential SSM and the theory developed in the previous sections applies. This motivates the definition of the {\bf constraintedness} of a n.n. $G$-SFT $\Omega$ as the connective constant of $\Gamma_{\Omega_{n_u}}$, i.e.,
$$
\mu(\Omega) := \mu(\Gamma_{\Omega_{n_u}}),
$$
which can be regarded as a measure of how much constrained is $\Omega$ (the higher $\mu(\Omega)$, the more constrained it is). Notice that if
$$
\frac{1}{n_s} < \lambda_c(\mu(\Omega)+1),
$$
then $(\Gamma_{\Omega_{n_u}},1/n_s)$ satisfies exponential SSM. In particular, $\Omega_{n_u}$ has a unique \emph{measure of maximal entropy} and therefore, also $\Omega$ has unique measure of maximal entropy, namely, the pushforward measure (see \cite{2-burton,1-haggstrom}). Moreover, the topological topological entropy of $\Omega_{n_u}$ has an arboreal representation and can be approximated efficiently. Since $\mu(\Omega) \leq \Delta(\Gamma_{\Omega_{n_u}}) - 1$, we have that it suffices that
$$
\frac{1}{n_s} < \lambda_c(\Delta(\Gamma_{\Omega_{n_u}})),
$$
For example, the n.n. $G$-SFT $\Omega$ represented in Figure \ref{fig:3cols} satisfies that $\Delta(\Gamma_{\Omega_{n_u}}) = 6$ and $\lambda_c(6) = \frac{5^5}{4^6} = \frac{3125}{4096}$; then, if $n_s > \frac{4096}{3125} = 1,31072$, we see that it suffices to have 2 copies of the safe symbol $0$ in order to have exponential SSM.
In general, since each vertex of the fundamental domain is connected to $n_u -1$ vertices in the clique and to at most $n_u$ vertices for each element $s$ in the generating set $S$, we see that each vertex in $\Gamma_{\Omega_{n_u}}$ is connected to at most $(n_u - 1) + |S|n_u$ other vertices. Then, we can estimate that
$$
\Delta(\Gamma_{\Omega_{n_u}}) \leq (|S|+1)n_u - 1,
$$
so, in particular, if
$$
\frac{1}{n_s} < \lambda_c((|S|+1)n_u - 1),
$$
exponential SSM holds (and therefore, again, uniqueness of measure of maximal entropy). This last equation and its relationship with the constraintedness of $\Omega$ has a similar flavor to the relationship between the percolation threshold $p_c(\mathbb{Z}^d)$ of the $\mathbb{Z}^d$ lattice and the concept of \emph{generosity} for $\mathbb{Z}^d$-SFTs introduced in \cite{1-haggstrom} by H\"aggstr\"om.
\begin{remark}
It may be the case that a n.n. $G$-SFT $\Omega \subseteq \Sigma^G$ with a safe symbol could be represented by a graph $\Gamma$ which is better in terms of connectedness or maximum degree compared with the canonical representation $\Gamma_\Omega$, since we could encode $\Sigma$ using other fundamental domains, with a lower connectivity than the complete graph. For example, the n.n. $\mathbb{Z}^2$-SFT $\Omega_\Gamma$ corresponding to the graph $\Gamma$ on the left in Figure \ref{fig:2graphs} has $7$ symbols (the $7$ independent sets of the $4$-cycle), including a safe one. However, the canonical graph representation of $\Omega_\Gamma$, i.e., the graph $\Gamma_{\Omega_\Gamma}$, has a fundamental domain consisting of a clique with $6$ vertices, without considering extra connections. In particular, we see that both, $\Gamma$ and $\Gamma_{\Omega_\Gamma}$ represent $\Omega$, but $\Delta(\Gamma) = 3 < 6 \leq \Delta(\Gamma_{\Omega_\Gamma})$. This motivates a finer notion of constraintedness, namely,
$$
\tilde{\mu}(\Omega) = \inf\{\mu(\Gamma): \Gamma \text{ represents } \Omega_{n_u}\},
$$
and the aforementioned results would still hold if we replace $\mu(\Omega)$ by $\tilde{\mu}(\Omega)$. Notice that a fundamental domain $U_0$ has at least $|U_0|+1$ independent sets (the empty one and all the singletons). In particular, this implies that $\tilde{\mu}(\Omega)$ is a minimum, since we only need to optimize over graphs $\Gamma$ with a fundamental domain $U_0$ such that $X(\Gamma[U_0]) = |\Sigma_{n_u}|$.
\end{remark}
\begin{figure}[ht]
\centering
\includegraphics[scale = 2.4]{linegraph.pdf}
\caption{On the left, an almost transitive graph $\Gamma$ with $\Gamma[U_0] \cong C_4$, the $4$-cycle. On the right, a portion of the line graph of $\mathrm{Cay}(\mathbb{Z}^2,\{\pm(1,0),\pm(0,1)\})$.}
\label{fig:2graphs}
\end{figure}
\subsection{The monomer-dimer model and line graphs}
Given a graph $\Gamma = (V,E)$, we say that two different edges $e_1,e_2 \in E$ are \emph{incident} if they have one vertex in common. A {\bf matching} in $\Gamma$ is a subset $M$ of $E$ without incident edges. In a total parallel with the hardcore model case, we can represent a matching with an indicator function $m: E \to \{0,1\}$, denote the set of matchings of $\Gamma$ by $X^e(\Gamma)$, and define the associated partition function for some activity function $\lambda: E \to \mathbb{R}_{> 0}$ as
$$
Z_\Gamma^{e}(\lambda) = \sum_{m \in X^e(\Gamma)} \prod_{e \in E}\lambda(e)^{m(e)}.
$$
The pair $(\Gamma,\lambda)$ is called the {\bf monomer-dimer model} and, as for the case of the hardcore model, we can define its associated free energy and Gibbs measures for a Gibbs specification adapted to this case.
An important feature of the monomer-dimer model is that, despite all its similarities with the hardcore model, it exhibits the SSM property for all values of $\lambda$ \cite{1-bayati} and, in particular, there is no phase transition \cite{1-heilmann}.
Considering this, most of the results presented in this paper, in particular the ones related to representation and approximation, can be adapted to counting matchings (see \cite{1-gamarnik} for a particular case), and there will not be a phase transition. One way to see this is through the \emph{line graph} $L(\Gamma)$ of the given graph $\Gamma$. Indeed, if we define $L(\Gamma)$ as the graph with set of vertices $E$ and set of edges containing all the adjacent edges in $E$, it is direct to see that there is a correspondence between matchings in $\Gamma$ and independent sets in $L(\Gamma)$, i.e.,
$$
Z_\Gamma^{e}(\lambda) = Z_{L(\Gamma)}(\lambda)
$$
In particular, this tell us that all the results in our paper that involve some restriction on $\lambda$, apply to every graph that can be obtained as a line graph of another one without restriction on $\lambda$. For example, the graph $\Gamma$ represented on the right in Figure \ref{fig:2graphs} corresponds to the line graph of the Cayley graph of $\mathbb{Z}^2$ with canonical generators, i.e.,
$$\Gamma = L(\mathrm{Cay}(\mathbb{Z}^2,\{\pm(1,0),\pm(0,1)\})).$$
Then, this observation implies that we can represent and approximate $f_{\mathbb{Z}^2}(\Gamma,\lambda)$ for every $\mathbb{Z}^2$-invariant activity function $\lambda$ on $\Gamma$.
\subsection{Spectral radius of matrices and occupation probabilities on trees}
A curious consequence of the hardcore model representation of a n.n. $G$-SFT with a safe symbol is that when $G = \mathbb{Z}$ and $S = \{1\}$, then $h_\mathbb{Z}(\Omega)$ has a well known characterization in terms of the transition matrix $M = M_1$ \cite{1-lind}.
If $M$ is irreducible and aperiodic, there is always a unique stationary Markov chain $\mathbb{P}_M$ associated to $M$ such that $\log \lambda_M = h_\mathbb{Z}(\Omega_M)$, where $\lambda_M$ denotes the Perron eigenvalue of $M$ and we consider the natural invariant order in $\mathbb{Z}$.
Now, if $M$ is a matrix such that the $i$th row and the $i$th column have no zeros, then $M$ is irreducible and aperiodic, and in fact, the $i$th symbol, let's call it $a$, is a safe symbol. In such case, we have that
$$
\log \lambda_M = h_\mathbb{Z}(\Omega_M) = -\log \mathbb{P}_M([a^{0}] \vert [a^{-\mathbb{N}}]) = -\log \mathbb{P}_M([a^{0}] \vert [a^{-1}]),
$$
Therefore, $\lambda_M = \frac{1}{\mathbb{P}_M(a^{0} \vert a^{-1})}$ and to compute the spectral radius of $M$ reduces to compute $\mathbb{P}_M(a^{0} \vert a^{-1})$. For example, consider the following matrix
$$
M =
\left(\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & 1 & 0 & 1 \\
1 & 0 & 1 & 0 \\
1 & 1 & 1 & 0 \\
\end{array}\right),
$$
where $a$ is the symbol associated to the first row. Given such matrix $M$, we can always construct a graph representation $\Gamma_{\Omega_M}$ of $\Omega_M$ as in Figure \ref{fig:graph1d}.
\begin{figure}[ht]
\centering
\includegraphics[scale = 2.4]{1dSFTshort.pdf}
\caption{A graph representation of the $0$-$1$ matrix $M$.}
\label{fig:graph1d}
\end{figure}
Now, it is known that $|\mathbb{P}_M(0^{0} \vert 0^{-1}) - \mathbb{P}_M(0^{0} \vert 0^{\{n,-1\}})|$ goes to zero exponentially fast as $n$ goes to infinity and, since every Markov chain is a Markov random field, we have that
\begin{align*}
\mathbb{P}_M(a^{0} \vert a^{\{n,-1\}}) & = \mathbb{P}_{\Gamma_{\Omega_M}[\{0,\dots,n-1\}U_0],1}(0^{U_0}) \\
& = \sum_{i=1}^{U_0}\mathbb{P}_{\Gamma_{\Omega_M}[\{0,\dots,n-1\}U_0],1}(0^{v_i} \vert 0^{\{v_1,\dots,v_{i-1}\}}) \\
& = \sum_{i=1}^{U_0}\mathbb{P}_{\Gamma_{\Omega_M}[\{0,\dots,n-1\}U_0 \setminus \{v_1,\dots,v_{i-1}\}],1}(0^{v_i}) \\
& = \sum_{i=1}^{U_0}\mathbb{P}_{T_\mathrm{SAW}\Gamma_{\Omega_M}[\{0,\dots,n-1\}U_0 \setminus \{v_1,\dots,v_{i-1}\}],1}(0^{v_i}) \\
\end{align*}
This gives us an arboreal representation and a method to compute the spectral radius of any such matrix $M$, that we believe could be of independent interest.
\subsection{Final comments}
We have established a computational phase transition for computing the free energy of hardcore models on almost transitive amenable graphs. To do this, the techniques presented here could be broken down in mainly two aspects: \emph{representation} and \emph{approximation}.
By representation, we mean all the preprocessing done to $f_G(\Gamma,\lambda)$ in Theorem \ref{thm:smb}, Theorem \ref{thm:rep1}, and Theorem \ref{thm:rep2}. The first theorem does not use much more than the definition of Gibbs measure and it is possible to prove that holds in other contexts (for example, see \cite{1-gurevich}). The second theorem requires SSM to hold, a property that other models exhibit and it is possible to extend it to such other cases (see \cite{2-briceno}). On the other hand, in \cite{1-adams} it was proven that analogue formulas could be obtained in the non-uniqueness regime for the hardcore model and other classical ones. The third theorem is very particular to the hardcore model, but the representation of marginal probabilities using trees of self-avoiding walks can be extended to any 2-spin model with isotropic nearest-neighbor constraints and interactions; in particular, this includes the Ising model case \cite{3-sinclair}.
Later, in Theorem \ref{thm:phase}, that involves the approximation aspects of our work, we gave an approximation algorithm for $f_G(\Gamma,\lambda)$ based on computing the occupation probability on a special tree satisfying an exponential decay of correlations. We believe that other techniques could be explored such as the ones developed by Barvinok \cite{1-barvinok} or the transfer matrix method \cite{2-marcus} adapted to other groups besides $\mathbb{Z}^d$. In this sense, it does not seem unlikely that our results could be extended to other values of $\lambda$ outside of the interval $[0,\lambda_c(\Delta))$.
\section*{Acknowledgements}
I would like to thank Brian Marcus for his valuable suggestions after reading a first draft of this work and Tom Meyerovitch for helpful discussions about invariant random orders.
\bibliographystyle{abbrv}
|
1,941,325,220,572 | arxiv | \section{Introduction}
\label{sec1}
The Index Coding problem, introduced by Birk and Kol \cite{BK} considers a system with a source and multiple receivers communicating via a noiseless broadcast channel where each receiver can have some prior knowledge about a few source symbols before the start of transmission.
The rate of an index code is the ratio of the length of the transmitted codeword to the length of the message symbols. An optimal index code is a transmission scheme that ensures decodability at receivers with minimal rate, known as the \textit{broadcast rate}.
Upper and lower bounds for the broadcast rate of index coding have been proposed through several graph-theoretic methods such as clique number \cite{BK}, chromatic number \cite{YBJ} and also via matrix methods \cite{YBJ}.
In particular, the broadcast rate of index coding is sandwiched between the clique number $\omega(\overline{G})$ and chromatic number $\chi(\overline{G})$ of the complement of an associated graph called the \textit{side-information graph} $G$. Several approaches via linear programming \cite{BKE}, interference-alignment \cite{MCJ1} were also used to address this problem in new directions.
The gap between the broadcast rate of index coding and various graph theoretic parameters were investigated for instance in \cite{HN},\cite{BKE}.
There exists a class of undirected graphs for which the clique number equals the chromatic number for every induced subgraph $H \subseteq G $. Such graphs are termed as \textit{perfect} graphs. \emph{Imperfect graphs} are those graphs which are not perfect. It was shown in \cite{CRST} that an undirected graph $G$ is perfect if and only if $G$ contains no odd cycle of length greater than five or its complement as an induced subgraph. For a perfect graph $G$, the broadcast rate is known to be equal to $\omega(\overline{G})$. Besides perfect graphs, there are only a few classes of graphs for which the broadcast rate is known. This includes graphs with broadcast rate $2$ \cite{BKE}
and graphs with neighbouring side-information \cite{MCJ1}.
The contributions and organization of this paper are as follows. Our first contribution is regarding the broadcast rate of \textit{circular perfect graphs}, which were introduced by Zhu \cite{ZHU} and contains the class of perfect graphs as a strict subset. After a brief review of index coding in Section \ref{sec2}, for the case of a side-information graph $G$ which is a complement of a circular perfect graph, we show that the broadcast rate is equal to a parameter known as the \textit{circular clique number} of $\overline{G}$ (Section \ref{sec3}). In the process we also obtain a new lower bound for the broadcast rate for undirected graphs, namely the \textit{circular clique number} of $\overline{G}$. Using our result, we give the broadcast rate for several new classes of graphs in Section \ref{subsecclasses}.
Our second contribution is regarding the relationship between the broadcast rate of a graph and its complement. Bounds on the product and sum of the chromatic numbers of a graph $G$ and its complement, which are together known as \emph{complementary graphs}, were derived by Nordhaus and Gaddum in \cite{NoG}. We present Nordhaus-Gaddum type upper and lower bounds for the product of the vector linear broadcast rates of $G$ and that of $\overline{G}$ (Section \ref{sec4}). The lower bound holds for all directed side-information graphs as well. We then show that these bounds are met with equality, even for some perfect graphs. We also curate some results from prior literature to show that there exists a class of circular perfect graphs (which include imperfect graphs) which satisfy the Nordhaus-Gaddum type lower bound with equality. We end the paper in Section \ref{discussion} with further directions of research.
\textit{{Notations and Terminology}}: Throughout the paper the following notations are used. A graph $G$ is an undirected simple graph with vertex set $V(G)$ and edge set $E(G)$. We specifically mention that a graph is directed if required. $K_n$ denotes the complete graph with $n$ vertices. For a positive integer $m$, $[m]$ denotes the set $\{1,2,\cdots,m\}$. The complement of graph $G$ is denoted by $\overline{G}$. A clique of $G$ is a set of vertices adjacent to each other. The \textit{clique number} $\omega(G)$ denotes the size of the largest clique of $G$. The \textit{chromatic number} $\chi(G)$ of $G$ is the minimum number of colors in a proper coloring of $G$. The \textit{fractional chromatic number} of $G$ is denoted as $\chi_f(G)$. For graphs $H$ and $G$, the notation $H\subseteq G$ denotes that $H$ is an induced subgraph of $G$. For more basic terminology related to graph theory, the reader is referred to \cite{Die}.
For a set $A,B$ $A\backslash B$ denotes the elements in $A$ but not in $B$. For some element $i$, We also denote $A\backslash \{i\}$ by $A\backslash i$. The finite field with $q$ elements is ${\mathbb F}_q$. The span of a set of vectors $U$ is denoted as $span(U)$.
\section{Definitions}
\label{sec2}
The single unicast index coding problem consists of a broadcast channel along with the following:
\begin{itemize}
\item A source possessing a set of $n$ messages $\left\{\boldsymbol{x_i} : i \in [n]\right\},$ where each $\boldsymbol{x_i}$ is a $t$-length vector over $\mathbb {F}_q$ denoted by $\boldsymbol{x_i}=(x_{i1},\hdots,x_{it})$ where $x_{ij}\in {\mathbb F}_q$.
\item $n$ receivers, with the receiver $i \in [n]$ demanding $\boldsymbol{x_i}$.
\item The receiver $i$ knows a subset of the messages corresponding to the indices $S_i\subset[n]\backslash i$ denoted by $\{\boldsymbol{x_j}: j\in S_i\}$. These are called the \textit{side information} at receiver $i$.
\end{itemize}
The $nt$ message symbols are encoded to an $l$-length codeword $\boldsymbol{c}\in{\mathbb F}_q^l$ by an encoding function and broadcasted to the receivers, such that for each $i\in[n]$, $\boldsymbol{x_i}$ can be decoded at the receiver $i$ using $\boldsymbol{c}$ and $\{\boldsymbol{x_j}:j\in S_i\}$. The \textit{rate} of such an \textit{index code} is then defined as $\frac{l}{t}$
Any single unicast problem can be defined by a directed graph $G$ called the \textit{side information graph} where $V(G)=[n]$. For $u,v\in V(G)$, there exists a directed edge in $G$ from $u$ to $v$ if $u \in S_v$. Index coding problems in which the side information sets are symmetric, i.e. $i \in S_j$ if and only if $j \in S_i, \forall i, j \in V$ and $i \neq j $, can be represented by undirected graphs (by replacing the directed edges between $i,j$ by undirected edges).
An index code is said to be \textit{linear} if the encoding operation is linear. If messages are of length $t=1$ the index code is called a \textit{scalar} code else the index code is known as a \textit{vector} index code. Let $\beta_q(t,G)$ denote the minimum length of any index code for a given $t$ for an index coding problem defined by a side information graph $G$. The \textit{broadcast rate} of $G$ \cite{BKE} is given as $\beta_q(G) = \lim\limits_{t\rightarrow \infty}\frac{\beta_q(t,G)}{t},$ where the limit is known to exist due to subadditivity of $\beta_q$. In a similar fashion, let $\beta_{vl}(G)$ be the minimum rate of vector linear index codes for $G$, and $\beta_{sl}(G)$ be the minimum rate over all possible scalar linear index codes. It is known that for a given single unicast setting $G$,
$\omega(\overline{G}) \leq \beta(G) \leq \beta_{vl}(G) \leq \chi_f(\overline{G}) \leq \chi(\overline{G})$ and $\beta_{vl}(G) \leq \beta_{sl}(G) \leq \chi(\overline{G})$ from previous studies \cite{YBJ},\cite{BKE} on bounds for $\beta(G)$.
For an index coding problem with $n$ messages, a linear index code of length $l$ encoding $t$-length messages can be represented using a matrix $B$ (over ${\mathbb F}_q$) of size $l\times nt$ such that the transmitted codeword is $B\boldsymbol{x^T},$ where $\boldsymbol{x}=(\boldsymbol{x_1},\hdots,\boldsymbol{x_n})\in {\mathbb F}_q^{nt}$ is the cumulated message vector. Assume that the columns of $B$ be denoted as $B^{ij}:i\in [n], j\in [t].$ Thus $B^{ij}$ denotes the precoding vector of the message $x_{ij}$. The following simple well known lemma (see \cite{MCJ1}, for instance, for a proof) will be used in this work to prove some results.
\begin{lemma}
\label{IAapproach}
A matrix $B$ with elements from ${\mathbb F}_q$ of size $l\times nt$ is a valid encoding matrix ensuring decoding at all receivers if and only if
{\small
\[
B^{ij}\notin span(\{B^{ij'}:j'\in[t]\backslash j\}\cup\{B^{i'j}: i'\in[n]\backslash \{S_i \cup \{i\}\} , \forall j\}).
\]}
\end{lemma}
We also need the concept of a \textit{confusion graph} of an index coding problem.
Consider the index coding instance defined by a (possibly directed) side information graph $G$. The \textit{confusion graph} \cite{ArK1} of $G$ for $t-$length message vectors is an undirected graph ${\Gamma_t(G)}$ with vertex set ${\mathbb F}_q^{nt}$ representing the cumulated message vectors $\{(\boldsymbol{x_1},\hdots,\boldsymbol{x_n}):\boldsymbol{x_i}\in{\mathbb F}_q^t\}$. Two vertices $\boldsymbol{x}=(\boldsymbol{x_1},\hdots,\boldsymbol{x_n}),\boldsymbol{y}=(\boldsymbol{y_1},\hdots,\boldsymbol{y_n})\in{\mathbb F}_q^{nt}$ are connected by an edge in $\Gamma_t(G)$ if there exists some $i\in[n]$ such that $x_{ik} \neq y_{ik}$ while $\boldsymbol{x_j}=\boldsymbol{y_j}, \forall j \in S_i $.
\section{The Broadcast Rate of Circular Perfect Graphs}
\label{sec3}
In this section, we obtain the broadcast rate for the complements of a class of graphs known as circular perfect graphs, and in the process refine (albeit mildly) the $\omega(\overline{G})$ lower bound for the broadcast rate of any graph $G$. We first recall some of the basic definitions and results related to circular perfect graphs.
Zhu \cite{ZHU} introduced \textit{circular perfect graphs} based on concept of a \textit{circular coloring} \cite{V}. For positive integers $k$ and $d$, a $(k,d)$ \textit{circular coloring} of graph $G$ is a function
$f: V\rightarrow \{0,1,\cdots k-1\}$ with $d \leq |f(u)-f(v)|\leq (k-d)$ if $(u,v) \in E$. The \textit{circular chromatic number} $\chi_c(G) $ is defined as follows.
\[
\chi_c(G) = min\left\{\frac{k}{d}: G \text{ has } (k,d) \text{ circular coloring }\right\}.
\]
A \textit{circular clique} $K_{\frac{k}{d}} (k \geq 2d)$ is a graph $G$ with $V(G)=\{0,\hdots,k-1\}$ and $d \leq |u-v|\leq k-d$ if and only if $(u,v)\in E(G)$. The \textit{circular clique number} $\omega_c(G)$ is defined as
\[
{\cal \omega}_c(G)=max\left\{\frac{k}{d} : K_{\frac{k}{d}} \subseteq G \text{ and } \gcd(k,d)=1\right\}.
\] An example of a circular clique is shown in Fig. \ref{fig:my_label}.
\begin{figure}[h!]
\begin{center}
\includegraphics [width=1.2in]{4.pdf}
\label{fig:my_label
\end{center}
\caption{A circular perfect graph: The circular clique $K_{9/3}$}
\label{fig:my_label}
\end{figure}
Generally $\omega(G)\leq \omega_c(G) \leq\chi_f(G)\leq \chi_{c}(G) \leq \chi(G)$ for any graph $G$. Furthermore it is known that $\omega(G)=\lfloor\omega_c(G)\rfloor$ and $\chi(G)=\lceil\chi_c(G)\rceil$.
Graphs for which $ \omega_c(H)=\chi_c(H)$ for every $H \subseteq G $ are called \textit{circular perfect}. By definition, the class of perfect graphs are a subclass of the class of circular perfect graphs. However, some imperfect graphs, such as odd cycles and their complements, are also circular perfect. In Section \ref{subsecclasses}, we give many examples of graph classes which are circular perfect. For a nice survey on circular perfect graphs and some of the abovementioned results, the reader is referred to \cite{ZHU1,BJH}.
We now prove that the broadcast rate of \textit{any} graph $G$ is bounded from below by the circular clique number $\omega_c(\overline{G})$. To show this, we need the following lemma which is known in index coding literature. Here we give a formal proof of the same as it seems missing in the literature to the best of our knowledge.
\begin{lemma}
\label{lowerboundconf}
Let $H$ be a induced subgraph of a graph $G$. Then $\beta(G)\geq \beta(H)$.
\end{lemma}
\begin{IEEEproof}
Consider a graph $G$ with vertex set $[n]$. Suppose $H$ be an induced sub graph with $k$ vertices say $[k]$ . We first show that for any $t$, the confusion graph $\Gamma_t(H)$ is isomorphic to a subgraph of $\Gamma_t(G).$ This would mean that $\omega(\Gamma_t(H))\leq \omega(\Gamma_t(G))$. The result will then follow from the fact shown in \cite{ArK1} (Theorem 2) where it is shown that \[
\beta(G)=\lim_{t\rightarrow \infty}\frac{1}{t}log(\omega(\Gamma_t(G))).
\]
We now show that $\Gamma_t(H)\subseteq \Gamma_t(G)$. To do this, to each vertex $\boldsymbol{v}=(\boldsymbol{v_1},\hdots,\boldsymbol{v_k})$ of $\Gamma_t(H)$ (where $\boldsymbol{v_i}\in {\mathbb F}_q^{t}$), associate the vertex $\boldsymbol{v'}=(\boldsymbol{v_1},\hdots,\boldsymbol{v_k},\boldsymbol{z_{k+1}},\hdots,\boldsymbol{z_n})\in {\mathbb F}_q^{nt}$ of $\Gamma_t(G)$ by appending some fixed $((n-k)t)$-length vector $(\boldsymbol{z_{k+1}},\hdots,\boldsymbol{z_n})$, where $\boldsymbol{z_i} \in {\mathbb F}_q^{t}.$ Now note that if $(\boldsymbol{v},\boldsymbol{w})\in E(\Gamma_t(H)),$ then $(\boldsymbol{v'},\boldsymbol{w'})\in E(\Gamma_t(G)).$ This follows from the definition of edge set of confusion graph. Thus $\Gamma_t(H)\subseteq \Gamma_t(G)$. Hence the claim follows.
\end{IEEEproof}
Using Lemma \ref{lowerboundconf}, we now show the following result which gives a new lower bound of the broadcast rate of undirected side-information graphs.
\begin{lemma}
\label{lowerboundbetacirccliq}
Let $G$ be an undirected side-information graph. Then the broadcast rate of $G$ is lower bounded by the circular clique number of $\overline{G},$ i.e., $\beta(G)\geq \omega_c(\overline{G})$.
\end{lemma}
\begin{IEEEproof}
By definition, if $\omega_c(\overline{G})=\frac{k}{d}$, then $\overline{K_{\frac{k}{d}}}$ is an induced subgraph of $G$. The broadcast rate of $\overline{K_{\frac{k}{d}}}$ is known (Theorem VI.3, \cite{BKE}) as $\beta(\overline{K_{\frac{k}{d}}})=\frac{k}{d}.$ Using Lemma \ref{lowerboundconf}, we thus have $\beta(G)\geq \beta(\overline{K_{\frac{k}{d}}})=\frac{k}{d}=\omega_c(\overline{G}).$ This concludes the proof.
\end{IEEEproof}
Observe that for any graph $G$, $\beta(G)\leq \chi_c(\overline{G})$, since $\beta(G)\leq \chi_f(\overline{G})$ and $\chi_f(\overline{G})\leq \chi_c(\overline{G})$. Combining this observation with Lemma \ref{lowerboundbetacirccliq}, we have the following result.
\begin{theorem}
\label{circperfectthm}
Let $G$ be a side-information graph such that $\overline{G}$ is circular perfect. Then $\beta(G)=\omega_c(\overline{G})=\chi_c(\overline{G}).$
\end{theorem}
The following lemma ensures that if $G$ is the complement of a circular perfect graph, and a $(k,d)$ circular coloring of $\overline{G}$ with $\frac{k}{d}=\chi_c(\overline{G})$ is known, then the optimal code for $G$ can be obtained in polynomial time.
\begin{lemma}
\label{construction}
For any index coding problem defined by a side information graph $G$, there exists an achievable index code with rate $\chi_c(\overline{G})$.
\end{lemma}
\begin{IEEEproof}
Let $\chi_c(\overline{G})=\frac{k}{d}$ and $C_i$ be the set of vertices colored with color $i$, for $i\in \{0,1,\cdots,k-1\}$ in a $(k,d)$ circular coloring of $\overline{G}$. We construct a linear code of rate $\frac{k}{d}$ over any field ${\mathbb F}_q$.
Suppose that each message is a vector of length $d$.
Let the message demanded by receiver $v\in G$ be $\boldsymbol{x_v}=(x_{v,0}, x_{v,1},\cdots, x_{v,d-1})\in{\mathbb F}_q^d$.
For each color class $C_i$, and for each $ j \in \{0,1,\cdots, d-1\}$, define
$X_{ij}\triangleq \sum_{v\in C_i} x_{v,j}$. Note that the receiver $v\in C_i$ which demands $x_{v,j}$ has all other messages $x_{v',j}$ for any other $v' \in C_i$ in its side information since all vertices in $C_i$ are colored with same color $i$, which means they are non-adjacent in $\overline{G}$.
We now describe the transmission scheme. For each $l \in \{0,1,\cdots, k-1\},$ define by $A_l$ the set $\{k-(d-1)+l, k-(d-2)+l, \hdots,l\}(mod~k)$. For each $l$, the symbol $T_l$ defined as below is transmitted.
\[
T_l \triangleq \sum_{i\in A_l} X_{ij_i},~where~ j_i=(k-i+l)(mod~k).
\]
We claim that the transmissions $T_l, l\in\{0,\hdots,k-1\}$ is a valid index code. To see this, note that for any $i,i' \in A_l$, $ k-(d-1) \leq |i-i'|\leq (d-1)~(mod~k)$.
By definition of circular coloring, the vertices $i\in A_l$ involved in transmission $T_l$ are non-adjacent to each other in $\overline{G}$ which ensures decodability of the messages in $T_l$ at the respective receivers. Also note that by definition of $T_l$, each message $x_{v,j}$ is a summand of $X_{ij}$ (such that $v\in C_i$), which is a summand in $T_{(i+j)(mod k)}$ . Note that the rate of the code is $\frac{k}{d}$. This proves the lemma.
\end{IEEEproof}
As a result of Theorem \ref{circperfectthm} and Lemma \ref{construction}, the broadcast rate and achievability of complements of circular perfect graphs are characterized. We remark that there is no known polynomial time algorithm to compute the circular chromatic number (or the circular clique number) of circular perfect graphs in general. Therefore the circular chromatic number (and the circular clique number) are known for a few specific classes of circular perfect graphs only. However it is known from \cite{PeW} that $\omega(G)$ and $\chi(G)$ for the circular perfect graphs can be obtained in polynomial time. As $\beta(G)=\chi_c(\overline{G})$ if $\overline{G}$ is circular perfect and $\chi(\overline{G})=\lceil\chi_c(\overline{G})\rceil$, a close approximation can be obtained for the broadcast rate of all complements of circular perfect graphs. In the following subsection, we give the circular perfect graphs for which $\chi_c(G)$ is known.
\subsection{Classes of circular perfect graphs}
\label{subsecclasses}
We provide here some known classes of circular perfect graphs whose circular chromatic numbers are known or can be computed in polynomial time. Thus, this enlarges the set of graphs for which the broadcast rates are known. These graphs include outerplanar \cite{PW}, convex-round graphs\cite{BJH}, interlacing graphs \cite{BSBL} and few classes of concave-round graphs \cite{SC}.
\textit{Convex-round graphs} are circular perfect\cite{BJH}. A graph is convex-round if vertices of $G$ can be circularly ordered i.e $V=\{v_1,v_2,\cdots,v_n\}$ such that the neighbourhood of each vertex $v_i$ is contiguous in the same ordering. In \cite{BJH}, it is shown that $\chi_c(G)$ for convex-round graphs can be computed in polynomial time. The class of side information graphs whose complements are convex round graphs includes the class of undirected \textit{symmetric neighbouring side information graphs}, studied for instance in \cite{MCJ1}.
Complements of convex-round graphs are called \textit{concave-round graphs}. Some classes of circular perfect concave round graphs are shown in \cite{SC}. \textit{Webs}, which are complements of circular cliques $K_{\frac{p}{q}}$, come under the class of concave-round graphs. For $\frac{p}{q} \geq 2$, $\chi_c(\overline{K_{\frac{p}{q}}})= \frac{p}{\lfloor{\frac{p}{q}}\rfloor}$. It is shown in \cite{YC} that a web $\overline{K_{\frac{p}{q}}}$ is circular perfect if and only if $q=2$ or $p=2q$ or $p=2q+1$ or $q=3,p=3k$. . In \cite{VaR}, the capacity of index coding problems with symmetric neighbouring interference is discussed. A \textit{symmetric neighbouring interference graph} is a side information graph in which there is an ordering of the $k$ vertices such that each receiver's side information set does not include $D$ messages `below' and $U$ messages `above' its desired message in the ordering. If $U=D$, then the complement of the neighbouring interference graph is a web $\overline{K_{\frac{k}{D+1}}}$ and as indicated above, we know that $\chi_c(\overline{K_{\frac{k}{D+1}}})= \frac{k}{\lfloor{\frac{k}{D+1}}\rfloor}$ for any $\frac{k}{D+1}\geq 2$. The construction for index code given in \cite{VaR} for such neighbouring interference graphs achieves a rate of $\frac{k}{\lfloor{\frac{k}{D+1}}\rfloor}$ and is therefore optimal when the complement is circular perfect.
Outerplanar graphs are circular perfect and the circular chromatic number of outer planar graph is $2$ when the graph has only even cycles and is $2+\frac{1}{d}$ when $2d+1$ is length of the shortest odd cycle\cite{PW}.
\textit{Interlacing graphs} are circular perfect \cite{BSBL}.
Consider a circle with points $\{1,2,\cdots,n\}$ in clockwise order. An interlacing graph is a graph with vertices $v$ corresponding to $k$- sized subsets of $\{1, ..., n\}$ such that any two distinct circle-points in a vertex $v$ have distance at least $r$ around the circle, and an edge exists between two vertices $v,v'$ if they \textit{interlace}, i.e. after removing the points in $v$ from the circle, the remaining points in $v'$ are in different connected components. The circular chromatic number of such an interlacing graph is $\frac{n}{k}$.
\section{On Linear Broadcast Rates for complementary Graphs}
\label{sec4}
The previous section addressed index coding problems on graphs whose complements are circular perfect. As the class of circular perfect graphs is not closed with complements, this means the broadcast rate of index coding problems for side-information graphs which themselves are circular perfect is still open. In this section, we relate the vector linear broadcast rate of a graph and its complement. Though these relationships do not seem to specialize particularly for circular perfect graphs, they seem important in the sense of providing an understanding of the broadcast rates for complementary index coding problems. These relationships can be viewed as Nordhaus-Gaddum type bounds for the vector linear broadcast rate, following the work of \cite{NoG} for $\chi(G)$.
We first prove a lower bound on the product of vector linear rates of complementary graphs. Towards that end, we need the following lemma. The proof of this lemma may be skipped by the reader who wants to quickly move to a main result in this section, which is Theorem \ref{mainthmlowerbound}.
\begin{lemma}
\label{lemmatensor}
Let $\boldsymbol{v}$ and $\boldsymbol{w}$ be non-zero vectors in a finite dimensional vector space $V$ over some field ${\mathbb F}$ such that $\boldsymbol{v}\notin span(A_1)$ and $\boldsymbol{w}\notin span(A_2)$, for some sets of vectors $A_1$ and $A_2$ from $V$. Let $D$ and $E$ be non-empty subsets of vectors from $V$. Then the tensor product,
\begin{align}
{\boldsymbol v}\otimes{\boldsymbol w}\notin span (\{\boldsymbol{a}\otimes \boldsymbol{d}&:\boldsymbol{a}\in span(A_1), \boldsymbol{d}\in D\}
\\
&\cup\{\boldsymbol{e}\otimes \boldsymbol{b}:\boldsymbol{b}\in span(A_2), \boldsymbol{e}\in E\}).
\end{align}
\end{lemma}
\begin{IEEEproof}
Without loss of generality, we assume that $D$ and $E$ are subspaces of $V$. If this is not the case, then we can use this proof to the statement of the lemma for $span(D)$ and $span(E)$ in the place of $D$ and $E$, and naturally recover our desired result. Let the dimension of $D$ and $E$ be $m$ and $n$ respectively, and their bases be $\boldsymbol{d_i}:i=1,..,m$ and $\boldsymbol{e_j}:j=1,...,n$ respectively. We prove the statement by induction on $n$.
We consider the base case to be $n=0$, i.e. $E=\{\boldsymbol{0}\}$. Suppose $n=0$ and the lemma is false. Then we have
\begin{align*}
{\boldsymbol v}\otimes{\boldsymbol w}=\sum_{i=1}^m\boldsymbol{a_i}\otimes{\boldsymbol{d_i}}+\sum_{j=1}^n\boldsymbol{e_j}\otimes{\boldsymbol{b_j}}=\sum_{i=1}^m\boldsymbol{a_i}\otimes{\boldsymbol{d_i}},
\end{align*}
for some $\boldsymbol{a_i}\in span(A_1)$ and some $\boldsymbol{b_j}\in span(A_2)$. Since $\boldsymbol{w}$ is a non-zero vector, from the above equation, there exists some non-zero component of the vector $\boldsymbol{w}$, say the $r^{th}$ component $w_r$, such that
\[
w_r\boldsymbol{v}=\sum_{i=1}^m d_{ir}\boldsymbol{a_i},
\]
where $d_{ir}$ is the $r^{th}$ component of $\boldsymbol{d_i}$. This contradicts the given statement that $\boldsymbol{v}\notin span(A_1).$ This proves the base case of $n=0.$
Now suppose that the statement of the lemma is true for the dimension of $E$ being $n-1$. We want to prove that it is true when the dimension is $n$.
Suppose not, then we have
\begin{equation}
\label{eqn801}
{\boldsymbol v}\otimes{\boldsymbol w}=\sum_{i=1}^m\boldsymbol{a_i}\otimes{\boldsymbol{d_i}}+\sum_{j=1}^n\boldsymbol{e_j}\otimes{\boldsymbol{b_j}}.
\end{equation}
As before, for some $w_r\neq 0,$ we have by the definition of the tensor product,
$
w_r\boldsymbol{v}=\sum_{i=1}^m d_{ir}\boldsymbol{a_i}+\sum_{j=1}^n b_{jr}\boldsymbol{e_j}.
$
Now since $\boldsymbol{v}\notin span(A_1),$ we must have some $j_1$ such that $b_{j_1r}$ is non-zero in the above equation. Thus we must have,
\[
\boldsymbol{e_{j_1}}=l_1\boldsymbol{v}+\sum_{i=1}^mg_i\boldsymbol{a_i}+\sum_{j=1,j\neq j_1}^{n}h_j\boldsymbol{e_j},
\]
for some scalars $l_1,~
g_i:i\in\{1,...,m\},~h_j:j\in\{1,...,n\}\backslash j_1.$ Using the above in (\ref{eqn801}), we have
\begin{align*}
{\boldsymbol v}\otimes{\boldsymbol w}&=\sum_{i=1}^m\boldsymbol{a_i}\otimes{\boldsymbol{d_i}}+\sum_{j=1,j\neq j_1}^n\boldsymbol{e_j}\otimes{\boldsymbol{b_j}}+\boldsymbol{e_{j_1}}\otimes\boldsymbol{b_{j_1}}\\
&=\sum_{i=1}^m\boldsymbol{a_i}\otimes{\boldsymbol{d_i}}+\sum_{j=1,j\neq j_1}^n\boldsymbol{e_j}\otimes{\boldsymbol{b_j}}+l_1\boldsymbol{v}\otimes\boldsymbol{b_{j_1}}\\ &~~~~+\sum_{i=1}^mg_i\boldsymbol{a_i}\otimes\boldsymbol{b_{j_1}}+\sum_{j=1,j\neq j_1}^{n}h_j\boldsymbol{e_j}\otimes\boldsymbol{b_{j_1}}.
\end{align*}
%
Thus by rearranging the terms, we have
\begin{align}
\nonumber
{\boldsymbol v}\otimes&({\boldsymbol w}-l_1\boldsymbol{b_{j_1}})\\
\nonumber
&\\
\label{eqn802} &=\sum_{i=1}^m\boldsymbol{a_i}\otimes(\boldsymbol{d_i}+g_i\boldsymbol{b_{j_1}})+ \sum_{j=1,j\neq j_1}^n\boldsymbol{e_j}\otimes(\boldsymbol{b_j}+h_j\boldsymbol{b_{j_1}}).
\end{align}
Observe with respect to (\ref{eqn802}), we have
\begin{itemize}
\item ${\boldsymbol v}\notin span(\{\boldsymbol{a_i}:i=1,...,m\})$ (by given condition).
\item $({\boldsymbol w}-l_1\boldsymbol{b_{j_1}})\notin span(\{\boldsymbol{b_j}+h_j\boldsymbol{b_{j_1}}: j\in\{1,..,n\}\backslash j_1\})$, since $\boldsymbol{w}\notin span(A_2)$ (given).
\end{itemize}
Now define $E$ to be the $(n-1)$-dimensional space spanned by $\{\boldsymbol{e_j}: j\in\{1,...,n\}\backslash j_1\}$, and $D$ to be the space spanned by the vectors $\{\boldsymbol{d_i}+g_i\boldsymbol{b_{j_1}}: i=1,...,m\}$. However, in that case, the equation (\ref{eqn802}) is a contradiction, since the induction hypothesis holds for $E$ being $(n-1)$-dimensional. Thus (\ref{eqn801}) must be false. This proves the lemma.
\end{IEEEproof}
We now show that the product of the vector linear broadcast rates of complementary graphs is bounded from below by the number of vertices of the graph. The analogous bound for scalar linear rates is shown in \cite{LS}.
The following lower bound holds even if the graph $G$ is directed, in which case $\overline{G}$ denotes the undirected complement.
\begin{theorem}
\label{mainthmlowerbound}
Let $G$ be any (possibly directed) graph with $n$ vertices. Then $\beta_{vl}(G)\beta_{vl}(\overline{G})\geq n$.
\end{theorem}
\begin{IEEEproof}
Let $\beta_{vl}(G)=\frac{l_1}{t_1}$ and $\beta_{vl}(\overline{G})=\frac{l_1}{t_2}.$ For simplicity, we only prove the statement for the case when $t_1=t_2=t$, i.e., for the case the message lengths are same in $G$ and $\overline{G}$. If not, we can assume message lengths $t_1'=t_2'=t_1t_2$, with code lengths $l_1'=l_1t_2$ and $l_2'=l_2t_1$ and continue with the proof (by time-sharing between multiple copies of the same codes, respectively for $G$ and $\overline{G}$).
Let $B$ and $C$ be matrices of size $l_1\times nt$ and $l_2\times nt$ denoting the index coding matrices for $G$ and $\overline{G}$ respectively. In $B$, for $i\in[n], j\in [t]$, let $B^{ij}$ denote the precoding vector corresponding to the $j^{th}$ symbol of the $i^{th}$ message. Similarly we have the columns indexed in $C$. Consider the matrix $F$ formed with the columns $\{B^{ij}\otimes C^{i,j_1}: i\in[n], j,j_1\in[t]\}.$ The matrix $F$ has dimensions $l_1l_2\times nt^2$. We want to show that the rank of this matrix is $nt^2$. The theorem will then follow, since then the number of rows $l_1l_2\geq nt^2,$ and thus $\beta_{vl}(G)\beta_{vl}(\overline{G})\geq n.$
Let $S_i$ be the side-information of the receiver $i$ which demands the message $\boldsymbol{x_i},$ in the index coding problem given by $G$. For some $i\in[n]$ and $j\in[t]$, define the sets
\begin{align*}
A_{1,ij}=\{B^{i'j'}:&i'\in [n]\backslash \{S_i\cup\{i\}\},\forall j'\in[t]\}\\
&\cup\{B^{ij'}: \forall j'\in[t]\backslash j\}. \\
A_{2,ij}=\{C^{i'j'}:&i'\in S_i,\forall j'\in[t]\}\cup\{C^{ij'}: \forall j'\in[t]\backslash j\}.
\end{align*}
Since $B$ and $C$ are valid index coding solutions for $G$ and $\overline{G}$ respectively, by Lemma \ref{IAapproach}, for each $i\in[n]$ and for each $j,j_1\in[t]$, it holds that $B^{ij}\notin span(A_{1,ij})$ and $C^{ij_1}\notin span(A_{2,ij_1})$ are true. By Lemma \ref{lemmatensor}, for any non-empty sets of vectors $D$ and $E,$ we have that
\begin{align*}
B^{ij}\otimes C^{ij_1}\notin
span&\left(\{\boldsymbol{a}\otimes \boldsymbol{d}:\boldsymbol{a}\in span(A_{1,ij}), \boldsymbol{d}\in D\}\right.
\\
&~~~\left.\cup\{\boldsymbol{e}\otimes \boldsymbol{b}:\boldsymbol{b}\in span(A_{2,ij_1}), \boldsymbol{e}\in E\}\right).
\end{align*}
Taking $D$ to be the columns of matrix $C$ and $E$ to be the columns of matrix $B$, we see that the following equation is a weaker statement of what is ensured by Lemma \ref{lemmatensor}.
\begin{multline*}
B^{ij}\otimes C^{ij_1}\\
\notin
span\Big(\{B^{i'j'}\otimes C^{i'j_1'}:i'\in[n]\backslash i,~ j',j_1'\in[t]\}\\
\cup\{B^{ij'}\otimes C^{ij_1'}:j'\in[t]\backslash j,j_1'\in[t]\backslash j_1\}
\\
\cup\{B^{ij}\otimes C^{ij_1'}:j_1'\in[t]\backslash j_1\} \\
\cup\{B^{ij'}\otimes C^{ij_1}: j'\in[t]\backslash j\}
\Big).
\end{multline*}
Note that the above equations hold for all $i\in[n]$ and for all $j,j_1\in[t]$. This shows that each column in the matrix $F$ is independent of all the other columns. Hence the rank of $F$ is $nt^2.$ The theorem follows from the arguments in the second paragraph of the proof. \end{IEEEproof}
\begin{remark*}
The bound in Theorem \ref{mainthmlowerbound} for $\beta_{vl}(G)\beta_{vl}(\overline{G})$ holds tight as it is clearly true if $G$ is a complete graph. Note that these graphs are perfect. However, for some circular-perfect graphs $G$ which are imperfect, the bound holds with equality. For instance, the undirected graphs with \textit{symmetric neighbouring side-information} considered in \cite{MCJ1} form complements of a subclass of convex round graphs which are circular perfect (and imperfect for certain values of $n$). Such a graph with $n$ vertices and $d-1$ symmetric side-information is shown in \cite{MCJ1} to have broadcast rate $\frac{n}{d}$. The complement of such a graph is a \textit{symmetric neighbouring interference graph} with $d-1$ symmetric non-side-information symbols. For some special cases of $n,d$, the broadcast rate of such graphs is known from \cite{VaR} as $d$. Thus these complementary graphs satisfy Theorem \ref{mainthmlowerbound} with equality. However, in general this bound does not hold with equality even for perfect graphs (for instance take any bipartite graph on 3 vertices with a single edge).
\end{remark*}
The upper bound on $\beta_{vl}(G)\beta_{vl}(\overline{G})$ is given by the following lemma.
\begin{lemma}
For any $G$, $\beta_{vl}(G)\beta_{vl}(\overline{G}) \leq (\frac{n+1}{2
})^2$ and this is tight even for a class of perfect graphs.
\end{lemma}
\begin{IEEEproof}
The proof of the first part follows from the fact that $\chi(G)\chi(\overline{G})\leq (\frac{n+1}{2
})^2$ (see \cite{NoG}) and since $\beta_{vl}(G)\leq \chi(\overline{G})$. We show that there exists a class of perfect graphs for which this bound holds with equality. Consider the perfect graph $G$ with vertices $[2n-1]$ ($n\geq 2$) obtained by taking the union of $K_n$ and $\overline{K_n}$ with one common vertex. Note that $\omega(\overline{G})=n$ and also $\chi(\overline{G})=n$. Hence $\beta_{vl}(G) = n$. Similarly we can show that $\beta_{vl}(\overline{G})=n$. Therefore, $\beta_{vl}(G)\beta_{vl}(\overline{G})= n^2$,satisfying the bound with equality.
\end{IEEEproof}
\begin{lemma}
For any $G$, $ {2}\sqrt{n} \leq \beta_{vl}(G)+\beta_{vl}(\overline{G}) \leq n+1$.\end{lemma}
\begin{IEEEproof}
The lower bound can be shown using Theorem \ref{mainthmlowerbound}, ($\beta_{vl}(G)\beta_{vl}(\overline{G}) \geq n)$ and the fact that $(\beta_{vl}(G)-\beta_{vl}(\overline{G}))^2 \geq 0$. Since, $\chi(G)+\chi(\overline{G}) \leq n+1$ is true by \cite{NoG} and $\beta_{vl}(G)\leq \chi(\overline{G})$ for any $G$, the upper bound holds too.
\end{IEEEproof}
\section{Discussion}
\label{discussion}
In this work we have obtained the broadcast rate of complements of circular perfect graphs. In order to tackle the question of the broadcast rate of circular perfect graphs themselves, we looked at the relationship between the broadcast rate of a graph and its complement. We obtain upper and lower bounds for the product and sum of the broadcast rates of complementary graphs. We give some example graphs for which these bounds are met with equality, but it would be worthwhile to investigate more such classes of such graphs. Another interesting line of direction is to look at broadcast rates of further generalizations of perfect graphs and their complements, beyond circular perfect graphs.
\vspace{-0.3cm} |
1,941,325,220,573 | arxiv | \section{\textbf{Introduction}}
C.R. Rao introduced his famous metric \cite{CRRao1949} in 1949 for
measuring distances between probability densities arising from population
parameters. This was later called by others the Rao distance (see,
for example, \cite{AtkinsonSankhya1981,Rios-}). There are several
articles available for the technicalities of Rao distance (see for
example, \cite{Amari-book,Jimenez,Chaudhuri,Chen,Nielsen-Arxiv})
and its applications (see for example, \cite{Rao-Krantz-Virtual,Taylor,Maybank}).
An elementary exposition of the same appeared during his centenary
in \cite{Planstino-Significance}. Rao distances and other research
contributions of renowned statistician C.R. Rao were recollected by
those who celebrated his 100th birthday during 2020 (see for example,
\cite{Efron-Amari-Rubin-Arni-Cox,BLSPRao et al,BLSP_ParthPP}). A
selected list of Rao's contributions in R programs was also made available
during his centenary (\cite{VinodHD}).
Rao distances are constructed under the framework of a quadratic differential
metric, Riemannian metric, and differential manifolds over probability
density functions and the Fisher information matrix. C.R. Rao considered
populations as abstract spaces which he called population spaces \cite{CRRao1949},
and then he endeavored to obtain topological distances between two
populations.
In the next section, we will describe manifolds. Section 3 will highlight
technicalities of Rao distances and Section 4 will treat conformal
mappings and basic constructions. Section 5 will conclude the chapter
with applications in virtual tourism.
\section{\textbf{Manifolds}}
Let $\mathbf{Df(a)}$ denotes the derivative of \textbf{$\mathbf{f}$
}at $\mathbf{a}$ for $\mathbf{a}\in\mathbb{R}^{n}$ and $\mathbf{f}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}.$
A function $\mathbf{f}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ is
\textit{differentiable} at $\mathbf{a}\in\mathbb{R}^{n}$ if there
exists a linear transformation $\mathbf{J}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$
such that
\begin{equation}
\lim_{h\rightarrow0}\frac{\left\Vert \mathbf{f(a}+\mathbf{h})-\mathbf{f(a)}-\mathbf{J}(\mathbf{h})\right\Vert }{\left\Vert \mathbf{h}\right\Vert }=0.\label{eq:1}
\end{equation}
Here $\mathbf{h}\in\mathbb{R}^{n}$ and $\mathbf{f(a+h)-f(a)-J(h)}\in\mathbb{R}^{n}.$
If $\mathbf{f}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ is differentiable
at $a,$ then there exists a unique linear transformation $\mathbf{J}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$
such that (\ref{eq:1}) holds. The $m\times n$ matrix created by
$\mathbf{Df(a)}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ is the Jacobian
matrix, whose elements are
\[
\begin{array}{cc}
\mathbf{Df(a)}= & \left[\begin{array}{cccc}
D_{1}f_{1}(a) & D_{2}f_{1}(a) & \cdots & D_{n}f_{1}(a)\\
D_{1}f_{2}(a) & D_{2}f_{2}(a) & \cdots & D_{n}f_{2}(a)\\
\vdots & \vdots & & \vdots\\
D_{1}f_{m}(a) & D_{2}f_{m}(a) & \cdots & D_{n}f_{m}(a)
\end{array}\right]\end{array}
\]
That is, $\mathbf{J}(h)=\mathbf{Df(a)}.$ Since ${\bf J}$ is linear,
we have $\mathbf{J}(b_{1}\lambda_{1}+b_{2}\lambda_{2})=b_{1}\mathbf{J}(\lambda_{1})+b_{2}\mathbf{J}(\lambda_{2})$
for every $\mathbf{\lambda_{1},\lambda_{2}\in}\mathbb{R}^{n}$ and
every pair of scalars $b_{1}$ and $b_{2}.$ Also, the directional
derivative of \textbf{$\mathbf{f}$} at $\mathbf{a}$ in the direction
of $\mathbf{v}$ for $\mathbf{v}\in\mathbb{R}^{n}$ is denoted by
$\mathbf{D(f,v)}$ is given by
\begin{equation}
\mathbf{D(f,v)}=\lim_{h\rightarrow0}\frac{\left\Vert \mathbf{f(a+h{\bf v})}-\mathbf{f(a)}\right\Vert }{\left\Vert \mathbf{h}\right\Vert }\label{eq:2}
\end{equation}
\noindent provided \emph{R.H.S. }of (\ref{eq:2}) exists. When $\mathbf{f}$
is linear $\mathbf{D(f,v)=f(v)}$ for every $v$ and every $\mathbf{a}.$
Since $J(\mathbf{h})$ is linear, we can write
\begin{equation}
\mathbf{f(a+u)=f(a)+D(f,u)+\left\Vert u\right\Vert \Delta_{a}(u)},\label{eq:3}
\end{equation}
where
\begin{align*}
\mathbf{u} & \in\mathbb{R}^{n}\text{ with}\left\Vert \mathbf{u}\right\Vert <r\text{ for }r>0,\text{so that }\mathbf{a+u}\in\mathbf{B}(a;r)\\
& \text{ \ensuremath{\qquad} for an \ensuremath{n-}ball }\mathbf{B}(a;r)\in\mathbb{R}^{n},
\end{align*}
\[
\mathbf{\Delta_{a}(u)}=\frac{\left\Vert \mathbf{f\mathbf{(a+h)-f(a)}}\right\Vert }{\left\Vert \mathbf{h}\right\Vert }-\mathbf{f'(a)}\text{ if }\mathbf{h}\neq0,
\]
\[
\mathbf{\Delta_{a}(u)}\rightarrow0\text{ as }\mathbf{u}\rightarrow0.
\]
When $\mathbf{u}=h\mathbf{v}$ in (\ref{eq:3}), we have
\begin{equation}
\mathbf{f(a+hv)-f(a)=hD(f,u)+}\left\Vert h\right\Vert \left\Vert \mathbf{v}\right\Vert \mathbf{\Delta_{a}(u)}\label{eq:4}
\end{equation}
For further results on the Jacobian matrix and differentiability properties,
refer to \cite{Tu-manifolds,Spivak-manifolds,Apostol-book}.
Consider a function $f=u+iv$ defined on the plane $\mathbb{C}$ with
$u(z),$ $v(z)\in\mathbb{R}$ for $z=(x,y)\in\mathbb{C}.$ If there
exists four partial derivatives
\begin{equation}
\frac{\partial u(x,y)}{\partial x},\frac{\partial v(x,y)}{\partial x},\frac{\partial u(x,y)}{\partial y},\frac{\partial v(x,y)}{\partial y},\label{eq:four partial derivatives}
\end{equation}
\noindent and these partial derivatives satisfy \emph{Cauchy-Riemann
}equations (\ref{eq:Cauchy-Riemann equations})
\begin{equation}
\frac{\partial u(x,y)}{\partial x}=\frac{\partial v(x,y)}{\partial y}\text{ and }\frac{\partial v(x,y)}{\partial x}=-\frac{\partial u(x,y)}{\partial y},\label{eq:Cauchy-Riemann equations}
\end{equation}
then
\[
\mathbf{D}f\mathbf{(a)}=\frac{\partial u(x,y)}{\partial x}+i\frac{\partial v(x,y)}{\partial x}\text{ for }u,v\in\mathbb{R}.
\]
\begin{thm}
Let $f=u(x,y)+iv(x,y)$ for $u(x,y),v(x,y)$ defined on a subset $B_{\mathbf{\delta}}(\mathbf{c)}$
$\subset\mathbb{C}$ for $\mathbf{\delta,c},(x,y)\in\mathbb{C}.$
Assume $u(x,y)$ and $v(x,y)$ are differentiable at an interior $\mathbf{a}=(a_{1},a_{2})\subset B_{\mathbf{\delta}}(\mathbf{c)}.$
Suppose the partial derivatives $\lim_{(x,y)\rightarrow\mathbf{a}}\frac{u(x,y)-u(\mathbf{a)}}{(x,y)-\mathbf{a}}$
and $\lim_{(x,y)\rightarrow\mathbf{a}}\frac{v(x,y)-v(\mathbf{a)}}{(x,y)-\mathbf{a}}$
exists for $\mathbf{a}$ and these partial derivatives satisfy Cauchy-Riemann
equations at $\mathbf{a.}$ Then
\[
\mathbf{D}f\mathbf{(a)}=\lim_{\mathbf{(u,v)\rightarrow(a_{1},a_{2})}}\frac{f(u,v)-f(\mathbf{a)}}{(u,v)-\mathbf{a}}
\]
exists, and
\[
\mathbf{D}f\mathbf{(a)}=\lim_{(x,y)\rightarrow\mathbf{a}}\frac{u(x,y)-u(\mathbf{a})}{(x,y)-\mathbf{a}}+i\left[\lim_{(x,y)\rightarrow\mathbf{a}}\frac{v(x,y)-v(\mathbf{a})}{(x,y)-\mathbf{a}}\right].
\]
\end{thm}
If $\mathbf{D}f\mathbf{(a)}$ exists for every $B_{\mathbf{\delta}}(\mathbf{c)}$
$\subset\mathbb{C}$ then we say that $f$ is holomorphic in $B_{\mathbf{\delta}}(\mathbf{c)}$
and is denoted as $H(B_{\mathbf{\delta}}(\mathbf{c)}).$ Readers are
reminded that when $f$ is a complex function in $B_{\mathbf{\delta}}(\mathbf{c)}$
$\subset\mathbb{C}$ that has a differential at every point of $B_{\mathbf{\delta}}(\mathbf{c)}$,
then $f\in H(B_{\mathbf{\delta}}(\mathbf{c)})$ if, and only if, the
Cauchy-Riemann equations (\ref{eq:Cauchy-Riemann equations}) are
satisfied for every $\mathbf{a}\in B_{\mathbf{\delta}}(\mathbf{c)}.$
Refer to \cite{Krantz-CAGeometric,Krantz-guide,Rudin-RealComplex,Apostol-book,Krantz-MATLAB}
for other properties of holomorphic functions and their association
with Cauchy-Riemann equations.
\subsection{Conformality between two regions}
Holomorphic functions discussed above allows us to study conformal
equivalences (i.e. angle preservation properties). Consider two regions
$B_{\mathbf{\delta}}(\mathbf{c)},B_{\alpha}(\mathbf{d)}\subset\mathbb{C}$
for some $\mathbf{c,d,}\delta,\alpha\in\mathbb{C}.$ These two regions
are conformally equivalence if there exists a function $g\in H(B_{\mathbf{\delta}}(\mathbf{c)})$
such that $g$ is one-to-one in $B_{\mathbf{\delta}}(\mathbf{c)}$
and such that $g(B_{\mathbf{\delta}}(\mathbf{c)})=B_{\alpha}(\mathbf{d)}.$
This means $g$ is conformally one-to-one mapping if $B_{\mathbf{\delta}}(\mathbf{c)}$
onto $B_{\alpha}(\mathbf{d)}$. The inverse of $g$ is holomorphic
in $B_{\alpha}(\mathbf{d)}$.
This implies $g$ is a conformal mapping of $B_{\alpha}(\mathbf{d)}$
onto $B_{\mathbf{\delta}}(\mathbf{c)}.$ We will introduce conformal
mappings in the next section. The two regions $B_{\mathbf{\delta}}(\mathbf{c)}$
and $B_{\alpha}(\mathbf{d)}$ are homeomorphic under the conformality.
The idea of manifolds is more general than the concept of a complex
plane. It uses the concepts of the Jacobian matrix, diffeomorphism
between $\mathbb{R}^{m}$ and $\mathbb{R}^{n}$, and linear transformations.
A set $M\subset\mathbb{R}^{n}$ is called a manifold if for every
$a\in M$, there exists a neighborhood $\mathbf{U}$ (open set) containing
$\mathbf{a}$ and a diffeomorphism $f_{1}:\mathbf{U}\rightarrow\mathbf{V}$
for $V\subset\mathbb{R}^{n}$ such that
\begin{equation}
\mathbf{f_{1}}(\mathbf{U}\cap M)=\mathbf{V}\cap\left(\mathbb{R}^{k}\times\{0\}\right)\label{eq:manifoldf1}
\end{equation}
The dimension of $M$ is $k.$ See \cite{Spivak-manifolds,Tu-manifolds}
for other details on manifolds. Further for an open set $\mathbf{V_{1}}\subset\mathbb{R}^{k}$
and a diffeomorphism
\begin{equation}
\mathbf{f_{2}:V_{1}}\rightarrow\mathbb{R}^{n}\label{eq:f2}
\end{equation}
\noindent such that $\mathbf{Df_{2}(b)}$ has rank $k$ for $\mathbf{b\in}\mathbf{V}_{\mathbf{1}}.$
\begin{rem}
There exists a diffeomorphism as in (\ref{eq:f2}) such that $\mathbf{f_{2}}:\mathbf{V_{1}}\rightarrow\mathbf{f(V_{1})}$
is continuous.
\end{rem}
\section{\textbf{Rao distance}}
A Riemannian metric is defined using an inner product function, manifolds,
and the tangent space of the manifold considered.
\begin{defn}
\textbf{Riemannian metric: }Let $a\in M$ and $T_{a}M$ be the tangent
space of $M$ for each $a.$ A Riemannian metric $\mathcal{G}$ on
$M$ is an inner product
\[
\mathcal{G}_{a}:T_{a}M\times T_{a}M\rightarrow\mathbb{R}^{n}
\]
\noindent constructed on each $a.$ Here $(M,\mathcal{G})$ forms
Riemannian space or Riemannian manifold. The tensor space can be imagined
as collection of all the multilinear mappings from the elements in
$M$ as shown in Figure 3.1. For general references on metric spaces
refer to \cite{Kobayashi-Nomizu-book,Ambrisio-Luigi-book}.
\noindent
\begin{figure}
\includegraphics{Figuremanifoldtrace.eps}
\caption{\label{fig:MultilinearMappinginM}Mapping of elements in the manifold
$M$ in a metric space $X$ to the tensor space $T_{a}M.$}
\end{figure}
\end{defn}
Let $p(x,\theta_{1},\theta_{2},...,\theta_{n})$ be the probability
density function of a random variable $X$ such that $x\in X,$ and
$\theta_{1},\theta_{2},...,\theta_{n}$ are the parameters describing
the population. For different values of $\theta_{1},\theta_{2},...,\theta_{n}$
we will obtain different populations. Let us call $P(x,\Theta_{n})$
the population space created by $\Theta_{n}$ for a chosen functional
form of $X.$ Here $\Theta_{n}=\left\{ \theta_{1},\theta_{2},...,\theta_{n}\right\} .$
Let us consider another population space $P\left(x,\Theta_{n}+\Delta\right)$,
where
\[
\Theta_{n}+\Delta=\left\{ \theta_{1}+\delta\theta_{1},\theta_{2}+\delta\theta_{2},...,\theta_{n}+\delta\theta_{n}\right\} .
\]
Let $\phi\left(x,\Theta_{n}\right)dx$ be the probability differential
corresponding to $P(x,\Theta_{n})$ and $\phi\left(x,\Theta_{n}+\Delta\right)dx$
be the probability differential corresponding to $P\left(x,\Theta_{n}+\Delta\right).$
Let
\begin{equation}
d\phi\left(\Theta_{n}\right)\label{eq:dphithea}
\end{equation}
be the differences in probability densities corresponding to $\Theta_{n}$
and $\Theta_{n}+\Delta.$ In (\ref{eq:dphithea}), C.R. Rao considered
only the first order differentials \cite{CRRao1949,Burbea-Rao,Micheccli-Noakes}.
The variance of the distribution of $\frac{d\phi}{\phi}$ is given
by
\begin{equation}
d\left[\frac{d\phi}{\phi}\right]^{2}=\sum\sum F_{ij}d\theta_{i}d\theta_{j}\label{eq:rao-metric}
\end{equation}
where $F_{ij}$ is the \emph{Fisher information matrix} for
\[
F_{ij}=E\left[\left(\frac{1}{\phi}\frac{\partial\phi}{\partial\theta_{i}}\right)\left(\frac{1}{\phi}\frac{\partial\phi}{\partial\theta_{j}}\right)\right]\text{ (for }E\text{ the expectation)}.
\]
Constructions in (\ref{eq:rao-metric}) and other measures between
probability distributions by C.R. Rao has played an important role
in statistical inferences.
Let $f_{3}$ be a measurable function on $X$ with differential $\phi\left(x,\Theta_{n}\right)dx.$
This implies that $f_{3}$ is defined on an interval $S\subset\mathbb{R}$
and there exists a sequence of step-functions $\{s_{n}\}$ on $S$
such that
\[
\lim_{n\rightarrow\infty}s_{n}(x)=f_{3}(x)\text{ almost everywhere on }S
\]
for $x\in X.$
If $f_{3}$ is a $\sigma$-finite measure on $X$, then it satisfies
\[
\frac{d}{d\theta_{i}}\int_{S}P\left(x,\Theta_{n}\right)d\mu=\int_{S}\frac{dP\left(x,\Theta_{n}\right)}{d\theta}d\mu
\]
and
\[
\frac{d}{d\theta_{i}}\int_{S}P\left(x,\Theta_{n}\right)d\mu=\frac{d}{d\theta_{i}}\int_{S}\frac{P'\left(x,\Theta_{n}\right)}{P\left(x,\Theta_{n}\right)}P\left(x,\Theta_{n}\right)d\mu.
\]
\begin{rem}
Since the random variable $X$ can be covered by the collection of
sets $T_{n}$ such that
\[
\bigcup_{n=1}^{\infty}T_{n}=X,
\]
$\mu$ is the $\sigma$-finite measure, and
\[
f_{2}(x)>0\text{ and }\int f_{2}(x)\mu(dx)<\infty.
\]
\begin{figure}
\includegraphics{Figureblanketsonthreed.eps}
\caption{\label{fig:Two-2D-shaped}Two 2D-shaped spreadsheets on 3D objects.
Metrics between such 2D-shaped spreadsheets can be studied based on
Rao distances. The distance between the space of points of $X_{a}$
located on the 3D shape $A$ to the space of $X_{b}$ located on the
3D object $B$ can be measured using population spaces conceptualized
in Rao distance.}
\end{figure}
\end{rem}
The idea of Rao distance can be used to compute the geodesic distances
between two 2D spreadsheets on two different 3D objects as shown in
Figure \ref{fig:Two-2D-shaped}. Burbea-Rao studied Rao distances
and developed $\alpha$- order entropy metrics for $\alpha\in\mathbb{R}$
\cite{Burbea-Rao}, given as
\begin{equation}
d\left[\frac{d\phi}{\phi}\right]_{\alpha}^{2}(\theta)=\sum_{i,j}^{n}=\mathcal{G}_{ij}^{(\alpha)}d\theta_{i}d\theta_{j}\label{eq:Burbea-Raometric}
\end{equation}
where
\begin{equation}
\mathcal{G}_{ij}^{(\alpha)}=\int_{X}P(x,\Theta_{n})^{\alpha}\left(\partial_{\theta_{i}}\log P\right)\left(\partial_{\theta_{j}}\log P\right)d\mu.\label{eq:Burbea-Rao-G}
\end{equation}
For the case of $P(x,\Theta_{n})$ as a multinomial distribution where
$x\in X$ for a sample space $X=\{1,2,...,n\},$ Burbea-Rao \cite{Burbea-Rao}
showed that
\begin{equation}
\mathcal{G}_{ij}^{(\alpha)}(\theta)=\int_{X}P(x,\Theta_{n})^{\alpha-2}\left(\partial_{\theta_{i}}\log P\right)\left(\partial_{\theta_{j}}\log P\right)d\mu.\label{eq:multinomial-tensor}
\end{equation}
The tensor of the metric in (\ref{eq:multinomial-tensor}) is of rank
$n.$
\section{\textbf{Conformal Mapping}}
The storyline of this section is constructed around Figure \ref{fig:Mapping-of-pointsbasics}
and Figure \ref{fig:3D-objects-and}. First let us consider Figure
\ref{fig:Mapping-of-pointsbasics} for our understanding of conformal
mapping property. Let $z(t)$ be a complex-valued function for $z(t)=a\leq t\leq b$
for $a,b\in\mathbb{R}.$ Suppose $\gamma_{1}$ is the \emph{arc} constructed
out of $z(t)$ values. Suppose an arc $\Gamma_{1}$ is formed by the
mapping $f_{4}$ with a representation
\[
f_{5}(t)=f_{4}\left(z(t)\right)\text{ for }a\leq t\leq b.
\]
Let us consider an arbitrary point $z(c)$ on $\gamma_{1}$ for $a\leq c\leq b$
at which $f_{4}$ is holomorphic and $f_{4}^{'}\left(z(c)\right)\neq0.$
Let $\theta_{1}$ be the angle of inclination at $c$ as shown in
Figure \ref{fig:Mapping-of-pointsbasics}, then we can write $\text{arg }z^{'}(c)=\theta_{1}.$
Let $\alpha_{1}$ be the angle at $f_{4}^{'}\left(z(c)\right)$, i.e.
\[
\text{arg }f_{4}^{'}\left(z(c)\right)=\alpha_{1}.
\]
By this construction,
\begin{equation}
\text{arg }f_{5}^{'}(c)=\alpha_{1}+\theta_{1},\text{ (because arg }f_{5}^{'}(c)=\text{arg }f_{4}^{'}\left(z(c)\right)+\text{ arg }z^{'}(c))\label{eq:arg f5c}
\end{equation}
\noindent where $\text{arg }f_{5}^{'}(c)$ is the angle at $f_{5}^{'}(c)$
corresponding $\Gamma_{1}.$ Suppose that $\gamma_{2}$ is another
{arc} passing through $z(c)$ and $\theta_{2}$ be the angle of
inclination of the directed tangent line at $\gamma_{2}.$ Let $\Gamma_{2}$
be the {arc} corresponding to $\gamma_{2}$ and $\text{arg }f_{6}^{'}(c)$
be the corresponding angle at $f_{6}^{'}(c).$ Hence the two directed
angles created corresponding to $\Gamma_{1}$ and $\Gamma_{2}$ are
\begin{landscape}
\begin{figure}
\includegraphics{FigurebasicsConformality.eps}
\caption{\label{fig:Mapping-of-pointsbasics}Mapping of points from the real
line to an arc in the complex plane. Suppose $\gamma_{1}$ is the
\emph{arc} constructed out of $z(t)$ values. An arbitrary point $z(c)$
on $\gamma_{1}$ for $a\protect\leq c\protect\leq b$ at which $f_{4}$
is holomorphic and $f_{4}^{'}\left(z(c)\right)\protect\neq0.$ Let
$\theta_{1}$ be the angle of inclination at $c.$ When we denote
$\text{arg }f_{4}^{'}\left(z(c)\right)=\alpha_{1},$ it will lead
to $\text{arg }f_{5}^{'}(c)=\alpha_{1}+\theta_{1}$}
\end{figure}
\end{landscape}
\begin{align*}
\text{arg }f_{5}^{'}(c) & =\alpha_{1}+\theta_{1}\\
\text{arg }f_{6}^{'}(c) & =\alpha_{2}+\theta_{2}
\end{align*}
This implies that
\begin{equation}
\text{arg }f_{6}^{'}(c)-\text{arg }f_{5}^{'}(c)=\theta_{2}-\theta_{1}.\label{eq:anglespreserved}
\end{equation}
The angle created from $\Gamma_{2}$ to $\Gamma_{1}$ at $f_{4}(z(c))$
is the same as the angle created at $c$ on $z(t)$ due to passing
of two {arcs} $\gamma_{1}$ and $\gamma_{2}$ at $c.$
Let $A,B,$ and $C$ be three $3D$ objects as shown in Figure \ref{fig:3D-objects-and}.
Object $A$ has a polygon-shaped structure with a pointed top located
at $A_{0}.$ A pyramid-shaped structure $B$ is located near object
$A$ and a cylinder-shaped object $C.$ Object $B$ has a pointed
top located at $B_{0}.$ Let $C_{0}$ be the nearest distance on $C$
from $B_{0}$ and \textbf{$C_{1}$ }be the farthest distance $C$
from $B_{0}.$ The norms of $A_{0},$ $B_{0},$ $C_{0}$, $C_{1}$
are all assumed to be different. Suppose $A_{0}=(A_{01,}A_{02},A_{03}),$
$B_{0}=(B_{01},B_{02},B_{03}),$ $C_{0}=(C_{01},C_{02},C_{03})$,
$C_{1}=(C_{11,}C_{12},C_{13}).$ Various distances between these points
are defined as below:
\begin{align}
A_{0}C_{0} & =\left\Vert A_{0}-C_{0}\right\Vert =\left[\sum_{i=1}^{3}\left(A_{0i}-C_{0i}\right)^{2}\right]^{1/2}\nonumber \\
A_{0}C_{1} & =\left\Vert A_{0}-C_{1}\right\Vert =\left[\sum_{i=1}^{3}\left(A_{0i}-C_{1i}\right)^{2}\right]^{1/2}\nonumber \\
B_{0}A_{0} & =\left\Vert B_{0}-A_{0}\right\Vert =\left[\sum_{i=1}^{3}\left(B_{0i}-A_{0i}\right)^{2}\right]^{1/2}\nonumber \\
B_{0}C_{0} & =\left\Vert B_{0}-C_{0}\right\Vert =\left[\sum_{i=1}^{3}\left(B_{0i}-C_{0i}\right)^{2}\right]^{1/2}\nonumber \\
B_{0}C_{1} & =\left\Vert B_{0}-C_{1}\right\Vert =\left[\sum_{i=1}^{3}\left(B_{0i}-C_{1i}\right)^{2}\right]^{1/2}\label{eq:fivedistances}
\end{align}
Let $\alpha$ be the angle from the ray $A_{0}C_{1}$ to the ray $A_{0}C_{0}$
with reference to the point $A_{0}$, $\beta_{1}$ be the angle from
the ray $B_{0}C_{1}$ to the ray $B_{0}C_{1}$ with reference to the
point $B_{0}$, and $\beta_{2}$be the angle from the ray $B_{0}A_{0}$
to the ray $B_{0}C_{0}$ with reference to the point $B_{0}$.
\begin{landscape}
\begin{figure}
\includegraphics{Figureconformthreed.eps}
\caption{\label{fig:3D-objects-and}3D objects and conformality with respect
to different viewpoints. The angles $\theta_{1}$, $\theta_{2}$,...
$\alpha$, $\beta_{1}$, $\beta_{2}$ are all measured. The distances
of the rays $A_{0}C_{0}$, $A_{0}C_{1}$, $B_{0}A_{0}$, $B_{0}C_{0}$,
$B_{0}C_{1}$ by assuming they are situated in a single $\mathbb{R}^{3}$
structure and also assuming they are situated in five different complex
planes is computed. By visualizing the three objects are replicas
of an actual tourist spot an application to virtual tourism is discussed
in section 5.}
\end{figure}
\end{landscape}
\begin{prop}
\label{prop:1}All the four points $A_{0},$ $B_{0},$ $C_{0}$, $C_{1}$
of Figure \ref{fig:3D-objects-and} can not be located in a single
Complex plane. These points could exist together in $\mathbb{R}^{3}.$
\end{prop}
\begin{proof}
Suppose the first coordinate of the plane represents the distance
from $x-$axis, the second coordinate is the distance from $y-$axis,
and the third coordinate represents the height of the 3D structures.
Even if $A_{03}=B_{03}=C_{03}$, still all the four points cannot
be on the same plane because $C_{03}$ cannot be equal to $C_{13}.$
Hence al the four points cannot be situated within a single complex
plane. However, by the same construction, they all can be situated
within a single 3D sphere or in $\mathbb{R}^{3}.$
\end{proof}
\begin{prop}
Suppose the norms and the third coordinates of $A_{0},$ $B_{0},$
$C_{0}$, $C_{1}$ are all assumed to be different. Then, it requires
five different complex planes, say, $\mathbb{C}_{1},$ $\mathbb{C}_{2}$,
$\mathbb{C}_{3}$, $\mathbb{C}_{4},$ and $\mathbb{C}_{5}$ such that
$A_{0},C_{0}\in\mathbb{C}_{1}$, $A_{0},C_{1}\in\mathbb{C}_{2}$,
$A_{0},B_{0}\in\mathbb{C}_{3}$, $B_{0},C_{0}\in\mathbb{C}_{4}$,
$B_{0},C_{1}\in\mathbb{C}_{5}.$
\end{prop}
\begin{proof}
By Proposition \ref{prop:1} all the four points $A_{0},$ $B_{0},$
$C_{0}$, $C_{1}$ cannot be in a single complex plane. Although the
third coordinates are different two out of four points can be considered
such that they fall within a same complex plane. Hence, the five rays
$A_{0}C_{0}$, $A_{0}C_{1}$, $B_{0}A_{0}$, $B_{0}C_{0}$, $B_{0}C_{1}$
can be accommodated in five different complex planes.
\end{proof}
\begin{prop}
The angles $\alpha$, $\beta_{1}$, $\beta_{2}$ and five distances
of (\ref{eq:fivedistances}) are preserved when $A_{0},$ $B_{0},$
$C_{0}$, $C_{1}$ are situated together in $\mathbb{R}^{3}.$
\end{prop}
\begin{proof}
The angle $\alpha$ is created while viewing the $3D$ structure $C$
from point $A_{0}.$ The angle $\beta_{1}$ is created while viewing
the $3D$ structure $C$ from the point $B_{0}.$ The angle $\beta_{2}$
is created while viewing the $3D$ structure $C$ from the point $A_{0}.$
These structures could be imagined to stand on a disc within a $3D$
sphere or in $\mathbb{R}^{3}$ even proportionately mapped to $\mathbb{R}^{3}.$
Under such a construction, without altering the ratios of various
distances, the angles remain the same in the mapped $\mathbb{R}^{3}.$
\end{proof}
Let us construct an arc $A_{0}C_{0}(t_{1})=a_{1}\leq t_{1}\leq b_{1}$
from the point $A_{0}$ to $C_{0}$ and call this arc $C_{1}.$ Here
$a_{1},b_{1}\in\mathbb{R}$ and $A_{0},C_{0}\in\mathbb{C}_{1}$. The
points of $C_{1}$ are $A_{0}C_{0}(t_{1}).$ The values of $t_{1}$
can be generated using a parametric representation which could be
a continuous random variable or a deterministic model.
\begin{equation}
t_{1}=\psi_{1}(\tau)\text{ for }\alpha_{1}\leq\tau\leq\beta_{1}.\label{eq:parametric_t1}
\end{equation}
Then the arc length $L(C_{1})$ for the arc $C_{1}$ is obtained through
the integral
\begin{equation}
L(C_{1})=\int_{\alpha_{1}}^{\beta_{1}}\left|A_{0}C_{0}^{'}\left[\psi_{1}(\tau)\right]\right|\psi_{1}^{'}(\tau)d\tau.\label{eq:L(c1)}
\end{equation}
Likewise, the arc lengths $L(C_{2}),$ $L(C_{3})$, $L(C_{4})$, $L(C_{5})$
for the arcs $C_{2}$, $C_{3}$, $C_{4}$, $C_{5}$ are constructed
as follows:
\begin{equation}
L(C_{2})=\int_{\alpha_{2}}^{\beta_{2}}\left|A_{0}C_{1}^{'}\left[\psi_{2}(\tau)\right]\right|\psi_{2}^{'}(\tau)d\tau,\label{eq:L(c2)}
\end{equation}
where $A_{0}C_{1}(t_{2})=a_{2}\leq t_{2}\leq b_{2}$ for $a_{2},b_{2}\in\mathbb{R}$
and $A_{0},C_{1}\in\mathbb{C}_{2}$ and with parametric representation
$t_{2}=\psi_{2}(\tau)\text{ for }\alpha_{2}\leq\tau\leq\beta_{2}.$
\begin{equation}
L(C_{3})=\int_{\alpha_{3}}^{\beta_{3}}\left|B_{0}A_{0}^{'}\left[\psi_{3}(\tau)\right]\right|\psi_{3}^{'}(\tau)d\tau,\label{eq:LC3}
\end{equation}
where $B_{0}A_{0}(t_{3})=a_{3}\leq t_{3}\leq b_{3}$ for $a_{3},b_{3}\in\mathbb{R}$
and $B_{0},A_{0}\in\mathbb{C}_{3}$ and with parametric representation
$t_{3}=\psi_{3}(\tau)\text{ for }\alpha_{3}\leq\tau\leq\beta_{3}.$
\begin{equation}
L(C_{4})=\int_{\alpha_{4}}^{\beta_{4}}\left|B_{0}C_{0}^{'}\left[\psi_{4}(\tau)\right]\right|\psi_{4}^{'}(\tau)d\tau,\label{eq:LC4}
\end{equation}
where $B_{0}C_{0}(t_{4})=a_{4}\leq t_{4}\leq b_{4}$ for $a_{4},b_{4}\in\mathbb{R}$
and $B_{0},C_{0}\in\mathbb{C}_{4}$ and with parametric representation
$t_{4}=\psi_{4}(\tau)\text{ for }\alpha_{4}\leq\tau\leq\beta_{4}.$
\begin{equation}
L(C_{5})=\int_{\alpha_{5}}^{\beta_{5}}\left|B_{0}C_{1}^{'}\left[\psi_{5}(\tau)\right]\right|\psi_{5}^{'}(\tau)d\tau,\label{eq:Lc5}
\end{equation}
where $B_{0}C_{1}(t_{5})=a_{5}\leq t_{5}\leq b_{5}$ for $a_{5},b_{5}\in\mathbb{R}$
and $B_{0},C_{1}\in\mathbb{C}_{5}$ and with parametric representation
$t_{5}=\psi_{5}(\tau)\text{ for }\alpha_{5}\leq\tau\leq\beta_{5}.$
\begin{rem}
One could also consider a common parametric representation
\[
\psi_{i}(\tau)=\psi(\tau)\text{ for }i=1,2,...,5
\]
if that provides more realistic situation of modeling.
\end{rem}
\section{\textbf{Applications}}
The angle preservation approach can be used in preserving the angles
and depth of 3D images for actual 3D structures. Earlier Rao \& Krantz
\cite{Rao-Krantz-Virtual} proposed such measures in the virtual tourism
industry.
Advanced virtual tourism technology is in the early stage of development
and it occupies a small fraction of the total tourism-related business.
Due to the pandemics and other large-scale disruptions around tourist
locations, there will be a high demand for virtual tourism facilities.
One such was visualized during COVID-19 (\cite{Rao-Krantz-Virtual}).
Let us consider a tourist location that has three 3D structured buildings
as in Figure 4.2. When a tourist visits the location in person then
such scenery can be seen directly from the ground level by standing
in between the three structures or standing beside one of the structures.
It is not always possible to see those features when standing above
those buildings. Suppose a video recording is available that was recorded
with regular video cameras; then the distances $A_{0}C_{0}$, $A_{0}C_{1}$,
$B_{0}A_{0}$, $B_{0}C_{0}$, $B_{0}C_{1}$ and angles $\alpha,$
$\beta_{1}$, $\beta_{2}$ would not be possible to capture. That
depth of the scenery and relative elevations and distances would not
be accurately recorded. The in-person virtual experience at most can
see the distance between the bottom structures of the tourist attractions.
The same scenery of Figure 4.2, when watched in person at some time
of the day, would be different when it is watched at a different time
due to the differences between day and night visions. The climatic
conditions and weather would affect the in-person tourism experiences.
All these can be overcome by having virtual tourism technologies proposed
for this purpose \cite{Rao-Krantz-Virtual}. The new technology called
LAPO (live-streaming with actual proportionality of objects) would
combine the pre-captured videos and photos with live-streaming of
the current situations using advanced drone technology. This would
enhance the visual experience of live videos by mixing them with pre-recorded
videos. Such technologies will not only enhance the visualizations
but also help in repeated seeing of the experiences and a closer look
at selected parts of the videos. Mathematical formulations will assist
in maintaining the exactness and consistency of the experiences. We
hope that the newer mathematical constructions, theories, and models
will also emerge from these collaborations.
The line integrals $L(C_{i})$ for $i=1,2,...,5$ are computed and
the angles between the structures can be practically pre-computed
for each tourist location so that these can be mixed with the live
streaming of the tourist locations. The angle preservation capabilities
to maintain the angles between various base points can be preserved
with actual measurements that will bring a real-time experience of
watching the monuments.
The virtual tourism industry has many potential advantages if it is
supported by high-end technologies. Viewing the normal videos of tourist
attractions through the internet browser could be enriched with the
new technology proposed \cite{Rao-Krantz-Virtual}. These new technologies
combined with more accurate preservations of the depth, angles, and
relative distances would enhance the experiences of virtual tourists.
Figure 4.2 could be considered as a view of a tourist location. There
are more realistic graphical descriptions available to understand
the proposed technology LAPO using the information geometry and conformal
mapping \cite{Rao-Krantz-Virtual}.
Apart from applying mathematical tools, there are advantages of virtual
tourism. Although this discussion is out of scope for this article,
we wish to highlight below a list of advantages and disadvantages
of new virtual tourism technology taken from \cite{Rao-Krantz-Virtual}.
\medskip{}
\\
\textsc{Advantages:}
\begin{enumerate}
\item[\textbf{(a)}] Environmental protection around ancient monuments;
\item[\textbf{(b)}] Lesser disease spread at the high population density tourist locations;
\item[\textbf{(c)}] Easy tour for physically challenged persons;
\item[\textbf{(d)}] Creation of newer employment opportunities;
\item[\textbf{(e)}] The safety of tourists;
\item[\textbf{(f)}] The possibility of the emergence of new software technologies.
\end{enumerate}
\vspace*{0.15in}
\textsc{Disadvantages:}
\begin{enumerate}
\item[\textbf{(a)}] Possible abuse of the technology that can harm the environment around
the tourist locations;
\item[\textbf{(b)}] Violation of individual privacy;
\item[\textbf{(c)}] Misuse of drone technology.
\end{enumerate}
\vspace*{0.15in}
Overall there are plenty of advantages of developing this new technology
and implementing it with proper care taken for protection against
misuse. The importance of this technology is that it will have deeper
mathematical principles and insights that were not utilized previously
in the tourism industry. When the population mobility reduces due
to pandemics the hospitality and business industry was seen to have
severe financial losses. In such a situation, virtual tourism could
provide an alternative source of financial activity.
There are of course several advantages of real tourism too, like understanding
the actual physical structures of the monuments, touching of the monuments
(trees, stones, water, etc.,), and feeling real climatic conditions.
We are not describing here all the possible advantages and disadvantages
between virtual versus real tourism experiences.
The concept of Rao distance constructed on population spaces can be
used to measure distances between two probability densities. One possible
application is to virtual tourism. This article is anticipated to
help understand various technicalities of Rao distances and conformal
mappings in a clear way.
\subsection*{Acknowledgements: }
ASRS Rao thanks to his friend Padala Ramu who taught him complex analysis
and to all the students who had attended ASRSR's courses on real and
complex analysis.
|
1,941,325,220,574 | arxiv | \section{Introduction: the penguin role}
Since the relevance of the penguin amplitudes was well established by the
CLEO experiment about ten years ago~\cite{cleo}, penguin--dominated processes
become more and more attractive despite their theoretical
complexity and experimental rareness.
The possibility to access to New Physics (NP)
through new virtual
particles in the loops makes them an opportunity to be
exploited rather than a limitation.
Non--leptonic two--body charmless decays of neutral $B$ mesons,
with their sensitivity to penguin amplitudes,
play a major role. They can be used to determine the values of the CKM related
quantities, which may differ from the ones extracted from
tree-level dominated processes, possibly indicating non-Standard Model
contributions.
Comparison between $\gamma$ determinations
from charmed tree--dominated decays and from charmless \ensuremath{B_{(s)}^0 \to h^+h'^-}\ probes possible presence of
NP in the penguin amplitude. Similarly, comparison of the mixing phase
between the tree--process \ensuremath{B_s^0 \to J\!/\!\psi \phi}\
and the penguin \ensuremath{B_s^0 \to \phi \phi}\ charmless mode
could allow to
disentangle NP contribution in mixing and decay.
Usually optimal sensitivity is obtained through
a full tagged and time--dependent analysis
(as the case of \ensuremath{B_s^0 \to J\!/\!\psi \phi}\ above).
However, this requires significant statistics of decays
that are typically rare.
Nevertheless, several strategies have been proposed that do not
require identification
of production flavor and provide already some sensitivity
to NP contributions
from measurements of averaged branching fractions, polarization
amplitudes and time--integrated
\ensuremath{\mathsf{CP}}--asymmetries.
The CDF experiment provides a joint access to large samples of
$B^0$ and $B^0_s$ mesons, which allows to study charmless decays into charged
final--state particles. This has been explored since the early 2000's with a pioneering and very rewarding
program (for instance~\cite{prog,BR1}); here,
the latest measurements of the \ensuremath{B_s^0 \to \phi \phi}\ decay and an overview of the
results of the \ensuremath{B_{(s)}^0 \to h^+h'^-}\ analysis are reported, discussing finally some prospects. All conjugate modes are
implied and branching fractions are \ensuremath{\mathsf{CP}}--averages; for each result, the first uncertainty
is statistical and the second one is the systematic error.
\section{Charmless $B$ decays in hadron collisions}
CDF is a multipurpose solenoidal spectrometer with calorimeters and
muon detectors,
located at one of the two interaction points of the Tevatron
collider. The good performances of the Tevatron
(peak luminosity $\simeq 4 \times 10^{32}$ cm$^{-2}$ s$^{-1}$) and the
high data--taking efficiency allow to store on tape
about 50 pb$^{-1}$ of data every week.
Despite the higher production rate of $B$ mesons with respect to the
$B$--factories (\ensuremath{\mathcal{O}}(10$^{3}$) more), the production cross section for a $b\bar{b}$ quarks pair at 1.96 TeV
(the Tevatron center of mass energy)
is a permille fraction of the total. Thus, a highly--discriminating
signature is needed to distinguish relevant events of $b$--physics.
This is provided by the characteristic long lifetime of
$b$--hadrons:
they are typically produced with transverse momentum of a few GeV,
thus flying about 0.5 mm in the detector and resulting
in secondary vertices displaced from the $p\bar{p}$ collision point.
Triggering on those vertices is challenging. First,
it requires a high resolution tracking detector; this is given by
double-sided silicon microstrips arranged in five cylindrical layers
and an open cell drift chamber with
$96$ sense wires, all immersed in a 1.4 T solenoidal magnetic
field~\cite{CDF}.
Second, it needs to read out all the silicon detector (212~000 channels)
and do pattern recognition and track
fitting online. In CDF this is done by the Silicon Vertex Trigger in 25 $\mu$s
with a resolution of the track impact parameter (IP), 48 $\mu$m,
comparable with off-line measurements~\cite{Luciano}.
A sample enriched with $b$--flavor particles is then
provided by a specific three--level trigger, based on the requirement
of two displaced tracks (\emph{i.~e.},\ with IP larger than
\ensuremath{\mathcal{O}}(120 $\mu$m)) with opposite charge.
\section{\ensuremath{B_s^0 \to \phi \phi}\ Polarization}
The $\ensuremath{B_s^0 \to \phi \phi}$ decay proceeds through a $b\rightarrow s\bar{s}s$ quark level
process, whose dominant SM diagram is the $b\rightarrow s$
penguin. The rich dynamics of decay of pseudoscalar meson
in two vector particles involves
three different amplitudes corresponding to the
polarization states. Hence, the \ensuremath{B_s^0 \to \phi \phi}\ channel is
attractive to test the theoretical predictions for these polarization
amplitudes~\cite{BVV_th}, which have shown several discrepancies with measurements of
similar penguin decays~\cite{BVV_exp}, raising considerable attention on the so--called
``polarization puzzle''.
The first evidence for the $\ensuremath{B_s^0 \to \phi \phi}$ decay has been reported by CDF
in 2005 with 8 events in a data sample corresponding to integrated
luminosity of 180 pb$^{-1}$~\cite{BR1}.
An updated analysis recently improved the measurement of branching
fraction using 2.9 fb$^{-1}$ of data and allows the world's first
decay polarization
measurement~\cite{phiphi}. We reconstruct the \ensuremath{B_s^0 \to J\!/\!\psi \phi}\ decay in the same dataset, and use this decay
as a normalization for the branching ratio measurement and as a
control sample for the polarization analysis.
Signal candidates are reconstructed by detecting \ensuremath{\phi \to K^+K^-}\ and \ensuremath{ J\!/\!\psi \to \mu^+\mu^-}\
decays and are formed by fitting four tracks to a common vertex.
Combinatorial background is reduced by exploiting several
variables sensitive to the long lifetime and relatively
hard \ensuremath{p_T}\ spectrum of $B$ mesons.
The requirements on the discriminating variables are optimized by maximizing
$\mathcal {S/\sqrt{S+B}}$, where
the signal ($\mathcal{S}$) is derived from a
Monte Carlo (MC) simulation and
the background ($\mathcal{B}$) is represented by appropriately normalized
data sampled from the sideband mass regions.
Two sources of background are expected
for both \ensuremath{B_s^0 \to \phi \phi}\ and \ensuremath{B_s^0 \to J\!/\!\psi \phi}:
a dominant and smooth combinatorial background
and a physics component, which
is given by \ensuremath{B^0 \to \phi K^{\ast 0}}\ (\ensuremath{B^0 \to J/\psi K^{\ast 0}}) decays in the case of
\ensuremath{B_s^0 \to \phi \phi}\ (\ensuremath{B_s^0 \to J\!/\!\psi \phi}) and it is estimated by simulation not to exceed a
3\% fraction of the signal. Signals of $295\pm20$ events of \ensuremath{B_s^0 \to \phi \phi}\ and $1766\pm 48$ events of \ensuremath{B_s^0 \to J\!/\!\psi \phi}\ are obtained
by fitting the mass distributions (see
fig.~\ref{fig:peaks}).
\begin{figure}
\begin{center}
\begin{overpic}[width=0.45\columnwidth]{PhiPhiconv.pdf}
\end{overpic}
\begin{overpic}[width=0.45\columnwidth]{JPsiPhiconv.pdf}
\end{overpic}
\end{center}
\caption{\label{fig:peaks}
$m_{KKKK}$ (left) and $m_{J/\Psi KK}$ (right) distributions
for \ensuremath{B_s^0 \to \phi \phi}\ and \ensuremath{B_s^0 \to J\!/\!\psi \phi}\ candidates with fit projections overlaid.}
\end{figure}
The relative \ensuremath{B_s^0 \to \phi \phi}\ decay rate is calculated using
\begin{eqnarray}
\label{eq:BRphiphi}
\frac{\ensuremath{\mathcal{B}}\left(\ensuremath{B_s^0 \to \phi \phi}\right)}
{\ensuremath{\mathcal{B}}\left(\ensuremath{B_s^0 \to J\!/\!\psi \phi}\right)} =
\frac{N_{\phi\phi}}{N_{\psi\phi}}
\frac{\ensuremath{\mathcal{B}}\left(\ensuremath{ J\!/\!\psi \to \mu^+\mu^-} \right)}{\ensuremath{\mathcal{B}} \left(\ensuremath{\phi \to K^+K^-} \right)}
\frac{\epsilon_{\psi\phi}}{\epsilon_{\phi\phi}}\, \epsilon^{\mu}_{\psi\phi}\ ,
\end{eqnarray}
where $N_{\phi\phi}$ ($N_{\psi\phi}$) is the number of \ensuremath{B_s^0 \to \phi \phi}\
(\ensuremath{B_s^0 \to J\!/\!\psi \phi}) events, $\epsilon_{\phi\phi}/\epsilon_{\psi\phi}=0.939\pm 0.030 \pm 0.009$ is
the relative trigger and selection efficiency extracted from
simulation; $\epsilon^{\mu}_{\psi\phi} = 0.8695 \pm 0.0044$, the
efficiency for identifying a muon, is obtained from data using inclusive
\ensuremath{ J\!/\!\psi \to \mu^+\mu^-}\ decays as a function of muon \ensuremath{p_T}. By using the known values for the branching
fractions \ensuremath{\mathcal{B}}(\ensuremath{\phi \to K^+K^-}), \ensuremath{\mathcal{B}}(\ensuremath{ J\!/\!\psi \to \mu^+\mu^-}) and \ensuremath{\mathcal{B}}(\ensuremath{B_s^0 \to J\!/\!\psi \phi}) ~\cite{PDG} (updated to current values of
$f_s/f_d$), we determine $\ensuremath{\mathcal{B}}(\ensuremath{B_s^0 \to \phi \phi})=(2.40 \pm 0.21 \pm 0.27 \pm 0.82)
\times 10^{-5}$\footnote{The
last uncertainty is given by the uncertainty in \ensuremath{\mathcal{B}}(\ensuremath{B_s^0 \to J\!/\!\psi \phi}).}
in agreement with the previous determination~\cite{BR1}.
We measure the polarization from the angular distributions of decay products, expressed as a function of helicity angles,
$\vec{\omega}=(\cos\vartheta_1,\cos\vartheta_2,\Phi$).
The total decay width is composed of three polarization amplitudes:
$A_0$, $A_\parallel$ and $A_\perp$.
Taking the untagged decay rate integrated in time and
neglecting \ensuremath{\mathsf{CP}}-violating differences between $\bar{B}^0_s$ and $B^0_s$
both in mixing and in decay, the differential decay rate is a function of the
helicity angles and depends on the polarization amplitudes at $t=0$ and on
the light and heavy $B_s^0$ mass-eigenstate lifetimes, $\tau_{L}$ and $\tau_{H}$
respectively, as follows:
\begin{equation}
\label{eq:decay_rate}
\frac{d^3\Gamma}{d\vec{\omega}} \propto \, \tau _{L}\big(
|A_0|^2f_1 + |A_\parallel|^2 f_2
+ |A_0||A_\parallel|\cos\delta_\parallel f_5 \big) + \tau_{H}|A_\perp|^2f_3,
\end{equation}
where $\delta_\parallel=\arg (A_0^\star A_\parallel)$ and the
$f_i =f_i(\vec{\omega})$ are functions of the helicity angles only.
We perform an unbinned maximum likelihood fit to the reconstructed
mass $m$ of the $B_s^0$ candidates and the helicity
angles. The mass distribution provides discrimination between signal and background.
At this stage only the combinatorial component is considered as background,
accounting for the small contamination of other decays to the
systematic uncertainties.
Fixing $\tau_{L}$ and $\tau_{H}$ to the known
values~\cite{PDG}, the polarization amplitudes are extracted
using directly Eq.~\ref{eq:decay_rate} as the angular
probability density function for the signal, corrected by
an acceptance factor given by a full MC simulation.
The angular background is modeled
on sidebands data with polynomials and
fitted in the whole mass range.
The fitter is extensively tested and validated through
statistical trials; the robustness and reliability of the approach is
tested in data as well by measuring the polarization of the \ensuremath{B_s^0 \to J\!/\!\psi \phi}\
in the sample used in the branching fraction update described above; the results,
$|A_0|^2=0.534\pm0.019$ and
$|A_\parallel|^2=0.220\pm0.025$, are in very good agreement
with~\cite{jpsiphi}.
\begin{figure}[b]
\begin{overpic}[width=0.31\columnwidth]{CosT1conv.pdf}
\end{overpic}
\begin{overpic}[width=0.31\columnwidth]{CosT2conv.pdf}
\end{overpic}
\begin{overpic}[width=0.31\columnwidth]{Phiconv.pdf}
\end{overpic}
\caption{\small Projections
of the fit for \ensuremath{B_s^0 \to \phi \phi}\ onto the helicity angles.}
\label{fig:ang}
\end{figure}
Several sources of systematic uncertainty are considered
and only the main two are reported here.
We account for the physics background effects through simulated
samples. In addition to the \ensuremath{B^0 \to \phi K^{\ast 0}}\ decay, we consider
other two contributions, the $B^0_s \to \phi
f_0$ decay and the non--resonant $B^0_s \to \phi(K^+K^-)$:
all the background decays give an additional 1.5\% uncertainty on the polarization
estimates. Then, the biases (1\%) introduced by the time integration
of the decay rate are
examined with MC simulation. We check the tiny
impact (0.2\%) of our assumption of neglecting \ensuremath{\mathsf{CP}}-violating effects.
The angular
fit projections are shown in fig.~\ref{fig:ang}.
The results of the polarization observables for the \ensuremath{B_s^0 \to \phi \phi}\
candidates are $|A_0|^2=0.348 \pm 0.041 \pm 0.021$,
$|A_\parallel|^2=0.287 \pm 0.043 \pm 0.011$,
$|A_\perp|^2=0.365 \pm 0.044 \pm 0.027$ and
$\cos\delta_\parallel = -0.91^{+0.15}_{-0.13} \pm 0.09$.
The measured amplitudes result in a
smaller longitudinal fraction with respect to the naive expectation,
$f_L=|A_0|^2 /(|A_0|^2+|A_\parallel|^2+|A_\perp|^2)=0.348\pm0.041\pm0.021$,
like previously found in
other similar $b \to s$ penguin decays~\cite{BVV_exp}.
