From ecc063dcfd7f6a9d9958af24583df6c59f192452 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 10 Feb 2022 00:29:38 +0100
Subject: More work on the matrix ensemble section.

---
 stokes.tex | 128 ++++++++++++++++++++++++++++++++++++++-----------------------
 1 file changed, 81 insertions(+), 47 deletions(-)

diff --git a/stokes.tex b/stokes.tex
index 1715160..33f2f5b 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -830,8 +830,10 @@ distribution in one complex variable $Z$ whose variances are
 $\overline{Z^*Z}=\overline{|Z|^2}=\Gamma$ and $\overline{Z^2}=C$. $\Gamma$ is
 positive, and $|C|\leq\Gamma$. The special case of $C=\Gamma$, where the
 variance of the complex variable and its covariance with its conjugate are the
-same, reduces to the ordinary normal distribution. Its probability density
-function is defined by
+same, reduces to the ordinary normal distribution. The case where $C=0$ results
+in the real and imaginary parts of $Z$ being uncorrelated, in what is known as
+the standard complex normal distribution. Its probability density function is
+defined by
 \begin{equation}
   p(z\mid\Gamma,C)=
   \frac1{\pi\sqrt{\Gamma^2-|C|^2}}\exp\left\{
@@ -841,41 +843,83 @@ function is defined by
   \right\}
 \end{equation}
 
-We will consider an ensemble of random matrices $A=A_0+\lambda_0I$, where the
-entries of $A_0$ are complex-normal distributed with variance $\Gamma=1/N$ and
-$\lambda_0$ is some constant shift to its diagonal. The eigenvalue distribution
-of these matrices is already known to take the form of an elliptical ensemble,
-with constant support inside the ellipse defined by
+We will consider an ensemble of random $N\times N$ matrices $B=A+\lambda_0I$, where the
+entries of $A$ are complex-normal distributed with variances
+$\overline{|A|^2}=A_0/N$ and $\overline{A^2}=C_0/N$, and $\lambda_0$ is a
+constant shift to its diagonal. The eigenvalue distribution of these matrices
+is already known to take the form of an elliptical ensemble, with constant
+support inside the ellipse defined by
 \begin{equation} \label{eq:ellipse}
-  \left(\frac{\operatorname{Re}(\lambda e^{i\theta})}{\Gamma+|C|}\right)^2+
-  \left(\frac{\operatorname{Im}(\lambda e^{i\theta})}{\Gamma-|C|}\right)^2
-  <1
+  \left(\frac{\operatorname{Re}(\lambda e^{i\theta})}{1+|C_0|/A_0}\right)^2+
+  \left(\frac{\operatorname{Im}(\lambda e^{i\theta})}{1-|C_0|/A_0}\right)^2
+  <A_0
 \end{equation}
-where $\theta=\frac12\arg C$ \cite{Nguyen_2014_The}. The eigenvalue
-spectrum of $A$ is therefore constant inside the same ellipse
+where $\theta=\frac12\arg C_0$ \cite{Nguyen_2014_The}. The eigenvalue
+spectrum of $B$ is therefore constant inside the same ellipse
 translated so that its center lies at $\lambda_0$.  Examples of these
 distributions are shown in the insets of Fig.~\ref{fig:spectra}.
 