\section{\ensuremath{B_{(s)}^0 \to h^+h'^-}\ decays}
Two-body decays mediated by the $b \to u$ quark level transition have
amplitudes sensitive to $\gamma$ CKM angle and to NP too, being
affected by significant contributions from penguin transitions~\cite{bhh_th}. A variety of open
channels with similar final states characterize this transition: this
allow cancellation of many common systematic effects and provides
key information to improve effective models of low--energy QCD.
In 1~fb$^{-1}$ of data, CDF observes four new modes of
these decays and has unique access to direct \ensuremath{\mathsf{CP}}\ violation in \ensuremath{B_s^0}\
and $\Lambda^0_b$ decays; in addition, our measurements of direct \ensuremath{\mathsf{CP}}\ violation
in the $B^0$ sector are competitive with the $B$--factories.
\begin{wrapfloat}{figure}{4}{0pt}
\includegraphics[width=0.38\columnwidth]{bhh.pdf}
\caption{\small Distribution of $\pi\pi$--mass for \ensuremath{B_{(s)}^0 \to h^+h'^-}\ candidates
with fit projections overlaid.}
\label{fig:bhh}
\end{wrapfloat}
An optimized offline selection isolates a clear signal of roughly 7000 events over
a smooth background (fig.~\ref{fig:bhh}).
A five-dimensional likelihood fit relying on kinematics differences between decays and particle
identification from the measurement of specific ionization in the
drift chamber allow statistical determination of the individual
contributions.
Event fractions for each channel are corrected for trigger and selection efficiencies
(from simulation and data) to extract the decay--rates. We report the first observation
of the decays $\ensuremath{B_s^0}\to K^-\pi^+$, $\Lambda^0_b \to p \pi^-$, $\Lambda^0_b \to p K^-$,
and world-leading measurements of (upper
limits on) the $\ensuremath{B_s^0}\to K^+K^-$ ($\ensuremath{B_s^0}\to \pi^+\pi^-$) branching
fractions: in unit of $10^{-6}$, $\ensuremath{\mathcal{B}}(\ensuremath{B_s^0}\to K^-\pi^+)=5.0\pm 0.7 \pm 0.8$,
$\ensuremath{\mathcal{B}}(\Lambda^0_b \to p \pi^-)=3.5\pm 0.6\pm 0.9$,
$\ensuremath{\mathcal{B}}(\Lambda^0_b \to p K^-)=5.6\pm 0.8\pm 1.5$,
$\ensuremath{\mathcal{B}}(B^0 \to \pi^+\pi^-)=5.02 \pm 0.33 \pm 0.35$,
$\ensuremath{\mathcal{B}}(\ensuremath{B_s^0} \to K^+K^-)=24.4 \pm 1.4 \pm 3.5$,
and $\ensuremath{\mathcal{B}}(\ensuremath{B_s^0}\to \pi^+\pi^-)<1.2$,
$\ensuremath{\mathcal{B}}(B^0\to K^+K^-)<0.7$, both at 90\% C.L. The measured ratio
$\ensuremath{\mathcal{B}}(\Lambda^0_b \to p \pi^-)/\ensuremath{\mathcal{B}}(\Lambda^0_b \to p K^-)=0.66\pm
0.14\pm 0.08$ shows that significant penguin contributions compensate
the Cabibbo (and kinematic) suppression expected at tree level.
We also report first measurements
of the following direct \ensuremath{\mathsf{CP}}--violating asymmetries: $A_\ensuremath{\mathsf{CP}}(\ensuremath{B_s^0}\to K^-\pi^+) = (39\pm15\pm8)\%$,
$A_\ensuremath{\mathsf{CP}}(\Lambda^0_b \to p \pi^-) = (3\pm 17\pm 5)\%$, and
$A_\ensuremath{\mathsf{CP}}(\Lambda^0_b \to p K^-) = (37 \pm 17 \pm 3)\%$. Direct \ensuremath{\mathsf{CP}}\ violation
in the $B^0\to K^+\pi^-$ mode is measured as $A_\ensuremath{\mathsf{CP}}(B^0\to K^+\pi^-) =
(-8.6\pm 2.3 \pm 0.9)\%$, consistent with the final $B$--factories
results.
The analysis of the 6 fb$^{-1}$
sample is now in progress. We expect to have more than 16 000 \ensuremath{B_{(s)}^0 \to h^+h'^-}\ decays where new
modes could be observed and \ensuremath{\mathsf{CP}}-violating asymmetries measured with doubled precision.
\section{Conclusions and Outlook}
Since the beginning of Run II, CDF is leading a rich program on
\ensuremath{B_s^0}\ meson physics, whose two--body
charmless decays are a key part, given their sensitivity to NP through rare penguin transitions.
We report the update of the branching fraction and the first
polarization measurement for the \ensuremath{B_s^0 \to \phi \phi}\ decay, which
provides useful information on
the puzzling scenario of $B$ to vector--vector decays.
Having a self-conjugate final state, the \ensuremath{B_s^0 \to \phi \phi}\ can be used to measure the \ensuremath{B_s^0}\ decay width
difference (\ensuremath{\Delta \Gamma_s}), and it is sensitive to the \ensuremath{\mathsf{CP}}\ violation in the decay
and/or mixing,
supplementing the
analogous measurements in tree-dominated \ensuremath{B_s^0 \to J\!/\!\psi \phi}\ decay.
In the \ensuremath{B_{(s)}^0 \to h^+h'^-}\ analysis, $B^0_s \to K^-\pi^+$, $\Lambda^0_b \to p \pi^-$,
and $\Lambda^0_b \to p K^-$ decays were newly observed and their
branching fractions and \ensuremath{\mathsf{CP}}-asymmetries measured.
Updates beyond 1~fb$^{-1}$ of this analysis will provide stringent
model--independent constraints on non--SM contributions and exploration of
time--dependent \ensuremath{\mathsf{CP}}--asymmetries, increasing our knowledge of
the $\gamma$ angle.
CDF collected to date 8~fb$^{-1}$\ of physics-quality data, which will
reach 10~fb$^{-1}$\ by October 2011.
Additional 6~fb$^{-1}$\ will be collected if the proposed three-year extension will be funded.
\small{
|
1,941,325,220,575 | arxiv | \section{Introduction}
\subsection{Algebraic solution to the discrete Toda equation}
The Toda equation
\[
\frac{d}{dt^2}\log(1+V_n(t))=V_{n+1}(t)-2V_n(t)+V_{n-1}(t)
\]
was proposed as a model equation of motion in a one-dimensional lattice of particles with the nearest neighbor interaction~\cite{toda1967wave}. Today the equation is known as a good example of an integrable equation owing to its rich structures.
In this paper, we intend to study time-discretization of the Toda equation\footnote{There exist at least two 'discrete Toda equations' which are famous and well-investigated (as far as we know).
See Remark \ref{rem:no.1}.}, which was originally given by Hirota~\cite{hirota1977}:
\begin{equation}\label{eq:originalform}
\frac{u_n^{t-1}u_n^{t+1}}{(u_n^{t})^2}=\frac{(1-\delta ^2+\delta^2u_{n-1}^{t})(1-\delta ^2+\delta^2u_{n+1}^{t})}{(1-\delta ^2+\delta^2u_n^{t})^2}.
\end{equation}
The above equation (\ref{eq:originalform}) boils down to the following bilinear form.
\begin{equation}\label{eq:bilinear}
(\tau_n^t)^2+\tau_{n-1}^t\tau_{n+1}^t=\tau_n^{t-1}\tau_n^{t+1}
\end{equation}
through the variable transformation given by
\[
u_n^t=
\frac{1-\delta^2}{\delta^2}\frac{\tau_{n}^{t+1}\tau_{n+2}^{t+1}}{(\tau_{n+1}^{t+1})^2}.
\]
Moreover, if we define two new variables given by
\begin{equation}\label{eq:a-and-b}
a_n^t:=\frac{\tau_n^t\tau_{n+1}^{t+1}}{\tau_n^{t+1}\tau_{n+1}^{t}},\qquad
b_n^t:=\frac{\tau_n^t\tau_{n+2}^{t+1}}{\tau_{n+1}^t\tau_{n+1}^{t+1}},
\end{equation}
the equation (\ref{eq:bilinear}) can be rewritten as
\begin{equation}\label{eq:local}
a_n^{t+1}+b_{n-1}^{t+1}=a_n^t+b_n^t,\qquad
a_{n+1}^{t+1}b_n^t=a_n^tb_n^{t+1}.
\end{equation}
One can recover (\ref{eq:originalform}) from (\ref{eq:local}) by using the following relation.
\begin{equation}\label{eq:u}
u_n^t=\frac{1-\delta^2}{\delta^2}
\frac{b_{n}^{t}}{a_{n}^{t}}.
\end{equation}
We consider the simultaneous equations (\ref{eq:local}) with the boundary condition
\begin{equation}\label{eq:boundarycondition}
a_0^t=a_{N+1}^t=1,\qquad b_0^t=b_{N}^t=0
\end{equation}
for some integer $N>0$.
Let
\[
X^t=
\left(
\begin{array}{cccc}
a^t_1 & 1 & & \\
& a^t_2 & \ddots & \\
& & \ddots & 1 \\
& & & a^t_N
\end{array}
\right),\qquad
Y^t=
\left(
\begin{array}{cccc}
1 & & & \\
-b^t_1 & 1 & & \\
& \ddots & \ddots & \\
& & -b_{N-1}^t & 1
\end{array}
\right)
\]
be $N\times N$ matrices.
The simultaneous equation (\ref{eq:local}), therefore, admits the use of the discrete Lax formulation given as follows:
\begin{equation}\label{eq:Lax}
X^{t+1}Y^t=Y^{t+1}X^t.
\end{equation}
We define $L^t:=(Y^t)^{-1}X^t$.
Thus, (\ref{eq:Lax}) could be written as
\begin{equation}\label{eq:LaxMatrixForm}
Y^tL^{t}=L^{t+1}Y^t\qquad \mbox{or}\qquad X^tL^t=L^{t+1}X^t.
\end{equation}
Similar to other classical integrable systems (see, for example, \cite{hirota1980direct, hirota2004direct, KOSTANT1979195, sklyanin2013bispectrality}), one can construct algebraic solutions for arbitrary initial values (values at time $t=0$) of the discrete Toda equation via the Lax representation (\ref{eq:LaxMatrixForm}).
\begin{rem}\label{rem:no.1}
There exist at least two famous 'discrete Toda equations'.
One is the equation $(\ref{eq:originalform})$, which we investigate in this study, and the other is the one expressed by the following bilinear form
\begin{equation}\label{eq:otherform}
(\tau_n^t)^2+\tau_{n-1}^{t+1}\tau_{n+1}^{t-1}
=\tau_n^{t+1}\tau_n^{t-1}.
\end{equation}
(Compare with $(\ref{eq:bilinear})$.)
\end{rem}
\subsection{Algebraic solution and dual Grothendieck polynomials}
Obtaining an algebraic solution to the discrete Toda equation, through use of the Lax formulation (\ref{eq:LaxMatrixForm}), with an arbitrary (generic) initial condition is relatively straightforward.
(See \S \ref{sec:2.1}--\S \ref{sec:2.3} for details.)
If all eigenvalues of the Lax matrix $L^t$ degenerate to one value (as (\ref{eq:degenetate}) in \S \ref{sec:2.4}), there exists a natural algebraic relation between these solutions and the dual Grothendieck polynomials, which are essentially $K$-theoretic analogues of Schur polynomials~\cite{lam2007combinatorial}.
Interestingly, 'the other' discrete Toda equation (\ref{eq:otherform}) corresponds to usual Schur polynomials (Remark \ref{rem:comparement}).
Recently, several researchers have reported interesting relations between '$K$-theoretic' objects and classical integrable systems. Motegi and Sakai~\cite{motegi2013vertex} discovered a remarkable relation between Grothendieck polynomials and algebraic solutions to certain integrable systems (TASEP). Very recently, Ikeda, Iwao, and Maeno \cite{ikeda2017peterson} constructed a ring isomorphism between the quantum $K$-theory of the complex flag variety $Fl_n$ and the $K$-theory of the affine Grassmannian $Gr_{SL_n}$ by using the mechanics of the relativistic Toda equation.
From this, it may be inferred that the discrete Toda equation $(\ref{eq:originalform})$ is amenable to {\it Grothendieck polynomial-type solutions}, while 'the other' discrete Toda equation $(\ref{eq:otherform})$ is amenable to {\it Schur polynomial-type solutions}.
\subsection{Ultradiscretization}
Another topic discussed in this paper is ultradiscretization.
Let us introduce the transformations $u_n^t=e^{\frac{U_n^t}{\varepsilon }}$ and $\delta =e^{-\frac{L}{2\varepsilon }}$ with a parameter $\ep>0$ and a positive constant $L>0$ for (\ref{eq:originalform}).
Then, by applying $\ep \log$ to both sides and taking the limit $\ep\to 0^+$, we obtain the ultradiscrete Toda equation~\cite{matsukidaira1997toda} given by
\begin{equation} \label{eq:uToda}
U_{n}^{t+1}-2U_{n}^{t}+U_{n}^{t-1} = \max[ 0, U_{n+1}^{t}-L]-2\max[ 0, U_{n}^{t}-L]+\max[ 0, U_{n-1}^{t}-L].
\end{equation}
In \cite{matsukidaira1997toda}, Matsukidaira {\it et al.}\ derived an soliton solution to (\ref{eq:uToda}) by ultradiscretizing the soliton solution {to} the discrete Toda equation (\ref{eq:originalform}).
Their solution is expressed by
\begin{equation} \label{Nsolitonsolution}
\begin{aligned}
&U^t_n = T^t_{n+1}-2T^t_n+T^t_{n-1}, \\
&T^t_n = \max_{\mu _j\in \{0, 1 \}}\left[ \sum _{j=1}^N \mu _j S_j(t, n) -\sum _{1\le i<j\le N} \mu _i \mu_j (P_j+\sigma _i\sigma _j Q_j)\right], \\
&S_j(t, n) = P_jn -\sigma _j Q_j t +C_j, \\
&0\le P_1\le P_2\le \dots \le P_N, \quad Q_j= \max[0, P_j-L ], \quad \sigma _j\in \{-1, 1 \}.
\end{aligned}
\end{equation}
Here $N$, $L$ are positive integers, and $P_j$, $C_j$ are arbitrary parameters.
The operator $\max_{\mu _j\in \{0, 1 \}}f(\mu _1, \mu _2, \dots, \mu _N)$ denotes the maximum value amongst $2^N$ possible values of $f(\mu _1, \mu _2, \dots, \mu _N)$ obtained by replacing each $\mu _j$ by $0$ or $1$.
It is known that the solution $U^t_n$ defined by (\ref{Nsolitonsolution}) possesses properties of solitary waves and soliton interactions~\cite{matsukidaira1997toda}.
It can be verified that $U_n^t$ always takes non-negative values. On the other hand, Hirota proposed another solution to (\ref{eq:uToda}), which is called 'the static-soliton'~\cite{hirota2009new}, and is expressed by
\begin{equation}\label{eq:Hirotasolution}
U^t_n = T^t_{n+1}-2T^t_n+T^t_{n-1}, \quad
T^t_n = C \sum _{j=j_0}^{j_1} \min[0, n-j],
\end{equation}
where $C$ is a positive parameter and $j_0\le j_1$ are integers. One can verify that now $U_n^t$ (\ref{eq:Hirotasolution}) may take negative values, which implies that the equation (\ref{eq:uToda}) should be amenable to different solutions.
It should be natural to expect the existence of solutions to the discrete Toda equation (\ref{eq:local}) whose ultradiscretization yields static-solitons.
(Here we would like to note that Hirota~\cite{hirota2009new} showed there is no time independent solution, to the discrete Toda equation, other than the trivial solution.)
In this paper, we aim to provide an answer to the question by constructing {\it tropical permanent solutions} for arbitrary initial values of the ultradiscrete Toda equation.
More precisely, we intend to show that the tropical tau function $T_n^t$ (\ref{eq:tropicalT}) can be used to solve (\ref{eq:uToda}) by setting
\[
U_n^t=A_n^t-B_n^t+L,\quad
A_n^t=T^t_n+ T^{{t+1}}_{n+1}-T^{t+1}_{n}-T ^t_{n+1}, \quad B_n^t=T^t_n+T^{{t+1}}_{n+2}-T^t_{n+1}- T^{{t+1}}_{n+1}.
\]
See \S \ref{sec:3.2} for details.
The new parameters $A_n^t,B_n^t$ represent the ultradiscretization of $a_n^t,b_n^t$, which satisfy the evolution equation (\ref{eq:udToda}).
A new cellular automaton realization of the system $\{A_n^t,B_n^t\}$ is proposed in \S \ref{sec:3.3}.
\subsection{Organization of the paper}
In Section \ref{sec:2}, we provide an algebraic solution to the discrete Toda equation (\ref{eq:local}) with the boundary condition (\ref{eq:boundarycondition}) using the Lax formulation.
Although this sequence of calculations is an established practice, we, nonetheless, have provided its details in \S \ref{sec:2.1}--\S \ref{sec:2.3} in order to make this paper self-contained.
In \S \ref{sec:2.4}, dual Grothendieck polynomials, as special solutions to the discrete Toda equation (\ref{eq:originalform}), have been derived.
In Section \ref{sec:3}, the tropical permanent solution to the ultradiscrete Toda equation, obtained by ultradiscretizing the algebraic solution defined in the previous section, has been provided.
The solution realizes the behavior of the solution given in \cite{matsukidaira1997toda,hirota2009new}.
A new cellular automaton realization of the system is proposed in \S \ref{sec:3.3}.
A concrete example and concluding remarks are contained in Section \ref{sec:4}.
\section{Solution to the discrete Toda equation}\label{sec:2}
\subsection{Lax formulation and the spectrum problem}\label{sec:2.1}
An algebraic solution to the discrete Toda equation via Lax formulation (\ref{eq:LaxMatrixForm}) is derived as follows.
Let
\[
f(\lambda):=\det(\lambda E_N-L^t)=\lambda^N-I_1\lambda^{N-1}+\cdots+(-1)^NI_N
\]
be the characteristic polynomial of $L^t$, which is $t$-invariant due to (\ref{eq:LaxMatrixForm}).
Define the $\CC$-algebra $\mathcal{O}=\CC[\lambda]/(f(\lambda))$, which is $N$ dimensional as a $\CC$-vector space.
At the same time, the spectral problem
\begin{equation}\label{eq:spectrum}
L^t\V{v}^t=\lambda \V{v}^t,\qquad (\V{v}^t\in \CC^N),
\end{equation}
which is equivalent to
\begin{equation}\label{eq:spectrum2}
(\lambda Y^t-X^t)\V{v}^t=\V{0},
\end{equation} has been considered.
We denote $(i,j)^{\mathrm{th}}$ minor of the matrix $\lambda Y^t-X^t$ by $M_{i,j}$ and define
\[
\Delta_{i,j}:=(-1)^{i+j}\det M_{i,j}.
\]
Therefore, the vectors
\[
\V{p}:=(\Delta_{N,1},\Delta_{N,2},\dots,\Delta_{N,N})^T,\qquad
\V{q}:=(\Delta_{1,1},\Delta_{1,2},\dots,\Delta_{1,N})^T
\]
satisfy the following properties.
\begin{enumerate}
\item Property A
\begin{itemize}
\item The $i^{\mathrm{th}}$ entry of $\V{p}$, which is denoted by $\V{p}_i$, is a monic polynomial of degree $(i-1)$ in $\lambda$.
\item The $i^{\mathrm{th}}$ entry of $\V{q}$, which is denoted by $\V{q}_i$, is of the form $\lambda^{N-1}\times (\mbox{a polynomial of degree $(N-i)$ in $\lambda^{-1}$})$.
\item If $\lambda$ is a root of $f(\lambda)$, both $\V{p}$ and $\V{q}$ are eigenvectors of $L^t$.
In other words, $\V{p}$ and $\V{q}$ are solutions of the spectral problem (\ref{eq:spectrum}).
\end{itemize}
\item Property B
\begin{itemize}
\item The constant term of $\V{p}_i$ is $(-1)^{i-1}a_1^ta_2^t\cdots a_{i-1}^t$.
\item The coefficient of $\lambda^{N-1}$ of $\V{q}_i$ is $b_1^tb_2^t\cdots b_{i-1}^t$．
\end{itemize}
\end{enumerate}
Thus, it can be directly proved that as a matrix over $\mathcal{O}$, the co-rank of $\lambda Y^t-X^t$ is $1$.
Hence, the eigenvector of $L$ must be unique up to a constant multiple.
In terms of $\mathcal{O}$, we have the following lemma.
\begin{lemma}
Let $\CC[\lambda]\to \mathcal{O}$ be the natural surjection and $\overline{\V{p}},\overline{\V{q}}\in \mathcal{O}^N$ be the image of $\V{p},\V{q}\in \CC[\lambda]^N$.
Thus, there exists some $F\in \mathcal{O}$ such that
\begin{equation}\label{eq:inverseproblem}
\overline{\V{q}}=-F\cdot \overline{\V{p}}.
\end{equation}
$($The minus sign is used for the convenience of the calculation below$)$
\end{lemma}
Now, let us consider the inverse problem.
\begin{center}
for given $F\in \mathcal{O}$, recover $\V{p}$ and $\V{q}$ with (\ref{eq:inverseproblem}).
\end{center}
The answer is as follows.
Equation (\ref{eq:inverseproblem}) is rewritten as
\begin{equation}\label{eq:linearproblem}
\left(
\begin{array}{cccc@{\ \ \vrule\ \ }cccc}
\ast & \ast & \cdots & \beta_1 & 1 & & & \\
& \ast & \cdots & \beta_2 & \alpha_2 & 1 & & \\
& & \ddots & \vdots & \vdots & \vdots & \ddots \\
& & & \beta_N& \alpha_N & \ast & \cdots & 1
\end{array}
\right)
\Array{1\\\lambda\\ \vdots\\ \lambda^{N-1}\\ F\\F\lambda\\\vdots\\F\lambda^{N-1}}
\equiv \V{0}\qquad \mod f(\lambda).
\end{equation}
Each $\ast$ is a coefficient of $\V{p}_i$ or $\V{q}_i$ as a polynomial in $\lambda$.
Therefore, we have
\begin{equation}\label{eq:atoalpha}
\alpha_i=(-1)^{i-1}a_1^ta_2^t\cdots a_{i-1}^t,\quad
\beta_i=b_1^tb_2^t\dots b_{i-1}^t.
\end{equation}
By applying Cramer's rule to the matrix equation (\ref{eq:linearproblem}), one can express the entries of the matrix to be ratio of determinants.
Let
$$
\V{c}:\mathcal{O}\to \CC^N
$$
be an arbitrary linear isomorphism.
We, therefore, have
\begin{equation}\label{eq:alpha}
\alpha_i=(-1)^{i-1}\frac{
\zet{\V{c}(\lambda^{i-1}),\V{c}(\lambda^{i}),\dots,\V{c}(\lambda^{N-1}),
\V{c}(F\lambda),\V{c}(F\lambda^{2}),\dots,\V{c}(F\lambda^{i-1})
}
}
{
\zet{\V{c}(\lambda^{i-1}),\V{c}(\lambda^{i}),\dots,\V{c}(\lambda^{N-1}),
\V{c}(F),\V{c}(F\lambda),\dots,\V{c}(F\lambda^{i-2})
}
}
\end{equation}
and
\begin{equation}\label{eq:beta}
\beta_i=(-1)^i
\frac{
\zet{\V{c}(\lambda^{i-1}),\V{c}(\lambda^{i}),\dots,\V{c}(\lambda^{N-2}),
\V{c}(F),\V{c}(F\lambda),\dots,\V{c}(F\lambda^{i-1})
}
}
{
\zet{\V{c}(\lambda^{i-1}),\V{c}(\lambda^{i}),\dots,\V{c}(\lambda^{N-1}),
\V{c}(F),\V{c}(F\lambda),\dots,\V{c}(F\lambda^{i-2})
}
}.
\end{equation}
Note that they are invariant under any exchange of $\V{c}$.
As long as the denominators are not $0$, they recover values of $a_n^t$ and $b_n^t$.
\begin{rem}\label{rem:propor}
The expressions $(\ref{eq:alpha},\ref{eq:beta})$ are invariant also under the transform $F\mapsto cF$ $(c\in \CC^\times)$.
Thus, we may assume $F$ to be an element of $\mathcal{O}/\CC^\times$ without any loss of generality.
\end{rem}
In the remaining part of this section, we denote $F=F^t$, $\V{p}=\V{p}^t$, and $\V{q}=\V{q}^t$, {\it etc.}\ to emphasize the $t$-dependencies of these quantities.
From the discrete Lax equation (\ref{eq:LaxMatrixForm}) and (\ref{eq:spectrum}), we have a spectral problem at time $t+1$ given by
\begin{equation}\label{eq:timeevolution}
L^{t+1}(X^t\V{v}^t)=\lambda (X^t\V{v}^t),\qquad
L^{t+1}(Y^t\V{v}^t)=\lambda (Y^t\V{v}^t).
\end{equation}
Let $\V{p}':=Y^t\V{p}^t$ and $\V{q}'=X^t\V{q}^t$.
Due to the shapes of the matrices $X^t$, $Y^t$ and Property A, the pair $(\V{p}',\V{q}')$ also satisfies Property A.
Because any pair of vectors with Property A uniquely restores the Lax matrix $L$ with Property B, we have
\[
\V{p}^{t+1}=Y^t\V{p}^t,\qquad \V{q}^{t+1}=\theta X^t\V{q}^t,\quad (\exists \theta\in \CC^\times).
\]
Substituting them into (\ref{eq:inverseproblem}), we obtain
\[
\theta\overline{X^t\V{q}^t}=-F^{t+1}\overline{Y^t\V{p}^t},
\]
which implies
\[
\theta \lambda \overline{\V{q}^t}=\theta \overline{(Y^t)^{-1}X^t\V{q}^t}=-F^{t+1}\overline{\V{p}^t}.
\]
Comparing with (\ref{eq:inverseproblem}), we finally derive
\begin{equation}\label{eq:timeevoofF}
F^{t+1}=\lambda\theta \cdot F^t.
\end{equation}
Although this equation contains the unknown constant $\theta$，the expression (\ref{eq:timeevoofF}) still determines the time evolution of $L^t$ without any ambiguity.
See Remark \ref{rem:propor}.
\subsection{The determinantal formula for the tau function}
From (\ref{eq:timeevoofF}) and the remark \ref{rem:propor}, one may identify $F=F^t$ with $\lambda^tF^0$ without any problem.
Hereafter, we assume any root of $f(\lambda)$ to be non-zero.
Let $M_\lambda:\CC^N\to \CC^N$ represent the $\CC$-linear map$\colon$
\[
\CC^N\stackrel{\V{c}^{-1}}{\longrightarrow}\mathcal{O}\stackrel{\times \lambda}{\longrightarrow} \mathcal{O}\stackrel{\V{c}}{\longrightarrow}\CC^N.
\]
By assumption, this is invertible.
Let $D=\det{M_\lambda}\neq 0$.
The numerator of (\ref{eq:alpha}) can be rewritten as
\begin{align*}
&\zet{\V{c}(\lambda^{i-1}),\V{c}(\lambda^{i}),\dots,\V{c}(\lambda^{N-1}),
\V{c}(F\lambda),\V{c}(F\lambda^{2}),\dots,\V{c}(F\lambda^{i-1})
}\\
&=D\cdot
\zet{\V{c}(\lambda^{i-2}),\V{c}(\lambda^{i-1}),\dots,\V{c}(\lambda^{N-2}),
\V{c}(F),\V{c}(F\lambda),\dots,\V{c}(F\lambda^{i-2})
}.
\end{align*}
By putting
\begin{equation}\label{eq:tau}
\tau^t_n:
=
\zet{\V{c}(\lambda^{{n-1}}),\V{c}(\lambda^{n}),\dots,\V{c}(\lambda^{N-1}),
\V{c}(F^0\lambda^t),\V{c}(F^0\lambda^{t+1}),\dots,\V{c}(F^0\lambda^{t+n-2})
},
\end{equation}
we get
\begin{equation}\label{eq:modfy-a-and-b}
a_n^t=\frac{\tau^t_n\tau^{{t+1}}_{n+1}}{\tau^{t+1}_{n}\tau^t_{n+1}},\qquad
b_n^t=-\frac{\tau^t_n\tau^{{t+1}}_{n+2}}{\tau^t_{n+1}\tau^{{t+1}}_{n+1}}
\end{equation}
from (\ref{eq:atoalpha}).
\subsection{Double Casorati determinant}\label{sec:2.3}
By choosing a specific isomorphism $\V{c}:\mathcal{O}\to \CC^N$, one can derive an explicit formula for the tau function.
One typical example is the double Casorati determinant formula, which has been given below.
Assume that all eigenvalues of $f(\lambda)$ are distinct. Thus,
\[
f(\lambda)=\textstyle\prod_{i=1}^{N}(\lambda-\lambda_i),\qquad (i\neq j\Rightarrow\lambda_i\neq \lambda_j).
\]
By the Chinese remainder theorem, the following map represents linear isomorphism.
\begin{equation}\label{eq:linear_isom_c}
\V{c}:\mathcal{O}\to \CC^N;\qquad \varphi(\lambda)\mod{f(\lambda)}\mapsto (\varphi(\lambda_1),\dots,\varphi(\lambda_N)).
\end{equation}
For this $\V{c}$, the tau function $\tau_n^t$ can be expressed as
\begin{equation}\label{eq:explicit-form-of-tau}
\tau_n^t=
\left|
\begin{array}{cccc@{\ \vline \ }ccccc}
\lambda_1^{n-1} & \lambda_1^n & \cdots & \lambda_1^{N-1} & f_1\lambda_1^t & f_1\lambda_1^{t+1} & \cdots & f_1\lambda_1^{t+n-2} \\
\lambda_2^{n-1} & \lambda_2^n & \cdots & \lambda_2^{N-1} & f_2\lambda_2^t & f_2\lambda_2^{t+1} & \cdots & f_2\lambda_2^{t+n-2} \\
\vdots & \vdots & \cdots & \vdots & \vdots & \vdots & \cdots &\vdots \\
\lambda_N^{n-1} & \lambda_N^n & \cdots & \lambda_N^{N-1} & f_N\lambda_N^t & f_N\lambda_N^{t+1} & \cdots & f_N\lambda_N^{t+n-2}
\end{array}
\right|,
\end{equation}
where $\V{c}(F^0)=(f_1,\dots,f_N)^T$.
By applying the Laplace expansion along the columns between $1^{\mathrm{st}}$ and $(N-n+1)^{\mathrm{th}}$ position, we have
\begin{align} \label{eq:expansionoftau}
\tau_n^t=
\sum_{\genfrac{}{}{0pt}{1}{[N]=\V{j}\sqcup\V{i}}{\sharp\V{i}=n-1}}
(-1)^{\ep}
\prod_{j\in \V{j}}\lambda_j^{n-1} \cdot
\prod_{i\in \V{i}}f_i\lambda_i^{t}\cdot
\prod_{\{j_1<j_2\}\subset \V{j}}(\lambda_{j_1}-\lambda_{j_2})\cdot
\prod_{\{i_1<i_2\}\subset \V{i}}(\lambda_{i_1}-\lambda_{i_2}),
\end{align}
where $\ep=i_1+\dots+i_{n-1}+\frac{(2N-n+2)(n-1)}{2}$.
\subsection{The dual stable Grothendieck polynomial}\label{sec:2.4}
In some special cases, the tau function is naturally related to the dual stable Grothendieck polynomial.
Let us consider the most degenerate case, wherein
\begin{equation}\label{eq:degenetate}
f(\lambda)=(\lambda-\gamma)^N,\qquad (\gamma\neq 0).
\end{equation}
(This assumption is valid in \S \ref{sec:2.4} only.)
The $\CC$-algebra $\mathcal{O}$, therefore is expressed as
\[
\mathcal{O}=\CC[\lambda]/((\lambda-\gamma)^N)=\CC[\mu]/(\mu^N),
\]
where $\mu:=\lambda-\gamma$.
The $\CC$-valued functions $c_0,\dots,c_{N-1}$ over $\mathcal{O}$ could be defined as
\[
c_i(\alpha_0+\alpha_1\mu+\alpha_2\mu^2+\cdots+\alpha_{N-1}\mu^{N-1}):=\alpha_i.
\]
Let $\CC[\mathcal{O}]$ be the polynomial ring over $\mathcal{O}\colon\CC[\mathcal{O}]=\CC[c_0,\dots,c_{N-1}]$.
Next, we set $\beta:=\gamma^{-1}$.
Let
\[
\tau_n=\zet{\V{c}(\lambda^{{n-1}}),\V{c}(\lambda^{n}),\dots,\V{c}(\lambda^{N-1}),
\V{c}(F\lambda),\V{c}(F\lambda^2),\dots,\V{c}(F\lambda^{n-1})
}
\]
be the $n^{\mathrm{th}}$ tau function corresponding to $F\in \mathcal{O}$, which can be rewritten as follows.
\begin{align*}
\tau_n
&=D^{n-1}
\zet
{
\V{c}(1),\V{c}(\lambda),\dots,\V{c}(\lambda^{N-n}),
\V{c}(F\lambda^{-(n-2)}),\dots,\V{c}(F\lambda^{-1}),\V{c}(F)
}\\
&=D^{n-1}
\zet
{
\V{c}(1),\V{c}(\lambda),\dots,\V{c}(\lambda^{N-n}),
\V{c}(F(\tfrac{\gamma-\lambda}{\gamma\lambda})^{n-2}),\dots,\V{c}(F\tfrac{\gamma-\lambda}{\gamma\lambda}),\V{c}(F)
}\\
&=D^{n-1}
\zet
{
\V{c}(1),\V{c}(\mu+\gamma),\dots,\V{c}((\mu+\gamma)^{N-n}),
\V{c}(F(\tfrac{-\mu}{\gamma(\mu+\gamma)})^{n-2}),\dots,\V{c}(F\tfrac{-\mu}{\gamma(\mu+\gamma)}),\V{c}(F)
}\\
&=D^{n-1}\beta^{\frac{(n-1)(n-2)}{2}}
\zet
{
\V{c}(1),\V{c}(\mu),\dots,\V{c}(\mu^{N-n}),
\V{c}(F),\V{c}(F\tfrac{\mu}{\mu+\gamma}),\dots,\V{c}(F(\tfrac{\mu}{\mu+\gamma})^{n-2})
}.
\end{align*}
We fix the linear isomorphism $\V{c}:\mathcal{O}\to \CC^N$ as
\[
\V{c}:=c_0\V{e}_1+c_1\V{e}_2+\dots+c_{N-1}\V{e}_N,\qquad\quad \V{e}_i=(0,\dots,\mathop{\widehat{1}}^i,\dots,0)^T.
\]
Thus, we have
\begin{gather*}
\V{c}(\mu^{i-1})=\V{e}_i,\quad \V{c}(F(\tfrac{\mu}{\mu+\gamma})^{p-1})=(\kappa_{p,1},\dots,\kappa_{p,N})^T,
\end{gather*}
where
\[
\textstyle \kappa_{p,q}=\sum_{i=0}^\infty {1-p\choose i}\gamma^{1-p-i}c_{q-p-i}(F)
=\beta^{p-1}\sum_{i=0}^\infty {1-p\choose i}\beta^{i}c_{q-p-i}(F).
\]
The tau function $\tau_n$, as an element of $\CC[\mathcal{O}]$, can be expressed as
\begin{align*}
\tau_n
&=D^{n-1}\beta^{\frac{(n-1)(n-2)}{2}}
\zet
{
\V{e}_1,\V{e}_2,\dots,\V{e}_{N-n+1},
\V{c}(F),\V{c}(F\tfrac{\mu}{\mu+\gamma}),\dots,\V{c}(F(\tfrac{\mu}{\mu+\gamma})^{n-2})
}\\
&=
D^{n-1}\beta^{\frac{(n-1)(n-2)}{2}}\det(\kappa_{p,N-n+1+q})_{p,q=1}^{n-1}\\
&=
D^{n-1}
\det\left(\textstyle\sum_{i=0}^\infty {1-p\choose i}\beta^{i}c_{N-n+1+q-p-i}(F)\right)_{1\leq p,q\leq n-1}.
\end{align*}
Under natural identification\footnote{
The identification $c_i/c_0\leftrightarrow h_i$ (or $c_i/c_0\leftrightarrow (-1)^ih_i$) appears in \cite{ikeda2017peterson} in order to relate the geometrical information of $Fl_n$ with symmetric polynomials.
The origin of this technique dates back to Fulton's historical work \cite[Part III]{fulton_1996}.
On the other hand, this identification can be understood in terms of the {\it Boson-Fermion correspondence}.
See \cite[Section 6]{ikeda2017peterson}.
}
$c_i/c_0\leftrightarrow h_i$, where $h_i=h_i(x_1,x_2,\dots)$ is the $i^{\mathrm{th}}$ complete symmetric polynomial in infinitely many variables $x_1,x_2,\dots$, and the tau function $\tau_n$ is proportional to
\[
\det\left(\textstyle\sum_{i=0}^\infty {1-p\choose i}\beta^{i}h_{N-n+1+q-p-i}\right)_{1\leq p,q\leq n-1}.
\]
According to~\cite{Shimozono2003stable,lascoux2014finite}, one finds that this expression exactly coincides with the Jacobi--Trudi type formula for the dual $\beta$-Grothendieck polynomial
\[
g^{(\beta)}_{R_{n-1}},
\]
where $R_k$ is the Young diagram $R_k=((N-k)^{k})$.
\begin{rem}
The dual $\beta$-Grothendieck polynomial reduces to the Schur polynomial when $\beta=0\colon$
$g^{(0)}_{R_k}=s_{R_k}$.
\end{rem}
\begin{rem}\label{rem:comparement}
By a similar method, one can derive the tau function for 'the other' discrete Toda equation $(\ref{eq:otherform})$.
In fact, it is described as
\begin{equation}\label{eq:anothertau}
\tau_n=\zet{\V{b}(1),\V{b}(\lambda),\dots,\V{b}(\lambda^{N-n}),
\V{b}(G),\V{b}(G\lambda),\dots,\V{b}(G\lambda^{n-2})
},
\end{equation}
where $G\in \mathcal{O}':=\CC[\lambda]/(g(\lambda))$, $g(\lambda)$ is the characteristic polynomial of the Lax matrix of $(\ref{eq:otherform})$ $($see, for example, \cite{hirota1995conserved}$)$ and $\V{b}:\mathcal{O}'\to \CC^N$ is an arbitrary linear isomorphism.
Note that $(\ref{eq:anothertau})$ is invariant under the transformation $\lambda\mapsto \lambda+\gamma$, while $(\ref{eq:tau})$ is not.
This implies that one cannot derive the dual Grothendieck polynomial from this expression.
In fact, the tau function $(\ref{eq:anothertau})$ is naturally related to the determinant
\[
\det(h_{N-n+1+q-p})_{1\leq p,q\leq n-1},
\]
which is the Jacobi--Trudi formula for the Schur polynomial $s_{R_{n-1}}$.
\end{rem}
\section{Solution to the ultradiscrete Toda equation}\label{sec:3}
\subsection{A new ultradiscrete evolution equation}\label{sec:3.1}
Equation (\ref{eq:local}) is equivalent to the following.
\begin{equation}\label{eq:subfreeform}
a_{n+1}^{t+1}=\frac{a_{n+1}^t+b_{n+1}^t}{a_{n}^t+b_{n}^t}a_n^t,\qquad
b_{n}^{t+1}=\frac{a_{n+1}^t+b_{n+1}^t}{a_{n}^t+b_{n}^t}b_n^t.
\end{equation}
Putting $a_n^t=e^{-\frac{A_n^t}{\ep}}$, $b_n^t=e^{-\frac{B_n^t}{\ep}}$, and taking the limit $\ep\to 0^+$, one derives the ultradiscrete evolution equation given by
\begin{equation}\label{eq:udToda}
\begin{cases}
A_{n+1}^{t+1}=(\min[A_{n+1}^t,B_{n+1}^t]-\min[A_{n}^t,B_{n}^t])+A_n^t,\\
B_{n}^{t+1}=(\min[A_{n+1}^t,B_{n+1}^t]-\min[A_{n}^t,B_{n}^t])+B_n^t,\\
A_0^t=A_{N+1}^t=0,\qquad B_0^t=B_N^t=+\infty.
\end{cases}
\end{equation}
\begin{prop}
Let $U_n^t:=A_n^t-B_n^t+L$.
$(U_0^t=U_N^t:=-\infty.)$
Then, $(\ref{eq:udToda})$ refers to the ultradiscrete Toda equation $(\ref{eq:uToda})$.
\end{prop}
\begin{proof}
We set $\Delta_n:=\max[0,U_{n}^{t}-L]-\max[0,U_{n-1}^t-L]$.
Equation (\ref{eq:udToda}) is equivalent to\footnote{Note the obvious relation $\max[X,Y]=-\min[-X,-Y]$.}
\begin{gather*}
\Delta_n=A_{n}^{t}-A_{n}^{t+1},\qquad
A_{n+1}^{t+1}-A_n^t=B_n^{t+1}-B_n^t.
\end{gather*}
Therefore, we have
\begin{align*}
&U_{n}^{t+1}-2U_n^t+U_n^{t-1}=A_n^{t+1}-2A_n^{t}+A_{n}^{t-1}-B_n^{t+1}+2B_n^t-B_n^{t-1}\\
&=A_n^{t+1}-2A_n^{t}+A_{n}^{t-1}-(A_{n+1}^{t+1}-A_n^{t})+(A_{n+1}^{t}-A_n^{t-1})\\
&=\Delta_{n+1}-\Delta_n,
\end{align*}
which implies the desired result.
\end{proof}
\begin{rem}
Only when $U_1^t\leq L$, one can recover $A_n^t$ and $B_n^t$ from $U_n^t$ using the formula
\[
A_n^t=\sum_{k=1}^{n-1}(U_{n-k}^{t-1}-U_{n-k}^{t})+C,\qquad
B_n^t=L-U_n^t+\sum_{k=1}^{n-1}(U_{n-k}^{t-1}-U_{n-k}^{t})+C,
\]
where $C$ is an arbitrary number.
In fact, we have $A_n^t-A_n^{t+1}=\sum_{k=1}^{n-1}(U_{n-k}^{t+1}-2U_{n-k}^{t}+U_{n-k}^{t-1})=
\sum_{k=1}^{n-1}(\Delta_{n-k+1}-\Delta_{n-k})=\Delta_{n}-\Delta_1=\Delta_n$, where the last equality follows from $U_1^t\leq L\Rightarrow\Delta_1=0$.
\end{rem}
\subsection{Tropical permanent solution}\label{sec:3.2}
In this section, we derive a {\it tropical permanent solution} to (\ref{eq:udToda}).
Let $P=(p_{i,j})_{1\leq i,j\leq N}$ $(p_{i,j}\in \RR\cup \{+\infty\})$ be an $N\times N$ matrix.
The tropical permanent
$\mathrm{TP}\zet{P}$ is an element of $\RR\cup \{+\infty\}$ defined by the formula
$$
\mathrm{TP}\zet{P}:=\min_{\sigma\in \mathfrak{S}_n}\left[
p_{1,\sigma(1)}+p_{2,\sigma(2)}+\dots+p_{N,\sigma(N)}
\right].
$$
We start with the determinantal solution (\ref{eq:explicit-form-of-tau}) to the discrete Toda equation, wherein we define the tropical permanent $T_n^t$ associated with $\tau_n^t$ by
\begin{equation}\label{eq:tropicalT}
\begin{aligned}
T_n^t:=\mathrm{TP}
&\left|
\begin{array}{cccc@{\ , \ }ccccc}
(n-1)\Lambda_1 & n\Lambda_1 & \cdots & (N-1)\Lambda_1
\\
(n-1)\Lambda_2 & n\Lambda_2 & \cdots & (N-1)\Lambda_2
\\
\vdots & \vdots & \cdots & \vdots
\\
(n-1)\Lambda_N & n\Lambda_N & \cdots & (N-1)\Lambda_N
\end{array}
\right.\\
&\hspace{20pt}
\left.
\begin{array}{ccccc}
F_1+t\Lambda_1 & F_1+(t+1)\Lambda_1 & \cdots & F_1+(t+n-2)\Lambda_1 \\
F_2+t\Lambda_2 & F_2+(t+1)\Lambda_2 & \cdots & F_2+(t+n-2)\Lambda_2 \\
\vdots & \vdots & \cdots &\vdots \\
F_N+t\Lambda_N & F_N+(t+1)\Lambda_N & \cdots & F_N+(t+n-2)\Lambda_N
\end{array}
\right|.
\end{aligned}
\end{equation}
\begin{prop}\label{prop:positive_roots}
Let $a_n^t=a_n^t(\ep)$, $b_n^t=b_n^t(\ep)$ be real analytic functions of $\epsilon>0$ with
$$
\epsilon\ll1 \quad\Rightarrow\quad a_n^t,\ b_n^t>0
$$
and
\[
-\lim\limits_{\ep\to 0^+}\ep\log a_n^t=A_n^t,\qquad
-\lim\limits_{\ep\to 0^+}\ep\log b_n^t=B_n^t.
\]
The following two claims, therefore, hold$\colon$
\begin{enumerate}
\item All eigenvalues of the Lax matrix $L^t$ are distinct and positive.
\[
f(\lambda)=\prod_{i=1}^{N}(\lambda-\lambda_i),\qquad
\epsilon\ll1 \quad\Rightarrow\quad
0< \lambda_N<\dots<\lambda_1.
\]
\item Under linear isomorphism $(\ref{eq:linear_isom_c})$, the image of $F^t\in \mathcal{O}$ (\S \ref{sec:2.1}) satisfies
\[
\epsilon\ll1 \quad\Rightarrow\quad(-1)^{n}f_n^t>0,\qquad \mbox{where}\quad \V{c}(F^t)=(f_1^t,\dots,f_n^t)^T.
\]
\end{enumerate}
\end{prop}
\begin{proof}
(1). The distinctness and the positivity of the eigenvalues are direct consequences of the fact that $L^t$ is a {\it totally non-negative and irreducible} matrix (if $\ep\ll 1$).
One may refer the textbook~\cite[Section 5]{pinkus2010totally}.
(2). Comparing the $1^\mathrm{st}$ components on the both sides of (\ref{eq:inverseproblem}), we have $F^t=-\det(\lambda Y^t-X^t)_{1,1}$, where $M_{i,j}$ is the $(i,j)^{\mathrm{th}}$ minor of the matrix $M$.
Using the Cauchy--Binet formula, we derive
\begin{align*}
F^t&=-\det\{(\lambda E -X^t(Y^t)^{-1})\cdot Y^t\}_{1,1}
=-\det(\lambda E -X^t(Y^t)^{-1})_{1,1}\cdot \det Y^t_{1,1}\\
&=-\det(\lambda E-L^{t+1})_{1,1}.
\end{align*}
(The second equality follows from the fact that $Y^t$ is the lower triangle.)
Generally, it is known~\cite[\S 5.3]{pinkus2010totally} that for any totally non-negative and irreducible matrix $A$, the principal minor $p(\lambda)=\det(\lambda E-A)_{1,1}$ satisfies $p(\lambda_1)>0$, $p(\lambda_{2})<0$, $p(\lambda_{3})>0$,\dots, where $0<\lambda_N<\dots<\lambda_1$ are eigenvalues of $A$.
The claim naturally follows from the fact that $L^t$ is totally non-negative and irreducible.
\end{proof}
\begin{prop}\label{prop:Udofdet}
Under the assumptions in proposition \ref{prop:positive_roots},
we write $\Lambda_i=-\lim\limits_{\ep\to 0^+}\ep\log\lambda_i$ and $F_n^t=-\lim\limits_{\ep\to 0^+}\ep\log\zet{f_n^t}$.
Let $\tau_n^t$ be the tau function $(\ref{eq:explicit-form-of-tau})$ and $T_n^t$ be the tropical permanent $(\ref{eq:tropicalT})$.
If $\lambda_1,\dots,\lambda_N$ satisfy the condition\footnote{
If we take $a_n^t,b_n^t$ generically, this condition holds automatically.
}
\begin{equation}\label{eq:generic_condition}
-\lim\limits_{\ep\to 0^+}\ep\log\zet{\lambda_i-\lambda_j}=\min[\Lambda_i,\Lambda_j]
\end{equation}
for any $\lambda_i,\lambda_j$ ($i\neq j$), we have
\[
-\lim\limits_{\ep\to 0^+}\ep \log \zet{\tau_n^t}=T_n^t.
\]
(This implies that the ultradiscretization of $\tau_n^t$ coincides with $T_n^t$.)
\end{prop}
\begin{proof}
From (\ref{eq:expansionoftau}) and the proposition \ref{prop:positive_roots} (2), we have
\begin{align*}
\zet{\tau_n^t}=
\sum_{\genfrac{}{}{0pt}{1}{[N]=\V{j}\sqcup\V{i}}{\sharp\V{i}=n-1}}
\prod_{j\in \V{j}}\lambda_j^{n-1} \cdot
\prod_{i\in \V{i}}|f_i|\lambda_i^{t}\cdot
\prod_{\{j_1<j_2\}\subset \V{j}}(\lambda_{j_1}-\lambda_{j_2})\cdot
\prod_{\{i_1<i_2\}\subset \V{i}}(\lambda_{i_1}-\lambda_{i_2}).
\end{align*}
Thus, we can derive
\begin{equation*}
-\lim\limits_{\ep\to 0^+}\ep \log \zet{\tau_n^t}= \min_{\genfrac{}{}{0pt}{1}{[N]=\V{j}\sqcup\V{i}}{\sharp\V{i}=n-1}} \left[ \sum _{k=1}^{N-n+1-k} (N-k)\Lambda _{j_k}+\sum _{l=1}^{n-1} (F_{i_l}+(t+n-1-l)\Lambda _{j_l} \right]
\end{equation*}
using the formula
\begin{equation*}
-\lim _{\ep \to 0^+}\ep\log\prod_{\{j_1<j_2\}\subset \V{j}}(\lambda_{j_1}-\lambda_{j_2}) = \sum _{k=1}^{N-n+1}(N-n+1-k)\Lambda _{j_k}.
\end{equation*}
(This formula follows from the condition (\ref{eq:generic_condition}).)
On the other hand, by applying the Laplace expansion for the tropical permanent along the rows between $1^{\mathrm{st}}$ and $(N-n+1)^{\mathrm{th}}$ positions~\cite{nagai2011ultradiscrete}, we get
\begin{align*}
T_n^t=&
\min_{\genfrac{}{}{0pt}{1}{[N]=\V{j}\sqcup\V{i}}{\sharp\V{i}=n-1}}
\Biggl[
\mathrm{TP}
\begin{vmatrix}
(n-1)\Lambda_{j_1} & n\Lambda_{j_1} & \cdots & (N-1)\Lambda_{j_1} \\
(n-1)\Lambda_{j_2} & n\Lambda_{j_2} & \cdots & (N-1)\Lambda_{j_2} \\
\vdots & \vdots & \cdots & \vdots \\
(n-1)\Lambda_{j_{N-n+1}} & n\Lambda_{j_{N-n+1}} & \cdots & (N-1)\Lambda_{j_{N-n+1}}
\end{vmatrix} \\
&\hspace{10pt}
+ \mathrm{TP}
\begin{vmatrix}
F_{i_1}+t\Lambda_{i_1} & F_{i_1}+(t+1)\Lambda_{i_1} & \cdots & F_{i_1}+(t+n-2)\Lambda_{i_1} \\
F_{i_2}+t\Lambda_{i_2} & F_{i_2}+(t+1)\Lambda_{i_2} & \cdots & F_{i_1}+(t+n-2)\Lambda_{i_2} \\
\vdots & \vdots & \cdots & \vdots \\
F_{i_{n-1}}+t\Lambda_{i_{n-1}} & F_{i_{n-1}}+(t+1)\Lambda_{i_{n-1}} & \cdots & F_{i_{n-1}}+(t+n-2)\Lambda_{i_{n-1}}
\end{vmatrix}
\Biggr]\\
=& \min_{\genfrac{}{}{0pt}{1}{[N]=\V{j}\sqcup\V{i}}{\sharp\V{i}=n-1}}
\Biggl[
\sum _{k=1}^{N-n+1} (N-k)\Lambda _{j_k}+\sum _{l=1}^{n-1} (F_{i_l}+(t+n-1-l)\Lambda _{j_l}
\Biggr].
\end{align*}
Herein, we use the formula
\begin{align*}
\mathrm{TP}
\begin{vmatrix}
\Lambda_{1} & 2\Lambda_{1} & \cdots & N\Lambda_{1} \\
\Lambda_{2} & 2\Lambda_{2} & \cdots & N\Lambda_{2} \\
\vdots & \vdots & \cdots & \vdots \\
\Lambda_{N} & 2\Lambda_{N} & \cdots & N\Lambda_{N}
\end{vmatrix} =& \sum _{k=1}^N (N+1-k)\Lambda _k,
\end{align*}
where $\Lambda _1\leq \Lambda _2\leq \dots \leq \Lambda _N$. This completes the proof.
\end{proof}
\begin{rem}
Under the assumptions in the proposition \ref{prop:positive_roots},
\begin{equation}
a_n^t=\frac{|\tau^t_n| |\tau^{{t+1}}_{n+1}|}{|\tau^{t+1}_{n}| |\tau^t_{n+1}|},\qquad
b_n^t=\frac{|\tau^t_n| |\tau^{{t+1}}_{n+2}|}{|\tau^t_{n+1}| |\tau^{{t+1}}_{n+1}|}
\end{equation}
holds. Thus, the relationships between $T_n^t$ and $A_n^t$, $B_n^t$ are given by
\begin{equation}\label{eq:solution}
A_n^t=T^t_n+ T^{{t+1}}_{n+1}-T^{t+1}_{n}-T ^t_{n+1}, \qquad B_n^t=T^t_n+T^{{t+1}}_{n+2}-T^t_{n+1}- T^{{t+1}}_{n+1}.
\end{equation}
\end{rem}
\subsection{The cellular automaton realization}\label{sec:3.3}
The ultradiscrete evolution equation (\ref{eq:udToda}) can be realized in the form of a cellular automaton as follows.
Consider $N$ cells numbered from $1$ to $N$ (see Figure \ref{fig:ex1}).
At time $t\in \ZZ$, the $n^\mathrm{th}$ cell contains $A_n^t$ 'kickers' and $B_n^t$ 'balls'.
The state at time $t+1$ is obtained by the following rules.
$\bullet$ A kicker kicks out one ball, if it exists, to the cell neighbor to the left.
$\bullet$ A kicker who has no ball to kick out moves to the cell neighbor to the right.
Figure \ref{fig:ex1} illustrates a typical example.
The solution to the ultradiscrete Toda equation (\ref{eq:uToda}) associated with the above scenario is given in Figure \ref{fig:ex2}.
\begin{figure}[htbp]
\begin{center}
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\multiput(0,0)(0,20){9}{\line(1,0){260}}
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\put(27,165){$2$}
\put(47,165){$3$}
\put(67,165){$4$}
\put(85,165){$\cdots$}
\put(205,165){$\cdots$}
\put(225,165){$12$}
\put(245,165){$13$}
\put(-25,145){$t=0$}
\put(-8,126){$1$}
\put(-8,106){$2$}
\put(-8,86){$3$}
\put(-8,66){$\vdots$}
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}
\end{picture}
\end{center}
\caption{Consider an example of the time evolution (\ref{eq:udToda}), where $N=13$.
Arabic numbers ($1,2,3,\dots$) denote the number of kickers, and Roman numbers ($\mathrm{i},\mathrm{ii},\mathrm{iii},\dots$) denote the number of balls.
A dot denotes $0$.
}\label{fig:ex1}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\verb|..31..2......|\\
\verb|...4..11.....|\\
\verb|...13..2.....|\\
\verb|....22.11....|\\
\verb|.....31.2....|\\
\verb|......4.11...|\\
\verb|......13.2...|\\
\verb|.......2211..|
\end{center}
\caption{Solution to the ultradiscrete Toda equation associated with the example in Figure \ref{fig:ex1}, where $L=1$.
Two solitons can be observed to be proceeding from left to right.
}\label{fig:ex2}
\end{figure}
Another example is given in Figure \ref{fig:ex3}, where a traveling soliton and a static soliton~\cite{hirota2009new} interact.
\begin{figure}[htbp]
\begin{center}
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\b{.}{i}\b{.}{i}\b{.}{i}\b{.}{i}\b{1}{.}\b{2}{iii}\b{.}{i}\b{.}{i}\b{.}{i}\b{.}{\infty}\\
\b{.}{i}\b{.}{i}\b{.}{i}\b{.}{i}\b{.}{ii}\b{3}{i}\b{.}{i}\b{.}{i}\b{.}{i}\b{.}{\infty}\\
\b{.}{i}\b{.}{i}\b{.}{i}\b{.}{i}\b{.}{iii}\b{1}{.}\b{2}{i}\b{.}{i}\b{.}{i}\b{.}{\infty}\\
\b{.}{i}\b{.}{i}\b{.}{i}\b{.}{i}\b{.}{iii}\b{.}{i}\b{2}{.}\b{1}{i}\b{.}{i}\b{.}{\infty}\\
\b{.}{i}\b{.}{i}\b{.}{i}\b{.}{i}\b{.}{iii}\b{.}{i}\b{.}{i}\b{3}{.}\b{.}{i}\b{.}{\infty}
}
\end{picture}
\hspace{20pt}
\begin{picture}(100,170)
\def\bal#1#2#3
\dimen0=8pt
\dimen1=20pt
\multiply \dimen0 by #
\multiply \dimen1 by #
\put(0,145){\hbox{\lower\dimen1 \hbox{\hspace{\dimen0}$#3$}}
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\d{.}\d{.}\d{3}\d{1}\d{.}\d{\u{2}}\d{.}\d{.}\d{.}\d{.}\\
\d{.}\d{.}\d{.}\d{4}\d{.}\d{\u{2}}\d{.}\d{.}\d{.}\d{.}\\
\d{.}\d{.}\d{.}\d{1}\d{3}\d{\u{2}}\d{.}\d{.}\d{.}\d{.}\\
\d{.}\d{.}\d{.}\d{.}\d{2}\d{.}\d{.}\d{.}\d{.}\d{.}\\
\d{.}\d{.}\d{.}\d{.}\d{\u{1}}\d{3}\d{.}\d{.}\d{.}\d{.}\\
\d{.}\d{.}\d{.}\d{.}\d{\u{2}}\d{2}\d{2}\d{.}\d{.}\d{.}\\
\d{.}\d{.}\d{.}\d{.}\d{\u{2}}\d{.}\d{3}\d{1}\d{.}\d{.}\\
\d{.}\d{.}\d{.}\d{.}\d{\u{2}}\d{.}\d{.}\d{4}\d{.}\d{.}
}
\end{picture}
\end{center}
\caption{Collision between a traveling soliton and a static soliton. ($L=1$).
An underlined number denotes a negative integer ($\underline{2}=-2$).
}\label{fig:ex3}
\end{figure}
\section{Example and Concluding Remarks}\label{sec:4}
\subsection{Example}
As seen above, one can construct the tropical permanent solution to the ultradiscrete Toda equation for {\it any} initial state.
The method we use here can be referred to as the {\it tropical inverse scattering method}.
In this section, we demonstrate the method to construct the tropical permanent solution through an example.
Let us consider the initial state ($N=8$, $L=1$)
\begin{gather*}
(A_1^0,A_2^0,\dots,A_8^0)=(1,1,1,1,0,0,0,0),\quad
(B_1^0,B_2^0,\dots,B_7^0)=(2,2,2,4,1,1,1).
\end{gather*}
Let
$q:=e^{-\frac{1}{\ep}}$.
Set the initial values
\begin{gather*}
(a_1^0,a_2^0,\dots,a_8^0)=(q,2q,3q,4q,1,2,3,4),\quad
(b_1^0,b_2^0,\dots,b_7^0)=(q^2,q^2,q^2,q^4,q,q,q)
\end{gather*}
for the discrete Toda equation.
The characteristic polynomial of the Lax matrix $L^0$ is calculated as
$f(\lambda)=\sum_{i=0}^8(-1)^iI_i\lambda^{8-i}$, where
\begin{gather*}
I_0=1,\quad
I_1=10+13q+3q^2+q^4,\quad
I_2=35+115q+96q^2+24q^3+10q^4+8q^5+2q^6,\\
I_3=50+368q+605q^2+360q^3+102q^4+63q^5+41q^6+8q^7,
\cdots,I_8=576q^4.
\end{gather*}
For sufficiently small $q>0$, the roots $0<\lambda_8<\dots<\lambda_1$ of $f(\lambda)$ could be expanded as
\begin{gather*}
\textstyle
\lambda_1=4+4q-4q^2+\cdots,\quad
\textstyle
\lambda_2=3+{{9q^2}\over{2}}-{{81q^3}\over{4}}+\cdots,\\
\textstyle
\lambda_3=2-2q^2+6q^3+\cdots,\quad
\textstyle
\lambda_4=1-q+{{3q^2}\over{2}}+\cdots,\\
\textstyle
\lambda_5=4q+4q^2-4q^3+\cdots,\quad
\lambda_6=3q+\frac{9}{2}q^3+\cdots,\quad
\lambda_7=2q+\cdots,\quad
\lambda_8=q+\cdots.
\end{gather*}
(Higher-order terms have been omitted owing to limitations of space.
If one needs to execute all calculations below, many higher-order terms would be required.
For example, $\lambda_1$ must be calculated up to the term of $q^{13}$, whose coefficient is ${{27560920906072627}\over{11337408}}$.)
For each $\lambda_i$, $f_i=-\det(\lambda_i Y-X)_{1,1}$ is calculated as follows$\colon$
\begin{gather*}
\textstyle
f_1=-\frac{32}{3}q^{13}+\cdots,\quad
f_2=\frac{9}{2}q^{12}+\cdots,\quad
f_3=-4q^{11}+\cdots,\quad
f_4=6q^{10}+\cdots,\\
f_5=-256q^{6}+\cdots,\quad
f_6=108q^5+\cdots,\quad
f_7=-96q^4+\cdots,\quad
f_8=144q^3+\cdots.
\end{gather*}
Therefore, we have
\[
(\Lambda_1,\dots,\Lambda_8)=(0,0,0,0,1,1,1,1),\qquad (F_1,\dots,F_8)=(13,12,11,10,6,5,4,3).
\]
Substituting these datum values to the tropical permanent $T_n^t$ (\ref{eq:tropicalT}), we obtain:
\def\Trop{}
\begin{gather*}
T_1^t=6,\quad
T_2^t=\min[20,t+9],\quad
T_3^t=\min[35,t+22,2t+13],\\
T_4^t=\min[51,t+36,2t+25,3t+18],\quad
T_5^t=\min[68,t+51,2t+38,3t+29,4t+24],\\
T_6^t=\min[t+67,2t+52,3t+41,4t+34],\quad
T_7^t=\min[2t+67,3t+54,4t+45],\\
T_8^t=\min[3t+68,4t+57],\quad
T_9^t=4t+70.
\end{gather*}
Further, substituting them into the formulas
$A_n^t=T_n^t+T_{n+1}^{t+1}-T_n^{t+1}-T_{n+1}^t$,
$B_n^t=T_n^t+T_{n+2}^{t+1}-T_{n+1}^{t+1}-T_{n+1}^t$, and
$U_n^t=L+A_n^t-B_n^t=1+A_n^t-B_n^t$,
we obtain the solution to the ultradiscrete Toda equation (\ref{eq:uToda}), as represented in Figure \ref{fig:4}.
\begin{figure}[htbp]
\begin{center}
\begin{picture}(50,150)
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\dimen0=6pt
\dimen1=14pt
\multiply \dimen0 by #
\multiply \dimen1 by #
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\def\balset#1
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#
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\hbox to 0pt{$t=0$ -- $9$}
\d{.}\d{.}\d{.}\d{\u{2}}\d{.}\d{.}\d{.}\\
\d{.}\d{.}\d{.}\d{\u{1}}\d{.}\d{.}\d{.}\\
\d{.}\d{.}\d{.}\d{.}\d{.}\d{.}\d{.}\\
\d{.}\d{.}\d{.}\d{1}\d{.}\d{.}\d{.}\\
\d{.}\d{.}\d{.}\d{2}\d{.}\d{.}\d{.}\\
\d{.}\d{.}\d{1}\d{1}\d{1}\d{.}\d{.}\\
\d{.}\d{.}\d{2}\d{.}\d{2}\d{.}\d{.}\\
\d{.}\d{1}\d{1}\d{1}\d{1}\d{1}\d{.}\\
\d{.}\d{2}\d{.}\d{2}\d{.}\d{2}\d{.}\\
\d{1}\d{1}\d{1}\d{1}\d{1}\d{1}\d{1}
}
\end{picture}
\hspace{60pt}
\begin{picture}(50,150)
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\setcounter{xax}{0
\addtocounter{yax}{1
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\underline{##1
#
\egroup
}
\balset
\hbox to 0pt{$t=10$ -- $19$}
\d{2}\d{.}\d{2}\d{.}\d{2}\d{.}\d{2}\\
\d{1}\d{1}\d{1}\d{1}\d{1}\d{1}\d{1}\\
\d{.}\d{2}\d{.}\d{2}\d{.}\d{2}\d{.}\\
\d{.}\d{1}\d{1}\d{1}\d{1}\d{1}\d{.}\\
\d{.}\d{.}\d{2}\d{.}\d{2}\d{.}\d{.}\\
\d{.}\d{.}\d{1}\d{1}\d{1}\d{.}\d{.}\\
\d{.}\d{.}\d{.}\d{2}\d{.}\d{.}\d{.}\\
\d{.}\d{.}\d{.}\d{1}\d{.}\d{.}\d{.}\\
\d{.}\d{.}\d{.}\d{.}\d{.}\d{.}\d{.}\\
\d{.}\d{.}\d{.}\d{\u{1}}\d{.}\d{.}\d{.}
}
\end{picture}
\end{center}
\caption{Example of the solution to the ultradiscrete Toda equation. ($N=8,L=1$).}
\label{fig:4}
\end{figure}
\subsection{Concluding remarks}
The primary objective of this paper has been to understand the algebraic structure and positivity of the discrete Toda equation with boundary conditions.
The solution itself is constructed in a straightforward manner.
Under certain natural identifications, it is possible to obtain a family of special solutions, which correspond to dual $\beta$-Grothendieck polynomials, which is the $K$-theoretic analogue of Schur polynomials.
One could infer that the discrete Toda equation is of the 'Grothendieck' polynomial type ($\simeq K$-theoretical), while the other discrete Toda equation is of the 'Schur' polynomial type.
This result can be expected to clarify deeper structures of the two discrete Toda equations.
The ultradiscrete analogues of the Toda equation have also been studied.
It is proved that the ultradiscrete Toda equation reduces to a new evolution equation, which is the ultradiscretization of the Lax formula.
The tropical permanent solutions have also been given.
Our result shows the correspondence between the determinant solution to the discrete Toda equation and the tropical permanent solution to the ultradiscrete Toda equation. Moreover, the cellular automaton realization of the Toda equation is proposed.
The proposed method has the inherent advantage of being applicable to arbitrary initial values.
Especially, we have generalized Hirota's formula for static-solitons.
\section*{Acknowledgment}
One of the authors (S.I.) is partially supported by KAKENHI (26800062).
We would like to thank Editage (www.editage.jp) for English language editing.
\bibliographystyle{abbrv-iwao}
|
1,941,325,220,576 | arxiv | \section{Introduction}
\label{sec:intro}
Low-frequency continuum radio emission from galaxies originates in the
synchrotron process, with the two sources of energy for the required
high-energy electrons and positrons being supernovae and their
remnants (in star-forming galaxies) or the activity of radio-loud
active galactic nuclei, which drive relativistic jets of magnetized
plasma into the external medium. In principle, both of these processes
provide us with information that cannot be accessed in any other way.
The inferred cosmic-ray population of star-forming galaxies stores
some fraction of the energy deposited by supernova and supernova
remnant activity, albeit
with an integration timescale that depends on radiative losses and
transport processes in the host galaxy, and thus depends on the
time-integrated star-formation rate, giving rise to the well-known
radio/far-infrared correlation \citep{vanderKruit71, deJong+85,
Helou+85, Yun+01, Ibar+08, Murphy09, Jarvis+10,
Ivison+10,Ivison+10b, Lacki+10, Smith+14}. The luminosity and other
properties (structure, spectrum, and polarization) of radio emission
from radio-loud AGN offer us the only method, in the absence of deep
X-ray observations for every target, of assessing the kinetic
luminosity produced by the AGN, or jet power, and the radio luminosity
alone is widely used for this purpose \citep{Willott+99} although
there are serious uncertainties in applying this method to individual
objects \citep{Hardcastle+Krause13}. In the local Universe, there is a
large population of radio-loud AGN which exhibit no signatures of
conventional thin-disc accretion, generally referred to as
low-excitation radio galaxies or jet-mode objects
\citep{Hardcastle+07,Hardcastle+09}: radio observations represent the
only way to study the accretion onto the central supermassive black
hole in these objects and by far the most efficient way (in the
absence of sensitive X-ray observations for large samples) to
constrain their effects on the external medium, the so-called feedback
process thought to be responsible for preventing massive star
formation from the hot phase of the intergalactic medium
\citep{Croton+06}.
Wide-area, sensitive radio surveys, in conjunction with wide-area
optical photometric and spectroscopic surveys such as the Sloan
Digital Sky Survey (SDSS: \citealt{Eisenstein+11}) provide us with the
ideal way to study both these processes in a statistical way in the
local Universe. Sensitive surveys are required to detect the radio
emission expected from low-level star formation, which can be faint;
star-forming objects start to dominate the radio-emitting population
at luminosities below about $10^{23}$ W Hz$^{-1}$ at 1.4 GHz
\citep{Mauch+Sadler07}, corresponding to 4 mJy for a source redshift
of 0.1 and 0.3 mJy at $z=0.3$. Wide-area surveys are required in order
to find statistically meaningful samples of powerful AGN that are
close enough to be optically identified and have their redshifts
determined using available optical data. A key problem, however, is
{\it distinguishing} between radio emission driven by low-level
star-formation activity and that powered by low-luminosity AGN \citep{Ibar+09}. In an
era where radio survey capabilities are expected to become vastly more
powerful, it is important to develop diagnostics that will help us to
understand this problem, or at least to understand its true extent. To
do this we need to calibrate the radio properties of identified radio
sources against their instantaneous star-formation rates and
star-formation histories obtained by other means. This motivates radio
observations of wide regions of the sky with good constraints on
star-formation activity.
One widely used diagnostic of star formation is the luminosity and
temperature of cool dust, heated by young stars. The ability of the
{\it Herschel} satellite \citep{Pilbratt+10} to make sensitive
far-infrared observations over a broad bandwidth made it exquisitely
sensitive to this particular tracer of star-formation activity. The
{\it Herschel}-ATLAS survey (H-ATLAS: \citealt{Eales+10}) carried out
wide-area surveys of several large areas of the sky in northern,
equatorial and southern fields using the PACS \citep{Poglitsch+10} and
SPIRE \citep{Griffin+10} instruments (at wavelengths of 100, 160, 250,
350 and 500 $\mu$m), allowing investigations of the relationship
between star formation and radio emission \citep{Jarvis+10} and
between star formation and AGN activity of various types
\citep{Serjeant+10,Hardcastle+10b,Bonfield+11,Hardcastle+13,Kalfountzou+14,Gurkan+15}.
However, these studies have been limited by the availability of
high-quality radio data, as they rely on the 1.4-GHz VLA surveys Faint
Images of the Radio Sky at Twenty-cm \citep[FIRST,][]{Becker+95} and
NRAO VLA Sky Survey \citep[NVSS;][]{Condon+98}, and, while these
surveys have proved extremely valuable, they have inherent weaknesses
when it comes to studying faint star formation and distant AGN. NVSS
is sensitive to all the radio emission from sources extended on scales
of arcminutes, but its resolution and sensitivity are low (resolution
of 45 arcsec; rms noise level $\sim 0.5$ mJy beam$^{-1}$) which means that
it can only detect luminous or nearby objects, and has difficulty
identifying them with optical counterparts. FIRST is higher-resolution
(5 arcsec) and more sensitive ($\sim 0.15$ mJy beam$^{-1}$) but its
lack of short baselines means that it resolves out extended emission
on arcmin scales, often present in nearby radio-loud AGN. Constructing
samples of radio-loud AGN from these surveys with reliable
identifications and luminosities involves a painstaking process of
combining the two VLA surveys \citep[e.g.][]{Best+05,Virdee+13}, and
good imaging of the sources is often not possible.
The Low-Frequency Array (LOFAR: \citealt{vanHaarlem+13}) offers the
opportunity to make sensitive surveys of large areas of the northern
sky with high rates of optical counterpart detection because of its
combination of collecting area, resolution (up to 5 arcsec with the
full Dutch array) and field of view. The LOFAR Surveys Key Science
Project \citep{Rottgering+06} aims to conduct a survey (the `Tier 1'
High Band Array survey, hereafter referred to as `Tier 1': Shimwell
{et al}\/.\ in prep.) of the northern sky at 5-arcsec resolution to an rms
noise at 150 MHz of $\sim 100$ $\mu$Jy beam$^{-1}$, which for a
typical extragalactic source with spectral index\footnote{Here and
throughout the paper spectral index is defined in the sense $S_\nu
\propto \nu^{-\alpha}$.} $\alpha = 0.7$ implies a depth 7 times
greater than FIRST's for the same angular resolution. Crucially, LOFAR
has good $uv$ plane coverage on both long and short baselines, and so
is able to image all but the very largest sources at high resolution
without any loss of flux density, limited only by surface brightness
sensitivity. Deep observations at these low frequencies are rare, and
the previous best large-area survey at frequencies around those of the
LOFAR High Band Array (HBA) is the TIFR GMRT Sky
Survey\footnote{\url{http://tgss.ncra.tifr.res.in/}} (TGSS), full data
from which were recently released \citep{Intema+16}: however, this has
a best resolution around 20 arcsec, which is substantially lower
than the $\sim 5$ arcsec that LOFAR can achieve, and, with an rms
noise of $\sim 5$ mJy beam$^{-1}$, significantly lower sensitivity
than will be achieved for the LOFAR Tier 1 survey.
AGN selection at the lowest frequencies has long been recognised to
provide the most unbiased AGN samples, because the emission is
dominated by unbeamed radiation from the large-scale lobes, a fact
which has ensured the long-term usefulness of low-frequency-selected
samples of AGN such as 3CRR, selected at 178 MHz \citep{Laing+83} or
samples derived from the 151-MHz 6C and 7C surveys
\citep[e.g.][]{Eales85,Rawlings+01,Willott+02,Cruz+06}. These AGN
surveys, however, have had little or no bearing on star-formation
work, since the flux density limits of the surveys exclude all but a few
bright nearby star-forming objects. The relationship between star
formation and radio luminosity at low frequencies is essentially
unexplored. LOFAR observations of {\it Herschel}-ATLAS fields
therefore offer us the possibility both to accumulate large, unbiased,
well-imaged, samples of radio-loud AGN {\it and} to study the
radio/star-formation relation in both radio-loud and radio-quiet
galaxies in the nearby Universe.
In this paper we describe an exploratory LOFAR HBA observation, of the
H-ATLAS North Galactic Pole (NGP) field, a rectangular contiguous area
of sky in the SDSS sky area covering $\sim 170$ square degrees around
$\mathrm{RA}= 13.5$ h and $\mathrm{Dec} = 30^\circ$, and therefore
well positioned in the sky for LOFAR, with a substantial overlap with
the position of the Coma cluster at low $z$. Our survey prioritizes
sky coverage over uniform sensitivity but achieves depth comparable to
the eventual Tier 1 LOFAR survey. We describe the imaging, cataloguing
and source identification process and the tests carried out on the
resulting catalogues. We then present some first results on the
radio/far-infrared relation observed in the fields and the properties
of optically identified radio sources, together with number counts for
star-forming sources at 150 MHz and a first $z=0$ 150-MHz luminosity
function. A subsequent paper (G\"urkan {et al}\/.\ in prep.) will explore
the 150-MHz radio/star-formation relation derived from LOFAR and
H-ATLAS data and we expect to carry out further analysis of the bright
AGN population.
Throughout the paper we assume a cosmology in which $H_0 = 70$ km
s$^{-1}$, $\Omega_{\rm m} = 0.3$ and $\Omega_\Lambda = 0.7$.
\section{Observations}
\begin{table*}
\caption{LOFAR observations of the NGP field}
\label{obslist}
\begin{tabular}{llllr}
\hline
Field name&RA&Dec&Start date/time&Duration (h)\\
\hline
Central&13h24m00s&+27d30m00s&2013-04-26 17:42:15&9.7\\
NW&13h00m00s&+31d52m00s&2014-04-22 18:30:30&8.0\\
SW&13h04m00s&+25d40m00s&2014-04-25 18:17:00&8.0\\
NE&13h34m00s&+32d18m00s&2014-07-15 13:28:38&8.0\\
\hline
\end{tabular}
\end{table*}
The NGP field was observed in four separate pointings, chosen to
maximise sky covered, with the LOFAR HBA (Table \ref{obslist}) as part
of the Surveys Key Science project. Observations used the
HBA\_DUAL\_INNER mode, meaning that the station beams of core and
remote stations roughly matched each other and giving the widest
possible field of view. The first observation, which was made early on
in LOFAR operations, was of slightly longer duration ($\sim 10$ h)
than the others ($\sim 8$ h). International stations were included in
some of the observations in 2014 but were not used in any of our
analysis, which uses only the Dutch array.
In each case, the observations of the field were preceded and followed
by short, 10-minute observations of calibrator sources (3C\,196 at the
start of the run and 3C\,295 at the end). Each observation used the
full 72 MHz of bandwidth provided by the HBA on the target field,
spanning the frequency range 110 to 182 MHz. As LOFAR is a software
telescope, multiple beams can be formed on the sky, and the total
bandwidth that can be processed by the correlator exceeds the total
bandwidth available from the HBA: this allowed us to observe an
additional 24 MHz spread throughout this frequency range on an
in-field calibrator, the bright point source 3C\,287, which lies in
the SE corner of the NGP field. The original intention was to use this
calibrator pointing for determination of the clock offsets between the
core and remote stations, but this proved unnecessary, as we shall see
below. As data with non-contiguous frequency coverage could not easily
be analysed using the facet calibration method (see below) at the time
of our analysis, we do not consider the 3C\,287 observations further.
After observation, the data were averaged by the observatory to 4
channels per sub-band (an HBA sub-band has a bandwidth of 195.3 kHz) and
a 5-second integration time. No `demixing' of bright off-axis sources
was carried out -- this was deemed unnecessary
given the sky positions of bright objects like Cyg A and Cas A --
and all further processing was carried out by us using the University
of Hertfordshire high-performance computing facility.
\section{Data processing, imaging and cataloguing}
\label{sec:data}
\subsection{Facet calibration}
The data were processed using techniques which are described in detail
by \cite{vanWeeren+16} (hereafter vW16) and \cite{Williams+16}
(hereafter W16), implemented by us in a
way which was intended to maximise the data processing efficiency on
the Hertfordshire cluster\footnote{The facet calibration scripts may
be found at \url{https://github.com/tammojan/facet-calibration}; the code
for the implementation described in this paper is available at
\url{https://github.com/mhardcastle/surveys-pipeline} .}. Here we give a
brief overview of the processes, highlighting steps in which our
approach differs from that of vW16 and W16.
Initial flagging using {\sc rficonsole} was done on each sub-band of
the target and calibrator observation (we used 3C\,196 as the primary
calibrator for all four observations) and we then solved for per
station for amplitude, phase and `rotation angle' -- a
term that accounts for differential Faraday rotation within a
sub-band, as described in
section 4.3 of vW16 -- on the calibrator observations using the `Black
Board Self-Calibration' ({\sc bbs}) software \citep{Pandey+09}, making
use of a high-resolution model of 3C\,196 kindly supplied by
V.N.\ Pandey. Because each sub-band was treated independently, we were
able to efficiently run many of these steps in parallel. We then
combined the complex gain solutions on the calibrator for each
sub-band, using tools in the LoSoTo
package\footnote{\url{https://github.com/revoltek/losoto}}. Bad
stations or sub-bands could be identified at this point by looking for
large rms values or gain outliers: when one or more stations were
identified as bad, we flagged them throughout the observation and
re-ran the calibration.
With all bad data removed from the calibrator observations, we then
fitted the phase solutions at all frequencies with a model intended to
solve for the effects of clock offsets (which introduce a phase offset
which is linear in observing frequency $\nu$) and the differential
total ionospheric electron content, or TEC (which introduces phase offsets which
go as $\nu^{-1}$). This so-called clock-TEC separation can only be run on
the calibrator observations, because of their high signal-to-noise
ratio, and must be run over as broad a bandwidth as possible to
maximize the effectiveness of the fitting process. The result was a
set of per-station clock offset values which we transferred, along
with the gain amplitudes, to the data for the target field, again for
each sub-band. (The ionospheric TEC values are not transferred to the
target because the ionosphere toward the target will be different, and
in general variable: the clock offsets, on the other hand, are not
expected to vary significantly over the observation.)
The clock-corrected sub-bands on the target field were now
concatenated into `bands' of ten sub-bands each: such a band has a
bandwidth of just under 2 MHz. This concatenation gives sufficient
signal-to-noise on the target field for per-band phase calibration.
The next stage was to generate a model to allow us to calibrate the
fields. To do this we phase calibrated band 20 (data at frequencies of
150-152 MHz, around the frequency range where the HBA is most
sensitive) using the observatory-provided global sky model, which is
based on low-resolution observations from other telescopes. We then
imaged this dataset with {\sc awimager} \citep{Tasse+13} at $\sim 10$-arcsec
resolution, derived a list of Gaussians from the image with {\sc
pybdsm}\footnote{\url{http://www.astron.nl/citt/pybdsm/}}
\citep{Mohan+Rafferty15} and cross-matched them with the FIRST
catalogue to obtain a list of sources in the field detected by both
LOFAR and FIRST, with their 151-MHz and 1.4-GHz flux densities. This
catalogue, while shallow, is virtually free from artefacts because of
the FIRST cross-matching and has excellent positional and structural
information from FIRST; this is achieved at the price of omitting only
a few bright steep-spectrum or resolved objects that are not seen in
FIRST.
For each band, the data were run through {\sc rficonsole} again (in
order to catch low-level RFI with the improved signal-to-noise of the
broader bandwidth) and then averaged by a factor 2 in time and
frequency, i.e. down to a 10-s integration time and 20 channels per
band. This averaging was selected for computational speed in the facet
calibration process, though it produces moderate bandwidth and
time-averaging smearing, increasing the apparent size of sources while
preserving their total flux density, beyond 2-3 degrees from the
pointing centre at the full LOFAR imaging resolution of $\sim 5$
arcsec. (In this paper we work at resolutions somewhat lower than this
full resolution, around $10 \times 6$ arcsec, so that smearing
effects are less important, but not negligible; we would expect a drop
in peak flux density by a factor 0.8 and a broadening by a factor 1.2
by 4 degrees from the pointing centre, which is more or less the
largest pointing centre offset considered in this work.) The
cross-matched catalogue was then used (scaling appropriately using the
LOFAR/FIRST spectral index) to provide the sky model for an initial
phase-only calibration for each band using {\sc bbs}, dividing the
thousand or so cross-matched sources in each field into about 100
discrete sky regions or patches to reduce computing time. As the
objective is only to provide phase calibration good enough to start
the facet calibration process, small defects in the sky model are not
important and in principle should not affect the final result. Again,
it was possible to apply this process to all bands in parallel.
With the initial phase-only calibration computed, we corrected the
data for the effect of the element beam and array factor using {\sc
bbs}, and all further imaging work until the very end of the process
was then carried out using apparent flux densities (i.e. no primary beam
correction was carried out in the imaging). At this point we could
image each band [we now used {\sc wsclean} \citep{Offringa+14} for
imaging, since we no longer required the primary beam correction
abilities of {\sc awimager}] and subtract the detected sources in
the two-stage manner (first using `high-resolution' images with
approximately 30 arcsec beam size, then using a lower-resolution $\sim
100$-arcsec image) described by vW16. Once each band had had the
sources subtracted, and we had a per-band sky model describing the
subtraction that has been done, we were in a position to start the
facet calibration itself.
\begin{table}
\caption{Spectral windows, frequencies and LOFAR band/sub-band
numbers used}
\label{tab:frequencies}
\begin{tabular}{rrrr}
\hline
Spectral&Frequency range&Band&Sub-band\\
window&(MHz)&numbers&numbers\\
\hline
1&126-134&8-11&80-119\\
2&134-142&12-15&120-159\\
3&142-150&16-19&160-169\\
4&150-158&20-23&200-239\\
5&158-166&24-27&240-279\\
6&166-173&28-31&280-319\\
\hline
\end{tabular}
\end{table}
Our approach to facet calibration is slightly different from that of
vW16. We are interested in imaging in several separate frequency
ranges (which we refer to hereafter as `spectral windows'), since we
would like to be able to measure in-band spectral indices for detected
sources. In addition, facet calibrating in different spectral windows
can be done in parallel, speeding the process up considerably.
Accordingly, we chose to facet calibrate with six spectral windows,
each made up of four bands and thus containing about 8 MHz of
bandwidth\footnote{When fitting over these narrow bandwidths, fitting
for differential TEC as described by vW16 becomes, effectively,
fitting for phase tied together over all four datasets. However, the
key point is the gain in signal to noise of a factor 2 derived from
the joint fit to all four bands.} (Table \ref{tab:frequencies}). We
intentionally did not include in this spectral range the very lowest
frequencies in the data, below 126 MHz, as they have significantly
worse sensitivity than the higher HBA frequencies, and also discarded
frequencies above 173 MHz, which are badly affected by RFI even after
flagging: thus our final images contain about 48 MHz of bandwidth out
of a possible total 72~MHz, but would probably gain little in
sensitivity by including the missing data.
Facets were defined using our knowledge of the bright source
distribution from the `high-resolution' (30 arcsec) images used for
subtraction: we aimed for between 20 and 30 facets per pointing in
order to sample the ionosphere as well as possible, with the number
actually used being determined by the number of bright sources
available. Calibration positions in the facets were defined, as
described by vW16, by selecting square regions (of size less than $50
\times 50$ arcmin, and normally several times smaller than that)
containing sources with a total flux density normally greater than 0.4
(apparent) Jy at 150~MHz. The boundaries between each facet were set
using Voronoi tessellation. Each facet was then calibrated and imaged
in the manner described by vW16, runnning in parallel over all six
spectral windows. In practice, we always started the run for the data
at 150--158~MHz before the other five, so that problems with, for
example, the definition of the facets could be ironed out using a
single dataset; such problems generally showed up as a failure of
phase or amplitude self-calibration, resulting in poor solutions
and/or increased residuals after subtraction, and were dealt with by
increasing the size of the calibration region to include more sources
or changing the phase/amplitude solution intervals. The whole facet
calibration process, for the six spectral windows, took between one
and two weeks per field, depending on the number of facets and the
number of problems encountered; typically the self-calibration,
imaging and subtraction step for one facet in one spectral window took
4--5 hours on a 16-core node with 64 GB RAM and 2.2-GHz clock speed.
The run time per facet was shortened by a factor $\sim 2$ with respect
to earlier implementations of facet calibration by the use of {\sc
wsclean} both for the final facet imaging and for the `prediction'
of the visibilities to subtract from the un-averaged data, by blanking
of the images given to {\sc pybdsm} for mask generation to reduce the
area that it searched for sources, and by some alterations to {\sc
pybdsm} to improve efficiency, particularly in using the efficient
FFTW algorithm for the Fourier transforms involved in cataloguing
extended sources. The use of {\sc wsclean} rather than a {\sc bbs}
subtract meant that care was necessary with the placement of facets in
the final images to avoid aliasing of sources near the field edge in
the prediction step, resulting in negative `sources' outside the facet
after subtraction. A few negative sources generated in this way
propagate through into the final images, but have no significant
effect on the final science results as they are not detected by the
source finding algorithms.
Once the facet calibration and imaging process was complete, the
resulting images for each spectral window were mosaiced into a single
large image for each field, masking each facet to ensure that only the
good portion of the image was used. Previous imaging of each band with
{\sc awimager} was used to make images of the primary beam for each
band and we divided through by these images (regridding them with {\sc
montage} to the scale of the facet calibration images and averaging
appropriately for each spectral window) to make maps of true rather
than apparent flux density. (To maximize the area covered by the survey we
image further down the primary beam than was attempted by e.g. W16:
primary beam corrections at the edge of the largest fields can
approach a factor 5.)
\subsection{Flux calibration}
\label{sec:fluxcal}
LOFAR flux density calibration is in practice somewhat problematic:
effectively, the gain normalizations transferred from the calibrator
to the source are only valid if the elevation of the source and the
calibrator are the same (implying the same effective telescope beam on
the sky), which can only be guaranteed for a snapshot observation and
is not generally true even then. This error manifests itself (after
facet calibration, which removes time-dependent effects) as a
frequency-dependent error in the flux scale. Currently the only
method available to ensure consistency in the flux scale and
correctness of in-band spectral index is to derive correction factors
as a function of frequency using flux densities from other
low-frequency surveys. Data available for our field include the VLSS
at 74 MHz, the TGSS, 7C and 6C surveys at 150 MHz, the WENSS survey at
327 MHz and the B2 survey at 408 MHz. Of these, we elected not to use
the 150-MHz data, although catalogues were available, in order to be
able to use them as comparison datasets later (see below). The VLSS
covers the whole survey area and should be properly calibrated on the
scale of \cite{Scaife+Heald12}, hereafter SH (we use the VLSSr
catalogue of \citealt{Lane+14}). The B2 survey also covers the whole
survey area, and SH report that it needs no systematic correction to
their flux scale; we assign errors to the flux densities based on the
recipes given by \cite{Colla+73}. WENSS is more problematic. Firstly,
the flux density of 3C\,286 in WENSS is a factor 1.23 above the SH
value. SH report an overall scaling factor for WENSS of 0.90, but in
order to ensure that 3C\,286's flux density is correct we scale all
WENSS flux densities for NVSS objects down by a factor 0.81. Secondly,
WENSS only goes down to a declination of $30^\circ$, leaving us with
almost no coverage for the SW field. We therefore supplemented our
catalogues with one derived using {\sc pybdsm} from the 350-MHz WSRT
survey\footnote{Data kindly provided by Shea Brown.} of the Coma
cluster by \cite{Brown+Rudnick11}, having verified that these data
were on the SH scale by comparison to a less sensitive VLA map at 330
MHz reduced by us from the VLA archive. We refer to this survey as
WSRT-Coma in what follows.
Our corrections are based on an initial catalogue of LOFAR sources
which was generated by concatenating the six beam-corrected spectral
window images, convolved to the same resolution using {\sc miriad},
into a single data cube and then using the spectral index mode of {\sc
pybdsm} to extract flux densities for each `channel' of the cube. We
used the stacked broad-band image without beam correction as the
detection image since this should have roughly uniform noise across
the field. Using the spectral index mode, rather than making
independent catalogues for each image, ensures that flux densities are
measured from matched apertures in each spectral window. The catalogue
returned at this point is of course expected to have incorrect flux
calibration, which affects both the total flux densities for each
source measured from the frequency-averaged images and the individual
channel flux densities.
We then filtered this catalogue to include only bright LOFAR sources
($>0.1$ Jy) and cross-matched positionally using {\sc stilts} with the
VLSS, B2, WENSS and WSRT-Coma catalogues, requiring that all LOFAR
sources should be detected in VLSS and at least one of the three
higher-frequency catalogues for further consideration. This gave us
50-70 sources per field with flux densities spanning the frequency
range between 74 and a few hundred MHz, all, of course, being
relatively bright and with high signal to noise in the LOFAR data: the
lowest flux densities at 74 MHz were around 0.4 Jy, corresponding to
around 0.2 Jy at LOFAR frequencies and 0.1 Jy by 300--400 MHz.
Integrated flux densities were used in all cases. We then used Markov-Chain
Monte Carlo methods, implemented in the {\it emcee} {\sc python}
package \citep{Foreman-Mackey+13}, to fit for the LOFAR flux correction factors for each
frequency. The likelihood function was calculated from the total
$\chi^2$ for a given selection of correction factors for power-law
fits\footnote{We investigated the use of models with spectral
curvature, e.g. quadratics in log space, but the parameters of these
were poorly constrained because the fit can trade off curvature
against correction factors to some extent; power laws should be
adequate for most sources over the less than one decade in frequency
that we use here.} to all frequencies of the data (i.e. both the
unscaled VLSS and higher-frequency data and the scaled LOFAR
measurements), with the normalization and power-law index for each
source being free parameters which are determined with a standard
Levenberg-Marquart $\chi^2$ minimization. We used a Jeffreys prior for
the scale factors. In the first round of this fitting we included all
sources, but in the second round we removed sources that were outliers
in the $\chi^2$ distribution before re-fitting, which eliminates any
sources that might be erroneous matches or heavily affected by
resolution effects or might have spectra that are intrinsically poorly
fit by a power law over the band, e.g. because of spectral curvature
or variability; in practice this second round generally eliminated at
most around 10 per cent of sources and gave results very similar to
the first round. The final flux calibration factors obtained in this
way for each field are tabulated in Table \ref{tab:factors}. Nominal
MCMC-derived credible intervals (errors) on these correction factors
are small, $<1$ per cent in all cases; the dominant source of error is
of course the real deviation from a power-law model of the sources
selected as calibrators, which is hard to quantify.
\begin{table}
\caption{Correction factors applied per field and spectral window,
and mean calibrator and target elevations}
\label{tab:factors}
\begin{tabular}{lrrrrrrrr}
\hline
Field&\multicolumn{6}{c}{Spectral window
number}&\multicolumn{2}{c}{Mean elevation}\\
&1&2&3&4&5&6&3C\,196&Field\\
\hline
Central&1.02&1.08&1.17&1.29&1.42&1.52&85.3&51.6\\
NW &0.97&1.03&1.07&1.14&1.21&1.30&82.2&58.6\\
SW &0.93&1.00&1.09&1.17&1.29&1.34&82.4&53.4\\
NE &0.72&0.75&0.79&0.84&0.89&0.92&78.2&58.9\\
\hline
\end{tabular}
\end{table}
Once this correction had been applied to each spectral window, it was
possible to cross-check the flux scale between fields by comparing
sources in the overlap regions. By this method we established that the
flux scale for the Central, NW and SW fields was in good agreement.
The NE field appeared to have a systematic offset with respect to the
others, which we attribute to the fact that the correction factor fits
are strongly affected by 3C\,286, which is rather poorly
facet-calibrated in this field (that is, relatively large artefacts
remain around the source, so that its flux density is probably poorly
measured). The flux densities in the NE field were systematically
high. To correct for this we scaled all correction factors in the NE
field by a further factor of 0.8, determined by cross-checking the
flux densities with the overlapping fields, which brings the flux scales in line
across the four fields; this factor is included in the numbers presented in
Table \ref{tab:factors}. We comment on the effectiveness of these
corrections in allowing us to recover reliable flux densities and
in-band spectral indices in the next section.
These corrections were then applied to the pre-existing images, and
new combined images at an effective frequency of 150 MHz, again
convolved to a matching common resolution, were made by averaging the
images from the 6 spectral windows for each field. These images, for
each field, are our deepest view of the data, and attain an rms noise
of 100 $\mu$Jy beam$^{-1}$ in the centre of the best fields. Their
resolution (Table \ref{tab:field-results}) is determined by the
Gaussian fit by {\sc wsclean} to the $uv$ plane coverage in spectral
window 1, which is slightly different for each field, and particularly
different for the Central field, which was observed for a longer time
but without all the long baselines available in later years. As
convolving to a common resolution would reduce the resolution of all
the images, we elected to retain these slightly different resolutions
between fields.
We then generated source catalogues for the corrected, matched-resolution
data with {\sc pybdsm} in its spectral index mode in the same way as
described above. {\sc pybdsm}'s ability to detect sources on multiple
scales and to associate several Gaussian components as single sources
were enabled in this cataloguing step, and sources were only
catalogued if they are detected with a peak above the local $5\sigma$
value. We also make use of {\sc pybdsm}'s ability to generate a map of
the rms of the four 150-MHz images, referred to as the rms map in what
follows.
\subsection{Image quality}
In general facet calibration worked well in these fields, succeeding
in its two design aims of allowing greatly improved subtraction of
bright sources and of making high-fidelity imaging of the data at
close to the full resolution of LOFAR tractable. The resulting images
are not artefact-free, but this is a consequence of not being able to
amplitude self-calibrate every bright source: artefacts manifest
themselves as dynamic range limitations around bright sources
(particularly those that were {\it not} in the calibration sub-region
of the facet) and should have only a limited effect on the quality of
the final catalogue. Where facet calibration fails, it generally does
so by having poor signal to noise on the calibrator, leading to poor
phase and/or amplitude solutions and, generally, no improvement in
self-calibration: if these poor solutions are then applied to the data
we see a facet with much higher rms noise than would be expected. As
discussed above, this problem can sometimes be solved by increasing
the calibration region size or the interval for phase or amplitude
self-calibration, but in extreme cases the facet simply has to be
abandoned: we discarded a few facets at the edges of the NE and
Central fields, typically at $>3$ degrees and so substantially down
the primary beam, where it was not possible to obtain good
self-calibration solutions (in addition to the fact that the
calibrator sources are generally fainter when they are further down
the primary beam, bandwidth and time-averaging smearing also start to
affect the quality of the results). Other facets with poor solutions
remain in the images and give rise to high-noise regions in the rms
map.
\begin{table*}
\caption{Basic field image and catalogue properties for the combined
150-MHz images}
\label{tab:field-results}
\begin{tabular}{lrrrrr}
\hline
Field&Area covered¢ral rms&Median rms&Image resolution&Catalogued\\
&(sq. deg.)&($\mu$Jy)&($\mu$Jy)&(arcsec)&sources\\
\hline
Central&44.5&223&782&$9.62 \times 7.02$&2,473\\
NW&34.7&104&296&$10.01 \times 5.54$&5,335\\
SW&36.1&111&319&$9.87 \times 5.49$&5,747\\
NE&40.6&100&382&$11.02 \times 4.68$&4,172\\
\hline
\end{tabular}
\end{table*}
\begin{figure}
\includegraphics[width=1.0\linewidth]{noisehist.pdf}
\caption{Cumulative histogram of the area below a given rms value,
calculated from the rms noise map derived from {\sc pybdsm} for the
four fields. Note that, since the rms values are corrected for the
telescope beam, the shape of this distribution depends on the
placement of the facets as well as the intrinsic noise qualities of
the image.}
\label{fig:rms}
\end{figure}
In Table \ref{tab:field-results} we tabulate the areal coverage,
central rms, and median rms of the four fields for the broad-band
150-MHz images, along with the final resolution achieved. The median rms is the rms below which the best half of
the field falls: this clearly depends on the placement of facets in
the beam as well as on the image quality. The area-rms distribution of
the fields is illustrated in Fig.\ \ref{fig:rms}. These rms values may
be compared to the FIRST and NVSS rms values converted to 150 MHz for
a source with $\alpha = 0.75$, which are 0.8 and 2.4 mJy beam$^{-1}$
respectively. Thus, purely considering rms levels, the LOFAR survey is
better than NVSS even far down the beam and always deeper than FIRST
in the central 50 per cent of each field. The LOFAR data are also
significantly better, in these terms, than the GMRT survey of the
equatorial H-ATLAS fields \citep{Mauch+13}, which has a best rms level
of 1 mJy beam$^{-1}$ at 325 MHz, corresponding to 1.8 mJy beam$^{-1}$
at 150 MHz. (We draw attention to the very different resolutions
of these comparison surveys: FIRST has a resolution of 5 arcsec,
NVSS 45 arcsec, and the GMRT survey between 14 and 24 arcsec. In
terms of resolution, our data are most comparable to FIRST.)
The Central field, the first that was observed and the only one to be
taken in Cycle 0 with the original correlator, was by far the worst of
the fields in terms of noise despite its slightly longer observing
duration. In this field the initial subtraction was simply not very
good, suggesting large amplitude and/or phase errors in the original
calibration, although it was carried out in exactly the same way as
for the other fields. As a consequence, facet calibration did not
perform as well as in the other fields (presumably because the
residuals from poorly subtracted sources acted as additional noise in
the visibilities) and there are more bad facets than in any other
field, together with a higher rms noise even in the good ones. It
should be noted that this is also the worst field in terms of
positioning of bright sources, with 3C\,286, 3C\,287 and 3C\,284 all a
couple of degrees away from the pointing centre: we do not know
whether this, the fact that the observation was carried out early on
in the commissioning phase, poor ionospheric conditions, or a
combination of all of these, are responsible for the poor results. The
other three fields are of approximately equal quality, with rms noise
values of 100~$\mu$Jy beam$^{-1}$ in the centre of the field and each
providing around 31 square degrees of sky with rms noise below 0.8 mJy
beam$^{-1}$ after primary beam correction. The worst of these three
fields, the NE field, which has some facets where facet
calibration worked poorly or not at all, was observed partly in
daytime due to errors at the observatory, which we would expect would
lead to poorer ionospheric quality which may contribute to the lower
quality of the data.
\begin{figure*}
\includegraphics[width=1.0\linewidth]{hatlas-rms-shrink.pdf}
\caption{Map showing the sky coverage and rms values of the four
fields, constructed as described in the text. Colour levels run from
100 $\mu$Jy to 2 mJy beam$^{-1}$. The green square shows the
approximate boundary of the {\it Herschel} survey. The LOFAR survey
is deeper (in rms terms) than FIRST, the previous most sensitive
radio survey of this area, in the blue regions of the image. The
many `point sources' in the image are the result of dynamic range
limitations around bright objects, rather than the objects
themselves: the pixel size in this image is 20 arcsec, significantly
larger than the image resolution.}
\label{fig:rmsmap}
\end{figure*}
A map of the sky coverage of the images and the rms levels, generated
by resampling the full-resolution rms maps onto a grid with 20-arcsec
cell size, is shown in Fig. \ref{fig:rmsmap}. Where two fields
overlap, the best rms value is shown, for reasons explained in the
following section.
\subsection{Catalogue generation and completeness}
\label{sec:completeness}
The final source catalogue is made by combining the four per-field
catalogues. Ideally we would have combined the images of each field
and done source finding on a mosaiced image, but this proved
computationally intractable given the very large image cubes that
result from having six spectral windows. We therefore merged the
catalogues by identifying the areas of sky where there is overlap
between the fields and choosing those sources which are measured from
the region with the best rms values. This should ensure that there are
no duplicate sources in the final catalogue. The final master
catalogue contains 17,132 sources and is derived from images covering
a total of 142.7 square degrees of independently imaged sky, with
widely varying sensitivity as discussed above. Total HBA-band
(150-MHz) flux densities of catalogued sources detected using {\sc
pybdsm} and a $5\sigma$ detection threshold range from a few hundred
$\mu$Jy to 20 Jy, with a median of 10 mJy.
For any systematic use of the catalogue it is necessary to investigate
its completeness. In the case of ideal, Gaussian noise and a catalogue
containing purely point sources this could simply be inferred from the
rms map, but neither of these things is true of the real catalogue. In
particular, the distribution of fitted deconvolved major axes in the
source catalogue shows a peak around 10 arcsec. This is probably the
result of several factors, including a certain fraction of genuinely
resolved sources, but we suspect that at least some of the apparent
broadening of these sources is imposed by the limitations of the
instrument and reduction and calibration procedure rather than being
physical. Part may be due to residual bandwidth or time-averaging
smearing in the individual facet images, though our lower angular
resolution (relative to the similar work of W16) helps to mitigate
these effects. We suspect that a significant fraction of the
broadening comes from residual phase errors in the facet-calibrated
images, particularly away from the calibration regions. This may be
compounded in our case by the effects of combining our multiple
spectral windows in the image plane -- no attempt was made to align
the images other than the self-calibration with an identical sky model
before facet calibration, and phase offsets between the spectral
windows will lead to blurring of the final image. Whatever the origin
of these effects, the fact that most sources are not pointlike in the
final catalogues needs to be taken into account in estimating the true
sensitivity of the data.
\begin{figure}
\includegraphics[width=1.0\linewidth]{rmscompleteness.pdf}
\caption{Completeness function for the whole survey compared with the
expectations from the rms map alone. The points are the results
of simulations (Poisson errors, though present, are generally
smaller than symbols and are not plotted) while the smooth curve
shows the best-fitting 5th-order polynomial in log space used to
approximate the completeness curve and interpolate to un-simulated
flux densities.}
\label{fig:completeness}
\end{figure}
To assess this we therefore carried out completeness simulations in
the standard way in the image plane\footnote{In principle we should
simulate the process all the way from the original observations,
injecting sources in the $uv$ plane, corrupting them with simulated
ionospheric and beam effects and repeating the facet calibration and
imaging many times. However, although this would be a valuable
exercise, it is computationally infeasible at present for the
purposes of completeness simulation, and challenging even for a
verification of the facet calibration process. Work being carried
out along these lines in the Key Science Project will be described
elsewhere.} [see, e.g., \cite{Heald+15} and W16] by adding in
simulated sources to the residual map for each field and recovering
them with {\sc pybdsm} with the same settings as used for the real
cataloguing. In our case, we assumed sources to be uniformly
distributed at random across the whole NGP area, and placed them on
the residual maps for the individual pointings based on the rms map
used for cataloguing. However, rather than placing point sources (i.e.
Gaussians with the parameters of the beam), we broadened the simulated
sources using a Gaussian blur where the broadening $\sigma$ was itself
drawn from an appropriate Gaussian distribution, chosen so as to
approximately reproduce in the extracted (output) catalogues the low
end of the observed distribution in deconvolved major and minor axes.
The use of the residual maps also naturally takes account of artefacts
around bright sources and other non-Gaussian features in the images,
such as any negative holes due to {\sc wsclean} aliasing effects.
We ran a number of simulations for each of a range of input source
flux densities, using between 10,000 and 30,000 simulated sources per run to
improve the statistics. We consider a source to be
matched if a source in the derived catalogue agrees with one in the
input catalogue to within 7 times the nominal error in RA and Dec and
20 times the nominal error in flux density. These criteria are deliberately
generous to reflect the fact that the errors on flux density and position from
off-source noise are generally underestimates. Noise peaks from the
residual map are removed from the catalogue before this comparison is
made to avoid false positives. It is also possible to recover false
detection rates in this way, but these are known to be very low (W16)
and so we do not discuss them further here.
The results are shown in Fig.\ \ref{fig:completeness}, where, for
comparison, the $5\sigma$ detection level for pure point sources based
on the rms map and the assumption of Gaussian noise is also shown. It
can be seen that the various effects we simulate have a strong effect
on completeness. The survey is complete, in the sense that a source of a
given flux density can be detected essentially anywhere, only above a
comparatively high flux density of $\sim 20$ mJy. At lower flux
densities, the completeness curve drops more steeply than the rms map
would imply. At 1 mJy, for example, the completeness curve implies a
probability of detection (for a source placed at random in the field)
ten times lower than would be inferred from the rms map. The curves
intersect again at very low flux densities ($\sim 0.5$ mJy), but we
suspect that the detection fraction here is artificially boosted by
Eddington bias (i.e. simulated sources placed on noise peaks in the
residual map are more likely to be recovered). The slight errors in
the completeness curve resulting from this are not problematic given
that there are so few sources with these flux densities in any case.
Also plotted in Fig.\ \ref{fig:completeness} is the best-fitting
5th-order polynomial in log space fitted to the results of the
simulations (taking account of the Poisson errors): this function
gives an adequate approximation to and interpolation of the
completeness curve, which we will make use of in later sections.
It is important to note that much of this incompleteness results from
the sparse sky coverage of the observations for this project, and the
poor quality of the Cycle 0 central field data. It is not
representative of the expectations for the Tier 1 (wide-area) LOFAR
sky survey: see W16 for a more representative completeness curve.
\subsection{Association, artefact rejection and optical identification}
\label{sec:optid}
The source catalogue was the starting point for our source association
and optical identification processes, which were carried out in
parallel. Optical identification was carried out using images and
catalogues from SDSS Data Release 12 \citep{Alam+15}, hereafter DR12.
Initially, we carried out a simple positional crossmatch for low-$z$
galaxies, selecting compact (deconvolved size $<10$ arcsec) LOFAR
sources whose position matched that of an optical source from the
MPA-JHU\footnote{The MPA-JHU catalogue is the Max Planck Institute for
Astrophysics/Johns Hopkins University catalogue of bright SDSS Data
Release 7 galaxies with spectroscopic redshifts: see
\url{http://wwwmpa.mpa-garching.mpg.de/SDSS/DR7/}. This catalogue
was used because the MPA-JHU catalogue forms the basis of the work
on the radio/star-formation relation to be described by G\"urkan
{et al}\/.} catalogue within 8 arcsec (chosen based on the distribution
of offsets). This identified 1,048 LOFAR sources, of which we would
expect around 30 to be chance coincidences given the number of MPA-JHU
sources in the survey area. We then visually inspected the LOFAR,
SDSS, FIRST and NVSS images for {\it all} the 16,084 remaining
sources, initially with a single author (one of GG, MJH or SCR)
inspecting each source. The person carrying out the visual inspection
was asked to associate individually detected LOFAR sources, i.e. to
say whether s/he believed that they were physically associated, to
identify any artefacts, and, for real sources, to specify any
plausible optical identification for the radio source. The NVSS images
were used only to confirm the reality of faint extended LOFAR sources,
which often show up well in the low-resolution NVSS data, but the
FIRST images had a more important role, as they turn out often to show
the flat-spectrum core of an extended LOFAR source making optical
identification far more robust. Identifications by one author were
cross-checked against those of another to ensure consistency and a
subset (consisting of a few hundred large, bright sources) of the
first pass of identifications were re-inspected visually by several
authors and some (a few per cent) corrected or rejected from the final
catalogue. The final outcomes of this process were (a) an associated,
artefact-free catalogue of 15,292 sources, all of which we believe to
be real physical objects, and (b) a catalogue of 6,227 objects with
plausible, single optical identifications with SDSS sources,
representing an identification fraction of just over 40 per cent.
(Note that around 50 sources with more than one equally plausible
optical ID are excluded from this catalogue; further observation would
be required to disambiguate these sources.) This identification
fraction is of interest because we can expect to achieve very similar
numbers in all parts of the Tier 1 LOFAR survey where SDSS provides
the optical catalogue. Forthcoming wide-area optical surveys such as
Pan-STARRS1 and, in the foreseeable future, LSST (for equatorial
fields), will improve on this optical ID rate.
Optical identification using shallow optical images can lead, and
historically has led, to misidentifications, where a plausible
foreground object is identified as the host instead of a true unseen
background source. This is particularly true when the LOFAR source is
large and no FIRST counterpart is seen. It is difficult to assess the
level of such misidentifications in our catalogue [likelihood-ratio
based methods, such as those of \cite{Sutherland+Saunders92},
require information about where plausible optical IDs could lie in a
resolved radio source that is hard to put in quantitative form] but
as our resolution is relatively high, so that most sources are not
large in apparent angular size and do not have more than one plausible
optical ID, we expect it to be low. Sensitive high-frequency imaging
over the field, and/or deeper optical observations, would be needed to
make progress.
In what follows we refer to the raw, combined output from {\sc pybdsm}
as the `source catalogue', the product of the association process as
the `associated catalogue' and the reduced catalogue with SDSS optical
IDs as the `identified catalogue'. Sources in the associated or
identified catalogues that are composed of more than one source in the
source catalogue are referred to as `composite sources' (in total
2,938 sources from the original catalogue were associated to make
1,349 composite sources). The process of association renders the {\sc
pybdsm}-derived peak flux densities meaningless (they are suspect in
any case because of the broadening effects discussed in the previous
subsection) and so in what follows unless otherwise stated the flux
density of a LOFAR source is its total flux density, derived either
directly from the source catalogue or by summing several associated
sources.
\section{Quality checks}
\label{sec:quality}
In this section we describe the tests carried out on the catalogues to
assess their suitability for further scientific analysis. From here
on, except where otherwise stated, we use only the associated and
identified catalogues.
\subsection{Flux scale tests: 7C crossmatch}
An initial check of the flux scale was carried out by crossmatching
the associated catalogue with the 7C catalogue \citep{Hales+07} over
the field. Unfortunately the NGP spans the southern boundary of 7C, so
we do not have complete coverage, though there is substantial overlap.
The crossmatching uses the same algorithm as that described by
\cite{Heald+15}, i.e. we use a simple maximum likelihood crossmatch
taking account of the formal positional errors in both catalogues and
using the correct (Rayleigh) distribution of the offsets, but not
taking into account any flux density information. Since 7C sources are
very sparse on the sky, any more complicated procedure is probably
unnecessary. Over the intersection of the 7C and LOFAR/NGP survey
areas, there are 735 7C sources, 694 of which (94 per cent) are
detected in the LOFAR images, with a mean positional offset of
$\delta_{RA} = 0.87 \pm 0.34$ arcsec and $\delta_{\rm Dec} = 0.28 \pm
0.32$ arcsec. The flux limit of 7C is a few hundred mJy, so we would
expect all 7C sources to be detected by LOFAR: in fact, the few
nominally unmatched 7C sources are either at the edges of one of the LOFAR
fields, where the sensitivity is very poor, or are actually close to a
LOFAR source but with discrepant co-ordinates, which could be
attributed to the very different resolutions of the surveys -- 7C has
a resolution of $70 \times 140$ arcsec at this declination. 7C is
complete above $\sim 0.4$ Jy at 150 MHz, and for sources above this
flux limit the mean ratio between 7C and LOFAR 150-MHz total flux
densities is $1.00 \pm 0.01$, showing excellent agreement between the
7C and LOFAR flux scales, though the scatter is larger than would be
expected from the nominal flux errors. We can conclude that there are
no serious global flux scale errors in the catalogue, at least in the
region covered by 7C (essentially the NE and NW fields).
\subsection{Flux scale tests: NVSS crossmatch}
\label{sec:nvss-flux}
\begin{figure}
\includegraphics[width=1.0\linewidth]{nvss-ratio-radius.pdf}
\caption{NVSS/LOFAR flux ratio as a function of radial distance from
the pointing centre of each field. Points show individual matched
sources, solid lines show the median in radial bins and its error
on the approximation of Gaussian statistics.}
\label{fig:ratio-radius}
\end{figure}
The most suitable high-frequency survey for a direct comparison with
the LOFAR results is NVSS, which is sensitive to large-scale
structure, although its resolution is much lower than that of the
LOFAR images. To generate a suitable catalogue we extracted the NVSS
images from the image server and mosaiced them into a large image
covering the whole field. We then applied {\sc pybdsm} to this mosaic
with exactly the same settings as were used for the LOFAR catalogue;
this procedure allows us to measure accurate total flux densities for
extended sources, rather than inferring them from the peak flux
densities and Gaussian parametrization provided in the NVSS catalogue.
Filtering our {\sc pybdsm} catalogue to match the area coverage of the
LOFAR survey, we found 5,989 NVSS sources. These were then
crossmatched to the LOFAR data as for the 7C data, but adding a
Gaussian term to the likelihood crossmatching to favour sources where
the flux densities are consistent with the expected power law of
$\alpha \approx 0.7$ (i.e. a term proportional to $\exp(-(S_{\rm
LOFAR} - (1400/151)^{0.7} S_{\rm NVSS})^2/\sigma^2)$: this helps to
reduce the incidence of spurious crossmatches) and also excluding
associations with a separation between NVSS and LOFAR positions of
greater than 1 arcmin. We obtained 4,629 matches: that is, as
expected, the vast majority of the NVSS sources have LOFAR
counterparts, with a mean positional offset of $0.4 \pm 0.1$ arcsec in
RA and $0.05 \pm 0.1$ in Dec. Counterparts are genuinely missing at
the edges of the LOFAR field, where the noise is high, but we have
verified by visual inspection that the comparatively large number of
`unmatched' sources within the field are the result of disagreements
about source position (e.g. arising from structure resolved by LOFAR
but unresolved by NVSS) rather than from genuinely missing sources.
Similarly, most bright LOFAR sources have an NVSS counterpart. We
therefore do not regard the match rate of only 77 per cent as
problematic: visual inspection of the images could probably bring it
close to 100 per cent.
Our expectation is that NVSS should be uniformly calibrated (VLA flux
calibration uncertainties are 2-3 per cent, which should not introduce
much scatter into this comparison), so the flux ratios between NVSS
and the LOFAR catalogue allow an accurate check of the flux scale,
subject only to the possibility that the fields have genuinely
intrinsically different spectral index distributions, which could
happen, for example, if the SW field were affected by the presence of
the Coma cluster. For a further check of the flux scale and also its
dependence on radius we computed the median NVSS/LOFAR flux ratio for
all matched sources (median rather than mean to avoid effects from
strong outliers which might arise from misidentifications or extreme
intrinsic spectral indices) and also its dependence on distance from
the pointing centre for each facet in bins of 0.5 degree in radius. We
see (Fig.\ \ref{fig:ratio-radius}) that there are no significant flux
scale (or, equivalently, spectral index) offsets between fields. The
scatter is large, but much of this is imposed by the known dispersion
in spectral index (see below, Section \ref{sec:outofband}).
An encouraging result from the radial plot is that there is also no
significant systematic difference with radius, within the
uncertainties imposed by the scatter in the data. This suggests (a)
that the primary beam correction applied is adequate, and (b) that
bandwidth and time-averaging smearing at the edge of the field, beyond
2-3 degrees, do not seem to be having any detectable effect on the
LOFAR total flux densities. The same comparison was also carried out
using the peak flux densities of the LOFAR images and those of the
cross-matched FIRST sources (see below), which should be more
sensitive to smearing effects, again with no discernible radial
dependence of the ratios.
\subsection{Flux scale tests: TGSS crossmatch}
\label{sec:tgss}
\begin{figure}
\includegraphics[width=1.0\linewidth]{lofar-tgss-flux-shrink.pdf}
\caption{TGSS against LOFAR flux densities, colour-coded by field.
Error bars are plotted for all data points, but in many cases are
smaller than the symbols. The central line shows the median ratio
between the flux densities (1.08) and the dispersion that would be
implied by 10 per cent calibration uncertainties on top of this. The
per-field flux scale offsets between LOFAR and TGSS data can be seen
as colour gradients across the plot.}
\label{fig:tgss-lofar}
\end{figure}
For comparison with a deeper survey than 7C at 151 MHz we make use of
the data from the TGSS survey made with the GMRT with a resolution of
$25 \times 25$ arcsec \citep{Intema+16}. As with the NVSS data, we
made a single large mosaic of the images, extracted flux densities
using {\sc pybdsm}, and then cross-matched positionally with the LOFAR
data. There are 2896 TGSS sources in the LOFAR field, of which almost
all (2449) can be cross-matched with LOFAR data. Surprisingly, given
the good agreement between LOFAR and NVSS flux scales (Section
\ref{sec:nvss-flux}), we see non-negligible per-field differences in
the mean LOFAR and TGSS flux densities. There is no overall flux scale offset
(as measured from median ratios of all matched sources), but the
median TGSS/LOFAR ratios for the individual fields vary between 0.86
and 1.10. GMRT flux calibration is itself not reliable to better than
about 10 per cent, and the overall medians will be dominated by the
sources close to the centre of each field, so it is perfectly possible
that much of this scatter comes from GMRT calibration uncertainties.
In addition, the GMRT's flux scale can be adversely affected by bright
sources in the field, and this is apparent, for example, in the flux
for the calibrator source 3C\,286, which is significantly offset in
the GMRT catalogue from the reference 150-MHz value of SH. We
therefore do not attempt to use the TGSS images to derive further
corrections to the per-field flux scale, but simply report the TGSS
comparison here for the benefit of future workers. Plotting the LOFAR
(corrected) total flux densities against TGSS flux densities (we
restrict the comparison to sources that should be unresolved to TGSS)
shows a good correlation, but, as with 7C, the scatter is larger than
would be expected from the nominal errors
(Fig.\ \ref{fig:tgss-lofar}), indicating some residual calibration
errors in either or both of the TGSS and LOFAR datasets. In the
absence of a detailed study of the TGSS flux calibration, we cannot
establish whether one or both of the datasets are responsible for
this.
\subsection{Positional accuracy tests: FIRST crossmatch}
\label{sec:first-crossmatch}
The 7C crossmatch shows that there are no gross astrometric errors in
the catalogue, but to investigate positional accuracy in more detail
we need a larger sample with higher resolution. For this purpose we
cross-matched the source catalogue with the FIRST survey data in the
field. There are 9,856 FIRST sources in the survey area, after
filtering out sources with FIRST sidelobe probability (i.e.
probability of being an artefact) $>0.05$. We restricted
the crossmatching to compact LOFAR sources (fitted size less than 15
arcsec) with well-determined positions (nominal positional error less
than 5 arcsec). 3,319 LOFAR/FIRST matches were obtained by this
method, with a mean offset over the whole field of $\delta_{\rm RA} =
0.16 \pm 0.01$ arcsec and $\delta_{\rm DEC} = 0.01 \pm 0.01$ arcsec.
Using these matches, we can determine the mean LOFAR/FIRST offset
within each facet, shown in Fig.\ \ref{fig:offset-facet}. Some facets
have relatively few matches, so the results should be treated with
caution, but a couple of points are fairly clear. Firstly, the typical
offsets are small, a couple of arcsec at most: given that any offsets
are likely introduced by the phase self-calibration in the facet
calibration process, we would not expect them to be much larger than
the pixel size of 1.5 arcsec, as is observed. Secondly, fields in
which we had worse results with facet calibration also show larger
offsets; by far the largest offsets are seen in some facets of the
Central field, which, as discussed above, also has significantly
higher noise. This is consistent with the idea that the quality of the
initial direction-independent phase calibration has a strong effect on
the final facet calibration results: if the initial phase calibration
is poor, we expect offsets in the initial images for the first
(phase-only) facet self-calibration step, and we will never be able to
recover from these completely without an external reference source.
\begin{figure}
\includegraphics[width=1.0\linewidth]{offsets-per-facet.pdf}
\caption{FIRST/LOFAR offsets in the field. The mean offset for each
facet is plotted individually. Error bars show the nominal errors on
the mean offsets.}
\label{fig:offset-facet}
\end{figure}
\subsection{In-band spectral index}
\label{sec:inband}
\begin{figure}
\includegraphics[width=1.0\linewidth]{ibspix.pdf}
\caption{In-band spectral index determined from LOFAR data. The overall histogram is shown together with the
histograms for each field.}
\label{fig:ibspix}
\end{figure}
We fitted power laws in frequency to the total flux densities for each source
in the associated catalogue. The in-band spectral index is a sensitive
test of the validity of the correction factors applied to the flux densities
in each field prior to combination, as even small calibration errors
will lead to large biases in in-band spectral index over the
relatively narrow HBA band alone. Many sources have poor $\chi^2$
values (suggesting that the errors in the catalogue are
underestimated) or large errors on the spectral index (estimated from
the fitting covariance matrix). The in-band spectral index
distribution for the overall associated catalogue and the four fields
is shown in Fig.\ \ref{fig:ibspix}, where we plot only sources with
nominal $1\sigma$ spectral index errors of $<0.2$ and exclude the
highest $\chi^2$ values ($\chi^2 > 80$). It can be seen that, although
the overall in-band spectral index distribution is reasonable and
peaked around the expected value (0.6--0.7), the catalogues for the
four fields have rather different distributions. The central field, in
particular, shows a peak at flat spectral index values which must be
the result of the generally poorer quality of the data in this field,
while the NE field has an excess of steep-spectrum sources.
By contrast, the normalization of the power-law fits at 150 MHz is
generally in good agreement with the broad-band total flux density we
measure. We conclude that in-band spectral indices cannot be reliably
compared between fields in this dataset, though sources with extreme
apparent in-band spectral index remain interesting topics for further
investigation. Reliable absolute in-band spectral index measurements
will require the LOFAR gain transfer problems to be solved by the use
of a correctly normalized beam model.
\begin{figure*}
\includegraphics[width=0.48\linewidth]{errplot_bin.pdf}
\includegraphics[width=0.48\linewidth]{errplot_centre_bin.pdf}
\caption{Density plot of errors on the in-band spectral index as a
function of total LOFAR flux density. Left: all sources and fields.
Right: only the central 2 degrees of the NW, NE and SW fields are
plotted.}
\label{fig:spixerr}
\end{figure*}
We can in addition comment on the errors on the in-band spectral index
to be expected from HBA data. Fig.\ \ref{fig:spixerr} shows the error
on in-band spectral index as a function of flux density for the
associated catalogue, both for the whole catalogue and for the inner 2
degrees of the three best fields, which should be more representative
of Tier 1 quality. It can be seen that errors are typically less than
$\pm 0.1$ only for bright sources, with flux densities $>100$ mJy,
even in the centres of the best fields. For almost all sources,
therefore, a much cleaner spectral index determination will be
obtained by comparing with NVSS, which will detect all but the
steepest-spectrum LOFAR sources with LOFAR flux densities above a few
tens of mJy. It will be possible to use in-band spectral index to
select sources which are extremely steep-spectrum (and so undetected
in NVSS) but this will only be reliable, even after LOFAR gain
calibration problems are solved, if they are also bright.
\subsection{Out-of-band spectral index}
\label{sec:outofband}
We use the NVSS/LOFAR crossmatch described above (Section
\ref{sec:nvss-flux}) to construct a distribution of spectral indices
between 150 MHz and 1.4 GHz (Fig.\ \ref{fig:nvhist}). The median
NVSS/LOFAR spectral index is 0.63, with almost no differences seen
between fields. It is important to note that the effective flux
density limit of $\sim 2.5$ mJy for point sources in the NVSS data
biases the global spectral index distribution to low (flat) values --
only flat-spectrum counterparts can be found to faint LOFAR sources
(Fig. \ref{fig:nvhist}). If we restrict ourselves to sources where
this bias is not significant, with LOFAR flux densities above $\sim
30$~mJy, the median spectral index becomes $0.755 \pm 0.005$ (errors
from bootstrap), in good agreement with other determinations of the
spectral index distribution around these frequencies \citep[e.g.,][and
references therein]{Mauch+13}. Deeper 1.4-GHz data with comparable
$uv$ plane coverage to LOFAR's are required to investigate the
spectral index distribution of faint sources.
\begin{figure*}
\includegraphics[width=0.33\linewidth]{nvspix.pdf}
\includegraphics[width=0.33\linewidth]{nvspix_scatter.pdf}
\includegraphics[width=0.33\linewidth]{nvspix_bright.pdf}
\caption{Left: histogram of LOFAR/NVSS spectral indices for all
sources and for the individual fields. Centre: the relationship
between spectral index and flux density: the solid line shows the region
to the top left that cannot be populated by point sources given the NVSS
sensitivity limit. Right: the histogram of spectral index for bright
($S_{150} > 30$ mJy) sources only.}
\label{fig:nvhist}
\end{figure*}
With both the in-band and LOFAR/NVSS spectral indices in hand, we can
compare the two, and this comparison is shown in
Fig.\ \ref{fig:spixcompare}. Here we plot the $\sim 2,000$ sources
that have LOFAR flux density $>30$ mJy and also satisfy the
requirement that the nominal error on the in-band spectral index is $<0.1$
and the fit is acceptable. A general tendency for the in-band spectral
index to be flatter than the LOFAR/NVSS index is observed,
unsurprisingly, but many sources exhibit unrealistically steep (in the
NE field) or flat (in the Central field) in-band indices, and in
general the scatter in the plot is probably dominated by the known
per-field biases in in-band index. It is
possible to identify in this plot some individual sources
that plausibly have interestingly steep, inverted or curved spectra,
but the unreliability of the in-band index limits its use.
\begin{figure}
\includegraphics[width=1.0\linewidth]{inband.pdf}
\caption{The in-band and NVSS/LOFAR spectral indices compared. The solid
line shows equality between the two spectral indices. In general
increasing spectral steepness with frequency means that we
would expect points to lie above this line.}
\label{fig:spixcompare}
\end{figure}
\subsection{The optical identifications}
As noted above, 6,227, or approximately 40 per cent, of the sources in
the associated catalogue have optical identifications with either
galaxies or point-like objects (presumably quasars) from the SDSS DR12
{\it photoobj} table. Of these, 1,934 have spectroscopic redshifts in
the {\it specobj} table and an additional 3,660 have photometric (but
not spectroscopic) redshifts, leaving 633 with no redshift information
(we discard objects with nominal errors $>0.3$ on the
photometric redshift). 263 objects are classed as pointlike in the
photometry catalogue based on the {\it prob\_psf} field, of which 89 have spectroscopic redshifts; the
pointlike objects with spectroscopic redshifts
are likely almost all quasars and we refer to them as quasars in what follows.
The highest spectroscopic redshift in the sample is for a quasar at
$z=5.2$, but no object that is not a quasar has a redshift much
greater than 1, as expected given the magnitude limits of SDSS; the
sharp cutoff in photometric redshifts at $z \approx 1$ is presumably a
consequence of the absence of $z>1$ objects from the training sets used in
SDSS photo-$z$ determination \citep{Beck+16}, but the locus of
magnitudes of radio-galaxy hosts with spectroscopic redshifts clearly
intercepts the SDSS $r$-band magnitude limit of 22.2 at this redshift
in any case. Detecting higher-redshift radio galaxies will require
deeper optical data. The spectroscopic coverage of the galaxies that
we do detect is excellent due to the presence of spectra from the
Baryon Oscillation Spectroscopic Survey (BOSS: \citealt{Dawson+13}) in
DR12, and as a result the number of objects with spectroscopic
redshifts is comparable to that in the FIRST/Galaxy and Mass Assembly
(GAMA: \citealt{Driver+09,Driver+11})-based sample of
\cite{Hardcastle+13}, although the distribution of redshifts is rather
different. The WEAVE-LOFAR
project\footnote{\url{http://star.herts.ac.uk/~dsmith/weavelofar.html}.}
\citep{Smith15} aims to obtain spectra and redshifts for essentially
all of the radio sources in the field.
\begin{figure*}
\includegraphics[width=0.48\linewidth]{maghist.pdf}
\includegraphics[width=0.48\linewidth]{magz.pdf}
\caption{Left: the distribution of identified sources with spectroscopic,
photometric, and no redshifts as a function of $r$-band magnitude.
The second peak of spectroscopic redshifts at $r \approx 20$ is
due to BOSS selection.
Right: $r$-band magnitude as a function of redshift. Redshift is
plotted on a $\log(1+z)$ scale but labelled with $z$ values.}
\label{fig:magdist}
\end{figure*}
In Fig. \ref{fig:magdist} we plot the Petrosian $r$ magnitude from
SDSS for the optically identified sample, showing objects with
spectroscopic, photometric or no redshift. We see that the sample is
virtually spectroscopically complete at $r<17.7$ mag, and almost all
sources have a spectroscopic or photometric redshift at $r<19$ mag. A
clear lower limit in magnitude at a given redshift is seen, expected
since radio-loud AGN tend to be the most massive galaxies at any
redshift; the very few sources with an apparent magnitude too bright
for their redshift are likely to be due to erroneously high
photometric redshifts, but these are too small in number to
significantly affect our analyis. We also note a small population of
objects that are very faint in $r$, due presumably to SDSS photometric
errors in the $r$-band -- most of these objects have more reasonable
magnitudes in other SDSS bands.
\begin{figure*}
\includegraphics[width=0.48\linewidth]{zdist.pdf}
\includegraphics[width=0.48\linewidth]{Lz.pdf}
\caption{Left: redshift and right: radio luminosity distributions for the
optically identified galaxy catalogue and the subsample with
spectroscopic redshifts. Redshift is
plotted on a $\log(1+z)$ scale but labelled with $z$ values.}
\label{fig:zdist}
\end{figure*}
Fig. \ref{fig:zdist} shows the distribution of spectroscopic and
photometric redshifts in the galaxy sample, and the corresponding
radio luminosities (where we use a single spectral index of $\alpha =
0.7$ for K-correction). We see that the radio luminosities of the
optically identified sample span the range from $10^{21}$ W Hz$^{-1}$
(where we would expect star formation to be the dominant process)
through to well above $10^{26}$ W Hz$^{-1}$ (the nominal FRI/FRII
break luminosity at 150 MHz) even for the spectroscopic subsample. The
wide area and high sensitivity provided by LOFAR coupled with the
availability of spectroscopy for a large number of faint galaxies in
SDSS DR12 drives the wide range in radio luminosity that we observe.
\section{Initial science results}
\label{sec:science}
In this section we discuss some scientific conclusions that can
easily be drawn from the various catalogues that we have constructed.
Detailed analyses of all these topics will be presented in later papers.
\subsection{Source counts}
\label{sec:counts}
The associated catalogue allows us to construct the standard
Euclidean-normalized differential source counts plot for the LOFAR
sample, and this is shown in Fig.\ \ref{fig:sourcecounts}. For
comparison at the bright end, we plot the 6C 151-MHz source counts of
\cite{Hales+88}. There is excellent agreement between the
normalization and slope of the 6C and LOFAR data where they overlap,
given the Poisson uncertainties on numbers of sources at the bright
end in the LOFAR data. Our source counts are corrected for
completeness (Section \ref{sec:completeness}) and of course take
account of physical associations between objects in the original
catalogue, but are not corrected for any other effects. W16, in their
similar but higher-resolution study, suggest that resolution bias,
i.e. the fact that resolved sources are less likely to be detected,
affects the counts significantly below a few mJy, where the SNR is
low, and this can be seen affecting the sub-mJy flux counts in the
comparison of their results with ours in Fig.\ \ref{fig:sourcecounts}; more detailed completeness simulations taking into account the
intrinsic distribution of source sizes would be necessary to have
confidence in the source counts at the very faint end of this plot.
Elsewhere our results are close to, but generally slightly above,
those of W16, which may be a result of our different approach to
completeness corrections.
\begin{figure}
\includegraphics[width=1.0\linewidth]{sourcecounts_areafn_v2.pdf}
\caption{Euclidean-normalized differential source counts from the
LOFAR associated catalogue after completeness correction.
Overplotted are the 6C counts from \protect\cite{Hales+88},
the LOFAR counts from W16, the
counts for the identified catalogue, and the counts for objects
classed as star-forming (see Section \ref{sec:sfg}) with the
corresponding counts for SFGs from the simulations of
\protect\cite{Wilman+08}. For clarity the very few points on the
simulated SFG line with total flux density greater than 0.25 Jy are not
plotted. Error bars are suppressed when there is only one count in
the corresponding bin.}
\label{fig:sourcecounts}
\end{figure}
\subsection{Cross-match with H-ATLAS}
\label{sec:herschel}
The H-ATLAS project produces maps and catalogues following the methods
described by \mbox{\cite{Pascale+11}} (SPIRE mapping), \cite{Ibar+10}
(PACS mapping) and \cite{Rigby+11} (cataloguing). An up-to-date
description of the process for the public data, shortly to be
released, will be provided by Valiante {et al}\/.\ (in prep.) and
descriptions of the NGP maps and catalogues will be provided by Smith
{et al}\/.\ (in prep.) and Maddox {et al}\/.\ (in prep.) respectively.
The currently available H-ATLAS catalogue of the NGP field contains
539,757 sources detected at approximately $2\sigma$ significance, of
which 443,500 overlap with the LOFAR images. For the purposes of flux
comparisons we restrict ourselves to sources with 250-$\mu$m
signal-to-noise (taking account of confusion noise) $> 4.0$, of which
there are 94,008 in the LOFAR field; this is a similar significance
level to the cut that will be applied in the forthcoming NGP data
release, and implies a typical 250-$\mu$m flux density limit of around
30 mJy. Clearly only a small fraction of these {\it Herschel} sources
are detected with LOFAR. We cross-matched on both LOFAR positions and
the positions of optical identifications, using the same
maximum-likelihood crossmatch as described above for radio catalogue
matches, with a maximum permitted offset of 8 arcsec. To do this we
take the error on {\it Herschel} positions to go as $1/SNR$,
normalizing to a positional error of 2.4 arcsec for a SNR of 4.5 based
on the results of Bourne {et al}\/.\ (in prep.) on the optical
crossmatching to the Phase 1 H-ATLAS data release. We find 2,994
matches to LOFAR positions and 1,957 matches to optical positions ---
the latter being more reliable as the optical positions are better
determined, but representing a smaller number of LOFAR objects as not
all have optical IDs. A flux-flux plot (Fig. \ref{fig:herschel}) shows
the expected two branches, one where there is a good correlation
between the radio and {\it Herschel} flux densities, and one where
there is none, representing respectively star-forming galaxies and
radio-loud AGN (some, but not all, of which will be detected in the
H-ATLAS images due to their star-formation activity). The flux-flux
relationship for the {\it detected} star-forming objects appears
approximately linear and could be represented by $S_{\rm 250\ \mu m}
\approx 20 S_{\rm 150\ MHz}$, as shown on Fig.\ \ref{fig:herschel};
such a relationship is consistent with the $z=0$ radio/far-infrared
(FIR) correlation observed at 1.4 GHz for sources detected in both
bands \citep{Jarvis+10,Smith+14} where the parameter $q_{250} =
\log_{10}(S_{\rm250\ \mu m}/S_{\rm 1.4\ GHz}) \approx 2.0$, assuming a
spectral index of 0.7 for these objects.
The LOFAR detection fraction (Fig.\ \ref{fig:herschel}) is low for all
{\it Herschel} flux densities after the brightest ones, but certainly
lower for the fainter objects, as would be expected given the
flux-flux relationship and the fact that the sensitivity of the LOFAR
images is not constant across the sky. It is interesting to ask
whether such an explanation {\it quantitatively} predicts the
detection fraction, which we can do if we assume that the flux-flux
relationship estimated above holds good for all H-ATLAS sources. We
can then use the LOFAR completeness curve to estimate which of the
H-ATLAS sources should have been detected in the LOFAR band. In fact
(Fig.\ \ref{fig:herschel}) we would expect to detect many more sources
(the simulations show this number to be around 12,000) than we
actually do if $S_{\rm 150\ MHz}$ were equal to $0.05\, S_{\rm
250\ \mu m}$ for all {\it Herschel} sources. While the flux-flux
correlation we see in the data must be correct for the brightest
sources (we would be able to detect sources with, for example,
250-$\mu$m flux densities at the Jy level and mJy-level LOFAR flux
densities, but none exist) the true flux-flux relationship for the
bulk of {\it Herschel} sources needs to be at least a factor 2 below
the naive estimate derived from the correlation seen for the brightest
sources in order to come close to reproducing the actual detection
statistics. This is again consistent with the results of
\cite{Smith+14}, who showed that stacking radio luminosities including
sources not detected in the radio gave rise to $q_{250}$ values $\sim
2.5$. The implications here are important: even without analysing the
luminosity distribution, we can see that radio/FIR relations derived
from samples flux-limited in both radio and FIR are likely to be
strongly biased unless non-detections are taken into account, with
implications for the radio emission expected to be seen from
star-forming objects in the distant Universe. Here we do not speculate
whether this bias arising from the combined radio-FIR selection is due
to true radio deficiency in some star-forming galaxies or to other
effects such as the differing dust temperatures of objects selected at
250 $\mu$m \citep{Smith+14}. Later papers (G\"urkan et al in prep.;
Read et al in prep.) will discuss the relationship between radio
emission and star formation in more detail.
\begin{figure*}
\includegraphics[width=0.48\linewidth]{hfluxflux.pdf}
\includegraphics[width=0.48\linewidth]{dethist.pdf}
\caption{Results of cross-matching {\it Herschel} and {\it LOFAR}
sources. Left: LOFAR/H-ATLAS flux density plot showing positional
crossmatches to the LOFAR positions (red) and positions of optical
counterparts (blue). Right: detection histogram, showing the
distribution of 250-$\mu$m flux densities in the whole area-matched
sample, the simulated number of detections if $S_{\rm
150\ MHz}$ were $0.05\, S_{\rm 250\ \mu m}$ for all sources, the distribution of flux densities of sources matched to
LOFAR positions and the distribution for sources
matched to SDSS sources. Note the logarithmic axes: at
$S_{250} < 0.1$~Jy fewer than ten per cent of the {\it Herschel}
sources are detected by LOFAR.}
\label{fig:herschel}
\end{figure*}
\subsection{AGN and star formation in the optically identified sample}
\label{sec:sfg}
\begin{figure}
\includegraphics[width=\linewidth]{llir-v2.pdf}
\caption{250-$\mu$m far-infrared luminosity, estimated as described
in the text, as a function of LOFAR radio luminosity. Objects are
colour coded depending on whether they have temperature
measurements, are detected at the $2\sigma$ level at 250 $\mu$m but without a
valid temperature measurement, or are $2\sigma$ upper limits at
250 $\mu$m. The
division into detected and non-detected sources at this level is
for plotting purposes only and plays no part in the analysis.
Solid lines show $q'_{250} = 20$ and $q'_{250} = 5$, where
$q'_{250}$ is as defined in the text.}
\label{fig:llirplt}
\end{figure}
We made use of the {\it Herschel} data to separate AGN and star
formation in the optically identified sample. To do this we measured
{\it Herschel} flux densities from all five bands directly from the
H-ATLAS maps at the positions of all optical identifications with
redshifts in the manner described by \cite{Hardcastle+13}. We then
fitted modified black-body models with $\beta=1.8$ [the best-ftting
value derived by \cite{Hardcastle+13} and \cite{Smith+13}] to all
objects with a $2\sigma$ detection in more than one {\it Herschel}
band, accepting fits with good $\chi^2$ and well-constrained
temperature, in the manner described by \cite{Hardcastle+13}. This
process gives us 1,434 dust temperatures and luminosities, with a mean
dust temperature of 24.5 K. For the remaining objects, we estimate the
250-$\mu$m IR luminosity, $L_{250}$, from the 250-$\mu$m flux density
alone, K-correcting using $\beta = 1.8$ and $T = 25$ K; we calculate a
luminosity in this way for all objects, including non-detections. The
temperature and $\beta$ parameters are only used here to provide a
K-correction at 250 $\mu$m, rather than to calculate an integrated
luminosity, and so the effects on the data should be very limited at
the low redshifts of the majority of objects in our sample. The
resulting radio-FIR luminosity plot is shown in
Fig.\ \ref{fig:llirplt}. A clear sequence of the radio-FIR correlation
can be seen, driven mostly by detected objects, as expected given the
results of the previous subsection; the correlation may be slightly
non-linear but at low luminosities/redshifts is broadly consistent
with a constant ratio of about a factor 20 between the two
luminosities. [It would not be surprising to see some non-linearity
given the dependence of the radio-FIR correlation on dust
temperature discussed by \cite{Smith+14}: once again, we defer
detailed discussion of the radio/FIR correlation to G\"urkan et
al.\ (in prep.).] Radio-loud AGN lie to the right of this
correlation, i.e. they have an excess in radio emission for a given
FIR luminosity. The scattering of points at high luminosities comes
from the high-$z$ quasar population, where the K-corrections almost
certainly break down to a large extent and where there may be some
contamination of the FIR from synchrotron emission.
\begin{figure*}
\includegraphics[width=0.32\linewidth]{asf_z.pdf}
\includegraphics[width=0.32\linewidth]{asf_petroMag_r.pdf}
\includegraphics[width=0.32\linewidth]{asf_L_150.pdf}
\caption{Distributions of key quantities for the whole sample and
after AGN/star-formation separation as described in the text. In
each histogram white shows the distribution of the parent sample,
red star-forming objects, and blue AGN, with the overlapping
regions of the two coloured histograms appearing in purple.}
\label{fig:twopop}
\end{figure*}
To make a quantitative separation between the two classes of object we
define the quantity $q'_{250} = L_{250} / L_{150}$ -- we take the
ratio here rather than its log, as is more conventional, to allow for
the negative values of $L_{250}$ which may be assigned to {\it Herschel}
non-detections. We use the value of $L_{250}$ derived from temperature
fitting where available and from the 250-$\mu$m flux density otherwise. Then
we take a source to be an AGN if $q'_{250}<5$, and a star-forming
object otherwise (the division being indicated by a line on
Fig.\ \ref{fig:llirplt}). By this classification, 3,900 of the objects
with redshifts are AGN and the remaining 1,667 are star-forming
galaxies (SFGs). Consistent with expectation, these two populations
have very different distributions in redshift, galaxy magnitude and
150-MHz luminosity (Fig.\ \ref{fig:twopop}). The dividing line used
here is, of course, arbitrary, though it is chosen so as to isolate
the radio/FIR relation at low luminosities. We do not expect a clear
separation between the two classes in $q_{250}$ since radio-loud AGN
may occur in strongly star-forming galaxies. However, we checked the
classification by testing what fraction of sources in the two classes
are morphologically complex, using as a proxy for this multi-component
sources with a maximum component separation of $>20$ arcsec (to avoid
sources that are only moderately resolved by LOFAR). We find that of
the 275 such sources, all but 4 are in the AGN class, and of the four
extended objects classed as SFGs, 3 are genuinely extended very nearby
galaxies; only one is a clear double which should be classified as an
AGN, and that turns out to be one of the quasars that contaminate the
high-luminosity end of Fig.\ \ref{fig:llirplt}, 14 of which have
$q'_{250}$ above the SF threshold. These objects are easily excluded
from our SF catalogue and, apart from them, we do not appear to be
including in the SF class any significant number of double AGN,
suggesting, at least, that the SFG class is not strongly contaminated
by AGN. The fraction of morphologically complex sources increases
immediately below $q'_{250} = 5$, consistent with the idea that this
is a useful dividing line.
The source counts of objects classed as SFGs
(Fig.\ \ref{fig:sourcecounts}) show good agreement with the SKADS
model counts of \cite{Wilman+08} in both normalization at the lowest
flux densities and slope (i.e. flat when Euclidean-normalized): the
only difference is that we lack the sky area to find extremely bright
SFGs, and that we find slightly higher numbers of SFGs at flux
densities of 2--4 mJy. This suggests that we are correctly classifying
the vast majority of both SFGs and AGN. Misclassification of SFGs
would lead to inconsistencies in normalization; contamination of the
SFGs with AGN would lead to inconsistencies in slope. Residual
differences may be due to cosmic variance -- our sky area is
considerably smaller than that simulated by \cite{Wilman+08}. Our
results contrast with those of \cite{Simpson+12} and
\cite{Lindsay+14}, who both found a deficit of faint objects at low
$z$ at 1.4~GHz compared to the SKADS models; it is possible that this
is evidence that LOFAR's short baselines allow it to pick up a
population of low-z SF sources resolved out by high-resolution VLA
surveys. We note (Fig.\ \ref{fig:twopop}) that we continue to find
objects classed as SF up to the highest redshifts in our sample, and
up to radio luminosities of $10^{25}$ W Hz$^{-1}$; these objects must,
if correctly classified, be strongly star-forming galaxies with SFR of
hundreds of solar masses per year.
We conclude that {\it Herschel} data, where available, offer a
reliable and simple method of carrying out AGN/star-formation
separation in LOFAR data at Tier 1 depth.
\subsection{Luminosity functions}
\label{sec:lf}
We construct the low-redshift 150-MHz luminosity function from sources
with $r<19$ mag, excluding quasars. Below this limit, 1809 of our 1917
candidate identifications (95 per cent) have redshifts (1190
spectroscopic) and so we are able to construct a luminosity function
without much normalization uncertainty. 1017 of the 1809 are classed
as SFGs by the $q'_{250}$ criterion, the rest are AGN. We drop at this
point 2 AGN with photometric redshifts that are clear outliers on the
$r$-$z$ plot (Fig.\ \ref{fig:magdist}) leaving a sample of 1017 SFGs
and 790 AGN with a maximum redshift just over $0.4$. This large
redshift range means that we may be somewhat affected by cosmological
evolution; the median SFG redshift is 0.12 and for AGN it is 0.24. We
return to this point below. We expect that there are very few
unidentified radio sources which should in fact be identified with
$r<19$ galaxies, setting aside small gaps in the DR12 photometric
catalogues around bright stars and the like, so that a luminosity
function with these constraints should be representative of the true
source population.
In order to calculate the luminosity function we must deal with the
effects of K-correction in the optical. We first of all calculated
absolute $g$ and $r$ magnitudes for our targets using the methods of
\cite{Chilingarian+10}, correcting for an average Galactic reddening
using {\sc astroquery} to query the IRSA Dust Extinction
Service\footnote{\url{http://irsa.ipac.caltech.edu/applications/DUST/index.html}}
and retrieve dust extinction calculated according to
\cite{Schlafly+Finkbeiner11}. A colour-magnitude diagram constructed
for the $r<19$ objects shows a good separation into red sequence and
blue cloud largely dominated by AGN and star-forming objects
respectively, as expected.
\begin{figure*}
\includegraphics[width=0.48\linewidth]{lf.pdf}
\includegraphics[width=0.48\linewidth]{lf_sf_z.pdf}
\caption{Left: the 150-MHz luminosity function in the H-ATLAS field
dividing the sources into SFG and AGN as described in the text.
Solid lines show the LOFAR-derived luminosity function: light
points with error bars
show the 1.4-GHz luminosity function from
\protect\cite{Mauch+Sadler07} and the 325-MHz luminosity function
of \protect\cite{Prescott+16}, scaled assuming a constant
$\alpha =0.7$. Error bars are Poissonian from number counts in
bins only. Right: the same plot, but for the SFG only, dividing
into three redshift bins.}
\label{fig:lf}
\end{figure*}
We then computed the luminosity function in the standard way, i.e. by
binning $1/V_{\rm max}$, where $V_{\rm max}$ is the maximum volume out
to which a source can be seen given the radio and optical limits. We
calculate $V_{\rm max}$ separately for optical and radio and adopt the
smaller of the two. For the radio, $V_{\rm max}$ is calculated as
$\int d_{\rm max} \mathrm{d}A$, where we use the completeness function
described in Section \ref{sec:completeness} to compute the area over
which sources can be seen out to a given depth. For the optical, the
survey is assumed to be uniform, but we invert the approximations of
\cite{Chilingarian+10} to derive K-corrections as a function of
redshift and intrinsic ($M_g - M_r$) colour.
The results for the whole sample are shown in Fig.\ \ref{fig:lf} (left
panel). For AGN, we see overall good agreement with a scaled ($\alpha
= 0.7$) version of the luminosity functions of \cite{Mauch+Sadler07}
at 1.4 GHz or \cite{Prescott+16} at 325 MHz, implying little variation
in the spectral index as a function of radio luminosity. However, the
luminosity function for SFG clearly has an excess with respect to the
literature catalogues at higher luminosities. We attribute this to
redshift evolution of the SFG population. This can clearly be seen in
the right-hand panel of Fig.\ \ref{fig:lf} where we plot the SFG
luminosity function only, broken down into redshift bins. A strong
positive luminosity function evolution with redshift is evident, and
the lowest-redshift luminosity function is now in excellent agreement
with that of \cite{Mauch+Sadler07}. By contrast, we have verified that
the AGN luminosity function shows no significant variation out to
$z=0.4$, the limit of our magnitude-limited sample, presumably because
the AGN in our sample are largely low-excitation radio galaxies which
are expected to show only weak cosmological evolution
\citep{Best+14,Pracy+16}. A future paper (Williams {et al}\/.\ 2016b) will
discuss the luminosity function evolution for radio-loud AGN, both
low-excitation and high-excitation using the deeper optical data in
the Bo\"otes field.
The strong radio luminosity function evolution we see for SFGs is
striking. We naturally expect some evolution given the known overall
evolution of the star-formation density of the universe
\citep[e.g.,][]{Madau+96,Hopkins+Beacom06}. In wide-area surveys hints
of positive evolution have been seen for some time
\citep{Machalski+Godlowski00,Condon+02}. However, most work in this
area has focussed on deep fields, and this has shown that the
luminosity function \citep[e.g.][]{Haarsma+00,Smolcic+09,McAlpine+13},
the specific star-formation rate estimated from the radio
\citep[e.g.][]{Karim+11,Zwart+14} and the total radio-estimated
star-formation rate density \citep[e.g][]{Seymour+08} all evolve
positively in the redshift range $z=0$ -- 1. What is unusual about our
sample, other than the fact that it is calculated at 150 MHz, is that
it has the area to see this evolution directly at low redshift,
coupled with the ability of the H-ATLAS data to allow AGN/SF
separation over such a wide area. Although the error bars are large,
Fig.\ \ref{fig:lf} implies that pure luminosity evolution has the form
$\sim (1+z)^5$, which is steeper than the $\sim (1+z)^{2.5}$ found in
most earlier work on the radio luminosity, suggesting either some
difference in our selection or a real change in the redshift
dependence at low radio luminosity and $z$. The corresponding positive
evolution at low $z$ in the far-IR is relatively well known
\citep{Dye+10,Magnelli+13,Gruppioni+13} and seems to imply a similarly
strong evolution with redshift \citep{Dye+10}, but, unlike the far-IR
where dust mass evolution may also be implicated \citep{Dunne+11}, the
radio data -- if contaminating AGN can be removed -- provide an
unambiguous tracer of star formation evolution comparable to the
ultraviolet or H$\alpha$. More optical identifications and
spectroscopic redshifts for objects in the NGP field, and LOFAR
observations of the equatorial H-ATLAS fields, will enable us to
investigate this evolution of the low-frequency luminosity function to
higher redshift in future, and to compare to the results at 1.4 GHz
and to the evolution of other star formation tracers \citep[see,
e.g.,][]{Mancuso+15}.
\vspace{10pt}
\subsection{The power/linear-size diagram and the incidence of giant sources}
\label{sec:pdd}
\begin{figure}
\includegraphics[width=1.0\linewidth]{pdd.pdf}
\caption{The power/linear-size diagram for AGN in our survey and for
3CRR objects. For the LOFAR AGN, we plot separately sources with
angular sizes smaller than 15 arcsec as measured by {\sc pybdsm},
where the physical sizes should probably be considered upper
limits; sources with larger sizes, probably at least somewhat
meaningfully measured by {\sc pybdsm}: and composite sources, whose
largest component separation is used as a proxy for their size.
3CRR sizes are all measured from high-resolution radio maps. Boxes
indicate the region where some or all LOFAR sources in the range
$z=0$ to $z=0.8$ with uniform surface brightness would drop below
the detection threshold for our images. The vertical dashed line
indicates our giant-radio-galaxy selection criterion (see the text
for details). The grey curves represent tracks in the diagram for
a source with a jet power of $10^{38}$ W, in various environments,
derived from the modelling of \protect\cite{Hardcastle+Krause14}.}
\label{fig:pdd}
\end{figure}
The radio power-linear size plot or $P$-$D$ diagram for radio-loud
AGN, introduced by \cite{Baldwin82}, is an important diagnostic of
radio galaxy evolution. A new-born radio source will start at $P=0$,
$D=0$ and (barring strong interactions with the external medium) is
expected to have ${\rm d}D/{\rm d}t > 0$ throughout its active
lifetime, as the expansion of the source is driven by the ram pressure
at the head of the jets: thus linear size is an important proxy of
age, though the relationship between the two is determined by the
source environment. The radio power is a function of the energy
density in electrons and magnetic field in the lobes and of their
volume, and so has a more complex relationship with source age,
particularly when the effects of radiative losses are taken into
account. Theoretical or numerical models of radio galaxy evolution
(e.g., \citealt{Kaiser+97}; \citealt{Blundell+99};
\citealt{Manolakou+Kirk02}; \citealt{Hardcastle+Krause13}; \citealt{Turner+Shabala15})
predict tracks in the $P$-$D$ diagram for individual sources,
depending on source environment and jet power; survey observations
provide an instantaneous snapshot of the positions of many sources on
their $P$-$D$ tracks. Observations of large samples can constrain
models directly if they contain sources that are extreme with respect
to the predicted tracks, e.g. very large or very powerful objects;
more importantly, we may hope in future to use observed
(multifrequency) radio power and size in large samples to infer
properties such as jet power, age and environment on a per-source
basis from the theoretical predictions. Inferring these properties for
the large samples of radio-loud AGN expected to be generated by
next-generation radio surveys, including those with LOFAR, will provide crucial
input into our understanding of the `feedback' processes believed to
control the evolution of the most massive galaxies, as discussed in
Section \ref{sec:intro}.
To construct a $P$-$D$ diagram clearly estimates of both $P$ and $D$
are necessary, i.e. in observational terms we need measurements of
radio flux density, largest angular size and redshift. We have
estimates of the angular sizes of our sources from the cataloguing
process, but these need to be treated with caution for several
reasons. As discussed in Section \ref{sec:completeness}, the
deconvolved major axes estimated by {\sc pybdsm} for objects fitted
with a single Gaussian tend to be overestimates, as small residual
phase errors or offsets between the different spectral windows will
make a source that is really unresolved to LOFAR appear marginally
resolved in the images. For this reason, we consider all deconvolved
sizes $<15$ arcsec to be unreliable, where the threshold is chosen
based on the distribution of apparent source sizes and on visual
inspection of the images. For sources where {\sc pybdsm} associates
more than one component, the size estimates are probably slightly more
reliable, but these are a minority. For sources associated by us, we
record the largest angular separation between the {\sc pybdsm}
positions of any pair of components, but this is only a crude estimate
of the true angular size, being most reliable for edge-brightened
FRII-type sources. All of these automatically generated sizes would
benefit from verification by human inspection. Nevertheless they
provide an interesting starting point for consideration of the sample
power/linear-size plot. This is shown for the sources classed as AGN
in Fig.\ \ref{fig:pdd}. We overlay on this plot the equivalent values
for the 3CRR sample\footnote{Data from
\url{http://3crr.extragalactic.info/}.} \citep{Laing+83}, which,
with its flux density limit of 10.9 Jy at 178 MHz, represents the most
luminous radio AGN in the Universe at any particular redshift; it can
be seen that there is significant overlap between the two,
unsurprising since a number of 3CRR sources are present in our survey,
but that, also as expected, the LOFAR survey picks up many more
low-luminosity AGN. We also overlay, for illustrative purposes only,
the theoretical tracks for a source with a jet power of $10^{38}$ W in
various different environments for sizes between 10 and 600 kpc,
taking account of radiative losses, derived from the MHD simulations
of \cite{Hardcastle+Krause14}, which, if accurate, indicate that the
most luminous large sources seen in the LOFAR survey have jet powers
around the value simulated in that work; however, some of the
luminous, compact sources we see in the LOFAR surveys with $D \la 20$
kpc, $P \approx 10^{27}$ W Hz$^{-1}$ may well be young sources with
significantly higher jet power that will eventually evolve, if their
jets remain active, to 3CRR-like luminosities of $10^{28}$ --
$10^{29}$ W Hz$^{-1}$. Thus we see the potential of the LOFAR data to
allow us to construct a true jet kinetic luminosity function over
several orders of magnitude in jet power, something we expect to
return to in future papers.
Also plotted in Fig.\ \ref{fig:pdd} are the expected regions where
sources cannot be detected, given the surface brightness limits at
full resolution and sensitivity, and on the assumption of uniform
source surface brightness, considering the redshift range 0--0.8 in
which most of our sources lie. We see that we are capable of detecting
(and do detect) sources with $\sim 100$-kpc sizes down to almost the
lowest radio luminosities at which we detect AGN, but we expect to be
significantly biased against low-luminosity large sources because of
our surface brightness limitations. At high radio luminosities, we
would expect to be able to detect all but the most extreme giant
radio galaxies, where we adopt the standard definition in which the
projected linear size of a giant is $>1$ Mpc. Giant radio galaxies are
of particular interest because they represent one of the extremes in
$P$-$D$ space: they must be particularly long-lived sources and their
very existence places constraints on models of e.g. the possible
active lifetime of jets. For this reason a number of searches for
giant sources in existing low-frequency surveys have been carried out
\citep[e.g.][]{Cotter+96,Lara+01,Machalski+01,Schoenmakers+01}. Such
searches have generally used large {\it angular} size as a proxy for
giant status, and then followed up optically and/or with spectroscopy
to identify physically large sources, thus favouring low-redshift
giants. For example, \cite{Lara+01} find a sky density of 1 giant per
$\sim 300$ square degrees, with an initial selection criterion of an
angular size exceeding 4 arcmin in NVSS.
Our optically identified sample and the fact that we are capable of
detecting powerful giants allows a direct approach to the problem. As
Fig.\ \ref{fig:pdd} shows, we do detect 7 sources with sizes $\sim 1$
Mpc: in counting these we use a selection at 900 kpc to allow for the
fact that the angular size values from component association are
generally slightly underestimated, since they are the separations
between the centres of the associated components, not their edges. Of
these 7, all but one seem likely to be bona fide powerful giants (see
Appendix \ref{app:tmbg}), implying a density of such $P \ga 3\times
10^{26}$ W Hz$^{-1}$ sources on the sky of at least 1 per 20 square
degrees. These are, of course, only the optically identified giants,
and we would expect to be biased against optical IDs of luminous
sources, which will tend to be at high redshift, as well as of large
sources, which are inherently difficult to identify. Our smallest (in
angular size) giant is 140 arcsec in length, and there are a further
10 composite sources with sizes $>2$~arcmin in the associated
catalogue, many of which may be high-$z$ giants. Although the numbers
are small, these are substantially higher sky densities than were
found by \cite{Lara+01}, and suggest that the Tier 1 surveys will be a
fruitful hunting ground for giant sources.
The surface brightness limitations in these full-resolution
observations suggest that it would be useful to re-image the
facet-calibrated data at low resolution (20--30 arcsec) to allow a
search for low-surface brightness sources: \cite{Saripalli+12} have
found a high detection rate of relatively low-luminosity large sources in a
small sky area with good surface brightness sensitivity. As noted above,
measurements of the numbers of giants as a function of radio
luminosity and redshift provide important constraints on models of
radio source evolution, and we plan to revisit the implications of the
population of large sources in the LOFAR surveys in a future paper.
\subsection{Remnant AGN}
\label{sec:remnant}
One of the key uncertainties in AGN evolution models is what happens
when the jets are switched off. At this point, about half the energy
that has ever been transported up the jets remains in the lobes, at
least for powerful double objects \citep{Hardcastle+Krause13} and so
the question of whether, and where, that energy is transferred to the
external medium is one of great interest. However, the detection of
sources in the post-switch-off phase, so-called remnant or relic AGN,
has been surprisingly difficult. There are some well-known objects
that appear to have no current AGN activity, for example B2 0924+30
\citep{Cordey87} or 3C\,319 \citep{Hardcastle+97}, showing no
flat-spectrum arcsec-scale core (the self-absorbed base of a currently
active jet) and, where data are available, no AGN activity at any
other waveband. But such objects are rare \citep{Giovannini+88},
making up no more than 7 per cent of the low-frequency classical
double (FRII) population selected from 3CRR at $z<1.0$, for example
\citep{Mullin+08}, though this fraction may be environment-dependent
\citep{Murgia+11}. In fact true remnants, where AGN activity has
completely ceased, seem to be somewhat rarer than double-double or
restarting radio galaxies \citep{Schoenmakers+00,Saripalli+12}
despite the fact that double-doubles should be a fairly short-lived
phenomenon as the newly active lobes will merge into the pre-existing
plasma \citep{Konar+13,Konar+Hardcastle13}, implying a very rapid
fading process for remnants (cf.\ \citealt{Kaiser+Cotter02}). To date,
however, statistical information on the remnant population has mainly
come from studies of bright flux-limited samples like 3CRR, and as
radio galaxies are expected to fade significantly as they age due to
the effects of adiabatic expansion and radiative losses, it is clear
that such samples may be biased against remnant sources.
It has long been suggested that remnant or relic AGN\footnote{Relic
AGN should not be confused with the `radio relics' found in clusters
of galaxies, whose origin is for the most part not directly in AGN
activity; we use the term `remnant', as adopted by
\cite{Brienza+16a}, to avoid this confusion.} would have steep
spectra \citep[e.g.][]{Parma+07,Murgia+11} and so would be detectable
in greater numbers in sensitive surveys of the low-frequency sky.
LOFAR should be extremely sensitive to remnant AGN, which are also
expected to be physically large. \cite{Brienza+16b} set out possible
methods for identifying remnant AGN in LOFAR fields, which include
spectral selection (i.e., looking for steep-spectrum sources),
morphological selection (looking for sources with little or no compact
structure) and what might be termed core selection (looking for
sources with no identifiable radio core). Each of these methods has
its advantages and disadvantages. \cite{Brienza+16b} show that
morphological selection appears to be more efficient in the Lockman
Hole field than spectral selection, for the criteria they use, and one
morphologically selected remnant in another LOFAR field has been
followed up in detail, confirming its remnant nature
\citep{Brienza+16a}. However, it is possible that remnants with
recently switched off AGN would be missed by both the morphological
and spectral selection methods; by contrast, all genuine remnant
sources, whatever their age, would be expected to have no nuclear jet
and so no arcsec-scale, flat-spectrum core. As an initial test for the
efficiency of remnant surveys using this core selection criterion at
Tier 1 depth and resolution, we have selected from the identified
catalogue all sources which are (1) bright (total flux density $>80$
mJy at 150 MHz), ensuring that the sample is flux-complete across the
survey region and that there is a good chance of seeing a core at high
frequency, (2) well resolved ($>40$ arcsec), and (3) classed as AGN on
the radio/FIR relation, for an initial visual search for remnants.
Of 127 such objects (after removing a few objects where it is doubtful
that they are truly extended), we can see no evidence for a currently
active core in the FIRST images in 38, a potential remnant fraction of
30 per cent. Examples of candidate remnant sources, together with some
comparable sources where a FIRST core is seen, are shown in Appendix
\ref{app:remnant}. We do not include in our remnant count any source where
FIRST emission is coincident with the optical ID, even if there is no
clear evidence of a point source; so the true remnant fraction could
be slightly higher than we quote above. Sources without a core in
FIRST are also less likely to be optically identified, hence again
biasing our estimate low. On the other hand, we do not exclude sources
on the basis of showing apparent compact hotspots in the FIRST images.
Even if truly compact, something we cannot really assess on the basis
of the FIRST images, such features may persist for more than a
light-travel time along the lobes after the jet turns off, and so do
not imply that the jet is still active. The fact that the fraction we
measure is higher than for 3CRR sources is consistent with the idea
that remnants might be more detectable in more sensitive surveys,
although it is clear that remnants do not dominate the LOFAR sky at
these flux density levels.
The main limitation on this conclusion is the fact that the FIRST
images are not particularly sensitive to cores. If we define the core
prominence as the ratio of core flux density at 1.4 GHz [the exact
frequency is unimportant since radio galaxy core spectra are flat up
to high frequencies: \citep[e.g.][]{Hardcastle+Looney08,Whittam+13}]
to total flux density at 150 MHz, then FIRST's $3\sigma$ upper limit
on core prominence for the faintest objects corresponds to $0.4/80 = 5
\times 10^{-3}$, while we know that the {\it median} core prominence
for 3CRR objects with detected cores is $\sim 3 \times 10^{-4}$
\citep{Mullin+08}. 3CRR objects are selected to be the brightest
low-frequency sources on the sky and would be expected to have
systematically low prominences, so this is an unfair comparison, but
clearly it is possible that even moderately faint radio cores are
escaping detection in our calculation of the remnant fraction above.
We therefore regard the remnant fraction we have derived above as an
upper limit. Even so, such a limit is interesting, as it requires the
typical fading timescale for remnants at 150 MHz to be at most $\sim
30$ per cent of the active time. Sensitive radio followup of remnant
candidates will be necessary to constrain the remnant fraction
further.
A power/linear-size diagram for the 80-mJy, 40-arcsec subsample
(Fig.\ \ref{fig:pdd-80}) shows that the remnant candidates do not
occupy any particularly special position with respect to the cored
sources, but there is a slight tendency for them to have lower radio
luminosity for a given size, in the sense that all remnant candidates
with sizes $>300$ kpc, and many of those below that size, lie at the
very lowest end of the radio luminosity distribution. This would
support the idea that remnants fade rapidly even at low frequencies
once the jets switch off. There is no apparent difference between the
in-band spectral indices of the two samples, but as noted above, these
are very unreliable, and some remnants by our definition would be
expected to have flat radio spectra anyway. We defer an investigation
of the LOFAR/NVSS spectral index for the bright identified sample to a
future paper, as this will require individual measurements from LOFAR
and NVSS maps for all sources.
\begin{figure}
\includegraphics[width=1.0\linewidth]{pdd-80mJy.pdf}
\caption{The power/linear size diagram for the bright, resolved
subsample discussed in the text.}
\label{fig:pdd-80}
\end{figure}
\section{Summary and conclusions}
We have presented details of the observations, data reduction and
quality assessment for a survey consisting of four LOFAR HBA pointings
in the H-ATLAS NGP area. Although this survey does not have uniform
noise, it is otherwise expected to be reasonably representative in
data quality of the much larger `Tier 1' survey currently being
carried out by the LOFAR Surveys Key Science project. Key points from
the discussion of the data reduction and quality assessment are as
follows:
\begin{itemize}
\item We have been able to image over 140 square degrees at 150 MHz
at a resolution of better than 10 arcsec and with rms noise in the
broad-band 150-MHz maps
ranging between 100 $\mu$Jy beam$^{-1}$ and $\sim 2$ mJy
beam$^{-1}$, thus covering almost all of the NGP field as surveyed
by H-ATLAS. The fact that this can be done in a total of 34 hours'
observing illustrates the capabilities of LOFAR for wide-area,
deep surveys.
\item The LOFAR flux scale remains problematic. The method we have
developed to calibrate flux densities in the field -- by
cross-matching a large number of objects to existing low-frequency
catalogues (Section \ref{sec:fluxcal}) -- works well so long as there
are enough objects in existing catalogues to make a statistical
comparison possible, but this is not the case for the whole sky.
\item Even after correcting as well as possible for the flux scale
issues, HBA in-band spectral indices are unreliable because of the
limited frequency range spanned by the data, to the extent that
these are unlikely to be useful for all but the brightest sources
(Section \ref{sec:inband}).
However, reasonable HBA/NVSS spectral index distributions are
obtained for compact sources (Section \ref{sec:outofband}).
\item Per-facet positional offsets introduced by the
self-calibration in facet calibration are small but can be
significant in poor-quality data. Perhaps more significant is the
effect that we take to be the residual blurring of sources by
inadequate phase calibration, leading to a loss of peak flux density and
incompleteness at low flux densities (Section
\ref{sec:completeness}). This may be exacerbated for us by the
image-plane combination of multiple spectral windows.
\end{itemize}
For data reduction of the Tier 1 surveys it would probably be
preferable to carry out our flux scaling method {\it before} facet
calibration, which would then allow the facet calibration to be run
using all bands simultaneously as described by vW16 while still having
correct flux densities. In-band spectral indices, if desired, could be derived
by re-imaging after facet calibration. It is not yet clear whether the
facet-based approach of vW16 is fundamentally limited in terms of the
calibration quality that can be achieved away from the calibration
point, i.e. whether some sort of phase screen interpolation or an
algorithm that can fit to much smaller facets will be required.
Nevertheless, the technique represents a significant advance towards
the exploitation of LOFAR's capabilities and in this paper we have
demonstrated that applying it to large areas is technically and
computationally feasible.
We have also constructed an optically identified catalogue using SDSS
galaxy catalogues and spectroscopy, and have used it to investigate
the science that can be done with a combination of LOFAR, SDSS and
{\it Herschel} data. Some key points from this analysis are as
follows:
\begin{itemize}
\item We achieve a roughly 40 per cent optical identification rate
for the LOFAR catalogue using SDSS together with FIRST to assist
with identifications (Section \ref{sec:optid}). This is the result
of a labour-intensive process in which the vast majority of
sources were visually inspected in several bands [a similar
approach for the Bo\"otes field will be described by Williams
{et al}\/.\ (2016b)]. Clearly if this
process is to be scaled up to the many hundreds of LOFAR pointings
already in hand in Tier 1, it will require significant automation.
On the other hand, the catalogues would benefit from further
visual inspection, for example to provide quality checks on large
sources found by {\sc pybdsm} and to measure flux density and
source size more accurately. Generating high-quality catalogues
from Tier 1 data will remain labour-intensive even if some of the
optical identification process can be streamlined.
\item It is important to note that there are many resolved sources
without optical IDs, and, as discussed in Section
\ref{sec:remnant}, many of these may not have radio cores in
FIRST. This may represent a challenge for spectroscopic followup
projects such as WEAVE-LOFAR \citep{Smith15} which rely on
accurate positions for their targets.
\item Comparing with the H-ATLAS images, we recover the well-known
radio/FIR relation in flux/flux and luminosity/luminosity plots
using the 250-$\mu$m {\it Herschel} data
(Sections \ref{sec:herschel}, \ref{sec:sfg}) but see evidence that
there is a population of radio-faint SFGs with radio flux densities well
below their expected values on this correlation, consistent with
earlier empirical work \citep{Smith+14}. On the other hand, we are
able to use the fact that sources with larger radio luminosities
than would be expected from their FIR emission must be radio-loud
AGN to perform an efficient SFG/AGN separation in the optically
identified catalogue. Radio source count analysis not only shows
consistency with existing 150-MHz determinations where available
(Section \ref{sec:counts}) but, for the SFGs, shows good agreement
(Section \ref{sec:sfg}) with the widely used models of
\cite{Wilman+08}.
\item We present a 150-MHz radio luminosity function derived from
$\sim 2,000$ objects with $r<19$ mag, which shows good agreement
with the expectations from higher frequencies (Section
\ref{sec:lf}). Strong luminosity function evolution with redshift
is seen for the SFGs.
\item The power/linear-size diagram for the overall sample (Section
\ref{sec:pdd}) shows that we are still insensitive to very large
sources at low radio luminosities, something which may need to be
addressed in Tier 1 data processing by an additional imaging step
at low resolution. However, we measure a sky density of genuine
optically identified powerful giant radio galaxies ($L_{\rm 150}
\ga 10^{26}$ W Hz$^{-1}$, $D \ga 1$ Mpc) which is
high compared to some estimates in the literature, thanks to our
good optical identification rate out to relatively high redshifts.
\item We carry out an initial search for candidate remnant sources,
where the jets have switched off, in a bright, resolved subsample
(Section \ref{sec:remnant}). Up to 30 per cent of the sample
sources show no FIRST core, which might imply a lifetime in the
remnant phase comparable to that in the active phase. However,
many of these remnant candidates may have radio cores below the
FIRST detection limit: sensitive high-frequency observations will
be necessary to refine the upper limit on remnants in LOFAR
samples.
\end{itemize}
Subsequent papers will address many of these points in more detail.
\section*{Acknowledgments}
We thank Shea Brown for providing the WSRT image of the Coma cluster,
V. N. Pandey for providing the model of 3C\,196, and
Michal Michalowski and Eduardo Ibar for helpful comments on the paper.
We thank the anonymous referee and scientific editor, whose
comments improved the paper's presentation.
MJH and WLW acknowledge support from the UK Science and Technology
Facilities Council [ST/M001008/1]. GG thanks the University of
Hertfordshire for a research studentship. PNB is grateful for support
from the UK STFC via grant ST/M001229/1. RM gratefully acknowledges
support from the European Research Council under the European Union's
Seventh Framework Programme (FP/2007-2013) ERC Advanced Grant
RADIOLIFE-320745. TWS acknowledges support from the ERC Advanced
Investigator programme NewClusters 321271. LD and SJM acknowledge
support from ERC Advanced and Consolidator grants Cosmic ISM and
Cosmic Dust. GJW gratefully acknowledges
support from the Leverhulme Trust.
This research has made use of the University of Hertfordshire
high-performance computing facility
(\url{http://stri-cluster.herts.ac.uk/}) and the LOFAR-UK computing
facility located at the University of Hertfordshire and supported by
STFC [ST/P000096/1]. This research
made use of {\sc Astropy}, a community-developed core Python package
for astronomy \citep{AstropyCollaboration13} hosted at
\url{http://www.astropy.org/}, of {\sc APLpy}, an open-source
astronomical plotting package for Python hosted at
\url{http://aplpy.github.com/}, and of {\sc topcat} and {\sc stilts} \citep{Taylor05}.
LOFAR, the Low Frequency Array designed and constructed by ASTRON, has
facilities in several countries, that are owned by various parties
(each with their own funding sources), and that are collectively
operated by the International LOFAR Telescope (ILT) foundation under a
joint scientific policy.
The {\it Herschel}-ATLAS is a project with {\it Herschel}, which is an ESA space
observatory with science instruments provided by European-led
Principal Investigator consortia and with important participation from
NASA. The H-ATLAS website is \url{http://www.h-atlas.org/}.
Funding for SDSS-III has been provided by the Alfred P. Sloan
Foundation, the Participating Institutions, the National Science
Foundation, and the U.S. Department of Energy Office of Science. The
SDSS-III web site is \url{http://www.sdss3.org/}.
SDSS-III is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS-III Collaboration including the
University of Arizona, the Brazilian Participation Group, Brookhaven
National Laboratory, Carnegie Mellon University, University of
Florida, the French Participation Group, the German Participation
Group, Harvard University, the Instituto de Astrofisica de Canarias,
the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins
University, Lawrence Berkeley National Laboratory, Max Planck
Institute for Astrophysics, Max Planck Institute for Extraterrestrial
Physics, New Mexico State University, New York University, Ohio State
University, Pennsylvania State University, University of Portsmouth,
Princeton University, the Spanish Participation Group, University of
Tokyo, University of Utah, Vanderbilt University, University of
Virginia, University of Washington, and Yale University.
The National Radio Astronomy Observatory (NRAO)
is a facility of the National Science Foundation operated under
cooperative agreement by Associated Universities, Inc.
\bibliographystyle{mnras}
\renewcommand{\refname}{REFERENCES}
|
1,941,325,220,577 | arxiv | \section{Introduction}
\label{sec:intro}
\vspace{-0.35cm}
Acoustic-to-articulatory inversion (AAI) \cite{C1} mainly solves the problem of deriving the pronunciation parameters of key organs from speech audio. In recent years, it has played an important role in many fields, such as pronunciation guidance \cite{C2} and speech recognition \cite{C4,C48,C50}, so it has attracted many researchers to devote themselves to this field.
Different deep learning based models and acoustic representations have been proposed to carry out the AAI task. In the early stage, codebook \cite{C7} was used for voice inversion, but the performance highly relied on the quality of codebook. Later, The data-driven voice inversion models were presented, such as hidden Markov model (HMM) \cite{C8,C49}, mixed Gaussian network (GMM) \cite{C38}, deep neural networks (DNNs) \cite{C39} and so on. At present, the most commonly used models are the recurrent neural network (RNN) and its variants, such as long-short term memory (LSTM) \cite{C14, C34}. In \cite{C14, C40, C41}, the different speech representations such as line spectral frequencies (LSF), Mel-frequency cepstral coefficients (MFCC) and filter bank energies (FBE) were used. In our work, we take MFCC and phonemes as the input of our model.
Up to now, there are two main challenges in AAI. One is that the available datasets are very limited, because we need to record the voice and pronunciation parameters at the same time, which is not only difficult to collect, but also expensive. The most commonly used public datasets are MOCHA-TIMIT \cite{C35}, MNGU0 \cite{C21}, and HPRC \cite{C36}. Another challenge is to improve the performance in speaker-independent scenarios.
\begin{figure*}[h]
\centerline{\includegraphics[width=0.90\linewidth]{ov.jpg}}
\vspace{-0.3cm}
\caption{The framework of SPN. The P is the phoneme stream feature and the S is the speech stream feature.}
\label{fig:ov}
\vspace{-0.4cm
\end{figure*}
For the first challenge, \cite{C44} proposed the method of using the cross corpus data to solve the problem of limited data volume. For the second challenge, \cite{C14} used the vocal tract normalization to map each speaker's pronunciation space to the same space, so as to improve the performance of that model, but it led to the loss of personalized information of speech. It is worth noting that a self-supervised pretraining model was proposed to solve the above challenges and achieved the best performance in \cite{C42}. However, this work only used MFCC as the network input which may limit the performance of AAI and used 1DCNNs to extract speech features which may result in the loss of global information of speech.
In order to solve the above two challenges, we propose a novel network that consists of two parts, speech stream network and phoneme stream network, which we call SPN in brief. 1DCNNs were used to extract speech features in \cite{C42}. But it was pointed out that CNN only extracts the local features in \cite{C45}, so we add an multi-head attention model to extract the global features to better represent the voice information. In addition, we propose a new phoneme stream network. More precisely, we use transcribed phonemes to perform phoneme inversion, then take the results of phoneme stream network as the phoneme features to perform voice inversion.
The motivation is that phonemes only represent the content information of the speech instead of the identity information. Therefore the phoneme features obtained by phoneme stream network are speaker-independent, which can improve the performance of the speaker-independent experiments.
In summary, there are three contributions of this work.
\begin{itemize}
\item In order to better represent voice information, we extract the local features and the global features through 1DCNNs and multi-head attention module respectively.
\item We propose a new phoneme stream network to gain the phoneme features to improve the performance of speaker-independent experiment.
\item Based on the experimental results, it is shown that the proposed model outperforms SOTA obviously on public HPRC dataset which decreases by 0.18mm on RMSE and increases by 6\% on PCC.
\end{itemize}
\vspace{-0.5cm}
\section{PROPOSED APPROACH}
\vspace{-0.25cm}
\label{sec:method}
\subsection{Overall Framework}
\vspace{-0.1cm}
As shown in Fig.~\ref{fig:ov}, the SPN we proposed is composed of two core modules, speech stream network and phoneme stream network. We get global and local features through speech stream network, then we feed them into the SAFN \cite{C42} network to obtain the speech features, and gain the phoneme features through phoneme stream network. Later, the integrated speech features and the phoneme features are fed to the articulatory inversion network (AIN) \cite{C42} to obtain the parameters of the key organs.
\subsection{Speech Stream Network}
\vspace{-0.1cm}
It was shown in \cite{C45} that CNN can only extract local features of speech, so to better represent voice information, a speech stream network is proposed. MFCC is fed into cascaded 1DCNNs and the multi-head attention module respectively to obtain the local features and the global features, which input into SAFN as speech features.
\begin{figure}[htb]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=8.5cm,height=4cm]{FFN2}}
\end{minipage}
\caption{ The workflow of the Speech Stream Network.}
\label{fig:FFN2}
\vspace{-0.4cm
\end{figure}
As shown in Fig.~\ref{fig:FFN2}, the speech stream network is mainly divided into two parts, one is the local feature extraction module, the other is the global feature extraction module. In the local feature extraction module, we choose the cascaded 1DCNNs used in \cite{C42}. We send the MFCC into the 1DCNNs with five different convolution kernels, whose sizes are 1, 3, 5, 7 and 9, respectively, to obtain the local features. For the global feature extraction module, we use the multi-head attention module because it can pay attention to the relationship between each frame and other frames of the speech. The attention module can be described as obtaining the outputs from a set of queries, keys and values, where outputs, queries, keys and values are the corresponding vectors. Specifically, the global feature extraction module includes six layers of multi-head attention. Each layer has eight heads in total and the dimension of keys and values is both 64. Then, we feed the features into a layerNorm layer to get the global features. Next, the local features and global features are fed into two fully connected layers which has 300 units in each layer to get the speech features. Finally we feed the speech features to the SAFN.
The computation of local features is formulated as:
\begin{equation}
\boldsymbol{y}_{i, j}^{\mathrm{local}}=b_{j}+\sum_{k=1}^{L_{i-1}} \mathbf{W}_{i} * \boldsymbol{y}_{i-1, k}^{\mathrm{local}}
\end{equation}
where * means the convolution operation and ${y}_{i, j}$ represents the feature map of $j$-th channel in $i$-th convolution layer, $b_{j}$ is the bias of $j$-th channel and $\mathbf{W}_{i}$ means the weights of $i$-th convolution layer, $L_{i-1}$ means the length of $\boldsymbol{y}_{i-1}^{\mathrm{local}}$.
\vspace{-0.3cm}
\subsection{Phoneme Stream Network}
\vspace{-0.1cm}
Inspired by the \cite{C43}, we use the outputs of phoneme stream network as the phoneme features to assist voice inversion. The reason for this is the following: phoneme itself only encodes the content information of the speech, instead of the identity information which is speaker-independent. We use the Penn phonetics lab forced aligner \cite{C46} to extract the phoneme frames. Each phoneme frame is a 39 dimensional one-hot vector.
In this module, we use three-layer BLSTM, and finally feed it to the two fully connected layers to get the pronunciation parameters as the phoneme features. The three-layer BLSTM has the same setting, which has 150 activation units in each layer, and there are 300 activation units for the fully connected layer. The output are the 12 dimensional pronunciation parameters, then we feed the pronunciation parameters as phoneme features and the speech features obtained from the speech stream network module into the AIN network to perform voice inversion. The core of the phoneme stream network is expressed as :
\begin{equation}
\setlength{\abovedisplayskip}{3pt}
s_{i} =f\left(U x_{i}+W s_{i-1}+b\right), \notag
\end{equation}
\begin{equation}
\setlength{\abovedisplayskip}{3pt}
s_{i}^{\prime} =f\left(U^{\prime} x_{i}+W^{\prime} s_{i+1}+b^{\prime}\right),
\end{equation}
\begin{equation}
o_{i} =g\left(V s_{i}+V^{\prime} s_{i}^{\prime}+c\right),\notag
\end{equation}
where $o_{i}$ is the phoneme features estimated at the frame $i$. ${x}_{i}$ is the input phoneme sequence at frame $i$, $s_{i}$ is the temporary state at frame $i$ and $U, W, U^{\prime}, W^{\prime}$ are the corresponding transformation matrices. $b, b^{\prime}$ are the biases.
\vspace{-0.35cm}
\section{EXPERIMENTS}
\vspace{-0.25cm}
\subsection{Experimental Setup}
\vspace{-0.1cm}
\textbf{Dataset.} In this work, the dataset we used is HPRC. There are six locations: T1, T2, T3, UL, LL and LI. To be consistent with the SOTA, we only use the locations of tongue organs in X and Z directions as our experimental predicted labels (\textit{e.g}., T1, T2, T3). The HPRC \cite{C36} dataset has eight native American English speakers (four males and four females), and each speaker data is made up of 720 utterances.
\textbf{Performance Metrics.} The performance of the experiment is measured by mean square error and Pearson correlation coefficient \cite{C11}. The first indicator measures the error between the predicted label and the real label and the second indicator represents the similarity of the two labels.
\textbf{Implementation Details.} The network based on speech decomposition and auxiliary feature proposed in \cite{C42} is the SOTA in AAI. Besides, the settings of SAFN and AIN modules in the network architecture diagram are the same as those of SOTA. In this work, we trained the model for 20 epochs and used the Adam optimizer. The learning rate is 1e-4, and the batch size is 5.
\vspace{-0.3cm}
\subsection{Comparisons with the SOTA}
To check the effectiveness of our proposed SPN network, we set three different experimental scenarios according to Table.~\ref{tab:TAB1}. S1 represents that we only train the phoneme stream network using phonemes as input to conduct speaker-independent experiments. S2 represents that we take phonemes and MFCC as inputs to conduct speaker-independent experiments. In this scenario, we take phoneme stream network as a pretraining model, and freeze the parameters while training the whole SPN. S3 represents speaker-independent experiments with phonemes and MFCC as network inputs. Unlike S2, we train phoneme stream network with the whole network. The loss function of our network is the weighted sum of the two parts which are the L2 loss of SPN and L2 loss of phoneme stream network. The formula is given as:
\begin{equation}
\setlength{\abovedisplayskip}{3pt}
\setlength{\belowdisplayskip}{3pt}
{L}_{joint} = \sum_{i=0}^{k}\left(y_{m}^{i}-\hat{y}_{m}^{i} \right)^{2} + \sum_{i=0}^{k}\left(y_{n}^{i}-\hat{y}_{n}^{i} \right)^{2},
\end{equation}
where $\hat{y}_{m}^{i}$, $\hat{y}_{n}^{i}$ represent the ground-truth of EMA. ${y}_{m}^{i}$, ${y}_{n}^{i}$ are the corresponding predicted labels. $k$ means the length of the speech feature.
More specifically, we calculated the performance of phoneme stream network and SPN at the same time in Scene 3. S3(P) represents the result of phoneme stream network and S3(S) represents the results of SPN.
It is worth noting that our experiment was conducted on the Haskins dataset and in each scenario, we trained a separate model for each speaker. We collect all the training set (80\%), validation set (20\%) from seven speakers' data and test on the left one speaker data (100\%). Finally, we take the average of the results of the eight speakers as the final results of our experiments.
\begin{table}[h]
\caption{ Experimental setup for 3 different scenarios. * means taking the proportion of data from each speaker. P and M represent phoneme and MFCC, respectively.
\label{tab:TAB1}
\centering
\begin{threeparttable}
\scriptsize
\setlength{\tabcolsep}{1.8mm}{
\begin{tabular}{ccccccc}
\toprule
\textbf{Scenarios}&\textbf{Input}&\textbf{\#Speaker}&\textbf{Train}&\textbf{Validation}&\textbf{Test}\\
\midrule
\multirow{2}{*}{\textbf{S1}}&\multirow{2}{*}{P}&N-1&80\%*&20\%*&-{}-{}-\\
&&1&-{}-{}-&-{}-{}-&100\%\\
\midrule
\multirow{2}{*}{\textbf{S2}}&\multirow{2}{*}{P, M}&N-1&80\%*&20\%*&-{}-{}-\\
&&1&-{}-{}-&-{}-{}-&100\%\\
\midrule
\multirow{2}{*}{\textbf{S3}}&\multirow{2}{*}{P, M}&N-1&80\%*&20\%*&-{}-{}-\\
&&1&-{}-{}-&-{}-{}-&100\%\\
\bottomrule
\end{tabular}}
\end{threeparttable}
\end{table}
\begin{table}[h]
\centering
\vspace{-0.4cm
\begin{threeparttable}
\caption{RMSE and PCC for SPN and SOTA
\label{tab:TAB3}
\centerin
\scriptsize
\begin{spacing}{0.85}
\setlength{\tabcolsep}{0.8mm}{
\begin{tabular}{ccccccccccc
\toprul
\textbf{Scenarios}&\textbf{F01}&\textbf{F02}&\textbf{F03}&\textbf{F04}&\textbf{M01}&\textbf{M02}&\textbf{M03}&\textbf{M04}&\textbf{RMSE}&\textbf{PCC} \\
\midrul
{\textbf{S1}}&2.279&2.182&1.681&2.322&1.753&2.546&2.036&1.898&2.087&0.755\\
\midrul
{\textbf{S2}}&2.720&2.740&2.088&2.759&2.352&3.064&2.395&2.528&2.580&0.798\\
\midrul
{\textbf{S3(P)}}&2.169&2.084&1.618&2.172&1.693&2.440&1.961&1.812&1.993&0.773\\
\midrul
{\textbf{S3(S)}}&2.634&2.665&2.100&2.602&2.235&3.195&2.368&2.504&2.537&0.810\\
\midrul
{\textbf{SOTA}}&-&-&-&-&-&-&-&-&2.721&0.751\\
\midrul
\end{tabular}}
\end{spacing}
\end{threeparttable}
\vspace{-0.5cm
\end{table}
\begin{figure}[h]
\centering
\includegraphics[width=0.8\linewidth]{draw1.png}
\caption{Comparisons of the predicted results of the tongue movement between SPN and SOTA. In each box, the blue line is real label, the red dashed line is predicted by SPN and the green dashed line is predicted by SOTA method.}
\label{fig:draw1}
\vspace{-0.4cm
\end{figure}
Table.~\ref{tab:TAB3} shows RMSE and PCC value in three scenarios in Haskins dataset. Because the previous SOTA works did not give the average RMSE on each speaker, we set the corresponding positions as '-' in Table.~\ref{tab:TAB3}. Basically, we can clearly observe that compared with the SOTA, our proposed SPN network decreases 0.141mm on RMSE and increases almost 5\% on PCC in scenario 2, decreases 0.184mm on RMSE and increases almost 6\% on PCC in scenario 3 which uses the joint-training strategy. It shows that the local features and global features we extracted can better represent speech information. After adding the phoneme features, we can further effectively improve the generalization ability of this model.
More interestingly, from the experimental results, we can see that in the case of scenario 3, the performance of phoneme stream network under joint training (S3(P)) is better than that training alone (S1), about 0.1mm on RMSE and almost 2\% on PCC. This shows that the performance of SPN and phoneme stream network both improved, indicating the effectiveness of joint-training strategy.
Qualitative comparisons between SPN and SOTA are shown in Fig.~\ref{fig:draw1}. It is obvious that the predicted tongue articulators generated by SPN are more similar to ground-truth than the SOTA, especially in the red circle of the Fig.~\ref{fig:draw1}. It indicates that the phoneme stream network and speech stream network have a positive effect on the reconstruction of tongue articulatory movements.
\vspace{-0.25cm}
\subsection{Ablation Study}
\vspace{-0.1cm}
To prove the effectiveness of the proposed models, we conduct the ablation experiment according to Table.~\ref{tab:TAB2}, and the results are presented in Table.~\ref{tab:TAB4}.
\begin{table}[h]
\scriptsize
\vspace{-0.5cm
\caption{\textbf\tiny{Ablation experiment proves the effectiveness of each module. SPN-S represents the model that combines the speech stream network and SOTA model.}
\centering
\label{tab:TAB2}
\begin{threeparttable}
\begin{spacing}{0.8}
\setlength{\tabcolsep}{5.0mm}{
\begin{tabular}{cccc}
\toprule
\textbf{}&\textbf{speech stream network}&\textbf{phoneme stream network}\\
\midrule
\textbf{SOTA}&\textbf{$\times$}&\textbf{$\times$}\\
\textbf{SPN-S}&\textbf{\checkmark}&\textbf{$\times$}\\
\textbf{SPN}&\textbf{\checkmark}&\textbf{\checkmark}\\
\bottomrule
\end{tabular}}
\end{spacing}
\end{threeparttable}
\end{table}
Obviously, the models with speech stream network (SPN, SPN-S) outperform that without the speech stream network (SOTA). Besides, the model with phoneme stream network (SPN) outperforms that without phoneme stream network (SPN-S). From the experimental results, we can clearly observe that in the speaker-independent experiment, the speech stream network improves the performance by extracting the local features and global features to better represent the speech information and after adding the phoneme features obtained by phoneme stream network brings a further large gain on the generalization ability of our model.
\begin{table}[h]
\scriptsize
\vspace{-0.35cm}
\caption{\textbf\tiny{RMSE and CC in three scenarios.}
\centering
\label{tab:TAB4}
\begin{threeparttable}
\begin{spacing}{0.8}
\setlength{\tabcolsep}{5.0mm}{
\begin{tabular}{cccc}
\toprule
\textbf{}&\textbf{RMSE}&\textbf{PCC}\\
\midrule
\textbf{SOTA}&\textbf{2.721}&\textbf{0.751}\\
\textbf{SPN-S}&\textbf{2.664}&\textbf{0.787}\\
\textbf{SPN}&\textbf{2.537}&\textbf{0.810}\\
\bottomrule
\end{tabular}}
\end{spacing}
\end{threeparttable}
\end{table}
\vspace{-0.7cm}
\section{Conclusion}
\vspace{-0.2cm}
To improve the speaker-independent performance and pay more attention to the global speech information, we propose a new network, including two parts. One is the speech stream network, which uses 1DCNNs and multi-head attention model to extract the local features and global features of speech to better represent the voice information. At the same time, we use the pronunciation parameters obtained by the phoneme stream network as the phoneme features to help predict the pronunciation parameters of voice inversion. The experimental results prove the effectiveness of our proposed network. In the future work, meta-learning will be explored into the field of AAI.
\vspace{-0.4cm}
\section{Acknowledgement}
\vspace{-0.2cm}
This work is supported by the National Natural Science
Foundation of China (No. 61977049), the National Natural Science Foundation of China (No. 62101351), and the GuangDong Basic and Applied Basic Research Foundation
(No.2020A1515110376).
\bibliographystyle{IEEEbib}
\section{Introduction}
\end{document}
|
1,941,325,220,578 | arxiv |
\section{Introduction}
Many physical processes can naturally be represented as \emph{passive systems}, i.e., dynamical systems that do not internally produce energy. This system class has been studied in great detail in the seminal works \cite{Wil71,Wil72,Wil72a} where various system theoretic characterizations have been given, among them one based on the well-known \emph{Kalman-Yakubovich-Popov} linear matrix inequality (KYP-LMI). Despite its long history, the interest in passivity-based control technique is still unabated, see, e.g., \cite{vdS96,vdSJ14} for a detailed introduction and an overview of existing results. More recently, a particular focus has been on the port-Hamiltonian (pH) representation of passive systems which not only paves the way for an especially targeted analysis of classical control techniques but also for compositional modelling, see, e.g., \cite{DuiMSB09,vdSM13} or specifically robust port-Hamiltonian representations \cite{BeaMV19,MehMS16}.
Port-Hamiltonian modeling has been first developed to provide a unified framework for systems belonging to different physical domains, by using energy as their `lingua franca'.
Even without taking into consideration multi-physics systems, port-Hamiltonian modeling offers several advantages, including passivity and stability properties as consequence of the underlying structure, structure-preserving interconnection allowing for modularized modeling, and structure-preserving methods for space- and time-discretization \cite{BruPM20,Cel17,KotL19,MehM19,HaMS19}.
While a port-Hamiltonian representation naturally allows for passivity-based control strategies such as \emph{control by interconnection} \cite{vdS96}, it is well-known that general controllers do not preserve the port-Hamiltonian structure.
For example, a classical linear quadratic Gaussian (LQG) controller will generally not preserve stability or passivity such that the weighting and covariance matrices have to be modified accordingly, see \cite{Hal94,LozJ88}.
The obvious downside of these specific choices is that the original interpretation of the matrices is lost and these rather serve as additional degrees of freedom to preserve the underlying structure. Based on the detailed structural analysis from \cite{Wu16}, in \cite{WuHLM14,Wuetal18} the authors have proposed different choices of weighting matrices in order to enforce LQG controllers to be port-Hamiltonian.
Since passive or port-Hamiltonian systems often are inherently infinite-dimensional \cite{JacZ12}, an efficient implementation of control techniques makes the use of reduced-order surrogate models inevitable.
For a general introduction to the field of model order reduction, we refer to, e.g., \cite{Ant05a}.
Similar to control approaches, classical model reduction techniques will, in general, not preserve the structure within the reduction process.
As a remedy, a variety of structure-preserving model reduction approaches have been suggested in the literature.
Since an exhaustive overview of all methods is out of the scope of this article, we only refer to \cite{EggKLMM18,Gugetal12,PolvdS12,WolLEK10} which are most relevant for our presentation.
Let us however also point to the recent thesis \cite{Lil20} for a more detailed overview of structure-preserving port-Hamiltonian model reduction.
Let us also mention that for passive systems, the method of positive real balanced truncation is known to be structure-preserving and, additionally, allows for an a priori error bound \cite{GuiO13,McFDG90}.
On the other hand, classical LQG balancing \cite{Cur03,Mey90} will, similar to the controller itself, not lead to a reduced-order port-Hamiltonian system.
While remedies exist \cite{PolvdS12,Wuetal18}, enforcing structure-preservation will typically destroy computable a priori error bounds.
This is also true for structure-preserving modifications of the classical balanced truncation method.
In \cite[Section 3.4.1]{Wu16}, the author has derived an error bound for effort-constraint balanced truncation, but the error bound relies on an auxiliary system and cannot be computed a priori.
Also structure-preserving balanced truncation for the different but still related class of second-order systems, cf.~\cite{ChaLVV06,ReiS08}, lacks computable a priori error bounds.
The paper is organized as follows. In Section 2, we recall the necessary background on port-Hamiltonian systems as well as balancing-based model order reduction. Section 3 studies (reduced-order) LQG controller design for port-Hamiltonian systems. In Section 4, we combine results on standard LQG balanced truncation with a recent observation on error bounds in the context of Lyapunov inequalities. Since in the particular case considered here, the particular pH representation is relevant, we further show how to minimize these error bounds in terms of an extremal solution of the KYP-LMI. We briefly discuss some new consequences for classical balanced truncation of port-Hamiltonian systems. Specifically, we derive an error bound for certain spectral factors of the Popov function and subsequently explain the influence of the particular pH representation, leading to a (theoretical) way to minimize these error bounds. Based on some numerical examples, in Section 5 we illustrate our main theoretical findings.
Appendix A reviews the recent approach from \cite{Wuetal18} and provides two examples showing that the resulting controller will generally not be port-Hamiltonian.
Furthermore, we compare the reduced-order models and controllers obtained by the new approach with the ones obtained by the approach from \cite{Wuetal18} and observe that the reduced-order models are equivalent, whereas the reduced-order controllers differ.
\section{Preliminaries}
\label{sec:preliminaries}
In this section, we collect some well-known results on port-Hamiltonian systems as well as balancing-based model order reduction. For a more detailed discussion of these topics, we refer to other references such as, e.g., \cite{Ant05a,BeaMV19,GugA04,PolvdS12,Wil71,Wil72,Wil72a}.
\subsection{Port-Hamiltonian systems}\label{subsec:pH}
Let us consider a pH system of the form
\begin{equation}\label{eq:pH}
\begin{aligned}
\dot{x}&= \underbrace{(J-R)Q}_{:=A}x+ Bu,\quad x(0)=0, \\
y&=\underbrace{B^\top Q}_{:=C} x ,
\end{aligned}
\end{equation}
where $J,R,Q\in \mathbb R^{n\times n}$, $B\in \mathbb R^{n \times m}$ and $ J^\top = -J$, $R=R^\top \succeq 0 $, $Q=Q^\top\succ 0$, together with a Hamiltonian function $\mathcal{H}(x)=\frac{1}{2}x^\top Qx$. Throughout this article, we assume the system \eqref{eq:pH} to be minimal, i.e., the matrix pairs $(A,B)$ and $(A,C)$ to be controllable and observable, respectively. In particular, this implies that $(A,B)$ and $(A,C)$ are stabilizable and detectable, respectively.
It is well-known that, by taking the Hamiltonian as an energy storage function, for $t_1\ge 0 $ the following \emph{dissipation inequality} holds:
\begin{equation}
\label{eq:dissipationIneq}
\mathcal{H}(x(t_1))-\mathcal{H}(x(0 )) = \int_{0 }^{t_1}\big(y(s)^\top u(s)-x(s)^\top QRQx(s) \big)\, \mathrm{d}s \le \int_{0 }^{t_1} y(s)^\top u(s) \, \mathrm{d}s,
\end{equation}
for arbitrary trajectories $x(\cdot)$ of \eqref{eq:pH}. Systems which possess a storage function satisfying such an inequality are called \emph{passive} and cannot produce energy internally and, in particular, are stable (though not necessarily asymptotically stable).
Furthermore, passive systems are \emph{dissipative} with supply rate $y^\top u$, see \cite{Wil72,Wil72a} for more details on dissipative systems.
Another important property of port-Hamiltonian systems with Hamiltonian function $\frac{1}{2}x^\top Qx$ is that $X=Q$ is a solution to the KYP-LMI:
\begin{align}\label{eq:kyp-lmi}
W(X)=\begin{bmatrix} -A^\top X - XA & C^\top - XB \\ C - B^\top X & 0 \end{bmatrix} \succeq 0, \quad X=X^\top \succeq 0,
\end{align}
since
\begin{align*}
W(Q) =\begin{bmatrix} -A^\top Q - QA & C^\top - QB \\ C - B^\top Q & 0 \end{bmatrix}
= \begin{bmatrix} 2QRQ & 0 \\ 0 & 0 \end{bmatrix} \succeq 0.
\end{align*}
On the other hand, if $X=X^\top\succ 0$ is a solution of the KYP-LMI \eqref{eq:kyp-lmi}, then the system \eqref{eq:pH} can be equivalently written as
\begin{equation}\label{eq:pHX}
\begin{aligned}
\dot{x}&= (J_X-R_X)Xx+ Bu,\quad x(0)=0, \\
y&= B^\top X x ,
\end{aligned}
\end{equation}
where $J_X=\frac{1}{2}(AX^{-1}-X^{-1}A^\top)=-J_X^\top$ and $R_X=-\frac{1}{2}(AX^{-1}+X^{-1}A^\top)=R_X^\top\succeq 0$.
This system is thus again port-Hamiltonian, but with respect to the Hamiltonian function $\mathcal H_X(x)=\frac{1}{2}x^\top Xx$, that is in general different from $\mathcal H(x)$.
Therefore, the representation of passive systems via a pH structure \eqref{eq:pH} is not unique.
In particular, there exist minimal $X_{\mathrm{min}}$ and maximal $X_{\mathrm{max}}$ solutions s.t.\@ for every solution $X$ to \eqref{eq:kyp-lmi}, it holds that $0\preceq X_{\mathrm{min}}\preceq X \preceq X_{\mathrm{max}}$.
While the system is equivalent, the new Hamiltonian $\mathcal H_X(x)$ may not have the same relevance as one associated with a particular solution of \eqref{eq:kyp-lmi}. For example, in \cite{BeaMV19} the authors investigate a maximally robust (w.r.t.\@ the passivity radius) representation based on what is called the \emph{analytic center}. As will be shown later, for the purpose of model reduction, the extremal solutions $X_{\min}$ and $X_{\max}$ are of special interest.
Another important property of port-Hamiltonian systems is that they maintain their structure when a state space transformation is applied.
In fact, if $T\in\mathbb{R}^{n\times n}$ is an invertible matrix, then by applying the change of variable $\tilde x=Tx$ we obtain the equivalent system
\begin{equation}\label{eq:pHCoV}
\begin{aligned}
\dot{\tilde{x}} &= (\tilde{J}-\tilde{R})\tilde{Q}\tilde{x} + \tilde{B}u,\quad \tilde x(0)=0, \\
y&= \tilde B^\top \tilde Q \tilde x ,
\end{aligned}
\end{equation}
where $\tilde J=TJT^\top=-\tilde J^\top$, $\tilde R=TRT^\top=\tilde R^\top\succeq 0$, $\tilde Q=T^{-\top}QT^{-1}=\tilde Q^\top\succ 0$, and $\tilde B=TB$.
We will then have $\tilde A=(\tilde J-\tilde R)\tilde Q=TAT^{-1}$.
Note that the Hamiltonian function $\tilde{\mathcal H}(\tilde x)=\frac{1}{2}\tilde x^\top\tilde Q\tilde x=\frac{1}{2}x^\top Qx$ is still the same.
Thus, state space transformations do not destroy the port-Hamiltonian structure and we will make use of this property when discussing balancing methods in the upcoming sections.
Similarly, a Petrov--Galerkin projection $\mathbb P=VW^\top$ of the form $A_p=W^\top AV$, $B_p=W^\top B$, $C_p=CV$ with $QV=WQ_p$ for a matrix $Q_p=Q_p^\top\succ 0$ and $W^\top V=I_p$ leads to a projected port-Hamiltonian system of the form
\begin{equation}\label{eq:pHPG}
\begin{aligned}
\dot x_p &= (J_p-R_p)Q_p x_p + B_p u,\quad x_p(0)=0, \\
y&= B_p^\top Q_p x_p ,
\end{aligned}
\end{equation}
where $x\approx Vx_p$, $J_p=W^\top JW$, $R_p=W^\top RW$, and $B_p=W^\top B$, with Hamiltonian function $\mathcal H_p(x_p)=\frac{1}{2}x_p^\top Q_px_p$.
Due to the minimality of \eqref{eq:pH}, passivity is equivalent to \emph{positive realness}, i.e., the transfer function $G(s)=C(sI_n-A)^{-1}B$ is analytic in the open right half plane $\mathbb C_+$ and satisfies
\begin{align}\label{eq:popov_fct}
G(s)^* + G(s)\succeq 0 \quad \text{for all } s \in \mathbb C_+.
\end{align}
If all eigenvalues of $A$ are in the open left half plane, then positive realness can be equivalently characterized by the positive semidefiniteness of the so-called Popov function $\Phi(s)=G(-s)^\top+G(s)$ on the imaginary axis.
Let us consider the following factorization of the Popov function $\Phi(s)$ (see, e.g., \cite{Wil72a}):
\begin{align}\label{eq:popov_aux}
\Phi(s)= \begin{bmatrix} B^\top (-sI_n-A^\top)^{-1} & I_m \end{bmatrix} W(X) \begin{bmatrix} (sI_n-A)^{-1}B \\ I_m \end{bmatrix}.
\end{align}
In particular, if $X$ is a solution to \eqref{eq:kyp-lmi}, then there exists $L_X\in \mathbb R^{n\times k},k\le n$ such that
\begin{align}\label{eq:popov_factor}
\Phi(s)= \begin{bmatrix} B^\top (-sI_n-A^\top)^{-1} & I_m \end{bmatrix} \begin{bmatrix} L_X^\top \\ 0 \end{bmatrix} \begin{bmatrix} L_X & 0 \end{bmatrix} \begin{bmatrix} (sI_n-A)^{-1}B \\ I_m \end{bmatrix}.
\end{align}
We will later on focus on a specific factorization of the form \eqref{eq:popov_factor} that will be associated with
\begin{equation}
\label{eq:dualKYPLMI}
\begin{bmatrix}
-AY - YA^\top & B-YC^\top \\
B^\top - CY & 0
\end{bmatrix}
\succeq 0,\quad Y=Y^\top \succeq 0,
\end{equation}
i.e., the dual version of \eqref{eq:kyp-lmi}.
\subsection{Balancing-based and effort-constraint model order reduction} \label{subsec:effortConstraint}
The concept of system balancing goes back to \cite{Moo81,MulR76} and relies on the infinite-time controllability and observability Gramians $\mL_{\mathrm{c}},\mM_{\mathrm{o}}$ associated with a linear (asymptotically) stable system characterized by $(A,B,C)$. These Gramians satisfy the following Lyapunov equations:
\begin{equation}\label{eq:Lyap_con_obs}
\begin{aligned}
A\mL_{\mathrm{c}} + \mL_{\mathrm{c}} A^\top + BB^\top &=0, \\
A^\top\mM_{\mathrm{o}} + \mM_{\mathrm{o}} A + C^\top C&=0.
\end{aligned}
\end{equation}
The main idea of balancing-based model order reduction now is to find a specific state space transformation $(A,B,C)\leadsto (TAT^{-1},TB,CT^{-1})$ that allows one to simultaneously measure the amount of controllability and observability in the system. This is possible since in the new coordinates, the Gramians are given by $T\mL_{\mathrm{c}} T^\top$ and $T^{-\top}\mM_{\mathrm{o}} T^{-1}$, allowing one to construct an appropriate contragredient transformation such that
\begin{align*}
T\mL_{\mathrm{c}} T^\top=T^{-\top}\mM_{\mathrm{o}} T^{-1}= \Sigma= \mathrm{diag}(\sigma_1,\dots,\sigma_n).
\end{align*}
In fact, the so-called square root balancing method (see, e.g., \cite{Ant05a,GugA04}) yields such a transformation via:
\begin{align*}
T=\Sigma^{-\frac{1}{2}} Z^\top L_{\mM_{\mathrm{o}}}, \quad T^{-1}=L_{\mL_{\mathrm{c}}}^\top U \Sigma^{-\frac{1}{2}},
\end{align*}
where we have the following singular value and Cholesky decompositions
\begin{align*}
L_{\mL_{\mathrm{c}}} L _{\mM_{\mathrm{o}}}^\top=U\Sigma Z^\top ,\quad \mL_{\mathrm{c}} = L_{\mL_{\mathrm{c}}}^\top L_{\mL_{\mathrm{c}}}, \quad \mM_{\mathrm{o}} = L_{\mM_{\mathrm{o}}}^\top L_{\mM_{\mathrm{o}}}.
\end{align*}
For an already balanced model $(A,B,C)$, consider then the partitioning:
\begin{align*}
A=\begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}, \quad
B=\begin{bmatrix} B_{1} \\ B_{2} \end{bmatrix},\quad C=\begin{bmatrix} C_1 & C_2 \end{bmatrix}.
\end{align*}
A reduced-order model $(A_r,B_r,C_r)$ is constructed by truncation, i.e., choosing $(A_{11},B_1,C_1)$. In the particular case of \eqref{eq:pH}, a straightforward implementation of this approach will generally not preserve the port-Hamiltonian structure. As a remedy, in \cite{PolvdS12}, the authors have proposed a so-called \emph{effort-constraint} reduction method which is based on a (balanced) partitioning of \eqref{eq:pH} of the form:
\begin{align}\label{eq:part}
J_\mathrm{b}=\begin{bmatrix} J_{11} & J_{12} \\ J_{21} & J_{22} \end{bmatrix}, \quad
R_\mathrm{b}=\begin{bmatrix} R_{11} & R_{12} \\ R_{21} & R_{22} \end{bmatrix}, \quad
Q_\mathrm{b}=\begin{bmatrix} Q_{11} & Q_{12} \\ Q_{21} & Q_{22} \end{bmatrix}, \quad
B_\mathrm{b}=\begin{bmatrix} B_{1} \\ B_{2} \end{bmatrix}.
\end{align}
The reduced-order model is then defined as
\begin{align}\label{eq:eff_const}
J_r = J_{11},\quad R_r = R_{11}, \quad Q_r=Q_{11}-Q_{12}Q_{22}^{-1} Q_{21}, \quad B_r = B_1, \quad C_r=B_r^\top Q_r.
\end{align}
Note in particular that with $Q=Q^\top \succ 0$ it also holds that the Schur complement is positive definite, i.e., $Q_r=Q_r^\top \succ 0$.
Furthermore, it can be noted that this reduced-order model can be obtained via Petrov--Galerkin projection $\mathbb{P}=V_\mathrm{b} W_\mathrm{b}^\top$ applied to the balanced system, with $W_\mathrm{b}^\top=\begin{bmatrix}I_r & 0\end{bmatrix}$ and $V_\mathrm{b}^\top=\begin{bmatrix}I_r & -Q_{12}Q_{22}^{-1}\end{bmatrix}$, satisfying $W_\mathrm{b}^\top V_\mathrm{b}=I_r$ and $Q_\mathrm{b} V_\mathrm{b}=W_\mathrm{b} Q_r$, or equivalently a Petrov--Galerkin projection $\mathbb{P}=VW^\top$ applied to the original system, with $W^\top=W_\mathrm{b}^\top T$ and $V=T^{-1}V_\mathrm{b}$ (see \autoref{subsec:pH}).
Hence, by construction the reduced-order model is port-Hamiltonian. This structure-preservation however comes with the loss of a (classical) $\mathcal{H}_\infty$-error bound. Let us remark that in \cite{Wuetal18}, the authors have derived an a posteriori error bound which makes use of an auxiliary system. Under certain assumptions on $R$ and $B$, in Section \ref{sec:classic_bt}, we modify the balancing approach such that an a priori error bound, analogous to the classical one, is valid. Furthermore, we establish a connection to balancing of a spectral factorization of the Popov function. Finally, we mention that the method of positive real balanced truncation \cite{HarJS83} aims at the preservation of passivity and relies on balancing the (dual) extremal solutions to \eqref{eq:kyp-lmi}.
\section{Port-Hamiltonian reduced LQG control design}\label{sec:pH-LQG}
In this section, given a port-Hamiltonian system \eqref{eq:pH},
our interest is the design of a controller
\begin{equation}\label{eq:controller}
\begin{aligned}
\dot{x}_{\mathrm{c}} &= A_{\mathrm{c}} x _c + B_{\mathrm{c}} u_{\mathrm{c}} , \quad x_{\mathrm{c}}(0)=0, \\
y_{\mathrm{c}} &= C_{\mathrm{c}} x_{\mathrm{c}},
\end{aligned}
\end{equation}
with $(A_{\mathrm{c}},B_{\mathrm{c}},C_{\mathrm{c}})\in \mathbb R^{n_{\mathrm{c}}\times n_{\mathrm{c}}}\times \mathbb R^{n_{\mathrm{c}} \times m} \times \mathbb R^{m \times n_{\mathrm{c}}}$ such that the dynamics $x(\cdot),x_{\mathrm{c}}(\cdot)$ satisfy a desired behavior, e.g., are associated with the solution of an optimal control problem.
It is well-known, see, e.g., \cite[Section 7]{vdS96}, that if $(A_{\mathrm{c}},B_{\mathrm{c}},C_{\mathrm{c}})$ is a pH system, the resulting closed loop system allows for a pH formulation if a \emph{power-conserving interconnection} of the form $u=-y_{\mathrm{c}}$ and $u_{\mathrm{c}}=y$ is used. Note however that classical LQG control design would result in
\begin{align}\label{eq:LQG_controllers}
A_{\mathrm{c}} = A-B\tilde{\mathcal R}^{-1}B^\top \mathcal{P}_\mathrm{c}-\mathcal{P}_\mathrm{f} C^\top \mathcal{R}_\mathrm{f}^{-1}C , \quad B_{\mathrm{c}}=\mathcal{P}_\mathrm{f} C^\top \mathcal{R}_\mathrm{f}^{-1}, \quad C_{\mathrm{c}}=\tilde{\mathcal R}^{-1} B^\top \mathcal{P}_\mathrm{c}
\end{align}
where $\mathcal{P}_\mathrm{c} $ and $\mathcal{P}_\mathrm{f}$ are solutions to the following control and filter Riccati equations
\begin{align}
A^\top \mathcal{P}_\mathrm{c} +\mathcal{P}_\mathrm{c} A - \mathcal{P}_\mathrm{c} B\tilde{\mathcal R}^{-1} B^\top \mathcal{P}_\mathrm{c} + \tilde{\mathcal Q}&=0 , \label{eq:lqg_care} \\
A \mathcal{P}_\mathrm{f} + \mathcal{P}_\mathrm{f} A^\top - \mathcal{P}_\mathrm{f} C^\top \mathcal{R}_\mathrm{f} ^{-1} C \mathcal{P}_\mathrm{f} + \mathcal{Q}_\mathrm{f} &=0 \label{eq:lqg_fare}.
\end{align}
Since such controllers generally do not preserve the pH structure (for a stability violation, we refer to \cite{Joh79}), we suggest a particular choice of the weighting matrices $\tilde{\mathcal R},\mathcal{R}_\mathrm{f},\tilde{\mathcal Q}$ and $\mathcal{Q}_\mathrm{f}$ and consider the solutions $\widehat{\mathcal{P}}_{\mathrm{c}}$ and $\widehat{\mathcal{P}}_{\mathrm{f}}$ of
\begin{align}
A^\top \widehat{\mathcal{P}}_{\mathrm{c}} +\widehat{\mathcal{P}}_{\mathrm{c}} A - \widehat{\mathcal{P}}_{\mathrm{c}} B B^\top \widehat{\mathcal{P}}_{\mathrm{c}} + C^\top C&=0 , \label{eq:lqg_care_ph} \\
A \widehat{\mathcal{P}}_{\mathrm{f}} + \widehat{\mathcal{P}}_{\mathrm{f}} A^\top - \widehat{\mathcal{P}}_{\mathrm{f}} C^\top C \widehat{\mathcal{P}}_{\mathrm{f}} + BB^\top +2R &=0 \label{eq:lqg_fare_ph}.
\end{align}
The choice $\tilde{\mathcal R}=\mathcal{R}_\mathrm{f}=I$ is mainly for simplicity and could be taken into account when a scaling of the input weights would be of further interest. Let us emphasize that our method is inspired by the approach proposed in \cite{Wuetal18} and the above modification of the LQG weights is based on ideas similar to those in \cite{LozJ88,Wu16}. In Appendix \ref{apdxA} we review the method from \cite{Wuetal18} and provide two examples which show that the resulting controller may generally not be realized as a port-Hamiltonian system.
With the particular relation between the covariance and weighting matrices from \eqref{eq:lqg_care_ph} and \eqref{eq:lqg_fare_ph}, the solution $\widehat{\mathcal{P}}_{\mathrm{c}}$ satisfies the KYP-LMI \eqref{eq:kyp-lmi} such that the controller is port-Hamiltonian.
\begin{theorem}\label{thm:cont_is_pH}
Let a minimal port-Hamiltonian system \eqref{eq:pH} be given by $(J,R,Q,B).$ If $\widehat{\mathcal{P}}_{\mathrm{c}}$ and $\widehat{\mathcal{P}}_{\mathrm{f}}$ are the unique stabilizing solutions of \eqref{eq:lqg_care_ph} and \eqref{eq:lqg_fare_ph}, respectively, then the associated LQG controller defined by
\begin{align*}
\widehat{A}_{\mathrm{c}}=A-BB^\top\widehat{\mathcal{P}}_{\mathrm{c}}-\widehat{\mathcal{P}}_{\mathrm{f}} C^\top C , \ \ \widehat{B}_{\mathrm{c}} = \widehat{\mathcal{P}}_{\mathrm{f}} C^\top, \ \ \widehat{C}_{\mathrm{c}}=B^\top \widehat{\mathcal{P}}_{\mathrm{c}}
\end{align*}
is port-Hamiltonian.
In particular, with the Hamiltonian function $\mathcal{H}_\mathrm{c}(x_{\mathrm{c}})=\frac{1}{2}x_{\mathrm{c}}^\top \widehat{\mathcal{P}}_{\mathrm{c}} x_{\mathrm{c}},$ a pH realization of the controller is as follows:
\begin{align*}
\widehat{J}_{\mathrm{c}} = \tfrac{1}{2}(\widehat{A}_{\mathrm{c}} \widehat{\mathcal{P}}_{\mathrm{c}}^{-1}-\widehat{\mathcal{P}}_{\mathrm{c}}^{-1}\widehat{A}_{\mathrm{c}}^\top ),\quad \widehat{R}_{\mathrm{c}} = \tfrac{1}{2}(\widehat{\mathcal{P}}_{\mathrm{c}}^{-1} Q + I_n)BB^\top (\widehat{\mathcal{P}}_{\mathrm{c}}^{-1} Q + I_n)^\top,\quad \widehat{Q}_{\mathrm{c}} = \widehat{\mathcal{P}}_{\mathrm{c}},\quad \widehat{B}_{\mathrm{c}} =B.
\end{align*}
\end{theorem}
\begin{proof}
Note that due to the stabilizability and detectability of the system, the equations \eqref{eq:lqg_care_ph} and \eqref{eq:lqg_fare_ph} have unique stabilizing solutions $\widehat{\mathcal{P}}_{\mathrm{c}}$ and $\widehat{\mathcal{P}}_{\mathrm{f}}$, respectively. In fact, it holds that $\widehat{\mathcal{P}}_{\mathrm{f}}=Q^{-1}$ since
\begin{align*}
&AQ^{-1} + Q^{-1} A^\top - Q^{-1} C^\top C Q^{-1} +2R+BB^\top \\
&= (J-R)QQ^{-1} + Q^{-1}Q^\top (J^\top -R^\top) - Q^{-1}QBB^\top Q Q^{-1}+2R +BB^\top =0.
\end{align*}
This, however, justifies the following derivations
\begin{align*}
\widehat{A}_{\mathrm{c}}^\top \widehat{\mathcal{P}}_{\mathrm{c}} + \widehat{\mathcal{P}}_{\mathrm{c}} \widehat{A}_{\mathrm{c}} &= (A-BB^\top\widehat{\mathcal{P}}_{\mathrm{c}}-\widehat{\mathcal{P}}_{\mathrm{f}} C^\top C)^\top \widehat{\mathcal{P}}_{\mathrm{c}} + \widehat{\mathcal{P}}_{\mathrm{c}} (A-BB^\top\widehat{\mathcal{P}}_{\mathrm{c}}-\widehat{\mathcal{P}}_{\mathrm{f}} C^\top C) \\
&= A^\top \widehat{\mathcal{P}}_{\mathrm{c}} + \widehat{\mathcal{P}}_{\mathrm{c}} A - \widehat{\mathcal{P}}_{\mathrm{c}} BB^\top \widehat{\mathcal{P}}_{\mathrm{c}} - \widehat{\mathcal{P}}_{\mathrm{c}} BB^\top \widehat{\mathcal{P}}_{\mathrm{c}} - C^\top C \widehat{\mathcal{P}}_{\mathrm{f}} \widehat{\mathcal{P}}_{\mathrm{c}} - \widehat{\mathcal{P}}_{\mathrm{c}} \widehat{\mathcal{P}}_{\mathrm{f}} C^\top C \\
&=-C^\top C -\widehat{\mathcal{P}}_{\mathrm{c}} BB^\top \widehat{\mathcal{P}}_{\mathrm{c}} - C^\top C Q^{-1}\widehat{\mathcal{P}}_{\mathrm{c}} - \widehat{\mathcal{P}}_{\mathrm{c}} Q^{-1} C^\top C \\
&= - (C^\top +\widehat{\mathcal{P}}_{\mathrm{c}} B)(C^\top +\widehat{\mathcal{P}}_{\mathrm{c}} B)^\top \preceq 0.
\end{align*}
We further obtain that
\begin{align*}
\widehat{B}_{\mathrm{c}}^\top \widehat{\mathcal{P}}_{\mathrm{c}} = (\widehat{\mathcal{P}}_{\mathrm{f}} C^\top )^\top \widehat{\mathcal{P}}_{\mathrm{c}} = C Q^{-1} \widehat{\mathcal{P}}_{\mathrm{c}} = B^\top \widehat{\mathcal{P}}_{\mathrm{c}} = \widehat{C}_{\mathrm{c}}.
\end{align*}
We conclude that $\widehat{A}_{\mathrm{c}}^\top \widehat{\mathcal{P}}_{\mathrm{c}} + \widehat{\mathcal{P}}_{\mathrm{c}} \widehat{A}_{\mathrm{c}}\preceq 0$ as well as $ \widehat{B}_{\mathrm{c}}^\top \widehat{\mathcal{P}}_{\mathrm{c}}=\widehat{C}_{\mathrm{c}}$ such that $\widehat{\mathcal{P}}_{\mathrm{c}}=\widehat{\mathcal{P}}_{\mathrm{c}}^\top \succeq 0$ satisfies
%
\begin{align*}
\begin{bmatrix} -\widehat{A}_{\mathrm{c}} ^\top \widehat{\mathcal{P}}_{\mathrm{c}} -\widehat{\mathcal{P}}_{\mathrm{c}} \widehat{A}_{\mathrm{c}} & \widehat{C}_{\mathrm{c}}^\top -\widehat{\mathcal{P}}_{\mathrm{c}} \widehat{B}_{\mathrm{c}} \\
\widehat{C}_{\mathrm{c}}-\widehat{B}_{\mathrm{c}}^\top \widehat{\mathcal{P}}_{\mathrm{c}} & 0 \end{bmatrix} \succeq 0 .
\end{align*}
%
Since $(A,C)$ is observable, it further holds that $\widehat{\mathcal{P}}_{\mathrm{c}} \succ 0$. Due to the discussion in \autoref{subsec:pH}, we can conclude that $(A_{\mathrm{c}},B_{\mathrm{c}},C_{\mathrm{c}})$ is port-Hamiltonian with respect to the Hamiltonian function $\mathcal H_{\mathrm{c}}(x_{\mathrm{c}})=\frac{1}{2}x_{\mathrm{c}}^\top\widehat{\mathcal{P}}_{\mathrm{c}}x_{\mathrm{c}}$.
In particular, the matrices $\widehat{J}_{\mathrm{c}}$ and $\widehat{R}_{\mathrm{c}}$ can be constructed as outlined in \autoref{subsec:pH}.
\end{proof}
\begin{remark}\label{rem:rel_between_Pf}
Note that since $\widehat{\mathcal{P}}_{\mathrm{f}}=Q^{-1}$ the controller is of the form:
\begin{align*}
\dot{x}_{\mathrm{c}} &= (A-BB^\top \widehat{\mathcal{P}}_{\mathrm{c}}- BC )x_{\mathrm{c}} +B u_{\mathrm{c}}, \quad x_{\mathrm{c}}(0)=0, \\
y_{\mathrm{c}}&= B^\top \widehat{\mathcal{P}}_{\mathrm{c}} x_{\mathrm{c}}.
\end{align*}
In particular, the system matrix $\widehat{A}_{\mathrm{c}}$ combines the feedback structure of a classical linear quadratic regulator $u_{\mathrm{c}}=-BB^\top \widehat{\mathcal{P}}_{\mathrm{c}} x_{\mathrm{c}}$ with that of a simple output feedback $u_{\mathrm{c}}=-BCx_{\mathrm{c}}$.
\end{remark}
\begin{remark}
\label{rem:modifyCARE}
Theorem~\ref{thm:cont_is_pH} states that the LQG controller based on the control Riccati equation \eqref{eq:lqg_care_ph} and the modified filter Riccati equation \eqref{eq:lqg_fare_ph} is port-Hamiltonian.
We note that a similar result can be obtained by modifying the control Riccati equation.
More precisely, by choosing the weighting matrices $\tilde{\mathcal Q} = C^\top C+2QRQ$ and $\mathcal{Q}_\mathrm{f} = BB^\top$, the resulting Gramian $\mathcal{P}_\mathrm{c}$ is simply given by $\mathcal{P}_\mathrm{c}=Q$ and the resulting LQG controller is port-Hamiltonian with Hamiltonian $\frac12 x_{\mathrm{c}}^\top \mathcal{P}_\mathrm{f}^{-1} x_{\mathrm{c}}$.
This choice of the weighting matrix $\tilde{\mathcal Q} = C^\top C+2QRQ$ also permits a physical interpretation as it corresponds to an optimal control problem where the LQR cost is given by the sum of the squared input norm, the squared output norm, and (twice) the dissipated energy, cf.~\eqref{eq:dissipationIneq}.
This cost function leads to an LQR control which is simply given by the output feedback $u = -B^\top\mathcal{P}_\mathrm{c} x = -B^\top Qx = -y$.
\end{remark}
\subsection{Structure-preserving LQG balanced truncation for port-Hamiltonian systems}
\label{subsec:modifiedLQG}
With regard to a numerical implementation for large-scale systems, let us further derive a reduced port-Hamiltonian controller which replaces $(\widehat{A}_{\mathrm{c}},\widehat{B}_{\mathrm{c}},\widehat{C}_{\mathrm{c}})$ by a surrogate model. This is done in three steps: first, the system is transformed into a balanced form \eqref{eq:part} such that $\widehat{\mathcal{P}}_{\mathrm{f}}=\widehat{\mathcal{P}}_{\mathrm{c}}=\Sigma=\mathrm{diag}(\sigma_1,\dots,\sigma_n).$
Subsequently, a reduced-order model is obtained by the effort-constraint method:
\begin{equation}\label{eq:rom_new}
\begin{aligned}
\dot{\widehat{x}}_{r} &= (\widehat{J}_r-\widehat{R}_r)\widehat{Q}_r \widehat{x}_{r}+\widehat{B}_ru,\quad \widehat{x}_{r}(0)=0, \\
\widehat{y}_r&= \widehat{C}_r \widehat{x}_{r},
\end{aligned}
\end{equation}
where $(\widehat{J}_r,\widehat{R}_r,\widehat{Q}_r,\widehat{B}_r,\widehat{C}_r)$ are as in \eqref{eq:eff_const}. Finally, a reduced controller $(\widehat{A}_{\mathrm{c}r},\widehat{B}_{\mathrm{c}r},\widehat{C}_{\mathrm{c}r})$ is constructed according to Theorem \ref{thm:cont_is_pH}. The entire procedure is summarized in Algorithm \ref{alg:new}.
\begin{algorithm}[H]
\caption{pH-preserving LQG reduced controller design}
\label{alg:new}
\begin{algorithmic}[1]
\REQUIRE $J, R,Q\in \mathbb R^{n\times n}, B\in \mathbb R^{n\times m}$ as in \eqref{eq:pH} with minimal $(A,B,C)$
\ENSURE Reduced-order LQG controller $(\widehat{A}_{\mathrm{c}r},\widehat{B}_{\mathrm{c}r},\widehat{C}_{\mathrm{c}r})$ \\
\STATE Compute $\widehat{\mathcal{P}}_{\mathrm{c}}$ and $\widehat{\mathcal{P}}_{\mathrm{f}}=Q^{-1}$ solving \eqref{eq:lqg_care_ph} and \eqref{eq:lqg_fare_ph}.\!\!\!
\STATE Compute $T\in \mathbb R^{n\times n}$ defined by $T\widehat{\mathcal{P}}_{\mathrm{f}} T^\top = T^{-\top} \widehat{\mathcal{P}}_{\mathrm{c}} T^{-1} = \mathrm{diag}(\sigma_1,\dots,\sigma_n)$.
\STATE Balance the system $J_\mathrm{b}=TJT^\top , R_\mathrm{b} = TRT^\top , Q_\mathrm{b} =T^{-\top } Q T^{-1} $ and $B_\mathrm{b}=TB$.
\STATE Obtain the reduced system $(\widehat{A}_r,\widehat{B}_r,\widehat{C}_r)$ by the effort-constraint method.
\STATE Compute the reduced-order LQG controller $(\widehat{A}_{\mathrm{c}r},\widehat{B}_{\mathrm{c}r},\widehat{C}_{\mathrm{c}r})$ based on Theorem \ref{thm:cont_is_pH} for $(\widehat{A}_r,\widehat{B}_r,\widehat{C}_r)$.
\end{algorithmic}
\end{algorithm}
It turns out that due to the particular choice of the weighting matrix $\mathcal{Q}_\mathrm{f} = BB^\top +2R$ and the resulting relation $\widehat{\mathcal{P}}_{\mathrm{f}}=Q^{-1}$, the reduced-order model \eqref{eq:rom_new} can equivalently be obtained by simple truncation.
\begin{theorem}\label{thm:rom_ph}
Let a minimal port-Hamiltonian system \eqref{eq:pH} be given by $(J,R,Q,B)$ and let $(\widehat{A}_r,\widehat{B}_r,\widehat{C}_r)$ be the corresponding reduced-order model obtained in Algorithm \ref{alg:new}.
Then $(\widehat{A}_r,\widehat{B}_r,\widehat{C}_r)$ has a port-Hamiltonian realization.
Moreover, $(\widehat{A}_r,\widehat{B}_r,\widehat{C}_r)$ can be obtained by simple truncation of a system that is balanced w.r.t.~$\widehat{\mathcal{P}}_{\mathrm{c}}$ and $\widehat{\mathcal{P}}_{\mathrm{f}}$ as in \eqref{eq:lqg_care_ph} and \eqref{eq:lqg_fare_ph}, respectively.
\end{theorem}
\begin{proof}
The fact that $(\widehat{A}_r,\widehat{B}_r,\widehat{C}_r)$ has a port-Hamiltonian realization immediately follows from its construction by the effort-constraint reduction technique. Note that for the balanced model it holds that $\widehat{\mathcal{P}}_{\mathrm{c}}=\widehat{\mathcal{P}}_{\mathrm{f}}=\Sigma$. As mentioned in the proof of Theorem \ref{thm:cont_is_pH}, we have that $\widehat{\mathcal{P}}_{\mathrm{f}}=Q_{\mathrm{b}}^{-1}$, i.e., $Q_{\mathrm{b}}=\mathrm{diag}(\frac{1}{\sigma_1},\dots,\frac{1}{\sigma_n})$. Hence, we conclude that $Q_{\mathrm{b}}$ is diagonal such that the Schur complement in \eqref{eq:eff_const} reads $Q_r=Q_{11}-Q_{12}Q_{22}^{-1}Q_{21}=Q_{11}$ which shows the second assertion.
\end{proof}
The relation $\widehat{\mathcal{P}}_{\mathrm{f}}=Q^{-1}$ yields some additional beneficial implications with regard to the computation of the reduced-order matrices $\widehat{J}_r$, $\widehat{R}_r$ and $\widehat{Q}_r$.
According to the general idea of square root balancing, let us assume that we have the decompositions
\begin{align*}
T=\Sigma^{-\frac{1}{2}} Z^\top L_{\widehat{\mathcal{P}}_{\mathrm{c}}}, \quad T^{-1} = L_{\widehat{\mathcal{P}}_{\mathrm{f}}}^\top U \Sigma^{-\frac{1}{2}},
\end{align*}
where
\begin{align}\label{eq:aux1}
L_{\widehat{\mathcal{P}}_{\mathrm{f}}} L _{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top = U\Sigma Z^\top , \quad Q^{-1}=\widehat{\mathcal{P}}_{\mathrm{f}} = L_{\widehat{\mathcal{P}}_{\mathrm{f}}}^\top L _{\widehat{\mathcal{P}}_{\mathrm{f}}},\quad
\widehat{\mathcal{P}}_{\mathrm{c}} = L_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top L _{\widehat{\mathcal{P}}_{\mathrm{c}}}.
\end{align}
The balanced system then is given by applying the change of variable $x_{\mathrm{b}}=Tx$ as in \eqref{eq:pHCoV}, leading to $A_{\mathrm{b}}=TAT^{-1}$, $B_\mathrm{b}=TB$, $C_{\mathrm{b}}=CT^{-1}$ and the port-Hamiltonian formulation:
\begin{align*}
J_\mathrm{b} = TJT^\top, \quad R_\mathrm{b} = TRT^\top, \quad Q_\mathrm{b} = T^{-\top}QT^{-1} = \Sigma^{-1}.
\end{align*}
We can therefore construct the reduced pH formulation by the Petrov--Galerkin projection $\mathbb P=VW^\top$ with $V^\top=\begin{bmatrix}I_r & 0\end{bmatrix}T^{-\top}$ and $W^\top=\begin{bmatrix}I_r & 0\end{bmatrix}T$ applied to $(A,B,C)$, i.e.,
\begin{equation}\label{eq:pH_red}
\begin{alignedat}{3}
\widehat{A}_r &= W^\top AV, \quad& \widehat{B}_r &= W^\top B, \quad& \widehat{C}_r &= CV, \\
\widehat{J}_r &= W^\top J W, \quad& \widehat{R}_r &= W^\top R W, \quad& \widehat{Q}_r &= (W^\top Q^{-1} W)^{-1} = \Sigma_1^{-1}.
\end{alignedat}
\end{equation}
This representation will be of particular interest in section \ref{sec:classic_bt}, where we provide an error bound that involves a reduced factorization of $\widehat{R}_r$, cf.\@ Remark \ref{rem:compare_spec_fac}.
\begin{remark}\label{rem:con_to_LQG}
With the previous considerations, we conclude that the reduced-order model $(\widehat{A}_r,\widehat{B}_r,\widehat{C}_r)$ results (and itself is balanced) from the truncation of a system that is balanced w.r.t.\@ the Gramians $\widehat{\mathcal{P}}_{\mathrm{c}}$ and $\widehat{\mathcal{P}}_{\mathrm{f}}$, respectively.
We point out that the representation corresponds to a rather simple modification of classical LQG balanced truncation (by adding the term $2R$) which will be utilized below to derive a suitably modified error bound in the gap metric.
\end{remark}
\subsection{PH formulation in co-energy variables}\label{sec:generalizedStateSpace}
For many applications, a pH formulation as in \eqref{eq:pH} may lead to limitations and it often turns out to be beneficial to consider a differential-algebraic formulation.
While a detailed analysis of the differential-algebraic case is out of the scope of this article, below we discuss required modifications of the Gramians defined in \eqref{eq:lqg_care_ph} and \eqref{eq:lqg_fare_ph} in the particular case of a formulation in the \emph{co-energy variables} $z\coloneqq Qx$.
Starting from the description in \eqref{eq:pH}, this leads to the formulation
\begin{equation}\label{eq:pH_gen}
\begin{aligned}
E\dot{z}&= (J-R)z+Bu, \ \ z(0)=0,\\
y&= B^\top z,
\end{aligned}
\end{equation}
of an implicitly given pH system, where $E=Q^{-1}=E^\top\succ 0$, with the Hamiltonian function $\tilde{\mathcal H}(z)=\frac{1}{2}z^\top Ez$. Note in particular that this formulation has the additional advantage of being linear in the defining matrices $(E,J,R).$ Based on well-known LQG control theory for generalized state space systems (e.g., \cite{Meh91}), it is natural to replace \eqref{eq:lqg_care_ph} by
\begin{align}\label{eq:lqg_care_ph_gen}
(J-R)^\top \widetilde{\mathcal{P}}_{\mathrm{c}} E + E^\top\widetilde{\mathcal{P}}_{\mathrm{c}} (J-R) - E ^\top\widetilde{\mathcal{P}}_{\mathrm{c}} BB^\top \widetilde{\mathcal{P}}_{\mathrm{c}} E + BB^\top &=0.
\end{align}
With regard to port-Hamiltonian structure of a controller, instead of \eqref{eq:lqg_fare_ph} we consider
\begin{align}\label{eq:lqg_fare_ph_gen}
(J-R) \widetilde{\mathcal{P}}_{\mathrm{f}} E^\top + E \widetilde{\mathcal{P}}_{\mathrm{f}} (J-R)^\top - E \widetilde{\mathcal{P}}_{\mathrm{f}} BB^\top \widetilde{\mathcal{P}}_{\mathrm{f}} E^\top + BB^\top +2R &=0.
\end{align}
On the one hand, after left- and right-multiplication of \eqref{eq:lqg_care_ph_gen} by $Q$, we get again \eqref{eq:lqg_care_ph}, thus $\widetilde{\mathcal{P}}_{\mathrm{c}}=\widehat{\mathcal{P}}_{\mathrm{c}}$.
On the other hand, it is clear that $\widetilde{\mathcal{P}}_{\mathrm{f}}=Q=E^{-1}$ is the solution of \eqref{eq:lqg_fare_ph_gen}.
With a slight modification of the arguments used in the proof of Theorem \ref{thm:cont_is_pH}, we have the following result.
\begin{corollary}\label{cor:cont_is_pH_gen}
Let $(E,J,R,B)$ define a minimal implicit port-Hamiltonian system of the form \eqref{eq:pH_gen}. If $\widetilde{\mathcal{P}}_{\mathrm{c}}$ and $\widetilde{\mathcal{P}}_{\mathrm{f}}$ are the unique stabilizing solutions of \eqref{eq:lqg_care_ph_gen} and \eqref{eq:lqg_fare_ph_gen}, respectively, then the associated LQG controller defined by
\begin{align*}
\widetilde{E}_{\mathrm{c}} = \widetilde{\mathcal{P}}_{\mathrm{c}}^{-1}, \ \ \widetilde{A}_{\mathrm{c}}=\widetilde{\mathcal{P}}_{\mathrm{f}}^{-1}E^{-1}(J-R-BB^\top\widetilde{\mathcal{P}}_{\mathrm{c}} E-E\widetilde{\mathcal{P}}_{\mathrm{f}} BB^\top)E^{-1}\widetilde{\mathcal{P}}_{\mathrm{c}}^{-1}, \ \ \widetilde{B}_{\mathrm{c}} =B, \ \ \widetilde{C}_{\mathrm{c}}=B^\top
\end{align*}
is a port-Hamiltonian system in co-energy variable formulation.
\end{corollary}
\begin{proof}
Since $\widetilde E_c=\widetilde{\mathcal{P}}_{\mathrm{c}}^{-1}=\widetilde E_c^\top\succ 0$, we only have to prove that $\widetilde{A}_{\mathrm{c}}+\widetilde{A}_{\mathrm{c}}^\top\preceq 0$. Since $\widetilde{\mathcal{P}}_{\mathrm{f}}=E^{-1}$, one easily deduces that
%
\begin{align*}
E\widetilde{\mathcal{P}}_{\mathrm{c}}(\widetilde{A}_{\mathrm{c}}+\widetilde{A}_{\mathrm{c}}^\top)\widetilde{\mathcal{P}}_{\mathrm{c}} E
&= E\widetilde{\mathcal{P}}_{\mathrm{c}}(J-R) + (J-R)^\top\widetilde{\mathcal{P}}_{\mathrm{c}} E -2E\widetilde{\mathcal{P}}_{\mathrm{c}} BB^\top\widetilde{\mathcal{P}}_{\mathrm{c}} E-E\widetilde{\mathcal{P}}_{\mathrm{c}} BB^\top-BB^\top\widetilde{\mathcal{P}}_{\mathrm{c}} E = \\
&= -E\widetilde{\mathcal{P}}_{\mathrm{c}} BB^\top\widetilde{\mathcal{P}}_{\mathrm{c}} E-BB^\top-E\widetilde{\mathcal{P}}_{\mathrm{c}} BB^\top-BB^\top\widetilde{\mathcal{P}}_{\mathrm{c}} E = \\
&= -(I_n+E\widetilde{\mathcal{P}}_{\mathrm{c}})BB^\top(I_n+E\widetilde{\mathcal{P}}_{\mathrm{c}})^\top\preceq0,
\end{align*}
%
therefore $\widetilde{A}_{\mathrm{c}}+\widetilde{A}_{\mathrm{c}}^\top\preceq 0$, as requested.
\end{proof}
Alternatively to the proof of Corollary \ref{cor:cont_is_pH_gen}, it can be easily shown that the associated LQG controller corresponds to a co-energy formulation of the LQG controller that we constructed in Theorem~\ref{thm:cont_is_pH}.
Thus, it is clearly port-Hamiltonian.
\section{An error bound for pH-LQG reduced-order controllers}
As mentioned in Remark \ref{rem:con_to_LQG}, the reduced model from Algorithm \ref{alg:new} can be interpreted as a variant of LQG balanced truncation where the constant term $BB^\top$ is extended by the term $2R.$
It is thus obvious to study possible error bounds w.r.t.\@ the gap metric. Some considerations in this direction have been given in \cite{Wu16}. In contrast to the latter work, here we follow the reasoning in \cite{DamB14} and therefore aim at showing that the reduced Gramians satisfy two Lyapunov \emph{inequalities} which will yield an error bound (for the closed loop dynamics).
Note that the idea of using Lyapunov inequalities instead of Lyapunov equations to derive an error bound can already be found for linear time-varying systems in \cite{SanR04}.
From now on we assume that $\Sigma$ is the balanced Gramian solving the equations \eqref{eq:lqg_care_ph} and \eqref{eq:lqg_fare_ph} such that the reduced-order model $(\widehat{A}_r,\widehat{B}_r,\widehat{C}_r)$ results from truncation of the associated balanced system $(A,B,C)$.
\subsection{Error bounds for standard LQG balanced truncation}
\label{sec:standardLQGBT}
While the gap metric is standard in the context of LQG balanced truncation, cf.~\cite{Cur03,McFDG90,Mey90,MoeRS11}, for a self-contained presentation, let us recall some well-known concepts regarding system norms and coprime factorizations that can be found in, e.g., \cite{Cur90,McFDG90,SefO93,Vid84,ZhoDG96}. The Hardy spaces $\mathcal{H}_\infty^{p,m}$ and $\mathcal{H}_2^{p,m}$ are defined by
\begin{align*}
\mathcal{H}_2^{p,m} &:=\left\{ F\colon \mathbb C^+ \to \mathbb C^{p\times m} \ | \ F \text{ is analytic, } \| F\|_{\mathcal{H}_2}:=\left( \sup\limits_{\sigma>0} \int _{-\infty}^{\infty} \| F(\sigma + \imath \omega )\| _{\mathrm{F}}^2 \, \mathrm{d}\omega \right)^{\frac{1}{2}} < \infty \right\}, \\
\mathcal{H}_{\infty}^{p,m} &:=\left\{ F\colon \mathbb C^+ \to \mathbb C^{p\times m} \ | \ F \text{ is analytic, } \| F\|_{\mathcal{H}_\infty}:=\sup\limits_{z \in \mathbb C^+} \| F(\sigma + \imath \omega )\| _2 < \infty \right\}.
\end{align*}
Let us further introduce $\mathcal{R}\mathcal{H}_{\infty}^{p,m}$ and $\mathcal{R}\mathcal{H}_{\infty}^{p,m}$ consisting of matrix valued real rational functions that additionally are in $\mathcal{H}_{\infty}^{p,m}$ and $\mathcal{H}_{2}^{p,m}$, respectively.
Based on the stabilizing solution of \eqref{eq:lqg_care_ph} we may construct a normalized right coprime factorization of the form
\begin{align}\label{eq:left_coprime_G}
G(s)=C(sI_n-A)^{-1}B=N(s)M(s)^{-1},
\end{align}
where $M\in \mathcal{R}\mathcal{H}_\infty^{m,m}$, $N\in \mathcal{R}\mathcal{H}_2^{m,m}$ and the (control) closed loop matrix $A_{\widehat{\mathcal{P}}_{\mathrm{c}}}$ are given as follows:
\begin{align*}
N(s)=C(sI_n-A_{\widehat{\mathcal{P}}_{\mathrm{c}}})^{-1}B, \quad M(s)=I_m-B^\top \widehat{\mathcal{P}}_{\mathrm{c}}(sI_n-A_{\widehat{\mathcal{P}}_{\mathrm{c}}})^{-1} B, \quad A_{\widehat{\mathcal{P}}_{\mathrm{c}}} = A-BB^\top \widehat{\mathcal{P}}_{\mathrm{c}}.
\end{align*}
Here, right coprimeness means that there exist transfer functions $X,Y\in \mathcal{R} \mathcal{H}_\infty^{m,m}$ such that
\begin{align*}
X(s)M(s)+Y(s)N(s)=I.
\end{align*}
The normalization property is understood as the identity
\begin{align*}
M(-s)^\top M(s)+N(-s)^\top N(s)=I.
\end{align*}
The relevance of such factorizations is that they relate to the graph of the transfer function $G(s)=N(s)M(s)^{-1}:$
\begin{align*}
\mathrm{im} \begin{bmatrix} M \\ N \end{bmatrix} = \left\{ \begin{bmatrix} Mf \\ Nf \end{bmatrix} \ \colon \ f \in \mathcal{H}_2^{m, 1} \right\} \subset \mathcal{H}_2^{2m,1}.
\end{align*}
Moreover, for two transfer functions $G_1,G_2$ with normalized coprime factorizations $\begin{bsmallmatrix}M_1 \\ N_1 \end{bsmallmatrix}$ and $\begin{bsmallmatrix} M_2\\ N_2 \end{bsmallmatrix}$, the gap metric can be defined (see \cite{SefO93}) as follows
\begin{align*}
\delta_{\mathrm{gap}}(G_1,G_2) =\max \left\{ \vec{\delta}_{\mathrm{gap}}(G_1,G_2),\vec{\delta}_{\mathrm{gap}}(G_2,G_1) \right\},
\end{align*}
where the directed gap $\vec{\delta}_{\mathrm{gap}}(G_1,G_2)$ is given by
\begin{align*}
\vec{\delta}_{\mathrm{gap}}(G_1,G_2)=\inf\limits_{\Pi\in \mathcal{H}_\infty^{m,m}} \left\|\begin{bmatrix} M_1 \\ N_1 \end{bmatrix} - \begin{bmatrix} M_2 \\ N_2 \end{bmatrix}\Pi \right\|_{\mathcal{H}_\infty}.
\end{align*}
For classical LQG balanced truncation (i.e.~when $R=0$ in \eqref{eq:lqg_fare_ph}), we have an error bound of the form
\begin{align}\label{eq:class_LQG_err_bnd}
\vec{\delta}_{\mathrm{gap}}(G,\widehat{G}_r)\le \left\|\begin{bmatrix} M \\ N \end{bmatrix} - \begin{bmatrix} \widehat{M}_r \\ \widehat{N}_r \end{bmatrix} \right\|_{\mathcal{H}_\infty}:=\sup_{z\in \mathbb C_{+}}\left\| \begin{bmatrix} M(z) \\ N(z) \end{bmatrix}- \begin{bmatrix} \widehat{M}_r(z) \\ \widehat{N}_r(z) \end{bmatrix}\right\|_2\le 2\sum_{i=r+1}^n \theta_i,
\end{align}
where $\theta_i=\frac{\sigma_i}{\sqrt{1+\sigma_i^2}}$ and $\sigma_i=\sqrt{\lambda_i(\widehat{\mathcal{P}}_{\mathrm{f}}\widehat{\mathcal{P}}_{\mathrm{c}})}$ are the so-called LQG characteristic values and where $\begin{bsmallmatrix} \widehat{M}_r\\ \widehat{N}_r \end{bsmallmatrix}$ is a normalized right coprime factorization of the transfer function of the reduced-order model.
Let us further emphasize that this gap metric error bound is particularly useful for analyzing the closed loop behavior of the associated reduced LQG controller. For more details on specific closed loop estimates, we refer to, e.g., \cite{Vid84}.
\subsection{Lyapunov inequalities for structure-preserving LQG balanced truncation}
Let us return to the general case where $R\succeq 0$ in \eqref{eq:lqg_fare_ph} and establish an error bound structurally identical to \eqref{eq:class_LQG_err_bnd}. For this purpose, in what follows we exploit an approach from \cite{DamB14} (similar to the arguments provided in \cite{SanR04}) where specific Lyapunov inequalities have been used to derive classical error bounds.
Note that $\widehat{\mathcal{P}}_{\mathrm{c}}$ satisfies
\begin{align}
A_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top \widehat{\mathcal{P}}_{\mathrm{c}} + \widehat{\mathcal{P}}_{\mathrm{c}} A_{\widehat{\mathcal{P}}_{\mathrm{c}}} + C^\top C +\widehat{\mathcal{P}}_{\mathrm{c}} BB^\top \widehat{\mathcal{P}}_{\mathrm{c}} =0.
\end{align}
In other words, $\widehat{\mathcal{P}}_{\mathrm{c}}$ coincides with the observability Gramian of the system
\begin{equation}\label{eq:coprime_real}
\begin{aligned}
\dot{w} &= A_{\widehat{\mathcal{P}}_{\mathrm{c}}} w + B u, \quad w(0)=0, \\
y&= \underbrace{\begin{bmatrix} -B^\top \widehat{\mathcal{P}}_{\mathrm{c}} \\ C \end{bmatrix}}_{C_{\widehat{\mathcal{P}}_{\mathrm{c}}}} w+ \begin{bmatrix} I_m \\ 0 \end{bmatrix} u
\end{aligned}
\end{equation}
which is a realization of $\begin{bsmallmatrix} M\\N \end{bsmallmatrix}$.
In the standard case, i.e.~using $\mathcal{Q}_\mathrm{f}=BB^\top$, the controllability Gramian of this system is given by $(I_n+\widehat{\mathcal{P}}_{\mathrm{f}}\widehat{\mathcal{P}}_{\mathrm{c}})^{-1}\widehat{\mathcal{P}}_{\mathrm{f}}$ (see, e.g., \cite{Cur03}) and, thus, balancing the matrices $\widehat{\mathcal{P}}_{\mathrm{f}}$ and $\widehat{\mathcal{P}}_{\mathrm{c}}$ will also transform the controllability and observability Gramians of $\begin{bsmallmatrix}M\\ N \end{bsmallmatrix}$ into diagonal (though not necessarily equal) form.
The following proposition shows that in our setting, i.e.~using $\mathcal{Q}_\mathrm{f}=BB^\top+2R$, the matrix $(I_n+\widehat{\mathcal{P}}_{\mathrm{f}} \widehat{\mathcal{P}}_{\mathrm{c}})^{-1}\widehat{\mathcal{P}}_{\mathrm{f}}$ is in general no longer the controllability Gramian of \eqref{eq:coprime_real}, but it still satisfies at least an associated Lyapunov inequality.
\begin{proposition}
Let $(J,R,Q,B)$ define a minimal port-Hamiltonian system of the form \eqref{eq:pH} and let $\widehat{\mathcal{P}}_{\mathrm{c}}$ and $\widehat{\mathcal{P}}_{\mathrm{f}}$ denote the unique stabilizing solutions to the control and filter Riccati equations \eqref{eq:lqg_care_ph} and \eqref{eq:lqg_fare_ph}, respectively.
For the system \eqref{eq:coprime_real} and $\mathcal{L}=(I_n+\widehat{\mathcal{P}}_{\mathrm{f}} \widehat{\mathcal{P}}_{\mathrm{c}})^{-1}\widehat{\mathcal{P}}_{\mathrm{f}}$ it holds that
\begin{align}\label{eq:obs_ineq_cp}
A_{\widehat{\mathcal{P}}_{\mathrm{c}}} \mathcal{L} + \mathcal{L} A_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top+BB^\top \preceq 0.
\end{align}
\end{proposition}
\begin{proof}
\newcommand{A_{\Phatf}}{A_{\widehat{\mathcal{P}}_{\mathrm{f}}}}
\newcommand{\Phatf}{\widehat{\mathcal{P}}_{\mathrm{f}}}
\newcommand{\Phatc}{\widehat{\mathcal{P}}_{\mathrm{c}}}
\newcommand{\tpc^{-1}}{\Phatc^{-1}}
Using $\widehat{\mathcal{P}}_{\mathrm{f}}=\widehat{\mathcal{P}}_{\mathrm{f}}^\top$ and $\widehat{\mathcal{P}}_{\mathrm{c}}=\widehat{\mathcal{P}}_{\mathrm{c}}^\top$, one easily verifies that $\mathcal{L}=\mathcal{L}^\top$. Hence, instead of \eqref{eq:obs_ineq_cp} we may show that
\begin{align*}
A_{\widehat{\mathcal{P}}_{\mathrm{c}}} \mathcal{L}^\top + \mathcal{L} A_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top+BB^\top \preceq 0.
\end{align*}
Since $I_n+\widehat{\mathcal{P}}_{\mathrm{f}} \widehat{\mathcal{P}}_{\mathrm{c}}$ is invertible, this is however equivalent to showing that
\begin{align*}
(I_n+\widehat{\mathcal{P}}_{\mathrm{f}} \widehat{\mathcal{P}}_{\mathrm{c}}) A_{\widehat{\mathcal{P}}_{\mathrm{c}}} \widehat{\mathcal{P}}_{\mathrm{f}}
%
+\widehat{\mathcal{P}}_{\mathrm{f}} A_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top (I_n+\widehat{\mathcal{P}}_{\mathrm{c}}\widehat{\mathcal{P}}_{\mathrm{f}})
%
+ (I_n+\widehat{\mathcal{P}}_{\mathrm{f}} \widehat{\mathcal{P}}_{\mathrm{c}}) BB^\top (I_n+\widehat{\mathcal{P}}_{\mathrm{c}} \widehat{\mathcal{P}}_{\mathrm{f}}) \preceq 0.
\end{align*}
We now obtain
\begin{align*}
& (I_n+\widehat{\mathcal{P}}_{\mathrm{f}} \widehat{\mathcal{P}}_{\mathrm{c}}) A_{\widehat{\mathcal{P}}_{\mathrm{c}}} \widehat{\mathcal{P}}_{\mathrm{f}}
%
+\widehat{\mathcal{P}}_{\mathrm{f}} A_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top (I_n +\widehat{\mathcal{P}}_{\mathrm{c}} \widehat{\mathcal{P}}_{\mathrm{f}})
%
+ (I_n+\widehat{\mathcal{P}}_{\mathrm{f}}\widehat{\mathcal{P}}_{\mathrm{c}}) BB^\top (I_n + \widehat{\mathcal{P}}_{\mathrm{c}} \widehat{\mathcal{P}}_{\mathrm{f}} ) \\
&= (I_n+\widehat{\mathcal{P}}_{\mathrm{f}} \widehat{\mathcal{P}}_{\mathrm{c}}) A \widehat{\mathcal{P}}_{\mathrm{f}} + \widehat{\mathcal{P}}_{\mathrm{f}} A^\top (I_n+\widehat{\mathcal{P}}_{\mathrm{c}} \widehat{\mathcal{P}}_{\mathrm{f}}) \\
%
&\quad - (I_n+\widehat{\mathcal{P}}_{\mathrm{f}} \widehat{\mathcal{P}}_{\mathrm{c}})BB^\top \widehat{\mathcal{P}}_{\mathrm{c}}\widehat{\mathcal{P}}_{\mathrm{f}} - \widehat{\mathcal{P}}_{\mathrm{f}} \widehat{\mathcal{P}}_{\mathrm{c}} BB^\top (I_n+\widehat{\mathcal{P}}_{\mathrm{c}} \widehat{\mathcal{P}}_{\mathrm{f}}) \\
%
&\quad + BB^\top + \widehat{\mathcal{P}}_{\mathrm{f}} \widehat{\mathcal{P}}_{\mathrm{c}} BB^\top+ BB^\top \widehat{\mathcal{P}}_{\mathrm{c}} \widehat{\mathcal{P}}_{\mathrm{f}} + \widehat{\mathcal{P}}_{\mathrm{f}} \widehat{\mathcal{P}}_{\mathrm{c}} BB^\top \widehat{\mathcal{P}}_{\mathrm{c}} \widehat{\mathcal{P}}_{\mathrm{f}} \\
%
&= A \widehat{\mathcal{P}}_{\mathrm{f}} + \widehat{\mathcal{P}}_{\mathrm{f}} A^\top + BB^\top \\
&\quad - \widehat{\mathcal{P}}_{\mathrm{f}} \widehat{\mathcal{P}}_{\mathrm{c}} BB^\top \widehat{\mathcal{P}}_{\mathrm{c}} \widehat{\mathcal{P}}_{\mathrm{f}}+\widehat{\mathcal{P}}_{\mathrm{f}} \widehat{\mathcal{P}}_{\mathrm{c}} A \widehat{\mathcal{P}}_{\mathrm{f}}+ \widehat{\mathcal{P}}_{\mathrm{f}} A^\top \widehat{\mathcal{P}}_{\mathrm{c}} \widehat{\mathcal{P}}_{\mathrm{f}} \\
&= -2R + \widehat{\mathcal{P}}_{\mathrm{f}}\left( C^\top C + \widehat{\mathcal{P}}_{\mathrm{c}} A +A^\top \widehat{\mathcal{P}}_{\mathrm{c}} - \widehat{\mathcal{P}}_{\mathrm{c}}^\top BB^\top \widehat{\mathcal{P}}_{\mathrm{c}} \right) \widehat{\mathcal{P}}_{\mathrm{f}}= -2R \preceq 0 . \qedhere
\end{align*}
\end{proof}
We are ready to state an error bound for a reduced-order model obtained either by Algorithm \ref{alg:new} or (equivalently) by balanced truncation w.r.t.\@ the Gramians $\mathcal{P}_\mathrm{c}$ and $\mathcal{P}_\mathrm{f}$ from Appendix \ref{apdxA}.
\begin{theorem}\label{thm:err_bnd_lqg}
Let $(J,R,Q,B)$ define a minimal port-Hamiltonian system of the form \eqref{eq:pH} which is balanced w.r.t.~the stabilizing solutions of \eqref{eq:lqg_care_ph} and \eqref{eq:lqg_fare_ph}.
Furthermore, let $(\widehat{A}_r,\widehat{B}_r,\widehat{C}_r)$ be a reduced-order model obtained by Algorithm \ref{alg:new}.
Denote the balanced Gramians $\widehat{\mathcal{P}}_{\mathrm{f}}=\widehat{\mathcal{P}}_{\mathrm{c}}=\Sigma=\mathrm{diag}(\Sigma_1,\Sigma_2)$, where $\Sigma_2=\mathrm{diag}(\sigma_{r+1},\dots,\sigma_n).$
If $\begin{bsmallmatrix} M\\ N \end{bsmallmatrix}$ and $\begin{bsmallmatrix} \widehat{M}_r\\ \widehat{N}_r \end{bsmallmatrix}$ are the associated right coprime factorizations as specified in \eqref{eq:coprime_real}, then
\begin{align}\label{eq:err_bnd_lqg}
\left\| \begin{bmatrix} M \\ N \end{bmatrix} - \begin{bmatrix} \widehat{M}_r \\ \widehat{N}_r \end{bmatrix}\right\|_{\mathcal{H}_\infty} \le 2 \sum_{i=r+1}^n \frac{\sigma_i}{\sqrt{1+\sigma_i^2}}.
\end{align}
\end{theorem}
\begin{proof}
The proof uses almost the exact same arguments as given in \cite{DamB14}.
In contrast to the latter reference, here the matrices $\widehat{\mathcal{P}}_{\mathrm{c}}$ and $\mathcal{L}$ satisfying the Lyapunov (in)equalities do not coincide.
While this only requires minor modifications in the reasoning from \cite{DamB14}, for self-consistency we provide a full proof of the assertion.
First note that since $(A,B,C)$ is balanced, from the above considerations, it follows that
\begin{align*}
\mathcal{L}=(I_n+\widehat{\mathcal{P}}_{\mathrm{f}} \widehat{\mathcal{P}}_{\mathrm{c}})^{-1}\widehat{\mathcal{P}}_{\mathrm{f}} = (I_n + \Sigma^2)^{-1}\Sigma = \mathrm{diag}\left(\frac{\sigma_i}{1+\sigma_i^2}\right), \ i=1,\dots,n,
\end{align*}
as well as
\begin{align}\label{eq:orig_lqg_bal}
A_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top \Sigma + \Sigma A_{\widehat{\mathcal{P}}_{\mathrm{c}}} + C_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top C_{\widehat{\mathcal{P}}_{\mathrm{c}}} = 0,\quad
A_{\widehat{\mathcal{P}}_{\mathrm{c}}} \mathcal{L} + \mathcal{L} A_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top +BB^\top \preceq 0.
\end{align}
Similarly, for the reduced-order surrogate of \eqref{eq:coprime_real} which is defined by
\begin{align*}
\widehat{A}_{\Sigma}&=\widehat{A}_r- \widehat{B}_r\widehat{B}_r^\top\Sigma_1, \quad \widehat{B}_{\Sigma}=\widehat{B}_r, \quad \widehat{C}_{\Sigma}= \begin{bmatrix} -\widehat{B}_r^\top \Sigma_1 \\ \widehat{C}_r \end{bmatrix}
\end{align*}
we conclude that
\begin{equation}\label{eq:red_lqg_bal}
\begin{aligned}
\widehat{A}_{\Sigma}^\top \Sigma_1 +\Sigma_1 \widehat{A}_{\Sigma}+\widehat{C}_{\Sigma}^\top \widehat{C}_{\Sigma} = 0,\quad
\widehat{A}_{\Sigma} \mathcal{L}_1 + \mathcal{L}_1 \widehat{A}_{\Sigma}^\top + \widehat{B}_{\Sigma} \widehat{B}_{\Sigma}^\top \preceq 0 .
\end{aligned}
\end{equation}
where $\mathcal{L}_1=\mathrm{diag}(\frac{\sigma_i}{1+\sigma_i^2}), i=1,\dots,r.$
Since the $\mathcal{H}_\infty$-norm is the $L_2$-$L_2$-induced norm between inputs and outputs (e.g., \cite[Section 5]{Ant05a}), we thus focus on the two systems
\begin{align*}
\dot{w}&=A_{\widehat{\mathcal{P}}_{\mathrm{c}}} w + B u, \quad w(0)=0, \quad y= C_{\widehat{\mathcal{P}}_{\mathrm{c}}}w+\begin{bmatrix}I_m \\ 0 \end{bmatrix} u, \\
\dot{\widehat{w}}_r&=\widehat{A}_{\Sigma} \widehat{w}_r + \widehat{B}_{\Sigma} u, \quad \widehat{w}_r(0)=0,\quad \widehat{y}_r= \widehat{C}_{\Sigma}\widehat{w}_r+\begin{bmatrix}I_m \\ 0 \end{bmatrix}u.
\end{align*}
Since by assumption $(A,B,C)$ is balanced, i.e., $\widehat{\mathcal{P}}_{\mathrm{f}}=\Sigma=\widehat{\mathcal{P}}_{\mathrm{c}}$ we also obtain
\begin{align*}
A_{\widehat{\mathcal{P}}_{\mathrm{c}}}&=A-BB^\top\Sigma =\begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}:= \begin{bmatrix} \widehat{A}_r & * \\ * & * \end{bmatrix} - \begin{bmatrix} \widehat{B}_r \\ B_2 \end{bmatrix} \begin{bmatrix} \widehat{B}_r \\ B_2 \end{bmatrix}^\top \begin{bmatrix} \Sigma_1 & 0 \\ 0 & \Sigma_2 \end{bmatrix} = \begin{bmatrix} \widehat{A}_r-\widehat{B}_r \widehat{B}_r^\top \Sigma_1 & * \\ * & * \end{bmatrix} \\
C_{\widehat{\mathcal{P}}_{\mathrm{c}}} &:= \begin{bmatrix} C_1 & C_2\end{bmatrix}= \begin{bmatrix} -\widehat{B}_r^\top \Sigma_1 & * \\ \widehat{C}_r & * \end{bmatrix}.
\end{align*}
In other words, the reduced right coprime factorization $(\widehat{A}_{\Sigma},\widehat{B}_{\Sigma},\widehat{C}_{\Sigma})$ is obtained by truncating the original right coprime factorization $(A_{\widehat{\mathcal{P}}_{\mathrm{c}}},B,C_{\widehat{\mathcal{P}}_{\mathrm{c}}})$.
With the state vector $w=\begin{bsmallmatrix} w_1 \\ w_2 \end{bsmallmatrix}$ being partitioned accordingly, we obtain
\begin{align*}
\frac{\, \text{d}}{\, \text{d} t} \langle w_1 - \widehat{w}_r,\Sigma_1(w_1 -\widehat{w}_r)\rangle &= 2 \langle w_1-\widehat{w}_r,\Sigma_1 [A_{11},A_{12}] \begin{bmatrix} w_1-\widehat{w}_r \\ w_2 \end{bmatrix} \rangle \\
\frac{\, \text{d}}{\, \text{d} t} \langle w_2 ,\Sigma_2 w_2\rangle &= 2\langle w_2,\Sigma_2 (A_{21}w_1 + A_{22}w_2 + B_2 u)\rangle.
\end{align*}
Note that it holds that $y-\widehat{y}_r=C_{\widehat{\mathcal{P}}_{\mathrm{c}}} \begin{bsmallmatrix} w_1-\widehat{w}_r \\ w_2 \end{bsmallmatrix}$ which we use to show
\begin{align*}
-\| y-\widehat{y}_r\|^2&=-\begin{bmatrix} w_1-\widehat{w}_r \\ w_2 \end{bmatrix}^\top C_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top C_{\widehat{\mathcal{P}}_{\mathrm{c}}} \begin{bmatrix} w_1-\widehat{w}_r \\ w_2 \end{bmatrix} = \begin{bmatrix} w_1-\widehat{w}_r \\ w_2 \end{bmatrix}^\top (A_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top \Sigma +\Sigma A_{\widehat{\mathcal{P}}_{\mathrm{c}}} ) \begin{bmatrix} w_1-\widehat{w}_r \\ w_2 \end{bmatrix} \\[1ex]
&=2 (w_1-\widehat{w}_r)^\top \Sigma_1 \begin{bmatrix}A_{11} & A_{12} \end{bmatrix} \begin{bmatrix} w_1-\widehat{w}_r \\ w_2 \end{bmatrix}
+2 w_2 ^\top \Sigma_2 \begin{bmatrix} A_{21} & A_{22} \end{bmatrix} \begin{bmatrix} w_1 - \widehat{w}_r \\ w_2 \end{bmatrix}.
\end{align*}
Combining all of the previous results leads to
\begin{align*}
-\| y-\widehat{y}_r\|^2 \frac{\, \text{d}}{\, \text{d} t} \langle w_1 - \widehat{w}_r,\Sigma_1(w_1 -\widehat{w}_r)\rangle + \frac{\, \text{d}}{\, \text{d} t} \langle w_2,\Sigma_2 w_2\rangle-2w_2^\top \Sigma_2 (A_{21}\widehat{w}_r+B_2u)
\end{align*}
and finally
\begin{align}\label{eq:aux2}
\int_0^T \|y(t)-\widehat{y}_r(t)\|^2 \, \text{d} t \le 2\int_0^T w_2(t)^\top \Sigma_2 (A_{21}\widehat{w}_r(t)+B_2u(t))\, \text{d} t.
\end{align}
Still following \cite{DamB14}, we want to show that
\begin{align}\label{eq:aux3}
4\int_0^T \| u(t)\| ^2 \, \text{d} t \ge 2 \int_0^T w_2(t)^\top \mathcal{L}_2^{-1}(A_{21}\widehat{w}_r(t)+B_2 u(t))\, \text{d} t),
\end{align}
where $\mathcal{L}_2=\mathrm{diag}(\frac{\sigma_i}{1+\sigma_i^2}),i=r+1,\dots,n.$
From \eqref{eq:orig_lqg_bal} it follows that
\begin{align*}
\mathcal{L}^{-1} A_{\widehat{\mathcal{P}}_{\mathrm{c}}}+ A_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top \mathcal{L}^{-1} + \mathcal{L}^{-1} BB^\top \mathcal{L}^{-1} \preceq 0
\end{align*}
which due to Schur complement properties also implies that
\begin{align*}
\begin{bmatrix} \mathcal{L}^{-1} A_{\widehat{\mathcal{P}}_{\mathrm{c}}}+ A_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top \mathcal{L}^{-1} & \mathcal{L}^{-1}B \\ B^\top \mathcal{L}^{-1} & -I_m \end{bmatrix} \preceq 0.
\end{align*}
This can be rewritten as follows
\begin{align*}
\begin{bmatrix} 0 & 0 \\ 0 & I_m \end{bmatrix}\succeq \begin{bmatrix}A_{\widehat{\mathcal{P}}_{\mathrm{c}}} & B \\ I_n & 0 \end{bmatrix}^\top \begin{bmatrix} 0 & \mathcal{L}^{-1} \\ \mathcal{L}^{-1} & 0 \end{bmatrix} \begin{bmatrix} A_{\widehat{\mathcal{P}}_{\mathrm{c}}} & B \\ I_n & 0 \end{bmatrix}.
\end{align*}
Let us multiply the previous inequality with $\begin{bmatrix} (w_1 + \widehat{w}_r)^\top & w_2^\top & 2 u^\top\end{bmatrix}$ and $\begin{bsmallmatrix} w_1 + \widehat{w}_r \\ w_2 \\ 2 u \end{bsmallmatrix}$ to obtain
\begin{align*}
4\| u \|^2 &\ge 2(w_1+\widehat{w}_r)^\top\mathcal{L}_1^{-1}(\begin{bmatrix} A_{11} & A_{12} \end{bmatrix} \begin{bmatrix} w_1 + \widehat{w}_r\\ w_2 \end{bmatrix}+2B_1u) \\
&\qquad + 2w_2^\top \mathcal{L}_2^{-1}(\begin{bmatrix} A_{21} & A_{22} \end{bmatrix} \begin{bmatrix} w_1 + \widehat{w}_r \\ w_2 \end{bmatrix}+2B_2 u ).
\end{align*}
As before, we obtain
\begin{align*}
\frac{\, \text{d}}{\, \text{d} t} (w_1+\widehat{w}_r)^\top \mathcal{L}_1^{-1} (w_1 + \widehat{w}_r) &= 2 (w_1+\widehat{w}_r)^\top \mathcal{L}_1^{-1} (\begin{bmatrix} A_{11} & A_{12} \end{bmatrix} \begin{bmatrix} w_1 + \widehat{w}_r \\ w_2 \end{bmatrix} + 2B_1 u ) \\
\frac{\, \text{d}}{\, \text{d} t} (w_2^\top \mathcal{L}_2^{-1} w_2) &= 2 w_2^\top \mathcal{L}_2^{-1}(A_{21}w_1 + A_{22}w_2 + B_2 u).
\end{align*}
Combining the last three (in)equalities leads to
\begin{align*}
4\|u\|^2 \ge \frac{\, \text{d}}{\, \text{d} t} ((w_1+\widehat{w}_r)^\top \mathcal{L}_1^{-1} (w_1 +\widehat{w}_r))+\frac{\, \text{d}}{\, \text{d} t}(w_2^\top \mathcal{L}_2^{-1}w_2)+2 w_2^\top \mathcal{L}_2^{-1}(A_{21}\widehat{w}_r+B_2 u).
\end{align*}
Integration and using $w(0)=0$, $\widehat{w}_r(0)=0$ yields \eqref{eq:aux3}.
If $\Sigma_2=\sigma_{\nu}$, then $\mathcal{L}_2=\frac{\sigma_{\nu}}{1+\sigma_{\nu}^2}$. Multiplication of \eqref{eq:aux3} with $\frac{\sigma_{\nu}^2}{1+\sigma_{\nu}^2}$ implies
\begin{align*}
\frac{4\sigma_{\nu}^2}{1+\sigma_{\nu}^2} \int_0^T \|u(t)\|^2 \, \text{d} t &\ge 2 \int_0^T w_2(t)^\top \frac{\sigma_{\nu}^2}{1+\sigma_{\nu}^2}\mathcal{L}_2^{-1}(A_{21}\widehat{w}_r(t)+B_2u(t))\, \text{d} t) \\
&= 2\int_0^T w_2(t)^\top \Sigma_2 (A_{21}\widehat{w}_r(t)+B_2u(t))\, \text{d} t) \ge \int_0^T \| y(t)-\widehat{y}_r(t)\|^2\, \text{d} t.
\end{align*}
We can now proceed recursively and truncate step by step one after the other state such that finally we obtain
\begin{align*}
\|y-\widehat{y}_r\|_{L^2([0,T];\mathbb R^m)} \le 2 (\theta_{r+1}+\cdots+\theta_{n})\| u\|_{L^2([0,T];\mathbb R^m)},
\end{align*}
where $\theta_i=\frac{\sigma_i}{\sqrt{1+\sigma_i^2}}$, for $i=r+1,\dots,n$.
As the previous bound is independent of the time $T$, we can consider its limit $T\to \infty$ which shows the assertion.
\end{proof}
\begin{remark}
Note that the final arguments remain valid if $\Sigma_2=\sigma_\nu I$, i.e., in the case where some of the characteristic values appear multiple times. In this sense, the original formulation from \cite{DamB14} is beneficial since the summation in the error bound only includes distinct characteristic values. For $R=0$, the two Gramians coincide with the ones used in classical LQG balanced truncation. While due to the special pH structure this would result in a purely Hamiltonian system, the right coprime framework could still be followed as long as $(J,B,C)$ is minimal.
\end{remark}
\begin{remark}
The error bound in Theorem~\ref{thm:err_bnd_lqg} scales with the $\sigma_i$ values which are the square roots of the eigenvalues of $\widehat{\mathcal{P}}_{\mathrm{f}}\widehat{\mathcal{P}}_{\mathrm{c}}$.
In particular, we observe that these values are invariant under similarity transformations, which follows from the fact that any similarity transformation based on an invertible matrix $T$ leads to the transformed Gramians
$\overline{\mathcal{P}}_{\mathrm{f}} = T\widehat{\mathcal{P}}_{\mathrm{f}} T^\top$ and $\overline{\mathcal{P}}_{\mathrm{c}} = T^{-\top}\widehat{\mathcal{P}}_{\mathrm{c}} T^{-1}$ which satisfy
\begin{equation*}
\overline{\mathcal{P}}_{\mathrm{f}} \overline{\mathcal{P}}_{\mathrm{c}} = T\widehat{\mathcal{P}}_{\mathrm{f}} \widehat{\mathcal{P}}_{\mathrm{c}} T^{-1}.
\end{equation*}
Thus, $\overline{\mathcal{P}}_{\mathrm{f}} \overline{\mathcal{P}}_{\mathrm{c}}$ and $\widehat{\mathcal{P}}_{\mathrm{f}} \widehat{\mathcal{P}}_{\mathrm{c}}$ have the same eigenvalues and, hence, the $\sigma_i$ values are invariant under similarity transformations.
For the special case $R=0$, these coincide with the classical LQG characteristic values, cf.~subsection~\ref{sec:standardLQGBT}.
\end{remark}
\subsection[Minimizing the error bound - choosing the right Q]{Minimizing the error bound - choosing the right $Q$}\label{subsec:choosing_Q}
While the error bound in Theorem \ref{thm:err_bnd_lqg} is structurally the same as in classical LQG balanced truncation, the characteristic values are derived from a filter Riccati equation including the additional term $2R.$ Since we have seen that this specific choice of the covariance matrix implies $\widehat{\mathcal{P}}_{\mathrm{f}}=Q^{-1}$, the question arises whether there exists a particularly \emph{good} pH representation in the sense that the error bound provided in Theorem \ref{thm:err_bnd_lqg} is minimized. With this in mind, first note that the error bound is determined by the eigenvalues of the matrix
\begin{align*}
\mathcal{L} \widehat{\mathcal{P}}_{\mathrm{c}} = (I_n+\widehat{\mathcal{P}}_{\mathrm{f}} \widehat{\mathcal{P}}_{\mathrm{c}})^{-1}\widehat{\mathcal{P}}_{\mathrm{f}} \widehat{\mathcal{P}}_{\mathrm{c}}= (I_n+Q^{-1}\widehat{\mathcal{P}}_{\mathrm{c}})^{-1}Q^{-1} \widehat{\mathcal{P}}_{\mathrm{c}}= (Q+\widehat{\mathcal{P}}_{\mathrm{c}})^{-1} \widehat{\mathcal{P}}_{\mathrm{c}}.
\end{align*}
Since $\widehat{\mathcal{P}}_{\mathrm{c}}=\widehat{\mathcal{P}}_{\mathrm{c}}^\top \succ 0$, we may consider a Cholesky decomposition $\widehat{\mathcal{P}}_{\mathrm{c}}=L_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top L _{\widehat{\mathcal{P}}_{\mathrm{c}}} $ such that:
\begin{align}\label{eq:lgq_char_val}
\theta_i=\sqrt{
\lambda_i( \mathcal{L}\widehat{\mathcal{P}}_{\mathrm{c}})}=\sqrt{\lambda_i(L_{\widehat{\mathcal{P}}_{\mathrm{c}}} \mathcal{L} \widehat{\mathcal{P}}_{\mathrm{c}} L_{\widehat{\mathcal{P}}_{\mathrm{c}}}^{-1} )}=\sqrt{\lambda_i(L_{\widehat{\mathcal{P}}_{\mathrm{c}}} (Q+\widehat{\mathcal{P}}_{\mathrm{c}})^{-1}L_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top)}.
\end{align}
Following \cite{BeaMV19}, let us assume that $X=X^\top \succ 0$ is a solution to \eqref{eq:kyp-lmi} and consider the associated alternative pH representation \eqref{eq:pHX}.
Since \eqref{eq:lqg_care_ph} only depends on $(A,B,C)$ its solution is independent of $R_X$ and still given by $\widehat{\mathcal{P}}_{\mathrm{c}}$. On the other hand, \eqref{eq:lqg_fare_ph} explicitly depends on $R_X,$ which for $X\neq Q$ will generally be affected by the previous transformation. In particular, for $\theta_i$, we have achieved the following transformation
\begin{align*}
\theta_i = \sqrt{\lambda_i(L_{\widehat{\mathcal{P}}_{\mathrm{c}}} (Q+\widehat{\mathcal{P}}_{\mathrm{c}})^{-1}L_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top)} \quad \to \quad \sqrt{\lambda_i(L_{\widehat{\mathcal{P}}_{\mathrm{c}}} (X+\widehat{\mathcal{P}}_{\mathrm{c}})^{-1}L_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top))} = \widehat{\theta}_i.
\end{align*}
Assume now that $X_{\mathrm{max}}=X_{\mathrm{max}}^\top \succ 0$ is the \emph{maximal} solution to \eqref{eq:kyp-lmi}. Hence, it holds that $X_{\mathrm{max}}\succeq Q$ as well as
\begin{align*}
X_{\mathrm{max}}+\widehat{\mathcal{P}}_{\mathrm{c}}\succeq Q+\widehat{\mathcal{P}}_{\mathrm{c}}\succ 0
\end{align*}
also implying that
\begin{align*}
0\prec ( X_{\mathrm{max}}+\widehat{\mathcal{P}}_{\mathrm{c}})^{-1} \preceq (Q+\widehat{\mathcal{P}}_{\mathrm{c}})^{-1}.
\end{align*}
Using the Courant-Fischer-Weyl min-max principle (\cite[Theorem 8.1.2]{GolV13}) allows us to conclude that
\begin{align*}
\theta_i^2 &= \lambda_i(L_{\widehat{\mathcal{P}}_{\mathrm{c}}} (Q+\widehat{\mathcal{P}}_{\mathrm{c}})^{-1}L_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top))\\
&=\min_{\substack{\mathcal{X}_i\subset \mathbb R^n\\ \mathrm{dim}(\mathcal{X}_i)=i}}\ \max_{\substack{z\in \mathcal{X}_i\\ \|z\|=1}} z^\top (L_{\widehat{\mathcal{P}}_{\mathrm{c}}} (Q+\widehat{\mathcal{P}}_{\mathrm{c}})^{-1}L_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top))z \\ & \ge \min_{\substack{\mathcal{X}_i\subset \mathbb R^n\\ \mathrm{dim}(\mathcal{X}_i)=i}}\ \max_{\substack{z\in \mathcal{X}_i\\ \|z\|=1}} z^\top (L_{\widehat{\mathcal{P}}_{\mathrm{c}}} (X+\widehat{\mathcal{P}}_{\mathrm{c}})^{-1}L_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top))z \\
&=\lambda_i(L_{\widehat{\mathcal{P}}_{\mathrm{c}}} (X+\widehat{\mathcal{P}}_{\mathrm{c}})^{-1}L_{\widehat{\mathcal{P}}_{\mathrm{c}}}^\top))=\widehat{\theta}_i^2.
\end{align*}
The previous considerations suggest to first replace \eqref{eq:pH} with the alternative representation \eqref{eq:pHX} by computing the maximal solution $X_{\mathrm{max}}$ to \eqref{eq:kyp-lmi} in order to minimize the error bound in Theorem \ref{thm:err_bnd_lqg}. Let us emphasize that such a computation is numerically demanding and the results first of all are of theoretical nature.
Furthermore, it should be noted that the transformation corresponds to choosing a different Hamiltonian, that might in turn have a different physical meaning. Whether this affects positively or negatively the method depends on the specific application.%
\begin{remark}
\label{rem:mixedBalancing}
In this subsection, we demonstrated that exchanging the weighting matrix $BB^\top+2R$ by $BB^\top+2R_{X_{\mathrm{max}}}$ yields the Gramian $\widehat{\mathcal{P}}_{\mathrm{f}} = X_{\mathrm{max}}^{-1}$ instead of $\widehat{\mathcal{P}}_{\mathrm{f}} = Q^{-1}$.
This approach allows for an alternative interpretation of the proposed balancing procedure.
To this end, we point out that the Gramian $\widehat{\mathcal{P}}_{\mathrm{f}} = X_{\mathrm{max}}^{-1}$ is not only the unique stabilizing solution of the modified filter algebraic Riccati equation \eqref{eq:lqg_fare_ph} with weighting matrix $BB^\top+2R_{X_{\mathrm{max}}}$, but it is also the minimal solution of the dual KYP-LMI \eqref{eq:dualKYPLMI}.
Thus, the proposed balancing procedure can be regarded as a mixture between classical LQG balanced truncation and positive real balanced truncation, in the sense that we balance the stabilizing solution of the algebraic Riccati equation \eqref{eq:lqg_care_ph} and the minimal solution of the dual KYP-LMI \eqref{eq:dualKYPLMI}.
A similar mixed approach has been proposed in \cite{UnnVE07}, where the solution of a Lyapunov equation and the solution of an algebraic Riccati equation have been balanced.
\end{remark}
\section{Consequences for standard balanced truncation}\label{sec:classic_bt}
In this section, we briefly discuss how the above framework can be used to modify standard balanced truncation for port-Hamiltonian systems such that analogous results and error bounds hold true.
Note that for balanced truncation realized within the effort-constraint reduction framework (see \cite{PolvdS12}), in \cite{Wuetal18} the authors have derived an error bound based on an auxiliary system.
Our approach is different in the sense that in specific situations, a classical balanced truncation error bound can be shown to hold true.
As is common in the context of balanced truncation, we assume in this section that $A=(J-R)Q$ is asymptotically stable, i.e., all eigenvalues of $A$ lie in the open left half plane.
The main idea is to transfer the previous concepts to the setting of standard balanced truncation.
In particular, we propose to replace the constant term $BB^\top$ by $2R$ in the standard controllability Lyapunov equation and, thus, to balance the system with respect to the solutions of the (modified) controllability and observability Lyapunov equations
\begin{equation}\label{eq:lyap}
\begin{aligned}
A \mL_{\mathrm{c}}+ \mL_{\mathrm{c}} A^\top + 2R &=0, \\
A^\top \mM_{\mathrm{o}} + \mM_{\mathrm{o}} A + C^\top C &=0.
\end{aligned}
\end{equation}
With this specific choice of the constant term $2R$ in the first equation, one immediately verifies that $\mL_{\mathrm{c}}=Q^{-1}$ for a pH system \eqref{eq:pH}.
Consequently, in balanced coordinates, it holds that
\begin{align*}
A_{\mathrm{b}} \Pi +\Pi A_{\mathrm{b}}^\top + 2 R_\mathrm{b}&=0, \\
A_{\mathrm{b}}^\top \Pi + \Pi A_{\mathrm{b}} + C_{\mathrm{b}}^\top C_{\mathrm{b}} &= 0,
\end{align*}
where $\Pi=\mathrm{diag}(\pi_1,\dots,\pi_n)$. In particular, we have $Q_\mathrm{b}=\Pi^{-1}$ such that an effort-constraint reduced-order model satisfies
\begin{align*}
Q_r=Q_{\mathrm{b}11}-Q_{\mathrm{b}12}Q_{\mathrm{b}22}^{-1} Q_{\mathrm{b}21}=Q_{\mathrm{b}11}=\mathrm{diag}\left(\frac1{\pi_1},\dots,\frac1{\pi_r}\right).
\end{align*}
Note that an effort-constraint reduced-order model automatically is port-Hamiltonian and is further obtained by simple truncation of a system balanced w.r.t.\@ modified system Gramians. With regard to an error bound as in Theorem \ref{thm:err_bnd_lqg}, we make an additional assumption on $R$ and $B.$ We summarize our findings in the following result.
\begin{corollary}\label{cor:bt}
Let $(J,R,Q,B)$ with asymptotically stable $A=(J-R)Q$ define a minimal port-Hamiltonian system of the form \eqref{eq:pH} which is balanced w.r.t.~the solutions of \eqref{eq:lyap}.
Furthermore, let $(\widehat{A}_r,\widehat{B}_r,\widehat{C}_r)$ be a reduced-order model obtained by truncation of $(A,B,C)$ and let $c>0$ be such that $c R\succeq \frac{1}{2}BB^\top$.
Besides, the balanced Gramians are denoted by $\mM_{\mathrm{o}}=\mL_{\mathrm{c}}=\Pi=\mathrm{diag}(\Pi_1,\Pi_2),$ where $\Pi_2=\mathrm{diag}(\pi_{r+1},\dots,\pi_n).$
If $G(\cdot)=C(\,\cdot\, I_n-A)^{-1}B$ and $\widehat{G}_r(\cdot)=\widehat{C}_r (\,\cdot\, I_r - \widehat{A}_r)^{-1} \widehat{B}_r$ are the associated transfer functions, then
\begin{align}\label{eq:Hinf_err_bnd}
\|G-\widehat{G}_r\|_{\mathcal{H}_\infty} \le 2\sqrt{c} \sum_{i=r+1}^n \pi_i.
\end{align}
Moreover, the reduced-order model $(\widehat{A}_r,\widehat{B}_r,\widehat{C}_r)$ is port-Hamiltonian.
\end{corollary}
\begin{proof}
Note that the assumption on $R$ and $B$ implies that for $\Lambda =c\Pi,$ we have
\begin{align*}
0=c (A\Pi + \Pi A^\top + 2R) = A\Lambda + \Lambda A^\top +2 cR\succeq A\Lambda + \Lambda A^\top + B B^\top.
\end{align*}
The proof then follows along the lines of the proof of Theorem \ref{thm:err_bnd_lqg} with $\mathcal{L}, \widehat{\mathcal{P}}_{\mathrm{f}} $ being replaced by $\mL_{\mathrm{c}},\mM_{\mathrm{o}},$ and is thus omitted here.
\end{proof}
We proceed with a discussion on the condition $cR\succeq\frac{1}{2}BB^\top$ that appears in Corollary \ref{cor:bt}.
First of all, we investigate necessary and sufficient conditions for the existence of such a $c>0$.
\begin{lemma}\label{lem:cRB_cond}
Let $R\in\mathbb{R}^{n\times n}$ with $R=R^\top\succeq0$ and $B\in\mathbb{R}^{n \times m}$ be two matrices.
Then there exists a constant $c>0$ such that $cR\succeq \frac{1}{2}BB^\top$ if and only if $\operatorname{Im}(B)\subseteq\ker(R)^\perp$ or, equivalently, $\operatorname{Im}(B)\subseteq\operatorname{Im}(R)$.
\end{lemma}
\begin{proof}
Since $R$ is symmetric, it is well-known that $\ker(R)^\perp=\operatorname{Im}(R)$. Therefore it is sufficient to prove the first statement.
Suppose first that there exists such a $c>0$. Let $z_1=Bu$ for some $u\in\mathbb{R}^m$ be a generic element of $\operatorname{Im}(B)$,
and let $z_2\in\ker(R)$: we need to show that $z_1^\top z_2=0$.
Since
%
\begin{equation*}
0 = z_2^\top R z_2 \geq \frac{1}{2c}z_2^\top BB^\top z_2 = \frac{1}{2c}\Vert B^\top z_2\Vert_2^2
\quad\Rightarrow\quad B^\top z_2=0,
\end{equation*}
%
we deduce that $z_1^\top z_2=u^\top B^\top z_2=0$, thus $\operatorname{Im}(B)\subseteq\ker(R)^\perp$.
Suppose now that $\operatorname{Im}(B)\subseteq\ker(R)^\perp$.
Note that $\ker(B^\top)=\ker(BB^\top)$, since $z^\top BB^\top z=\Vert B^\top z\Vert_2^2$ for all $z\in\mathbb{R}^n$.
In particular, $\operatorname{Im}(BB^\top)=\ker(BB^\top)^\perp=\ker(B^\top)^\perp=\operatorname{Im}(B)\subseteq\operatorname{Im}(R)$.
Let
%
\begin{equation*}
U^\top R U = \bmat{\Lambda & 0 \\ 0 & 0}, \quad
0\prec \Lambda \in\mathbb{R}^{k\times k} \text{ diagonal}, \quad
U^\top U=I_n
\end{equation*}
%
be the SVD of $R$, and let $\lambda_m>0$ be the minimum diagonal entry of $\Lambda$.
Since $\operatorname{Im}(BB^\top)\subseteq\operatorname{Im}(R)$, we clearly have
%
\begin{equation*}
U^\top BB^\top U = \bmat{S & 0 \\ 0 & 0}, \quad
S = S^\top \in \mathbb{R}^{k\times k}.
\end{equation*}
%
In particular, $\Vert S\Vert_2=\Vert BB^\top\Vert_2=\Vert B\Vert_2^{2}$, and for any $z=(z_1,z_2)\in\mathbb{R}^{n}=\mathbb{R}^{k+(n-k)}$ we have
%
\begin{align*}
z^\top U^\top\frac{1}{2}BB^\top Uz
&= \frac{1}{2}z_1^\top Sz_1 \leq \frac{1}{2}\Vert S\Vert_2 z_1^\top z_1 \leq \frac{\Vert B\Vert_2^{2}}{2\lambda_m}z_1^\top\Lambda z_1 = \frac{\Vert B\Vert_2^{2}}{2\lambda_m}z^\top U^\top RUz.
\end{align*}
%
Since $U$ is invertible, this is equivalent to $cR\succeq\frac{1}{2}BB^\top$, with $c=\frac{\Vert B\Vert_2^{2}}{2\lambda_m}$.
\end{proof}
In other words, Lemma \ref{lem:cRB_cond} states that the condition of Corollary \ref{cor:bt} is satisfied exactly when the input port (represented by $B$) interacts directly with the dissipation port (represented by $R$). In particular, intuition tells us that, if the condition holds, any small input will be damped by the dissipation of the system and will preserve the pH structure, while if the condition fails, there are small inputs that will insert new energy in the system, without that the dissipation can immediately act on that. We formalize this intuition with the following result.
\begin{proposition}\label{prop:cRB_cond}
Let a port-Hamiltonian system \eqref{eq:pH} be given by $(J,R,Q,B)$, and consider output feedbacks of the form $u=Fy=FB^\top Qx$ with a feedback matrix $F\in\mathbb{R}^{m\times m}$, leading to the closed loop system
%
\begin{equation}\label{eq:closed_loop}
\dot x = (J-R+BFB^\top)Qx.
\end{equation}
%
Then a constant $c>0$ satisfies the condition $cR\succeq \frac{1}{2}BB^\top$ if and only if \eqref{eq:closed_loop} is port-Hamiltonian with respect to $Q$ for all feedback matrices $F$ with $\Vert F\Vert_2\leq\frac{1}{2c}$.
\end{proposition}
\begin{proof}
Let $\alpha=\frac{1}{2c}>0$, so that the two conditions are $R\succeq\alpha BB^\top$ and $\Vert F\Vert_2\leq\alpha$.
Let $F$ be a generic feedback matrix with $\Vert F\Vert_2\leq\alpha$. If we split $F=F_{\mathrm{s}}+F_{\mathrm{k}}$ into its symmetric part $F_{\mathrm{s}}=\frac{1}{2}(F+F^\top)$ and skew-symmetric part $F_{\mathrm{k}}=\frac{1}{2}(F-F^\top)$, we can rewrite \eqref{eq:closed_loop} as
%
\begin{equation*}
\dot x = \big((J+BF_{\mathrm{k}} B^\top)-(R-BF_{\mathrm{s}} B^\top)\big)Qx = (J_F - R_F)Qx.
\end{equation*}
%
It is clear that this system is pH with respect to $Q$ if and only if $R_F=R_F^\top\succeq 0$.
Suppose first that $R\nsucceq\alpha BB^\top$: then the feedback matrix $F=F_{\mathrm{s}}=\alpha I_m$ satisfies $\Vert F\Vert_2=\alpha$ but produces a closed loop system that is not pH with respect to $Q$, since $R_F=R-\alpha BB^\top\nsucceq 0$.
Suppose now that $R\succeq\alpha BB^\top$. Note that $\Vert F_{\mathrm{s}}\Vert_2=\frac{1}{2}\Vert F+F^\top\Vert_2\leq\Vert F\Vert_2\leq\alpha$, therefore we can assume without loss of generality that $F=F^\top=F_{\mathrm{s}}$.
Since $F$ is symmetric, there exist a matrix $G=G^\top\succeq 0$ such that $F\preceq G$ and $\Vert G\Vert_2=\Vert F\Vert_2\leq\alpha$, that can be constructed by taking the spectral decomposition of $F$ and replacing all negative eigenvalues with their absolute value, and a matrix $S=S^\top\succeq 0$, such that $G=S^2$ and $\Vert S\Vert_2^2=\Vert G\Vert_2\leq\alpha$, that can be constructed by replacing the eigenvalues of $G$ with their square root.
In particular, we have
%
\begin{align*}
z^\top BFB^\top z &\leq z^\top BGB^\top z = z^\top BS^2B^\top z = \Vert SB^{\top}z\Vert_2^2 \leq \Vert S\Vert_2^2\Vert B^{\top}z\Vert_2^{2} \leq \alpha z^\top BB^\top z \leq z^\top Rz,
\end{align*}
%
i.e., $R_F=R-BFB^\top\succeq 0$, as requested.
\end{proof}
It should be stressed that Proposition \ref{prop:cRB_cond} does not provide necessary conditions for the port-Hamiltonian system to admit structure-preserving output feedback.
In fact, any output feedback of the form $u=Fy$ with $F+F^\top\preceq 0$ will lead to a closed loop port-Hamiltonian system of the form $\dot x=(\tilde J-\tilde R)Qx$, with $\tilde J=J+\frac{1}{2}B(F-F^\top)B^\top$ and $\tilde R=R-\frac{1}{2}B(F+F^\top)B^\top$ satisfying $\tilde J=-\tilde J^\top$ and $\tilde R=\tilde R^\top\succeq 0$, regardless of the form of $R$ and $B$, and of the magnitude of $\lVert F\rVert_2$.
In fact, Proposition \ref{prop:cRB_cond} provides conditions for the existence of output feedback with bounded norm that \emph{destroys} the port-Hamiltonian structure.
This can for example be useful in one of the following scenarios: if we expect disturbances in the output feedback, in which case $c$ can provide a measure of robustness, or if we are actually interested in destabilizing the system.
\begin{remark}
It is clear that, given two matrices $R=R^\top\succeq0$ and $B$, the set
%
\begin{equation*}
\Omega(R,B) = \left\{ c\in\mathbb{R} \mid cR \succeq \tfrac{1}{2}BB^\top \right\}
\end{equation*}
%
is either empty or a closed infinite left-bounded interval of the form $[\overline c,\infty)$, where $\overline{c}\geq 0$.
Note that in the cases we are interested in we actually have $\overline{c}>0$ since $\overline c=0$ would imply $B=0$.
Since the error bound in Corollary \ref{cor:bt} holds for any $c\in\Omega$, we are particularly interested in the optimal value $\overline{c}$, or equivalently $\overline{\alpha}=\frac{1}{2\overline{c}}$, for which the error bound is minimal.
One way to find the optimal $\overline{\alpha}$ is offered by Lemma \ref{lem:cRB_cond}.
Since $R=R^\top\succeq 0$, there exists an invertible matrix $P\in\mathbb{R}^{n\times n}$, such that
%
\begin{equation*}
P^\top RP = \bmat{I_k & 0 \\ 0 & 0}, \quad\text{with}\quad
P = U\bmat{\Lambda^{-\frac{1}{2}} & 0 \\ 0 & I_{n-k}}
\end{equation*}
%
where $U,\Lambda$ are as in the proof of Lemma \ref{lem:cRB_cond}.
If there exists a constant $c>0$ satisfying the condition, then because of the same Lemma we have
%
\begin{equation*}
P^\top BB^\top P = \bmat{ \Lambda^{-\frac{1}{2}}S\Lambda^{-\frac{1}{2}} & 0 \\ 0 & 0 }
= \bmat{ \widehat{S} & 0 \\ 0 & 0 },
\end{equation*}
%
where $S=S^\top\geq0$ is again as in the proof of Lemma \ref{lem:cRB_cond}.
Let now $\widehat{S}=V^\top\widehat{\Lambda}V$ be the spectral decomposition of $\widehat{S}$, where $V^\top V=I_n$ and $\widehat{\Lambda}\succeq 0$ is diagonal.
Then it is clear that
%
\begin{equation*}
R \succeq \alpha BB^\top \quad\iff\quad
P^\top RP \succeq \alpha P^\top BB^\top P \quad\iff\quad
I_k \succeq \alpha \widehat{S} \quad\iff\quad
I_k \succeq \alpha \widehat{\Lambda},
\end{equation*}
%
therefore the optimal $\overline{\alpha}$ is the reciprocal of the largest eigenvalue of $P^\top BB^\top P$.
The optimal value $\overline{\alpha}$ has in particular a nice interpretation. Due to Proposition \ref{prop:cRB_cond}: it represents the \emph{radius of structure-preservation} of the pH system \eqref{eq:pH} under output feedback, for a fixed $Q$.
A possible strategy to decrease the error bound is then to consider a different pH representation, given by a different matrix $X$ (see \eqref{eq:pHX}), such that the radius of structure-preservation is larger.
Changing $Q$ unfortunately will in general also change the values $\pi_i$, therefore it is not always clear whether maximizing the radius of structure-preservation is a good solution.
\end{remark}
When there is no $c>0$ such that the condition $cR\succeq\frac{1}{2}BB^\top$ is satisfied, we might want to look for a different error bound.
Similarly as in \eqref{eq:popov_aux}, for every $Y=Y^\top \in \mathbb R^{n\times n}$, we may decompose the Popov function as
\begin{align}
\Phi(s)=\begin{bmatrix} C(sI_n-A)^{-1} & I_m \end{bmatrix} \underbrace{\begin{bmatrix} -AY - YA^\top & B-YC^\top \\
B^\top - CY & 0 \end{bmatrix}}_{:=\widehat{W}(Y)} \begin{bmatrix} (-sI_n-A^\top)^{-1}C^\top \\ I_m \end{bmatrix}.
\end{align}
In particular, if $Y$ is such that $\widehat{W}(Y)\succeq 0,$ a Cholesky decomposition yields
\begin{align*}
\Phi(s)&=\begin{bmatrix} C(sI_n-A)^{-1} & I_m \end{bmatrix} \begin{bmatrix} L_Y^\top \\0 \end{bmatrix} \begin{bmatrix} L_Y & 0 \end{bmatrix}
\begin{bmatrix} (-sI_n-A^\top)^{-1}C^\top \\ I_m \end{bmatrix} \\
&= C(sI_n-A)^{-1}L_Y^\top L_Y(-sI_n-A^\top)^{-1}C^\top=V(s)V(-s)^\top,
\end{align*}
where $V(\cdot)=C(\cdot I_n-A)^{-1}L_Y^{\top}$ denotes a spectral factor of the Popov function $\Phi$. For a pH system \eqref{eq:pH}, we immediately have the solution $Y=Q^{-1}$ with the property:
\begin{align*}
-AY-YA^\top = -AQ^{-1}-Q^{-1}A^\top = -(J-R)-(J^\top-R)=2R.
\end{align*}
Hence, with the Cholesky decomposition $2R=L_R^\top L_R$, we obtain the spectral factor $V(\cdot) = C(\cdot I_n-A)^{-1} L_R^\top $. With regard to balancing of the solutions to \eqref{eq:lyap}, observe that this corresponds to classical balanced truncation of a system described by $V(\cdot)= C(\cdot I_n-A)^{-1} L_R^\top$. As a consequence, we have the following result.
\begin{corollary}\label{cor:spec_bt}
Let $(J,R,Q,B)$ with asymptotically stable $A=(J-R)Q$ and minimal $(A,R,C)$ define a port-Hamiltonian system of the form \eqref{eq:pH} which is balanced w.r.t.~the solutions of \eqref{eq:lyap}.
Furthermore, let $(\widehat{A}_r,\widehat{B}_r,\widehat{C}_r)$ be a reduced-order model obtained by truncation of $(A,B,C)$.
Besides, the balanced Gramians are denoted by $\mM_{\mathrm{o}}=\mL_{\mathrm{c}}=\Pi=\mathrm{diag}(\Pi_1,\Pi_2),$ where $\Pi_2=\mathrm{diag}(\pi_{r+1},\dots,\pi_n).$
If $V(\cdot)=C(\cdot I_n-A)^{-1}L_R^\top$ and $\widehat{V}_r(\cdot)=\widehat{C}_r (\cdot I_r - \widehat{A}_r)^{-1} \widehat{L}_{\widehat{R}_r}^\top$ are spectral factors of the associated Popov function, then
\begin{align}\label{eq:Hinf_err_bnd_V}
\|V-\widehat{V}_r\|_{\mathcal{H}_\infty} \le 2 \sum_{i=r+1}^n \pi_i.
\end{align}
Moreover, the reduced-order model $(\widehat{A}_r,\widehat{B}_r,\widehat{C}_r)$ is port-Hamiltonian.
\end{corollary}
\begin{remark}\label{rem:sr-balancing}
While the previous result does not require any condition for $B$, it is based on minimality of the triple $(A,R,C).$ This is however a less restrictive assumption than the one from Corollary \ref{cor:bt}. Indeed, if $(A,B,C)$ is minimal and if there exists $c>0$ as in Corollary \ref{cor:bt}, then with Lemma~\ref{lem:cRB_cond} it follows that $(A,R,C)$ is also minimal. Moreover, it is well-known that minimality can always be ensured by a \emph{low rank square root} balancing transformation, see, e.g., \cite[Theorem 2.2]{TomP87}, \cite[Section 4.1]{Pen06} and \cite{BenQ99}.
\end{remark}
\begin{remark}
The bound \eqref{eq:Hinf_err_bnd_V} is structurally similar to the well-known bounded real balanced truncation error bound for the stable minimum phase spectral factors, see \cite{OpdJ88}.
\end{remark}
\begin{remark}\label{rem:compare_spec_fac}
For the error bound \eqref{eq:Hinf_err_bnd_V}, it is crucial that $L_R^\top $ and $\widehat{L}_{\widehat{R}_r}^\top$ have the same number of columns since otherwise a comparison in the $\mathcal{H}_\infty$-norm does not make sense. A naive computation of $\widehat{L}_{\widehat{R}_r}^\top$ would limit the number of columns by the reduced system dimension and, hence, would generally be smaller than the number of columns of $L_R^\top$. On the other hand, from \eqref{eq:pH_red} it makes sense to define $\widehat{L}_{\widehat{R}_r}^\top$ via $ \widehat{L}_{\widehat{R}_r}^\top = W^\top L_R^\top$
which automatically ensures identical number of columns.
\end{remark}
Following the discussion in Section \ref{subsec:choosing_Q}, we may again ask whether there exists a particularly good pH representation such that the previous error bounds are minimized.
With this in mind, it is obvious to minimize the eigenvalues $\lambda_i(\mM_{\mathrm{o}} \mL_{\mathrm{c}}) = \lambda_i(\mM_{\mathrm{o}} Q^{-1})$.
If $Y=Q^{-1}$ is such that $\widehat{W}(Y)\succeq 0$, it then follows that $X=Y^{-1}=Q$ is a solution to \eqref{eq:kyp-lmi} and vice versa.
Consequently, if $X$ is a maximal solution to \eqref{eq:kyp-lmi} and the system is replaced by the alternative pH formulation \eqref{eq:pHX}, then the error bound will be minimal.
This will be further illustrated in the numerical examples.
Similarly as in Remark~\ref{rem:mixedBalancing}, we note that the approach outlined in the last paragraph can be viewed as a mixed balancing procedure, where the solution of the observability Lyapunov equation in \eqref{eq:lyap} is balanced with the minimal solution of the dual KYP-LMI \eqref{eq:dualKYPLMI}.
Consequently, this approach coincides with the balancing procedure presented in \cite{UnnVE07}.
In contrast to \cite{UnnVE07}, we are not only able to show that the reduced-order models are port-Hamiltonian and, thus, passive, but also to derive two computable a priori error bounds, which hold under some additional assumptions on the dissipation matrix $R$, cf.~Corollaries~\ref{cor:bt} and \ref{cor:spec_bt}.
\section{Numerical examples}
In this section, we provide two numerical examples that naturally lead to port-Hamiltonian systems.
Let us emphasize that the focus of this section is on illustrating the theory rather than demonstrating that the new methods outperform existing ones or than applying the methods to large-scale systems.
Thus, we only consider systems of moderate state space dimension and focus on the error bounds and on demonstrating the influence of the different choices for the Hamiltonian on these bounds.
All simulations were generated on an Intel i5-9400F @ 4.1 GHz x 6,
64 GB RAM, MATLAB\textsuperscript{\textregistered} \;version R2019b.
With regard to the implementation of the individual methods, the following remarks are in order:
\begin{itemize}
\item For obtaining the stabilizing solutions to Riccati equations, we relied on the MATLAB\textsuperscript{\textregistered}\; built-in routine \texttt{icare}. Similarly, for the computations of the $\mathcal{H}_{\infty}$-errors, we used the \texttt{Control System Toolbox}.
\item The extremal solutions to the KYP-LMI \eqref{eq:kyp-lmi} were computed by a regularization (see \cite[Theorem 2]{Wil72a}) approach based on an artificial feedthrough term $D+D^\top=10^{-12}I_m$ which allowed to replace the LMI
\begin{align*}
\begin{bmatrix} -A^\top X - XA & C^\top -XB \\ C-B^\top X & D+D^\top \end{bmatrix} \succeq 0 \hspace{2cm}
\intertext{by the Riccati equation associated with the Schur complement, i.e.,}
A^\top X + XA + (C^\top -XB) (D+D^\top)^{-1} (C-B^\top X) = 0.
\end{align*}
\item In order to avoid numerically ill-conditioned operations such as, e.g., the computation of the Schur complement in the effort-constraint reduction method, cf.\@ \eqref{eq:eff_const}, numerically minimal realizations have been computed in a preprocessing step.
For this, we utilized the approach from Section \ref{sec:classic_bt} due to its structure and Hamiltonian preserving nature with a truncation threshold $\varepsilon_{\mathrm{trunc}}=10^{-11}$, cf.~Remark \ref{rem:sr-balancing}.
\end{itemize}
\subsection{A scalable mass-spring-damper system} \label{sec:massSpringDamper}
The first example is a scalable multiple-input and multiple-output mass-spring-damper system that has been introduced in \cite{Gugetal12}. The system consist of two inputs $u_1,u_2$ acting as external forces applied to the first two masses of the system. In view of the port-Hamiltonian framework, the outputs $y_1,y_2$ are the velocities of the masses. We refrain from a more detailed discussion and instead refer to the original presentation in \cite{Gugetal12}. For our numerical simulations, we follow the parameters used in the latter reference and define the masses $m_i$, spring constants $k_i$ and damping constants $c_i$ as $m_i=4,k_i=4,$ and $c_i=1$ for all $i=1,\dots,n$. Here, we use a system consisting of $\ell=500$ masses, resulting in an original system of dimension $n=1000$ defined by the matrices $(J,R,Q,B)$.
In Figure \ref{fig:msd}, we show the results obtained for the method from Theorem \ref{thm:err_bnd_lqg} for three different port-Hamiltonian realizations as discussed in Section \ref{subsec:choosing_Q}. In particular, besides the canonical Hamiltonian function $\mathcal{H}(x)=\frac12 x^\top Qx$ resulting from the modelling, we also include the results for a Hamiltonian corresponding to the extremal solutions $X_{\mathrm{min}}$ and $X_{\mathrm{max}}$ to \eqref{eq:kyp-lmi}.
Furthermore, we include a comparison to the classical version of LQG balanced truncation.
The observations are as follows. As seen in Figure \ref{fig:msd_errorDecayOfTransferFunction}, the error bounds for $\left\|\begin{bsmallmatrix} M \\ N \end{bsmallmatrix}-\begin{bsmallmatrix} \widehat{M}_r \\ \widehat{N}_r \end{bsmallmatrix}\right\|_{\mathcal{H}_\infty}$ follow the actual behavior of the error. Moreover, as predicted by the discussion in Section \ref{subsec:choosing_Q}, the error bound clearly depends on the chosen Hamiltonian, with the maximal solution of \eqref{eq:kyp-lmi} being favorable compared to other choices. The minimal solution of \eqref{eq:kyp-lmi} yields a closed loop error (bound) that is almost stagnating. While error and error bound for LQG balanced truncation is smallest, let us recall that the reduced controllers will generally not be port-Hamiltonian.
For investigating the effect on the open-loop behavior, we further show in Figure~\ref{fig:msd_errorDecayOfSpectralFactors} the standard $\mathcal{H}_{\infty}$-error for the different methods. Note that neither for the method from Theorem \ref{thm:err_bnd_lqg} nor for LQG balanced truncation, an error bound is available for this open loop behavior. We also include reduced-order models obtained by effort-constraint balancing w.r.t.\@ the standard LQG Riccati equations, since these are guaranteed to be port-Hamiltonian as well. The conclusions are similar to the closed loop behavior, indicating that if the maximal solution to the KYP-LMI \eqref{eq:kyp-lmi} defines the Hamiltonian, the approximability of the systems is maximized.
\begin{figure}[tb]
\centering
\begin{subfigure}{\textwidth}
\centering
\input{comparisonMSD}
\caption{Decay of the $\mathcal{H}_{\infty}$-error of the coprime factors for the approach from subsection \ref{subsec:modifiedLQG} using three different choices for the Hamiltonian. The solid lines represent the respective error bounds as presented in Theorem \ref{thm:err_bnd_lqg}. As a reference, we also added the error decay for classical LQG balanced truncation and the corresponding error bound.}
\label{fig:msd_errorDecayOfTransferFunction}
\end{subfigure}
\\[0.5cm]
\begin{subfigure}{\textwidth}
\centering
\input{errorBoundMSD}
\caption{Decay of the $\mathcal{H}_\infty$ error of the transfer function for the approach from subsection \ref{subsec:modifiedLQG} using three different choices for the Hamiltonian. As a reference, we also added the error decay for classical LQG balanced truncation and for the effort-constraint reduction based on balancing the standard LQG Riccati equations.}
\label{fig:msd_errorDecayOfSpectralFactors}
\end{subfigure}
\caption{Comparison of closed and open loop errors for a mass-spring-damper system with $n=1000$.}
\label{fig:msd}
\end{figure}
\subsection{Damped wave propagation}
In this subsection we consider the set of linear partial differential equations
\begin{alignat}{2}
a\partial_t p(t,x) &= -\partial_x q(t,x),\qquad && \text{for all }(t,x)\in (0,t_{\mathrm{end}})\times(0,\ell), \label{eq:massBalance}\\
b\partial_t q(t,x) &= -\partial_x p(t,x)-dq(t,x),\qquad && \text{for all }(t,x)\in (0,t_{\mathrm{end}})\times(0,\ell),
\label{eq:momentumBalance}
\end{alignat}
which describe the damped propagation of pressure waves in a pipeline, see for instance \cite{EggKLMM18}.
The unknowns of this system are the mass flow $q$ and the relative pressure $p$, where the latter one represents the deviation of the absolute pressure $\tilde{p}$ to a constant reference pressure $\tilde{p}_0$, i.e., $p\vcentcolon=\tilde{p}-\tilde{p}_0$.
Furthermore, $a,b\in\R_{>0}$ are parameters derived from the properties of the fluid and from the geometry of the pipeline, cf{.} \cite{EggKLMM18}.
Besides, the dissipation within the pipeline is modelled by the damping parameter $d\in\R_{>0}$.
Finally, $t_{\mathrm{end}}$ denotes the length of the considered time interval and $\ell$ the length of the pipe.
In addition to the partial differential equations \eqref{eq:massBalance} and \eqref{eq:momentumBalance}, we consider homogeneous initial conditions as well as the boundary conditions
\begin{alignat*}{4}
p(t,0) &= u_1(t),\qquad &&p(t,\ell) &&= u_2(t),\qquad && \text{for all }t\in (0,t_{\mathrm{end}})
\end{alignat*}
with given functions $u_1$ and $u_2$.
For the semi-discretization in space we use the mixed finite element method as outlined in the appendix of \cite{EggKLMM18}.
To this end, we decompose the computational domain $[0,\ell]$ by an equidistant mesh with $N$ inner grid points and mesh width $h=\frac{\ell}{N+1}$.
Based on this mesh, the relative pressure $p(t,\cdot)$ is approximated by a piecewise constant function and the mass flow $q(t,\cdot)$ by a piecewise linear function.
As a result, we obtain after Galerkin projection the linear system of ordinary differential equations
\begin{equation}
\label{eq:semidiscreteISO2}
\begin{bmatrix}
aM_1 & 0\\
0 & bM_2
\end{bmatrix}
\begin{bmatrix}
\dot{p}_h(t)\\
\dot{q}_h(t)
\end{bmatrix}
=
\begin{bmatrix}
0 & -D\\
D^\top & -dM_2
\end{bmatrix}
\begin{bmatrix}
p_h(t)\\
q_h(t)
\end{bmatrix}
+
\begin{bmatrix}
0\\
B_2
\end{bmatrix}
u(t).
\end{equation}
Here, the vectors $p_h(t)\in\R^{N+1}$ and $q_h(t)\in\R^{N+2}$ contain the coefficients of $p(t,\cdot)$ and $q(t,\cdot)$ with respect to the corresponding finite element bases.
Furthermore, the coefficient matrices are given by $M_1=hI_{N+1}$,
\begin{alignat*}{2}
M_2 &= \frac{h}6
\begin{bmatrix}
2 & 1 & 0 & \cdots & 0 & 0\\
1 & 4 & 1 & \ddots & \vdots & \vdots\\
0 & 1 & 4 & \ddots & 0 & 0\\
\vdots & \ddots & \ddots & \ddots & 1 & 0\\
0 & \cdots & 0 & 1 & 4 & 1\\
0 & \cdots & 0 & 0 & 1 & 2
\end{bmatrix}
\in\R^{(N+2)\times(N+2)},\quad B_2 =
\begin{bmatrix}
1 & 0\\
0 & 0\\
\vdots & \vdots\\
0 & 0\\
0 & -1
\end{bmatrix}
\in\R^{(N+2)\times2},\quad\text{and}\\
D &=
\begin{bmatrix}
-1 & 1 & 0 & \cdots & 0\\
0 & -1 & 1 & \ddots & \vdots\\
\vdots & \ddots & \ddots & \ddots & 0\\
0 & \cdots & 0 & -1 & 1
\end{bmatrix}
\in\R^{(N+1)\times(N+2)}.
\end{alignat*}
By adding the output equation
\begin{equation}
\label{eq:outputEq}
y(t) =
\begin{bmatrix}
0 & B_2^\top
\end{bmatrix}
\begin{bmatrix}
p_h(t)\\
q_h(t)
\end{bmatrix}
,
\end{equation}
we observe that the system \eqref{eq:semidiscreteISO2} and the output equation \eqref{eq:outputEq} form a port-Hamiltonian system in generalized state space form as in \eqref{eq:pH_gen}.
We summarize this system as
\begin{subequations}
\label{eq:ISO2pHE}
\begin{alignat}{1}
E\dot{z}(t) &= (\tilde{J}-\tilde{R})z(t)+\tilde{B}u(t) \label{eq:ISO2pHE_state}\\
y(t) &= \tilde{B}^\top z(t) \label{eq:ISO2pHE_output}
\end{alignat}
\end{subequations}
with
\begin{equation*}
z(t) =
\begin{bmatrix}
p_h(t)\\
q_h(t)
\end{bmatrix}
,\quad E =
\begin{bmatrix}
aM_1 & 0\\
0 & bM_2
\end{bmatrix}
,\quad \tilde{J} =
\begin{bmatrix}
0 & -D\\
D^\top & 0
\end{bmatrix}
,\quad \tilde{R}=
\begin{bmatrix}
0 & 0\\
0 & dM_2
\end{bmatrix}
,\quad \text{and}\quad \tilde{B} =
\begin{bmatrix}
0\\
B_2
\end{bmatrix}
.
\end{equation*}
Finally, we transform \eqref{eq:ISO2pHE} to a pH system of the form \eqref{eq:pH} in the following way.
We first compute the Cholesky decomposition $E=LL^\top$, transform the state via $x=L^\top z$ and multiply \eqref{eq:ISO2pHE_state} from the left by $L^{-1}$.
This leads to the matrices $Q=I_{2N+3}$, $J=L^{-1}\tilde{J}L^{-\top}$, $R=L^{-1}\tilde{R}L^{-\top}$, and $B=L^{-1}\tilde{B}$.
Especially, we obtain a sparse matrix $Q$, but in general dense matrices $J$ and $R$.
However, since the first diagonal block of $\tilde{R}$ is $0$ and the second one a multiple of $M_2$, the matrix $R$ is also block diagonal with the diagonal blocks $0$ and $\frac{d}{b}I_{N+2}$.
Thus, the matrix $R$ is sparse for the example at hand and, furthermore, since the first block of $B$ is $0$ and since the second diagonal block of $R$ is positive definite, there exists $c\in\R_{>0}$ such that $c R\succeq \frac{1}{2}BB^\top$.
Hence, the assumptions of Corollary \ref{cor:bt} are satisfied and we can employ the corresponding error bound.
\begin{figure}[tb]
\centering
\begin{subfigure}{\textwidth}
\centering
\input{comparisonDampedWaveEquation}
\caption{Decay of the $\mathcal{H}_\infty$ error of the transfer function for the approach from Section \ref{sec:classic_bt} using three different choices for the Hamiltonian. The solid line represents the error bound from Corollary \ref{cor:bt} for the case of the canonical Hamiltonian. As a reference, we also added the error decay for effort-constrained reduction and for positive real balanced truncation.
}
\label{fig:dampedWave_errorDecayOfTransferFunction}
\end{subfigure}
\\[0.5cm]
\begin{subfigure}{\textwidth}
\centering
\input{errorBoundDampedWaveEquation}
\caption{Decay of the $\mathcal{H}_\infty$ error of the spectral factors for the approach from Section \ref{sec:classic_bt} using three different choices for the Hamiltonian. The solid lines represent the respective error bounds as presented in Corollary \ref{cor:spec_bt}.}
\label{fig:dampedWave_errorDecayOfSpectralFactors}
\end{subfigure}
\caption{Damped wave equation -- Error decay for the balancing-based model reduction approach introduced in Section \ref{sec:classic_bt}.}
\label{fig:dampedWave_errorDecay}
\end{figure}
For the numerical experiments we set $a=1$, $b=1$, $d=50$, and $\ell=1$.
Furthermore, the number of internal grid points is chosen as $N=500$ which results in a state space dimension of $1003$.
The resulting full-order model is used for illustrating the findings of Section \ref{sec:classic_bt}.
In particular, we investigate the influence of choosing different Hamiltonians, as mentioned in the paragraph after Remark \ref{rem:compare_spec_fac}, and we compare the errors with the proposed error bounds.
In particular, in the case of the canonical Hamiltonian function $\frac12x^\top Qx$, the constant $c$ from Corollary \ref{cor:bt} can be calculated by computing the largest eigenvalue of $\frac12BB^\top$ and dividing it by $\frac{d}{b}$.
Here, we exploited again that the second diagonal block of $R$ is a multiple of the identity matrix.
However, this reasoning does not apply when we change the Hamiltonian, and thus $R$, as discussed in subsection \ref{subsec:choosing_Q}.
In this case, the assumptions of Corollary \ref{cor:bt} may be violated and this is in fact what we observe numerically for the example at hand.
Thus, there is no error bound available for $\|G-\widehat{G}_r\|_{\mathcal{H}_\infty}$ in the cases where we replace $Q$ by $X_{\mathrm{max}}$ or $X_{\mathrm{min}}$ in the Hamiltonian.
In Figure \ref{fig:dampedWave_errorDecay} we depict the errors in the transfer function and in the spectral factors of the Popov function over the dimension of the reduced-order models.
When comparing the errors with the respective error bounds, we observe that there is a good agreement between their qualitative behaviors.
Furthermore, we find in Figure \ref{fig:dampedWave_errorDecayOfSpectralFactors} that the choice of the Hamiltonian plays an important role for the error bound as predicted by the theory.
Since the decay of the actual error resembles the decay of the corresponding error bound in this example, the choice of the Hamiltonian can also be observed to have a significant influence on the actual errors.
In addition to the approach introduced in Section \ref{sec:classic_bt}, we also show the corresponding $\mathcal{H}_\infty$ error decays for the effort-constraint reduction method and for positive real balanced truncation (positive real BT) in Figure \ref{fig:dampedWave_errorDecayOfTransferFunction}.
We choose them as reference methods since they are the schemes most related to our new approach in the sense that they are balanced-based model reduction techniques which preserve the port-Hamiltonian structure.
For the example at hand, we observe that the error decay of positive real balanced truncation is similar to the one by our approach when using the Hamiltonian corresponding to $X_{\mathrm{max}}$.
On the other hand, the effort-constraint reduction yields reduced-order models with larger errors which are of the same order of magnitude as the ones obtained by our approach using the Hamiltonian based on $X_{\mathrm{min}}$.
\section{Conclusion}
In this paper we propose new balancing-based methods for controller design and for model order reduction which preserve the structure of linear time-invariant port-Hamiltonian systems without algebraic constraints.
To this end, we first derive a modified LQG balancing approach which ensures that the resulting LQG controller is port-Hamiltonian, by properly choosing the weighting matrices in the algebraic Riccati equations.
Based on this balancing, we introduce a corresponding model reduction method to obtain a low-dimensional port-Hamiltonian controller.
In this context, we also present an a priori error bound in the gap metric, which is structurally the same as the one for standard LQG balanced truncation.
Moreover, we show that the error bound can be improved by replacing the canonical Hamiltonian by one which is based on the maximal solution of the associated KYP linear matrix inequality.
In addition to the modified LQG balancing approach, we also derive a modification of standard balanced truncation in order to ensure preservation of the port-Hamiltonian structure.
For this method, we derive an error bound w.r.t.\@ the spectral factors of the associated Popov function.
For the case that a certain condition on the dissipation port and the input port of the full-order model is satisfied, we also present an a priori bound for the $\mathcal{H}_\infty$ error of the transfer function.
Furthermore, we give an intuitive interpretation of the mentioned condition by showing its relation to the existence of a structure-preserving output feedback.
Finally, the new methods and the corresponding error bounds are illustrated by means of two numerical examples: A mass-spring-damper system and a semi-discretized damped wave equation.
Based on the findings of this paper, an interesting next step is the generalization to port-Hamiltonian differential-algebraic equation systems.
This would allow to also construct low-dimensional port-Hamiltonian controllers for systems with algebraic constraints.
Furthermore, we mainly focus on the theory in this paper, whereas an efficient and robust numerical implementation is not addressed.
These practical considerations are certainly an interesting research direction for the future in order to explore the applicability and competitiveness of the new approaches.
\subsection*{Acknowledgements}
We would like to thank Volker Mehrmann for comments on a previous version of this manuscript. We thank the Deutsche Forschungsgemeinschaft for their support within the project B03 in the Sonderforschungsbereich/Transregio 154 ``Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks".
Finally, we express our gratitude to the two anonymous reviewers for their comments and suggestions which helped to improve and clarify this manuscript.
|
1,941,325,220,579 | arxiv | \section{Introduction}
\IEEEPARstart{D}{ue} to the intrinsic characteristics of wireless channels, i.e., broadcast and superposition, wireless networks are vulnerable to jamming attacks (also referred as to interference attacks), and it is difficult to protect transmitted signals from unauthorized recipients~\cite{PLSsur1}. Intelligent reflecting surfaces (IRSs) has been an emerging wireless technology for 5G, 6G and beyond~\cite{IRSsur1,IRSsur2}. Legitimate IRSs can be used to provide an important approach for enhancing the physical layer security (PLS) in wireless networks~\cite{ResGroup,IRSPLSadd}.
Therefore, many previous studies have investigated the use of legitimate IRSs to improve PLS~\cite{ANIRS,ActiveJammerIRS}. In~\cite{ANIRS}, IRSs combined with artificial noise (AN) or friendly jamming at the access point (AP) are used for security enhancement in the presence of illegal eavesdroppers. In~\cite{ActiveJammerIRS}, the authors proposed an IRS-assisted anti-jamming scheme against jamming attacks, where a friendly IRS is used to prevent the illegal jammer (IJ) from jamming legitimate users (LUs). Note that the legitimate AP in the legitimate IRS aided scenario knows the legitimate IRS's information, like its location, and can control the reflecting phase shifts of the legitimate IRS.
In contrast, illegal IRSs (I-IRSs) represent the illegal deployment and utilization of IRSs~\cite{IIRSSur}, where the legitimate AP does not know the I-IRSs' information
and also can not control the I-IRSs.
Due to the passive nature, the I-IRSs are hard to be detect. Consequently, the I-IRSs impose a more serious harmful impact on PLS. For example, an I-IRS has been employed to deteriorate signals at LUs in the presence of jamming attacks~\cite{IIRSSur}, where the I-IRS aggravates the AN generated by the IJ to reduce the received signal-to-noise ratio (SNR) or the signal-to-interference-noise ratio (SINR). However, there are two requirements in existing methods to achieve the I-IRS-based IJ.
\textit{1) I-IRSs need to know the channel state information (CSI) of all channels involved.}
Yet, the uplink channel estimation for IRS-aided channels remains difficult due to the passive nature of IRSs~\cite{DaipartI}.
Acquiring the I-IRS-aided channels' CSI at IJ is too idealistic to implement in practice.
Although illegal jamming can be achieved without the CSI by broadcasting the AN~\cite{ANwoCSI}, the performance gain obtained by implementing an I-IRS, in this case, is limited as reflecting phase shifts of the I-IRS are hard to optimize without the CSI.
\textit{2) A large amount of power is needed to transmit jamming signals continuously.}
Even a few papers attempt to realize an I-IRS-based passive jammer (PJ) without the transmit power
for single-user systems~\cite{PassJamSU}, which minimizes the received power at the LU by destructively adding the signal from the AP-IRS-User channel.
However, this I-IRS-based PJ still requires the CSI of IRS-aided channels to optimize the I-IRS's reflecting phase shifts.
Limited by these two requirements above, especially the CSI acquisition, the I-IRS-based IJ seems to be difficult to implement in practical wireless networks. So in this paper, we try to answer the following research question: \textit{{Can IJs jam LUs without both the transmit power and the CSI?}}
To draw attention to the impact of I-IRSs on multi-user multiple-input single-output (MU-MISO) systems, we propose an I-IRS-based fully-passive jammer (FPJ) that can launch jamming attacks without relying on the transmit power and the CSI. To the best of our knowledge, it is the first time that an IJ can jam LUs without the CSI.
\begin{itemize}
\item An I-IRS is exploited to actively change wireless channels, and therefore, the orthogonality of the multi-user active beamforming vectors and the co-user channels is destroyed, which is referred to as \textbf{active channel aging}$\footnote{Channel aging is CSI inaccuracy due to time variation of wireless channels and delays in the computation~\cite{ChanAge}. In this work, we actively introduce CSI inaccuracy by using an I-IRS. To differentiate, we call it active channel aging.}$.
\item During \textit{the reverse pilot transmission (RPT) phase}, we randomly generate reflecting phase shifts for the I-IRS. During \textit{the data transmission (DT) phase}, we randomly generate other reflecting phase shifts. The I-IRS acts like a \textbf{``disco ball"} without optimizing its phase shifts based on the CSI. The resulting serious inter-user interference due to active channel aging jams the LUs effectively.
\end{itemize}
\emph{Notation:} We use bold capital type for a matrix, e.g., $\boldsymbol{\Phi}$, small bold type for a vector, e.g., $\boldsymbol{\varphi}$, and italic type for a scalar, e.g., $K$. Moreover, the superscripts $(\cdot)^{H}$ and $(\cdot)^{T}$ denote the Hermitian transpose and the transpose. Moreover, the symbols $|\cdot|$ and $\|\cdot\|$ denote the absolute value and the Frobenius norm.
\section{System Statement}
In this section, first, we describe the general mode of an MU-MISO system jammed by the I-IRS-based FPJ. Then, we give the optimization metric and state the two communications phases: \emph{the RPT phase} and \emph{the DT phase}.
\subsection{System Model and Channel Model}
\begin{figure}[!t]
\centering
\includegraphics[scale=0.65]{Fig1_1bit.eps}
\caption{Illustration of a MU-MISO system jammed by the I-IRS-based FPJ, where phase shifts of the I-IRS are randomly generated by the independent I-IRS controller. RPT: reverse pilot transmission; DT: data transmission.}
\label{fig1}
\end{figure}
Figure~\ref{fig1} schematically illustrates a MU-MISO system jammed by the I-IRS-based FPJ, where the legitimate AP is equipped with an $N_{\rm A}$-element uniform linear array (ULA) and communicates with $K$ single-antenna LUs termed ${\rm {LU}}_{1}, {\rm {LU}}_{2}, \cdots, {\rm {LU}}_{K}$. An I-IRS comprised of $N_{\rm I}$ one-bit controllable reflecting elements is deployed near the AP$\footnote{Based on the existing literature on the IRS's deployment location~\cite{IRSdeployment}, the IRS should be deployed as close to users or as close to the AP as possible to increase its effect. Yet, in the jamming scenario, we make the more robust assumption that the IJ does not know any information about LUs, for instance, LUs' locations and CSI. Therefore, we deploy the I-IRS near the AP.}$ to jam LUs.
When the data signal $s_k\in \mathbb{C}$ for ${\rm{LU}}_k$ $(1\ \leq k \leq K)$ is normalized to unit power, the signal received at ${\rm{LU}}_k$ is expressed as,
\begin{equation}
{y_k} = \boldsymbol{h}_{{\rm{com}},k}^H\!\sum_{u = 1}^K {{\boldsymbol{w}_u}{s_u}} + {n_k},
\label{eq1}
\end{equation}
where $\boldsymbol{h}^H_{\!{\rm{com}},k}\! = \!\left( \!{ {\boldsymbol{h}_{{\rm{I}},k}^H{{\bf{\Phi }}}{{\bf{G}}}} \!+\! \boldsymbol{h}_{{\rm{d}},k}^H} \right) \!\in\! {\mathbb{C}^{1\!\times\!{N_{\rm{A}}}}}$ denotes the combined channel between the legitimate AP and ${\rm{LU}}_k$, $\boldsymbol{h}_{{\rm{I}},k} \in\mathbb{C}^{N_{\rm{I}}\!\times\! 1}$ denotes the channel between the I-IRS and ${\rm{LU}}_k$, ${\bf{G}} \in\mathbb{C}^{N_{\rm{I}}\!\times\! N_{\rm{A}}}$ denotes the channel between the legitimate AP and the I-IRS, and $\boldsymbol{h}_{{\rm{d}},k} \in\mathbb{C}^{N_{\rm{A}}\!\times\! 1}$ denotes the direct channel between the legitimate AP and ${\rm{LU}}_k$.
In \eqref{eq1}, $\boldsymbol{\Phi}\!=\!{\rm{diag}}(\boldsymbol{\varphi})\in\mathbb{C}^{N_{\rm{I}}\!\times\! N_{\rm{I}}}$ represents the reflecting matrix of the I-IRS, where the one-bit reflecting vector ${\boldsymbol{\varphi}}$ is expressed as ${\boldsymbol{\varphi}} = \left[{e^{j{\varphi_{1}}}}, \cdots, {e^{j{\varphi_{{N\!_I}}}}}\right]^H$, and ${\varphi_{n}}\in \Omega = \left\{0, \pi \right\} (1\le n \le N_{\rm I})$ denotes reflecting phase shift of the $n$-th reflecting element. The independent I-IRS controller
generates ${\boldsymbol{\varphi}}$ and then controls the I-IRS to implement the corresponding phase shifts.
Besides, $\boldsymbol{w}_k$
denotes the active beamforming at the AP for ${\rm{LU}}_k$, and $n_k$ denotes the additive white Gaussian noise with 0 mean and $\sigma^2$ variance, i.e., $n_k\sim \mathcal{CN}\left(0,\sigma^2\right)$.
For ease of representation, we further define the multi-user direct channel between the AP and the LUs, the multi-user channel between the I-IRS and the LUs, as well as the multi-user combined channel between the AP and all LUs as ${\bf{H}}_{\rm{d}}^H \!=\!\left[{\boldsymbol{h}}_{{\rm{d}},1},{\boldsymbol{h}}_{{\rm{d}},2},\cdots,{\boldsymbol{h}}_{{\rm{d}},K}\right]^H$, ${\bf{H}}_{{\rm{I}}}^H \!=\!\left[\boldsymbol{h}_{{\rm{I}},1},{\boldsymbol{h}}_{{\rm{I}},2},\cdots,\boldsymbol{h}_{{\rm{I}},K}\right]^H$, and ${\bf{H}}_{{\rm{com}}}^H=\left[ {{\boldsymbol{h}_{{\rm{com}},1}},{\boldsymbol{h}_{{\rm{com}},2}},\cdots,{\boldsymbol{h}_{{\rm{com}},K}}} \right]^H$, respectively. Furthermore, the multi-user active beamforming at the AP is denoted as ${\bf{W}}\!=\!\left[{\boldsymbol{w}}_1,{\boldsymbol{w}}_2,\ldots,{\boldsymbol{w}}_K\right]$.
The multi-user direct channel ${\bf{H}}_{\rm{d}}$ follows Rayleigh fading, while the IRS-aided channels ${\bf{G}}$ and $\boldsymbol{h}_{{\rm{I}},k}$ follow Rician fading~\cite{AORIS}. Specifically, ${\bf{G}}$ and $\boldsymbol{h}_{{\rm{I}},k}$ are modeled as
\begin{alignat}{1}
&{{\bf{G}}}\! = \!{\mathscr{L}}_{\rm G} \!\left(\!\!{\sqrt {\frac{{{\kappa_{\rm \!G}}}}{{1\!+\!{\kappa_{\rm \!G}}}}} {\bf{G}}^{{\rm{LOS}}}\!+\!\sqrt {\frac{1}{{1 + {\kappa_{\rm \!G}}}}} {\bf{G}}^{{\rm{NLOS}}}}\!\!\right),\notag\\
&{\boldsymbol{h}_{{\rm{I}},k}}\! = \!{\mathscr{L}}_{{\rm I},k} \!\left(\!\!{\sqrt {\frac{{{\kappa_{{\rm{I}}}}}}{{1\!+\!{\kappa _{{\rm{I}}}}}}} \boldsymbol{h}_{{\rm{I}},k}^{{\rm{LOS}}}\!+\!\sqrt {\frac{1}{{1\!+\!{\kappa_{{\rm{I}}}}}}}\!\boldsymbol{h}_{{\rm{I}},k}^{{\rm{NLOS}}}}\!\!\right),
\label{eq2}
\end{alignat}
where ${\mathscr{L}}_{\rm G}$ and ${\mathscr{L}}_{{\rm I},k}$ represent the large-scale path loss between the AP and the I-IRS and that between the I-IRS and ${\rm{LU}}_k$, and ${\kappa _{{\rm{G}}}}$ and ${\kappa _{{\rm{I}}}}$ are the Rician factors of ${{\bf{G}}}$ and ${\boldsymbol{h}_{{\rm{I}},k}}$.
In \eqref{eq2}, ${\bf{G}}^{{\rm{LOS}}}$ and $\boldsymbol{h}_{{\rm{I}},k}^{{\rm{LOS}}}$ are the line-of-sight (LOS) components of ${{\bf{G}}}$ and ${\boldsymbol{h}_{{\rm{I}},k}}$, and ${\bf{G}}^{{\rm{NLOS}}}$ and $\boldsymbol{h}_{{\rm{I}},k}^{{\rm{NLOS}}}$ are non-line-of-sight (NLOS) components. The NLOS components follow Rayleigh fading, while the LOS components are~\cite{AORIS},
\begin{alignat}{1}
&{\bf{G}}^{{\rm{LOS}}} = \sqrt {{N_{\rm{I}}}{N_{\rm A}}} {\boldsymbol{\alpha} _{\rm{I}}}\left( {{\vartheta},{\theta}} \right)\boldsymbol{\alpha} _{\rm{A}}^H\left( {{\phi}} \right),\notag\\
&\boldsymbol{h}_{{\rm{I}},k}^{{\rm{LoS}}} = \sqrt {{N_{\rm{I}}}} {\boldsymbol{\alpha} _{\rm{I}}}\left( {{\vartheta _{{\rm{I}},k}},{\theta_{{\rm{I}},k}}} \right),
\label{eq3}
\end{alignat}
where $\boldsymbol{\alpha} _{\rm{A}}$ and $\boldsymbol{\alpha} _{\rm{I}}$ are the array responses~\cite{AORIS}.
\subsection{Wireless Communications: The RPT and DT Phases}\label{CommPhase}
In practice, the main aim of a MU-MISO system is to maximize a certain performance metric that generally is a strictly-increasing utility function of SINR~\cite{BF}. Specifically, a widely-used performance metric is the sum rate, which is expressed as ${R_{{\rm{sum}}}} = \sum\nolimits_{k = 1}^K {{R_k}} = \sum\nolimits_{k = 1}^K {{{\log }_2}\left( {1 + {\gamma _k}} \right)}$. According to~\eqref{eq1}, the received SINR $\gamma_k$ at ${\rm{LU}}_k$ is stated as,
\begin{equation}
{\gamma _k} = \frac{{{{\left| {\boldsymbol{h}_{{\rm{com}},k}^H{\boldsymbol{w}_k}} \right|}^2}}}{{\sum\limits_{u \ne k} {{{\left| {\boldsymbol{h}_{{\rm{com}},k}^H{\boldsymbol{w}_u}} \right|}^2} + {\sigma ^2}} }}.
\label{eq6}
\end{equation}
\subsubsection{Acquiring CSI During The RPT Phase}
From \eqref{eq6}, it can be seen that the optimization of multi-user active beamforming ${\bf{W}} = \left[{\boldsymbol{w}}_1,{\boldsymbol{w}}_2,\ldots,{\boldsymbol{w}}_K\right]$ at the AP aims to maximize the signal term $\left| {\boldsymbol{h}_{{\rm{com}},k}^H{\boldsymbol{w}_k}} \right|$ while minimizing the inter-user interference term ${\sum\nolimits_{u \ne k}{\left| {\boldsymbol{h}_{{\rm{com}},k}^H{\boldsymbol{w}_u}} \right|}}$. In order to optimize ${\bf{W}}$, the CSI of $\bf{H}_{{\rm{com}}}$ must be obtained at the AP$\footnote{In the MU-MISO system under I-IRS-based jamming attacks, it is impractical to acquire the CSI of IRS-aided channels and the direct channel, respectively. The legitimate AP cannot know any information about the I-IRS, like its location, much less jointly train the IRS-based channels with the I-IRS. Namely, the legitimate AP can only obtain the CSI of $\bf{H}_{{\rm{com}}}$. Note that the CSI of $\bf{H}_{{\rm{com}}}$ is easily obtained at the legitimate AP when ${\bf{\Phi}}$ is determined, which is the traditional MISO channel estimation. The phase shifts of the I-IRS are generated at random by the independent I-IRS controller, and therefore, ${\bf{\Phi}}$ is always determined for the legitimate AP, as shown in Fig.~\ref{fig1}.}$.
Generally, the CSI can be acquired during \textit{the RPT phase} according to the pilot estimation, as shown in Fig.~\ref{fig1}. More specifically, to acquire the CSI of $\boldsymbol{h}_{{\rm{com}},k}$, the ${\rm{LU}}_k$ sends pilot signals to the legitimate AP, and the AP then estimates $\boldsymbol{h}_{{\rm{com}},k}$ by certain traditional solutions, for instance, the least square (LS) algorithm~\cite{DaipartI}.
\subsubsection{Precoding During The DT Phase}
Based on the obtained CSI in \textit{the RPT phase}, the multi-user active beamforming used during \textit{the DT phase} can be designed. Generally, the multi-user active beamforming optimization problem is a nondeterministic polynomial-time (NP)-hard problem, and therefore, computing the optimal multi-user active beamforming is difficult. To this end, some heuristic beamforming designs, which can achieve near-optimal performance, have been investigated.
A widely known beamforming solution is the zero-forcing beamforming (ZFBF) algorithm~\cite{BF}, which causes zero inter-user interference. Specifically, the multi-user active beamforming ${\bf{W}}_{\rm{ZF}}$ computed via the ZFBF algorithm is written as
\begin{equation}
{{\bf{W}}_{{\rm{ZF}}}} = \frac{{{{\bf{H}}_{{\rm{com}}}}{{\left( {{\bf{H}}_{{\rm{com}}}^H{{\bf{H}}_{{\rm{com}}}}} \right)}^{ - 1}}{{\bf{P}}^{ \frac{1}{2}}}}}{{{{\left\| {{{\bf{H}}_{{\rm{com}}}}{{\left( {{\bf{H}}_{{\rm{com}}}^H{{\bf{H}}_{{\rm{com}}}}} \right)}^{ - 1}}} \right\|}^2}}},
\label{eq7}
\end{equation}
where ${{\bf{P}}^{\frac{1}{2}}} = {\rm{diag}}\left( {\sqrt {{p_1}} ,\sqrt {{p_2}} , \cdots ,\sqrt {{p_K}} } \right)$, and $p_k$ represents the transmit power allocated to ${\rm{LU}}_k$. The power allocation must satisfy the constraint that $\sum\nolimits_{k = 1}^K {{p_k}} \le {P_0}$, where $P_0$ is the total transmit power at the AP. The optimal power allocation can be calculated by the water-filling algorithm~\cite{BF}.
\subsubsection{Orthogonal Interference Subspace}
According to \eqref{eq6}, the ratio of inter-user interference to noise (I/N) ${\mathscr{I}}$ is equal to
\begin{equation}
{\mathscr{I}} = \sum\limits_{k = 1}^K \sum\limits_{u \ne k} \frac{{{{\left| {\boldsymbol{h}_{{\rm{com}},k}^H{\boldsymbol{w}_u}} \right|}^2} }}{\sigma^2} .
\label{eq8}
\end{equation}
Incorporating \eqref{eq7} into \eqref{eq8}, it is clear that ${\mathscr{I}} = 0$ due to the presence of the pseudoinverse ${{\left( {{\bf{H}}_{{\rm{com}}}^H{{\bf{H}}_{{\rm{com}}}}} \right)}^{ - 1}}$. In other words, ZFBF causes zero inter-user interference by projecting the user channel $\boldsymbol{h}_{{\rm{com}},k}$ onto the subspace that is orthogonal to the co-user channels $\boldsymbol{h}_{{\rm{com}},1},\cdots,\boldsymbol{h}_{{\rm{com}},k-1},\boldsymbol{h}_{{\rm{com}},k+1},\cdots,\boldsymbol{h}_{{\rm{com}},K}$, i.e., the orthogonal interference subspace.
\section{I-IRS-based Fully-Passive Jammer via Active Channel Aging}\label{PassJamm
To raise concerns about the potential threat that an I-IRS could launch jamming attacks without the transmit power or even the CSI, we introduce a CSI-based PJ without the transmit power in Section~\ref{PJwCSI}, i.e., the extension of~\cite{PassJamSU}. Furthermore, the results from the CSI-based PJ are used as benchmarks.
In Section~\ref{proposed}, we propose an I-IRS-based FPJ via active channel aging. By destroying the orthogonality of the multi-user active beamforming vectors and the co-user channels, the proposed I-IRS-based FPJ can jam LUs without the transmit power and the CSI.
\subsection{CSI-Based Jamming Attacks Without Power}\label{PJwCSI}
To implement the extension of~\cite{PassJamSU}, it is necessary to consider the most ideal case for jamming attacks: the legitimate AP only knows the CSI of ${\bf{H}}_{\rm d}$ and then calculates the multi-user active beamforming ${\bf{W}}_{\rm{d}}$ via the ZFBF algorithm, while the independent I-IRS controller knows the CSI of ${\bf{H}}_{\rm{d}}$, ${\bf{H}}_{{\rm{I}}}$, and ${\bf{G}}$ as well as ${\bf{W}}_{\rm{d}}$.
The CSI-based PJ can launch jamming attacks without the transmit power, where the reflecting vector for the I-IRS is optimized by minimizing a certain performance metric. Taking the example of minimizing the sum rate $R_{\rm{sum}}$ received at LUs, the optimization of the one-bit reflecting vector is mathematically represented as
\begin{alignat}{1}
&\mathop {\min }\limits_{\boldsymbol{\varphi}} {R_{{\rm{sum}}}} = \mathop {\min }\limits_{\boldsymbol{\varphi}} \sum\limits_{k = 1}^K {{{\log }_2}}\!\! \left( {1 + \frac{{{{\left| {{\boldsymbol{h}}_{{\rm{com}},k}^H{{\boldsymbol{w}}_{{\rm{d}},k}}} \right|}^2}}}{{\sum\limits_{u \ne k} {{{\left| {{\boldsymbol{h}}_{{\rm{com}},k}^H{{\boldsymbol{w}}_{{\rm{d}},u}}} \right|}^2}} + {\sigma ^2}}}} \right)\label{addeq81}\\
&{\rm{s}}.{\rm{t}}.\;\;{\varphi _n} \in \Omega, n = 1,2, \cdots ,{N_{\rm{I}}}.
\label{addeq82}
\end{alignat}
The phase shift optimization problem in~\eqref{addeq81} can be solved by enumerating all possible $\left\{ {{\varphi _n}} \right\}_{n = 1}^{{N_{\rm{I}}}}$ combinations.
However, there are $2^{N_{\rm I}}$ different combinations, and thus the computational complexity is large.
To this end, we first relax the discrete phase shift constraint in~\eqref{addeq82} to a continuous constraint. Mathematically, the reflecting vector optimization is relaxed to
\begin{alignat}{1}
& \mathop {\max }\limits_{\bar{\boldsymbol{\varphi}}} \! \sum\limits_{k = 1}^K \!-{{{\log }_2}}\!\! \left( \!{1 +\! \frac{{{{\left|\!{
\left({{{\bar{\boldsymbol{\varphi}}}{{\rm{diag}}(\boldsymbol{h}_{{\rm{I}},k}^H)}{{\bf{G}}}} \!+\! \boldsymbol{h}_{{\rm{d}},k}^H}\!\right)\!{{\boldsymbol{w}}_{{\rm{d}},k}}}\! \right|}^2}}}{{\sum\limits_{u \ne k} {{{\left| {\left(\!{{{\bar{\boldsymbol{\varphi}}}{{\rm{diag}}(\boldsymbol{h}_{{\rm{I}},k}^H)}{{\bf{G}}}} \!+ \! \boldsymbol{h}_{{\rm{d}},k}^H}\!\right)\!{{\boldsymbol{w}}_{{\rm{d}},u}}} \right|}^2}} \!\! +\! {\sigma ^2}}}}\! \right)\label{addeq911}\\
&{\rm{s}}.{\rm{t}}.\;\;{\bar {\varphi}_n} \in \left[ {0,2\pi } \right], n = 1,2, \cdots ,{N_{\rm{I}}}.
\label{addeq922}
\end{alignat}
The objective function in~\eqref{addeq911} is then a continuous and differentiable function of $ \bar {{\boldsymbol{\varphi}}}$,
and the constraint in~\eqref{addeq922} creates a complex circle manifold. Therefore, the optimization problem in~\eqref{addeq911} can be computed by the Riemannian conjugate gradient (RCG) algorithm~\cite{AORCG}. After computing the continuous reflecting vector $\bar{\boldsymbol{\varphi}}$, the discrete reflecting vector is obtained by
\begin{alignat}{1}
&\mathop {\min }\limits_{{\boldsymbol{\varphi}}} {\left\| {{{\boldsymbol{\varphi}}} - {\bar{\boldsymbol{\varphi}}} } \right\|^2} \label{addeq101}\\
\nonumber
&{\rm{s}}.{\rm{t}}.\;\;\eqref{addeq82}.
\end{alignat}
The complexity of the benchmarking CSI-based PJ is ${\mathcal O}\!\left(I_{\rm R}K^2N^2_{\rm I} \right)$, where $I_{\rm R}$ represents the iteration times of the RCG algorithm. In each iteration, the complexity comes mainly from calculating the Euclidean gradient~\cite{AORCG}. Specifically, the complexity of the Euclidean gradient calculation is ${\mathcal O}\!\left( K^2N^2_{\rm I} \right)$. Moreover, the complexity of the discreteization of $\bar{\boldsymbol{\varphi}}$ expressed by~\eqref{addeq101} is ${\mathcal O}\!\left( 2 N_{\rm I} \right)$. When the number of reflecting elements packed on the I-IRS is large ($N_{\rm I} \gg 1$), the complexity of the discreteization, i.e., ${\mathcal O}\!\left( 2 N_{\rm I} \right)$, can be ignored.
\subsection{I-IRS-Based Jamming Attacks Without Power and CSI}\label{proposed}
Although the CSI-based PJ proposed in Section~\ref{PJwCSI} can jam without the transmit power, the CSI of all channels needs to be obtained at the independent I-IRS controller, which is difficult to satisfy in practice. In wireless communications, the AP needs to obtain the CSI during \textit{the RPT phase} before \textit{the DT phase}, as stated in Section~\ref{CommPhase}.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.6]{Fig22.eps}
\caption{I-IRS-based FPJ via {\textbf{active channel aging}}, where the I-IRS acts like a {\textbf{``disco ball"}} and ZFBF cannot project the user channel to the orthogonal interference subspace.}
\label{fig2}
\end{figure}
\subsubsection{The RPT Phase}
During \textit{the RPT phase}, the one-bit reflecting vector for the I-IRS is generated by tuning the $n$-th reflecting element to a random phase shift belonging to $\Omega$, i.e., ${{\varphi}}_n^1 \sim \mathcal{U}\left(\Omega\right)$. More particularly, the reflecting vector ${\boldsymbol{\varphi}}^1$ follows the uniform distribution denoted ${\boldsymbol{\varphi}}^1 \sim \mathcal{U}\left(\Omega^{N_{\rm I}}\right)$. It is worth noting that the independent I-IRS controller in the proposed scheme does not need to optimize the reflecting phase shifts of the I-IRS.
Consequently, the multi-user combined channel estimated by the AP is written as $({{\bf{H}}_{{\rm{com}}}^1} )^H = {\bf{H}}_{\rm I}^H{\rm{diag}}\left({\boldsymbol{\varphi}}^1\right){\bf{G}}+{\bf{H}}_{\rm d}^H = \left[{{\boldsymbol{h}}_{{\rm{com}},1}^1}, {{\boldsymbol{h}}_{{\rm{com}},2}^1}, \cdots, {{\boldsymbol{h}}_{{\rm{com}},K}^1}\right]^H$.
Based on ${{\bf{H}}_{{\rm{com}}}^1}$, the AP can compute the multi-user active beamforming used in \textit{the DT phase} that is expressed as
\begin{equation}
{\!{\bf{W}}_{{\!\rm{ZF}}}^1} \!=\! \frac{{\!{{\bf{H}}_{\!{\rm{com}}}^1}{{\left( {(\!{{\bf{H}}_{\!{\rm{com}}}^1} \!)^{\!H}{\!{\bf{H}}_{\!{\rm{com}}}^1}} \right)}^{\! - 1}}{\!{\bf{P}}^{\! \frac{1}{2}}}}}{{{{\left\|\! {{\bf{H}}_{\!{\rm{com}}}^1}{{\left( {(\!{{\bf{H}}_{\!{\rm{com}}}^1} \!)^{\!H}{\!{\bf{H}}_{\!{\rm{com}}}^1}} \right)}^{\! - 1}} \!\right\|}^2}}} \! = \!\left[{\!{\boldsymbol{w}}_{\!{\rm{ZF}},1}^1}, {\!{\boldsymbol{w}}_{\!{\rm{ZF}},2}^1}, \!\cdots\!, {\!{\boldsymbol{w}}_{\!{\rm{ZF}},\!K}^1} \!\right],
\label{ZF1}
\end{equation}
where ${{\boldsymbol{w}}_{{\rm{ZF}},k}^1}$ is orthogonal to the subspace of co-user channels $\boldsymbol{h}_{{\rm{com}},1}^1,\cdots,\boldsymbol{h}_{{\rm{com}},k-1}^1,\boldsymbol{h}_{{\rm{com}},k+1}^1,\cdots,\boldsymbol{h}_{{\rm{com}},K}^1$
\subsubsection{The DT Phase}
Then, during \textit{the DT phase}, the one-bit reflecting vector of the I-IRS is formed according to another reflecting vector ${\boldsymbol{\varphi}}^2$ that also follows the uniform distribution in $\Omega$, i.e., ${\boldsymbol{\varphi}}^2 \sim \mathcal{U}\left(\Omega^{N_{\rm I}}\right)$. Therefore, during \textit{the DT phase}, the multi-user combined channel is changed to
\begin{equation}
(\!{{\bf{H}}_{{\rm{com}}}^2} \!)^{\! H} \!\!=\! {\!\bf{H}}_{\rm I}^H{\!\rm{diag}}\!\left(\!{\boldsymbol{\varphi}}^2\!\right){\!\bf{G}}\!+{\!\bf{H}}_{\!\rm d}^H \!=\! \left[{\!{\boldsymbol{h}}_{\!{\rm{com}},1}^2}, {\!{\boldsymbol{h}}_{\!{\rm{com}},2}^2}, \!\cdots\!, {\!{\boldsymbol{h}}_{\!{\rm{com}},K}^2}\!\right]^{\! H}.
\label{addHcom2}
\end{equation}
Including~\eqref{ZF1} and~\eqref{addHcom2} into~\eqref{eq6}, the actual received SINR ${{\bar{\gamma}}_k}$ at ${\rm{LU}}_k$ during \textit{the DT phase} is
\begin{equation}
{{\bar{\gamma}}_k} = \frac{{{{\left| {({{\boldsymbol{h}}_{{\rm{com}},k}^2})^H{\boldsymbol{w}_{{\rm {ZF}},k}^1}} \right|}^2}}}{{\sum\limits_{u \ne k} {{{\left| {({{\boldsymbol{h}}_{{\rm{com}},k}^2})^H{\boldsymbol{w}_{{\rm {ZF}},u}^1}} \right|}^2} + {\sigma ^2}} }}.
\label{eq9}
\end{equation}
The complexity of our proposed scheme comes from randomly generating the two reflecting vectors used in \textit{the RPT phase} and \textit{the DT phase}, which is only ${\mathcal O}\!\left( 2 N_{\rm I} \right)$. Compared with the benchmarking CSI-based PJ, the I-IRS's controller in the proposed I-IRS-based FPJ not only does not require the CSI of all channels involved, but also the complexity of the proposed I-IRS-based FPJ is much lower.
\subsubsection{Active Channel Aging}
Based on~\eqref{ZF1} and~\eqref{addHcom2}, the reflecting vectors for the I-IRS are different and random during \textit{the RPT phase} and \textit{the DT phase} (like a ``disco ball" shown in Fig.~\ref{fig2}), which destroys the orthogonality generated from ZFBF due to active channel aging.
The ${{\boldsymbol{w}}_{{\rm{ZF}},k}^1}$ is only orthogonal to the subspace of co-user channels $\boldsymbol{h}_{{\rm{com}},1}^1,\cdots,\boldsymbol{h}_{{\rm{com}},k-1}^1,\boldsymbol{h}_{{\rm{com}},k+1}^1,\cdots,\boldsymbol{h}_{{\rm{com}},K}^1$, and thus ${\mathscr{I}}$ in \eqref{eq8} is then equal to $\sum\nolimits_{k = 1}^K{\sum\nolimits_{u \ne k} \frac{{{{\left| {(\boldsymbol{h}_{{\rm{com}},k}^2)^H{\boldsymbol{w}^1_{{\rm{ZF}},u}}} \right|}^2} }}{\sigma^2}}$, which is no longer zero due to active channel aging.
As a result, the actual received SINR ${{\bar{\gamma}}_k}$ in~\eqref{eq9} achieved under the proposed I-IRS-based FPJ is dramatically reduced compared to that without attacks.
We stated that the reflecting vector for the I-IRS is different during \textit{the RPT phase} and \textit{the DT phase}. In fact, there is no need for precise synchronization in practical implementation. Assuming that the periods of \textit{the RPT phase} and \textit{the DT phase} are $T_{\rm{r}}$ and $T_{\rm{d}}$ ($T_{\rm{r}} \le T_{\rm{d}}$), the reflecting vector changes randomly with a period of no more than $T_{\rm{r}}$, and active channel aging then occurs.
\section{Simulation Results and Discussion}\label{ResDis}
Consider a MU-MISO system with four single-antenna LUs, where the legitimate AP is equipped with a 12-element ULA~\cite{BF} and an I-IRS contains 1$,$024 reflecting elements ($N_{\rm{I},y}=N_{\rm{I},z}=32$).
Moreover, the AP is located at (0m, 0m, 0m) and the four LUs are randomly distributed in a circle centered at (200m, 0m, 0m) with a radius of 10m, while the I-IRS is deployed at (5m, 5m, 2m).
Most of the existing performance-enhancing IRS-aided systems make the assumption that ${\bf{H}}_{\rm d}$ has significant large-scale path loss or is blocked, while the large-scale path losses of $\bf{G}$ and ${\bf{H}}_{\rm I}$ are much smaller~\cite{AORIS,AORCG}. However, this assumption is too idealistic for jamming attacks.
According to the 3GPP propagation environment~\cite{3GPP}, the large-scale path losses ${\mathscr{L}}_{k}$, ${\mathscr{L}}_{\rm G}$ and ${\mathscr{L}}_{\rm{I},k}$ are set as ${\mathscr{L}}_{k}\!=\!32.6\!+\!22{\log _{10}}({d_{k}})$, ${\mathscr{L}}_{\rm G} \!=\!35.6\!+\!20{\log _{10}}({d_{{\rm{G}}}})$ and ${\mathscr{L}}_{{\rm I},k}\!=\!35.6\!+\!22{\log _{10}}({d_{{\rm I},k}})$, where $d_{k}$ is the distance between the AP and ${\rm{LU}}_k$, $d_{\rm G}$ is the distance between the AP and and the I-IRS, and ${d_{{\rm{I}},k}}$ is the distance between the I-IRS and ${\rm{LU}}_k$ $(1\!\leq\! k\!\leq\!4)$. Moreover, $\sigma^2\!=\!-170\!+\!10\log _{10}(BW)$ dBm, where $BW$ denotes the transmission bandwidth and $BW\!=\!180$ kHz~\cite{AORCG}.
We compare the proposed I-IRS-based FPJ with three benchmarks.
\textit{1) Benchmark 1:} The average sum rates without IJ (w/o IJ) are computed based on the multi-user direct channel, where the received SINR $\gamma_k$ at ${\rm{LU}}_k$ is $\gamma_k = \frac{{{{\left| {{\boldsymbol{h}}_{{\rm{d}},k}^H{\boldsymbol{w}_{{\rm{d}},k}}} \right|}^2}}}{{\sum\nolimits_{u \ne k} {{{\left| {\boldsymbol{h}_{{\rm{d}},k}^H{\boldsymbol{w}_{{\rm{d}},u}}} \right|}^2}} + {\sigma ^2}}}$.
\textit{2) Benchmark 2:} The average sum rates under the active jammer (w/ AJ/N) with different ratios of the jamming power to the noise power (AJ/N) at each LU. More specifically, the received SINR $\gamma_k$ at ${\rm{LU}}_k$ under active jamming is expressed as $\gamma_k = \frac{{{{\left| {{\boldsymbol{h}}_{{\rm{d}},k}^H{\boldsymbol{w}_{{\rm{d}},k}}} \right|}^2}}}{{\sum\nolimits_{u \ne k} {{{\left| {\boldsymbol{h}_{{\rm{d}},k}^H{\boldsymbol{w}_{{\rm{d}},u}}} \right|}^2}} + {P_J} + {\sigma ^2}}}$, where AJ/N = ${P_J/ {\sigma ^2}}$=5 dB and 10 dB, respectively.
\textit{2) Benchmark 3:} The CSI-based PJ in Section~\ref{PJwCSI}, i.e., the extension of~\cite{PassJamSU}.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.68]{Fig3R2.eps}
\caption{Average sum rates (left, solid lines) and I/N (right, dash-dot lines) of different schemes vs total transmit power.}
\label{fig3}
\end{figure}
Fig.~\ref{fig3} illustrates the average sum rates via the proposed FPJ and the above three benchmarks, where ${\mathscr{I}}$ generated from them is also given. By destroying the orthogonality of the multi-user active beamforming vectors and the co-user channels, the inter-user interference becomes significant due to active channel aging. The reflecting vector in the proposed FPJ affects both the multi-user combined channel and the multi-user active beamforming, while the reflecting vector in the CSI-based PJ just impacts the multi-user combined channel. As shown in Fig.~\ref{fig3}, ${\mathscr{I}}$ from the proposed FPJ is more serious than that from the CSI-aided PJ (Benchmark 3). Therefore, the proposed FPJ can jam LUs without the transmit power and the CSI, even more effectively than the CSI-aided PJ.
From Fig.~\ref{fig3}, one can see that the sum rate of Proposed FPJ is smaller than that of Benchmark 2 with 5 dB AJ/N when the total transmit power is greater than 0 dBm. In contrast to the active jamming, the jamming launched by the proposed FPJ cannot be mitigated by increasing the total transmit power.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.68]{Fig4R2.eps}
\caption{Influence of quantization reflecting phase shift bits.}
\label{fig4}
\end{figure}
To show the influence of the number of quantization reflecting phase shift bits, the relationships between the average sum rates and quantization bits are illustrated in Fig.~\ref{fig4}. One can see that the proposed FPJ is robust to the quantization bits since the reflecting vector is randomly generated. Based on the proposed FPJ, the one-bit I-IRS is enough to launch effective jamming attacks on LUs.
The greater the number of quantization bits, the smaller the difference ${\left\| {{{\boldsymbol{\varphi}}} - {\bar{\boldsymbol{\varphi}}} } \right\|^2}$ in~\eqref{addeq101} is.
Although the sum rate achieved by Benchmark 3 decreases with the number of quantization bits, the high-bit I-IRS requires high physical implementation costs.
Moreover, Fig.~\ref{fig5} shows the relationship between the sum rates and the number of reflecting elements as well as that between I/N and the number of reflecting elements. The difference between the sum rates achieved by Benchmark 3 and Proposed FPJ increases with the number of reflecting elements. On the one hand, active channel aging becomes more significant with the number of reflecting elements, and thus the corresponding jamming attacks are more effective. On the other hand, the minimum value of ${\left\| {{{\boldsymbol{\varphi}}} - {\bar{\boldsymbol{\varphi}}} } \right\|^2}$ in~\eqref{addeq101} gets bigger with the number of reflecting elements. In practice, an IRS generally consists of massive reflecting elements, which is beneficial to the proposed I-IRS-based FPJ.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.68]{Fig5R2.eps}
\caption{Influence of the number of reflecting elements.}
\label{fig5}
\end{figure}
\section{Conclusions}
In this letter, we investigated the impact of I-IRSs on MU-MISO systems, where an I-IRS-based FPJ was proposed. By exploiting an I-IRS to cause active channel aging, we have demonstrated that the proposed FPJ can jam without relying on the transmit power and the CSI.
Due to the impacts on both the multi-user combined channel and the multi-user active beamforming, the jamming launched by the proposed FPJ is even more effective than that launched by the CSI-aided PJ.
Meanwhile, the proposed FPJ is robust to the number of quantization reflecting phase shift bits. Different from the active jamming attacks, the jamming attacks launched by the proposed FPJ cannot be mitigated by increasing the total transmit power at the legitimate AP. When the legitimate AP has large transmit power, the proposed FPJ can jam LUs more effectively.
Moreover, the proposed FPJ can be perfectly hidden in wireless environments because it does not require additional transmit power and instantaneous CSI.
|
1,941,325,220,580 | arxiv | \section{Introduction}
The characterization and quantification of quantum correlations in composite quantum systems is one of the most challenging topics in quantum information theory \cite{Nielsen,Bellac}. The interest in this field is motivated by the fact that entanglement can be used as useful resource for fundamental study in quantum mechanics and for applications in quantum teleportation \cite{Bouwmeester,Braunstein1998}, dense coding \cite{Mattle1996,Li2002} and quantum key distribution \cite{Bennett1992,Daoud2011}. The entanglement expresses the non local character of quantum mechanics theory \cite{Einstein1935,Bell1966}. To quantify the amount of entanglement, various measures have been proposed. The most familiar ones are the concurrence \cite{Yu2009,Wootters2001}, entanglement of formation \cite{BennettA1996,Popescu1997}, linear entropy \cite{Bose2000}, entanglement of distillation \cite{Bennett1996} and the negativity \cite{Peres1996,Vidal2002}. Some results have shown that quantum correlations can not only be limited to entanglement, especially in mixed states, since separable quantum states can also have non-classical correlations. This yielded many works dedicated to introduce quantum correlation quantifiers beyond entanglement. In 2001, Ollivier and Zurek and independently Henderson and Vedral introduced the concept of entropic quantum discord as a quantifier of quantum correlations in bipartite quantum systems \cite{Ollivier2001,Henderson2001}. It is defined as the difference between the quantum mutual information and the classical correlations existing in a bipartite system \cite{Yurischev2015,Chakrabarty2011}. For pure bipartite states, the quantum discord coincides with entanglement of formation. The entropic quantum discord of any 2-qubit rank-two state can be calculated exactly. Unfortunately, the situation becomes more complicated for states with rank large than two. To overcome this problem, a geometric variant of quantum discord was introduced to provide an alternative way to deal with non classical correlations in bipartite systems \cite{Dakic2010,Girolami2011}. This geometric measure is defined as Hilbert-Schmidt distance between the considered state $\rho$ and its closest classical state $\chi$. It must be emphasized that the geometric measure of quantum discord by using the Hilbert-Schmidt norm can exhibit less robustness than entanglement in two-qubit systems under special dissipative effects. This unexpected result was reported in \cite{Huang2016}. Furthermore, now it is well established that geometric discord based on Hilbert-Schmidt \cite{Daoud2015,Daoud2014} is not a faithful measure of quantum correlations \cite{Piani2012}. In fact, this quantifier can increase under local quantum operations acting on the unmeasured qubit. Other geometric quantifiers have been introduced such as trace distance discord \cite{Paula2013} and trace norm measurement-induced nonlocality \cite{LuoFu2011} to get analytical expressions of quantum discord. A geometric interpretation of one-norm geometric quantum discord for a class of two-qubit X-states is studied in \cite{Huang22016}. Also, the dynamics of entanglement and trace norm measurement-induced nonlocality between two mutually independent atoms interacting with a thermal bath of scalar particles has been investigated in \cite{Huang2018}.\\
Recently, the local quantum uncertainty was introduced by Girolami et al \cite{Girolami2013} (see also the references quoted in \cite{Slaoui2018}) as another quantifier of quantum correlation. This measure is easy to compute analytically for a generic quantum state. It is based on the notion of skew information, introduced by Wigner and Yanase in 1963 \cite{Wigner1963}. The skew information was originally used to describe the information content of mixed states. It plays a fundamental role in quantifying the uncertainty measurements of observables \cite{Luo2005}. It is also related to the concept of Fisher information which is useful in quantum metrology \cite{Luo2003,Frieden2004,Slaoui2019}. Another important feature of the skew information is related to the distinguishability of quantum states \cite{Luo2012}. Moreover, we can use the skew information as a quantifier of quantum coherence \cite{Girolami2014,Du2015,Baumgratz2014}.\\
The aim of this paper is to develop an analytical method to evaluate
quantum correlations in three qubit systems by means of the concept
of local quantum uncertainty. The paper is structured as follows. In section 2, we analyze the quantum correlations in three-qubit $X$ states by employing the concept of local quantum uncertainty. We provide an analytical expression for this quantum correlations quantifier. The obtained results are compared with the tripartite negativity and
entropic quantum discord for some special three-qubit $X$ states. The section 3 is devoted to the dynamics of local quantum uncertainty under decoherence effects and its robustness in some special situations is discussed. In fact, we shall consider the study of the decoherence effects induced by dephasing, depolarizing and phase reversal environments. These different decoherence scenarios are described by employing the Kraus formalism \cite{Kraus1983,Nielson2002}. In section 4, we discuss the monogamy properties of local quantum uncertainty. This is essential to understand the distribution of quantum correlation between the three-qubits family and the whole quantum system. Concluding remarks are given in the last section.
\section{The analytical expression of local quantum uncertainty for three-qubit $X$ states}
\subsection{Definition}
Classically, it is possible to measure any two observables with arbitrary precision. However, for quantum systems, the uncertainty relation imposes a fundamental limit on the precision with which certain pairs of physical properties of a particle can be measured. There are several ways to quantify this uncertainty. In quantum mechanics, the uncertainty of an observable $H$, in a quantum state, is usually quantified by the variance $V\left( {\rho ,H} \right): = {\rm tr}\rho {H^2} - {\left( {\rm tr}\rho H \right)^2}$. However, this relation may exhibits, especially in mixed states, contributions of quantum and classical nature. To deal only with the quantum part of the variance, Wigner and Yanase introduced the notion of skew information. They have shown that the quantum uncertainty relation can be also described in terms of skew information as \cite{Wigner1963}
\begin{equation}\label{skew}
I\left( {\rho ,H} \right): = - \frac{1}{2}{\rm tr}{\left[\sqrt {{\rho}} ,H \right]^2}.
\end{equation}
This quantity gives the uncertainty of the observable $H$ in the state $\rho$. It reduces to the variance $V\left( {\rho ,H} \right)$ when $\rho$ is a pure state. The skew information is non-negative and vanishes if and only if the density matrix and the observable commute. It is convex. In fact, it does not increase under classical mixing and satisfies the inequality $I\left( {\sum\limits_i {{\lambda _i}{\rho _i},H} } \right) \le \sum\limits_i {{\lambda _i}I\left( {{\rho _i},H} \right)} $ for all quantum states ${{\rho _i}}$ and positive constants ${{\lambda _i}}$ satisfying $\sum\limits_i {{\lambda _i} = 1} $. Based on the concept of skew information, the local quantum uncertainty was introduced recently as a new kind of quantum correlations quantifiers \cite{Girolami2013}. For a qubit-qudit system, the local quantum uncertainty is defined as the minimum skew information achievable by a single local measurement. For a qubit $A$ and a qudit $B$, it writes as \cite{Girolami2013,Slaoui22018}
\begin{equation}
\mathcal{U}\left( {{\rho _{AB}}} \right) = 1 -
{\lambda _{\max }}\left( {{W_{AB}}} \right), \label{lquA}
\end{equation}
where ${\lambda _{\max }}\left( {{W_{AB}}} \right)$ denotes the maximum eigenvalue of the $3 \times 3$ symmetric matrix ${W_{AB}}$ whose elements are given by
\begin{equation}
w_{ij} = {\rm tr}\left\{ {\sqrt {{\rho _{AB}}} \left( {{\sigma _{i}}
\otimes {\openone_{d}}} \right)\sqrt {{\rho _{AB}}} \left( {{\sigma _{j}}
\otimes {\openone_{{d}}}} \right)} \right\}, \label{w}
\end{equation}
where $\sigma _{i}\left(i=1,2,3\right) $ denote the usual Pauli matrices. Several works were devoted to this type of quantum correlations measure and its dynamics under noisy effects. These works were essentially motivated with the aim to understand how to reduce the decoherence effects and to provide an adequate scheme to protect the quantum correlations for a given decoherence scenario. In this sense, an interesting protocol proposed in \cite{Huang2017} to protect measurement-induced nonlocality and local quantum uncertainty in a two-qubit system passing through an amplitude damping channel. For pure bipartite states $\rho = \left| {{\psi _{AB}}} \right\rangle \langle {\psi _{AB}}|$, the local quantum uncertainty reduces to the linear entropy of entanglement \cite{Sen2015}
\begin{equation}
{\mathcal{U}}\left( {\left| {{\psi _{AB}}} \right\rangle \left\langle {{\psi _{AB}}} \right|} \right) = 2\left( {1 - {\rm tr}\left( {{\rho _A}^2} \right)} \right).
\end{equation}
\subsection{Derivation of local quantum uncertainty in three qubit $X$ states}
Two qubit states, with non-zero density matrix elements only along the diagonal and anti-diagonal, are called $X$ states because of their visual form resembling the letter $X$. The extension to multi-qubit states was discussed in \cite{Vinjanampathy2011}. In this paper we shall mainly focus on three-qubit states whose density matrices are $X$-shaped. Thus, we consider the family of $X$ states having the following form
\begin{equation}
{\rho _{123}} = \left( {\begin{array}{*{20}{c}}
{{\rho _{11}}}&0&0&0&0&0&0&{{\rho _{18}}} \\
0&{{\rho _{22}}}&0&0&0&0&{{\rho _{27}}}&0 \\
0&0&{{\rho _{33}}}&0&0&{{\rho _{36}}}&0&0 \\
0&0&0&{{\rho _{44}}}&{{\rho _{45}}}&0&0&0 \\
0&0&0&{{\rho _{54}}}&{{\rho _{55}}}&0&0&0 \\
0&0&{{\rho _{63}}}&0&0&{{\rho _{66}}}&0&0 \\
0&{{\rho _{72}}}&0&0&0&0&{{\rho _{77}}}&0 \\
{{\rho _{81}}}&0&0&0&0&0&0&{{\rho _{88}}}
\end{array}} \right), \label{1}
\end{equation}
in the computational basis $\left\{ {\left| {000} \right\rangle ,\left| {010} \right\rangle ,\left| {100} \right\rangle ,\left| {110} \right\rangle ,\left| {001} \right\rangle ,\left| {011} \right\rangle ,\left| {101} \right\rangle ,\left| {111} \right\rangle } \right\}$. The density matrix ${\rho _{123}}$ can be rewritten as
\begin{equation}
{\rho _{123}} = \sum\limits_{i,j=0,1} {{\rho ^{ij}}} \otimes \left| i \right\rangle \left\langle j \right|, \label{2}
\end{equation}
where the density matrices ${{\rho ^{ij}}}$ are defined by
\begin{equation}
{\rho ^{ii}} = \left( {\begin{array}{*{20}{c}}
{{\rho _{1 + 4i\hspace{0.05cm} 1 + 4i}}}&{0}&{0}&{0} \\
{0}&{{\rho _{_{2 + 4i\hspace{0.05cm} 2 + 4i}}}}&{0}&{0} \\
{0}&{0}&{{\rho _{_{3 + 4i \hspace{0.05cm}3 + 4i}}}}&{0} \\
{0}&{0}&{0}&{{\rho _{_{4 + 4i\hspace{0.05cm} 4 + 4i}}}}
\end{array}} \right) \hspace{2cm} i = 1,2, \label{rhoii}
\end{equation}
\begin{equation}
{\rho ^{ij}} = \left( {\begin{array}{*{20}{c}}
{0}&{0}&{0}&{{\rho _{1 + 4i \hspace{0.05cm}4 + 4j}}} \\
{0}&{0}&{{\rho _{2 + 4i \hspace{0.05cm}3 + 4j}}}&{0} \\
{0}&{{\rho _{3 + 4i \hspace{0.05cm}2 + 4j}}}&{0}&{0} \\
{{\rho _{4 + 4i \hspace{0.05cm} 1 + 4j}}}&{0}&{0}&{0}
\end{array}} \right) \hspace{2cm} i \ne j. \label{rhoij}
\end{equation}
In the Fano-Bloch representation, the three-qubit state (\ref{1}) writes also as
\begin{equation}
{\rho _{123}} = \frac{1}{8}\sum\limits_{\alpha \beta \gamma } {{R_{\alpha \beta \gamma }}{\sigma _\alpha } \otimes {\sigma _\beta }} \otimes {\sigma _\gamma },
\end{equation}
where $\alpha $, $\beta $ and $\gamma $ take the values 0, 1, 2 and 3 and the correlation matrix elements ${{R_{\alpha \beta \gamma }}}$ are given by
\begin{equation}
{R_{\alpha \beta \gamma }} = {\rm tr}\left( {{\rho _{123}}\left( {{\sigma _\alpha } \otimes {\sigma _\beta } \otimes {\sigma _\gamma }} \right)} \right). \label{R}
\end{equation}
For states of type (\ref{1}), the non-zero correlation matrix elements ${{R_{\alpha \beta \gamma }}}$, are those with the triplet $(\alpha \beta \gamma$) belonging to the following set
\begin{equation*}
\{(000),(003),(030),(033),(300),(303),(330),(333),
\end{equation*}
\begin{equation*}
(111),(112),(121),(122),(211),(212),(221),(222)\}.
\end{equation*}
The Fano-Bloch representations of the states ${\rho ^{ii}}$ $(i= 1,2)$ (equation (\ref{rhoii})) and ${\rho ^{ij}}$ ($i=1, j=0 $ or $i=0, j=1$) (equation (\ref{rhoij})) are
\begin{equation}
{\rho ^{ii}} = \frac{1}{4}\sum\limits_{\alpha \beta } {R_{_{\alpha \beta }}^{ii}{\sigma _\alpha } \otimes {\sigma _\beta }},
\end{equation}
with $R_{_{\alpha \beta }}^{ii} = {\rm tr}\left( {{\rho ^{ii}}{\sigma _\alpha } \otimes {\sigma _\beta }} \right)$. The non-zero correlation tensor components are given by
\begin{equation}
\begin{array}{l}
R_{00}^{ii} = {\rho _{1 + 4i1 + 4i}} + {\rho _{2 + 4i2 + 4i}} + {\rho _{3 + 4i3 + 4i}} + {\rho _{4 + 4i4 + 4i}}\\
R_{30}^{ii} = {\rho _{1 + 4i1 + 4i}} + {\rho _{2 + 4i2 + 4i}} - {\rho _{3 + 4i3 + 4i}} - {\rho _{4 + 4i4 + 4i}}\\
R_{03}^{ii} = {\rho _{1 + 4i1 + 4i}} - {\rho _{2 + 4i2 + 4i}} + {\rho _{3 + 4i3 + 4i}} - {\rho _{4 + 4i4 + 4i}}\\
R_{33}^{ii} = {\rho _{1 + 4i1 + 4i}} - {\rho _{2 + 4i2 + 4i}} - {\rho _{3 + 4i3 + 4i}} + {\rho _{4 + 4i4 + 4i}}.
\end{array}
\end{equation}
Thus, we have
\begin{equation*}
{\rho ^{ii}} = \frac{1}{4}\left[ {R_{00}^{ii}{\sigma _0} \otimes {\sigma _0} + R_{30}^{ii}{\sigma _3} \otimes {\sigma _0} + R_{03}^{ii}{\sigma _0} \otimes {\sigma _3} + R_{33}^{ii}{\sigma _3} \otimes {\sigma _3}} \right].
\end{equation*}
For the anti-diagonal matrix ${\rho ^{ij}}$ ($i \ne j$), we get
\begin{equation}
{\rho ^{ij}} = \frac{1}{4}\left[ {R_{11}^{ij}{\sigma _1} \otimes {\sigma _1} + R_{12}^{ij}{\sigma _1} \otimes {\sigma _2} + R_{21}^{ij}{\sigma _2} \otimes {\sigma _1} + R_{22}^{ij}{\sigma _2} \otimes {\sigma _2}} \right],
\end{equation}
with
\begin{equation}
R_{11}^{ij} = {\rho _{1 + 4i 4 + 4j}} + {\rho _{4 + 4i1 + 4j}} + {\rho _{2 + 4i 3 + 4j}} + {\rho _{3 + 4i 2 + 4j}},
\end{equation}
\begin{equation}
R_{12}^{ij} = i\left( {{\rho _{1 + 4i4 + 4j}} - {\rho _{4 + 4i1 + 4j}} - {\rho _{2 + 4i3 + 4j}} + {\rho _{3 + 4i2 + 4j}}} \right),
\end{equation}
\begin{equation}
R_{21}^{ij} = i\left( {{\rho _{1 + 4i4 + 4j}} - {\rho _{4 + 4i1 + 4j}} + {\rho _{2 + 4i3 + 4j}} - {\rho _{3 + 4i2 + 4j}}} \right),
\end{equation}
\begin{equation}
R_{22}^{ij} = {\rho _{2 + 4i3 + 4j}} + {\rho _{3 + 4i2 + 4j}} - {\rho _{1 + 4i4 + 4j}} - {\rho _{4 + 4i1 + 4j}}.
\end{equation}
The elements of the tripartite correlations matrix ${R_{\alpha \beta \gamma }}$ (\ref{R}) can be written in terms of bipartite correlation parameters $R_{_{\alpha \beta }}^{ij}$. Indeed, it is simple to see that the density matrix $\rho_{123}$ can be written as
\begin{equation*}
{\rho _{123}} = \frac{1}{2}\left( {{\rho ^{00}} + {\rho ^{11}}} \right) \otimes {\sigma _0} + \frac{1}{2}\left( {{\rho ^{01}} + {\rho ^{10}}} \right) \otimes {\sigma _1} + \frac{i}{2}\left( {{\rho ^{01}} + {\rho ^{10}}} \right) \otimes {\sigma _2} + \frac{1}{2}\left( {{\rho ^{00}} - {\rho ^{11}}} \right) \otimes {\sigma _3}.
\end{equation*}
Furthermore, one verifies that
\begin{equation}
\frac{1}{2}\left( {{\rho ^{00}} + {\rho ^{11}}} \right) = \frac{1}{8}\left[ {R_{00}^{ + + }{\sigma _0} \otimes {\sigma _0} + R_{03}^{ + + }{\sigma _0} \otimes {\sigma _3} + R_{30}^{ + + }{\sigma _3} \otimes {\sigma _0} + R_{33}^{ + + }{\sigma _3} \otimes {\sigma _3}} \right],
\end{equation}
\begin{equation}
\frac{1}{2}\left( {{\rho ^{00}} - {\rho ^{11}}} \right) = \frac{1}{8}\left[ {R_{00}^{ - - }{\sigma _0} \otimes {\sigma _0} + R_{03}^{ - - }{\sigma _0} \otimes {\sigma _3} + R_{30}^{ - - }{\sigma _3} \otimes {\sigma _0} + R_{33}^{ - - }{\sigma _3} \otimes {\sigma _3}} \right],
\end{equation}
\begin{equation}
\frac{1}{2}\left( {{\rho ^{01}} + {\rho ^{10}}} \right) = \frac{1}{8}\left[ {R_{11}^{ + - }{\sigma _1} \otimes {\sigma _1} + R_{12}^{ + - }{\sigma _1} \otimes {\sigma _2} + R_{21}^{ + - }{\sigma _2} \otimes {\sigma _1} + R_{22}^{ + - }{\sigma _2} \otimes {\sigma _2}} \right],
\end{equation}
\begin{equation}
\frac{i}{2}\left( {{\rho ^{01}} - {\rho ^{10}}} \right) = \frac{1}{8}\left[ {R_{11}^{ - + }{\sigma _1} \otimes {\sigma _1} + R_{12}^{ - + }{\sigma _1} \otimes {\sigma _2} + R_{21}^{ - + }{\sigma _2} \otimes {\sigma _1} + R_{22}^{ - + }{\sigma _2} \otimes {\sigma _2}} \right].
\end{equation}
with
\begin{equation}
R_{\alpha \beta }^{ + + } = R_{\alpha \beta }^{00} + R_{\alpha \beta }^{11}, \hspace{1cm} R_{\alpha \beta }^{ - - } = R_{\alpha \beta }^{00} - R_{\alpha \beta }^{11} \hspace{0.5cm}{\rm for}\hspace{0.5cm} \alpha ,\beta = 0,3, \label{eq22}
\end{equation}
and
\begin{equation}
R_{\alpha \beta }^{ + - } = R_{\alpha \beta }^{01} + R_{\alpha \beta }^{10}, \hspace{1cm} R_{\alpha \beta }^{ - - } = i\left( {R_{\alpha \beta }^{01} - R_{\alpha \beta }^{10}} \right) \hspace{0.5cm}{\rm for}\hspace{0.5cm} \alpha ,\beta = 1,2. \label{eq23}
\end{equation}
In this picture, we get
\begin{equation*}
{\rho _{123}} = \frac{1}{8}\left[ {\sum\limits_{\alpha ,\beta = 0,3} {{R_{\alpha \beta 0}}{\sigma _\alpha } \otimes {\sigma _\beta } \otimes {\sigma _0} + {R_{\alpha \beta 3}}{\sigma _\alpha } \otimes {\sigma _\beta } \otimes {\sigma _3}} + \sum\limits_{\alpha ,\beta = 1,2} {{R_{\alpha \beta 1}}{\sigma _\alpha } \otimes {\sigma _\beta } \otimes {\sigma _1} + {R_{\alpha \beta 2}}{\sigma _\alpha } \otimes {\sigma _\beta } \otimes {\sigma _2}} } \right],
\end{equation*}
where the matrix elements $R_{\alpha \beta \gamma}$ are given by
\begin{equation}
{R_{\alpha \beta 0}} = R_{\alpha \beta }^{ + + }, \hspace{1cm} {R_{\alpha \beta 3}} = R_{\alpha \beta }^{ - - } \hspace{0.5cm}{\rm for}\hspace{0.5cm} \alpha ,\beta = 0,3,
\end{equation}
and
\begin{equation}
{R_{\alpha \beta 1}} = R_{\alpha \beta }^{ + - }, \hspace{1cm}{R_{\alpha \beta 2}} = R_{\alpha \beta }^{ - + } \hspace{0.5cm}{\rm for}\hspace{0.5cm}\alpha ,\beta = 1,2,
\end{equation}
in term of the correlations matrix elements (\ref{eq22}) and (\ref{eq23}). The three qubit system described by the density matrix (\ref{1}) may be viewed as $2\times4$ quantum systems. The first sub-system is a qubit $(d=2)$ and the second sub-system is a quartet $(d=4)$. In this partitioning scheme, the matrix elements (\ref{w}) write as
\begin{equation}
{w_{ij}} = {\rm tr}\left( {\sqrt {\rho _{123}} \left( {{\sigma _i} \otimes {\sigma _0} \otimes {\sigma _0}} \right)\sqrt {{\rho _{123}}} \left( {{\sigma _j} \otimes {\sigma _0} \otimes {\sigma _0}} \right)} \right),\label{W for X-three}
\end{equation}
where $i$ and $j$ take the values 1, 2, 3. To evaluate $w_{ij}$ (\ref{W for X-three}), we write the matrix $\sqrt {\rho _{123}}$ in the Fano-Bloch representation as $\sqrt {{\rho _{123}}} = \sum\limits_{\chi \delta \eta } {{T_{\chi \delta \eta }}} {\sigma _\chi } \otimes {\sigma _\delta } \otimes {\sigma _\eta }$. The correlation tensor elements $T_{\chi \delta \eta }$ are given in the appendix. After some algebra, one shown that the matrix elements (\ref{W for X-three}) are explicitly given by
\begin{equation}
{w_{ij}} = \frac{1}{8}\left[ {\left( {{S_{00}} - \sum\limits_k {{S_{kk}}} } \right){\delta _{ij}} + 2{S_{ij}}} \right], \label{eq27}
\end{equation}
where the quantities $S_{ij}$ are given by
\begin{equation}
S_{00} = \sum\limits_{\chi \delta } {T_{0\chi \delta }{T_{0\chi \delta }}}, \label{S00}
\end{equation}
\begin{equation}
{S_{kk}} = \sum\limits_{\chi \delta } {{T_{k\chi \delta }}{T_{k\chi \delta }}} \hspace{1cm}{\rm for}\hspace{1cm} k = 1,2,3, \label{SKK}
\end{equation}
and
\begin{equation}
{S_{ij}} = \sum\limits_{\chi \delta } {{T_{i\chi \delta }}{T_{j\chi \delta }}}. \label{Sij}
\end{equation}
The matrix elements $w_{ij}$ (\ref{eq27}) can be alternatively expanded as
\begin{equation}
{w_{11}} = \frac{1}{8}\left[ {{S_{00}} + {S_{11}} - {S_{22}} - {S_{33}}} \right]
\end{equation}
\begin{equation}
{w_{22}} = \frac{1}{8}\left[ {{S_{00}} - {S_{11}} + {S_{22}} - {S_{33}}} \right]
\end{equation}
\begin{equation}
{w_{33}} = \frac{1}{8}\left[ {{S_{00}} - {S_{11}} - {S_{22}} + {S_{33}}} \right]
\end{equation}
\begin{equation}
{w_{13}} = {w_{31}} = {w_{23}} = {w_{32}} = 0,
\end{equation}
where the quantities $S_{ij}$ are given by (\ref{S00}), (\ref{SKK}) and (\ref{Sij}). The explicit form of the matrix elements $w_{ij}$ in terms of the density matrix elements is given in the appendix.
\subsection{Illustration}
To exemplify the results obtained above, we consider the special cases of mixed {\rm GHZ} states and three-qubit states of Bell type.
\subsubsection{Mixed {\rm GHZ}-states}
We consider first the three-qubit mixed states of {\rm GHZ} type given by
\begin{equation}
{\rho _{ GHZ}} = \frac{p}{8}\openone_{3} + \left( {1 - p} \right)\left| {\rm GHZ} \right\rangle \left\langle {\rm GHZ} \right|, \label{GHZ}
\end{equation}
where $\left| {\rm GHZ} \right\rangle$ denotes the usual $GHZ$ state: $\left| {\rm GHZ} \right\rangle = \frac{1}{{\sqrt 2 }}\left( {\left| {000} \right\rangle + \left| {111} \right\rangle } \right)$. In the computational basis, the state (\ref{GHZ}) takes the form
\begin{equation}
{\rho _{GHZ}} = \frac{1}{8}\left( {\begin{array}{*{20}{c}}
{4 - 3p}&0&0&0&0&0&0&{4\left( {1 - p} \right)} \\
0&p&0&0&0&0&0&0 \\
0&0&p&0&0&0&0&0 \\
0&0&0&p&0&0&0&0 \\
0&0&0&0&p&0&0&0 \\
0&0&0&0&0&p&0&0 \\
0&0&0&0&0&0&p&0 \\
{4\left( {1 - p} \right)}&0&0&0&0&0&0&{4 - 3p}
\end{array}} \right).
\end{equation}
Using the equation (\ref{W11}), (\ref{W22}) and (\ref{W33}), the matrix elements $w_{ij}$ needed to evaluate the local quantum uncertainty, write as
\begin{equation}
{w_{11}} = {w_{22}} = \frac{1}{2}\left( {p + \sqrt {\frac{{p\left( {4 - 3p + \sqrt {p\left( {8 - 7p} \right)} } \right)}}{2}} } \right), \hspace{1cm} {w_{33}} = \frac{{8\sqrt {p\left( {8 - 7p} \right)} - 31{p^2} + 38p + 1}}{{8\left( {4 - 3p + \sqrt {p\left( {8 - 7p} \right)} } \right)}}.
\end{equation}
The off diagonal matrix elements $w_{ij}$ are zero. It is simple to verify that ${w_{33}} \geqslant {w_{11}}$ for any value of $ p$. This implies that ${\lambda _{\max }}\left( W \right) = {w_{33}}$. Therefore, the local quantum uncertainty is simply given by
\begin{equation}
{\mathcal{U}}\left( {{\rho _{GHZ}}} \right) = \frac{{31{{\left( {p - 1} \right)}^2}}}{{8\left( {4 - 3p + \sqrt {p\left( {8 - 7p} \right)} } \right)}}. \label{lqughz}
\end{equation}
The variation of local quantum uncertainty versus the parameter $p$ is reported in figure 2 and will compared with two other quantum correlations quantifies: the tripartite negativity \cite{Vidal2002} and quantum discord \cite{Giorgi2011}.
\subsubsection{Three-qubit state of Bell type}
As a second instance of three-qubit X-states, we consider states of Bell type \cite{Cabello2002}
\begin{equation}
\rho _B = \frac{1}{8}\left( {{\sigma _0} \otimes {\sigma _0} \otimes {\sigma _0} + \sum\limits_{i = 1}^3 {{c_i}{\sigma _i} \otimes {\sigma _i} \otimes {\sigma _i}} } \right), \label{Bellstate}
\end{equation}
with $0 \le {c_i} \le 1$ and $c_{1}^{2}+c_{2}^{2}+c_{3}^{2}\le 1$. In the computational basis, $\rho _B$ takes the form
\begin{equation}
\rho _B = \frac{1}{8}\left( {\begin{array}{*{20}{c}}
{1 + {c_3}}&0&0&0&0&0&0&{{c_1} + i{c_2}} \\
0&{1 - {c_3}}&0&0&0&0&{{c_1} - i{c_2}}&0 \\
0&0&{1 - {c_3}}&0&0&{{c_1} - i{c_2}}&0&0 \\
0&0&0&{1 + {c_3}}&{{c_1} + i{c_2}}&0&0&0 \\
0&0&0&{{c_1} - i{c_2}}&{1 - {c_3}}&0&0&0 \\
0&0&{{c_1} + i{c_2}}&0&0&{1 + {c_3}}&0&0 \\
0&{{c_1} + i{c_2}}&0&0&0&0&{1 + {c_3}}&0 \\
{{c_1} - i{c_2}}&0&0&0&0&0&0&{1 - {c_3}}
\end{array}} \right).
\end{equation}
In this case also the matrix $W$, whose elements are given by (\ref{W for X-three}), is diagonal. The diagonal elements write
\begin{equation}
{w_{11}} = 1 - \frac{{{c_2}^2 + {c_3}^2}}{{1 + \sqrt {1 - {c_1}^2 - {c_2}^2 - {c_3}^2} }}, \hspace{0.3cm} {w_{22}} = 1 - \frac{{{c_1}^2 + {c_3}^2}}{{1 + \sqrt {1 - {c_1}^2 - {c_2}^2 - {c_3}^2} }}, \hspace{0.3cm} {w_{33}} = 1 - \frac{{{c_1}^2 + {c_2}^2}}{{1 + \sqrt {1 - {c_1}^2 - {c_2}^2 - {c_3}^2} }}.
\end{equation}
Therefore, the local quantum uncertainty in the state ${\rho _B}$ is
given by
\begin{equation}
{\mathcal{U}}\left( {{\rho _B}} \right) = \left\{ \begin{gathered}
\frac{{{c_2}^2 + {c_3}^2}}{{1 + \sqrt {1 - {c_1}^2 - {c_2}^2 - {c_3}^2} }}, \hspace{0.5cm}{\rm if} \hspace{0.5cm} {c_1} \geqslant {c_2}\hspace{0.25cm}{\rm and} \hspace{0.25cm} {c_1} \geqslant {c_3} \hfill \\
\frac{{{c_1}^2 + {c_3}^2}}{{1 + \sqrt {1 - {c_1}^2 - {c_2}^2 - {c_3}^2} }}, \hspace{0.5cm}{\rm if}\hspace{0.5cm}{c_2} \geqslant {c_1}\hspace{0.25cm}{\rm and} \hspace{0.25cm} {c_2} \geqslant {c_3} \hfill \\
\frac{{{c_1}^2 + {c_2}^2}}{{1 + \sqrt {1 - {c_1}^2 - {c_2}^2 - {c_3}^2} }}, \hspace{0.5cm}{\rm if}\hspace{0.5cm}{c_3} \geqslant {c_1}\hspace{0.25cm} {\rm and} \hspace{0.25cm} {c_3} \geqslant {c_2}. \hfill \\
\end{gathered} \right. \label{lquBell}
\end{equation}
To simplify our numerical analysis and the comparison with other discord-like quantifies, we consider the situation where $c_1 = c_2 = c_3= c$. In this case the equation (\ref{lquBell}) gives
\begin{equation}
{\mathcal{U}}\left( {{\rho _B}} \right) = \frac{{2{c^2}}}{{1 + \sqrt {1 - 3{c^2}} }}.
\end{equation}
The local quantum uncertainty is depicted in Fig 1.
\subsection{Comparison with quantum discord and negativity}
\subsubsection{Tripartite Negativity}
The entanglement properties of three qubit $X$-states was discussed
in several works (see for instance \cite{Sabin2008,Weinstein2009}).
To decide about entanglement in a tripartite system, the tripartite
negativity ${N^{\left( 3 \right)}}\left( \rho_{123} \right)$ was
introduced in \cite{Vidal2002}. It is given by
\begin{equation}
{N^{\left(3 \right)}}\left(\rho_{123}\right):= \sqrt[3]{N(\rho_{123}^{T_1})N(\rho_{123}^{T_2})N(\rho_{123}^{T_3})}, \label{eq44}
\end{equation}
where $N\left( {{\rho_{123}^{{T_1}}}} \right)$ denotes the bipartite
negativity between the qubit $1$ and the subsystem $2$ and $3$. It
is defined by $N(\rho_{123}^{T_1})= \sum\limits_i {\left| {{\lambda
_i}\left( {{\rho_{123}^{{T_1}}}} \right)} \right|} - 1$ where
${{\rho_{123}^{{T_1}}}}$ is the partial transpose of $\rho_{123}$
with respect to the subsystem 1 and ${{\lambda _i}\left(
{{\rho_{123}^{{T_1}}}} \right)}$ are the eigenvalues of
${{\rho_{123}^{{T_1}}}}$. Similar definitions hold for
$N(\rho_{123}^{T_2})$ and $N(\rho_{123}^{T_3})$. The negativity can
be equivalently interpreted as the most negative eigenvalue of the
partial transpose of the density matrix with respect to qubit $1$.
For tripartite quantum systems with permutation invariance, ${N^{\left( 3\right)}}\left( \rho_{123}\right)$ reduces to the bipartite negativity of any bipartition of the system. This writes as
\begin{equation}
{N^{\left( 3 \right)}}(\rho_{123}) = N(\rho_{123}^{T_1}) = N(\rho_{123}^{T_2}) = N(\rho_{123}^{T_3}). \label{Negativity}
\end{equation}
Thus, for the three-qubit state of Bell type (\ref{Bellstate}), the tripartite negativity vanishes
\begin{equation}
{N^{\left( 3 \right)}}\left(\rho _B \right) = \frac{1}{2}\left|1 - \sqrt 3 c \right| + \frac{1}{2}\left|1 + \sqrt 3 c \right| - 1=0.
\end{equation}
\begin{figure}[h]
\centerline{\includegraphics[width=12cm]{NDUB.eps}}
{\bf Figure 1.} {\sf The local quantum uncertainty and negativity in three-qubit state of Bell type versus the parameter $c$.} \label{fg8}
\end{figure}
\subsubsection{Tripartite Quantum Discord}
According to the reference \cite{Giorgi2011}, the genuine tripartite total correlations $T^{\left( 3\right)}\left( \rho\right) $ in a mixed three-qubit state $\rho_{123}$ is defined by
\begin{equation}
T^{\left( 3\right) }\left( \rho_{123}\right)= T\left( \rho_{123}\right)-T^{\left( 2\right) }\left( \rho_{123}\right), \label{totatcorrelation}
\end{equation}
where $T\left( \rho_{123}\right)= S\left( \rho_{1}\right)+S\left( \rho_{2}\right)+S\left( \rho_{3}\right)-S\left( \rho_{123}\right)$ is the quantum extension of Shannon classical mutual information and $T^{\left( 2\right) }\left( \rho\right)$ is the maximum of the pairwise correlations in the quantum system
\begin{equation}
T^{\left( 2\right) }\left( \rho_{123}\right) ={\rm max}[I^{\left(
2\right) }\left( \rho_{1|2}\right),I^{\left( 2\right) }\left(
\rho_{1|3}\right),I^{\left( 2\right) }\left( \rho_{2|3}\right)]
\end{equation}
with ${I^{\left( 2 \right)}}\left( {{\rho _{i|j}}} \right) = S\left( {{\rho _i}} \right) + S\left( {{\rho _j}} \right) - S\left( {{\rho _{ij}}} \right)$. Here $\rho_{i}\left( i=1,2,3\right) $ is the reduced density matrix for the subsystem $i$ and $S\left( \rho\right) =-{\rm tr}[\rho\log\left( \rho\right) ]$ is the von Neumann entropy. The total tripartite correlation (\ref{totatcorrelation}) rewrites also as
\begin{equation}
T^{\left( 3\right) }\left(\rho_{123} \right)={\rm min}[I^{\left( 2\right) }\left( \rho_{1|23}\right),I^{\left( 2\right) }\left( \rho_{2|13}\right),I^{\left( 2\right) }\left( \rho_{3|12}\right)].
\end{equation}
Analogously, the genuine tripartite classical correlations $J^{\left( 3\right) }\left( \rho_{123}\right)$ is defined as
\begin{equation}
J^{\left( 3\right) }\left( \rho_{123}\right)=J\left( \rho_{123}\right) -J^{\left( 2\right)}\left( \rho_{123}\right),
\end{equation}
where
\begin{equation}
J\left( \rho_{123}\right) = \mathop {\max }\limits_{i,j,k \in \left\{ {1,2,3} \right\}} \left[ {S\left( {{\rho _i}} \right) - S\left( {{\rho _{i|j}}} \right) + S\left( {{\rho _k}} \right) - S\left( {{\rho _{k|ij}}} \right)} \right],
\end{equation}
and
\begin{equation}
{J^{\left( 2 \right)}}\left( \rho_{123}\right) = \max \left[ {{J^{\left( 2 \right)}}\left( {{\rho _{1|2}}} \right),{J^{\left( 2 \right)}}\left( {{\rho _{1|3}}} \right),{J^{\left( 2 \right)}}\left( {{\rho _{2|3}}} \right)} \right].
\end{equation}
As for two-qubit systems \cite{Ollivier2001,Henderson2001}, the tripartite quantum discord ${D^{\left( 3 \right)}}\left( \rho_{123}\right)$ can be expressed as the difference between the genuine total correlations and the genuine classical correlations. This is given by
\begin{equation}
{D^{\left( 3 \right)}}\left( \rho_{123}\right) = {T^{\left( 3 \right)}}\left( \rho_{123}\right) - {J^{\left( 3 \right)}}\left( \rho_{123}\right).
\end{equation}
For three qubit states which are invariant under permutations symmetry, the total and classical correlations can be expressed, respectively, as
\begin{equation}
{T^{\left( 3 \right)}}\left( \rho_{123} \right) = S\left( {{\rho _1}} \right) + S\left( {{\rho _{1|2}}} \right) - S\left( \rho_{123} \right),
\end{equation}
and
\begin{equation}
{J^{\left( 3 \right)}}\left(\rho_{123}\right) = S\left(\rho _1 \right) - S\left( \rho _{1|23} \right).
\end{equation}
In this case, the tripartite quantum discord reduces to
\begin{equation}
{D^{\left( 3 \right)}}\left(\rho_{123}\right) = S\left(\rho _{1/23} \right) + S\left( {{\rho _{1,2}}} \right) - S\left( \rho \right), \label{tripartiteDiscord}
\end{equation}
where $S\left( {{\rho _{1|23}}} \right) = \mathop {\min }\limits_{\left\{ {E_{ij}^{23}} \right\}} \sum\limits_{ij} {{p_{ij}}S\left( {{\rho _{1/E_{ij}^{23}}}} \right)} $ is the relative entropy of the qubit $"1"$ when the measurement is carried out on the subsystem (23), the operators ${E_{ij}^{23}}$ are positive-operator-valued measures (POVMs) that act on the qubits $2$ and $3$ (see \cite{Hamieh2004}), ${p_{ij}} = {\rm tr}\left[ {\left( {{\openone^1} \otimes E_{ij}^{23}} \right)\rho } \right]$ is the probability to obtain the $\left(i,j\right) $ outcomes. The density matrix of the system after the measurement is given by ${\rho _{1/E_{ij}^{23}}} = {{{\rm tr}_{23}\left[ {\left( {{\openone^1} \otimes E_{ij}^{23}} \right)\rho } \right]} \mathord{\left/{\vphantom {{T{r_{BC}}\left[ {\left( {{1^A} \otimes E_{ij}^{BC}} \right)\rho } \right]} {{p_{ij}}}}} \right. \kern-\nulldelimiterspace} {{p_{ij}}}}$. For three qubit $X$ states, the quantum discord is given by \cite{Buscemi2013,Beggi2014}
\begin{align}\label{D}
{D^{\left( 3 \right)}}\left( \rho_{123}\right) &= S\left( {{\rho _{1|23}}} \right) - \frac{1}{3}\left( {1 + 4{\rho _{11}}} \right)\log \left( {2 + 8{\rho _{11}}} \right) - \frac{2}{3}\left( {1 - 2{\rho _{11}}} \right)\log \left( {4 - 8{\rho _{11}}} \right) - 1 + 2{\rho _{11}}\log \left( 3 \right) \notag\\ &+ \left( {{\rho _{11}} - {\rho _{18}}} \right)\log \left( {8{\rho _{11}} - 8{\rho _{18}}} \right)
+ \left( {{\rho _{11}} + {\rho _{18}}} \right)\log \left( {8{\rho _{11}} + 8{\rho _{18}}} \right) +\\ & \frac{1}{2}\left( {1 - 2{\rho _{11}} - 6{\rho _{27}}} \right)\log \left( {4 - 8{\rho _{11}} - 24{\rho _{27}}} \right)\notag + \frac{1}{2}\left( {1 - 2{\rho _{11}} + 6{\rho _{27}}} \right)\log \left( {4 - 8{\rho _{11}} + 24{\rho _{27}}} \right),
\end{align}
where the relative entropy $S\left( {{\rho _{1|23}}} \right) = \min \left\{ {{S_1},{S_3}} \right\}$ if $\left| {3{\rho _{18}}} \right| \ge \left| {{\rho _{27}}} \right|$ and ${\rho _{18}}{\rho _{27}}<0$, otherwise it is given by $S\left( {{\rho _{1|23}}} \right) = \min \left\{ {{S_1},{S_2}} \right\}$. The quantities $S_{1}$, $S_{2}$ and $S_{3}$ are given by
\begin{equation}
{S_1} = 1 - \frac{1}{{12}}F\left( 1 - 8{\rho _{11}} \right), \hspace{0.5cm} {S_2} = 1 - \frac{1}{2}G\left( {6{\rho _{27}} + 2{\rho _{18}}} \right), \hspace{0.5cm} {S_3} = 1 - \frac{1}{2}G\left(\sqrt {\frac{{\left( \rho _{18} - {\rho _{27}} \right)^3}}{\rho _{18}}} \right), \label{76}
\end{equation}
where the functions $F\left( x \right)$ and $G\left( x \right)$ are defined by
\begin{equation}
G\left( x \right) = \left( {1 + x} \right)\log \left( {1 + x} \right) + \left( {1 - x} \right)\log \left( {1 - x} \right),
\end{equation}
\begin{equation}
F\left( x \right) = \left( {3 + x} \right)\log \left( {3 + x} \right) + \left( {3 - 3x} \right)\log \left( {3 - 3x} \right) - 2\left( {3 - x} \right)\log \left( {3 - x} \right).
\end{equation}
Using the expressions (\ref{tripartiteDiscord}), (\ref{D}) and (\ref{76}), the quantum discord in the {\rm GHZ} state (\ref{GHZ}) is given by
\begin{equation}
{D^{\left( 3 \right)}}\left( {{\rho _{GHZ}}} \right) = \left( {\frac{{3p - 4}}{4}} \right)\log \left( {4 - 3p} \right) + \frac{p}{8}\log \left( p \right) + \frac{{8 - 7p}}{8}\log \left( {8 - 7p} \right).
\end{equation}
The amount of quantum correlation is compared, in Fig.2, to the local quantum uncertainty (\ref{lqughz}) and the negativity
\begin{equation}
{N^{\left( 3 \right)}}\left( {{\rho _{GHZ}}} \right) = \left| {\frac{p}{2}} \right| + \left| {\frac{{12 - 9p}}{8}} \right| + \left| {\frac{{5p - 4}}{8}} \right| - 1,
\end{equation}
which is simply obtained from the expressions (\ref{eq44}) and (\ref{Negativity}).
\begin{figure}[H]
\centerline{\includegraphics[width=9cm]{LQUGHZZ.eps}}
{\bf Figure 2.} {\sf The quantum correlations in tripartite mixed GHZ state measured by local quantum uncertainty, quantum discord and negativity.}
\label{Fig.1}
\end{figure}
The results plotted in Fig 2 show that local quantum uncertainty and quantum discord present similar behaviour. It is also noticed that like quantum discord, the local quantum uncertainty captures the classical correlations that cannot be revelead by the negativity. Indeed for $0.8 \le p \le 1$, the negativity is zero and cannot be employed as a faithful measure of quantum correlations. Our results agree with the results reported in \cite{Huang2016} for special mixed three-qubit states in which the discord-like measures reveal more quantum correlations than entanglement. Moreover, this shows that local quantum uncertainty constitutes a good quantum correlation quantifier for three-qubit systems.
\section{Local quantum uncertainty in three-qubit {\rm GHZ} state under decoherence channels}
The Markov dynamics of states evolving in noisy environments is modeled by the quantum operation $\varepsilon :\rho \to \varepsilon \left( \rho \right)$. The channel action on a tripartite state $\rho_{123}$ can be completely characterized in the Kraus representation as follows
\begin{equation}
\varepsilon \left( \rho_{123} \right) = \sum\limits_{ijk} {\left( {{K_i} \otimes {K_j} \otimes {K_k}} \right)\rho_{123} } {\left( {{K_i} \otimes {K_j} \otimes {K_k}} \right)^\dag },
\end{equation}
where ${{K_i}}$ are the local Kraus operators describing the decoherence of a single qubit. For several decoherence scenarios, the action of the
decoherence channel is generally parameterized by the decoherence probability $q = 1 - \exp \left( { - \kappa t} \right)$ with $\kappa$ is the decay parameter. The Kraus operators satisfy the closure condition ${\sum\limits_i {{K_i}{K_i}} ^\dag } = 1$. In this section we shall discuss the effects of three different environments on the mixed {\rm GHZ}-state (\ref{GHZ}). The dynamics of the quantum correlations is fully characterized by the decoherence parameter $q$.
\subsection{Dephasing environment}
We first consider the situation where each qubit is submitted to a dephasing effect induced by the environment. The Kraus operators representing this effect are
\begin{equation}
{K_1} = \left( {\begin{array}{*{20}{c}}
1&0 \\
0&{\sqrt {1 - q} }
\end{array}} \right) \hspace{1cm} {K_2} = \left( {\begin{array}{*{20}{c}}
0&0 \\
0&{\sqrt q }
\end{array}} \right).
\end{equation}
Under this effect, the tripartite state ${\rho _{123}}$ (\ref{1}) remains of $X$-type. The evolved matrix density denoted by $\varepsilon_{DE} \left( \rho_{123} \right)=\rho_{123}^{DE}$ has the same diagonal elements as $\rho_{123}$ while the anti-diagonal elements are multiplied by the factor ${\left( {1 - q} \right)^{\frac{3}{2}}}$. The local quantum uncertainty $\mathcal{U}\left( \rho_{123}^{DE} \right) = 1 -
{\lambda _{\max }}\left( W^{DE} \right)$ can be evaluated using the results reported in the section 2. For the {\rm GHZ}-state (\ref{GHZ}), we shows that the non vanishing matrix elements of the matrix ${W^{DE}}$ (\ref{eq27}), associated with the density matrix $\rho_{123}^{DE}$, are given by
\begin{equation}
w_{11}^{DE} = w_{22}^{DE} = \frac{1}{2}\left( {p + \sqrt {\frac{{p\left( {4 - 3p + \sqrt {{{\left( {4 - 3p} \right)}^2} - 16{{\left( {1 - p} \right)}^2}{{\left( {1 - q} \right)}^3}} } \right)}}{2}} } \right),
\end{equation}
\begin{equation}
w_{33}^{DE} = \frac{{32\left( {1 - p} \right)\left( {1 + \left( {p - 1} \right){{\left( {1 - q} \right)}^3}} \right) + {{\left( {1 + p} \right)}^2} + 4\left( {p + 2\sqrt {{{\left( {4 - 3p} \right)}^2} - 16{{\left( {1 - p} \right)}^2}{{\left( {1 - q} \right)}^3}} } \right)}}{{8\left( {4 - 3p + \sqrt {{{\left( {4 - 3p} \right)}^2} - 16{{\left( {1 - p} \right)}^2}{{\left( {1 - q} \right)}^3}} } \right)}}.
\end{equation}
It is simple to check that $w_{33}^{DE} \ge w_{11}^{DE}$. This gives ${\lambda _{\max }}\left( {{W^{DE}}} \right) = w_{33}^{DE}$ and in this case the local quantum uncertainty is
\begin{equation}
{\mathcal{U}}\left( {\rho _{GHZ}^{DE}} \right) = \frac{{{{\left( {1 - p} \right)}^2}\left( {32{{\left( {1 - q} \right)}^3} - 1} \right)}}{{8\left( {4 - 3p + \sqrt {{{\left( {4 - 3p} \right)}^2} - 16{{\left( {1 - p} \right)}^2}{{\left( {1 - q} \right)}^3}} } \right)}}.
\end{equation}
The local quantum uncertainty is plotted in Fig.3 for different values of mixedness parameter $p$ and the decoherence parameter $q$ which reflects the degradation of quantum correlations under decoherence effects. To investigate the robustness of local quantum uncertainty in comparison with other quantifiers, we analysed also the dynamics of quantum discord and negativity. Using the results (\ref{Negativity}) and (\ref{D}), the negativity in the state ${\rho _{GHZ}^{DE}}$ writes
\begin{equation}
{N^{\left( 3 \right)}}\left( {\rho _{GHZ}^{DE}} \right) = \left| {\frac{p}{2}} \right| + \left| {\frac{{4 - 3p}}{4}} \right| + \left| {\frac{{p - 4\left( {1 - p} \right){{\left( {1 - q} \right)}^{\frac{3}{2}}}}}{8}} \right| + \left| {\frac{{p + 4\left( {1 - p} \right){{\left( {1 - q} \right)}^{\frac{3}{2}}}}}{8}} \right| - 1,
\end{equation}
and the quantum discord (\ref{D}) is given by
\begin{align}
{D^{\left( 3 \right)}}\left( {\rho _{GHZ}^{DE}} \right) &= \frac{{3p - 4}}{4}\log \left( {4 - 3p} \right) + \frac{1}{8}\left( {4 - 3p - 4\left( {1 - p} \right){{\left( {1 - q} \right)}^{\frac{3}{2}}}} \right)\log \left( {4 - 3p - 4\left( {1 - p} \right){{\left( {1 - q} \right)}^{\frac{3}{2}}}} \right){\rm{ }} \notag\\ &+ \frac{1}{8}\left( {4 - 3p + 4\left( {1 - p} \right){{\left( {1 - q} \right)}^{\frac{3}{2}}}} \right)\log \left( {4 - 3p + 4\left( {1 - p} \right){{\left( {1 - q} \right)}^{\frac{3}{2}}}} \right).
\end{align}
\begin{figure}[H]
\centering
\begin{minipage}[t]{3in}
\centering
\includegraphics[width=3.1in]{LQUDE.eps}
\end{minipage}
\begin{minipage}[t]{3in}
\centering
\includegraphics[width=3.1in]{DDE.eps}
\end{minipage}
{\bf Figure 3.} {\sf The local quantum uncertainty and the quantum discord versus the dephasing parameter $q$. }
\end{figure}
\begin{figure}[H]
\centerline{\includegraphics[width=9cm]{NDE.eps}}
{\bf Figure 4.} {\sf The negativity versus the dephasing parameter $q$.}
\label{Fig.6}
\end{figure}
The entropic quantum discord in mixed {\rm GHZ} states shows more
robustness against the dephasing effects in comparison with local
quantum uncertainty. On the other hand, the negativity is more
robust than local quantum uncertainty and quantum discord (see
figure 4). Indeed, for $0.8<q<1$, the local quantum uncertainty and
quantum discord vanished whereas the negativity is non-zero for a mixing parameter $p$ taking the values between $0$ and $0.6$. This is an unusual and unexpected important behavior of the negativity in noisy environment in comparison with quantum discord and local quantum uncertainty believed stranger and robust that entanglement.
\subsection{Depolarizing environment}
The depolarization effect of a single qubit in the Kraus representation is given by
\begin{equation}
{K_1} = \sqrt {1 - \frac{{3q}}{4}} \left( {\begin{array}{*{20}{c}}
1&0 \\
0&1
\end{array}} \right), \hspace{0.5cm} {K_2} = \frac{{\sqrt q }}{2}\left( {\begin{array}{*{20}{c}}
0&1 \\
1&0
\end{array}} \right), \hspace{0.5cm} {K_3} = \frac{{\sqrt q }}{2}\left( {\begin{array}{*{20}{c}}
0&{ - i} \\
i&0
\end{array}} \right), \hspace{0.5cm} {K_4} = \frac{{\sqrt q }}{2}\left( {\begin{array}{*{20}{c}}
1&0 \\
0&{ - 1}
\end{array}} \right).
\end{equation}
In this case, the evolved state is also a three-qubit state of $X$ type. The non-zero matrix elements of the density matrix ${\rho _{GHZ}^{PE}}$ are given
\begin{equation}
\rho{'_{11}} = \rho' _{88} = \frac{{4 - 3p}}{8}\left( {1 - \frac{{3q}}{2} + \frac{{3{q^2}}}{4}} \right) + \frac{{3p}}{8}\left( {\frac{q}{2} - \frac{{{q^2}}}{4}} \right),
\end{equation}
\begin{equation}
\rho'_{22} =\rho'_{33} = \rho'_{44} = \rho'_{55} = \rho'_{66} = \rho'_{77} = \frac{p}{8}\left( {1 - \frac{{3q}}{2} + \frac{{3{q^2}}}{4}} \right) + \frac{{4 - p}}{8}\left( {\frac{q}{2} - \frac{{{q^2}}}{4}} \right),
\end{equation}
\begin{equation}
\rho'_{18} = \rho'_{81} = \frac{{\left( {1 - p} \right){{\left( {1 - q} \right)}^3}}}{2},
\end{equation}
in the computational basis. The elements of the matrix (\ref{W for X-three}) write as
{\footnotesize \begin{align}
w_{11}^{PE}= w_{22}^{PE} = &\sqrt {\frac{1}{8} (p + q (1 - p) (2 - q))\left( 4 - 3p + 3q (1 - p) (q - 2) + \sqrt {\left( 4 - 3p + 3q\left( 1 - p \right)\left( {q - 2} \right) \right)^2 - 16{{\left( 1 - p \right)}^2}{\left( 1 - q \right)^6}} \right)} \notag \\ & + \frac{1}{2}\left( p + q\left(1 - p \right)\left(2 - q \right) \right),
\end{align}}
\begin{align}
w_{33}^{PE} &= \frac{{{{\left( {1 - p} \right)}^2}{{\left( {1 - q} \right)}^2}\left( {1 - 16{{\left( {1 - q} \right)}^4}} \right)}}{{8\left( {4 - 3p + 3q\left( {1 - p} \right)\left( {q - 2} \right) + \sqrt {{{\left( {4 - 3p + 3q\left( {1 - p} \right)\left( {q - 2} \right)} \right)}^2} - 16{{\left( {1 - p} \right)}^2}{{\left( {1 - q} \right)}^6}} } \right)}} \notag \\ &+ \frac{1}{8}\left( {4 + 3p + 3q\left( {1 - p} \right)\left( {q - 2} \right) + \sqrt {{{\left( {4 - 3p + 3q\left( {1 - p} \right)\left( {q - 2} \right)} \right)}^2} - 16{{\left( {1 - p} \right)}^2}{{\left( {1 - q} \right)}^6}} } \right).
\end{align}
Since $w_{11}^{PE} \leqslant w_{33}^{PE}$, the local quantum uncertainty is given by ${\mathcal{U}}\left( {\rho _{GHZ}^{PE}} \right) = 1 - w_{33}^{PE}$ and one gets
\begin{align}
{\mathcal{U}}\left( {\rho _{GHZ}^{PE}} \right) &= \frac{{{{\left( {1 - p} \right)}^2}{{\left( {1 - q} \right)}^2}\left( {16{{\left( {1 - q} \right)}^4} - 1} \right)}}{{8\left( {4 - 3p + 3q\left( {1 - p} \right)\left( {q - 2} \right) + \sqrt {{{\left( {4 - 3p + 3q\left( {1 - p} \right)\left( {q - 2} \right)} \right)}^2} - 16{{\left( {1 - p} \right)}^2}{{\left( {1 - q} \right)}^6}} } \right)}} \notag \\ &+ \frac{1}{8}\left( {4 - 3p - 3q\left( {1 - p} \right)\left( {2 - q} \right) - \sqrt {{{\left( {4 - 3p + 3q\left( {1 - p} \right)\left( {q - 2} \right)} \right)}^2} - 16{{\left( {1 - p} \right)}^2}{{\left( {1 - q} \right)}^6}} } \right).
\end{align}
The negativity in the evolved state ${\rho _{GHZ}^{PE}}$ is
\begin{align}
{N^{\left( 3 \right)}}\left( {\rho _{GHZ}^{PE}} \right){\rm{ }} &= \frac{1}{2}\left| {p - 2pq + p{q^2} + 2q - {q^2}} \right| + \frac{1}{4}\left| {4 - 6q + 3{q^2} - 3p + 6pq - 3p{q^2}} \right|- 1 + \\&\notag \frac{1}{8}\left| {p - 2pq + p{q^2} + 2q - {q^2} + 4(1 - p){{(1 - q)}^3}} \right| + \frac{1}{8}\left| {p - 2pq + p{q^2} + 2q - {q^2} - 4(1 - p){{(1 - q)}^3}} \right| ,
\end{align}
and the quantum discord is given by the following expression
{\small\begin{align}
D^{\left( 3 \right)}\left( \rho _{GHZ}^{PE} \right){\rm{ }} = -& \frac{1}{4}\left( {\left( {4 - 3p} \right)\left( {1 - \frac{{3q}}{2} + \frac{{3{q^2}}}{4}} \right) + 3p\left( {\frac{q}{2} - \frac{{{q^2}}}{4}} \right)} \right)\log \left( {\left( {4 - 3p} \right)\left( {1 - \frac{{3q}}{2} + \frac{{3{q^2}}}{4}} \right) + 3p\left( {\frac{q}{2} - \frac{{{q^2}}}{4}} \right)} \right) \notag \\ & +\frac{1}{8}{\beta _ + }\left( {p,q} \right)\log \left( {{\beta _ + }\left( {p,q} \right)} \right){\rm{ + }} \frac{1}{8}{\beta _ - }\left( {p,q} \right)\log \left( {{\beta _ - }\left( {p,q} \right)} \right),
\end{align}}
where
\begin{equation}
\beta _ \pm \left( {p,q} \right) = \left( {4 - 3p} \right)\left( {1 - \frac{{3q}}{2} + \frac{{3{q^2}}}{4}} \right) + 3p\left( {\frac{q}{2} - \frac{{{q^2}}}{4}} \right) \pm 4\left( {1 - p} \right){\left( {1 - q} \right)^3}.
\end{equation}
\begin{figure}[H]
\centering
\begin{minipage}[t]{3in}
\centering
\includegraphics[width=3.1in]{LQUDPP.eps}
\end{minipage}
\begin{minipage}[t]{3in}
\centering
\includegraphics[width=3.1in]{DDP.eps}
\end{minipage}
{\bf Figure 5.} {\sf The local quantum uncertainty and the quantum discord versus the depolarizing strength $q$. }
\end{figure}
\begin{figure}[H]
\centerline{\includegraphics[width=9cm]{NDP.eps}}
{\bf Figure 6.} {\sf The negativity versus the depolarizing strength $q$.}
\end{figure}
Under depolarizing effects, the quantum discord is more robust in comparison with local quantum uncertainty. However, like for the dephasing effects, it is remarkable that the negativity shows more robustness than the local quantum uncertainty and the quantum discord.
\subsection{Phase reversal environment:}
Phase reversal environment leaves the state invariant $\left| 0 \right\rangle$ with the probability $q$ and changes the state $\left| 1 \right\rangle$ to $-\left| 1 \right\rangle$ with the probability $\left( {1 - q} \right)$. The corresponding Kraus operators are
\begin{equation}
{K_1} = \sqrt {1 - q} \left( {\begin{array}{*{20}{c}}
1&0 \\
0&1
\end{array}} \right) \hspace{1cm} {K_2} = \sqrt q \left( {\begin{array}{*{20}{c}}
1&0 \\
0&{ - 1}
\end{array}} \right).
\end{equation}
Under the phase reversal effect, the matrix elements of the evolved density matrix $\rho_{GHZ}^{PRE}$ are $\rho_{ij}^{PRE}=(1 + 10{q^2} - 6q\left( {1 + {q^2}} \right))\rho_{ij}$ for $i \ne j$ and $\rho_{ii}^{PRE}=\rho_{ii}$. In this case, the eigenvalues of the matrix $W^{PRE}$ obtained from the equation (\ref{W for X-three}) are given by
\begin{equation}
w_{11}^{PRE} = w_{22}^{PRE} = \frac{1}{2}\left( {p + \sqrt {\frac{{p\left( {4 - 3p + \sqrt {{{\left( {4 - 3p} \right)}^2} - 16{{\left( {1 - p} \right)}^2}\left( {32\left( {1 + 10{q^2} - 6q\left( {1 + {q^2}} \right)} \right) - 1} \right)} } \right)}}{2}} } \right),
\end{equation}
and
\begin{align}
w_{33}^{PRE} = &\frac{{32\left( {1 - p} \right)\left( {1 + \left( {p - 1} \right)\left( {32\left( {1 + 10{q^2} - 6q\left( {1 + {q^2}} \right)} \right) - 1} \right)} \right) + {{\left( {1 + p} \right)}^2}}}{{8\left( {4 - 3p + \sqrt {{{\left( {4 - 3p} \right)}^2} - 16{{\left( {1 - p} \right)}^2}\left( {32\left( {1 + 10{q^2} - 6q\left( {1 + {q^2}} \right)} \right) - 1} \right)} } \right)}} + \notag \\
&\frac{{\left( {p + 2\sqrt {{{\left( {4 - 3p} \right)}^2} - 16{{\left( {1 - p} \right)}^2}\left( {32\left( {1 + 10{q^2} - 6q\left( {1 + {q^2}} \right)} \right) - 1} \right)} } \right)}}{{2\left( {4 - 3p + \sqrt {{{\left( {4 - 3p} \right)}^2} - 16{{\left( {1 - p} \right)}^2}\left( {32\left( {1 + 10{q^2} - 6q\left( {1 + {q^2}} \right)} \right) - 1} \right)} } \right)}}.
\end{align}
Here also we have $w_{11}^{PRE} \leqslant w_{33}^{PRE}$
and the local quantum uncertainty is simply given by
${\mathcal{U}}\left( {\rho _{GHZ}^{PRE}} \right) = 1 -
w_{33}^{PRE}$. This can be written as
\begin{equation}
{\mathcal{U}}\left( {\rho _{GHZ}^{PRE}} \right) = \frac{{{{\left( {1 - p} \right)}^2}\left( {32\left( {1 + 10{q^2} - 6q\left( {1 + {q^2}} \right)} \right) - 1} \right)}}{{8\left( {4 - 3p + \sqrt {{{\left( {4 - 3p} \right)}^2} - 16{{\left( {1 - p} \right)}^2}\left( {1 + 10{q^2} - 6q\left( {1 + {q^2}} \right)} \right)} } \right)}}.
\end{equation}
For the evolved state $\rho _{GHZ}^{PRE}$, the negativity is given by
\begin{align}
{N^{\left( 3 \right)}}\left( {\rho _{GHZ}^{PRE}} \right) = &\left| {\frac{p}{2}} \right| + \left| {\frac{{4 - 3p}}{4}} \right| + \left| {\frac{{p - 4\left( {1 - p} \right)\left( {1 + 10{q^2} - 6q\left( {1 + {q^2}} \right)} \right)}}{8}} \right|+ \notag\\& \left| {\frac{{p + 4\left( {1 - p} \right)\left( {1 + 10{q^2} - 6q\left( {1 + {q^2}} \right)} \right)}}{8}} \right| - 1,
\end{align}
and the quantum discord writes as
\begin{align}
{D^{\left( 3 \right)}}\left( {\rho _{GHZ}^{PRE}} \right) &=\frac{1}{8}\left( {4 - 3p - 4\left( {1 - p} \right)\left( {1 + 10{q^2} - 6q\left( {1 + {q^2}} \right)} \right)} \right)\log \left( {4 - 3p - 4\left( {1 - p} \right)\left( {1 + 10{q^2} - 6q\left( {1 + {q^2}} \right)} \right)} \right) \notag\\
& +\frac{1}{8}\left( {4 - 3p + 4\left( {1 - p} \right)\left( {1 + 10{q^2} - 6q\left( {1 + {q^2}} \right)} \right)} \right)\log \left( {4 - 3p + 4\left( {1 - p} \right)\left( {1 + 10{q^2} - 6q\left( {1 + {q^2}} \right)} \right)} \right) \notag\\
& +\frac{3p - 4}{4}\log \left( 4 - 3p \right).
\end{align}
\begin{figure}[H]
\centering
\begin{minipage}[t]{3in}
\centering
\includegraphics[width=3.1in]{LQUDFF.eps}
\end{minipage}
\begin{minipage}[t]{3in}
\centering
\includegraphics[width=3.1in]{DDF.eps}
\end{minipage}
{\bf Figure 7.} {\sf The local quantum uncertainty and the quantum discord versus the decoherence parameter $q$. }
\end{figure}
\begin{figure}[H]
\centerline{\includegraphics[width=9cm]{NDF.eps}}
{\bf Figure 8.} {\sf The negativity versus the decoherence parameter $q$.}
\end{figure}
For a mixing parameters $p$ such that $0<p<0.6$, the local quantum
uncertainty shows a revival phenomenon under the phase reversal
environment. It increases to become maximal for higher values of
decoherence parameter $q$. It seems that the phase reversal tends to
enhance the amount of quantum correlation in the system. Also for
states with a mixing parameter $p$ such that $0.7<p<1$, the local
quantum uncertainty tends to be frozen. The entropic quantum discord
does not exhibit this phenomenon. The negativity and entropic
quantum discord confirm the revival of non-classical correlation in
three-qubit {\rm GHZ} under phase reversal effects (see figures 7
and 8).
\section{Monogamy of local quantum uncertainty}
The free shareability relation of classical correlations is no longer valid in the quantum case. Indeed, it has been shown that there are several limitation in sharing quantum correlations between the components of a multi-partite quantum system. These limitations are expressed by the so-called monogamy relation \cite{Kim2012,Streltsov2012}. This limitation was equaled by Coffman, Kundu and Wootters in 2001 for the concurrence and was extended since then to other quantum correlations quantifiers \cite{Coffman2000}. The concept of monogamy can be expressed as follows. Let us denote by $Q_{1|2}$ the shared correlations between the qubits $1$ and $2$, and $Q_{1|3}$ the amount of quantum correlations between the qubits $1$ and $3$. The monogamy constraint imposes that the quantum correlation $Q_{1|23}$ (between the qubit $1$ and the sub-system comprising the qubits $2$ and $3$) is always greater than the sum of $Q_{1|2}$ and $Q_{1|3}$:
\begin{equation} \label{monogamy}
Q_{1|23}\ge Q_{1|2} + Q_{1|3}.
\end{equation}
Extending this inequality to local quantum uncertainty, the monogamy holds when
\begin{equation}
{{\mathcal{U}}_{1|23}} \ge {{\mathcal{U}}_{1|2}} + {{\mathcal{U}}_{1|3}}.
\end{equation}
We first consider the monogamy of local quantum uncertainty in the mixed {\rm GHZ} state. To do this, we determine the bipartite local quantum uncertainty in the subsystems containing the qubits $2$ and $3$ and the qubits $1$ and $3$. The corresponding reduced density matrices are
\begin{equation}
\rho _{GHZ}^{23} = \rho _{GHZ}^{13} = \frac{1}{4}\left( {\begin{array}{*{20}{c}}
{2 - p}&0&0&0\\
0&p&0&0\\
0&0&p&0\\
0&0&0&{2 - p}
\end{array}} \right).
\end{equation}
Using the equation (\ref{lquA}), the elements of the matrix $W$ (\ref{w}) are simply given by
\begin{equation}
w_{11} = w_{22} = \sqrt {p\left( {2 - p} \right)} \hspace{1cm}{\rm and}\hspace{1cm} w_{33} = 1.
\end{equation}
We have $w_{33} \ge w_{11}$ and the local quantum uncertainty is
\begin{equation}
{\mathcal{U}}\left( {\rho _{GHZ}^{23}} \right) = {\mathcal{U}}\left( {\rho _{GHZ}^{13}} \right) = 0.
\end{equation}
It is clear that the local quantum uncertainty in the ${{\rho _{GHZ}}}$ satisfies the monogamy constraint
\begin{equation}
{\mathcal{U}}\left( {\rho _{GHZ}} \right) \ge {\mathcal{U}}\left( {\rho _{GHZ}^{23}} \right) + {\mathcal{U}}\left( {\rho _{GHZ}^{13}} \right).
\end{equation}
Similarly, we consider the monogamy property of local quantum uncertainty in the three-qubit Bell states (\ref{Bellstate}). In this case we have $\rho _B^{23} = \rho _B^{13} = \frac{1}{4}{\openone_{4 \times 4}}$ and $w_{11} = w_{22} = {w_{33}} = 1$. Thus, we obtains
\begin{equation}
{\mathcal{U}}\left( {\rho _B^{23}} \right) = {\mathcal{U}}\left( {\rho _B^{13}} \right) = 0,
\end{equation}
and this implies that the local quantum uncertainty in the Bell states (\ref{Bellstate}) is monogamous.
\section{Conclusion}
The analytical expression of local quantum uncertainty is derived
for three-qubit $X$ states. As illustration, we computed the non
classical correlation in mixed {\rm GHZ} state and Bell-type
three-qubit state by employing this quantum correlation quantifier.
The obtained results are compared with ones obtained by means of
entropic quantum discord and negativity. The amount of quantum
correlations quantified by local quantum uncertainty is almost
similar to one measured by entropic quantum discord. This indicates
that local quantum uncertainty constitutes an appropriate quantifier
to deal with quantum correlation in multi-qubit systems. This is
essentially due to its easiness computability. Also, it goes beyond
the negativity and offers the tool to quantify the non-classical
correlations contained in multi-qubit separable states. The
evolution of local quantum uncertainty in different noisy
environments is also discussed in this work. Different aspects of
the local quantum uncertainty are compared to the evolution of
entropic quantum discord and negativity. In particular, we have
shown that under phase reversal effect, the local quantum
uncertainty show revival and frozen phenomena. We also investigated
the monogamy property in three qubit state of {\rm GHZ} and Bell
type. We have shown that the monogamy constraint (\ref{monogamy}) is
satisfied. As prolongation of the results obtained
in this work, we believe that it will be interesting to study the
local quantum uncertainty for other three-qubit states, which are
not of $X$-type, as for instance mixed $W$-states of type ${\rho _W}
= \alpha \left| W \right\rangle \left\langle W \right| + \frac{{1 -
\alpha }}{8}{\openone_8}$ where $\alpha \in {\bf R} $ and
$\left| W \right\rangle = \frac{1}{{\sqrt 3 }}\left( {\left| {001}
\right\rangle + \left| {010} \right\rangle + \left| {100}
\right\rangle } \right)$ denotes a three-qubit $W$-state. We hope to
report on this question in another work.
\section*{Appendix}
In this appendix, we give the necessary tools to compute the local
quantum uncertainty for three-qubit $X$ states. First, the
eigenvalues corresponding to density matrix ${\rho _{123}}$ of the
form (\ref{1}) are given by
\begin{equation}
\begin{gathered}
\lambda _1^ \pm = \frac{1}{2}{t_1} \pm \frac{1}{2}\sqrt {{t_1}^2 - 4{d_1}}, \hfill \\
\lambda _2^ \pm = \frac{1}{2}{t_2} \pm \frac{1}{2}\sqrt {{t_2}^2 - 4{d_2}}, \hfill \\
\lambda _3^ \pm = \frac{1}{2}{t_3} \pm \frac{1}{2}\sqrt {{t_3}^2 - 4{d_3}}, \hfill \\
\lambda _4^ \pm = \frac{1}{2}{t_4} \pm \frac{1}{2}\sqrt {{t_4}^2 - 4{d_4}}, \hfill \\
\end{gathered}
\end{equation}
where
\begin{equation}
\begin{array}{l}
t_1 = \rho _{11} + \rho _{88},\\
t_2 = \rho _{22} + \rho _{77},\\
t_3 = \rho _{33} + \rho _{66},\\
t_4 = \rho _{44} + \rho _{55},
\end{array} \hspace{2cm} \begin{array}{l}
d_1 = \rho _{11}\rho _{88} - \rho _{18}\rho _{81},\\
d_2 = \rho _{22}\rho _{77} - \rho _{27}\rho _{72},\\
d_3 = \rho _{33}\rho _{66} - \rho _{36}\rho _{63},\\
d_4 = \rho _{44}\rho _{55} - \rho _{45}\rho _{54}.
\end{array}
\end{equation}
The matrix $\sqrt {{\rho _{123}}}$ is also $X$-shaped and has the form
\begin{equation*}
\sqrt {{\rho _{123}}} = \left( {\begin{array}{*{20}{c}}
{\frac{{{\rho _{11}} + \sqrt {{d_1}} }}{{\sqrt {{t_1} + 2\sqrt {{d_1}} } }}}&0&0&0&0&0&0&{\frac{{{\rho _{18}}}}{{\sqrt {{t_1} + 2\sqrt {{d_1}} } }}} \\
0&{\frac{{{\rho _{22}} + \sqrt {{d_2}} }}{{\sqrt {{t_2} + 2\sqrt {{d_2}} } }}}&0&0&0&0&{\frac{{{\rho _{27}}}}{{\sqrt {{t_2} + 2\sqrt {{d_2}} } }}}&0 \\
0&0&{\frac{{{\rho _{33}} + \sqrt {{d_3}} }}{{\sqrt {{t_3} + 2\sqrt {{d_3}} } }}}&0&0&{\frac{{{\rho _{36}}}}{{\sqrt {{t_3} + 2\sqrt {{d_3}} } }}}&0&0 \\
0&0&0&{\frac{{{\rho _{44}} + \sqrt {{d_4}} }}{{\sqrt {{t_4} + 2\sqrt {{d_4}} } }}}&{\frac{{{\rho _{45}}}}{{\sqrt {{t_4} + 2\sqrt {{d_4}} } }}}&0&0&0 \\
0&0&0&{\frac{{{\rho _{54}}}}{{\sqrt {{t_4} + 2\sqrt {{d_4}} } }}}&{\frac{{{\rho _{55}} + \sqrt {{d_4}} }}{{\sqrt {{t_4} + 2\sqrt {{d_4}} } }}}&0&0&0 \\
0&0&{\frac{{{\rho _{63}}}}{{\sqrt {{t_3} + 2\sqrt {{d_3}} } }}}&0&0&{\frac{{{\rho _{66}} + \sqrt {{d_3}} }}{{\sqrt {{t_3} + 2\sqrt {{d_3}} } }}}&0&0 \\
0&{\frac{{{\rho _{72}}}}{{\sqrt {{t_2} + 2\sqrt {{d_2}} } }}}&0&0&0&0&{\frac{{{\rho _{77}} + \sqrt {{d_2}} }}{{\sqrt {{t_2} + 2\sqrt {{d_2}} } }}}&0 \\
{\frac{{{\rho _{81}}}}{{\sqrt {{t_1} + 2\sqrt {{d_1}} } }}}&0&0&0&0&0&0&{\frac{{{\rho _{88}} + \sqrt {{d_1}} }}{{\sqrt {{t_1} + 2\sqrt {{d_1}} } }}}
\end{array}} \right).
\end{equation*}
In the Fano-Bloch representation, the matrix $\sqrt {{\rho _{123}}}
$ rewrites as
\begin{equation}
\sqrt {{\rho _{123}}} = \sum\limits_{\chi \delta \eta } {{T_{\chi \delta \eta }}} {\sigma _\chi } \otimes {\sigma _\delta } \otimes {\sigma _\eta },
\end{equation}
where $\chi ,\delta ,\eta = 0,1,2,3$ and the Fano-Bloch parameters are defined by ${T_{\chi \delta \eta }} = {\rm tr}\left( {\sqrt {{\rho _{123}}} {\sigma _\chi } \otimes {\sigma _\delta } \otimes {\sigma _\eta }} \right)$.\\
The non vanishing elements ${T_{\chi \delta \eta }}$ are given by
\begin{align}
{T_{111}} = &\frac{{{R_{111}} - {R_{221}} - {R_{122}} - {R_{212}}}}{{4\sqrt {{t_1} + 2\sqrt {{d_1}} } }} + \frac{{{R_{111}} - {R_{221}} + {R_{122}} + {R_{212}}}}{{4\sqrt {{t_2} + 2\sqrt {{d_2}} } }} + \\ \notag
& \frac{{{R_{111}} + {R_{221}} + {R_{122}} - {R_{212}}}}{{4\sqrt {{t_3} + 2\sqrt {{d_3}} } }} + \frac{{{R_{111}} + {R_{221}} - {R_{122}} + {R_{212}}}}{{4\sqrt {{t_4} + 2\sqrt {{d_4}} } }},
\end{align}
\begin{align}
{T_{211}} = &\frac{{{R_{112}} + {R_{121}} + {R_{211}} - {R_{222}}}}{{4\sqrt {{t_1} + 2\sqrt {{d_1}} } }} + \frac{{{R_{222}} + {R_{121}} + {R_{211}} - {R_{112}}}}{{4\sqrt {{t_2} + 2\sqrt {{d_2}} } }} + \\ \notag
& \frac{{{R_{121}} - {R_{211}} - {R_{112}} - {R_{222}}}}{{4\sqrt {{t_3} + 2\sqrt {{d_3}} } }} + \frac{{{R_{121}} + {R_{112}} + {R_{222}} - {R_{211}}}}{{4\sqrt {{t_4} + 2\sqrt {{d_4}} } }},
\end{align}
\begin{align}
{T_{121}} =& \frac{{{R_{112}} + {R_{121}} + {R_{211}} - {R_{222}}}}{{4\sqrt {{t_1} + 2\sqrt {{d_1}} } }} + \frac{{{R_{112}} - {R_{222}} - {R_{121}} - {R_{211}}}}{{4\sqrt {{t_2} + 2\sqrt {{d_2}} } }} + \\ \notag
&\frac{{{R_{211}} + {R_{112}} + {R_{222}} - {R_{121}}}}{{4\sqrt {{t_3} + 2\sqrt {{d_3}} } }} + \frac{{{R_{121}} + {R_{112}} + {R_{222}} - {R_{211}}}}{{4\sqrt {{t_4} + 2\sqrt {{d_4}} } }},
\end{align}
\begin{align}
{T_{221}} = &\frac{{{R_{221}} + {R_{122}} + {R_{212}} - {R_{111}}}}{{4\sqrt {{t_1} + 2\sqrt {{d_1}} } }} + \frac{{{R_{111}} - {R_{221}} + {R_{122}} + {R_{212}}}}{{4\sqrt {{t_2} + 2\sqrt {{d_2}} } }} + \\ \notag
&\frac{{{R_{111}} + {R_{221}} + {R_{122}} - {R_{212}}}}{{4\sqrt {{t_3} + 2\sqrt {{d_3}} } }} + \frac{{{R_{122}} - {R_{111}} - {R_{221}} - {R_{212}}}}{{4\sqrt {{t_4} + 2\sqrt {{d_4}} } }},
\end{align}
\begin{align}
{T_{112}} = &\frac{{{R_{112}} + {R_{121}} + {R_{211}} - {R_{222}}}}{{4\sqrt {{t_1} + 2\sqrt {{d_1}} } }} + \frac{{{R_{222}} + {R_{121}} + {R_{211}} - {R_{112}}}}{{4\sqrt {{t_2} + 2\sqrt {{d_2}} } }} + \\ \notag
&\frac{{{R_{211}} + {R_{112}} + {R_{222}} - {R_{121}}}}{{4\sqrt {{t_3} + 2\sqrt {{d_3}} } }} + \frac{{{R_{211}} - {R_{121}} - {R_{112}} - {R_{222}}}}{{4\sqrt {{t_4} + 2\sqrt {{d_4}} } }},
\end{align}
\begin{align}
{T_{122}} = &\frac{{{R_{221}} + {R_{122}} + {R_{212}} - {R_{111}}}}{{4\sqrt {{t_1} + 2\sqrt {{d_1}} } }} + \frac{{{R_{111}} - {R_{221}} + {R_{122}} + {R_{212}}}}{{4\sqrt {{t_2} + 2\sqrt {{d_2}} } }} + \\ \notag
&\frac{{{R_{212}} - {R_{111}} - {R_{221}} - {R_{122}}}}{{4\sqrt {{t_3} + 2\sqrt {{d_3}} } }} + \frac{{{R_{111}} + {R_{221}} - {R_{122}} + {R_{212}}}}{{4\sqrt {{t_4} + 2\sqrt {{d_4}} } }},
\end{align}
\begin{align}
{T_{212}} =& \frac{{{R_{221}} + {R_{122}} + {R_{212}} - {R_{111}}}}{{4\sqrt {{t_1} + 2\sqrt {{d_1}} } }} + \frac{{{R_{221}} - {R_{111}} - {R_{122}} - {R_{212}}}}{{4\sqrt {{t_2} + 2\sqrt {{d_2}} } }} + \\ \notag
&\frac{{{R_{111}} + {R_{221}} + {R_{122}} - {R_{212}}}}{{4\sqrt {{t_3} + 2\sqrt {{d_3}} } }} + \frac{{{R_{111}} + {R_{221}} - {R_{122}} + {R_{212}}}}{{4\sqrt {{t_4} + 2\sqrt {{d_4}} } }},
\end{align}
\begin{align}
{T_{222}} =& \frac{{{R_{222}} - {R_{112}} - {R_{121}} - {R_{211}}}}{{4\sqrt {{t_1} + 2\sqrt {{d_1}} } }} + \frac{{{R_{222}} + {R_{121}} + {R_{211}} - {R_{112}}}}{{4\sqrt {{t_2} + 2\sqrt {{d_2}} } }} +\\ \notag
& \frac{{{R_{211}} + {R_{112}} + {R_{222}} - {R_{121}}}}{{4\sqrt {{t_3} + 2\sqrt {{d_3}} } }} + \frac{{{R_{121}} + {R_{112}} + {R_{222}} - {R_{211}}}}{{4\sqrt {{t_4} + 2\sqrt {{d_4}} } }},
\end{align}
\begin{align}
{T_{000}} = \sqrt {{t_1} + 2\sqrt {{d_1}} } + \sqrt {{t_2} + 2\sqrt {{d_2}} } + \sqrt {{t_3} + 2\sqrt {{d_3}} } + \sqrt {{t_4} + 2\sqrt {{d_4}} },
\end{align}
\begin{align}
{T_{030}} = &\frac{{{R_{030}} + {R_{300}} + {R_{003}} + {R_{333}}}}{{4\sqrt {{t_1} + 2\sqrt {{d_1}} } }} + \frac{{{R_{333}} + {R_{003}} - {R_{030}} - {R_{300}}}}{{4\sqrt {{t_2} + 2\sqrt {{d_2}} } }} + \\ \notag
&\frac{{{R_{300}} + {R_{003}} - {R_{030}} - {R_{333}}}}{{4\sqrt {{t_3} + 2\sqrt {{d_3}} } }} + \frac{{{R_{030}} - {R_{300}} + {R_{003}} - {R_{333}}}}{{4\sqrt {{t_4} + 2\sqrt {{d_4}} } }},
\end{align}
\begin{align}
{T_{300}} = &\frac{{{R_{030}} + {R_{300}} + {R_{003}} + {R_{333}}}}{{4\sqrt {{t_1} + 2\sqrt {{d_1}} } }} - \frac{{{R_{333}} + {R_{003}} - {R_{030}} - {R_{300}}}}{{4\sqrt {{t_2} + 2\sqrt {{d_2}} } }} -\\ \notag
& \frac{{{R_{300}} + {R_{003}} - {R_{030}} - {R_{333}}}}{{4\sqrt {{t_3} + 2\sqrt {{d_3}} } }} + \frac{{{R_{030}} - {R_{300}} + {R_{003}} - {R_{333}}}}{{4\sqrt {{t_4} + 2\sqrt {{d_4}} } }},
\end{align}
\begin{equation}
{T_{330}} = \sqrt {{t_1} + 2\sqrt {{d_1}} } - \sqrt {{t_2} + 2\sqrt {{d_2}} } - \sqrt {{t_3} + 2\sqrt {{d_3}} } + \sqrt {{t_4} + 2\sqrt {{d_4}} },
\end{equation}
\begin{align}
{T_{003}} =& \frac{{{R_{030}} + {R_{300}} + {R_{003}} + {R_{333}}}}{{4\sqrt {{t_1} + 2\sqrt {{d_1}} } }} + \frac{{{R_{030}} + {R_{300}} - {R_{333}} - {R_{003}}}}{{4\sqrt {{t_2} + 2\sqrt {{d_2}} } }} + \\ \notag
&\frac{{{R_{300}} + {R_{003}} - {R_{030}} - {R_{333}}}}{{4\sqrt {{t_3} + 2\sqrt {{d_3}} } }} + \frac{{{R_{300}} + {R_{333}} - {R_{003}} - {R_{030}}}}{{4\sqrt {{t_4} + 2\sqrt {{d_4}} } }},
\end{align}
\begin{equation}
{T_{033}} = \sqrt {{t_1} + 2\sqrt {{d_1}} } - \sqrt {{t_2} + 2\sqrt {{d_2}} } + \sqrt {{t_3} + 2\sqrt {{d_3}} } - \sqrt {{t_4} + 2\sqrt {{d_4}} },
\end{equation}
\begin{equation}
{T_{303}} = \sqrt {{t_1} + 2\sqrt {{d_1}} } + \sqrt {{t_2} + 2\sqrt {{d_2}} } - \sqrt {{t_3} + 2\sqrt {{d_3}} } - \sqrt {{t_4} + 2\sqrt {{d_4}} },
\end{equation}
\begin{align}
{T_{333}} =& \frac{{{R_{030}} + {R_{300}} + {R_{003}} + {R_{333}}}}{{4\sqrt {{t_1} + 2\sqrt {{d_1}} } }} + \frac{{{R_{333}} + {R_{003}} - {R_{030}} - {R_{300}}}}{{4\sqrt {{t_2} + 2\sqrt {{d_2}} } }} + \\ \notag
&\frac{{{R_{030}} + {R_{333}} - {R_{300}} - {R_{003}}}}{{4\sqrt {{t_3} + 2\sqrt {{d_3}} } }} + \frac{{{R_{300}} + {R_{333}} - {R_{003}} - {R_{030}}}}{{4\sqrt {{t_4} + 2\sqrt {{d_4}} } }}.
\end{align}
The eigenvalues $\sqrt {\lambda _1^ \pm } $,$\sqrt {\lambda _2^ \pm
} $,$\sqrt {\lambda _3^ \pm } $ and $\sqrt {\lambda _4^ \pm } $ of
the matrix $\sqrt {{\rho _{123}}} $ are given by
\begin{equation}
\begin{array}{l}
\sqrt {\lambda _1^ \pm } = \frac{1}{2}\sqrt {{t_1} + 2\sqrt {{d_1}} } \pm \sqrt {{t_1} - 2\sqrt {{d_1}} }, \\
\sqrt {\lambda _2^ \pm } = \frac{1}{2}\sqrt {{t_2} + 2\sqrt {{d_2}} } \pm \sqrt {{t_2} - 2\sqrt {{d_2}} }, \\
\sqrt {\lambda _3^ \pm } = \frac{1}{2}\sqrt {{t_3} + 2\sqrt {{d_3}} } \pm \sqrt {{t_3} - 2\sqrt {{d_3}} }, \\
\sqrt {\lambda _4^ \pm } = \frac{1}{2}\sqrt {{t_4} + 2\sqrt {{d_4}} } \pm \sqrt {{t_4} - 2\sqrt {{d_4}} }.
\end{array}
\end{equation}
The elements of the matrix W defined by (\ref{w}) are given in terms
of ${R_{\alpha \beta \gamma }}$ and $\sqrt {\lambda _{'i}^ \pm } $
by
\begin{align*}
{w_{12}} = {w_{21}} =& \frac{{\left( {{R_{111}} - {R_{212}}} \right)\left( {{R_{121}} - {R_{222}}} \right) + \left( {{R_{221}} + {R_{122}}} \right)\left( {{R_{112}} + {R_{211}}} \right)}}{{8\left( {\sqrt {\lambda _1^ + } + \sqrt {\lambda _1^ - } } \right)\left( {\sqrt {\lambda _3^ + } + \sqrt {\lambda _3^ - } } \right)}} +\\
& \frac{{\left( {{R_{111}} + {R_{212}}} \right)\left( {{R_{121}} + {R_{222}}} \right) + \left( {{R_{122}} - {R_{221}}} \right)\left( {{R_{112}} - {R_{211}}} \right)}}{{8\left( {\sqrt {\lambda _2^ + } + \sqrt {\lambda _2^ - } } \right)\left( {\sqrt {\lambda _4^ + } + \sqrt {\lambda _4^ - } } \right)}}
\end{align*}
{\footnotesize{\begin{align*}
{w_{11}} = &\left( {\sqrt {\lambda _1^ + } + \sqrt {\lambda _1^ - } } \right)\left( {\sqrt {\lambda _3^ + } + \sqrt {\lambda _3^ - } } \right) + \left( {\sqrt {\lambda _2^ + } + \sqrt {\lambda _2^ - } } \right)\left( {\sqrt {\lambda _4^ + } + \sqrt {\lambda _4^ - } } \right) + \\
&\frac{{{{\left( {{R_{300}} + {R_{003}}} \right)}^2} + {{\left( {{R_{111}} - {R_{212}}} \right)}^2} - {{\left( {{R_{221}} + {R_{122}}} \right)}^2} + {{\left( {{R_{112}} + {R_{211}}} \right)}^2} - {{\left( {{R_{121}} - {R_{222}}} \right)}^2} - {{\left( {{R_{030}} + {R_{333}}} \right)}^2}}}{{16\left( {\sqrt {\lambda _1^ + } + \sqrt {\lambda _1^ - } } \right)\left( {\sqrt {\lambda _3^ + } + \sqrt {\lambda _3^ - } } \right)}} +\\
& \frac{{{{\left( {{R_{111}} + {R_{212}}} \right)}^2} - {{\left( {{R_{122}} - {R_{221}}} \right)}^2} + {{\left( {{R_{112}} - {R_{211}}} \right)}^2} - {{\left( {{R_{121}} + {R_{222}}} \right)}^2} + {{\left( {{R_{003}} - {R_{300}}} \right)}^2} - {{\left( {{R_{333}} - {R_{030}}} \right)}^2}}}{{16\left( {\sqrt {\lambda _2^ + } + \sqrt {\lambda _2^ - } } \right)\left( {\sqrt {\lambda _4^ + } + \sqrt {\lambda _4^ - } } \right)}},
\end{align*}}}
{\footnotesize{\begin{align*}
{w_{22}} = &\left( {\sqrt {\lambda _1^ + } + \sqrt {\lambda _1^ - } } \right)\left( {\sqrt {\lambda _3^ + } + \sqrt {\lambda _3^ - } } \right) + \left( {\sqrt {\lambda _2^ + } + \sqrt {\lambda _2^ - } } \right)\left( {\sqrt {\lambda _4^ + } + \sqrt {\lambda _4^ - } } \right) +\\
& \frac{{{{\left( {{R_{221}} + {R_{122}}} \right)}^2} - {{\left( {{R_{111}} - {R_{212}}} \right)}^2} + {{\left( {{R_{121}} - {R_{222}}} \right)}^2} - {{\left( {{R_{112}} + {R_{211}}} \right)}^2} + {{\left( {{R_{300}} + {R_{003}}} \right)}^2} - {{\left( {{R_{030}} + {R_{333}}} \right)}^2}}}{{16\left( {\sqrt {\lambda _1^ + } + \sqrt {\lambda _1^ - } } \right)\left( {\sqrt {\lambda _3^ + } + \sqrt {\lambda _3^ - } } \right)}} + \\
&\frac{{{{\left( {{R_{122}} - {R_{221}}} \right)}^2} - {{\left( {{R_{111}} + {R_{212}}} \right)}^2} + {{\left( {{R_{121}} + {R_{222}}} \right)}^2} - {{\left( {{R_{112}} - {R_{211}}} \right)}^2} + {{\left( {{R_{003}} - {R_{300}}} \right)}^2} - {{\left( {{R_{333}} - {R_{030}}} \right)}^2}}}{{16\left( {\sqrt {\lambda _2^ + } + \sqrt {\lambda _2^ - } } \right)\left( {\sqrt {\lambda _4^ + } + \sqrt {\lambda _4^ - } } \right)}},
\end{align*}}}
\begin{align*}
{w_{33}}& =\frac{1}{2}\left[ {{{\left( {\sqrt {\lambda _1^ + } + \sqrt {\lambda _1^ - } } \right)}^2} + {{\left( {\sqrt {\lambda _3^ + } + \sqrt {\lambda _3^ - } } \right)}^2} + {{\left( {\sqrt {\lambda _2^ + } + \sqrt {\lambda _2^ - } } \right)}^2} + {{\left( {\sqrt {\lambda _4^ + } + \sqrt {\lambda _4^ - } } \right)}^2}} \right] +\\
& \frac{1}{{32}}\left[ {\frac{{{{\left( {{R_{033}} + {R_{300}} + {R_{003}} + {R_{333}}} \right)}^2} - {{\left( {{R_{112}} + {R_{121}} + {R_{211}} - {R_{222}}} \right)}^2} - {{\left( {{R_{111}} - {R_{221}} - {R_{122}} - {R_{212}}} \right)}^2}}}{{{{\left( {\sqrt {\lambda _1^ + } + \sqrt {\lambda _1^ - } } \right)}^2}}}} \right] +\\
& \frac{1}{{32}}\left[ {\frac{{{{\left( {{R_{333}} + {R_{003}} - {R_{300}} - {R_{030}}} \right)}^2} - {{\left( {{R_{222}} + {R_{121}} + {R_{211}} - {R_{112}}} \right)}^2} - {{\left( {{R_{111}} - {R_{221}} + {R_{122}} + {R_{212}}} \right)}^2}}}{{{{\left( {\sqrt {\lambda _2^ + } + \sqrt {\lambda _2^ - } } \right)}^2}}}} \right] +\\
& \frac{1}{{32}}\left[ {\frac{{{{\left( {{R_{300}} + {R_{003}} - {R_{030}} - {R_{333}}} \right)}^2} - {{\left( {{R_{121}} - {R_{211}} - {R_{112}} - {R_{222}}} \right)}^2} - {{\left( {{R_{111}} + {R_{221}} + {R_{122}} - {R_{212}}} \right)}^2}}}{{{{\left( {\sqrt {\lambda _3^ + } + \sqrt {\lambda _3^ - } } \right)}^2}}}} \right] +\\
& \frac{1}{{32}}\left[ {\frac{{{{\left( {{R_{030}} - {R_{300}} + {R_{003}} - {R_{333}}} \right)}^2} - {{\left( {{R_{121}} - {R_{211}} + {R_{112}} + {R_{222}}} \right)}^2} - {{\left( {{R_{111}} + {R_{221}} - {R_{122}} + {R_{212}}} \right)}^2}}}{{{{\left( {\sqrt {\lambda _4^ + } + \sqrt {\lambda _4^ - } } \right)}^2}}}} \right].
\end{align*}
After some long but feasible calculations, we gets
\begin{equation}
{w_{12}} = {w_{21}} = \frac{{2i\left( {{\rho _{18}}{\rho _{72}} - {\rho _{81}}{\rho _{27}}} \right)}}{{\left( {\sqrt {\lambda _1^ + } + \sqrt {\lambda _1^ - } } \right)\left( {\sqrt {\lambda _3^ + } + \sqrt {\lambda _3^ - } } \right)}} + \frac{{2i\left( {{\rho _{36}}{\rho _{54}} - {\rho _{45}}{\rho _{63}}} \right)}}{{\left( {\sqrt {\lambda _2^ + } + \sqrt {\lambda _2^ - } } \right)\left( {\sqrt {\lambda _4^ + } + \sqrt {\lambda _4^ - } } \right)}},
\end{equation}
\begin{align}\label{W11}
{w_{11}} = &\left( {\sqrt {\lambda _1^ + } + \sqrt {\lambda _1^ - } } \right)\left( {\sqrt {\lambda _3^ + } + \sqrt {\lambda _3^ - } } \right) + \left( {\sqrt {\lambda _2^ + } + \sqrt {\lambda _2^ - } } \right)\left( {\sqrt {\lambda _4^ + } + \sqrt {\lambda _4^ - } } \right) + \notag \\
& \frac{{\left( {{\rho _{11}} - {\rho _{88}}} \right)\left( {{\rho _{22}} - {\rho _{77}}} \right) + 2\left( {{\rho _{18}}{\rho _{72}} + {\rho _{81}}{\rho _{27}}} \right)}}{{\left( {\sqrt {\lambda _1^ + } + \sqrt {\lambda _1^ - } } \right)\left( {\sqrt {\lambda _3^ + } + \sqrt {\lambda _3^ - } } \right)}} + \frac{{\left( {{\rho _{44}} - {\rho _{55}}} \right)\left( {{\rho _{33}} - {\rho _{66}}} \right) + 2\left( {{\rho _{36}}{\rho _{54}} + {\rho _{45}}{\rho _{63}}} \right)}}{{\left( {\sqrt {\lambda _2^ + } + \sqrt {\lambda _2^ - } } \right)\left( {\sqrt {\lambda _4^ + } + \sqrt {\lambda _4^ - } } \right)}},
\end{align}
\begin{align}\label{W22}
{w_{22}} =& \left( {\sqrt {\lambda _1^ + } + \sqrt {\lambda _1^ - } } \right)\left( {\sqrt {\lambda _3^ + } + \sqrt {\lambda _3^ - } } \right) + \left( {\sqrt {\lambda _2^ + } + \sqrt {\lambda _2^ - } } \right)\left( {\sqrt {\lambda _4^ + } + \sqrt {\lambda _4^ - } } \right) + \notag \\
& \frac{{\left( {{\rho _{11}} - {\rho _{88}}} \right)\left( {{\rho _{22}} - {\rho _{77}}} \right) - 2\left( {{\rho _{18}}{\rho _{72}} + {\rho _{81}}{\rho _{27}}} \right)}}{{\left( {\sqrt {\lambda _1^ + } + \sqrt {\lambda _1^ - } } \right)\left( {\sqrt {\lambda _3^ + } + \sqrt {\lambda _3^ - } } \right)}} + \frac{{\left( {{\rho _{44}} - {\rho _{55}}} \right)\left( {{\rho _{33}} - {\rho _{66}}} \right) - 2\left( {{\rho _{36}}{\rho _{54}} + {\rho _{45}}{\rho _{63}}} \right)}}{{\left( {\sqrt {\lambda _2^ + } + \sqrt {\lambda _2^ - } } \right)\left( {\sqrt {\lambda _4^ + } + \sqrt {\lambda _4^ - } } \right)}},
\end{align}
\begin{align}\label{W33}
{w_{33}} = &\frac{1}{2}\left( {1 + 2\sum\limits_{i = 1}^4 {\sqrt {{d_i}} } } \right) + \frac{{{{\left( {2{\rho _{11}} + {\rho _{66}} - {\rho _{88}} - {\rho _{77}} - {\rho _{55}}} \right)}^2} - 16{\rho _{18}}{\rho _{81}}}}{{8{{\left( {\sqrt {\lambda _1^ + } + \sqrt {\lambda _1^ - } } \right)}^2}}} + \frac{{{{\left( {{\rho _{33}} - {\rho _{66}}} \right)}^2} - 4{\rho _{36}}{\rho _{63}}}}{{2{{\left( {\sqrt {\lambda _4^ + } + \sqrt {\lambda _4^ - } } \right)}^2}}} \notag \\
& + \frac{{{{\left( {{\rho _{44}} - {\rho _{55}}} \right)}^2} + {{\left( {{\rho _{63}} - {\rho _{36}}} \right)}^2} - {{\left( {{\rho _{45}} + {\rho _{54}}} \right)}^2}}}{{2{{\left( {\sqrt {\lambda _2^ + } + \sqrt {\lambda _2^ - } } \right)}^2}}} + \frac{{{{\left( {{\rho _{22}} - {\rho _{77}}} \right)}^2} - 4{\rho _{27}}{\rho _{72}}}}{{2{{\left( {\sqrt {\lambda _3^ + } + \sqrt {\lambda _3^ - } } \right)}^2}}}.
\end{align}
with $\lambda^{\pm} (i=1,2, 3, 4)$ are the eigenvalues of the density matrix $\rho_{123}$ (\ref{1}).
|
1,941,325,220,581 | arxiv | \subsection{Generalised Wilson ratio}
Our results in the following reveal that
in both lattices and within the SL regime $s(T)$ and $\chi_0(T)$ are very
similar in a broad range of $T$. In this respect very convenient quantity is
$T$-dependent generalized Wilson ratio $R(T)$, defined as
\begin{equation}
R= 4 \pi^2 T \chi_0 / (3 s), \label{rw}
\end{equation}
which is equivalent (assuming theoretical units $k_B= g \mu_B =1$) to the
standard $T=0$ Wilson ratio in the case of Fermi-liquid
behavior where $s= C_V=\gamma T$. Definition Eq.(\ref{rw}) is more
convenient with respect to the standard one (with $C_V$ instead of $s$)
since it has meaningful $T$ dependence due to monotonously increasing
$s(T)$, having also finite high-$T$ limit
$R_\infty = \pi^2/(3 \ln2) = 4.75$. Moreover, it can differentiate
between quite distinct $T \to 0$ scenarios:
\noindent a) in the case of magnetic LRO at
$T \to 0$ one expects in 2D (isotropic HM)
$\chi_0(T\to 0) \sim \chi_0^0 >0$ but $s \propto T^2$~
\cite{manousakis91}, so that $R_0 = R(T \to 0) \to \infty$,
\noindent b) in a
gapless SL with large spinon Fermi surface one would expect
Fermi-liquid-like constant $R_0 \sim 1$ \cite{balents10,zhou17,law17},
\noindent c) $R_0 \ll 1$ or a decreasing $R(T \to 0) \to 0$ would indicate that
low-energy singlet excitations dominate over the triplet ones
\cite{balents10,lauchli19}.
In the following we find in the SL regime
numerical evidence for the last scenario, which within the EM we
attribute to low-lying chiral fluctuations being a hallmark of Heisenberg model
in the SL regime. It should be pointed out that the same property might be
very general property of SL models and it remains to be clarified in relation with
experiments on SL materials.
In Sec.~II we derive and discuss the form of the EM model for the $J_1$-$J_2$ Heisenberg
model on both TL and KL. In Sec.~III we present numerical methods employed to calculate
thermodynamic quantities in the EM as well as the in full models.
\section{Reduced effective model}
We consider the isotropic $S=1/2$ extended $J_1$-$J_2$ Heisenberg
model,
\begin{equation}
H= J_1 \sum_{\langle kl \rangle} {\bf S}_k \cdot {\bf S}_l + J_2 \sum_{\langle \langle kl \rangle \rangle}
{\bf S}_k \cdot {\bf S}_l, \label{hjj}
\end{equation}
on the TL and KL, where $J_1>0$ and $J_2 $ refer to nn and nnn
exchange couplings (see Fig.\ref{figsl0}), respectively. The role of $J_2 >0$ on TL is to
destroy the 120$^0$ LRO allowing for a SL
\cite{kaneko14,watanabe14,zhu15,iqbal16,prelovsek18}, while for KL it
has the opposite effect \cite{kolley15}. Further we set $J_1=J=1$ as an
energy scale.
\begin{figure}[h!]
\centering
\includegraphics[width=1.1\columnwidth]{shema.eps}
\vspace{-1cm}
\caption{(a) Triangular and (B) kagome lattice represented as the (triangular) lattices
of coupled basic triangles. Shown are also model exchange couplings on both lattices.} \label{figsl0}
\end{figure}
As shown in Fig.~\ref{figsl0} the model Eq.~(\ref{hjj}) on both lattices can be represented as
coupled basis triangles \cite{subrahmanyan95,mila98,budnik04,capponi04} where we keep
in the construction of the EM only four degenerate $S=1/2$ states (local
energy $\epsilon_0=-3/4$), neglecting higher $S=3/2$ states (local
$\epsilon_1=3/4$),
\begin{eqnarray}
|\uparrow \pm \rangle &=&\frac{1}{\sqrt{3}}[ | \downarrow\uparrow \uparrow \rangle +
\mathrm{e}^{\pm i \phi} | \uparrow\downarrow \uparrow \rangle +\mathrm{e}^{\mp i \phi} | \uparrow\uparrow \downarrow \rangle ],
\nonumber \\
|\downarrow \pm \rangle &=& \frac{1}{\sqrt{3}}[ | \uparrow\downarrow \downarrow \rangle + \mathrm{e}^{\mp i \phi} | \downarrow\uparrow \downarrow \rangle +
\mathrm{e}^{\mp i \phi} | \downarrow\downarrow \uparrow \rangle ], \label{updown}
\end{eqnarray}
where $\phi = 2 \pi/3$, $\uparrow, \downarrow$ are (new) spin states
and $\pm$ refer to local chirality. One can rewrite Eq.(\ref{hjj})
into the new basis acting between nn triangles $\langle i,j \rangle$.
The derivation is straightforward taking care that the matrix elements of the original model.
Eq.~(\ref{hjj}), within the new basis, Eq.~(\ref{updown}), are exactly
reproduced with new
operators. We follow the procedure through intermediate (single triangle) orbital spin operators
(see Fig.~\ref{figsl0}),
\begin{equation}
S_{(0,+,-)}=S_1+(1,\omega,\omega^*) S_2+ (1,\omega^*,\omega) S_3, \label{s0}
\end{equation}
with $\omega = \mathrm{e}^{ i \phi}$. Operators in Eq.~(\ref{s0}) have only few nonzero matrix elements within
the new basis, Eq.~(\ref{updown}), e.g.,
\begin{eqnarray}
&&\langle \uparrow + | S_0^z | \uparrow + \rangle = -\langle \downarrow + | S_0^z | \downarrow + \rangle = \frac{1}{2}, \nonumber \\
&& \langle \downarrow + | S_-^z | \downarrow - \rangle = - \langle \uparrow + | S_- ^z | \downarrow + \rangle = 1 , \nonumber \\
&&\langle \uparrow + | S_0^+ | \downarrow + \rangle = ~~\langle \uparrow - | S_0^+ | \downarrow - \rangle = -1, \nonumber \\
&&\langle \uparrow - | S_+^+ | \downarrow + \rangle = ~~\langle \downarrow + | S_-^- | \uparrow - \rangle = 2. \label{locs}
\end{eqnarray}
Such operators can be fully represented in terms of standard local $s=1/2$ spin operators ${\bf s}$,
and pseudospin (chirality) operators ${\bf \tau}$(again $\tau=1/2$),
\begin{eqnarray}
&&S^z_0= s^z, \quad S^\pm_0=-s^\pm , \nonumber \\
&& S^z_\pm= -2 s^z \tau^\mp, \quad S^\pm_+= 2 s^\pm \tau^-, \quad S^\pm_-= 2 s^\pm \tau^+, \label{effspin}
\end{eqnarray}
Since the original Hamiltonian, Eq.~(\ref{hjj}), acts only between two sites on the new
reduced (triangular) lattice, the effective model (EM), subtracting local $\epsilon_0$,
can be fully represented for both lattices
(as shown by an example further on) in terms of ${\bf s}_i$
and ${\bf \tau}_i$ operators, introduced in Eq.~(\ref{locs}),
\begin{eqnarray}
\tilde H &=& \frac{1}{2} \sum_{i ,d } {\bf s}_{i} \cdot {\bf s}_{j} ( D+ {\cal H}^d_{ij}) , \label{em} \\
{\cal H}^d_{ij} &=& F_d \tau^+_i \tau^-_j + P_d \tau^+_i + Q_d \tau^+_j + T_d \tau^+_i \tau^+_j + \mathrm{H.c}, \nonumber
\end{eqnarray}
where directions $d=1$--$6$ and $j=i+d$ run over all nn sites of site $i$,
and the new lattice is again TL. We note that Eq.~(\ref{em}) corresponds
to the one studied before for simplest KL \cite{subrahmanyan95,mila98,budnik04}, but it is
valid also for TL \cite{subrahmanyan95} and nnn $J_2$. It is remarkable that the spin part
remains $SU(2)$ invariant whereas the chirality part is not, i.e., it
is of the XY form.
It should be pointed out that EM, Eq.~(\ref{em}), is not based on a perturbation expansion assuming weak
coupling between triangles, although in the latter case it offers a full description of low-lying states and can be further
treated analytically in (rather artificial) strong breathing limit $|\tilde H| \ll |E_0|$
\cite{subrahmanyan95,mila98}. As for several other applications of reduced effective models (prominent
example being the $t$-$J$ model as reduced/projected Hubbard model), one expects that low-$T$ physics is (at least
qualitatively) well captured by the EM.
\subsection {Triangular lattice}
For the considered TL and KL we further present the actual parameters of the EM,
Eq.~(\ref{em}). The derivation is straightforward using the representation
Eqs.~(\ref{s0}) and (\ref{effspin}). As an example we present the $J_1$-term interaction in
Eq.~(\ref{hjj}) between (new) sites $0$ and $1$ (see Fig.~\ref{figsl0}a),
\begin{eqnarray}
&&~~~~\tilde H^1_{01}/J_1 = {\bf S}_{01} \cdot ( {\bf S}_{12} + {\bf S}_{13} ) = \nonumber \\
&& = \frac{1}{9}(S_{00}^z + S^z_{0+} + S^z_{0-})(2 S^z_{10} - S^z_{1+} - S^z_{1-} ) + \nonumber \\
&&+\frac{1}{18} [(S_{00}^+ + S^+_{0+} + S^+_{0-})(2 S^-_{10} - S^-_{1+} - S^-_{1-}) +
\mathrm{H.c.}] = \nonumber \\
&&= - \frac{4}{9} {\bf s}_0 \cdot {\bf s}_1 (\tau^-_0 + \tau^+_0 -\frac{1}{2})(\tau^-_1 + \tau^+_1 + 1).
\label{h01}
\end{eqnarray}
Deriving in the same manner also other $\tilde H^d$ terms, including now also $J_2$ term
(which are even simpler in TL), we can identify the parameters (with $J_1=1$) in the EM, Eq.(\ref{em}),
\begin{equation}
D= \frac{2}{9} + \frac{1}{3} J_2, \qquad F= - \frac{4}{9} + \frac{4}{3} J_2,
\label{df}
\end{equation}
and further terms depending explicitly on direction $d$,
\begin{eqnarray}
P_1&=& -\frac{4}{9},\quad P_2= \frac{2}{9} \omega^* ,
\quad P_3= -\frac{4}{9} \omega , \nonumber \\
P_4&=& \frac{2}{9} , \quad P_5= -\frac{4}{9} \omega^* ,
\quad P_6= \frac{2}{9} \omega, \nonumber \\
T_1&=& -\frac{4}{9} ,\quad T_2 = -\frac{4}{9} \omega , \quad T_3 = -\frac{4}{9} \omega^*
\end{eqnarray}
with $T_{d+3}=T_d$ and $Q_d = P_{d+3}$. It is worth noticing, that $J_2$ does not enter
in terms $P_d,Q_d,T_d$.
It can be also directly verified, that for the latter couplings, the average over all nn bonds vanish, i.e.,
\begin{equation}
\bar P = \frac{1}{6}\sum_d P_d = 0, \qquad \bar Q = \bar T =0,
\end{equation}
indicating possible minor importance of these terms. This is, however, only partially true
since such terms also play a role of distributing the increase of entropy $s(T)$ to a wider $T$
interval.
Eqs.~(\ref{em}),(\ref{df}) yield also some basic insight into the HM model
on TL, as well the similarity between models on TL and KL. While $\chi_0(T)$ is
governed entirely by ${\bf s}$ operators, low-$T$ entropy $s(T)$ (and
specific heat $C_V(T)$) involves also chirality ${\bf \tau}$ fluctuations. In
TL at $J_2=0$ $\tau$ coupling is ferromagnetic and favors spiral
$120^0$ LRO. $\tau$ fluctuations are enhanced via $J_2 >0$ reducing
$F$ and finally $F \to 0$ on approaching $J_2 \sim 0.3$. Still, before
such large $J_2$ is reached $P_d,Q_d,T_d$ terms become relevant and
stabilize SL at $J_2 \sim 0.1$ \cite{kaneko14,watanabe14,zhu15,hu15,iqbal16}. It
should be stressed that in TL a standard magnetic LRO requires LRO ordering of
both ${\bf s}$ and ${\bf \tau}$ operators.
\subsection{Kagome lattice.}
In analogy to the TL example, Eq.~(\ref{h01}), we derive also the corresponding terms for the case of KL.
Without loosing the generality we can include here also the
third-neighbor exchange term $J_3$,
see Fig.~\ref{figsl0}b, which also couples only neighboring triangles.
Then $D$ and $F_d$ couplings are given by
\begin{equation}
D = \frac{1}{9} + \frac{2}{9} J_2 + \frac{2}{9} J_3, \qquad F_1 =
\frac{4}{9} \omega + \frac{8}{9} \omega^* J_2 + \frac{4}{9} J_3,
\end{equation}
while $F_{d+1}= F_d^*$. In contrast to TL, in KL at $J_2=0$ the $F_d$ coupling is complex and alternating, with a
nonzero average, being real and negative, i.e. $\bar F = (1/6) \sum_d F_d < 0$.
Moreover, $| \mathrm{Im} F_d| < |\mathrm{Re} F_d |$, indicates the absence of
LRO. Here, $J_2 >0$ reduces $|\mathrm{Im} F_d|$ and on approaching
$J_2 \sim 0.5$ one reaches real $F_d <0$ connecting KL model to TL at
$J_2=0$ and related LRO, as observed in numerical studies
\cite{kolley15}.
Further terms are given by
\begin{eqnarray}
P_1&=& - \frac{2}{9} + \frac{2}{9} \omega^* J_2 - \frac{2}{9} \omega J_3, \quad
P_2 = - \frac{2}{9} \omega + \frac{2}{9} \omega^* J_2 - \frac{2}{9} J_3, \nonumber \\
P_3&=& - \frac{2}{9} \omega + \frac{2}{9} J_2 - \frac{2}{9} \omega^* J_3, \quad
P_4 = - \frac{2}{9} \omega^* + \frac{2}{9} J_2 - \frac{2}{9} \omega J_3, \nonumber \\
P_5&=& - \frac{2}{9} \omega^* + \frac{2}{9} \omega J_2 - \frac{2}{9} J_3, \quad
P_6 = - \frac{2}{9} + \frac{2}{9} \omega J_2 - \frac{2}{9} \omega^* J_3, \nonumber \\
\end{eqnarray}
with $Q_d=P_{d+3}$ and
\begin{eqnarray}
T_1&=& \frac{4}{9} \omega^* - \frac{4}{9} \omega^* J_2 + \frac{4}{9} J_3,
\quad T_2 = \frac{4}{9} - \frac{4}{9} J_2 + \frac{4}{9} J_3, \nonumber \\
T_3&=& \frac{4}{9} \omega - \frac{4}{9} \omega J_2 + \frac{4}{9} J_3, \qquad T_{d+3}=T_d .
\end{eqnarray}
Again, terms which do not conserve $\tau^z_{tot}$ have the property $\bar P =\bar Q = \bar T =0$.
\section{Numerical method}
In the evaluation of thermodynamical quantities we use the FTLM
\cite{jaklic94,jaklic00,prelovsek13}, which is based on the Lanczos
exact-diagonalization (ED) method \cite{lanczos50}, whereby the Lanczos-basis states
are used to evaluate the normalized thermodynamic sum
\begin{equation}
Z(T) = \mathrm{Tr} \exp[- (H-E_0)/T],
\end{equation}
(where $E_0$ is the ground state energy of a system). The FTLM is
particularly convenient to apply for the calculation of the conserved
quantities , i.e., operators $A$ commuting with the Hamiltonian
$[H,A]=0$. In this way we evaluate $Z$, the thermal average energy
$\Delta E =\langle H - E_0 \rangle $ and magnetization
$M = \langle s^z_{tot} \rangle$. From these
quantities we evaluate the thermodynamic observables of interest,
i.e. uniform susceptibility $\chi_0(T)$ and entropy density $s(T)$,
\begin{equation}
\chi_0= \frac{M^2}{N T}, \qquad s = \frac{T \ln Z + \Delta E}{N T} ,
\end{equation}
where $N$ is the number of sites in the original lattice.
We note that for above conserved operators $A$ there is no need to store Lanczos
wavefunctions, so the requirements are essentially
that of the g.s. Lanczos ED method, except that we need the summation over all
symmetry sectors and a modest sampling $N_s < 10$ over initial
wavefunctions is helpful. To reduce the Hilbert space of basis states $N_{st}$
we take into account symmetries, in particular the translation symmetry
(restricting subspaces to separate wavevectors $q$) and
$s^z_{tot}$ while the EM, Eq.~(\ref{em}), does not conserve $\tau^z_{tot}$.
In such a framework in the present study we are restricted to systems
with $N_{st} < 5 \cdot 10^6$ symmetry-reduced basis states, which means EM with up to
$N = 42$ sites. The same system size would require in the full HM
$N_{st} \sim 10^{10}$ basis states.
An effective criterion for the macroscopic relevance of FTLM results is
$Z(T) \gg 1$ (at least for system where gapless excitations are
expected), which in practice leads to a criterion
$Z>Z^*=Z(T_{fs}) \gg 1$ determining the finite-size temperature $T_{fs}$.
Taking $Z^* \sim 20$ implies (for $N=42$)
also the threshold entropy density $s(T_{fs}) \sim 0.07$,
independent of the model. It is then evident that $T_{fs}$ depends crucially
on the model, so that large $s(T)$ works in favor of
using FTLM for frustrated and SL systems. Here we do not present
the finite-size analysis of FTLM results within EM, but they are quite analogous to previous
application of the method to HM on KL \cite{schnack18} where similar low $T_{fs}$ was
established (but much higher $T_{fs}$ ) in unfrustrated lattices and models \cite{jaklic00}.
Moreover, in the case of models with a sizable gap, e.g. $\Delta > T_{fs}$ the results of FTLM
can remain correct even down to $T \to 0$. \cite{jaklic00}
\begin{figure}[h!]
\vspace{0.3cm}
\centering
\includegraphics[width=\columnwidth]{figsl1n.eps}
\caption{ Results for the effective model (EM) for the triangular lattice, compared with the
full HM. All results for are for $N=30$ sites \cite{prelovsek18}: a)
for entropy density $s(T)$ and b) uniform susceptibility
$\chi_0(T)$. c) and d) same quantities for the kagome lattice,
compared to full HM, all on $N=42$ sites \cite{schnack18}.} \label{figsl1}
\end{figure}
\section{Entropy and uniform susceptibility}
Let us first benchmark results within the EM with the existing results for
the full HM on TL and KL. In Figs.~\ref{figsl1}a,b we present $s(T)$ and
$\chi_0(T)$, respectively, as obtained on TL for $J_2=0.1$ on $N=30$
via FTLM on EM, compared with the full HM of the same size
\cite{prelovsek18}. The qualitative behavior of both quantities
within EM is quite similar at low $T$ to the full HM, in particular for $s(T)$,
although EM evidently misses part of $s(T)$ with increasing $T$ due to reduced
basis space. More pronounced is quantitative (but not qualitative) discrepancy
in $\chi_0(T)$ which can be attributed to missing higher spin states in EM.
Still, the peak in $\chi_0(T)$ and related spin (triplet) gap
$\Delta_t >0$ at low $T$) are reproduced well within EM.
Similar conclusions emerge from Figs.~\ref{figsl1}c,d
where corresponding results are compared for the KL, where full-HM
results for $s(T)$ and $\chi_0(T)$ are taken from study on $N=42$
sites \cite{schnack18}. Here, EM clearly reproduces reasonably not
only triplet gap $\Delta_t \sim 0.1$ but also singlet excitations
dominating $s(T \to 0)$, while apparently EM underestimates the value
of $\chi_0(T)$.
After testing with the full model, we present in Figs.~\ref{figsl2}
and \ref{figsl3} EM
results for $s(T)$ and $\chi_0(T)$ for both lattices as they vary with
$J_2 > 0$. In Fig.~\ref{figsl2}a,b we follow the behavior on TL for different
$J_2=0, 0.1, 0.15$. From the inflection (vanishing second derivative) point
of $s(T)$ defining singlet temperature $T=T_s$ one can speculate on the
coherence scale (in the case of LRO) or possible
(singlet) excitation gap $\Delta_s \lesssim T_s$ (in the case of SL),
at least provided that $T_s > T_{fs}$.
Although the influence of $J_2 >0$ does not appear large, it still
introduces a qualitative difference. From this perspective $s(T)$
within TL EM at $J_2=0$ reveals higher effective $T_s$
being consistent with $s(T<T_s) \propto T^2$ and a spiral LRO at $T=0$.
Still, we get in this case $T_s \sim T_{fs}$ within
EM, so we can hardly make stronger conclusions.
On the other hand, for TL and $J_2 = 0.1, 0.15$ where the SL can be expected
\cite{kaneko14,iqbal16,gong17} the EM reveals smaller $T_s \sim 0.05$
which is the signature of the singlet gap (which could still be
finite-size dependent). More important, results confirm large
residual entropy $s(T) \sim 0.1 = 0.14 s_{max}$ even at $T \sim 0.1$.
This is in contrast with $\chi_0(T)$ in Fig.~\ref{figsl2}b which reveal
$T$-variation weakly dependent on $J_2$. While for $J_2=0$ the drop
at $\chi_0(T < T_t)$ is the signature of the finite-size spin gap (where
due to magnetic LRO $\chi_0^0 = \chi_0( T \to 0) >0 $ is expected),
$J_2 =0.1, 0.15$ examples are different since vanishing $\chi_0^0$
could indicate the spin triplet gap $\Delta_t > 0.1$ beyond the
finite-size effects, i.e. $T_t \sim 0.1 > T_{fs}$.
\begin{figure}[h!]
\vspace{0.4cm}
\centering
\includegraphics[width=0.9\columnwidth]{figsl2n.eps}
\caption{ Results within EM on triangular lattice for different $J_2 =0, 0.1, 0.15$; a)
for $s(T)$, and b) $\chi_0(T)$.}
\label{figsl2}
\end{figure}
In Figs.~\ref{figsl3}a,b we present the same quantities for the case of KL,
now for $J_2=0, 0.2, 0.4$. The effect of $J_2>0$ is opposite, since
it is expected to recover the LRO at $J_2 \sim 0.4$ \cite{kolley15}, with
$120^0$ spin orientation analogous to $J_2=0$ TL. The largest low-$T$
entropy $s(T)$ is found for KL with $J_2=0$. Moreover, EM here yields
a quantitive agreement with the full HM \cite{schnack18}, revealing
large remanent $s(T)$ due to singlet (chirality) excitations down to
$T \sim 0.01$ \cite{lauchli19}. The evident
effect of $J_2 >0$ is to reduce $s(T)$ and finally leading
$s(T) \propto T^2$ at large $J_2 \sim 0.4$ which should be a regime of
magnetic LRO \cite{kolley15}. Again, at $J_2=0$ in contrast to entropy
$\chi_0(T)$ has well pronounced downturn at
$T \sim 0.1$ consistent with the triplet gap $\Delta_t \sim 0.1$ found
in most other numerical studies \cite{sindzingre00,misguich07,
bernu15,schnack18,lauchli19}. Introducing $J_2 > 0$ does not change
$\chi_0(T)$ qualitatively.
\begin{figure}[h!]
\vspace{0.4cm}
\centering
\includegraphics[width=0.9\columnwidth]{figsl3n.eps}
\caption{ Results within EM on kagome lattice
for different $J_2 = 0, 0.2, 0.4$; a) for $s(T)$, and b) $\chi_0(T)$.}
\label{figsl3}
\end{figure}
\section{Wilson ratio: results}
To calculate $R(T)$, Eq.~(\ref{rw}), let us first use available
results for the full HM for TL
\cite{prelovsek18} and KL \cite{schnack18}, comparing in Fig.~\ref{figsl4} also
the result for unfrustrated HM on a square lattice \cite{prelovsek18}.
Here, we take into account data for $T> T_{fs}$, acknowledging that
$T_{fs}$ are quite different (taking $s(T_{fs}) \sim 0.1$ as
criterion) for these systems, representative also for the degree of
frustration. Fig.~\ref{figsl4} already confirms different scenarios for
$R(T)$. In HM on a simple square lattice, starting from high-$T$ limit $R(T)$
reaches minimum at $T^* \sim 0.7$ and then increases, which is consistent with
$R(T\to 0) \to \infty$ for a 2D system with $T=0$ magnetic LRO. The same
behavior appears for TL at $J_2=0$ with a shallow minimum shifted to
$T^* \sim 0.3$. In contrast, results for KL as well as for TL with
$J_2=0.1$ do not reveal such increase, at least not for $T>T_{fs}$,
and they are more consistent with the interpretation that
$R(T\to 0) \to 0$.
\begin{figure}[h!]
\vspace{0.3cm}
\centering
\includegraphics[width=0.7\columnwidth]{figsl4.eps}
\caption{ Wilson ratio $R(T)$, evaluated from $s(T)$ and $\chi_0(T)$ for full HM
on square lattice, triangular lattice with $J_2=0, 0.1$ \cite{prelovsek18}, and kagome
lattice \cite{schnack18}. Results are presented for $T>T_{fs}$.}
\label{figsl4}
\end{figure}
Finally, results for $R(T)$ within EM are shown in Fig.~\ref{figsl5}a,b as
they follow from Figs.~\ref{figsl2} and \ref{figsl3} for different $J_2 \geq 0$. We recognize
that EM qualitatively reproduce numerical data within the full HM on
Fig.~\ref{figsl4}. Although for $J_2=0$ TL results in Fig.~\ref{figsl5}a fail to reveal
clearly the minimum down to $T_{fs}\sim 0.1$, there is still a marked
difference to the SL regime $J_2=0.1, 0.15$ where EM confirms
$R_0 \ll 1$. Results within EM for KL, as shown in
Fig.~\ref{figsl5}b, are even better demonstration for vanishing $R_0$. Here, for $J_2=0$ EM
yields quite similar $R(T)$, decreasing and tending towards
$R_0 \sim 0$. On the other hand, the effect of finite $J_2 >0$ is
well visible and leads towards magnetic LRO with $R_0 \to \infty$ for
$J_2 =0.4$.
\begin{figure}[h!]
\vspace{0.4cm}
\centering
\includegraphics[width=0.9\columnwidth]{figsl5n.eps}
\caption{ $R(T)$, evaluated within EM for: a) TL with $J_2=0,0.1,0.15$,
and b) KL for $J_2 = 0, 0.2, 0.4$.}
\label{figsl5}
\end{figure}
\section{Conclusions}
The main message (apparently not stressed in previous published studies of the SL
models) of presented results for
the thermodynamic quantities: the entropy $s(T)$,
susceptibility $\chi_0(T)$, and in particular the Wilson ratio $R(T)$,
behave similarity in the extended $J_1$-$J_2$ Heisenberg model
on TL and KL in their presumed SL regimes. Moreover, results on both lattices
follow quite analogous development by varying the nnn exchange $J_2 >0$, whereby
the effect is evidently opposite between TL and KL, regarding the magnetic LRO and SL
phases \cite{kolley15}.
While above similarities can be extracted already from full-model results obtained via
FTLM \cite{schnack18,prelovsek18}, the introduction of reduced effective model appears
crucial for the understanding and useful analytical insight. Apart from offering some
numerical advantages of reduced Hilbert space of basis states, essential for
ED methods, EM clearly puts into display two (separate) degrees of freedom: a)
the effective spin $s=1/2$ degrees ${\bf s}_i$,
determining $\chi_0(T)$, but as well the dynamical quantities, as e.g. the dynamical spin structure factor
$S({\bf q}, \omega)$ not discussed here \cite{prelovsek18},
b) chirality pseudospin degrees ${\bf \tau}_i$ which do
not contribute to $\chi_0(T)$, but enter the entropy $s(T)$ and related
specific heat. From the EM and its dependence on $J_2$ it
is also quite evident where to expect large fluctuations of ${\bf \tau}_i$
and consequently SL regime, which is not very apparent within the full HM.
As the EM is based on the direct reduction/projection of the basis (and not on perturbation
expansion) of the original HM, it is expected that the correspondence is qualitative and not fully
quantitative. In any case, the EM is also by itself a valuable and highly nontrivial model,
and could serve as such to understand better the onset and properties of SL.
The essential common feature of SL regimes in HM on
both lattices is a pronounced remanent $s(T)>0$ at $T \ll J_1$, which
within the EM has the origin in dominant low-energy chiral
fluctuations, well below the effective spin triplet gap $\Delta_t$
which is revealed by the drop of $\chi_0(T)$. As a consequence we
observe the vanishing of the Wilson ratio $R_0 = R(T\to 0) \to 0$,
which seems to be quite generic feature of 2D SL models \cite{prelovsek19}.
Clearly, due to finite-size
restrictions we could hardly distinguish a spin-gapped system from
scenarios with more delicate gap structure which could also lead to
renormalized $R_0 \ll 1$. Moreover, it is even harder to decide
beyond finite-size effects whether singlets excitations are gapless or with finite
singlet gap $\Delta_s >0$, which should be in any case very small $\Delta_s \ll J$ and from
results $R(T\to 0) \to 0$ have to be evidently smaller than a triplet one,
i.e. $\Delta_s<\Delta_t$. From our finite-size results it is also hard to exclude the
scenario of valence-bond ordered ground state (with broken translational symmetry),
although we do not see an indication for that and vanishing $R_0$, it is also
not easily compatible with the latter either.
Quantities discussed above are measurable in real
materials and have been indeed discussed for some of them. There are
evident experimental difficulties, i.e., $\chi_0(T)$ can have
significant impurity contributions while $s(T)$ may be masked by
phonon contribution at $T > 0$. The essential hallmark for material
candidates for the presented SL scenario should be a substantial
entropy $s(T)$ persisting well below $T \ll J_1$. There are indeed
several studies of $s(T)$ reported for different SL candidates (with
some of them revealing transitions to magnetic LRO at very low $T$), e.g., for
KL systems volborthite \cite{hiroi01}, YCu$_3$(OH)$_6$Cl$_3$
\cite{zorko19,arh19}, and recent TL systems 1T-TaS$_2$
\cite{kratochvilova17} and Co-based SL materials \cite{zhong19}.
While existing experimental results on SL materials do not seem to indicate
vanishing (or very small) $R_0$, it might also happen that
the (above) considered SL models are not fully capturing the low-$T$ physics.
In particular, there could be important role played by additional terms, e.g., the
Dzaloshinski-Moriya interaction \cite{cepas08,rousochatzakis09,arh19} and/or 3D
coupling, which can reduce $s(T)$ or even induce magnetic LRO at $T \to 0$.
\acknowledgments This work is supported by the program P1-0044 of the
Slovenian Research Agency. Authors thank J. Schnack for providing
their data for kagome lattice and J. Schnack, F. Becca,
A. Zorko, K. Morita and T. Tohyama for fruitful discussions.
|
1,941,325,220,582 | arxiv |
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submitted for publication, should be cited as ``unpublished'' \cite{b4}. Papers
that have been accepted for publication should be cited as ``in press'' \cite{b5}.
Capitalize only the first word in a paper title, except for proper nouns and
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citation first, followed by the original foreign-language citation \cite{b6}.
\section*{Acknowledgment}
This work has been funded by the CISCO Research Centre/Silicon Valley Community Foundation, CG\# 955107, project 'Fog to FIELD: Securing Wide-area Monitoring and Control for Critical Infrastructure 4.0'.
\section{Conclusions}
\label{sec:concl}
In this paper we created a model of a system using an ETC controller, and
a model of the LoRaWAN LPWA protocol.
We discussed the causes of delays and message losses in the LoRaWAN protocol,
and how these affect the stability and performance of the ETC controlled system.
Our results demonstrated the feasibility of the LoRaWAN protocol for certain
application scenarios and network parameter settings.
We also identified the limitations of LoRaWAN that prevent its wide adoption for use in certain classes of control-based application scenarios.
This paper represents a step forward in addressing the communication
and control systems co-design problem for long-range control over wireless
channel.
It is an ongoing process and there is no single widely accepted methodology
to address the development of protocol stacks suitable for WNCSs.
We showed that the current specification of LoRaWAN can be considered as
suitable for slow-changing physical processes where the data transmission
requirements are not high.
We will continue to address the coexistence of LPWA protocols and ETC control
by focusing on $n$ end-devices sharing the same set of channels.
We will also investigate the possibility of having priority slots, so even if
there is a duty cycle restriction in place there is a way to report a possible
failure due to the disturbance.
Additionally, we will exploit the possibility of using 'Listen before Talk'
feature of LoRa where by listening to the channel and sending if free the duty
cycle restriction can be bypassed.
\section{Event-Triggered Control Model}
\label{sec:sys_model}
Here we give a model of our event-triggered control scheme as a linear control
system of the form:
\begin{align} \label{eq:lin_sys}
\dot{\xi}(t)&=A\xi(t)+B\upsilon(t), \quad \xi(0)=\xi_0
\end{align}
where $\xi(t) \in \textrm{R}^n$ denotes the state input from the sensors
and $\upsilon(t) \in \textrm{R}^m$ denotes the control action sent to the
actuator at time $t$.
The matrix $A \in \textrm{R}^{n\times n}$ is the state matrix and
$B \in \textrm{R}^{n\times m}$ is the input matrix.
The system is connected via a communication channel to the controller.
In this paper, the communication channel uses LoRaWAN wireless protocol.
In a conventional discrete-time state feedback control,
the control input $\upsilon(t)$ is given by
\begin{align} \label{eq:feedback_gain}
\upsilon(t)&=K\xi(t_k), \quad \forall t \in [t_k,t_{k+1}]
\end{align}
where $t_k$, $k \in \mathrm{N}$ are the sampling instants which occur
periodically, i.e. $t_k=kh$ for $h>0$.
The feedback gain matrix is denoted as $K\in \textrm{R}^{n\times n}$.
In ETC schemes, the sampled state taken from the sensor, $\xi(t_k)$,
is used to evaluate a predefined triggering condition at each $t_k=kh$
for $k \in \mathrm{N}$, $h>0$.
In that way, the controller only updates and sends a control action
when the triggering condition is satisfied.
We refer to this as to an event and it can be expressed in terms of the
measurement error $\epsilon(t_k)=|\xi(t_k)-\xi(t)|\leq \eta$ for $t \in
[t_k,t_{k+1}]$ that exceeds the predefined threshold value $\eta$.
This approach differs from conventional periodic control schemes where
events are transmitted regularly regardless of the state of the plant.
ETC schemes only transmit an event if one actually occurs.
This difference reduces the computational load of the controller and
the amount of communication required between the sensors/actuators and
the controller.
The reduction of computation and communication equate to direct energy
savings for the control system.
This energy saving aspect is why ETC strategies are actively researched
for WNCSs.
In general, the controller is centralized and it works in
a sample-and-hold fashion:
\begin{align} \label{eq:c1_update}
\upsilon(t_k)=\left\{\begin{array}{l l}
K\xi(t_k), \quad \quad \textrm{if} \quad \theta_k=1 \\
K\xi(t_{k-1}), \quad \textrm{if} \quad \theta_k=0
\end{array}\right. .
\end{align}
The indicator function $\theta_k=1$ indicates that the triggering
condition is satisfied, $\theta_k=0$ indicates that the triggering
condition is not satisfied at time $t_k$.
Satisfaction of the triggering condition is based on measurements
received from each end-device.
With ETC control schemes in \cite{Mazo2014} the event-triggering
mechanism is distributed to the end-devices such that
$j\in\{1,\ldots, N_d\}$, where $N_d$ is the total number of end-devices
in the network.
End-devices work asynchronously.
Only the data of the end-device that measured a threshold violation is
sent to the gateway to update the control action in Eq.~\ref{eq:c1_update},
i.e. if $j\in\{1,\ldots,N_d\}$ and $\theta_{k_j}=1$, then
$\xi(t_k)=[\xi_1(t_{k-1}), \xi_2(t_{k-1}), \ldots, \xi_j(t_k), \ldots,
\xi_N(t_{k-1})]^T$.
The decentralisation of the event triggering mechanism further increases
energy efficiency by reducing the number of events.
There exists a non-zero minimum time that must always elapse after an
event and before the next event is triggered.
This lower bound on inter-arrival time can be explicitly computed
(see \cite{Tabuada2007}).
We show later that when an ETC system relies on a LoRaWAN,
there is a practical bound imposed by LoRaWAN on the event rate due
to the LoRaWAN protocol and the limitations of wireless communication.
This practical bounds imposed by LoRaWAN can be abstracted to event
delays, control action delays and event losses in the previously
discussed ETC implementation.
We define these as:
\begin{definition}
We denote by $\tau_k^{j\rightarrow C}$ the event delay of the
measurement $\xi(t_k^j)$ of the end-device $j$ to the controller
$C$ at time $t_k^j$.
Similarly, $\tau_k^{C\rightarrow j}$ is the control action delay of
the control action to the end-device $j$.
\end{definition}
\begin{definition}
We denote by $P_k^{i\rightarrow C}$ the number of successive event
losses in the transmission of the measurement $\xi(t_k^j)$ of end-device
$j$ to the controller $C$ at time $t_k^j$.
A loss of an event message also means the loss of the subsequent
control action message as it is dependant on the successful reception
of the event message by the controller.
\end{definition}
Event or control action delays or losses directly affect the ability
of the controller to maintain a stable plant.
They cause the absence of control actions.
The stability or level of performance of the plant is determined by
$\eta$, the size of the error that can be tolerated.
There is a maximum number of consecutive event losses and a maximum
tolerable delay for which the performance of the system can be guaranteed
(see \cite{Mazo2014,Guinaldo2012}).
In the next section we discuss LoRaWAN protocol, and its delays and losses.
\section{Evaluation}
\label{sec:evaluation}
In this section, we present a simulation based experiment to illustrate the
feasibility of using LoRaWAN with an ETC scheme to maintain the stability
of a WNCS and guarantee a certain level of performance.
First, we present a dynamical model with an asynchronous ETC scheme.
The example is motivated by a real-life water network that can be characterised
as a slow-rate physical process.
Then, using projected Round Trip Time delays and Blackout Periods of data loss from Fig.~\ref{fig:timeonair} and Fig.~\ref{fig:blackouttime} we show that LoRaWAN
is capable of meeting the system's performance goal for certain scenarios.
Finally, we characterise the limitations of using LoRaWAN protocol if the
application requirements increase or if the system is exposed to external
disturbances.
\subsection{Motivating Example}
Consider a hybrid linear model in the form of Eq.~\ref{eq:lin_sys} that
represents a water distribution network.
The state input vector $\xi(t) = [\xi_1(t) \enskip \xi_2(t) \enskip \xi_3(t)]^T$
represents the water levels in three different tanks.
Each tank is equipped with an end-device (sensor and actuator).
The model can work in two different modes for which the state matrix $A$ is
given as a zero matrix, i.e. $A_w=A_p=O_{3\times 3}$, and the input matrix
$B$ is given as
\small
\begin{align*}
B_w&=10^{-5} \times \left[\begin{array}{rrr}0.1436 & -0.0170 & -0.0164\\ -0.0098 & 0.1060 & -0.0100 \\ -0.0139 & -0.0139 & 0.1492\end{array}\right], \\
B_p&=10^{-5} \times \left[\begin{array}{rrr}0.7666 & -0.0493 & -0.0457\\ -0.0274 & 0.5848 & -0.0279 \\ -0.0393 & -0.0432 & 0.1492\end{array}\right].
\end{align*}
\normalsize
The control law in Eq.~\ref{eq:feedback_gain} works in a sample-and-hold fashion
with the feedback gain matrix $K$ given as
\small
\begin{align*}
K_w&=\left[\begin{array}{rrr}99950 & 3029 & 872\\ -3014 & 99940 & -1679 \\ -922 & 1652 & 99982\end{array}\right], \\
K_p&=\left[\begin{array}{rrr}9998.5 & 167.1 & 41.0\\ -166.6 & 9997.9 & -116.0 \\ -43.0 & 115.3 & 9999.2\end{array}\right].
\end{align*}
\normalsize
The \emph{'weak mode'} is represented by $(A_w, B_w, K_w)$ and it simulates
low water demand that a water network would experience during the night when
only the assistant pump is on.
The \emph{'powerful mode'} is represented by $(A_p, B_p, K_p)$ and it simulates
high water demand during the day when the powerful pump is enabled.
The model uses asynchronous ETC given in Sec.~\ref{sec:sys_model}.
The switching from powerful to weak mode is triggered by a function that maps
actuator saturation and quantization to represent the degree to which the
node's valve is open: $|S(-K_p\xi(t)+\alpha_p^{in})|_1<180^{\circ}$.
The valves themselves are discrete, and open and close in steps of $10^\circ$.
The term $|\cdot|_1$ is $\mathrm{L}^1$-norm, or sum of the entries, of the
resulting vector, and $\alpha_p^{in}$ denotes the degree that the in-valves
are open at equilibrium while in the powerful mode.
The switching from weak to powerful mode is triggered by $\xi_j(t)\leq h_{l_j}$
where $h_{l_j}$ is minimum water level for end-device $j$ and equals $0.03$m.
More details on the model can be found in \cite{Kartakis2017}.
The system response under normal operating conditions is given in
Fig.~\ref{fig:normal_operation}.
It takes on average $100$ seconds for the system to reach the steady-state
value and continues operating within the safe bounds (in our case between $0.03$m
and $0.06$m).
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.85\linewidth]{Figures/NormalOperation.png}
\caption{\label{fig:normal_operation} The system response under normal operating conditions (no delays nor event losses).}
\vspace{-5mm}
\end{center}
\end{figure}
\subsection{System Performance Analysis when using LoRaWAN}
We simulate the system for $600$s using a sampling rate of $1$ms.
Each end-device is allocated $N=3$ LoRaWAN channels with $1$\% duty
cycle restriction.
Our goal is to record overshoots and undershoots due to delays and
message losses caused by LoRaWAN limitations.
We measure this through the \emph{deviation metric} which indicates
the maximum system's deviation from the safe operating conditions which
is critical for water networks.
The behaviour of our system is understood and bounded.
Its acceptable deviation is up to $\pm15\%$.
We consider two cases:
1) \emph{Disturbance-Free System} -
We assume an ideal case when there is no disturbance to the system.
We investigate the maximum system deviation for all spreading factors
(from $7$ to $12$) and various message sizes (from $10$Bytes to $50$Bytes
in increments of $10$).
The results are presented in Fig.~\ref{fig:max_dev}.
We present the results for the tank $1$ only as it is the smallest size tank and
therefore the most sensitive to state changes for identical control input.
As it can be observed from Fig.~\ref{fig:max_dev}, for low SFs and message sizes
the system is well within the bounds.
This is due to low RTTs and BPs as demonstrated in Sec.~\ref{sec:lora}.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.85\linewidth]{Figures/MaxDeviation.png}
\caption{\label{fig:max_dev} Maximum deviation of water level in the tank~$1$ when the communication is supported by LoRaWAN with $N=3$ channels (y-axis uses logarithmic scale).}
\vspace{-2mm}
\end{center}
\end{figure}
2) \emph{System with Disturbance} - We model disturbance as a step impulse
in the water level state of the first tank, $\xi_1(t)$.
The magnitude of disturbance is equivalent to an increase in demand of $30\%$
at time $t$ or to a leak due to a pipe burst.
We vary the duration of the disturbance from $1$s to $10$s.
The results are presented in Table.~\ref{table:delays}.
These show that for severe disturbances (duration larger that $5$s), there
is a strict bound on SFs and message sizes that guarantee system performance
within desired levels.
To reject a disturbance, measurements need to be transmitted to the controller
in shorter time.
However, long BPs caused by LoRaWAN prevent reaction of the system at a rate that
can keep it stable.
\begin{table}[!t]
\caption{ Maximum deviation of water level (tank $1$) given in \% when there is an external step disturbance of $0.01$m ($N=3$ channels, SF8)}
\begin{center}
\begin{tabular}{l c c c}
\toprule
\textbf{Disturbance Duration} & 1s & 5s & 10s \\
\midrule
\textbf{10Bytes} & 0.00 & 8.67 & 16.33 \\
\textbf{20Bytes} & 1.67 & 9.23 & 18.00 \\
\textbf{30Bytes} & 0.67 & 10.33 & 18.00 \\
\textbf{40Bytes} & 0.67 & 10.33 & 22.67 \\
\textbf{50Bytes} & 10.73 & 16.87 & 28.80 \\
\bottomrule
\end{tabular} \vspace{-5mm}
\label{table:delays}
\end{center}
\end{table}
With this example we illustrated that LoRaWAN is able to support the classes
of applications where the underlying phenomenon changes at a slow rate.
The mode switching happens between every $50$s and $150$s.
LoRaWAN in such control system handles packet sizes of up to $50$Bytes for
SF7 and SF8 without affecting the system's performance and stability, as well
as the smallest packet size of up to $10$Bytes for SF7, SF8 and SF9.
LoRaWAN can be used for systems that are resilient to blackout period
s and delays of up to $10$s.
These limits become more conservative when the system is affected by significant
disturbances.
LoRaWAN protocol does not provide flexibility of sending time-critical data
if there is a duty cycle restriction in place at that moment.
\section{Introduction}\label{sec:introduction}}
\else
\section{Introduction}
\label{sec:intro}
\fi
The design of traditional periodic control systems \cite{Astrom1997} assumes
instantaneous and reliable communication of sensor data and control actions
between the sensors and the controllers.
This assumption is easy to meet when the sensors and controllers are connected
by cables or wires.
The controller will have relevant data to maintain the stability and the
desired level of performance of the controlled physical system.
Periodic transmissions at a high frequency can be achieved because wired
systems do not pose restrictions on the bandwidth available.
There is a current movement to instrument industrial control systems like
water/waste distribution, unmanned off-shore oil rigs, transportation networks,
and agricultural facilities with wireless sensors and actuators.
Wireless Networked Control Systems (WNCSs) are smarter, more responsive to user
demand, and more efficient in their use of resources such as energy
\cite{Rajkumar2010}.
These next generation industrial systems are composed of a network of sensor
nodes installed around a plant to measure its physical processes and transmit
the measurements (data) via a wireless network.
The controller receives the measurements and processes them according to
an underlying control scheme.
The result is a control action that is sent wirelessly to actuators to influence
the dynamics of the plant.
The challenge of this approach is that wireless radio networks are not reliable,
and the assumption of instantaneous and reliable communication no longer holds
\cite{Ali2015}.
There are two associated challenges with WNCSs. The first is to develop control
schemes that can tolerate a certain amount of delay and data loss caused by
unreliable wireless networks.
The second is to develop wireless network protocols that can provide the bounded
maximum delay and message loss required by the control scheme.
To provide the solution that addresses both challenges, the communication and
control systems have to be jointly designed.
This is known as the communication and control systems co-design problem
\cite{Park2017, Wu2017}.
An additional constraint on the communication and control systems design problem
is energy efficiency.
WNCS sensors are battery powered which introduces restrictions on the available
bandwidth.
They are often deployed in inaccessible locations which makes frequent battery
changes difficult.
In this paper, we investigate the feasibility of using the low-power wide-area
communication protocol LoRaWAN with an event-triggered control scheme.
Event-Triggered Control (ETC) schemes \cite{Tabuada2007} are a solution to the
high communication cost and frequent battery changes of traditional periodic
control schemes.
Traditional periodic control schemes send communication every fixed period of
time, even if there is no change in the underlying physical process.
ETC schemes save energy by only sending communication when an event occurs
and new action is needed.
An event is triggered when there is an indication that the stability or
performance of the system are about to be compromised.
Low-Power Wide-Area (LPWA) networks \cite{Raza2017} have been developed to
enable wireless communication over long ranges.
LPWA techniques enable long-range communication of up to $15$km's at a low
data rate of $0.3$-$37.5$Kbps.
This offers trade-off between communication coverage, and data rates when
compared to commonly used short-range protocols such as those used by
WirelessHART and ISA-100.a.
In this paper we provide the following contributions:
\begin{itemize}
\item We model the delays and message losses introduced by
the LPWA communication protocol LoRaWAN \cite{LoRaWAN2015} for different
rates and message sizes.
We analyse an application scenario where we model a linear ETC system in
Matlab.
We evaluate the effects of LoRaWAN delays and message loss rates on system
stability and performance guarantees for both cases, an ideal ETC system
and the ETC system that is exposed to external disturbances.
\item Most of the existing studies that address event-triggered controllers
with radio networks assume that bounds on communication delays and packet
losses are given a priori (e.g. \cite{Wang2011, Lehmann2012}).
However, it is difficult to obtain such properties in real applications.
In this paper we consider a practical communication protocol LoRaWAN.
LoRaWAN is an example of a long-range protocol whose use in control
scenarios has received very little attention \cite{Park2017}.
\end{itemize}
The rest of the paper is organized as follows: Sec.~2 presents the
problem formulation.
Sec.~3 presents the event-triggered control model.
Sec.~4 presents the LoRaWAN communication model.
We give the evaluation results in Sec.~5 and end the paper in Sec.~6
with brief concluding remarks.
\section{LoRaWAN Communication Model}
\label{sec:lora}
In this section we explain how LoRaWAN functions and discuss some
of its parameters.
We relate its parameters and performance to the event and control action delays
and losses discussed in the previous section.
\subsection{Network Architecture and Transmission Parameters}
LoRaWAN provides bidirectional communication with an uplink (end-device to
gateway) and a downlink (gateway to end-device).
LoRaWAN defines three classes of end-devices, classes A, B and C.
Class A end-devices only receive downlink communication from the gateway
after a successful uplink sent from the end-device to the same gateway.
There are two time periods (or windows) that are available to the gateway
for downlink transmissions at $1$s and $2$s.
Class B end-devices allow the gateway to schedule downlink communication
windows without a prior successful uplink transmission from an end-device.
Class C end-devices listen for transmissions from the gateway all the time
unless they are transmitting.
In this paper, we consider only Class A end-devices.
Class A devices are the most energy efficient, and are a perfect fit for
ETC because communication from the gateway is always triggered by an uplink
transmission from the end-device.
LoRaWAN end-devices transmit uplink packets using an ALOHA-based channel access
scheme.
The LoRa physical layer uses Chirp Spread Spectrum (CSS) modulation.
CSS signals are modulated by pulses that increase or decrease in frequency,
or chirps.
The number of chirps used to encode each symbol is given by $2^{\mathrm{SF}}$
where $\mathrm{SF}$ represents the Spreading Factor that varies between $7$
and $12$ in increments of $1$.
SF7 provides a data rate of $5.468$kbps, while SF12 provides $0.293$kbps.
The data rate in LoRa also depends on the channel bandwidth (we use $125$kHz)
and the code rate (we use $4/5$).
The trade-off in LoRa is that the higher SF has a lower data rate, but a longer
range and is more resilient to interference.
Lower SF has a higher data rate, but a shorter range and less resilience.
In LoRa, the use of retransmissions is optional.
The use of CSS makes the signals very robust to interference, and it has been
experimentally shown that LoRA is very reliable~\cite{cattani2017experimental}.
If retransmissions are used, the number of and timing is at the discretion of the
each end-device.
In this paper, we assume no use of retransmissions.
\subsection{Round Trip Time Delay of LoRaWAN}
Wireless technologies such as IEEE 802.15.4 have transmission times in
the range of $10$-$100$ms depending on the payload size \cite{Gutierrez2001}.
LoRaWAN networks have transmission times from $61$ms-$2.7$s depending on the
message size, as it can be seen in Fig.~\ref{fig:timeonair}.
We define the time needed for a message (sensor measurements) to be transmitted
from an end-device to the controller (an uplink transmission) as
$\mathrm{TimeOnAir}$.
Figure~\ref{fig:timeonair} shows the $\mathrm{TimeOnAir}$ of an uplink
transmission using LoRaWAN for a code rate of $4/5$, bandwidth of $125$kHz,
all SF$7$ through $12$ inclusive, and messages sizes of $10$, $20$, $30$, $40$,
and $50$Bytes of payload.
The messages sizes are shown without the header.
The message header adds $13$Bytes regardless of SF.
As can be seen from Fig.~\ref{fig:timeonair}, large SFs increase the
$\mathrm{TimeOnAir}$.
Large SFs also have an impact on duty cycling (channel availability) as shown
later.
\begin{figure}[!t]
\begin{center}
\includegraphics*[width=0.83\linewidth]{Figures/TimeOnAir.png}
\caption{\label{fig:timeonair}$\mathrm{TimeOnAir}$ of LoRaWAN for a code rate of $4/5$ and a bandwidth of $125$kHz.}
\vspace{-5mm}
\end{center}
\end{figure}
Downlink transmissions also use the LoRaWAN protocol.
These transmissions contain the control action sent from the controller to the
actuators connected to the end-devices as in Fig.~\ref{fig:architecture}.
With class A end-devices, downlink transmissions are possible only after
a successful uplink transmission.
We assume that the downlink payload size is the same as the uplink payload size
and both have the same $\mathrm{TimeOnAir}$.
In Def.~1 we defined event delays as $\tau_k^{j\rightarrow C}$ and control
action delays as $\tau_k^{C\rightarrow j}$.
These delays are a result of the used SF.
They are expressed in our LoRaWAN model as $\mathrm{TimeOnAir}$.
Additionally, the LoRaWAN specification \cite{LoRaWAN2015} introduces a
fixed time delay between an uplink and a downlink transmission.
This delay is $\mathrm{RECEIVE\_DELAY}$ and it equals $1$s.
We now define Round Trip Time ($\mathrm{RTT}$) as:
\begin{equation} \label{eq:RTT}
\mathrm{RTT} = 2\times\mathrm{TimeOnAir}+\mathrm{RECEIVE\_DELAY}.
\end{equation}
From Eq.~\ref{eq:RTT} and Fig.~\ref{fig:timeonair} it can be seen that the
$\mathrm{RTT}$ delay of a message sent from the end-device to the controller,
and the response from the controller back to the end-device using LoRaWAN
ranges from $1.12$s to $6.59$s.
We do not include controller processing time in $\mathrm{RTT}$ because
$\mathrm{RECEIVE\_DELAY}$ is almost certainly larger than the processing time.
An important observation to make is that $\mathrm{RTT}$ bounds the rate at
which events can be reported from the sensor to the controller.
An event that occurs less than $\mathrm{RTT}$ after a previous event will
be dropped due to the fact that the end-device will be awaiting a reply
from the controller.
\subsection{Duty Cycle Limitation of LoRaWAN}
LoRaWAN operates in the unlicensed frequency band $863$-$870$MHz in Europe
(where this study was conducted).
European regulations impose duty cycles on users of this band to ensure fair
usage.
When a LoRa end-device transmits a message, it can not use the same
channel for the length of its duty cycle.
The duty cycle ($\mathrm{DutyCycle}$) varies from $0.1$\% to $10$\% usage
time per node per channel.
For example, if an end-device spends $0.5$s transmitting a message on a
channel that specifies a $1$\% duty cycle, that channel will be unavailable
to the sending node for next $49.5$s.
Different channels can specify different duty cycles, as long as they remain
within the regulated duty cycle specifications.
We define the metric Blackout Period (BP) as
\begin{equation} \label{eq:toff}
\mathrm{BP} = \frac{\mathrm{TimeOnAir}}{\mathrm{DutyCycle}}-\mathrm{TimeOnAir}
\end{equation}
Blackout Period, $\mathrm{BP}$, is the time for which an end-device cannot
access a channel after it has sent a message due to duty cycle restriction.
We now relate $P_k^{i\rightarrow C}$, the event losses defined in Def.~2
of our ETC model, to the Blackout Period.
The cause of the BP, and therefore event losses, is the LoRaWAN duty cycle.
An end-device might have $N$ channels available which decreases the blackout
time.
For example, if the end-device can transmit on $3$ channels instead of one,
each individual channel is still occupied for $1$\%.
However, the device is now transmitting for $1$\% of time units in each
channel, giving it a duty cycle of $3$\% which reduces $\mathrm{BP}$ in
Eq.~\ref{eq:toff}.
However, Eq.~\ref{eq:RTT} introduces the bound on minimum inter-arrival time of
two consecutive messages that has to be taken into account.
Therefore, the BP for an end-device that has the access to $N$ channels is given by
\begin{equation}
\mathrm{BP_N} = \mathrm{BP}-(N-1)\times\mathrm{RTT}.
\end{equation}
According to the LoRaWAN specification~\cite{LoRaWAN2015} a minimum of $3$
channels must be available to all end-devices.
An individual network may provide more channels.
For example, The Things Network \cite{TheThingsN} allocates $8$ channels with
$1$\% duty cycle to each end-device.
We now examine the affect of multiple channels on the length of the BP.
Figure~\ref{fig:blackouttime}, part a) shows the BP of LoRaWAN end-device
when $3$ channels are available for various SFs and message sizes.
\begin{figure}[!t]
\begin{center}
\includegraphics*[width=0.83\linewidth]{Figures/BlackoutTime.png}
\caption{\label{fig:blackouttime}Blackout Period of LoRaWAN for a) $N=3$ channels and b) $N=8$ channels, a code rate of $4/5$ and a bandwidth of $125$kHz (the y-axis uses logarithmic scale).}
\vspace{-5mm}
\end{center}
\end{figure}
We can see that the BP varies from $3.8$s to $236.4$s when an end-device has
the access to only $3$ channels.
When the number of channels is increased to $8$, in Fig.~\ref{fig:blackouttime}, part b)
and a $1\%$ duty cycle is still maintained, the BP completely disappears for
SF$7$ payloads of $30$Bytes and less.
It is interesting to observe that the BP does not disappear for the other
SFs.
SF$10$, SF$11$, and SF$12$ only show a minimal improvement.
This shows that BPs can be reduced by adding more channels, but that it is
difficult to remove them completely, and that they get worse as the range
and robustness of the links improve.
In the next section we will see the impact of the RTTs and the BPs on the
ability of ETC control schemes to maintain system stability and performance levels.
\section{Wireless Networked Control System Problem Formulation}
In Fig.~\ref{fig:architecture} we present a diagram of the closed-loop
WNCS considered in this paper.
The WNCS consists of a large complex physical process (the plant) and
the control and management system which are connected via a LPWA
communication network.
The plant is a continuous-time physical system instrumented with the
set of sensors $\{\mathcal{S}_1, \mathcal{S}_2, \ldots, \mathcal{S}_N\}$
and the set of actuators $\{\mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}_M\}$.
We assume that all of the sensors and actuators are collocated on
LPWA-enabled end-devices which share same communication channels.
The end-devices communicate with the set of gateways
$\{\mathcal{G}_1, \mathcal{G}_2, \ldots, \mathcal{G}_P\}$ when a change in
the plant is detected.
Both, sensor-to-gateway and gateway-to-actuator communication is achieved
in a single-hop fashion via a LPWA network.
A single-hop network topology is far more favourable for control-based
systems due to its high reliability and low energy cost compared to multi-hop
networks.
The information exchange between gateways and the controller is achieved via
traditional wired communication and thus is instantaneous and reliable.
In the rest of this paper the terms the gateway and the controller will be
used interchangeably.
\begin{figure}[!t]
\begin{center}
\includegraphics*[width=0.9\linewidth]{Figures/Architecture.png}
\caption{\label{fig:architecture} The architecture of wireless networked control system.}
\vspace{-5mm}
\end{center}
\end{figure}
To describe the WNCS in Fig.~\ref{fig:architecture} we present two models:
\begin{enumerate}
\item \emph{Event-Triggered Control Model} -
A general model for event-triggered control of a linear plant.
\item \emph{LoRaWAN Communication Model} -
We explain how LoRaWAN works and present its main concepts and parameters.
\end{enumerate}
We use the notion of communication delays and message losses to investigate
the feasibility of using LoRaWAN with an event-triggered control system.
The LoRaWAN introduces delays and losses as part of the protocol.
In an ETC system there is a maximum bound on the delay and data loss for
which the system can remain stable and a certain level of performance is
guaranteed.
In this paper, we check if these bounds can be met by LoRaWAN.
|
1,941,325,220,583 | arxiv | \section{Introduction}\label{intro}
Multilevel control attracts intensive research as a systematic tool for control of real plants with respect to high-level target while adhering to the local constraints \citep{Tat08Advanced}. The upper levels are usually concerned with plant-wide steady state objectives with low rate sampling. The lower levels address fast dynamic control. There is a mature trend of applying advanced optimization packages to fill the gap between these two layers. Well-known industrial examples such as AspenOne and RHMPC use MPC as the core optimizer to deal with constraints \citep{Qin03survey,Fro06Model}. MPC is a constraint-handling optimization method where the core idea is based on the receding horizon control. At each sampling time, the current plant output/state is measured, and an optimal input is derived to minimize a performance index subject to state and input constraints. This desired inputs are sent to PID controllers to directly manipulate the actuators. These PIDs must be tuned to minimize the mismatch with the updated optimal input at each sampling step. The first objective of the paper aims to bypass this two-phase complication through direct optimization of the PID gains.
Currently, there are two approaches of MPC, using either \emph{online} implementation \citep{May00Constrained} for slow processes or \emph{offline} implementation \citep{Bem02explicit} for fast processes. The former control approach solves in real time an optimization problem, thus it is more flexible to system design changes. The latter approach solves the same problem offline for all feasible states, and obtains the optimal control law in real time by searching the current state over feasible regions. This scheme, named parametric MPC, can effectively facilitate a PID gain scheduling implementation. The resulting PID controller will deal with constraints by changing gains upon the transition of active constraint regions, not at each time step. This is the second and main objective: to develop a practical implementation of parametric MPC.
The PID realization of MPC can be achieved with its robustness property intact. In fact, a great number of research methods have carefully addressed the robustness of MPC for perturbations both along the trajectory (robust performance) and at steady state (robust stability). Polytopic uncertainty model is discussed in \citet{Gri03} with LMI and in \citet{Bem03Min,Nag04Open} where min-max solutions are formed; bounded disturbances addressed by tube-based MPC is proposed in \citet{Alv08,Mar11Stochastic}. The tradeoff lies in the complexity of the solutions. In this note, we are keen on observing the robust stability provided by the simple PID form of the proposed solution.
In the literature, many finite-horizon optimal PID designs for constrained multivariable systems have been attempted to deliver a systematic PID tuning. In \citet{Mor03}, the velocity form of PID prohibited the variable gain structure, thus a fixed PID gain must be used across the prediction horizon. As shown in \citet{Camacho03Model,Aro08,Sat12} the GPC-based PID results apply to the plants approximated by a first or second order model, thus limiting their applications to multivariable plants. The solution in \citet{Di10Model} partially solves the problem, but the two controllers MPC and PID must operate in parallel. A flexible framework for optimal PIDs is still under ongoing research.
Collectively through the two mentioned objectives, this paper seeks to improve the MPC-based PID scheme to further close the gap between MPC optimization and PID controllers. In Section II, we formulate the tracking problem and analyze the controllability and observability of the augmented system. In Section III, we describes the MPC formula and shows that either a new or existing multi-loop unconstrained PID designs can be adopted into the framework. For convenience, two methods are provided to calculate the PI/PID gains at the operating point so that the closed loop system is stable. The first method applies LQR on the PI state while the latter leads to linear matrix inequalities (LMI) with the size proportional to the number of tracked outputs. Section IV applies this PID design on the piecewise affine (PWA) solution of MPC, which suggests a distributed PID gain scheduling framework to deal with constraints. Fig. \ref{fig1} shows the involved levels within the plantwide structure.
\subsection*{Notation}
The operators $\sum, \Delta$ are the integral and differential terms. The notation $Q \succ 0$ denotes positive definiteness. $x$, $\hat{x}$ and $\tilde{x}$ denote the state, estimated state and state error; $u$ and $\tilde{u}$ denote the inputs for tracking and regulating problems, respectively. Subscript $i$ indicates matrix/vector component and $k$ the prediction step, superscript $i$ is the critical region index. $I_m$ is an identity matrix of order $m$.
\begin{figure}[t]
\centering
\includegraphics[width=3.2in]{ctrlDiag1-eps-converted-to.pdf}
\caption{Optimization control with multi-layers.}
\label{fig1}
\end{figure}
\section{Preliminaries}\label{preli}
To obtain a linear feedback involving proportional-integral-differential gains, it is necessary to form a system state that contains the corresponding variables. Provided that is the case, an optimal linear feedback gain is also an optimal PID gain. This section introduces the augmented PI/PID-state systems and covers the analysis of their controllability and observability.
\subsection{Plant Model}
Consider a linear time-invariant system
\begin{IEEEeqnarray}{rCl}
x(k+1)&=& Ax(k) + Bu(k)\nonumber \\
v(k)&=& C_vx(k)\nonumber \\
y(k) &=& Cx(k).\label{mdl}
\end{IEEEeqnarray}
subject to the constraint
\begin{equation}
Ex(k) + Fu(k) \leq G.\label{cons}
\end{equation}
In \eqref{mdl}, $x(k)\in \mathbb{R}^{n}, u(k)\in \mathbb{R}^{m}, v(k)\in \mathbb{R}^{q}\, (q\leq n)$, and $y(k)\in \mathbb{R}^{p}$ are the state, input, tracked output and measured output. Assume $(A,B)$ is controllable and $(A,C)$ is observable; $C,C_v$ having full row rank; $E, F, G$ are appropriate matrices defining the state and input constraints.
The plant model \eqref{mdl} is augmented with an integral of the tracked output $\sum{v(k)}$ to ensure zero offset during the steady state. The following PI-state model is used
\begin{IEEEeqnarray}{rCl}
\begin{bmatrix}x(k+1)\\ \sum{v(k+1)}\end{bmatrix}&=&
\begin{bmatrix}A& 0\\C_v& I_q\end{bmatrix}
\begin{bmatrix}x(k)\\ \sum{v(k)}\end{bmatrix}+
\begin{bmatrix}B\\ 0\end{bmatrix}u(k)\nonumber \\
y(k) &=& Cx(k).\label{augmdl}
\end{IEEEeqnarray}
In special cases, $C_v=I$ requires a full-state tracking while $C_v=C$ expects only output tracking.
\begin{proposition}
The PI-augmented system \eqref{augmdl} is detectable. Furthermore, it is controllable if and only if (A,B) is controllable and
\begin{equation}
rank \begin{bmatrix}A-I_n& B\\ C_v& 0\end{bmatrix}=n+q
\label{hauCtrl}
\end{equation}
\end{proposition}
\begin{proof}
The Hautus condition for observability is
\begin{equation}
rank\begin{bmatrix}A^T-\lambda I_n& C_v^T& C^T\\ 0& I_q-\lambda I_q& 0 \end{bmatrix}=n+q\text{\quad for all }\lambda\in \mathbb{C}.
\label{hauObsv}
\end{equation}
The condition \eqref{hauObsv} does not hold only at $\lambda=(1,0)$, but the unobservable integrating state can be controlled to decay to a constant so the system is detectable.
Similarly, \eqref{hauCtrl} follows directly from Hautus controllability where only the case of $\lambda=(1,0)$ is to check.
\end{proof}
By addition of the differential term, the PID-state system presents as
\begin{IEEEeqnarray}{rCl}
\begin{bmatrix}x(k+1)\\ \sum{v(k+1)}\\ \Delta v(k+1)\end{bmatrix}&=&
\begin{bmatrix}A& 0& 0\\ C_v& I_q& 0\\ C_v(A-I_n)& 0& 0\end{bmatrix}
\begin{bmatrix}x(k)\\ \sum{v(k)}\\ \Delta v(k)\end{bmatrix}+\begin{bmatrix}B\\ 0\\ C_vB\end{bmatrix}u(k)\nonumber \\
y(k) &=& Cx(k).\label{augmdl2}
\end{IEEEeqnarray}
This PID-augmented system is detectable and stabilizable. The proof is similar to Proposition \ref{hauCtrl}.
\begin{rmk}The number of tracked variables is presumed less than or equal to the number of manipulated variables ($q\leq m$ for PI case and $q\leq m/2$ for PID case); the other case was well treated in \citet{Mae09Linear}.
\end{rmk}
The objective is to design a finite-horizon optimal control based on the augmented system \eqref{augmdl} or \eqref{augmdl2} so that $v(k)$ tracks a piece-wise constant reference.
\subsection{Observer Design}
From the system detectability, an observer can make use of the system \eqref{mdl} to estimate the current state, and simply calculate the integral and differential state through a sum of the estimated $\hat{v}(k)=C_v\hat{x}(k)$ and its difference.
Since $(A,C)$ is observable, the observer is designed as
\begin{IEEEeqnarray}{rCl}
\hat{x}(k)&=&A\hat{x}(k-1)+Bu(k-1)\nonumber\\
&&+L_x[-y(k-1)+C\hat{x}(k-1)]\nonumber\\
\sum{\hat{v}(k)}&=& \sum{\hat{v}(k-1)}+C_v\hat{x}(k-1)\nonumber\\
&&+C_vL_x[-y(k-1)+C\hat{x}(k-1)]\nonumber\\
\Delta \hat{v}(k) &=& C_v(\hat{x}(k)-\hat{x}(k-1))
\end{IEEEeqnarray}
It is only necessary to design the observer gain $L_x$ as $eig (A+L_xC)<1$ so that $\hat{x}(k)-x(k)\rightarrow 0$. This automatically leads to $\Delta \hat{v}(k)$ being stable. The integral estimation error is not required to decay to zero, but a steady state because $\sum{\hat{v}(k)}-\sum{v(k)}\rightarrow const$ means $\hat{v}(k)-v(k)\rightarrow 0$.
\section{Controller Design}\label{MPCPID}
\subsection{MPC tracking structure}\label{MPC}
This section will outline the general MPC controller design for a state space model that results in PI/PID control implementation fulfilling the constraints.
Consider the linear system with constraints $z(k+1)=A_mz(k)+B_mu(k)$. Define the operating points $(z_s, u_s)$ and the deviation variables
\begin{IEEEeqnarray}{rCl}
\tilde{z}(k)&=&z_s-z(k)\nonumber \\
\tilde{u}(k)&=&u_s-u(k),
\label{dev}
\end{IEEEeqnarray}
\begin{IEEEeqnarray}{rCl}
\text{then\qquad}\tilde{z}(k+1)&=&A_m\tilde{z}(k)+B_m\tilde{u}(k)\nonumber\\
\tilde{y}(k) &=& C_m\tilde{z}(k).
\label{devmdl}
\end{IEEEeqnarray}
The finite-horizon quadratic optimal control problem is posed as
\begin{IEEEeqnarray}{lCl}
V_N^o(\tilde{z}_0,\tilde{U}&&)=\underset{\tilde{U}}{\operatorname{min.}}\,\tilde{z}_N^TP\tilde{z}_N\nonumber\\
&&\qquad+\sum_{k=0}^{N-1}({\tilde{z}_k}^TC_m^TQC_m\tilde{z}_k+\tilde{u}_k^TR\tilde{u}_k)\label{MPClaw}\\
subj.\,to \, && \tilde{z}_k\in X, \tilde{u}_k\in U \quad \forall k\in {0,...,N-1},\nonumber\\
&&\tilde{z}_0\in X_0,\,\tilde{z}_N\in X_f,\nonumber \\
&& \tilde{z}_{k+1}= A_m\tilde{z}_k+ B_m\tilde{u}_k,\, \nonumber
\end{IEEEeqnarray}
where $\tilde{U}=\{\tilde{u}_0,...,\tilde{u}_{N-1}\}$. Here $Q\geq 0, R\succ 0$ are the weighting matrices, $(Q^{1/2},A_m)$ is detectable; $P\geq 0$ is the terminal penalty matrix. $X_0, X_f$ are the initial feasible set and the terminal constraint set. Note that $X, U$ are translated constraints from \eqref{cons} through the transformation in \eqref{dev}. By the receding horizon policy, only $\tilde{u}_0$ is applied to the plant.
\begin{assumption}
The state and input constraints are not active for $k\geq N$. Also, $X_f$ contains the origin.
\end{assumption}
The optimizer $\tilde{U}$ stabilizes \eqref{devmdl} if the value function $V_N^o(\tilde{z})$ corresponds to a local Lyapunov function $V_f$ within the terminal set $X_f$. In addition, the decay rate of that Lyapunov function must be larger than the stage cost \citep{May00Constrained}. Under this setup, any admissible $\tilde{z}_0$ is steered to a level set of $V_f$ (and so $X_f$) within N steps, after which convergence and stability of the origin follows. In other words, $z_k$ is stable at $z_s$ for $k\geq N$.
Therefore, given the state and input weighting matrices $Q,R$, one would want to first compute an unconstrained stabilizing feedback $\tilde{u}=K\tilde{z}$ and its Lyapunov function $V(\tilde{z})$ that satisfy
\begin{IEEEeqnarray}{rCl}
V_f(\tilde{z})&=& \tilde{z}^TP\tilde{z}\geq 0,\nonumber\\
\Delta V_f(\tilde{z})&=& \tilde{z}^TA_K^TPA_K\tilde{z}- \tilde{z}^TP\tilde{z}\nonumber \\
&\leq &-\tilde{z}^TQ\tilde{z}-\tilde{z}^TK^TRK\tilde{z},\, \forall \tilde{z}\in X_f,\label{conMPC}
\end{IEEEeqnarray}
where $A_K=A_m+B_mK$. The other ingredients of MPC formula are then determined as follows.
\begin{itemize}
\item $X_f$ is the maximal positively invariant polyhedron of $\tilde{z}_{k+1}= A_m\tilde{z}_k + B\tilde{u}_k$ with respect to $\tilde{z}_k\in X, \tilde{u}_k\in U$. As commented in \citet{Rawlings09Model}, if $X_f$ is ellipsoidal, the problem is no longer a quadratic program but a convex program but can be solved with available softwares.
\item $X_0$ is the N-step stabilizable set of the system \eqref{MPClaw} with respect to $X_f$. $N$ is a trade-off value between the complexity of MPC problem and a larger set $X_0$ (i.e. larger initial error $\tilde{z}_0$).
\item $P$ is chosen as the solution of the equality in \eqref{conMPC}, the unique positive-definite solution of
a discrete Lyapunov equation once $K$ is known \citep{Gri05Stabilizing}.
\end{itemize}
$X_0, X_f$ can be calculated analytically using the method detailed in \citet{Bla99Set,Ale06}.
A popular choice for $K$ is obtained from the LQR gain with weighting matrices $Q, R$ \citep{Chm96constrained, Sco98Constrained}. However, in this note, it is left as a general stabilizing gain $K$ that will be computed in the next section.
\subsection{Computation of Stabilizing PI/PID}\label{PIDsec}
This session describes a method to compute an unconstrained feedback gain $K$ that is used to reconstruct the MPC formula \eqref{MPClaw}. It is because this gain would result in PI/PID controllers, as shown in the following theorem. For the general case, let $z=\begin{bmatrix}x^T& \sum{v^T}& \Delta v^T\end{bmatrix}^T$.
\begin{thm}\label{theo1}
A control law $\tilde{u}(k)=K\tilde{z}(k)$ implements PID control on the system state $x(k)$ which ensures robust tracking for $v(k)$.
\end{thm}
\begin{proof}
Because $\tilde{z}$ is the augmented state error, the control law is written as
\begin{IEEEeqnarray}{rCl}
\tilde{u}(k)&=& K\tilde{z}(k)\nonumber \\
&=& K_1\tilde{x}(k) + K_2\sum{\tilde{v}(k)}+ K_3\Delta\tilde{v}(k)\nonumber\\
&=& K_1\tilde{x}(k)+K_2C_v\sum{\tilde{x}(k)}+K_3C_v\Delta\tilde{x}(k).\IEEEeqnarraynumspace
\label{PID}
\end{IEEEeqnarray}
Since $rank(C_v)=q\leq n$, there are $m\times(n-q)$ P controllers and $m\times q$ PID controllers. In particular, PID control is applied to the state variables which influence the tracked output $v(k)$, so they are robust against disturbances.
\end{proof}
Let $(\bar{A}, \bar{B}, \bar{x})$ be the augmented model and state of \eqref{augmdl}.
\subsubsection{PI Controller ($K_3=0$)}
For this case, it is essential to obtain the feedback gain for $z = \bar{x}=\begin{bmatrix}x^T& \sum{v^T}\end{bmatrix}^T$. The PI control can be formulated by applying LQR to the error model of \eqref{augmdl}
to produce a PI control law $u(k)=K_{PI}z(k)$. From here simply take $(A_m,B_m,C_m)=(\bar{A},\bar{B},\bar{C}), K=K_{PI}$ and use \eqref{conMPC} to apply the MPC formula.
\subsubsection{PID Controller}
To get a non-trivial differential gain $K_3$, one can treat the differential term as an output feedback of the system \eqref{augmdl}. Define $\phi=\begin{bmatrix}x^T& \sum{v^T}& \phi_3^T\end{bmatrix}^T$ where $\phi_3 = \Delta v - C_vBu=C_v(Ax+Bu-x)-C_vBu = C_v(A-I_n)x$. Then
\begin{IEEEeqnarray}{rCl}
\bar{x}(k+1)&=& \bar{A}\bar{x}(k)+\bar{B}u(k)\nonumber \\
\phi(k) &=& \bar{C}\bar{x}(k)=\begin{bmatrix}I_n&0\\ 0& I_q\\ C_v(A-I_n)& 0\end{bmatrix}\bar{x}(k).
\label{sofmdl}
\end{IEEEeqnarray}
Design of static output feedback (SOF) $u(k)=F\phi(k)$ for the discrete time system above has been investigated in \citet{Gar03Robust,Bar05Static,Don07Static,He08Output} which use LMI conditions. There exists more outputs than inputs in this case, so we present a simple solution to determine $F$ in Theorem \ref{SOF} \citep{Bar05Static}. In that work, the solution can be extended to the $H_\infty$ design, but the detail is omitted here for simplicity (refer to Remark \ref{rmk3}).
\begin{thm}\label{SOF}
System \eqref{sofmdl} is stabilizable by a static output feedback if there exist a symmetric positive definite matrix $P_0\in\mathbb{R}^{(n+q)\times(n+q)}$ and a positive scalar $\sigma\in\mathbb{R}$ such that
\begin{equation}\label{LMI1}
\bar{A}^TP_0\bar{A}-P_0+\sigma \bar{B}\bar{B}^T\prec 0\\
\end{equation}
is satisfied. Furthermore, the SOF gain $F$ can be obtained by solving
\begin{equation}\label{LMI2}
(\bar{A}+\bar{B}F)^TP_0(\bar{A}+\bar{B}F)-P_0\prec 0.
\end{equation}
\end{thm}
Conditions \eqref{LMI1}, \eqref{LMI2} can be solved as two LMI problems. Once we have found a stabilizing output feedback $u(k)=F\phi(k)$ or equivalently $\tilde{u}(k)=F\tilde{\phi}(k)$, it can be rewritten in the PID form \cite{Zhe02design} as
\begin{IEEEeqnarray}{rCl}
\tilde{u}(k)&=&F_1\tilde{x}^T+F_2\sum{\tilde{v}^T}+F_3\Delta \tilde{v}+F_3C_vB\tilde{u}(k)\\
\text{so\ }\tilde{u}(k)&=&(I_n+F_3C_vB)^{-1}[F_1\tilde{x}(k)^
+F_2\sum{\tilde{v}(k)}+F_3\Delta \tilde{v}(k)]\nonumber\\
&=&K_{PID}\tilde{z}(k),
\end{IEEEeqnarray}
where $\tilde{z}=\left[\tilde{x}^T\ \sum{\tilde{v}^T}\ \Delta \tilde{v}^T\right]^T$. The invertibility of matrix $(I_n+F_3C_vB)$ is a necessary condition to render $K_{PID}$.
The MPC formula takes $\tilde{z}(k+1) = A_m\tilde{z}(k)+B_m\tilde{u}(k)$,
\begin{IEEEeqnarray}{l}\label{Am}
A_m=\begin{bmatrix}A& 0& 0\\ C_v& I_n& 0\\ C_v(A-I_n)& 0& 0\end{bmatrix}, B_m=\begin{bmatrix}B\\ 0\\C_vB\end{bmatrix}u(k)\IEEEeqnarraynumspace
\end{IEEEeqnarray}
and $K=K_{PID}$ to apply into \eqref{conMPC}.
\begin{rmk} \label{rmk2} Applying LQR directly to PID state for system \eqref{Am} will not result in a PID controller. In fact, since $A_m$ is no longer full rank, the optimal input $\tilde{u}(k)=(R+B_m^TQB_m)^{-1}B_m^TQA_m\tilde{z}(k)$ depends only on the first two components of $\tilde{z}(k)$, so it is not a full PID but a PI gain. However, we realize that increasing the weight on $\Delta v$ of $Q$ does reduce the overshoot and enhance the disturbance response of $v(k)$.
\end{rmk}
\begin{rmk} \label{rmk3} The PID design for multivariable systems used in this paper is not unique. It is possible to use other techniques such as \citet{Dic09parameter,Soy03Fast,Tos09Robust} to derive a robust PID gain before applying it into MPC.
\end{rmk}
\section{From parametric MPC to PID gain scheduling controllers}\label{fastMPC}
The result from Section III holds when it is applied to either an \emph{online} or \emph{offline} MPC formulation. In this section, we particularly use parametric MPC (offline) to demonstrate the PID gain scheduling realization.
\subsection{Parametric MPC}
Observe that the problem \eqref{MPClaw} minimizes a convex value function subject to a convex constraint set. We have the following definition
\begin{definition}[Critical Region]
A critical region is defined as the set of parameters $\tilde{z}$ for which the same set of constraints is active at the optimum $(\tilde{z},\bar{U}^0(\tilde{z}))$.
\end{definition}
In other words, if the constraints in \eqref{MPClaw} is presented as $G\bar{U}\leq S\tilde{z} + W$ and $A$ is an associated set of row index,
\begin{IEEEeqnarray}{l}
CR_A =\{\tilde{z}\in X_0\,|\,G_i\bar{U}^0=S_i\tilde{z}+W_i\text{ for all } i\in A\}\IEEEeqnarraynumspace
\label{eq:}
\end{IEEEeqnarray}
In \citet{Bao02,Ton03algorithm}, it is shown that these critical regions are a finite number of closed, non-overlapped polyhedra and they covers completely $X_0$. Since $\tilde{U}=\{\tilde{u}_0,...,\tilde{u}_{N-1}\}$, the same properties apply for $\tilde{u}^0_0$. Theorem \ref{theo2} states the key result (see \citet{Bem02explicit}).
\begin{thm}[Parametric solution of MPC]
\label{theo2}
The optimal control law $\tilde{u}^0_0=f(\tilde{z}_0),f:X_0\mapsto U$, obtained as a solution of \eqref{MPClaw} is continuous and piecewise affine on the polyhedra
\begin{equation}
f(\tilde{z})=F^i\tilde{z}+g^i\quad if\, \tilde{z}\in CR^i, i=1,...,N^r,
\label{mulctrl}
\end{equation}
where the polyhedral sets $CR^i\triangleq \{H^i\tilde{z}\leq k^i\}, i=1,...,N^r$ are a partition of the feasible set $X_0$.
\end{thm}
\subsection*{Tracking for piecewise constant setpoint}
Recall the admissible set $X_0$ the MPC controller can stabilize depends on the linearized model $x(k+1)=f(x(t))|_{x=x_s}$ and control horizon $N$. Tracking of a new setpoint can be done by increasing $N_2$ based on the new model so that a jump in reference $z_{s1}\rightarrow z_{s2}$ is feasible within $N_2$ steps.
\begin{figure}[!t]
\centering
\includegraphics[width=3.0in]{feas-eps-converted-to.pdf}
\caption{Feasibility check of new setpoint $z_{s2}$.}
\label{feas}
\end{figure}
In the case of fixed $N$, Corollary \ref{reffsb} states the necessary and sufficient condition for a new feasible setpoint
\begin{corollary}\label{reffsb}
With a fixed-horizon proposed controller, a change in setpoint $z_{s1}\rightarrow z_{s2}$ is feasible if and only if $z_{s1}-z_{s2}\in X_0(z_{s2})$.
\end{corollary}
\begin{proof}
The proof can be inferred from Fig \ref{feas}. If $z_{s1}$ is out of the maximal admissible region $X_0(z_{s2})$ constructed around $z_{s2}$, it is impossible to drive the current error $\tilde{z}=z_{s1}-z_{s2}$ to zero with the existing controller.
\end{proof}
Corollary \ref{reffsb} suggests a way to detect if a new setpoint is feasible so that the local optimization for steady state target can recalculate $z_s$ early before the infeasibility happens. One can use a single model and treat the model mismatch at a different operating point as disturbance, but generally $X_0$ still needs to be rebuilt through \eqref{dev} because the constraints change with setpoint relocation.
\subsection{PID Gain Scheduling Design}
The optimal input of MPC is applied for regions outside $X_f$. When $\tilde{z}(k)$ reaches $X_f$, the system will be stabilized by the pure gain $F^0=K$. Therefore, one practical way to design PID for constrained systems is designing a PID gain for its unconstrained region, which has been accomplished in Section \ref{MPCPID}, and applying these settings on the MPC formulation \eqref{MPClaw}.
\begin{figure}[t]
\centering
\includegraphics[width=3.6in]{ctrlDiag3_1-eps-converted-to.pdf}
\caption{Proposed PID gain scheduling structure.}
\label{ctrlDiag3}
\end{figure}
Fig. \ref{ctrlDiag3} shows a series of PIDs plus a single feedforward vector where the controller gains are determined from \eqref{mulctrl}. Each of the PIDs is fully flexible (might contain only P or PI components) and have its own look-up gain scheduling for different partition indexes. At each time step, the proposed scheme would look for the region in which the augmented error $\tilde{z}(k)$ lies in. This search engine would broadcast the region index $i$ to the PID network. The feedforward term associated with region $i$ is added to compensate the active constraints. Non-zero tracking accounts for the addition of steady state input and recovers the input delivered to the plant.
\begin{rmk}
As seen from Fig. \ref{ctrlDiag3}, the PID network consists of one-to-one mappings between each state variable error of the original state $x$ and an input. This fact results from equation \eqref{PID}.
\end{rmk}
\section{Example}
\begin{figure}[t]
\centering
\subfigure{
\label{x1}
\includegraphics[width=\columnwidth]{X_1-eps-converted-to.pdf}}
\subfigure{
\label{u1}
\includegraphics[width=\columnwidth]{X_2-eps-converted-to.pdf}}
\subfigure{
\label{x2}
\includegraphics[width=\columnwidth]{U_1-eps-converted-to.pdf}}
\subfigure{
\label{u2}
\includegraphics[width=\columnwidth]{U_2-eps-converted-to.pdf}}
\caption{State responses and control inputs under disturbances at transient and steady-state.}
\label{stateinput}
\end{figure}
The proposed control design was illustrated in the following example, generalized from \citet{Bem02explicit} with two inputs. Consider a continuous stirred-tank reactor model
\begin{IEEEeqnarray}{rCl}
A&=&\begin{bmatrix}0.7326& -0.0861\\0.1722& 0.9909\end{bmatrix}, B=\begin{bmatrix}0.0609& 0\\ 0& 0.0064\end{bmatrix},\nonumber\\
C&=&\begin{bmatrix}1& 0\\ 0& 1\end{bmatrix},\, X=\left\{x\in \mathbb{R}^2|\begin{bmatrix}-0.5\\ -0.5\end{bmatrix}\leq x \leq \begin{bmatrix}1.5\\ 2.5\end{bmatrix}\right\},\nonumber\\
U&=&\left\{u\in \mathbb{R} | \begin{bmatrix}-2\\ -2\end{bmatrix}\leq u \leq \begin{bmatrix}2\\ 2\end{bmatrix} \right\}.
\end{IEEEeqnarray
The task was to track the level 1 $x_1$ with the reference $x_{1s} = 1$. To observe the robustness of tested controllers, the disturbances $d_1=[1;-0.5]$ (impulse), $d_1'=[0.01;-0.01]$ (additive) within an active constrained region at $k=3$ and $d_2=[-0.15;0]$ (additive) at steady state $k=100$ were introduced.
The three following controllers were compared: simple parametric MPC (I), the whole state tracking with full PI (II) and $x_1$-tracking with partial PID (III). The prediction horizon (also control horizon in this case) is chosen as $N=2$.
Tuning weighting matrices for PID control had been discussed in \citet{Ngu11Enhanced}. For PI, $z=\left[x_1^T\ x_2^T\ \sum{x_1^T}\ \sum{x_2^T}\right]^T$ and $Q=diag(1,1,0.001,0.001)$, $R=0.01I_2$; for PID $z=\left[x_1^T\ x_2^T\ \sum{x_1^T}\ \Delta x_1^T\right]^T$, $Q=diag(1,1,0.001,0.1)$, $R=0.01I_2$. MATLAB LMI solver was used to obtain the unconstrained PID gain for case III, and Multiparametric toolbox \citep{mpt} was applied to obtain the gains under critical regions.
The unconstrained gain $K$ in the three cases were
\begin{IEEEeqnarray}{lCl}
K_I &=& \begin{bmatrix}4.501& 3.792\\0.711& 2.160\end{bmatrix},\nonumber\\
K_{II} &=& \begin{bmatrix}5.792& 6.353& 0.289& 0.469 \\1.103& 7.094& -0.579& 0.326\end{bmatrix},\nonumber \\
K_{III} &=& \begin{bmatrix}0.493& 1.399& 0.139& -0.392\\3.014& 18.545& 1.765& -0.766 \end{bmatrix},\IEEEeqnarraynumspace
\end{IEEEeqnarray}
and they resulted in control laws with $8,\,14,\,12$ critical regions, respectively.
\begin{figure}[!ht]
\centering
\subfigure[]{\label{state1}
\includegraphics[width=\columnwidth,height=2.0in]{S_1-eps-converted-to.pdf}}
\subfigure[]{\label{state2}
\includegraphics[width=\columnwidth,height=2.0in]{S_2-eps-converted-to.pdf}}
\subfigure[]{
\label{state3}
\includegraphics[width=\columnwidth,height=2.0in]{S_3-eps-converted-to.pdf}}
\caption{Controller partitions projected on subspace $[\tilde{x}_1,\tilde{x}_2]$ and the state trajectory with (a) Scheme I, (b) Scheme II (cut at $\sum{\tilde{x}_1}=\sum{\tilde{x}_2}=0$) and (c) Scheme III (cut at $\sum{\tilde{x}_1}=\Delta \tilde{x}_1=0$).}
\label{state}
\end{figure}
From the state response in Fig. \ref{stateinput}, we saw that the scheme I could not negate the additive disturbance happened either at an active constraint region or at steady state. It resulted in offset $\tilde{x}=\begin{bmatrix}0.06& -0.06\end{bmatrix}^T$ and $\tilde{x}=\begin{bmatrix}-0.05& -0.45\end{bmatrix}^T$, respectively. The scheme II could track both the state variables but with significant overshoot due to the regulation of $\sum{\tilde{v}}$ back to 0. That effect could be removed by tracking it to a constant (as a tuning parameter), but ignored in this example for simplicity. The scheme III tracked $x_1$ as required, and successfully forced the disturbance effect into $x_2$. The tracking under setpoint change and disturbance rejection also happened faster than scheme II. We stressed that all the three schemes were able to deal with the state and input constraints $x_1\leq 1.5$, $u\leq 3$ during transient stage because of the feedforward term $g^i$ in the parametric MPC law.
Fig. \ref{state} gave another perspective of the result. Provided that the impulse disturbance did not excite the current state out of the feasible region $X_0$, it was feasible to find an optimal input for all the three schemes. Secondly, scheme II hit on the outer constraint $\tilde{x}_1=-0.5\, (x_1=1.5)$ and took a long time to recover. Indeed, an integral windup happened at this upper output bound. The scheme III showed the full PID potential. It is known that the proportional-integral deals with the present and past behavior of the plant. The differential term predicts the plant behavior and can be used to stabilize the plant faster. This was in line with Remark \ref{rmk2}. The trajectory quickly returned to the origin in both cases of setpoint change and additive disturbance. Lastly, while scheme II regulated the state error back to the origin, scheme III only drove it to the axis $\tilde{x}_1=0$ as expected. It meant only $m\times q$ PIDs and $m\times (n-q)$ Ps were needed to track $q$ outputs.
In conclusion, it is observed that as long as the disturbance does not drive the equilibrium outside of the unconstrained region, output tracking using the integral state variables remains feasible. The robust stability during transient stage is inherent through the PID form. The robust stability around setpoint only concerns the PID control design described in Section \ref{PIDsec}, which can be improved further by $H_\infty$ approaches as stated in Remark \ref{rmk3}. Overall, extension to integral and differential terms is the natural to perform tracking control.
\section{Conclusion and Future work}
As it was never emphasized enough, the link between MPC and a robust linear controller at equilibrium is revisited in this paper. We modify the linear controller to be capable of offset-free tracking. The resultant control architecture is a PID gain scheduling network with a feedforward part to deal with state and input constraints. A simple test for setpoint tracking feasibility is also discussed. Finally, the example results show that the robustness stability of the proposed method is inherent within the PI/PID structure when disturbances arrives.
\bibliographystyle{elsarticle-harv}
|
1,941,325,220,584 | arxiv | \subsection*{Abstract}
Repackaging is a technique that has been increasingly adopted by authors of Android malware. The main problem facing the research community working on devising techniques to detect this breed of malware is the lack of ground truth that pinpoints the malicious segments grafted within benign apps. Without this crucial knowledge, it is difficult to train reliable classifiers able to effectively classify novel, out-of-sample repackaged malware. To circumvent this problem, we argue that reliable classifiers can be trained to detect repackaged malware, if they are allowed to request new, more accurate representations of an app's behavior. This learning technique is referred to as \emph{active learning}.
In this paper, we propose the usage of active learning to train classifiers able to cope with the ambiguous nature of repackaged malware. We implemented an architecture, {\fontfamily{cms}\selectfont Aion}, that connects the processes of stimulating and detecting repackaged malware using a feedback loop depicting active learning. Our evaluation of a sample implementation of {\fontfamily{cms}\selectfont Aion} using two malware datasets (\emph{Malgenome} and \emph{Piggybacking}) shows that active learning can outperform conventional detection techniques and, hence, has great potential to detect Android repackaged malware.
\section{Introduction}
\label{sec:intro}
Malware authors targeting Android have recently been adopting a more sophisticated technique to develop and distribute their malicious instances. In essence, they leverage the ease of decompiling, modifying the source code of, and recompiling Android applications (hereafter apps) to repackage legtimate, trusted apps with malicious payloads \cite{wei2017deep,li2017understanding,pan2017dark,tam2017evolution,zhou2012dissecting}. This breed of malware is often referred to as either \emph{piggybacked} apps \cite{li2017understanding, li2017automatically} or \emph{repackaged} malware \cite{zhou2012detecting, zhou2012dissecting}. In this paper, we use the two terms interchangeably to refer to the same type of Android malware: legitimate apps that have been grafted with malicious payloads.
Repackaged malware can be thought of as an evolution of Trojan horses. The two malware breeds differ in one key aspect viz., the originality of the app's benign functionality. On one hand, Trojan horses usually comprise benign functionalities that have been developed from scratch by the malware author for the sole purpose of presenting the app as a benign one. Furthermore, malware authors of Trojan horses seldom invest adequate amounts of time and effort in developing such fake, benign facade. The resulting benign segment is, therefore, of low quality and limited functionality (e.g., a Sudoku app with five puzzles). On the other hand, repackaged malware comprise pre-existing legitimate apps (e.g., Angry Birds), that have been amalgamated with a newly injected malicious payload (e.g., sending SMS messages to premium rate numbers \cite{fratantonio2016triggerscope}).
The primary threat that repackaged malware poses is undermining user trust in legitimate apps, their developers, and the app distribution infrastructure, which can potentially have devastating effects on the entire Android ecosystem. Unfortunately, malware authors have been increasingly adopting repackaging particularly targeting third-party marketplaces as their distribution platform. In \cite{zhou2012dissecting}, Zhou et al.\ studied 1260 Android malware instances, and concluded that more than 86\% of them were repackaged. In 2015, TrendMicro reported that 77\% of the top free 50 apps on the Google Play marketplace had fake versions, with a total of 890,482 fake apps being discovered 51\% of which were found to exhibit unwanted and/or malicious behaviors \cite{luo+2014}. More recently, Li et al.\ managed to gather piggybacked versions of around 1,400 legitimate apps \cite{li2017understanding}.
Consequently, the research community has been working towards devising methods to detect Android repackaged malware. To the best of our knowledge, the majority of such methods analyze, and perhaps execute, apps belonging to a dataset (e.g., \emph{Malgenome} \cite{zhou2012dissecting}), to extract numerical features used to train a machine learning classifier. The trained classifier is used to classify out-of-sample (hereafter test) apps as malicious and benign \cite{tam2017evolution,pan2017dark,arshad2016android}. One fundamental problem facing the research community working on detecting repackaged malware is the quality of the ground truth of the currently available datasets
That is to say, a repackaged app is merely labeled as malicious without the knowledge of which paths depict the grafted malicious behaviors, whether there are triggers that control the execution of such behaviors, and how sophisticated those triggers are. Lacking such knowledge hinders training reliable classifiers based on the apps' dynamic behaviors, which is necessary given that malware instances have increasingly been adopting techniques (e.g., obfuscation, dynamic code loading, usage of triggers, etc.), to evade detection by conventional static-based methods \cite{wei2017deep,li2017static,rasthofer2017making,zhauniarovich2014}.
We argue that, in essence, the problem of stimulating, analyzing, and detecting Android repackaged malware is a \emph{search problem}.
During the phase of training a classifier, we are in pursuit of the segments in the training apps' source code (e.g., paths in a CFG), that depict or include the injected malicious code. Finding and executing those segments facilitates the extraction of features that resemble their runtime behavior, and enables the classifier to have a collective, accurate view of the malicious behaviors exhibited by Android repackaged malware. Similarly, to reliably classify a test app, we need to execute multiple paths to increase the probability of finding (a representation of) the injected code prior to deciding upon the app's malignance.
Needless to say, the nature and locations of the grafted malicious code in repackaged apps might vary. Given that we lack such ground truth, the training and test processes need to adopt a \emph{trial-and-error} technique.
In the training phase, for example, if a feature vector ($\hat{x_i}$) representing the runtime behavior of a training app ($a_i$) is misclassified, we assume that such feature vector represents the execution path that does not reflect the app's true nature (i.e., malicious or benign). Thus, the app ($a_i$) needs to be re-executed to explore another path that yields a feature vector ($\hat{x^{\prime}_i}$) that might help the classifier assign the app to its proper class. This process can be iterated until the maximum training accuracy is achieved, which signals the best possible representation of benign and malicious behaviors embedded in the training apps.
The number of iterations needed to achieve such maximum accuracy depicts the number of times an app needs to be executed in order to have a comprehensive overview of its behavior. For repackaged malware, it can be interpreted as the number of executions within which the malicious code is likely to execute. The feature vectors corresponding to the different executions can be used to classify the app (e.g., using majority votes), as malicious or benign.
Within the context of machine learning, the aforementioned trial and error technique is referred to as \emph{active learning}. In this paper, we propose, implement, and evaluate an active learning architecture, called {\fontfamily{cmss}\selectfont Aion\footnote{Aion is a Greek deity associated with eternity and everlasting time. His orb, depicting repetition, resembles the behavior of our proposed active learning approach, where a feedback loop closes a circle between the processes of stimulation and detection of Android repackaged malware.}} \cite{aion2017}, to stimulate, analyze, and detect Android repackaged malware.
Our findings show that--even with primitive stimulation and classification techniques--training a classifier using active learning improves the performance of conventional dynamic-based detection methods on test datasets and can outperform their static-based counterpars. Furthermore, the conducted experiments helped us answer questions about the number of iterations needed to train the best performing classifier, the most suitable kind of features, and the type of the best performing classifier (i.e., Random Trees, K-Nearest Neighbors, Ensemble, etc.). We believe that our experiments provide insights on how to stimulate and detect Android repackaged malware using active learning, and a benchmark against which further enhancements of the architecture (e.g., via using more sophisticated stimulation techniques) can be compared.
The contributions of this paper are:
\begin{itemize}
\item We propose a novel architecture and approach to stimulate, analyze, and detect Android repackaged malware using active learning and investigate its applicability and performance.
\item We implemented a framework that utilizes the proposed architecture, and made the source code, the data (i.e., gathered during evaluation), and a multitude of interactive figures plotting our results available online \cite{aion2017}.
\item We evaluate {\fontfamily{cms}\selectfont Aion} on the recently analyzed and released \emph{Piggybacking} dataset instead of its obsolete \emph{Malgenome} counterpart, and use the former to set a classification benchmark of 72\%, similar to Zhou et al.'s 79.6\% benchmark on the latter in November 2011.
\end{itemize}
\noindent
\textbf{Organization} The rest of the paper is organized as follows. In section \ref{sec:background}, we briefly discuss fundamental concepts on which our research is built. In section \ref{sec:design} the methodology adopted during the implementation of {\fontfamily{cmss}\selectfont Aion}, its architecture, and internal structure are introduced. In section \ref{sec:evaluation}, we discuss how {\fontfamily{cmss}\selectfont Aion} was evaluated by introducing the datasets we used and the experiments conducted, and discuss the observed results in section \ref{sec:discussion}. Research efforts related to our work are enumerated in section \ref{sec:related}. Lastly, section \ref{sec:conclusion} concludes the paper.
\section{Background}
\label{sec:background}
In this section, we briefly discuss concepts fundamental to follow the remainder of the paper. We discuss our definition of repackaged malware and what it comprises, and introduce the notion of active learning and relate it to our work.
\subsection{Repackaged Malware}
\label{subsec:repackaged}
Android apps can be easily disassembled, reverse engineered, modified, and reassembled \cite{bartel2012improving}. Consequently, repackaging benign apps with malicious code segments is a straightforward process that typically involves the following activities. Firstly, a reverse engineering tool, such as {\fontfamily{lms}\selectfont Apktool} \cite{apktool2017}, is used to disassemble an app into a readable, intermediate representation (i.e., Smali). Secondly, a segment of Smali code that delivers some malicious functionality is added and merged with the original Smali code. Lastly, the app is reassembled, re-signed, and uploaded to an app marketplace, possibly with a different name
The malicious code segments injected into benign apps usually comprise two parts viz., a malicious payload and a trigger (or a hook \cite{li2017understanding}). The former part depicts the malicious intents of the malware author (e.g., sending SMS messages to premium numbers, deleting the user's contacts, leaking the current user's GPS location, etc.). In \cite{fratantonio2016triggerscope}, Fratantonio et al.\ argue that a malicious payload need not be disconnected from the original app logic; it can distort the actual app functionality, in what they refered to as logic bombs.
The trigger is meant to transfer control to the grafted malicious payload. We can define the trigger as a set of (boolean) conditions that need to hold in order for the malicious payload to execute. For authors of repackaged malware, designing a trigger is a compromise between the likelihood of execution and stealthiness of the injected malicious payload \cite{li2017understanding}. On one hand, some malware authors may elect to maximize the likelihood of the malicious payload being executed by unconditionally calling/branching to the malicious code (e.g., deployed in a separate method or component) \cite{li2017understanding}. In this case, the trigger is a \emph{null} condition. On the other hand, some authors may give stealthiness more priority, and develop more complex triggers that activate the grafted malicious functionality depending on specific system properties (e.g., date, time, GPS location, etc.), app/system intents and notifications (e.g., android.intent.action.BOOT\_COMPLETED), custom values forwarded to the app via its authors (e.g., via SMS messages \cite{rasthofer2015current}), or app logic (e.g., $if(sum\_of\_calculation==50)$). This trend of scheduling the triggering of malicious segments has been found to be adopted by the majority of Android malware \cite{wei2017deep}.
\subsection{Active Learning}
\label{subsec:activelearning}
To the best of our knowledge, the majority of Android malware detection techniques adopt the following process. Firstly, a dataset of benign and malicious apps are gathered. Secondly, the apps in the dataset are analyzed to extract numerical features from each app. That is to say, each app is represented by a vector of features ($x_i$) and a label ($y_i$) depicting its nature (i.e., malicious or benign). The extracted feature vectors, often organized in a matrix representation ($X$), are used to train a machine learning classifier ($C$) (e.g., a support vector machine). Given a feature vector of an app ($x^*$) that has not been used during the training phase (i.e., test app), the classifier ($C$), attempts to predict its corresponding label ($y^*$) effectively classifying it as malicious or benign. To asses the prediction capability of a classifier, the original labels of the test apps, known as \emph{ground truth}, are compared to those predicted by the classifier. The percentage of correctly predicted labels is known as the classification \emph{accuracy}.
Needless to say, a number of the test apps' labels are incorrectly predicted (i.e., the corresponding app is misclassified), which negatively affects the classification accuracy. If the classification accuracy is inplausible, other classifiers can be probed, the existing features extracted from the apps are further processed (e.g., to eliminate noisy, irrelevant features), a new method of feature extraction is adopted, or a combination of those methods. This setting is referred to as \emph{passive learning} \cite{tong2001active}. The passiveness dwells in the inability of the trained classifier to ask for more accurate representations of the misclassified apps in the form of different feature vectors.
In an active learning setting, the classifier is allowed to \emph{query} the sources from whence the feature vectors have been extracted for alternative representations of the same entities \cite{tong2001active}. Within the context of Android malware detection, if a feature vector of a test app ($x^*$) is misclassified, the classifier ($C$) is allowed to instruct the feature extraction mechanism to generate another feature vectors ($x^*_{new}$) for the same app. This process can be repeated until either the misclassified app is correctly classified, or the overall classification accuracy of ($C$) has converged to a maximum value.
\section{Design and Implementation}
\label{sec:design}
\subsection{Overview}
The typical malware detection process introduced in the previous section can be grouped into two main sub processes viz., \emph{Data Generation} and \emph{Data Inference}, as seen in figure \ref{fig:architecture} which depicts our proposed architecture {\fontfamily{cmss}\selectfont Aion}.
The former process commences with the acquisition of a dataset of malicious and benign Android apps. Using a \emph{Stimulation} method, the collected apps are analyzed statically and executed within a controlled, virtualized environment to gather information about the apps' runtime behavior. Within the domain of malware analysis, there are different non-/invasive approaches to stimulating an app \cite{xue2017malton}. For example, Abraham et al.\ \cite{abraham2015} implemented an approach meant to force the execution of particular branches within an app that contain suspicious API calls, whereas Rasthofer et al.\ \cite{rasthofer2017making} utilized symbolic execution to devise environments (i.e., a set of values) required to execute branches containing specific type of API calls. The monitoring component depicted in the figure is responsible for recording the the stimulated behavior for further analysis. It can be deployed on a remote server, on the environment within which an app is executed, embedded within the app itself (e.g., as logging statements), or a combination of those techniques. Lastly, the monitored behavior may need to be further processed to represent it in a parseable, understandable format. For example, the API calls issued by an app during runtime are usually reorganized in the format of comma-separated strings (i.e., traces)
The stimulated, monitored, and reconstructed representations of the apps' behaviors are usually raw, and need to be further processed before they are used for training a classifier. Data Inference is concerned with inferring relevant information from the raw data received from its Data Generation counterpart that might facilitate segregating the two classes of apps. The \emph{Projection} component is reponsible for processing the raw data and projecting it into a different representation or dimensionality. For example, a trace of API calls can be processed to omit those calls that do not manipulate sensitive system resources (e.g., camera) or extract numerical features (e.g., counts of different API calls). To further remove noise from projected data, the \emph{Inference} component attempts to extract patterns and information that might facilitate the classification of apps. Depending on the format of the projected data, this component can either apply feature selection techniques to rule out irrelevant features, or infer rules that depict characteristics unique to a class of apps. For instance, by studying the API call traces of apps, a rule can be inferred that malicious apps usually use a particular set/sequence of API calls. Lastly, the \emph{Training} component handles training a classifier using the processed raw data, validates the results (e.g., via a test dataset), and reports classification results.
The two dashed arrows looping out of and into the two processes imply that the operations within such processes can be perfomed multiple times before the process reports its final output. For example, the Data Generation process can continue to add new apps to its repository, stimulate them multiple times, monitor their runtime behaviors, and report an augmented version of the different stimulation sessions. Similarly, the Data Inference process is allowed to consult various combinations of projection, inference, and classification techniques in pursuit of the best classification accuracy.
\begin{figure}
\includegraphics[scale=0.75]{figs/architecture_functional_1}
\caption{An architecture that uses active learning to stimulate, analyze, and detect Android repackaged malware.}
\label{fig:architecture}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.6]{figs/cfg}
\caption{An example of a repackaged malware's control flow graph (CFG). The colored segment depicts the malicious payload injected into the app, whereas the dashed arrows represent different paths through the CFG.}
\label{fig:cfg}
\end{figure}
Lastly, the dashed arrow extending from the Data Inference process to its Data Generation counterpart depicts the active learning aspect of our proposed approach. We illustrate the functionality and significance of such mechanism using the control flow graph (CFG) in figure \ref{fig:cfg}. Consider such CFG as that of a benign app ($a_i$) that has been injected with malicious segments (colored blocks). Regardless of the technique it employs to target and execute specific branches, the stimulation component should devise test cases that execute the path ($P_2$) which includes the malicious segments. The feature vectors extracted from the representation of ($P_2$) (e.g., API call trace) should depict the malicious behavior grafted into the benign app and, in turn, help the trained classifier assign this app to the malicious class. Nevertheless, since the location of the injected malicious payload is unknown, and since the malicious payload is assumed to be smaller than the original benign code, the stimulation component is likely to target other branches/statements within the CFG, effectively executing other paths (i.e., $P_1$ or $P_3$). Feature vectors extracted from such paths are expected to represent benign behaviors, leading to the misclassification of the app. The feedback mechanism reports misclassification of ($a_i$) to the Data Generation process, particularly the stimulation component. Using this information, the latter attempts to target different branches/statements to execute different paths within the app's CFG, and forward the reconstructed behaviors of such newly-executed paths to the Data Generation process for re-training. In theory, ($a_i$) will continue to be misclassified until ($P_2$) is executed, which is expected to help the trained classifier deem the app as malicious. Consequently, the feedback mechanism will continue to instruct the stimulation component to target different code segments until such path is executed.
Given a dataset of training apps, the feedback mechanism is meant to provide the Data Generation module with the best representation of each app that resembles its nature (i.e., malicious or benign). A classifier trained with such representations is expected to possess a comprehensive view of the malicious and benign behaviors exhibited by the apps within the training dataset
However, the large combination of different locations of the injected malicious payloads, their contents and behaviors, and their triggers significantly impedes the process of executing multiple paths per app. In other words, it can take an unforeseeable amount of time to find the best representation of each app in the training dataset. Furthermore, achieving a perfect training accuracy is unlikely, especially since the behaviors of some apps can mimick those of apps belonging to the other class. That is to say, classification errors are inevitable. In this context, we design the feedback process to terminate once the best possible training accuracy is achieved, which is subject to two constraints. Firstly, the feedback process continues as long as the performance of the classifier is increases across two iterations or decreases by a small percentage (e.g., 1\%). Secondly, we specify a maximum number of iterations, after which the training process is terminated.
\subsection{Implementation}
\label{subsec:implementation}
The exact methods, techniques, and algorithms used by different components in the two processes of Data Generation and Data Inference can vary. In other words, the proposed architecture is completely agnostic to specific methods utilized by different components. However, the use of different stimulation approaches might affect the implementation of the Data Inference components and the feedback mechanism. In this paper, we focus on demonstrating the proposed architecture and investigating the applicability of active learning to the problem at hand. Consequently, we adopt an example in which we use basic techniques for stimulation, detection, and feedback mechanisms, and leave the utilization of more sophisticated stimulation/detection methods for future work.
The stimulation technique used in this paper is based on a non-invasive, UI-based tool, called {\fontfamily{lms}\selectfont Droidutan} \cite{droidutan2017}, which is designed to emulate the user-app interaction. The tool starts the main activity of an app, retrieves its UI elements, chooses a random element out of the retrieved ones, and interactes with it. The interaction hinges on the class of the chosen UI element. For example, if the UI element is a \emph{Button}, {\fontfamily{lms}\selectfont Droidutan} will tap it, if it is a \emph{TextField}, a random text will be typed into it, and so forth. The tool replicates the same behavior with any activity that may start during the stimulation period. To simulate the occurrence of system notifications or inter-app communication, the tool randomly broadcasts intents declared and registered to by the app in its manifest file
Interacting with an app generates a runtime behavior that we define in terms of API calls.
We use {\fontfamily{lms}\selectfont droidmon} as our monitoring component to intercept and record a particular set of API calls that interact with system resources such as the device's camera, the user's contacts, the package installer, the GPS module, etc.
The tool records the intercepted calls along with their arguments and return results, and writes them to the system log.
For each app, we retrieve the API calls from the log and represent the app's behavior as a series of calls, each having the format $[package.module.class].[method]$.
The APK archives of the apps under test and their corresponding traces retrieved from {\fontfamily{lms}\selectfont droidmon}'s output are used to extract static and dynamic features, respectively.
After surveying the literature, we concluded that static features can be categorized into basic metadata features extracted from the AndroidManifest.xml file \cite{sato2013detecting}, features related to the permissions required by the app \cite{sanz2013puma, huang2013performance}, and features related to the API calls found within the app's code \cite{wu2012droidmat}.
For each app, we extract static features depicting the aforementioned three categories. We refer to those features as \emph{basic}, \emph{permission}, and \emph{api} features, and we list them in appendix \ref{app:appendixa}.
The dynamic features extracted from each app's API calls trace is a feature vector of 37 attributes, each of which depicts the number of times a method belonging to one of the API packages hooked by {\fontfamily{lms}\selectfont droidmon}, listed in appendix \ref{app:appendixb}, has been encountered in the trace.
The static and dynamic feature vectors extracted from the apps' APK's and their traces are used to train a majority vote classifier (i.e., Ensemble classifier), using a total of 12 classifiers. The classifiers used to train the Ensemble classifier are K-Nearest Neighbors (KNN) with values of $K=\lbrace 10,25,50,100,250,500\rbrace$, random forests with values of $Estimators\ (E)=\lbrace 10,25,50,75,100\rbrace$, and a Support Vector Machine (SVM) with the linear kernel.
We assess the performance of different classifiers according to two metrics viz., F1 score and specificity. The first metric is defined as $F1=2\times \frac{precision\times recall}{precision+recall}$ such that precision $= \frac{TP}{TP+FP}$ and recall $=\frac{TP}{TP+FN}$. True positives (TP) denote malicious apps correctly classified as malicious, false positives (FP) denote benign apps mistakenly classified as malicious, and false negatives (FN) denote malicious apps mistakenly classified as benign. Thus, the F1 score is meant to assess the ability of a classifier to recognize malicious apps and correctly classify them. We also keep track of the classifier's performance on benign apps (i.e., whether it classifies them correctly as benign), via specificity $=\frac{TN}{TN+FP}$ where (TN) stands for true negatives: benign apps correctly classified as benign. We keep track of these two metrics to assess whether a classifier is biased towards one class in favor of the other. For example, if a classifier scores high F1 and low specificity scores, this signals its bias towards classifying the majority of apps as malicious. Detecting a classifier bias can be used to amend the methods adopted to train a classifier (e.g., the type of extracted features).
The apps misclassified by the Ensemble classifier should be, according to our proposed approach, re-stimulated to execute different segments within their code yielding different paths, different traces of API calls and, consquently, different feature vectors.
Given that {\fontfamily{lms}\selectfont Droidutan} is a random-based testing tool, each run of an app is likely to execute a different path.
Consequently, the feedback mechanism in this implementation of {\fontfamily{cms}\selectfont Aion} comprises merely re-running an app.
To ensure that random stimulation results into the execution of different paths, we kept track of the API traces recorded by {\fontfamily{lms}\selectfont droidmon} for each app.
We averaged the number of API calls that change in an apps trace across two consecutive stimulations, and found out that an average of 60 API calls (disregarding their arguments and return values) change with every stimulation.
\section{Evaluation}
\label{sec:evaluation}
In this section, we evaluate the proposed active learning based approach using two datasets of real world Android malware. The main hypothesis of this paper is that active learning enhances the performance of classifiers trained to effectively detect Android repackaged malware. Consequently, we aspire to answer the following research questions:
\noindent \textbf{Q1:} How effective and efficient are the classifiers trained using active learning in comparison to conventional analysis and detection techniques?\\
\noindent \textbf{Q2:} How reliable are the results achieved by such classifiers?\\
\noindent \textbf{Q3:} How well do active learning classifiers generalize on test datasets?\\
\noindent \textbf{Q4:} What is the type of features (e.g., static versus dynamic) and classifiers that yield the best detection accuracies on test datasets?\\
\subsection{Datasets}
We used two datasets of malicious and benign Android apps to evaluate our approach. The malicious and benign apps of the first dataset were gathered separately. The malicious apps of such dataset belong to the \emph{Malgenome} dataset \cite{zhou2012dissecting}. Malgenome originally comprised of more than 1200 malicious apps, almost 86\% of which were found to be repackaged malware instances. Prior to being discontinued in 2015, Malgenome was considered the de facto dataset of Android repackaged malware. Fortunately, the repackaged malware belonging to this dataset continue to exist within the \emph{Drebin} dataset \cite{arp2014drebin}, from whence we acquired them. To complement the first dataset with benign apps, we randomly selected and downloaded 1800 apps from Google Play store in December 2016. Consequently, we refer to this dataset as \emph{Malgenome+GPlay} in the following sections.
The second dataset, \emph{Piggybacking} \cite{li2017understanding}, is more recent and comprises around 1300 pairs of original apps along with their repackaged versions. The process of gathering the apps, labeling them as malicious and benign, and matching apps to their repackaged counterparts has been carried out between 2014 and 2017. The dataset we used contains 1355 original, benign apps and 1399 repackaged, malicious apps. The reason behind such imbalance is that some original apps have more than one repackaged version.
Lastly, the two datasets were used separately during evaluation. Consequently, some of the experiments we conducted have two variants (i.e., one for each dataset).
\subsection{Experiments}
To answer the aforementioned research questions, we designed two sets of experiments. The first set of experiments is meant to decide upon the datasets to use in evaluating the proposed active learning approach, and to set a performance benchmark against which the proposed approach is to be compared. The benchmark is set using results achieved from conventional analysis and detection methods that have been previously utilized to detect Android (repackaged) malware. We refer to this set of experiments as preliminary experiments. The second set of experiments evaluates the performance of our proposed active learning approach.
\begin{figure}
\centering
\includegraphics[scale=0.75]{figs/experiment_vertical}
\caption{The workflow of our experiments. Ellipses depict inputs and outputs, rectangles depict operations performed on such inputs and outputs, and rhombi depict boolean decisions. The edges between different nodes indicate the directionality of data flow. Lastly, the blue numbers are used as references to different stages of the experiment. Preliminary experiments, whether static or dynamic, conclude after stage (5) and are, hence, highlighted using the dashed gray box.}
\label{fig:experiment}
\end{figure}
Both experiment sets share a common workflow, seen in figure \ref{fig:experiment}. All experiments commence with randomly splitting a dataset of APK's into a training dataset ($T_R$) comprising two thirds of the total number of APK's and a test dataset ($T_E$) comprising the remaining one third. The second stage initializes different variables including an initial F1 score of $-1$ and a variable to keep track of the current iteration's number (i.e., $t$). The third stage is responsible for extracting a vector of numerical features ($\hat{x_i}$) for each APK ($a_i$) in both the training and test datasets ($T_R\cup T_E$). The method used to extract such numerical features hinges on the type of the experiment run. That is to say, if the experiment is \emph{dynamic}, then the APK's will be installed on an Android Virtual Device (AVD), run using {\fontfamily{lms}\selectfont Droidutan}, and monitored using {\fontfamily{lms}\selectfont droidmon}. However, if the experiment is \emph{static}, the APK's will be statically analyzed using {\fontfamily{lms}\selectfont androguard}. Note that for dynamic experiments, {\fontfamily{lms}\selectfont droidmon} may fail to produce API traces for an unforeseeable number of APK's due to either (a) the app crashing during runtime, or (b) the app not using any of the API calls monitored by the tool. The third phase concludes with producing feature vectors of numerical features for the APK's in ($T_R$) and ($T_E$). The feature vectors are stored in two datasets ($|X_R|\leq |T_R|$) and ($|X_E|\leq |T_E|$). Stage four uses the feature vectors in ($X_R$) to train a majority vote classifier ($E_t$: Ensemble classifier trained at iteration $t$), as discussed in section \ref{subsec:implementation}. In stage five, the trained classifier is validated using the vectors from ($X_R$) and tested using the vectors from ($X_E$) yielding corresponding training F1 score ($F1_{R_t}$) and test F1 score ($F1_{E_t}$).
Preliminary experiments conclude after stage five, storing the results into a SQLite database for future study. Active learning experiments compare the training accuracy ($F1_{R_t}$) scored at the current iteration ($t$) to that scored at the previous iteration ($F1_{R_{t-1}}$).
Test accuracies are not used in such comparison to prevent the test dataset (i.e., containing out-of-sample apps), from affecting the training process.
This comparison always evaluates to true after the first iteration, especially since the training F1 score is initialized to $-1$. For values of ($t> 1$), if the absolute difference between ($F1_{R_t}$) and ($F1_{R_{t-1}}$) is less than or equal to a threshold percentage ($E$) (default is $1\%$), the misclassified feature vectors will be re-run in a manner similar to stage three. Intuitively, we continue running the experiment as long as the current training F1 score ($F1_{R_t}$) is increasing. In other words, there is a potential for achieving better training, and perhaps test, scores. Furthermore, in order to accommodate for fluctuations in training accuracies, we allow the current training F1 score ($F1_{R_t}$) to drop for a maximum of ($E$). This means, however, that the experiment could go on for a prolonged period of time. To avoid this behavior, we set an upper bound ($t_{MAX}$) for the number of iterations the training phase is allowed to run. In evaluating our approach, we use the values of $10$ and $1\%$ for ($t_{MAX}$) and ($E$), respectively.
Regardless of the experiment set, a dataset is randomly split into training and test datasets at the beginning of each experiment. The training and test datasets persist throughout the experiment (i.e., through all iterations). Consequently, we run all experiments at least 25 times to increase the likelihood of different segments of the datasets being used as, both, training and test samples. Throughout implementing and evaluating {\fontfamily{cms}\selectfont Aion}, we generated a multitude of figures, which we could not include in this paper. We host and make available all our (interactive) figures, data, and source code on \cite{aion2017}.
\subsubsection{Preliminary Static Experiments}
\label{sec:exp_preliminary_static}
\begin{figure*}
\centering
\begin{subfigure}{0.49\textwidth}
\scalebox{.55}{\input{figs/exp_preliminary_static/Malgenome_static_F1Score_TEST_scatter.pgf}}
\caption{Malgenome+GPlay median F1 scores}
\label{fig:exp_pre_static_malgenomef1}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\scalebox{.55}{\input{figs/exp_preliminary_static/Piggybacking_static_F1Score_TEST_scatter.pgf}}
\caption{Piggybacking median F1 scores}
\label{fig:exp_pre_static_piggybackingf1}
\end{subfigure}
\caption{The median (after 25 runs) F1 scores (Y-axis) recorded by 13 classifiers (X-axis) using static features on the test datasets. The specificity scores are plotted in appendix \ref{app:appendixc}.}
\label{fig:exp_pre_static_f1}
\end{figure*}
In this set of experiments, only the three categories of static features, introduced in section \ref{subsec:implementation}, are used to classify apps as malicious and benign. That is to say, each app is represented by three feature vectors depicting such categories. Moreover, we combine the three feature vectors to come up with a fourth category of static features that we refer to as \emph{All}.
The scores in figure \ref{fig:exp_pre_static_f1} depict the median F1 scores recorded by different classifiers using all four categories of static features on the Malgenome+GPlay dataset and its more recent counterpart Piggybacking. To preserve space for more thorough discussion, we opted to move the specificity scores to appendix \ref{app:appendixc}.
We speculated that solely using static features to classify repackaged malware would yield mediocre classification accuracies, especially since repackaged malware usually poses as benign apps.
Nevertheless, as seen in figure \ref{fig:exp_pre_static_malgenomef1}, most classifiers perform well on the Malgenome+Play using static features, apart from permission-based features, which apparently are not informative enough to separate malicious and benign apps in this dataset.
We noticed, however, that the classifiers comparably underperform in terms of, both, the F1 and specificity scores upon applying the same method to classify apps in the Piggybacking dataset.
We believe there are two reasons behind such noticeable difference. Firstly, the malicious apps in our hybrid Malgenome+GPlay dataset have been gathered between August 2010 and October 2011 \cite{zhou2012dissecting}, whereas their benign counterparts were gathered in December 2016. Given that the Android ecosystem is continuously evolving, any two apps developed 5 years apart might look very different courtesy of newly-introduced technologies, programming interfaces, or even hardware capabilities. In fact, we noticed that there is a large difference between the sizes of APK archives belonging to apps in the Malgenome dataset (average size: 1.25MB) and those belonging to the benign apps we gathered from Google Play (average size: 16.35MB).
This difference implies, we believe, the existence of more components (e.g., activities and their classes) in recent Android apps.
Static features are built on counts and ratios of those components, the permissions they request, and the API calls they issue. Needless to say, the feature vectors extracted from apps developed in 2011 would look entirely different from those extracted from apps developed in 2015 or 2016, effectively facilitating segregating them in two classes which happen to be malicious and benign.
Secondly, despite being defined as repackaged malware, the malicious apps in Malgenome do not comply with the more recent definition of repackaged/piggybacked apps. In other words, we believe that the 86\% of apps comprising repackaged malware in the Malgenome dataset are, in fact, closer to being classic Trojan horses than being repackaged malware. The malicious apps in the Piggybacking dataset, however, comply with the definition of repackaged malware (i.e., standalone benign apps grafted with malicious payloads \cite{li2017understanding}). To empirically verify this claim, we used a compiler fingerprinting tool, {\fontfamily{lms}\selectfont APKiD}, to identify the compilers used to compile the apps in both datasets Malgenome+GPlay and Piggybacking along with a dataset of malicious apps called \emph{Drebin} \cite{arp2014drebin}. The distribution of compilers utilized by apps in such datasets is tabulated in table \ref{tab:apkid_compilers}. Using the list of compilers, we rely on the following hypothesis, originally made in \cite{rednaga2016}: if the compiler used to compile an APK is not the standard Android SDK compiler (dx) but rather a compiler used by reverse engineering tools (e.g., {\fontfamily{lms}\selectfont Apktool}) such as (dexlib), then an app is probably repackaged by a party other than the legitimate developers in possession of the app's original source code.
\begin{table}
\centering
\caption{Percentages of compilers used to compile APK's in different malicious/benign datasets as fingerprinted by {\fontfamily{lms}\selectfont APKiD}. The \emph{dexmerge} compiler is used by IDE's after \emph{dx} for incremental builds.}
\label{tab:apkid_compilers}
\begin{small}
\begin{tabular}{ccccc}
\hline
\textbf{Dataset} & \textbf{dx} & \textbf{dexmerge} & \textbf{dexlib 1.X/2.X} & \textbf{Total} \\ \hline
Drebin & 84\% & -- & 16\% & 4326 \\ \hline
Malgenome & 52\% & -- & 48\% & 1234 \\ \hline
Google Play & 61\% & 34\% & 5\% & 1882 \\ \hline
\begin{tabular}[c]{@{}c@{}}Piggybacking\\ (malicious)\end{tabular} & 22\% & 6\% & 72\% & 1399 \\ \hline
\begin{tabular}[c]{@{}c@{}}Piggybacking\\ (original)\end{tabular} & 61\% & 22\% & 17\% & 1355 \\ \hline
\end{tabular}
\end{small}
\end{table}
As seen in table \ref{tab:apkid_compilers}, the majority benign apps gathered from Google Play and the original apps in the Piggybacking dataset have been compiled using a compiler usually used within IDE's. This implies that the developers were likely in possession of the apps' source code. The same applies to the Drebin dataset which mainly comprises of malicious apps developed from scratch. However, the datasets that presumably comprise repackaged malware indicate a major difference. More than a half of the Malgenome dataset comprises apps that--according to the definitions in sections \ref{sec:intro} and \ref{sec:background}--are Trojans, whereas the majority of malicious apps in the Piggybacking dataset comply with the notion of a benign app being grafted with malicious payload and recompiled using non-standard compilers.
Such findings led us to deem the Malgenome dataset as obsolete and inaccurate.
In fact, during their analysis of a dataset of 24,650 Android malware samples, Wei et al.\ also deemed the Malgenome dataset outdated and not representative of the current Android malware landscape \cite{wei2017deep}.
Thus, devising methods to analyze and detect malicious apps in the Malgenome dataset has little benefit to the community.
Consequently, we opted to conduct the remainder of our experiments on the Piggybacking dataset, and only consider the static scores recorded on such dataset as our benchmark.
\subsubsection{Preliminary Dynamic Experiments}
\label{sec:exp_preliminary_dynamic}
The preliminary dynamic experiments are meant to resemble conventional dynamic approaches to stimulating and detecting Android repackaged malware. That is to say, apps are stimulated once prior to being processed for feature extraction and classification. In essence, such experiments depict the first iteration of our active learning experiments (i.e., without the feedback loop). The features used during such experiments are the dynamic features introduced in section \ref{subsec:implementation} along with a combination of the \emph{All} static and dynamic features that we refer to as \emph{hybrid features}.
Figure \ref{fig:exp_pre_dynamic_f1} depicts the median F1 scores achieved by different classifiers on the Piggybacking dataset after 25 runs using dynamic and hybrid features. As a reference to the performance of static features, we also plot the scores achieved by such classifiers using permission-based static features, which achieved the highest scores on the Piggbacking dataset. In figure \ref{fig:exp_pre_dynamic_f1}, dynamic features appear to be incapable of matching the detection ability of static features. One can also speculate that combining dynamic features with static features (i.e., hybrid features) decreases the latter's detection capability. Nevertheless, dynamic features perform better or slightly worse than other features in correctly classifying benign apps. To conclude, the dynamic preliminary experiments imply that dynamic features extracted after running the apps only once are initially biased towards classifying all apps as benign.
We argue that stimulating a repackaged malware once is unlikely to execute the paths under which the grafted malicious payload resides, ultimately leading to the misclassification of such app. In the next set of experiments, we attempt to verify whether re-stimulating the misclassified apps yields different paths that enhance the performance of the detection module and its classifiers.
\begin{figure}
\centering
\scalebox{.55}{\input{figs/exp_preliminary_dynamic/Piggybacking_dynamic_TEST_F1Score_scatter.pgf}}
\caption{The median (after 25 runs) F1 scores (Y-axis) recorded by 13 classifiers (X-axis) using dynamic and hybrid features versus the best scoring static features (permission-based) on the test datasets of Piggybacking. The specificity scores are plotted in appendix \ref{app:appendixc}.}
\label{fig:exp_pre_dynamic_f1}
\end{figure}
\subsubsection{Active Learning Experiments}
\label{sec:exp_active}
Using the Piggybacking dataset, we ran at total of 25 active learning experiments, during which 4 AVD's were simultaneously used to stimulate apps using {\fontfamily{lms}\selectfont Droidutan} each for 60 seconds. The misclassified apps were allowed to be re-stimulated until either the difference in the F1 score of the Ensemble classifier drops for more than a threshold of 1\% between two iterations, or an upper bound of re-stimulations is reached. We experimented with such upper bound, and found that with 10 iterations, an experiment takes on average 26 hours to complete. Given that increasing such number did not have a noticeable effect on the achieved scores, and that it substantially increased the time taken to complete one run of the experiment, we adopted an upper bound of 10 iterations.
\begin{figure*}
\centering
\begin{subfigure}{0.32\textwidth}
\scalebox{.36}{\input{figs/exp_active/Piggybacking_active_f1spec_KNN500_TEST_line.pgf}}
\caption{K-NN (K=500)}
\label{fig:exp_active_knn500}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\scalebox{.36}{\input{figs/exp_active/Piggybacking_active_f1spec_Ensemble_TEST_line.pgf}}
\caption{Ensemble}
\label{fig:exp_active_ensemble}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\scalebox{.36}{\input{figs/exp_active/Piggybacking_active_f1spec_Trees100_TEST_line.pgf}}
\caption{Random Forest (T=100)}
\label{fig:exp_active_trees100}
\end{subfigure}
\caption{The median (after 25 runs) F1 and specificity scores (Y-axis) recorded by 13 classifiers (X-axis) for each iteration using dynamic and hybrid features versus the best scoring static features (permission-based) on the test datasets of Piggybacking.}
\label{fig:exp_active}
\end{figure*}
The scores of the lowest scoring classifier, 500-Nearest Neighbors, the highest scoring classifier, Random forest of 100 trees, and the Ensemble classifier are plotted in figure \ref{fig:exp_active}. As seen in the preliminary dynamic experiments, the figures depict the median F1 and specificity scores achieved using dynamic, hybrid, and static permission-based features achieved by the classifiers at every iteration. The static scores are plotted as straight lines, because static experiments did not comprise any iterations. Thus, the scores achieved by a classifier persist across all iterations.
We made the following observations from the plotted figures. Firstly, regardless of the used classifier, dynamic features fail to match or overperform their static counterparts. However, combining static and dynamic features boosts the performance of the latter by around 7\%. In other words, dynamic features are incapable of matching static features on their own
Secondly, the iteration on which the maximum F1 score is achieved depends on the utilized classifier and features type. Furthermore, both the F1 and specificity scores fluctuate across iterations, and do not maintain any specific ascending or descending pattern.
Lastly, regardless of the utilized feature type, some classifiers maintain more stable performance on both the benign and malicious apps than others. For example, for the majority of iterations, the difference between the F1 and specificity scores achieved by the K-Nearest Neighbors classifier in figure \ref{fig:exp_active_knn500} is around 20\% and 25\% for the dynamic and hybrid features, respectively. This behavior indicates bias towards one class in favor of the other. The Ensemble and Random Forest classifiers maintain more balanced performances on malicious and benign apps, which manifests in closer scores across different iterations.
\section{Discussion}
\label{sec:discussion}
In this section, we attempt to draw conclusions from the conducted experiments to answer the research questions posed earlier.
With regard to research question \textbf{Q1}, we define the effectiveness of an active learning classifier in terms of the F1 and specificity scores it achieves using dynamic and hybrid features on the training datasets in comparison to the scores achieved by the same classifier during the static and dynamic preliminary experiments. We also consider the classifier's ability to avoid bias towards a specific class as a measure of its effectiveness. We noticed that the aforementioned scores differ from one classifier to another. For example, using dynamic features, the active learning trained 500-Nearest Neighbor classifier scored F1 and specificty scores significantly lower than what it scored during the static preliminary experiments. Furthermore, it was biased towards classifying training apps as benign, which is noticeable via its specificity score in comparison to the F1 score. However, the Ensemble and Random Forest classifiers managed to record F1 and specificity scores that are (a) greater than the scores they recorded during the static and dynamic preliminary experiments, and (b) show low bias given how close both scores are. The scores achieved by all classifiers during different types of experiments can be found on {\fontfamily{cms}\selectfont Aion}'s \href{https://aionexperiments.github.io/figures.html}{\emph{website}} \cite{aion2017}.
Similarly, the efficiency of active learning classifiers differed from one classifier to another. We define efficiency in terms of the number of iterations it takes a classifier to record its highest F1 and specificity scores on the training dataset. The 500-Nearest neighbors classifier, for instance, achieved its highest scores at the ninth iteration, whereas the Ensemble and Random Forest (with 100 Trees) classifiers recorded their highest scores at the second and third iterations, respectively. We averaged this number across all classifiers, and noticed that it takes around five iterations for an active learning classifier to achieve its maximum training accuracy. As discussed earlier, we consider this number as the number of iterations required by a classifier to have a comprehensive view of the malicious and benign behaviors within the training dataset.
The reliability of the training scores achieved by active learning classifiers, which is the concern of \textbf{Q2}, refers to the quality of the training data used to train such classifiers. In other words, we want to make sure that the classifiers are being trained using traces and features that reflect the malicious and benign behaviors exhibited by current Android repackaged malware and apps, respectively. Otherwise, the classifiers will be trained using inaccurate data that might yield misleading results.
To illustrate this concern, consider the CFG in figure \ref{fig:cfg}. We discussed that, ideally, a repackaged malware ($a_i$) should be classified as malicious if and only if the malicious path ($P_2$) was executed. With this research question, we reason about whether the high F1 scores achieved by classifiers such as Random Trees during training honor this condition. In other words, how likely is it that a classifier deems the app ($a_i$) as benign, given that a path other than ($P_2$) has executed. Unfortunately, as discussed earlier, we do not possess any ground truth about the paths that reveal the malicious behaviors injected within apps labeled as repackaged malware. Consequently, we can only speculate about the reliability of the results. However, we reasoned about the scenarios under which this behavior could be observed (i.e., $a_i$ is classified as malicious based on benign paths, such as $P_1$).
Firstly, and presumably less likely, the original labels of the apps in the dataset may turn out to be wrong (i.e., $a_i$ is a benign or a harmless app being mistakenly labeled as malicious).
Secondly, ($P_1$) could in fact turn out to be a malicious behavior that we were not aware of and assert as benign.
Thirdly, within ($a_i$), the path ($P_1$) could be benign; however, it may have been utilized within a malicious context in other apps in the training dataset, which obliges a classifier to deem it as malicious.
In the fourth scenario, the path ($P_1$) could be a benign path that is rarely used by benign apps (i.e., anomalous), which encourages the classifier to deem it as malicious as well.
Lastly, there is the possibility of utilizing poor classifiers, features, or learning processes, which yields mediocre, unreliable classification results.
Considering an extreme scenario, assume that all repackaged malware instances in a training dataset have been correctly classified despite the fact that none of their malicious paths has executed. For this condition to hold, we need to assume that all the benign paths deemed malicious by a classifier are anomalous (i.e., seldom encountered in benign apps), which is highly unlikely given the limited number of methods a task (e.g., sending a text message), can be achieved. Moreover, if a classifier mistakenly deems the majority of benign paths as malicious, this behavior should be traceable via the specificity scores which should plumet to minimum values.
To empirically complement the assumptions above, we manually went through the repackaged malware test traces generated by {\fontfamily{lms}\selectfont droidmon} during the $25^{th}$ run of the active learning experiments. Around $75\%$ of such test traces were correctly classified using the Random Forest (T=100) classifier trained using active learning. During our analysis, we found evidence that the majority of behaviors based on which the apps were classified as malicious are, in fact, intrinsically malicious. For example, we found that a repackaged version of a simulation gaming app (i.e., \emph{com.happylabs.happymall}), was classified as malicious based on a trace that included retrieving the device's IMEI, IMSI, and network operator. Furthermore, such trace included the decryption of a URL that pointed to \emph{http://csapi.adfeiwo.com:9999}. Needless to say, such behavior is not expected from a gaming app.
We also examined traces of repackaged malware misclassified as benign apps. We found that the majority of such misclassified traces exhibited benign behaviors. The trained classifier, thus, reasonably assigned them to the benign class. This observation is expected, given that the test apps were stimulated using the primitive tool {\fontfamily{lms}\selectfont Droidutan}, which is unlikely to execute the malicious behaviors within repackaged malware after one execution.
Nevertheless, upon submitting the hashes of the apps corresponding to the examined traces to VirusTotal \cite{virustotal2017}, we noticed that a noticeable number of the malicious apps classified as benign were also deemed as benign by the majority of antiviral software. For example, the repackaged apps \emph{com.sojunsoju.go.launcherex.theme.greenlantern} and \emph{com.gametowin.save\_monster\_Google} were labeled as potentially unwanted by only two out of more than 50 antiviral software, whereas a repackaged version of \emph{com.gccstudio.DiamondCaveLite} was labeled as benign by all antiviral software on VirusTotal. The latter observation raises the question about the accuracy of the labels in current Android repackaged malware datasets, as discussed earlier.
In absence of the ground truth necessary to verify the reliability of the achieved scores, and based on our previous discussion and observations, our answer to question \textbf{Q2} is that we believe that the scores achieved by the active learning classifiers are likely to be reliable.
The active learning process we adopt to train classifiers might lead to overfitting. In other words, the trained classifier would not be able to correctly classify a large number of test apps (i.e., apps not used during the training phase). Research question \textbf{Q3} focuses on this issue. Similar to the training phase, we noticed that some classifiers underperform in comparison to conventional training methods, while others can match and outperform them. For example, using hybrid features, Random Forest classifiers trained using active learning manage to outperform conventional training methods using static, dynamic, and hybrid features. Furthermore, it avoids bias towards a certain class by maintaining a good balance between the F1 and specificty scores.
Observing the performances of different active learning classifiers, we conclude that Random Forest classifiers are able to achieve the best F1 and specificty scores during, both, training and test phases specifically using hybrid features, which is the answer to \textbf{Q4}.
\section{Related Work}
\label{sec:related}
To the best of our knowledge, there are no research efforts that attempt to apply active learning to the problem of stimulating and detecting Android repackaged malware. However, there are indeed efforts that study repackaged malware/piggybacked apps and attempt to stimulate and detect it. Moreover, we managed to identify research efforts that apply active learning to detecting Android malware in general. Consequently, we divide our related work section into two sub sections that discuss those two dimensions.
\subsection{Repackaged Malware Analysis and Detection}
There are different efforts that attempt to detect repackaging in Android apps including \cite{shao2014towards,hanna2012juxtapp,zhou2012detecting,deshotels2014droidlegacy}. Repackaged apps need not necessarily be malicious, however. An app could be repackaged for the sole purpose of injecting advertisement modules or translation purposes. In this paper, we focus on detecting repackaged malware, or piggybacked apps, that conform with Li et al.'s definition: benign apps that has been grafted with malicious payloads that are possible protected with trigger conditions \cite{li2017understanding}.
In \cite{pan2017dark}, Pan et al.\ developed a program analysis technique that decides whether an app witholds hidden sensitive operations (HSO) (i.e., malicious payloads), effectively deeming it as repackaged malware. Their technique, called \emph{HSOMiner}, gathers static features from the app's source code that capture the code segments' relationship with their surrounding segments, the type of inputs they tend to check, and the operations they usually perform. The extracted features are used to train and validate a SVM that decides whether an app contains a HSO. Needless to say, if an app contains HSO, it probably is malicious.
Tian et al.\ implemented another static approach to detect Android repackaged malware based on \emph{code heterogeneity analysis} \cite{tian2016analysis}. The approach is based on extract features that depict the dependence of a code segments on one another. In essence, injected code segments should exhibit more heterogeneity to the rest of the segments. That is to say, the malicious payloads injected into benign apps should be logically and semantically independent of the original app portions. Similar to our approach, they used the extracted features to train and compare the performance of four different classifiers viz., K-Nearest Neighbors, Support Vector Machine, Decision Tree, and Random Forest.
Lastly, Shahriar et al.\ use a metric called Kullback-Leibler Divergence (KLD) to detect Android repackaged malware \cite{shahriar2015kullback}. KLD is a metric depicting the difference between two probability distributions. In their paper, Shahriar et al.\ analyzed the Smali code of benign and malicious apps, from the Malgenome dataset, to build probability distributions of different Smali instructions. Their detection approach is based on the assumption that repackaged malware would have different distributions for different instructions than its benign counterpart.
\subsection{Detection with Active Learning}
Active learning has been applied to the problem of malware detection on the Windows operating system \cite{nissim2014novel} and within the network intrusion detection \cite{stokes2008aladin} domain. In \cite{zhao2012robotdroid}, Zhao et al.\ attempt to apply the same technique to the problem of Android malware detection. Despite adopting a similar architecture that comprises a characteristic (monitoring and extraction) module and a detection module, their work differs from ours in two main aspects. Firstly, Zhao et al.\ do not consider repackaged malware, rather Android malware in general. Secondly, and more importantly, their definition and utilization of active learning is different from ours. In \cite{zhao2012robotdroid}, active learning is defined as a technique to reduce the size of the training samples used to train a SVM classifier that deem an Android app, represented by a behavioral signature (i.e., API call trace), malicious or benign. This reduction is carried out as follows. A pool of pre-generated and labeled behavioral signatures is built as a source for training data. Given an unlabeled, out-of-sample signature, a classifier is trained using the subset of the training pool that guarantees maximum classification accuracy. The main difference between this method and ours is that the classification accuracy in \cite{zhao2012robotdroid} solely depends on the combination of the training samples, whereas ours depends on the content of such samples, especially since the training vectors usually differ from one iteration to another. Furthermore, this approach to active learning is very likely to yield small training datasets that hinder the trained classifiers from achieving good classification accuracies on test datasets (i.e., \emph{generalization}).
\section{Conclusion}
\label{sec:conclusion}
The lack of ground truth about the nature and location of malicious payloads injected into benign apps continues to hinder the development of effective methods to stimulate, analyze, and detect Android repackaged malware. We argue that training an effective classifier can be achieved if the classifier is allowed to request better representations of the training apps (i.e., via active learning).
Using a sample implementation of our proposed active learning architecture {\fontfamily{cms}\selectfont Aion}, we used active learning to stimulate, analyze, and detect Android repackaged malware. In \cite{zhou2012detecting}, Zhou et al.\ managed to detect malware in the Malgenome dataset with a best case of 79.6\% accuracy \cite{zhou2012detecting}, highlighting the difficulty of detecting such breed of malware. In this paper, our active learning classifiers managed to achieve test F1 scores of 72\% using the more recent dataset Piggybacking \cite{li2017understanding}, which we consider a benchmark for future comparison.
We consider this effort as the first step towards designing better app stimulation, analysis, and detection techniques that focus on Android repackaged malware, which continues to pose a serious threat \cite{wei2017deep}. Our future plan to enhance {\fontfamily{cms}\selectfont Aion} has three dimensions. Firstly, we plan on using the number of iterations it takes to train a classifier, and verify whether it transfers to the number of executions needed to effectively classify a test app. Secondly, we plan to use more sophisticated stimulation engines and compare their effect on the training and test scores compared to the random, primitive technique used in this paper. Thirdly, we plan to utilize features and classifiers that capture the semantics of the runtime behaviors exhibited by under test.
{\footnotesize \bibliographystyle{acm}
|
1,941,325,220,585 | arxiv | \section*{Methods}
\subsection{Details of the experiment}
The granular medium was made of a number $N \in [300,2600]$ of spheres
of diameter $d=4$ mm made of non-magnetic steel (mass of each sphere:
$0.267$ g), glass (mass $0.0854$ g), or
delrin\textsuperscript{\textregistered} (mass $0.0462$ g). They were
housed in a plexiglas\textsuperscript{\textregistered} cylinder with a
conical-shaped floor (diameter $90$ mm, minimum height $28,5$ mm,
maximum height $47.5$ mm) in which a plexiglas vane (height $15$ mm,
width $6$ mm, length $35$ mm) was suspended in order to be in contact
with the granular medium and not with the container~\cite{gnoli14}. The container
was vertically vibrated by an electrodynamic shaker (LDS V450) fed
with an acceleration signal $a(t)$. In most of the experiments $a(t)$
is a white noise with a band-pass filter between $200$ Hz and $400$
Hz, while in the lowest $p_{00}$ case ($p_{00}=78$ Pa) we used a
sinusoidal signal at frequency $53$ Hz. This choice is motivated by
two empirical observations: 1) a lower number of particles (as in the
case of low $p_{00}$) requires a larger energy input to be
homogeneously fluidized and to reach the blade, and this can be
obtained by supplying energy through a sinusoidal signal
at low frequency; 2) in dense cases a sinusoidal signal induces
spurious resonances, while in diluted cases such resonances are never
observed. We have checked that performing the same experiments with
noise signal for $a(t)$ (pushing the shaker to its working limits)
gives flow curves with the same shape. An accelerometer placed on the
container side measured $a(t)$, allowing us to define
$\Gamma=\sqrt{2\langle a^2(t)\rangle}/g$.
The vane, mounted through its rotation axis to a rotary encoder, was
also connected to a dc motor (typical working voltage $12$ V) as the
source of the driving torque. The motor was directly fed by a dc
voltage supply in the range $0$ to $7$ V. No limit was set for the
maximum current absorbed by the motor that, averaged on the duration
of the experiment, was never higher than $450$ mA. A data acquisition
system collected data for the angular position/velocity of the vane,
the effective motor voltage, the current circulating in the motor and
the root-mean-square vertical acceleration of the container. A
procedure of calibration allowed us to translate average values of
current into average values of applied torque. The same procedure
helped determining the moment of inertia of the rotating block,
$3.2\times 10^2 \textrm{g mm}^2$ (the blade with its axis and the
gears linking it to the motor). The typical experiment, at a given
$\Gamma$ and applied motor voltage, was $3600$ s long, with the
granular always ``reset'' at the beginning of each run for $30$ s at
high shaking intensity ($\Gamma=42$) and motor off. This procedure -
together with periodic replacement of used spheres - guaranteed
reproducible results at a distance of several weeks. Packing fractions
was non-homogeneous (it was larger in regions far from the borders of
the container): its value at rest was estimated to be in the range
$55\% - 70\%$, while it decreased when vibration was switched on. In
the analysis we have identified the shear rate $\dot\gamma$ with the
average of the angular velocity $\omega(t)$ of the rotating blade,
i.e. $\dot\gamma=\langle \omega(t) \rangle$, while the shear stress
$\sigma$ is proportional to the average of the applied torque $T(t)$
through the shear stress constant $\kappa$, i.e. $\sigma=\kappa
\langle T(t) \rangle$ with $\kappa=2 \pi R^2 H$ (with $R$ and $H$ the
blade half-length and height, respectively)~\cite{dzuy83}.
\subsection{Details of data analysis}
Velocity power density spectra (VPDS) are defined as $S(f)=(1/(2
t_{TOT}) |\int_0^{t_{TOT}} \omega(t) e^{i(2 \pi f) t} dt|^2$ with
$t_{TOT}$ the time-length of an experiment ($=3600$ seconds). Some
examples of $S(f)$ curves are shown in the Fig.~1 of Supplementary
Information (SI). In~\cite{scalliet15} VPDS in a similar vibro-fluidized
experimental setup, without applied torque ($\sigma=0$), has been
investigated. In the dilute or gas-like limit, e.g. low number of
spheres at high shaking, the VPDS takes a simple Lorentzian shape
$S(f)=D/[1+(2 \pi f /f_{visc})^2]$, with $D$ the asymptotic (long
time) diffusivity and $f_{visc}$ the effective viscosity due to
granular gas-vane collisions. When the number of particles (density)
is increased and/or the intensity of shaking ($\Gamma$) is reduced,
the system approaches a slow liquid regime and the VPDS develops a
wide bump (or smooth peak) with a maximum near $f \sim 20$ Hz, which
is associated to oscillations of the velocity autocorrelation induced
by liquid cages. At much smaller frequencies the VPDS reaches a
plateau whose height, $\lim_{f \to 0} S(f)$, corresponds to diffusivity
$D$: indeed the blade is not trapped in a cage forever, eventually it
manages to explore a much larger phase space and reaches normal
diffusion. From the low frequencies plateau of VPDS we have extracted
values of $D$ for Figure~\ref{fig:fluct}. We have defined the
cage-exit frequency $f_{cage}$ as the x-position, in the VPDS plot, of the
minimum separating the cage bump from the low-frequency diffusive
plateau (see filled circles in Fig.~1 of SI).
\vspace{.5cm}
{\bf Acknowledgements}
We thank Giorgio Pontuale for useful discussions and suggestions. We
also thank Rachele De Felice and Francisco Vega Reyes for reading
the manuscript.
\vspace{.5cm}
{\bf Corresponding author}
Correspondence to: Andrea Puglisi, [email protected]
|
1,941,325,220,586 | arxiv | \section{Introduction}
Many e-commerce websites provide a wide range of products to the users. The users commonly have different needs and tastes based on which they buy the products. Providing the most appropriate products to the users make the buying process efficient and improves the user satisfaction. The enhanced user satisfaction keeps the user loyal to the website and improves the sales and thus profits to the retailers. Therefore, more retailers start recommending the products to the users which needs efficient analysis of user interests in products. E-commerce leaders like Amazon and Netflix use recommender systems to recommend products to the users. The standard recommendation algorithms like content-based filtering and collaborative filtering, model the ratings given by users to items as a matrix and predict the ratings of the customers to unrated items based on user/item similarity.
Recommender systems recommend items to users. Items include products and services such as movies, music, books, web pages, news, jokes and restaurants. The recommendation process utilizes data including user demographics, product descriptions and the previous actions of users on items like buying, rating, and watching. The information can be acquired explicitly by collecting ratings given by users on items or implicitly by monitoring user's behavior such as songs listened in music websites, news/movies watched in news/movies websites, items bought in e-commerce websites or books read in book-listing websites in the past. Usage of intelligent recommendation systems improved the revenue of Amazon by 35\%, caused a business growth of 24\% for BestBuy, increased 75\% of views on Netflix and 60\% views on YouTube. \cite{chui2017artificial}. Therefore, building a personalized recommendation system has a profound significance not only in commercial arena, but also in the fields like health care, news, food, academia and so many. Each domain needs to consider different features.
Recommender system is a typical application of link prediction in heterogeneous networks. Link Prediction problem infers future links in a network represented as graph where nodes are users and edges denote interactions between users. In the context of recommender systems, the nodes may be of two types: items and users. A transaction of a user buying an item can be shown as an edge from user node to item node. Recommendation problem can be viewed as a task of selecting unobserved links for each user node, and thus can be modeled as a link prediction problem.
In this work, we have applied various link prediction measures on the bipartite network in the context of recommender systems and verified the efficacy of those measures. We have chosen a medium sized dataset of MovieLens1M for experimentation.
The contributions made in this paper are
\begin{itemize}
\item Evaluated the efficacy of base line link prediction measures to the recommendation task.
\item Extended existing probabilistic measures called co-occurrence probability measure and temporal co-occurrence probability measure to make it suitable for recommendation problem.
\end{itemize}
\section{Problem Statement}
Schafer et al \cite{RS_base_2001} define the problem of recommender systems as follows:
Given a set of users $U=\lbrace u_1,u_2,\ldots u_m \rbrace $, and items $I= \lbrace I_1, I_2, \ldots ,I_n \rbrace $ and the ratings $R$ representing the ratings given by user $u_i$ to item $I_j$, the task of recommender systems is to recommend a new item $I_j$ to a user $u_i$ which the user $u_i$ did not buy.
For example, consider the matrix given in Fig.\ref{ratings_matrix}, where rows correspond to the users and the columns denote products. The matrix entry $R_{ij}$ represents the rankings given by user $u_i$ on item $I_j$. The main task of the recommender systems is to predict the unrated entries in the rating matrix.
\begin{figure}
\begin{equation*}
\mathbf{R}=
\begin{blockarray}{*{4}{c} l}
\begin{block}{*{4}{>{$\footnotesize}c<{$}} l}
$item_1$ & $item_2$ & $item_3$ & $item_4$ & \\
\end{block}
\begin{block}{[*{4}{c}]>{$\footnotesize}l<{$}}
5 & & 2 & & User 1 \\
4 & 4 & 2 & & User 2 \\
4 & & & 1 & User 3 \\
& 2 & & 2 & User 4 \\
\end{block}
\end{blockarray}
\end{equation*}
\caption{Example rating matrix denoting rating given by users to 4 products.}
\label{ratings_matrix}
\end{figure}
Recommender systems are natural examples of weighted bipartite networks. A \textbf{Bipartite network} contains exactly two types of nodes and single type of edges existing between different types of nodes. Bipartite network is defined as $G = (V_1 \cup V_2, \quad E)$, where $V_1$ and $V_2$ are sets of two types of nodes, $E$ represents the set of edges between nodes of type $V_1$ and $V_2$. Fig.\ref{user-item-net} depicts a sample scenario of users buying items in e-commerce sites such as Amazon, modeled as a bipartite network.
\begin{figure}
\label{user-item-net}
\centering
\includegraphics[width=0.4\textwidth]{UserItemBipartiteNet.jpg}
\caption{\small{An example of user-item Bipartite Network with edges denoting rating given by user on item.}} \label{user-item-net}
\end{figure}
\section{Related Literature}
The papers on recommender system in the literature are discussed from algorithmic as well as domain perspective in the next two sections. Fig. \ref{rs_taxonomy} gives a taxonomy of the literature.
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{RS_Taxonomy.png}
\caption{Taxonomy of recommender systems}
\label{rs_taxonomy}
\end{figure}
\subsection{Domain Perspective}
Product recommendation is the mostly explored domain. E-commerce sites like Amazon, Flipkart, Ebay etc use various collaborative filtering techniques for recommending products to their customers. Video recommendations like movies, TV shows and web series are increasing exponentially by companies like Netflix, YouTube. Netflix has launched a competition with 1Million dollars prize money for improving 10\% of RMSE in the movie recommendation, by releasing a 100 million customer ratings~\cite{bell2007lessons}. A news recommendation system has been implemented in~\cite{liu2010personalized} using google news, by constructing a Bayesian network for identifying the user interests and then applying content based as well as collaborative filtering and hybrid techniques. Research article recommendation, food recommendation and health recommendation are other domains among many.
Valdez et al. integrate recommender systems into individual medical records to help patients improve their autonomy. The authors have collected health related information from the patients' records and have performed text processing for extracting features and evaluated using metrics available in Information Extraction domain~\cite{valdez2016recommender}.
All these domains need different pre-processing techniques, algorithms, and evaluation metrics.
\subsection{Algorithmic perspective}
Recommender systems is seeing an explosive growth in the literature with many novel algorithms being proposed almost every day. Content-based recommendation, Collaborative Filtering (CF) and Hybrid approach are some of the popular approaches that solve the problem of recommender systems. Context aware recommender systems predict recommendations based on user context. Graph based recommender systems is main focus of this work. In graph based recommender system, the purchases are modeled as bipartite graph and various graph traversal techniques are applied to generate a set of co-purchased items/users.
\subsubsection{Content-based recommendation}
Content-based recommendation uses item descriptions and constructs user profiles which contain the information about user preferences~\cite{RS_Content_2007}. These user preferences may include a genre, director and actor for a movie, an author for book etc. The recommendation of an item to the user is then based on the similarity between the item description and the user profile. This method has the benefit of recommending items to users that have never been rated by them~\cite{RS_Content_2000}. Content-based recommendation systems require complete descriptions of items and detailed user profiles. This is one of the main limitations of such systems~\cite{RS_Content_2000}.
Content-based recommender systems are divided into three general approaches: Explicit, Implicit and Model-based. In explicit content based methods, profile information of the users is collected through questionnaires that ask questions about their preferences for the items. Implicit methods build the user profiles implicitly by searching for similarities in liked and disliked item descriptions by the user. The third class of methods, viz., model-based methods construct user profiles based on a learning method, using item descriptions as input features to be given to a supervised learning algorithm, and producing user ratings of items as output.
The item descriptions are often textual information contained in documents, web sites and news messages. User profiles are modeled as weighted vectors of these item descriptions. The advantage of this method is the ability to recommend newly introduced items to users~\cite{RS_Content_2000}. Content-based recommendation systems require the complete descriptions of items and detailed user profiles. This is the main limitation of such systems. Privacy issues such as users dislike to share their preferences with others is another limitation of content-based systems.
\subsubsection{Collaborative filtering}
Collaborative filtering systems, however, do not require such difficult to come by, well-structured item descriptions. Instead, they are based on users’ preferences for items, which can carry a more general meaning than is contained in an item description. Indeed, viewers generally select a film to watch based upon more criteria other than only its genre, director or actors.
Collaborative filtering (CF) techniques depend only on the user's past behavior and provide personalized recommendation of items to users~\cite{linden2003amazon}. CF techniques take a rating matrix as input where the rows of the matrix correspond to users, the columns correspond to items and the cells correspond to the rating given by the user to the item~\cite{RS_ullman_2014}. The rating by a user to an item represents the degree of preference of the user towards the item. The major advantage of the CF methods is they are domain independent. CF methods are classified into Model-based CF~\cite{RS_Koren_2009} and Memory-based CF. Model based techniques model ratings by describing both items and users on some factors induced by matrix factorization strategies. Data sparsity is a serious problem of CF methods.
Model based approaches such as dimensionality reduction techniques, Principal Component Analysis (PCA), Singular Value Decomposition (SVD) are some of the popular techniques that address the problem of sparsity. However, when certain users or items are ignored, useful information for recommendations may be lost causing reduction in the quality of recommendation~\cite{CF_Limitation}. The limitation of collaborative filtering systems is the cold start problem. i.e., these methods can not recommend new items to the existing users as there is no past buying history for the item. Similarly, it is difficult to recommend items to a new user without knowing the user's interests in the form of ratings.
Memory based methods compute a set of nearest neighbors for users as well as items using similarity measures like Pearson coefficient, cosine distance and Manhattan distance. Memory based CF techniques are further classified into user based and item based~\cite{RS_2017}.
\paragraph{User-based CF}\cite{RS_Taxonomy} User based methods compute the similarity between the users based on their ratings on items~\cite{huang2004graph}. This creates a notion of user neighborhoods. These methods associate a set of nearest neighbors with each user and then predict the user's rating for unrated items utilizing the information of the neighbor's rating for that item.
\paragraph{Item-based CF}~\cite{RS_Item_based}: Similarly, item neighborhood depicts the number of users who rate the same items~\cite{sarwar2001item}. The item rating for a given user can then be predicted based upon the ratings given in their user neighborhood and the item neighborhood. These methods associate an item with a set of similar neighbors, and then predict the user's rating for an item using the ratings given by the user for the nearest neighbors of the target item.
\subsubsection{Hybrid approach}
Hybrid approach uses both types of information, collaborative and content based. Content boosted CF algorithm~\cite{RS_Hybrid1}, uses the item profile information to recommend the items to new users.
\subsubsection{Context aware recommendation system}
Context aware recommendation system is able to label each user action with an appropriate context and effectively tailor the system output to the user in that given context~\cite{context_aware_RS}. In~\cite{kefalas2015graph}, the authors proposed a recommendation of an activity to a social network user based on demographic information. Kefalas et al. have represented their data as a k-partite network modeled as a tensor and proposed novel algorithms. In the same way, in~\cite{ali2020graph}, a citation recommendation is made to researchers also considering tags and time. Standard recommender system algorithms cannot address these issues.
\subsubsection{Graph based Recommender Systems}
Li et al. map transactions to a bipartite user–item interaction graph and thus converting recommendation into a link prediction problem. They propose a kernel-based recommendation approach which generate random walk between user–item pair and define similarities between user–item pairs based on the random walks. Li et al. use a kernel-based approach that indirectly examines customers and items related to user-item pair to foresee the existence of an edge between them. The paper uses a set of nodes representing two types of nodes users(U) and items(I) and edges denoting transactions by the user regarding the items. The graph kernel is defined on the user-item pairs context~\cite{li2013recommendation}.
Zhang et al. propose a model for music recommendation~\cite{zhang2020bipartite}. The authors represent the recommendation data as a bipartite graph with user and item nodes and the weight of the link is represented as a complex number depicting the dual preference of users in the form of like and dislike and improve the similarity of the users.
In graph-based recommender system, the task of recommendation is equivalent to predicting a link between user-item based on graphical analysis. Link prediction in graphs is commonly a classification task, that predicts existence of a link in future \cite{LibenNowellBasicLP2007}. But the problem of recommender systems is modeled as a regression task, which predicts the rating of a link. The predicted links with high ratings are generally recommended to users. The standard techniques of recommendation systems take the ratings matrix as input and predicts the empty cells in the matrix. This approach severely suffer with the sparsity in the matrix. Ranking-oriented collaborative filtering approach is more meaningful compared to the rating based approach because, the recommendation is a ranking task recommending top-k ranked items to a user ~\cite{RS_TopN_2010}. In some applications like recommending web pages, rating information may not be available. Graph-based recommendation can efficiently utilize the heterogeneous information available in the networks by expanding the neighborhoods and can compute the proximity between the users and items~\cite{RS_2017}.
A probabilistic measure for predicting future links in bipartite networks is given in~\cite{MyPAKDD}. Several link prediction measures in various kinds of networks have been summarized in~\cite{ASONAM2018}.
Our aim is to evaluate the efficacy of these measures in the context of recommender systems by modeling them as bipartite graphs. In this paper, we have applied various existing link prediction measures on bipartite graphs in order to build a recommender system. However, the link prediction approach does not give the actual rating, but gives the top k-item recommendations for a user. For experimentation, we have chosen the bench-mark 'Movielens' dataset. Next section gives an overview of application of link prediction measures on bipartite graphs.
\section{Link Prediction in bipartite graph}
Graph-based recommendation algorithms compute recommendations on a bipartite graph where nodes of the graph represent users and items and an edge forms between a user node and an item node when the user buys the item. Once the transactions are modeled as a graph, all the standard link prediction methods defined in \cite{LibenNowellBasicLP2007},\cite{LichtenwalterPropFlow2010} can be applied directly on the graph to predict the future links~\cite{RS_JC_2014}. All the measures used in \cite{LibenNowellBasicLP2007} are based on neighbors and paths. Especially, common neighbors, which lie on paths of length two play an important role. All the measures are extended to heterogeneous environment in \cite{ASONAM2018} with a mention of bipartite networks as special case.
We follow similar notation as in \cite{ASONAM2018} in this paper, which is given below.
\begin{itemize}
\item ($u$, $p$) is a pair of user and product nodes without an edge between them.
\item $k$ represents hop-distance between two nodes and $t$ denotes time.
\item $\Gamma_k(x)$ is the set of $k$-$hop$ neighbors of node $x$. $\Gamma_1(x)$ refers to the set of all nodes connected by an edge of any type to $x$ generally written as $\Gamma(x)$.
\item $\Gamma(u)\cap\Gamma(p)$ denotes the set of common neighbors between node $u$ and $p$. In bipartite graph, this set is empty.
\item $\Gamma_k(u)\cap\Gamma_k(p)$ contains all the common neighbors within $k$-hop distance between nodes $u$ and $p$. In bipartite graph, $k$ needs to be even to start and end path between same type of nodes and odd if path starts and ends between nodes of different types. As recommender system tries to recommend items to users, the odd length paths are meaningful.
\item $P_k(u,p)$ denotes the set of paths connecting $u$ and $p$ by at most $k$ edges.
\end{itemize}
\subsection{Baseline link prediction measures in bipartite graph}
In recommendation systems modeled as bipartite graph, the task is to recommend products to users. The product nodes and user nodes are connected with odd length paths. Even length paths connect the nodes of same type, which is not meaningful in this scenario. Keeping this in mind, the baseline link prediction measures for bipartite graphs are given below. We use suffix B with all link prediction measures to indicate they are used in context of bipartite network.
\begin{itemize}
\item \textbf{Common Neighbors~(CN)} :
The common neighbor measure in bipartite graph is given as follows:
\begin{equation} \label{CN}
\begin{split}
CN_B(u, p)&=|\Gamma_3(u) \cap \Gamma_3(p)|
\end{split}
\end{equation}
Jaccard Coefficient, AdamicAdar and Preferential Attachment measures are also neighborhood-based defined in a similar way as follows.
\item \textbf{Jaccard Coefficient~(JC)} : Jaccard Coefficient is the normalized $CN$ measure
\begin{equation} \label{JC}
\begin{split}
JC_B(u, p)= \frac{|\Gamma_3(u) \cap \Gamma_3(p)|}{|\Gamma_3(u) \cup \Gamma_3(p)|}
\end{split}
\end{equation}
\item \textbf{Adamic Adar~(AA)} : This measure gives importance to the common neighbors with low degree. The following definition for bipartite networks has been hinted at \cite{DarcyMRLP2013}.
\begin{equation} \label{AA}
\begin{split}
AA_B(u, p)=\displaystyle \sum_{z \in \Gamma_3(u) \cap \Gamma_3(p)} \frac{1}{log(|\Gamma_3(z)|)}
\end{split}
\end{equation}
\item \textbf{Preferential Attachment~(PA)} : This measure does not change in bipartite environment because the measure is concerned only about the degree of the node whatever the type of the node may be.
\begin{equation} \label{PA}
PA_B(u, p)=|\Gamma_1(u)| * |\Gamma_1(p)|
\end{equation}
\item \textbf{Katz~(KZ)} : This measure is based on the total number of paths between $u$ and $p$ bounded by a limit penalized by path length.
\begin{equation} \label{KZ}
\begin{split}
KZ_B(u, p)= \sum_{l} \beta ^l |P_l{(u, p)}^{l}|
\end{split}
\end{equation}
\item \textbf{Page Rank~(PR)} : This measure can be extended to heterogeneous network by including the heterogeneous edges in the random-walk.
\begin{equation}\label{PR}
\begin{split}
PR_B(u)=\frac{1-\alpha}{|E|}+\alpha\displaystyle \sum_{z \in \Gamma_3(u)}\frac{PR(z)}{|\Gamma_3(z)|}
\end{split}
\end{equation}
where $|E|$ is the total number of links in $G$.
\item \textbf{Rooted Page Rank~(RPR)} : In order to make the measure $PR$ symmetric, Rooted Page Rank is computed for an edge $(u, p)$ as follows.
\begin{equation}
RPR_B(u, p)=PR(u, p)+PR(p, u)
\end{equation}
\item \textbf{PropFlow~(PF)} : This is a random-walk beginning at node $u$ and ending at $p$ within $l$ steps. This random-walk from node $u$ to $p$ terminates either when it reaches $p$ or revisits any node. $PF_B(u, p)$ is the probability of information flow from $u$ to $p$ based on random transmission along all paths defined recursively as follows.
\begin{equation} \label{PF}
PF_B(u, p) = \displaystyle\sum_{l=2}^L \displaystyle\sum_{p \in paths_l(u, p)} \sum_{\forall (z_1,z_2) \in p} PF(z_1,z_2)
\end{equation}
where
\begin{equation} \label{PF_rec}
PF_B(z_1,z_2) = PF(a,z_1) * \frac{w(z_1,z_2)}{\displaystyle\sum_{z \in \Gamma(z_1)}w(z_1,z)}
\end{equation}
with $a$ as previous node of $z_1$ in the random-walk, $PF(a,z_1)$=1 if $a$ is the starting node and $paths_l(u, p)$ is the set of paths of length $l$ between $u$ and $p$. In bipartite network, $l$ is odd.
\begin{equation} \label{HPF}
PF_B(u, p) = \displaystyle\sum_{l=2}^L \displaystyle\sum_{p \in P_l(u, p)} \sum_{\forall(z_1,z_2) \in p} PF_B(z_1,z_2)
\end{equation}
where $PF_B(z_1,z_2) $ is as defined in Eq.\ref{PF_rec}.
\end{itemize}
\subsection{Temporal measures for link prediction in bipartite graph}
\subsubsection{Time-Score($TS_B$)}
We extend Time-Score measure proposed for homogeneous networks~\cite{TimeScore} to bipartite environment as follows:
\begin{equation}
\label{Hetero-TS}
TS_B(u,p)= \sum_{path \in P_3(u,p)}\frac{w(path)* \beta^{r(path)}}{|latest(path)-oldest(path)|+1}
\end{equation}
where $w(path)$ is equal to the harmonic mean of edge weights of edges in $path$, $\beta$ is a damping factor~(0<$\beta$<1),
$latest($path$)=\displaystyle \max_{e~on~P_3}(t(e))$, $oldest($path$)= \displaystyle \min_{e~on~P_3}(t(e))$ and $r$ is a recency factor, defined as~~ $r($path$)$= $current\_time-latest($path$)$.
\subsubsection{Link-Score($LS_B$)}
Choudhary et al. extend the Time-score measure to obtain a path based measure called $Link$-$score$~\cite{LinkScore} for homogeneous networks. To obtain the $Link$-$score$ between a pair of nodes $u$ and $p$ which are not directly connected, the authors define a Time Path Index~($TPI$) on each path $p$ between the nodes $u$ and $p$. $TPI$ evaluates path weight based on time stamps of links involved in a path. Link-score is the sum of $TPI$ of each path between the nodes $x$ and $y$.
\hspace{3em}We extend $Link$-$score$ to bipartite network by considering $paths$ of odd length between two nodes instead of paths containing any length. The modified definitions of $TPI$ and $Link$-$Score$ are given in equation~\ref{Hetero-LS}.
\begin{equation}
\label{Hetero-LS}
TPI_{path}= \frac{w(path)* \beta^{current\_time-avg(path)}} {|current\_time-latest(path)|+1}
\end{equation}
where $avg(path)$ is the average active year, which is the average of years of recent interaction of edges on odd length path and all others are as defined in equation~\ref{Hetero-TS}.
\begin{equation}
LS_B(u,p)=\displaystyle \sum_{l=2}^{L}\frac{Avg(TPI_{P_l(u,p)})}{l-1}
\end{equation}
where $L$ is the maximum length of paths between nodes $i$ and $j$.
\subsubsection{T\_Flow($TF_B$)}
T\_Flow~\cite{TFlow} is a random-walk based measure, which is an extension of PropFlow measure defined in~\cite{LichtenwalterPropFlow2010}. Munasinghe et.al~\cite{TFlow} define T\_Flow that computes the information flow between a pair of nodes $x$ and $y$ through all random-walks starting from node $u$ to node $p$ including link weights as well as activeness of links by giving more weight to recently formed links recursively and take the summation.
We extend T\_Flow measure to bipartite networks as follows:
\begin{equation}
\label{Hetero-TF}
TF_B(u,p)=\displaystyle\sum_{i=2}^l \displaystyle\sum_{path \in P_l(x,y)} \sum_{\forall e \in path} TF_B(z_1,z_2) * (1-\alpha)^{r(path)}
\end{equation}
If $(u,p)\in E$, then $TF_B(u,p)$ is given by
\begin{equation}
TF_B(u,p) = TF_B(a,u) * \frac{w(u,path)}{\displaystyle\sum_{z \in \Gamma(u)}w(u,z)}*(1-\alpha)^{r(path)}
\end{equation}
where $t_u$ is the time stamp of the link when the random walk visits the node $u$ and $t_p$ is the time stamp of the link when the random walk visits node $p$.
\subsection{Probabilistic measure for link prediction in bipartite graph}
The Probabilistic Graphical Model (PGM) represents the structure of the graph in a natural way by considering the nodes of graph as random variables and edges as dependencies between them. By representing a graph as a PGM, the problem of link prediction, which is calculation of the probability of link formation between two nodes $u,~p$ is translated to computing the joint probability of the random variables $U,~P$.
Recommender systems represented as bipartite networks are large in size. Therefore, finding the joint probabilities of link formation is intractable. But the links in the graphs are also sparse, with nodes generally directly connected to only a few other nodes. For example, in the e-commerce sites, out of lakhs of items, a user buys only a few items. This property allows the PGM distribution to be represented tractably. The model of this framework is simple to understand. Inference between two random variables is same as finding the joint probability between those two random variables in PGM. Many algorithms are available for computation of joint probability between variables, given evidence on others. These inference algorithms work directly on the graph structure and are generally faster than computing the joint distribution explicitly. With all these advantages, link prediction in PGM is more effective.
In most of the cases, the pair-wise interactions of entities are available in the event logs. For example, a user buying/rating three items say $p_1$, $p_2$ and $p_3$ is available in the corresponding transaction database. In order to model the unknown distribution of co-occurrence probabilities, events available in the event logs can be used as constraints to build a model for the unknown distribution. Probabilistic Graphical Models efficiently utilize this higher order topological information and thus are efficient in link prediction task~\cite{Cooccur:ICDM:2007}.
The probabilistic model helps in estimating the joint probability distribution over the variables of the model~\cite{JointPDF}. That means, a probabilistic model represents the probability distribution over all possible deterministic states of the model. This facilitates the inference of marginal probabilities of the individual variables of the model.
Wang et al. \cite{Cooccur:ICDM:2007} are among first researchers who modeled the problem of link prediction using MRFs. Kashima et al. \cite{Kashima} propose a probabilistic model of network evolution and apply it for predicting future links. The authors show that by intelligently selecting the parameters in an evolution model, the problem of network evolution reduces to the problem of link prediction. Clauset et al. \cite{Clauset_2008_Hierarchical} propose a probabilistic model based on hierarchical structure of the network. In hierarchical structure, the vertices are divided into groups that further subdivide into groups of groups and so on. The model infers hierarchical structure from network data and can be used for prediction of missing links. The learning task uses the observed network data and infers the most likely hierarchical structure through statistical inference.
The works of Kashima~\cite{Kashima} and Clauset~\cite{Clauset_2008_Hierarchical} are global models and are not scalable for large networks. The method of Wang et. al. \cite{Cooccur:ICDM:2007} uses local probabilistic information of graphs for link prediction and we adopt their method in our work. The following section explains the algorithm for link prediction using MRF proposed by Wang et al. \cite{Cooccur:ICDM:2007}
\section{Proposed approach: Link Prediction in bipartite networks induced as PGM}
Wang et al. \cite{Cooccur:ICDM:2007} propose a measure called Co-occurrence Probability (COP) to be computed between a pair of nodes without an edge between them. The procedure for computing COP has three steps. First step chooses a few set of nodes which may play a significant role in future link formation. Second step computes a Markov Random Field with the chosen nodes in the first step. Third step infers joint probability between the two nodes using the MRF constructed. The measure has been extended to temporal networks in ~\cite{TCOP}. Heterogeneous version of the measure is defined in ~\cite{MyPAKDD}. Though $Hetero$-COP has been defined in~\cite{MyPAKDD}, the algorithm has to be re-worked for the bipartite context. In this paper, we have extended COP measure to bipartite environment, named it as $B$-COP of a missing link $(u, p)$ in the lines of $COP$, which is given in Algorithm \ref{Hetero-COP-Algo}. The extended computation procedure for bipartite networks is given in subsequent sections.
\begin{algorithm}[H]
\caption{$B$-COP measure for Link Prediction in bipartite graphs}
\label{Hetero-COP-Algo}
\textbf{Input}: $G = (V, E),$ where \\
\hspace{\algorithmicindent} $V=\lbrace V_1 \cup V_2 \rbrace$ is the set of two types of nodes, \\
\hspace{\algorithmicindent} $E$ is the set of links from nodes of $V_1$ to $V_2$. \\
\textbf{Output}: $B$-COP$(u, p)$ where $(u, p)$ is a missing link.\\
\begin{algorithmic}
\STATE {\bf Step 1:} Extract $B$-cliques from $G$, from the event logs using Algorithm \ref{LP-based-RS}. Call it $BCliq$.
\STATE {\bf Step 2:} Compute \textbf{BCNS}, the central neighborhood set of $(u, p)$ using the Algorithm \ref{HCNS-Computation}.
\STATE {\bf Step 3:} Extract $B$-cliques formed with the nodes in BCNS and compute clique potentials. This forms the local MRF with the nodes in BCNS.
\STATE {\bf Step 4:} Return $B$-COP$(u, p)$ which is the joint probability of link $(u, p)$ using junction tree algorithm.
\end{algorithmic}
\end{algorithm}
\hspace{3em}A clique in bipartite graphs can be considered as complete bipartite sub-graph. A sample $B$-clique is shown in Fig.\ref{B-clique}.
\begin{figure}[h]
\centering
\includegraphics[width=0.25\textwidth]{general_bipartite_clique.jpg}
\caption{An example $B$-clique.}
\label{B-clique}
\end{figure}
\subsection{Computation of Central Neighborhood Set in Bipartite graph(BCNS)}
BCNS$(u,p)$ is computed using a breadth first search based algorithm on bipartite graph as follows: All paths between $u$ and $p$ are obtained using breadth first search (BFS) algorithm. Then, the frequency score of each path is computed by summing the occurrence count of nodes on the path. Occurrence count of a node is the number of times the node appears in all paths. The paths are now ordered in the increasing order of length with equal paths being ordered in decreasing order of frequency score. The size of the central neighborhood set is further restricted by considering only top $k$ nodes. The procedure of computing $BCNS(u,p)$ is described in the Algorithm~\ref{HCNS-Computation}.
\begin{algorithm}[!htb]
\caption{ {\bf Bipartite Central Neighborhood Set($G, u, p, l,maxSize$)}}
\label{HCNS-Computation}
\textbf{Input}: $G$: a graph; $u$: starting node; $p$: ending node; $l$:maximum path length; $maxSize$:Central Neighborhood Set size threshold \\
\textbf{Output}: $BCNS$, Bipartite Central Neighborhood Set between $u$ and $p$; \\
\begin{algorithmic}
\STATE {\bf Step 1:} \hspace{\algorithmicindent} Compute paths of length $\leq~ l$ between $u$ and $p$.
\STATE {\bf Step 2:} \hspace{\algorithmicindent} Find occurrence count $O_k$ of each node $k$ in paths between $u$ and $p$.
\STATE {\bf Step 3:}\hspace{\algorithmicindent} Compute $frequency$-$score$ $F_p$, of each path $p$ as follows: \\
\begin{center}
$F_p=\sum_{k \in p} O_k$ \\
\end{center}
\hspace{\algorithmicindent} $frequency$-$score$ of a path is the sum of the occurrence counts of all nodes along the paths.
\STATE {\bf Step 4:} \hspace{\algorithmicindent} Sort the paths in increasing order of path length and then in decreasing order of $frequency$-$score$. Let the ranked list of paths be $P$.
\STATE {\bf Step 5:} \hspace{\algorithmicindent}
\WHILE {size(BCNS) $\leq maxSize$}
\STATE Add nodes of path $p \in P$ to BCNS
\ENDWHILE
\RETURN BCNS
\end{algorithmic}
\end{algorithm}
The procedure of computation of BCNS for a toy example given in Fig.\ref{CNS-toy} is shown in Table~\ref{CNS-paths}.
\begin{minipage}{\textwidth}
\begin{minipage}[c]{0.39\textwidth}
\includegraphics[width=0.7\textwidth]{BCNS_Toy_Ex.jpg}
\captionof{figure}{\small{A toy example for illustrating computation of BCNS between nodes $u$ and $p$. Node weights are the occurrence counts.}}
\label{CNS-toy}
\end{minipage}
\hfill
\begin{minipage}[c]{0.59\textwidth}
\small{
\begin{tabular}{|l|l|} \hline
\textbf{path(p)} & \textbf{frequency-score($F_p$)} \\ \hline
$u-p_2-u_2-p$ & $O_{p_2} + O_{u_2}$ = 6 \\
$u-p_1-u_2-p$ & $O_{p_1} + O_{u_2}$ = 5 \\
$u-p_2-u_3-p$ & $O_{p_2} + O_{u_3}$ = 4 \\
$u-p_1-u_2-p_2-u_3-p$ & $O_{p_1} + O_{u_2} + O_{p_2} + O_{u_3}$ = 9 \\ \hline
\end{tabular}
}
\captionof{table}{All paths between nodes $u$ and $p$ in Fig.\ref{CNS-toy} and their frequency scores.}\label{CNS-paths}
\end{minipage}
\end{minipage}
Table~\ref{CNS-paths} shows all paths sorted in the increasing order of path length and frequency score, between the nodes $u$ and $p$ of the graph shown in Fig.\ref{CNS-toy} along with their $frequency$-$scores$.
One can observe that all these paths are of odd length. For the bipartite graph in Fig.\ref{CNS-toy}, BCNS$\left(u,p\right)= \lbrace u,p_2,u_2,p_1,p \rbrace$, if $maxSize$ is taken as 5.
\subsection{Construction of local MRF}
After computing the BCNS, the $B$-cliques containing only nodes of BCNS are extracted. This forms the clique graph of local MRF. MRF construction needs computation of clique potentials. The clique potential table of a $B$-clique is computed using the NDI.
In most of the cases, the information of homogeneous cliques containing all nodes of same type are available in the event logs. For example, in coauthorship networks, the group of authors who publish a paper together forms a homogeneous clique of author nodes and an author who publishes a paper in a conference forms a heterogeneous edge between the author node and conference node. But in the case of recommender systems, user cliques and item cliques are not readily available in the event logs. We propose an algorithm for extracting B-cliques shown in Algorithm~\ref{LP-based-RS}. Since the number of users is huge in comparison to the set of items which is a much smaller set, the extraction of $B$-cliques can start from the item cliques.
\begin{algorithm}
\caption{\bf Extraction of \textbf{B-Cliques} from user-item bipartite graph} \label{LP-based-RS}
\textbf{Input}: $G=(V,E)$ where $V = U \cup I$, $U$ is set of user nodes and $I$ is the set of item nodes and $E \subseteq U X I$ represents weighted edges. \\
\textbf{Output}: $Bcliq$, set of maximal B-cliques of $G$.
\begin{algorithmic}
\STATE $\textbf{Step 1:}$ Extract the set of all item cliques, $Item\_Cliq$ as follows:
\STATE $Item\_Cliq=\phi$
\FOR {each user $u \in U$}
\STATE $I_u=\phi $
\FOR {each $i \in I \text{~~and~~} (u,i)\in E$}
\STATE $I_u=I_u \cup \{ i \}$
\ENDFOR ~~//~~$I_u$ is formed by taking all the items to which $u$ gives a rating. \\
$Item\_Cliq=Item\_Cliq \cup I_u$
\ENDFOR
\STATE $\textbf{Step 2:}$ Extract the set of all user cliques, $User\_Cliq$ as follows:
\STATE $User\_Cliq=\phi$
\FOR {each item $i \in $}
\STATE $U_i=\phi $
\FOR {each $u \in U \text{~~and~~} (u,i)\in E$}
\STATE $U_i=U_i \cup \{ u \}$
\ENDFOR ~~//~~$U_i$ contains all users that have rated item $i$. \\
$User\_Cliq=User\_Cliq \cup U_i$
\ENDFOR
\STATE $\textbf{Step 3:}$
\FOR {each item $i$ in $I$ }
\STATE $J \leftarrow I$
\FOR {each user $v$ in $U_i$ }
\STATE $J \leftarrow J \cap ~I_v$
\ENDFOR
\STATE //~~ $J = \displaystyle\bigcap_{v \in U_i} I_v$ ~~//
\STATE $Bcliq_i = J \cup U_i$
\STATE $Bcliq = \cup_i ~ Bcliq_i$
\ENDFOR
\STATE return $Bcliq$
\\
\end{algorithmic}
\end{algorithm}
The B-clique extraction algorithm first extracts all homogeneous cliques of items $Icliq$ and users $Ucliq$. A homogeneous user clique is formed with all the users who rate/buy the same item. Similarly, a homogeneous clique of items is formed with all the items a user rate/buy. To extract a B-clique, first consider an item $i$. For each user $v$ who have rated $i$, compute the common items rated by user $v$. The union of $U_i$ along with all the common items rated by $U_i$ forms a $B$-clique. The process of extracting B-cliques for toy example in Fig.\ref{user-item-bipartite} is illustrated below:
\vspace{5em}
\begin{minipage}{\textwidth}
\begin{minipage}[l]{0.3\textwidth}
\begin{figure}[H]
\includegraphics[scale=0.55]{user_item_bipartite.jpg}
\caption{User-Item bipartite graph}
\label{user-item-bipartite}
\end{figure}
\end{minipage}
\hfill
\begin{minipage}[c]{0.68\textwidth}
U=$\lbrace u_1,u_2,u_3,u_4,u_5 \rbrace$ \hspace{2cm} I=$\lbrace i_1,i_2,i_3,i_4,i_5 \rbrace$ \\
Item cliques : \\
Item clique corresponding to user $u_1=\lbrace i_1, i_2 \rbrace$ \\
Item clique corresponding to user $u_2=\lbrace i_1, i_2 \rbrace$ \\
Item clique corresponding to user $u_3=\lbrace i_2, i_3, i_4, i_5 \rbrace$ \\
Item clique corresponding to user $u_4=\lbrace 1_3, i_4, i_5\rbrace$ \\
Item clique corresponding to user $u_5=\lbrace 1_3, i_4, i_5\rbrace$ \\
User cliques : \\
User clique corresponding to item $i_1=\lbrace u_1, u_2 \rbrace$ \\
User clique corresponding to item $i_2=\lbrace u_1, u_2, u_3 \rbrace$ \\
User clique corresponding to item $i_3=\lbrace u_3, u_4, u_5 \rbrace$ \\
User clique corresponding to item $i_4=\lbrace u_3, u_4, u_5 \rbrace$ \\
User clique corresponding to item $i_5=\lbrace u_3, u_4, u_5 \rbrace$ \\
The the extraction of B-cliques from the above homogeneous cliques is shown in Fig.\ref{extract-item-b-clique}.
\end{minipage}
\end{minipage}
\begin{figure}[H]
\centering
\includegraphics[width=1.2\textwidth]{user_item_cliq_illustration.jpg}
\caption{Illustration of extracting $B$-cliques from User-Item event logs}
\label{extract-item-b-clique}
\end{figure}
\subsection{Computation of $B$-COP score} Once the local MRF of a pair of nodes $u$ and $p$ is constructed, the $B$-COP score between the nodes is obtained using junction tree inference algorithm. Note that $B$-COP score for a link $u$ - $p$ cannot be computed if $u$ and $p$ are in disjoint cliques as there exists no path connecting these cliques.
The experimental evaluation of proposed approach is given in next section.
\section{Experimental Evaluation}
The experimentation is carried out on a benchmark MovieLens recommender system whose details are given below.
\subsection{Dataset}
This data set used for experimental evaluation contains more than ten million ratings given by 71,567 users on 10,681 movies of the online movie recommender service MovieLens~\cite{movielens}. In the graph recommendation systems, movies are considered as nodes and items are users and rating given by users to the movies are considered as the weights on links. In MovieLens dataset, the train-test splits are given in \cite{movielens}. The users who have rated at least 20 movies have been chosen randomly to be included in this dataset. The benchmark data set~\cite{movielens1} is given with 80\% - 20\% split with 80\% given as training set and test set containing 20\%. The training and test sets are formed by splitting the ratings data such that, for every user, 80\% of his/her ratings is taken in training and the rest are taken in the test set. In this experimentation, 5 fold cross validation is used. All the 5 sets of training and test datasets are made available at \cite{movielens1}.
The evaluation metrics used for recommender systems are given in the following section.
\subsection{Evaluation Metrics}
Evaluation metrics in recommender systems can be classified as
\begin{itemize}
\item Accuracy measures: Mean Absolute Error (MAE), Root of Mean Square Error (RMSE), Normalized Mean Average Error (NMAE).
\item Set recommendation metrics : Precision, Recall and Area Under Receiver Operating Characteristic (AUROC), Area Under Precision-recall curve (AUPR)
\item Rank recommendation metrics: Half-life, discounted cumulative gain and Rank-score
\end{itemize}
\hspace{3em}Most of the measures listed above, use rating to calculate the error and hence are not applicable in our context. We use AUROC, AUPR are used for evaluating performance.
\paragraph{Rank-score} Rank-score metric measures the ability of a recommendation algorithm to produce a ranked list of recommended items. The recommender system method is efficient, if the ranking given by the method matches with the user's order of buying the items in the recommended list. Rank-score is defined as follows : For a user $u$, the list of items $i$ recommended to $u$, that is predicted by the algorithm is captured by $rankscore_p$
\begin{center}
\begin{math}
\begin{aligned}
rankscore_{max} = \frac{1}{\sum_{j =1}^{|T|} 2^{\frac{j-1}{\alpha}}} \\
rankscore_p = \frac{1}{\sum_{j \in T} 2^{\frac{rank(j)-1}{\alpha}}} \\
Rankscore = \frac{rankscore_p}{rankscore_{max}} \\
\end{aligned}
\end{math}
\end{center}
where $rank(j)$ is the rank given by the recommender algorithm to item $j$. $T$ is the set of items of interest and $\alpha$ is ranking half-life, an exponential reduction factor and can be any numeric.
\subsection{Results and Discussion}
The prediction scores of all baseline link prediction measures are computed using the tool $LPMade$ \cite{lpmade}, with default parameters.
In the computation of $TS_B$, $LS_B$ and $TPI$, the damping factor $\beta$ is taken as 0.5. The decay factor $\alpha$ is considered as 0.1 in computation of $TF_B$. In $TCOP$, maximum path length of 10 is taken for computing BCNS. The accuracy of all these link prediction measures is compared with the standard User-based and Item-based collaborative filtering (CF) methods. Pearson correlation coefficient is used to find the similar users in User-based CF and cosine similarity is used for finding item similarity in Item-based CF.
The results obtained for AUROC, AUPR and Rank-score for MovieLens dataset are given in Table.\ref{movielens-results}.
\begin{table}[h]
\centering
\caption{Performance of LP measures for recommending movies to users in \textbf{MovieLens} Bipartite Network}
\label{movielens-results}
\begin{tabular}{|l||l|l|l|}
\hline
\textbf{LP measure} & \textbf{Rank-score} & \textbf{AUROC} & \textbf{AUPR} \\ \hline \hline
\multicolumn{4}{|c|}{\textbf{Non-temporal} LP measures} \\ \hline \hline
$CN_B$ & 2.0501 & 0.5123 & 0.0092 \\ \hline
$JC_B$ & 1.3615 & 0.4956 & 0.0085 \\ \hline
$AA_B$ & 1.7819 & 0.5749 & 0.0153 \\ \hline
$PA_B$ & 2.6804 & 0.6832 & 0.0251 \\ \hline
$KZ_B$ & 3.2546 & 0.6635 & 0.0193 \\ \hline
$PF_B$ & 3.2990 & 0.6846 & 0.0286 \\ \hline
$COP_B$ & 3.6661 & 0.7231 & 0.0613 \\ \hline \hline
\multicolumn{4}{|c|}{\textbf{Temporal} LP measures} \\ \hline \hline
$TS_B$ & 4.0015 & 0.7016 & 0.0365 \\ \hline
$LS_B$ & 5.2134 & 0.7340 & 0.0861 \\ \hline
$TF_B$ & 5.3684 & 0.7532 & 0.0960 \\ \hline
$TCOP_B$ & {\color{red}\textbf{8.6304}} & {\color{red}\textbf{0.8165}} & {\color{red}\textbf{0.2351}} \\ \hline \hline
\multicolumn{4}{|c|}{\textbf{Classical Collaborative Filtering} methods} \\ \hline \hline
User-based CF & 3.9942 & 0.6925 & 0.0415 \\ \hline
Item-based CF & 3.0274 & 0.7136 & 0.0491 \\ \hline
\end{tabular}
\end{table}
First observation in this experimentation is that some of the link prediction measures like Katz, PropFlow could produce better recommendation compared to Item-based CF. The usage of temporal measures seem to help in improving the quality of recommendations. The time of formation of link or the time of rating given by a user to movie plays crucial role, as the user's preferences change over time. The temporal measures TS, LS, TF and TCOP assign more weight to the recent ratings. Therefore, temporal measures performed better than all the other measures including User-based CF and Item-based CF. Fig.\ref{Movie_neg} depicts a situation where non-temporal measures predict a link between the nodes $1$ and $1080$ and temporal measures predict correctly that the link won't be formed. This is clearly evident from the fact that $B-Cliq 1$ is formed with very old links formed in the year 1996 compared to the prediction year 2008. Fig.\ref{Movie_pos} depicts the situation where the temporal links predict correctly. TCOP outperformed all the other link prediction measures and the user-based and item-based collaborative filtering methods. The recommendation performance is improved by 6\% in terms of AUROC over TF and nearly 10\% over CF methods. It can be observed from Table.\ref{movielens-results} that the AUPR score also have shown great improvement from 0.0960(of TF) to 0.2351. Similar trend is observed for the evaluation measure of Rank-score. All the temporal measures perform better than User-based CF and Item-based CF. TCOP is rated as highest by Rank-score.
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{movielens_neg_snapshot.jpg}
\caption{A snapshot from movielens dataset where the link predicted is not formed.}
\label{Movie_neg}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{movielens_pos_snapshot.jpg}
\caption{A snapshot from movielens dataset where the link predicted is formed.}
\label{Movie_pos}
\end{figure}
\section{Conclusion}
In this paper, the recommender systems problem is solved using link prediction approach. Link prediction approach is scalable as it is based on local neighborhood of the large sparse graph. The standard recommender systems approaches do not utilize the temporal information available on the link effectively. In this work, some extensions are proposed to existing temporal measures in bipartite graphs. One of the main contributions of this work is an algorithm for computing temporal cooccurrence probability measure on bipartite graphs and its application to movie recommendation system. Temporal measures for link prediction such as Time score, Link Score, T\_Flow and Temporal cooccurrence measure achieve improvement in recommendation quality by utilizing this temporal information more efficiently. However, link prediction approach to solve recommender systems do not address the cold start problem. In future, we would like to work on predicting actual rating of the link.
\bibliographystyle{aps-nameyear}
|
1,941,325,220,587 | arxiv | \section{Introduction}
Let us say that a transitive model of $\ZF$ is a \emph{$\ZF$-ground}, or simply a \emph{ground}, of $V$
if $V$ is a generic extension of the model.
If a ground satisfies $\AC$, the Axiom of Choice,
then it is a \emph{$\ZFC$-ground}.
In $\ZFC$, Fuchs-Hamkins-Reitz \cite{FHR} studied the structure of all $\ZFC$-grounds of $V$,
it is called \emph{set-theoretic geology}.
Their work was under $\AC$, and Usuba \cite{U2} tried to study \emph{set-theoretic geology without $\AC$};
The universe $V$ and each ground are not assumed to satisfy $\AC$.
These are attempts to investigate the nature of forcing method.
When we want to build choiceless models,
the method of \emph{symmetric submodel}, or \emph{symmetric extension}, is a very powerful and flexible tool.
For a generic extension $V[G]$ of $V$, a symmetric submodel of $V[G]$, or a symmetric extension of $V$,
is realized as a submodel of the generic extension $V[G]$.
The model $V[G]$ itself is a symmetric submodel of $V[G]$, so every generic extension is a symmetric extension.
Moreover symmetric extensions have many properties which are parallel to generic extensions,
such as forcing relation and forcing theorem.
We want to say that a model $W$ of $\ZF$ is a \emph{symmetric ground} of $V$ if
$V$ is a symmetric extension of $W$, that is, $V$ is a symmetric submodel of \emph{some} generic extension of $W$.
Now we can expect to extend standard set-theoretic geology to one which treats
symmetric extensions and symmetric grounds.
If $V$ is a symmetric extension of some model,
then $V$ could be a choiceless model.
So our base theory should be $\ZF$.
On the other hand, our definition of symmetric grounds causes a problem:
What is \emph{some} generic extension of $W$?
It would be possible that there is a $W$-generic $G$ which is
\emph{not} living in $V$ and in any generic extension of $V$, but $V$ is realized as a
symmetric submodel of $W[G]$.
Hence, in $V$, it is not clear if we can describe that ``$W$ is a symmetric ground of $V$''
and develop geology of symmetric grounds.
For this problem, using Grigorieff's work \cite{G},
we prove that the statement ``$W$ is a symmetric ground of $V$'' is actually describable
in $V$ by a certain first order formula of the extended language $\{\in, W\}$.
We also prove that, under a certain assumption,
all symmetric grounds are uniformly definable by a first order formula of set-theory.
Let $\CLS$ denote the assertion that there is a proper class of \LS cardinals (see Section 3).
\begin{thm}[in $\ZF$]
Suppose $\CLS$.
Then there is a first order formula $\varphi(x,y)$ of set-theory such that:
\begin{enumerate}
\item For every set $r$, $\overline{W}_r=\{x \mid \varphi(x,r)\}$ is a symmetric ground of $V$ with $r \in \overline{W}_r$.
\item For every symmetric ground $W$ of $V$, there is $r$ with
$W=\overline{W}_r$.
\end{enumerate}
\end{thm}
$\CLS$ follows from $\AC$.
Hence in $\ZFC$, all symmetric grounds are uniformly definable.
We also show that if $V$ satisfies $\AC$,
then $V$ is definable in any symmetric extension of $V$.
The above uniform definability of symmetric grounds allows us to investigate \emph{set-theoretic geology of symmetric grounds} in $\ZF$, which is a study of the structure of all symmetric grounds.
This paper is a first step in the set-theoretic geology of symmetric grounds.
In $\ZF$, it is consistent that there are two grounds which have no common ground (\cite{U2}).
We show that, if $\AC$ is forceable over $V$ then
all symmetric grounds are downward directed.
Moreover the intersection of all symmetric grounds is a model of $\ZFC$ if $\AC$ is forceable over $V$.
\section{Preliminaries}
\textbf{Throughout this paper, we do not assume $\AC$ unless otherwise specified.}
Forcing means a set forcing.
A \emph{class}
means a second order object in the sense of
Von Neumann-Bernays-G\"odel set-theory $\mathsf{NBG}$
unless otherwise specified.
We do not require that a class $M$ is definable in $V$ with some
parameters, but
we assume that $V$ satisfies the comprehension and replacement scheme
for the formulas of the language $\{\in, M\}$
(where we identify $M$ as a unary predicate).
Note that, if $M$ is a definable class by a formula of the language $\{\in\}$,
then $V$ satisfies the comprehension and replacement scheme
for the formulas of the language $\{\in, M\}$.
Hence every definable class is a \emph{class} in our sense.
We also note that $V$ is a class of any generic extension of $V$
by the forcing theorem.
Any theorems in this paper involving classes can be formalized
in $\NBG$, or some small extension of $\ZF$.\footnote{
For instance,
we extend the language of set-theory by adding unary predicates $M, N, W,\dotsc$
and let $T$ be the theory $\ZF$ together with the comprehension and replacement scheme
for the formulas of the language $\{\in, M, N,W,\dotsc\}$.}
To treat classes in generic extensions, we extend the forcing language and
relation as follows.
Let $M$ be a class.
First, we extend the forcing language
by adding the symbol $\check M$.
For a poset $\bbP$, $p \in \bbP$, and a $\bbP$-name $\dot x$,
we define $p \Vdash_\bbP \dot x \in \check M$ if
the set $\{q \le p \mid$ there is $x \in M$ with $q \Vdash_\bbP x =\dot x \}$
is dense below $p$.
$\check M$ can be treated as a class-sized canonical name, that is,
$\check M=\{\seq{\check x, p} \mid p \in \bbP, x \in M\}$,
where $\check x$ is the canonical name for $x$.
By the standard way, we define $p \Vdash_\bbP \varphi$ for every formula $\varphi$
of the extended forcing language.
Since $M$ is a class of $V$, we can easily check that the forcing theorem holds for the
extended forcing language and relation,
and $M=\{\dot x^G \mid p \Vdash_\bbP \dot x \in \check M$ for some $p \in G\}$.
Using the extended forcing theorem,
we have the following:
If $M$ is a class of $V$, then it is a class of any generic extension of $V$.
A model of $\ZF(\mathsf{C})$ means a transitive class model of $\ZF(\mathsf{C})$ containing all ordinals.
It is known that a transitive class $M$ containing all ordinals
is a model of $\ZF$ if and only if
$M$ is closed under the G\"odel operations and almost universal,
that is, for every set $X \subseteq M$ there is a set $Y \in M$ with
$X \subseteq Y$.
So we can identify the sentence ``$M$ is a model of $\ZF$''
with the conjunction of the following sentences of the language $\{\in, M\}$:
\begin{enumerate}
\item $M$ is transitive and contains all ordinals.
\item $M$ is closed under the G\"odel operations.
\item $M$ is almost universal.
\end{enumerate}
A model $W$ of $\ZF$ is a \emph{$\ZF$-ground}, or simply \emph{ground}, of $V$ if
there are a poset $\bbP \in W$ and
a $(W, \bbP)$-generic $G$ with $V=W[G]$.
If a $\ZF$-ground satisfies $\AC$, then it is a \emph{$\ZFC$-ground}.
\textbf{\textbf From now on, a ground means a $\ZF$-ground unless otherwise specified.}
For a model $M$ of $\ZF$ and an ordinal $\alpha$, let $M_\alpha=M \cap V_\alpha$,
the set of all $x \in M$ with rank less than $\alpha$.
Here we present a series of results by Grigorieff \cite{G},
which we will use frequently.
\begin{thm}[\cite{G}]
For models $M$, $N$ of $\ZF$,
if $M \subseteq N$, $M$ is a ground of $N$, and $N$ is a ground of $V$,
then $M$ is a ground of $V$.
\end{thm}
This means that the ``is a ground of'' relation is transitive in $\ZF$.
For a model $M$ of $\ZF$ and a set $X$,
let $M(X)=\bigcup_{\alpha \in \mathrm{ON}} L(M_\alpha \cup \{X\})$.
$M(X)$ is the minimal model of $\ZF$ with $M\subseteq M(X)$ and $X \in M(X)$.
Note that $M(X)$ is also a class of $V$.
\begin{thm}[Theorem B in \cite{G}]\label{1.2}
Let $M \subseteq V$ be a ground of $V$.
Let $N$ be a model of $\ZF$
and suppose $M \subseteq N \subseteq V$.
Then the following are equivalent:
\begin{enumerate}
\item $V$ is a generic extension of $N$.
\item $N$ is of the form $M(X)$ for some $X \in N$.
\end{enumerate}
\end{thm}
For a class $X$,
let $\HOD_X$ be the collection of all hereditarily definable sets
with parameters from $\ON \cup X$.
If $M$ is a model of $\ZF$,
it is known that
$\HOD_M$ is also a model of $\ZF$ with $M \subseteq \HOD_M$
\footnote{Note that $\HOD_M$ is not the relativisation of $\HOD$ to $M$.},
and $\HOD_M$ is a class of $V$ (e.g., see \cite{G}).
We note that $M$ is a class of $\HOD_M$ if $M$ is a model of $\ZF$.
\begin{thm}[9.3 Theorem 1 in \cite{G}]\label{1.3}
Let $M$ be a model of $\ZF$ such that
$V=M(X)$ for some $X \in V$.
Then $V$ is a ground of some generic extension of $\HOD_M$,
and $\HOD_M=M(Y)$ for some $Y \in V$ with $Y \subseteq M$.
\end{thm}
\begin{note}\label{1.4}
One may wonder what is ``some generic extension of $\HOD_M$''.
This theorem can be justified as follows:
There is a poset $\bbP$
such that, in $V$, $\bbP$ forces that ``$(\HOD_M)^{\check{}}$ is a ground of the universe''.
\end{note}
For a set $S$,
let $\col(S)$ be the poset consists of all finite partial functions from
$\om$ to $S$ ordered by reverse inclusion.
$\col(S)$ is weakly homogeneous, and if $S$ is ordinal definable then so is $\col(S)$.
\begin{thm}[4.9 Theorem 1 in \cite{G}]\label{1.5}
Let $\bbP$ be a poset, and $G$ be $(V, \bbP)$-generic.
Let $\alpha$ be a limit ordinal with $\alpha>\mathrm{rank}(\bbP)\cdot \om$.
Let $H$ be $(V[G], \col(V[G]_\alpha))$-generic.
Then there is a $(V, \col(V_\alpha))$-generic $H'$ with
$V[G][H]=V[H']$.
\end{thm}
The following is just rephrasing of this theorem:
\begin{lemma}\label{1.5+}
Let $M,N$ be grounds of $V$, and $\alpha$ a sufficiently large limit ordinal.
Then whenever $G$ is $(V, \col(V_\alpha))$-generic,
there are an $(M, \col(M_\alpha))$-generic $H$ and an $(N, \col(N_\alpha))$-generic $H'$ with
$V[G]=M[H]=N[H']$.
\end{lemma}
We recall some definitions and facts about symmetric submodels and extensions.
See Jech \cite{J} for more details.
Let $\bbP$ be a poset, and $\mathrm{Auto}(\bbP)$ the group of the automorphisms on $\bbP$.
For $\pi \in \mathrm{Auto}(\bbP)$,
$\pi$ can be extended to the (class) map from the $\bbP$-names to $\bbP$-names canonically.
Let $\calG$ be a subgroup of $\mathrm{Auto}(\bbP)$.
A non-empty family $\calF$ of subgroups of $\calG$ is a \emph{normal filter} on $\calG$ if:
\begin{enumerate}
\item $H, H' \in \calF \Rightarrow H \cap H' \in \calF$.
\item $H \in \calF$, $H'$ is a subgroup of $\calG$ with $H \subseteq H'$
then $H'\in \calF$.
\item For $H \in \calF$ and $\pi \in \calG$, we have
$\pi^{-1} H \pi \in \calF$.
\end{enumerate}
A triple $\seq{\bbP, \calG, \calF}$ is a \emph{symmetric system}
if $\calG$ is a subgroup of $\mathrm{Auto}(\bbP)$ and
$\calF$ is a normal filter on $\calG$.
A $\bbP$-name $\dot x$ is
\emph{$\calF$-symmetric} if $\{ \pi \in \calG \mid \pi(\dot x)=\dot x\} \in \calF$.
Let $\mathrm{HS}_{\calF}$ be the class of all hereditarily $\calF$-symmetric names.
If $G$ is $(V, \bbP)$-generic,
then $\mathrm{HS}_{\calF}^G=\{\dot x^G \mid \dot x \in \mathrm{HS}_\calF\}$
is a transitive model of $\ZF$ with $V \subseteq\mathrm{HS}_{\calF}^G$.
$\mathrm{HS}_{\calF}^G$ is called
a \emph{symmetric submodel of $V[G]$}, or a \emph{symmetric extension of $V$}.
If we want to specify a poset $\bbP$,
we say that $\mathrm{HS}_{\calF}^G$ is a symmetric extension of $V$ via a poset $\bbP$,
or a symmetric submodel of $V[G]$ via a poset $\bbP$.
We note that
$\mathrm{HS}_{\calF}^G$ is a class of $V[G]$, and $V$ is a class of $\mathrm{HS}_{\calF}^G$.
\begin{note}\label{2.7-}
Let $M$ be a model of of $\ZF$,
$V[G]$ a generic extension of $V$ via a poset $\bbP$,
and $V \subseteq M \subseteq V[G]$.
One may expect that if $M$ is a symmetric extension of $V$,
then $M$ is a symmetric submodel of $V[G]$
via a poset $\bbP$. However this is not correct.
This follows from the construction of the Bristol model (Karagila \cite{K}).
The Bristol model $M$ is an intermediate model between
$L$ and the Cohen forcing extension $L[c]$,
and $M$ cannot be of the form $L(X)$ for some set $X$.
$M$ is of the form $\bigcup_{\alpha \in \mathrm{ON}} L(M_{\alpha})$,
and each $L(M_{\alpha})$ is a symmetric extension of $L$,
but there is some $\beta$ such that for every $\alpha \ge \beta$,
$L(M_\alpha)$ is not a symmetric submodel of $L[c]$ via the Cohen forcing notion.
\end{note}
The following theorem tells us when $M \subseteq V[G]$ is a symmetric submodel of $V[G]$.
\begin{thm}[Theorem C in \cite{G}]\label{1.6}
Let $M \subseteq V$ be a model of $\ZF$, $\bbP \in M$ a poset, and
$G$ be $(V, \bbP)$-generic and suppose $V=M[G]$.
Let $N$ be a model of $\ZF$ with $M \subseteq N \subseteq V$.
Then $N$ is a symmetric submodel of $V$ via $\mathrm{ro}(\bbP)^M$ if and only if
$N$ is of the form $\mathrm{HOD}_{M(X)}$ for some
$X \in N$, where $\mathrm{ro}(\bbP)$ is the completion of $\bbP$.\footnote{We can take the completion of $\bbP$ without $\AC$,
see \cite{G}.}
\end{thm}
\section{Choiceless geology}
In this section, we recall some definitions and facts about set-theoretic geology without $\AC$.
See Usuba \cite{U2} for more information.
For \LS cardinals below, see also \cite{U3}.
We use the notion of \emph{\LS cardinal},
which was introduced in \cite{U2}.
The notion of \LS cardinal corresponds to the downward
\LS theorem in the context of $\ZFC$.
The downward \LS theorem is not provable from $\ZF$,
but under the existence of \LS cardinals,
we can take \emph{small} elementary submodels of $V_\alpha$ for every $\alpha$.
\begin{define}
An uncountable cardinal $\ka$ is a \emph{\LS cardinal}
if for every $\alpha \ge \ka$, $x \in V_\alpha$, and $\gamma<\ka$,
there are $\beta \ge \alpha$ and $X \prec V_\beta$ satisfying the following:
\begin{enumerate}
\item $x \in X$ and $V_\gamma \subseteq X$.
\item ${}^{V_\gamma} (X \cap V_\alpha) \subseteq X$.
\item The transitive collapse of $X$ belongs to $V_\ka$.
\end{enumerate}
\end{define}
Note that every limit of \LS cardinals
is a \LS cardinal as well.
\begin{thm}[\cite{U2}]\label{2.2}
Suppose $\ka$ is a limit of \LS cardinals.
Then for every poset $\bbP \in V_\ka$, $\bbP$ forces that
``$\ka$ is a \LS cardinal''.
Conversely, if $\ka<\la$ are cardinals,
$\bbP \in V_\ka$ a poset, and $\Vdash_\bbP$``$\ka,\la$ are \LS'',
then $\la$ is \LS in $V$.
\end{thm}
\begin{define}
For a set $X$, the \emph{norm} of $X$, denoted by $\norm{X}$, is the least ordinal $\alpha$
such that there is a surjection from $V_\alpha$ onto $X$.
\end{define}
\begin{define}
Let $M \subseteq V$ be a model of (a sufficiently large fragment of ) $\ZF$,
and $\alpha$ an ordinal.
\begin{enumerate}
\item $M$ has the \emph{$\alpha$-norm covering property} for $V$
if for every set $X \subseteq M$, if $\norm{X}<\alpha$ then
there is $Y \in M$ such that $X \subseteq Y$ and $\norm{Y}^M<\alpha$.
\item $M$ has the \emph{$\alpha$-norm approximation property} for $V$
if for every set $Y \subseteq M$, if $Y \cap a \in M$
for every $a \in M$ with $\norm{a}^M<\alpha$
then $Y \in M$.
\end{enumerate}
\end{define}
\begin{thm}[\cite{U2}]\label{2.5}
Suppose $\ka$ is a \LS cardinal and $\alpha<\ka$.
For all models $M, N$ of (a sufficiently large fragment of) $\ZF$,
if $M$ and $N$ have the $\alpha$-norm covering and the $\alpha$-norm approximation properties for $V$
and $M_\ka=N_\ka$,
then $M=N$.
\end{thm}
\begin{thm}[\cite{U2}]\label{2.6}
Let $\ka$ be a \LS cardinal, and $W \subseteq V$ a model of $\ZF$.
If there are a poset $\bbP \in W_\ka$ and
a $(W, \bbP)$-generic $G$ with $V=W[G]$,
then $W$ has the $\ka$-norm covering and the $\ka$-norm approximation properties for $V$.
\end{thm}
Let us say that a model $M$ of $\ZF$ is a \emph{pseudo-ground}
if $M$ has the $\alpha$-norm covering and the $\alpha$-norm approximation properties for $V$
for some $\alpha$.
The following is immediate:
\begin{lemma}\label{2.7}
Let $M \subseteq N \subseteq V$ be models of $\ZF$, and $\alpha$ an ordinal.
\begin{enumerate}
\item If $M$ has the $\alpha$-norm covering and the $\alpha$-norm approximation properties for $N$,
and $N$ for $V$,
then $M$ has the $\alpha$-norm covering and the $\alpha$-norm approximation properties for $V$.
\item If $M$ has the $\alpha$-norm covering and the $\alpha$-norm approximation properties for $V$,
then $M$ has the $\alpha$-norm covering and the $\alpha$-norm approximation properties for $N$.
\end{enumerate}
\end{lemma}
\begin{define}
Let $\CLS$ denote the assertion
that ``there is a proper class of \LS cardinals''.
\end{define}
By Theorem \ref{2.2},
$\CLS$ is absolute between $V$, all generic extensions, and all grounds of $V$.
\begin{thm}[\cite{U2}]\label{2.9}
Suppose $\CLS$.
Then all pseudo-grounds are uniformly definable:
There is a first order formula $\varphi'(x,y)$ of set-theory such that:
\begin{enumerate}
\item For every $r \in V$, the class $W_r'=\{x \mid \varphi'(x,r) \}$ is a pseudo-ground
of $V$ with $r \in W_r'$.
\item For every pseudo-ground $W$ of $\ZF$,
there is $r$ with $W=W_r'$.
\end{enumerate}
\end{thm}
We sketch the proof since we will need to know how to define $\varphi'$ and $W'_r$ later.
\begin{proof}[Sketch of the proof]
For $r \in V$, suppose $r$ fulfills the following properties:
\begin{enumerate}
\item $r$ is of the form $\seq{X, \ka, \alpha}$ where $\ka$ is a \LS cardinal,
$\alpha< \ka$, and $X$ is a transitive set with $X \cap \mathrm{ON}=\ka$.
\item For each cardinal $\la>\ka$, if $V_\la$ is a model of a sufficiently large
fragment of $\ZF$, then there is a unique transitive model $X^{r,\la}$ of
a sufficiently large fragment of $\ZF$ such that
$X^{r,\la} \cap \mathrm{ON}=\la$, $(X^{r,\la})_\ka=X$, and
$X^{r,\la}$ has the $\alpha$-norm covering and the $\alpha$-norm approximation properties for $V_\la$.
\end{enumerate}
Then let $W_r'=\bigcup_{\la>\ka} W^{r,\la}$.
Otherwise, let $W_r'=V$.
By Theorem \ref{2.5},
one can check that the collection $\{W_r' \mid r\in V\}$ is as required.
\end{proof}
Using this, we have the uniform definability of all grounds under $\CLS$.
\begin{thm}[\cite{U2}]\label{unif. def. grounds}
Suppose $\CLS$.
Then there is a first order formula $\varphi(x,y)$ of set-theory such that:
\begin{enumerate}
\item For every $r \in V$, the class $W_r=\{x \mid \varphi(x,r) \}$ is a ground
of $V$ with $r \in W_r$.
\item For every ground $W$ of $\ZF$,
there is $r$ with $W=W_r$.
\end{enumerate}
\end{thm}
\begin{proof}[Sketch of the proof]
Let $\{W_r' \mid r \in V\}$ be the definable collection of
pseudo-grounds of $V$ as in Theorem \ref{2.9}.
If $W_r'$ is a ground of $V$ then put $W_r=W_r'$,
and $W_r=V$ otherwise.
By Theorem \ref{2.6},
the definable collection $\{W_r \mid r \in \}$ comprises all grounds of $V$.
\end{proof}
\begin{note}\label{notedayo}
In addition, we have the following by Theorem \ref{2.2}:
Suppose $\CLS$. If the formula $\varphi$ defines all grounds as above,
then, in all generic extensions,
the formula $\varphi$ defines all its grounds by the same way.
\end{note}
\begin{lemma}\label{2.10}
Suppose $\CLS$.
Then there is a first order formula $\psi(x,y,z)$ in the language of set-theory
such that for every poset $\bbP$, $\bbP$-name $\dot s$, and a set $t$,
$\Vdash_\bbP$``\,$W_{\dot s}^{V[\dot G]} =\check W_t'$''
if and only if $\psi(\bbP, \dot s,t)$ holds.
Hence
the statement $\Vdash_\bbP$``\,$\check W_t'$ is a ground of the universe'' is
a first order assertion as $\exists \bbP \exists \dot s \psi(\bbP, \dot s, t)$.
\end{lemma}
\begin{proof}
Let $\psi(\bbP,\dot s,t)$ be the following sentence:
There are \LS cardinals $\ka<\la$ which are limits of \LS cardinals and $\bbP \in V_\ka$
such that:
\begin{enumerate}
\item $\Vdash_\bbP$``$W_{\dot s}^{V[\dot G]}$ is a ground of the universe $V[\dot G]$ via
a poset $\bbQ \in (W_{\dot s}^{V[\dot G]})_\ka$''.
\item For every $p \in \bbP$, there is $q \le p$ and a set $X$
such that $q \Vdash_\bbP$``$X=(W_{\dot s}^{V[\dot G]})_\la$'',
$X=(W_{u}')_\la$ where $u=\seq{X, \la, \ka}$,
and $W_{u}'=W_t'$.
\end{enumerate}
We see that this $\psi$ works.
First, suppose $\Vdash_\bbP$``$W_{\dot s}^{V[\dot G]} =\check W_t'$''.
By $\CLS$,
there are large \LS cardinals $\ka<\la$ which are limits of \LS cardinals
such that $\Vdash_\bbP$``$W_{\dot s}^{V[\dot G]}$ is a ground of the universe $V[\dot G]$ via
poset $\bbQ \in (W_{\dot s}^{V[\dot G]})_\ka$''.
We see that $\ka$ and $\la$ witness $\psi(\bbP, \dot s, t)$.
By Theorems \ref{2.2} and \ref{2.6}, the following hold:
\begin{enumerate}
\item $\Vdash_\bbP$``$\ka$ and $\la$ are \LS cardinals''.
\item $\Vdash_\bbP$``$W_{\dot s}^{V[\dot G]}$ has the $\ka$-norm covering and
the $\ka$-norm approximation properties for $V[\dot G]$''.
\item $\Vdash_\bbP$``$\check V$ has the $\ka$-norm covering and
the $\ka$-norm approximation properties for $V[\dot G]$''.
\end{enumerate}
Note that whenever $G$ is $(V,\bbP)$-generic and $s=\dot s^G$,
we have $W_t'=W_s^{V[G]} \subseteq V \subseteq V[G]$.
Since $W_t'=W_s^{V[G]}$ has the $\ka$-norm covering and the $\ka$-approximation properties for $V[G]$,
we know that $W_t'=W_s^{V[G]}$
has the $\ka$-norm covering and the $\ka$-approximation properties for $V$ by
Lemma \ref{2.7}.
Now take $p \in \bbP$.
Since $\Vdash_\bbP$``$W_{\dot s}^{V[\dot G]} =\check W_t' \subseteq \check V$'',
we can choose $q \le p$ and a set $X$ such that
$q \Vdash_\bbP$``$X= (W_{\dot s}^{V[\dot G]})_\la$''.
Put $u=\seq{X, \la, \ka}$.
$W_t'$ has the $\ka$-norm covering and the $\ka$-norm approximation properties for $V$
and $X=(W_t')_\la$.
Hence $W_t'=W_u'$ by the definition of $W_u'$ and Theorem \ref{2.5}.
For the converse,
suppose $\psi(\bbP, \dot s, t)$ holds.
Fix \LS cardinals $\ka<\la$ witnessing $\psi(\bbP, \dot s, t)$.
Fix $p \in \bbP$, and take $q \le p$ and a set $X$
such that
$q \Vdash_x$``$X=(W_{\dot s}^{V[\dot G]})_\la$''.
Let $u=\seq{X, \la,\ka}$.
We know that $W_t'=W_u'$, $(W_u')_\la=X$, and $W_t'=W_u'$ has the $\ka$-norm covering and the $\ka$-norm approximation properties for $V$.
Now take a $(V,\bbP)$-generic $G$.
In $V[G]$, $\ka$ and $\la$ remain \LS cardinals,
and $W_s^{V[G]}$ has the $\ka$-norm covering and the $\ka$-norm approximation properties
for $V[G]$.
Moreover $V$ has the $\ka$-norm covering and the $\ka$-norm approximation properties for $V[G]$.
Then so does $W_u'$ for $V[G]$ by Lemma \ref{2.7}.
Applying Theorem \ref{2.5},
we have $W_t'=W_u'=W_s^{V[G]}$.
\end{proof}
\begin{define}[Blass \cite{B}]
The principle $\SVC$ (Small Violation of Choice)
is the assertion that there is a set $S$ such that for every set $X$, there is an ordinal
$\alpha$ and a surjection from $S \times \alpha$ onto $X$.
\end{define}
\begin{thm}[\cite{B}]
The following are equivalent:
\begin{enumerate}
\item $\SVC$ holds.
\item $\AC$ is forceable, that is,
there is a poset $\bbP$ which forces $\AC$.
\end{enumerate}
\end{thm}
By this theorem, we know that
$\SVC$ is absolute between $V$, any ground, and any generic extension of $V$.
\begin{thm}[\cite{U2}]
Suppose $\SVC$ holds.
Then $\CLS$ holds, in particular all grounds of $V$ are uniformly definable.
\end{thm}
\section{Symmetric grounds}
In this section, we study a characterization of
symmetric grounds without taking \emph{some} generic extension,
and we prove the uniform definability of symmetric grounds under $\CLS$.
\begin{lemma}\label{3.1}
Let $M, N$ be models of $\ZF$ and suppose $M$ is a ground of $V$,
say $V=M[G]$.
If $M \subseteq N \subseteq V$ and $N$ is a symmetric submodel of $V=M[G]$,
then $N$ is of the form $M(X)$ for some $X$.
\end{lemma}
\begin{proof}
By Theorem \ref{1.6},
$N$ is of the form $\HOD_{M(Y)}$ for some $Y$.
Since $V=M[G]=M(Y)(G)$, we know that $N=\HOD_{M(Y)}$ is of the form $M(Y)(Z)$ for some $Z$ with $Z \subseteq M(Y)$ by Theorem \ref{1.3}. Then $N=M(\{Y,Z\})$.
\end{proof}
\begin{lemma}\label{3.2}
Let $M, N$ be models of $\ZF$ and suppose $M$ is a ground of $V$, say $V=M[G]$.
Suppose $M \subseteq N \subseteq V$,
$M$ is a class of $N$, and $N$ is of the form $M(X)$ for some $X$.
Then there is a large limit $\alpha$ such that
whenever $G'$ is $(N, \col(N_\alpha))$,
there is an $(M, \col(M_\alpha))$-generic $H$
such that $N[G']=M[H]$ and $N$ is a symmetric submodel of $M[H]$.
\end{lemma}
\begin{note}
As in Note \ref{1.4}, the conclusion of this proposition
can be read as follows:
There is some $\alpha$ such that,
in $N$, $\col(N_\alpha)$ forces ``
there is an $(\check M, \col(M_\alpha))$-generic $H$
such that $\check {M}[H]$ is the universe,
and $\check N$ is a symmetric submodel of $\check M[H]$''.
\end{note}
\begin{proof}
By Theorem \ref{1.2}, $V=M[G]$ is a generic extension of $N$.
Applying Lemma \ref{1.5+}, there is a large limit $\alpha$ such that
if $G$ is $(V, \col(V_\alpha))$-generic,
there is an $(M, \col(M_\alpha))$-generic $H$ and
an $(N, \col(N_\alpha))$-generic $H'$
with $V[G]=M[H]=N[H']$.
Then, in $N$, $\col(N_\alpha)$ forces that
``there is an $(\check M, \col(M_\alpha))$-generic $H$ such
that $\check M[H]$ is the universe''.
Hence whenever $G'$ is $(N, \col(N_\alpha))$-generic,
there is an $(M, \col(M_\alpha))$-generic $H$
such that $N[G']=M[H]$.
We shall see that $N$ is of the form $(\HOD_{M(X)})^{M[H]}$, hence $N$ is a symmetric submodel of $M[H]$ by Theorem \ref{1.6}.
Since $N=M(X)$, it is clear that $N \subseteq (\HOD_{M(X)})^{M[H]}$.
Because $\col(N_\alpha)$ is a weakly homogeneous ordinal definable poset in $N$,
we have $(\HOD_{M(X)})^{M[H]}=(\HOD_N)^{N[G']} \subseteq N$.
\end{proof}
We have the following observation.
Roughly speaking, it asserts that $W$ is a symmetric ground of $V$ if and only if
$W$ is a ground of \emph{some} generic extension of $V$.
\begin{prop}\label{3.4}
Let $M\subseteq N$ be models of $\ZF$, and suppose $M$ is a class of $N$.
\begin{enumerate}
\item Suppose $M$ is a ground of $V$, say $V=M[G]$.
If $N$ is a symmetric submodel of $M[G]=V$,
then there are a poset $\bbP\in N$ and $\bbQ \in M$
such that, in $N$, $\bbP$ forces ``there is an $(\check M, \bbQ)$-generic $H$
such that $\check M[H]$ is the universe''.
\item If there are a poset $\bbP \in N$ and $\bbQ \in M$
such that, in $N$, $\bbP$ forces ``there is an $(\check M, \bbQ)$-generic $H$
such that $\check M[H]$ is the universe'',
then $N$ is a symmetric submodel of some generic extension of $M$.
\end{enumerate}
\end{prop}
\begin{proof}
(1). By Lemma \ref{3.1},
$N$ is of the form $M(X)$ for some set $X \in N$.
Then by Lemma \ref{3.2},
there is some large $\alpha$
such that, in $N$, $\col(N_\alpha)$ forces that
``there is an $(\check M, \col(\check M_\alpha))$-generic $H$
such that $\check M[H]$ is the universe''.
(2). Take an $(N, \bbP)$-generic $G$.
In $N[G]$, there is $H \subseteq \bbQ \in M$ with $N[G]=M[H]$.
$M \subseteq N \subseteq N[G]=M[H]$,
hence $N$ is of the form $M(X)$ for some $X \in M[H]$
by Theorem \ref{1.2}, and $N$ is a symmetric submodel of some generic extension of $M[H]$ by Lemma \ref{3.2}.
\end{proof}
Using this proposition, we can obtain a formal definition of symmetric grounds.
\begin{define}\label{3.5}
Let $W$ be a model of $\ZF$. Let us say that $W$ is a \emph{symmetric ground} of $V$,
or $V$ is a \emph{symmetric extension} of $W$,
if there are posets $\bbP \in V$ and $\bbQ \in W$ such that
$\bbP$ forces that
``there is a $(\check W, \bbQ)$-generic $H$
such that $\check W[H]$ is the universe''.
\end{define}
By Proposition \ref{3.4},
our notion of symmetric grounds and extensions coincide with the standard definition of symmetric submodels
and extensions.
Moreover our notion of symmetric extension is equivalent to
quasi-generic extension in Grigorieff \cite{G}.
\begin{note}
In Definition \ref{3.5},
$V$ need not be a symmetric extension of $W$ via a poset $\bbQ$.
\end{note}
\begin{note}
If $W$ is a symmetric ground of $V$,
then $V=M(X)$ for some set $X$ by Lemma \ref{3.1}.
But the converse does not hold;
For instance, if $0^\#$ exists,
then $V=L(0^\#)=L[0^\#]$ is not a symmetric extension of $L$.
\end{note}
By Lemma \ref{1.5+}, posets $\bbP$ and $\bbQ$ can be
Levy collapsings.
\begin{lemma}\label{3.7+}
Let $W$ be a model of $\ZF$.
Then $W$ is a symmetric ground of $V$ if and only if
for every sufficiently large limit ordinal $\alpha$,
$\col(V_\alpha)$ forces ``there is a $(\check W, \col(\check W_\alpha))$-generic $H$
such that $\check{W}[H]$ is the universe''.
\end{lemma}
Now we obtain a first order definition of symmetric grounds under $\CLS$.
\begin{thm}\label{3.7}
Suppose $\CLS$ (e.g., $\SVC$ holds).
Then there is a first order formula $\varphi(x,y)$ of set-theory such that:
\begin{enumerate}
\item For every set $r$, $\overline{W}_r=\{x \mid \varphi(x,r)\}$ is a symmetric ground of $V$ with $r \in \overline{W}_r$.
\item For every symmetric ground $W$ of $V$, there is $r$ with
$W=\overline{W}_r$.
\end{enumerate}
\end{thm}
\begin{proof}
Let $\{W_r'\mid r \in V \}$ be the collection of all pseudo-grounds defined as in Theorem \ref{2.9}.
Define $\overline{W}_r$ as follows:
If there is a poset $\bbP$ such that
$\Vdash_\bbP$``$\check W_{r}'$ is a ground of the universe'',
then $\overline{W}_r=W_r'$. Otherwise $\overline{W}_r=V$.
By Lemma \ref{2.10},
$\{\overline{W}_r \mid r \in V\}$ is
a first order definable collection.
We check that $\{\overline{W}_r \mid r \in V\}$ is the collection of all symmetric grounds of $V$.
For $r \in V$, if $\overline{W}_r \neq V$ then $\overline{W}_r=W_r'$ and there is a poset $\bbP$
such that $\Vdash_\bbP$``$\check W_r'$ is a ground of the universe''.
Hence $W_r'$ is a ground of a generic extension of $V$ via $\bbP$,
so $\overline{W}_r=W_r'$ is a symmetric ground of $V$.
For the converse,
suppose $W$ is a symmetric ground of $V$.
We can choose a generic extension $V[G]$ of $V$ via poset $\bbP$
such that $W$ is a ground of $V[G]$.
We can take a large \LS cardinal $\ka$
such that $\ka$ is \LS in V[G],
and $W$ have the $\ka$-norm covering and the $\ka$-norm approximation properties for $V[G]$.
Since $W \subseteq V \subseteq V[G]$,
$W$ has the $\ka$-norm covering and the $\ka$-norm approximation properties for $V$.
Hence $W=W_r'$ for some $r \in V$, and in $V$ we can choose a poset $\bbQ$
with $\Vdash_\bbQ$``$\check W_r'$ is a ground of the universe'',
so $W=\overline{W}_r$.
\end{proof}
Hence in $\ZFC$, we can always define all symmetric grounds uniformly.
\section{Some properties of symmetric extensions and grounds}
In this section, we make some observations and prove some useful properties of
symmetric extensions and grounds.
The following is partially proved by Grigorieff (\cite{G}).
\begin{lemma}\label{4.3}
Let $M$ and $N$ be models of $\ZF$ with $M \subseteq N \subseteq V$.
Then any two of the following conditions imply the third:
\begin{enumerate}
\item $N$ is a symmetric extension of $M$.
\item $V$ is a symmetric extension of $N$.
\item $V$ is a symmetric extension of $M$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1)$\land$(2) $\Rightarrow$ (3).
By Lemma \ref{3.7+},
there is some large limit ordinal $\alpha$
such that:
\begin{itemize}
\item In $N$,
$\col(N_\alpha)$ forces that ``there is an $(\check M , \col(\check{M}_\alpha))$-generic $H$
such that $\check M[H]$ is the universe''.
\item In $V$,
$\col(V_\alpha)$ forces that ``there is an $(\check N, \col(\check{N}_\alpha))$-generic $H$
such that $\check N[H]$ is the universe''.
\end{itemize}
Take a $(V, \col(V_\alpha))$-generic $G$.
Then there is an $(N,\col(N_\alpha))$-generic $H$
with $V[G]=N[H]$,
and there is an $(M, \col(M_\alpha))$-generic $H'$
with $M[H']=N[H]$.
Hence $M$ is a ground of the generic extension $V[G]$ of $V$,
and $M$ is a symmetric ground of $V$.
(1)$\land$(3) $\Rightarrow$ (2).
Since $V$ is a symmetric extension of $M$,
there is a generic extension $M[G]$ of $M$ such that
$V \subseteq M[G]$ and $V=M(X)$ for some $X \in V$ by Lemma \ref{3.1}.
By the same argument, $N=M(Y)$ for some $Y \in N$.
We know $M \subseteq N=M(Y) \subseteq V=M(X) \subseteq M[G]$.
Then $M[G]$ is a generic extension of $N=M(Y)$ by
Theorem \ref{1.2}.
We know $V=N(X)$, hence $V$ is a symmetric extension of $N$ by
Lemma \ref{3.2}.
(2)$\land$(3) $\Rightarrow$ (1). By Lemma \ref{3.7+},
there is some large $\alpha$ such that
whenever $G$ is $(V, \col(V_\alpha))$-generic, we have that
$V[G]=N[H]=M[H']$ for some $H \subseteq \col(N_\alpha) \in N$ and $H' \subseteq \col(M_\alpha) \in M$.
Hence $M$ is a ground of $N[H]$, so $M$ is a symmetric ground of $N$.
\end{proof}
By this lemma,
the ``is a symmetric extension of'' relation is transitive.
It is known that if $M \subseteq N \subseteq M[G]$ are models of $\ZFC$ and
$M[G]$ is a generic extension of $M$,
then $M$ is a ground of $N$, and $N$ is of $M[G]$ (compare Theorem \ref{1.2}).
We prove a variant of this fact in our context.
\begin{lemma}\label{5.3}
Suppose $M$ is a symmetric ground of $V$.
Then for every model $N$ of $\ZF$ with $M \subseteq N \subseteq V$,
the following are equivalent:
\begin{enumerate}
\item $N$ is a symmetric ground of $V$.
\item $N$ is of the form $M(X)$ for some set $X \in V$.
\item $N$ is a symmetric extension of $M$.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) $\iff$ (3) is immediate from Lemma \ref{4.3}.
(2) $\Rightarrow$ (1). Since $M$ is a symmetric ground,
there is a generic extension $V[G]$ of $V$ such that
$M$ is a ground of $V[G]$.
Then $N=M(X)$ is a ground of $V[G]$ by Theorem \ref{1.2},
so $N$ is a symmetric ground of $V$.
(1) $\Rightarrow$ (2) is Lemma \ref{3.1}.
\end{proof}
It is also known that the Bristol model (\cite{K})
is an intermediate model $N$ between the constructible universe $L$
and the Cohen forcing extension of $L$ but $N$ does not satisfy (1)--(3) of Lemma \ref{5.3}.
We can characterize $\SVC$ in terms of symmetric grounds.
Note that under $\AC$, all grounds are uniformly definable.
\begin{thm}[\cite{U1}]\label{DDG}
Suppose $V$ satisfies $\AC$,
and let $\{W_r \mid r \in V\}$ be the collection of all $\ZFC$-grounds of $V$.
Then for every set $X$, there is
a set $s$ such that $W_s \subseteq \bigcap_{r \in X} W_r$.
\end{thm}
\begin{prop}\label{4.5}
The following are equivalent:
\begin{enumerate}
\item $\SVC$ holds.
\item There is a symmetric ground satisfying $\AC$.
\item There is a symmetric ground $W$ satisfying $\AC$ and $V=W(X)$ for some set $X$.
\end{enumerate}
\end{prop}
\begin{proof}
(1) $\Rightarrow$ (2). Take a poset $\bbP$ which forces $\AC$.
Take a $(V,\bbP \times \bbP)$-generic $G \times H$.
Both $V[G]$ and $V[H]$ are grounds of $V[G][H]$ and satisfy $\AC$.
By Theorem \ref{DDG}, there is a ground $W$ of $V[G][H]$ such that
$W$ satisfies $\AC$ and $W \subseteq V[G] \cap V[H]$.
Since $V[G] \cap V[H]=V$ (e.g., see \cite{FHR}), we have $W \subseteq V \subseteq V[G][H]$.
Then $V$ is of the form $W(X)$ for some set $X$ by Theorem \ref{1.2}.
Now $V$ is a symmetric extension of $W$ by Lemma \ref{5.3}.
(2) $\Rightarrow$ (1).
There is a generic extension $W[G]$ of $W$ such that
$V$ is a ground of $W[G]$.
$W[G]$ satisfies $\AC$, so $W[G]$ witnesses that $\SVC$ holds in $V$.
(2) $\Rightarrow$ (3).
If $W$ is a symmetric ground satisfying $\AC$,
then $W$ is a ground of some generic extension $V[G]$ of $V$.
Applying Theorem \ref{1.2},
we have that $V=W(X)$ for some set $X \in V$.
(3) $\Rightarrow$ (2) is trivial.
\end{proof}
Next we prove the absoluteness of
$\CLS$ and $\SVC$ between all symmetric grounds and symmetric extensions.
\begin{prop}\label{4.7}
Let $W$ be a symmetric ground of $V$.
Then $\CLS$ and $\SVC$ are absolute between $W$ and $V$.
\end{prop}
\begin{proof}
Since $V$ is a symmetric extension of $W$,
we have that $V$ is a ground of some generic extension $W[G]$ of $W$.
Since $\CLS$ and $\SVC$ are absolute between all grounds and generic extensions,
it is also absolute between $W$, $V$, and $W[G]$.
\end{proof}
By the absoluteness of $\CLS$ and Theorem \ref{3.7},
we have:
\begin{cor}
Suppose $\CLS$ holds.
Then $V$ is definable in its symmetric extensions with parameters from $V$.\footnote{A. Karagila independently obtained this result assuming that $V$ satisfies $\AC$.}
\end{cor}
In particular, under $\AC$, $V$ is always definable in its symmetric extensions.
\begin{note}
As the uniform definability of grounds,
under $\CLS$, if the formula $\varphi$ defines all symmetric grounds as in Theorem \ref{3.7},
then in all symmetric grounds and symmetric extensions of $V$,
$\varphi$ also defines all its symmetric grounds.
This follows from the absoluteness of $\CLS$ and the proof of Theorem \ref{3.7}.
\end{note}
Finally, we observe the structure of the symmetric models under $\AC$.
\begin{lemma}\label{5.8}
Suppose $V$ satisfies $\AC$. Then every symmetric ground satisfying $\AC$ is a ground.
\end{lemma}
\begin{proof}
Let $W$ be a symmetric ground of $V$.
Then $W$ is a ground of some generic extension $V[G]$ of $V$.
$V$, $V[G]$, and $W$ are models of $\ZFC$ with $W \subseteq V \subseteq V[G]$.
Then $W$ is a ground of $V$.
\end{proof}
\begin{cor}
Suppose $V$ satisfies $\AC$.
Then for every model $M$ of $\ZF$,
the following are equivalent:
\begin{enumerate}
\item $M$ is a symmetric ground of $V$.
\item $M$ is a ground of $V$.
\item There is a $\ZFC$-ground $W$ of $V$
and a set $X$ such that $M=W(X)$.
\end{enumerate}
\end{cor}
\begin{proof}
(3) $\Rightarrow$ (2). If $M=W(X)$ for some ground $W$ of $V$,
then $M$ is a ground by Theorem \ref{1.2}.
(2) $\Rightarrow$ (1) is trivial.
(1) $\Rightarrow$ (3). Suppose $M$ is a symmetric ground of $V$.
Since $V$ satisfies $\SVC$ trivially,
$\SVC$ holds in $M$ as well by Proposition \ref{4.7}.
Then $M$ has a symmetric ground $W$ satisfying $\AC$ and
a set $X$ with $M=W(X)$ by Proposition \ref{4.5}.
By Lemma \ref{4.3}, $W$ is a symmetric ground of $V$,
and in fact it is a ground by Lemma \ref{5.8}.
\end{proof}
We also note the following, which contrasts with Lemma \ref{5.3}.
\begin{cor}
Let $M$ be a symmetric ground of $V$.
Then for every model $N$ of $\ZF+\SVC$ with $M \subseteq N \subseteq V$,
$N$ is a symmetric ground of $V$, and $N$ is a symmetric extension of $M$.
\end{cor}
\begin{proof}
By Proposition \ref{4.5},
$N$ has a symmetric ground $W$ satisfying $\AC$ and $N$ is of the form $W(X)$.
Since $M$ is a symmetric ground of $V$,
there is a set $Y$ with $V=M(Y)$.
Then clearly $V=W(\{X,Y\})$, and $\SVC$ holds in $V$. Force with $\col(\mathrm{trcl}(\{X,Y\}))$ over $V$.
This produces a generic extension $V[G]$ of $V$
such that $V[G]$ which satisfies $\AC$.
Moreover $\SVC$ holds in $M$ as well by Proposition \ref{4.7}.
Thus there is a symmetric ground $U$ of $M$
such that $U$ satisfies $\AC$ by Proposition \ref{4.5} again.
Then $U$ is a symmetric ground of $V[G]$, hence is a ground of $V[G]$ by Lemma \ref{5.8}.
Now $U \subseteq W \subseteq V[G]$ and $V[G]$ is a generic extension of $U$,
so $W$ is a ground of $V[G]$.
Therefore $N=W(X)$ is a symmetric ground of $V[G]$, and
so $N$ is a symmetric ground of $V$.
In addition $N$ is a symmetric extension of $M$ by Lemma \ref{5.3}.
\end{proof}
\section{The downward directedness of symmetric grounds}
In $\ZFC$,
the collection of $\ZFC$-grounds is downward directed (Theorem \ref{DDG}).
Unlike in the $\ZFC$-context, under $\ZF$+$\SVC$,
it is possible that $V$ has two grounds which have no common ground
(see \cite{U2}).
On the other hand, we prove that under $\SVC$,
all symmetric grounds are downward directed.
\begin{prop}\label{6.1}
Suppose $\SVC$ holds (hence all symmetric grounds are uniformly definable),
and let $\{\overline{W}_r \mid r \in V\}$ be the collection of all symmetric grounds.
Then for every set $X$,
there is a symmetric ground $W$ of $V$
such that $W \subseteq \bigcap_{r \in X} \overline{W}_r$,
and $W$ is a symmetric ground of each $\overline{W}_r$ for $r \in X$.
\end{prop}
\begin{proof}
By Corollary \ref{4.7},
$\SVC$ holds in $\overline{W}_r$ for every $r \in X$.
By Proposition \ref{4.5},
each $\overline{W}_r$ has a symmetric ground satisfying $\AC$.
Every symmetric ground of $\overline{W}_r$ is a symmetric ground of $V$ as well
by Lemma \ref{4.3}.
Hence we can find a set $Y$ such that
for each $s \in Y$, $\overline{W}_s$ satisfies $\AC$,
and for every $r \in X$ there is $s \in Y$
with $\overline{W}_s \subseteq \overline{W}_r$.
Take a generic extension $V[G]$ of $V$ such that $\AC$ holds in $V[G]$.
By Lemma \ref{5.8}, for each $s \in Y$, $\overline{W}_s$ is a ground of $V[G]$.
By Theorem \ref{DDG},
there is a ground $W$ of $V[G]$
such that $W$ satisfies $\AC$ and $W \subseteq \overline{W}_{s}$ for every $s \in Y$.
Now, for each $s \in Y$,
we have $W \subseteq \overline{W}_s \subseteq V \subseteq V[G]$.
Since $V[G]$ is a generic extension of $V$ and $W$,
we know that $W$ is a symmetric ground of $V$.
By the choice of $Y$, we have that $W \subseteq \bigcap_{r \in X} \overline{W}_r$.
In addition, by Lemma \ref{4.3},
$W$ is a symmetric ground of $\overline{W}_r$ for every $r \in X$.
\end{proof}
\section{Mantles}
If all grounds are uniformly definable as in Theorem \ref{unif. def. grounds},
then we can define the intersection of all grounds.
We say that the intersection of all grounds is the \emph{$\ZF$-mantle}, or simply the \emph{mantle},
and the intersection of all $\ZFC$-grounds is the \emph{$\ZFC$-mantle}.
The mantle and $\ZFC$-mantle are important objects in set-theoretic geology.
Suppose $\CLS$, and let $\varphi$ be the formula
which defines all grounds uniformly as in Theorem \ref{unif. def. grounds}.
As mentioned in Note \ref{notedayo}, in all generic extensions of $V$,
$\varphi$ also defines all its grounds by the same way.
Then we can define the \emph{generic mantle},
which is the intersection of all grounds of all generic extensions.
\begin{define}[\cite{FHR}, \cite{U2}]
Suppose $\CLS$.
\begin{enumerate}
\item The \emph{$\ZF$-mantle}, or simply the \emph{mantle}, $\bbM^{\ZF}$ is the intersection of all grounds,
and the \emph{$\ZFC$-mantle} $\bbM^{\ZFC}$ is the intersection of all $\ZFC$-grounds.
\item The \emph{$\ZF$-generic mantle}, or the \emph{generic mantle}, $g\bbM^{\ZF}$ is the intersection of all grounds of
all generic extensions,
and the \emph{$\ZFC$-generic mantle} $g\bbM^{\ZFC}$
is the intersection of all $\ZFC$-grounds of
all generic extensions.
\end{enumerate}
\end{define}
It is known that the ($\ZFC$-)mantle and the ($\ZFC$-)generic mantle are parameter free definable
transitive classes (under $\CLS$).
In $\ZFC$, it is known that the $\ZFC$-mantle is a model of $\ZFC$,
and it coincides with the $\ZFC$-generic mantle (\cite{FHR}, \cite{U1}).
In $\ZF(+\CLS)$, the generic mantle is a model of $\ZF$, and it is a forcing invariant model
(\cite{FHR}, \cite{U2}).
\begin{thm}[\cite{U2}]
If $\SVC$ holds,
then the generic mantle $g\bbM^{\ZF}$ is a model of $\ZFC$.
Moreover, $g\bbM^{\ZF}$ coincides with $g\bbM^{\ZFC}$.
\end{thm}
Let us consider the intersection of all symmetric grounds.
\begin{define}
Suppose $\CLS$.
The \emph{symmetric mantle} $s\bbM$ is the intersection of all symmetric grounds.
\end{define}
As the mantle and the generic mantle,
the symmetric mantle is a parameter-free first order definable transitive class
containing all ordinals.
By the definitions, we have $g \bbM^{\ZF} \subseteq s\bbM \subseteq \bbM^{\ZF}$.
\begin{prop}
If $\SVC$ holds,
then the symmetric mantle coincides with the generic mantle.
In particular the symmetric mantle is a model of $\ZFC$.
\end{prop}
\begin{proof}
Take a set $x$ and suppose $x \notin g\bbM^{\ZF}$.
Then there is a generic extension $V[G]$ of $V$ and a ground $W$ of $V[G]$ such that
$x \notin W$. Note that $W$ need not be a symmetric ground of $V$,
but $W$ and $V$ are symmetric grounds of $V[G]$.
By Proposition \ref{6.1}, there is $M$ which is a common symmetric ground of $W$ and $V$.
Then $x \notin M$, hence $x \notin s\bbM$.
\end{proof}
However, beside the generic mantle, it is not known if $s \bbM$ and $\bbM^{\ZF}$ are always models of $\ZF$.
\section{Questions}
To conclude this paper let us pose some questions.
\begin{question}
Without any assumption, are all symmetric grounds uniformly definable?
\end{question}
This question is almost equivalent to the uniform definability of all grounds in $\ZF$,
this is also asked in \cite{U2}.
\begin{question}
Is the symmetric mantle a model of $\ZF$?
\end{question}
\begin{question}
Does the symmetric mantle always coincide with the generic mantle?
\end{question}
As in the context of $\ZFC$,
the downward directedness of symmetric grounds
yields a positive answer of this question.
\begin{question}
Without any assumption, are all symmetric grounds downward directed?
\end{question}
\begin{question}
Is it consistent that the symmetric mantle is strictly smaller than the mantle?
\end{question}
Under $V=L$, there is no proper symmetric ground.
However we do not know a choiceless model which has no proper symmetric ground.
\begin{question}
Is it consistent that $V$ does not satisfy $\AC$, and has no proper symmetric ground?
\end{question}
If there exists such a model, then $\SVC$ must fail in the model by Proposition \ref{4.5}.
A similar question is:
\begin{question}
Is it consistent that $V$ has a proper symmetric ground but has no
proper ground?
\end{question}
If such a model exists, then $\AC$ must fail in the model, and
the symmetric mantle of the model is strictly smaller than
the mantle.
\section*{Acknowledgments}
The author would like to thank Asaf Karagila for many valuable comments.
The author also thank the referee for many useful comments and suggestions.
This research was supported by
JSPS KAKENHI Grant Nos. 18K03403 and 18K03404.
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