-The eigenvalue spectrum of the Hessian of the real part is not the
-spectrum $\rho(\lambda)$ of $\partial\partial H$, but instead the
-square-root eigenvalue spectrum of $(\partial\partial H)^\dagger\partial\partial H$;
-in other words, the singular value spectrum $\rho(\sigma)$ of $\partial\partial
-H$. When $\kappa=0$ and the elements of $J$ are standard complex normal, this
-is a complex Wishart distribution. For $\kappa\neq0$ the problem changes, and
-to our knowledge a closed form is not in the literature.  We have worked out an
-implicit form for the singular value spectrum using the replica method.
+When $C=0$ and the elements of $A$ are standard complex normal, the singular
+value distribution of $B$ is a complex Wishart distribution. For $C\neq0$ the
+problem changes, and to our knowledge a closed form is not in the literature.
+We have worked out an implicit form for the singular value spectrum using the
+replica method.
+
+The singular values of $B$ correspond with the square-root of the eigenvalues
+of $B^\dagger B$, but also they correspond to the absolute value of the
+eigenvalues of the real $2N\times2N$ block matrix
+\begin{equation}
+  \left[\matrix{\operatorname{Re}B&-\operatorname{Im}B\cr-\operatorname{Im}B&-\operatorname{Re}B}\right]
+\end{equation}
+as we saw in \S\ref{sec:stationary.hessian}. The eigenvalue spectrum of this
+block matrix can be studied by ordinary means.  Defining the `partition function'
+\begin{equation}
+  Z(\sigma)=\int dx\,dy\,\exp\left\{
+    -\frac12\left[\matrix{x&y}\right]
+    \left(\sigma I-
+    \left[\matrix{\hphantom{-}\operatorname{Re}B&-\operatorname{Im}B\cr-\operatorname{Im}B&-\operatorname{Re}B}\right]
+  \right)
+    \left[\matrix{x\cr y}\right]
+  \right\}
+\end{equation}
+implies a Green function
+\begin{equation}
+  G(\sigma)=\frac\partial{\partial\sigma}\log Z(\sigma)
+\end{equation}
+This can be put into a manifestly complex form in the same way it was done in \S\ref{sec:stationary.hessian}, using the same linear transformation of $x$ and $y$ into $z$ and $z^*$. This gives
+\begin{eqnarray}
+  Z(\sigma)
+  &=\int dz\,dz^*\,\exp\left\{
+    -\frac12\left[\matrix{z^*&-iz}\right]
+    \left(\sigma I-
+    \left[\matrix{0&(iB)^*\cr iB&0}\right]
+  \right)
+    \left[\matrix{z\cr iz^*}\right]
+  \right\} \\
+  &=\int dz\,dz^*\,\exp\left\{
+    -\frac12\left(
+      2z^\dagger z\sigma-z^\dagger B^*z^*-z^TBz
+    \right)
+  \right\} \\
+  &=\int dz\,dz^*\,\exp\left\{
+    -z^\dagger z\sigma+\operatorname{Re}(z^TBz)
+  \right\}
+\end{eqnarray}
+which is a general expression for the singular values $\sigma$ of a symmetric
+complex matrix $B$.
+
 
 Introducing replicas to bring the partition function into the numerator of the
 Green function \cite{Livan_2018_Introduction} gives
 \begin{equation} \label{eq:green.replicas}
-  G(\sigma)=\lim_{n\to0}\int d\zeta\,d\zeta^*\,(\zeta_i^{(0)})^*\zeta_i^{(0)}
+  G(\sigma)=\lim_{n\to0}\int dz\,dz^*\,(z^{(0)})^\dagger z^{(0)}
     \exp\left\{
-    \frac12\sum_\alpha^n\left[(\zeta_i^{(\alpha)})^*\zeta_i^{(\alpha)}\sigma
-      -\operatorname{Re}\left(\zeta_i^{(\alpha)}\zeta_j^{(\alpha)}\partial_i\partial_jH\right)
+    -\sum_\alpha^n\left[(z^{(\alpha)})^\dagger z^{(\alpha)}\sigma
+      +\operatorname{Re}\left((z^{(\alpha)})^TBz^{(\alpha)}\right)
     \right]
   \right\},
 \end{equation}
-with sums taken over repeated Latin indices. The average is then made over
+The average is then made over
 $J$ and Hubbard--Stratonovich is used to change variables to the replica matrices
 $N\alpha_{\alpha\beta}=(\zeta^{(\alpha)})^\dagger\zeta^{(\beta)}$ and
 $N\chi_{\alpha\beta}=(\zeta^{(\alpha)})^T\zeta^{(\beta)}$, and a series of
@@ -885,8 +929,8 @@ $\chi_{\alpha\beta}=\chi_0\delta_{\alpha\beta}$. The result is
 \begin{equation}\label{eq:green.saddle}
   \overline G(\sigma)=N\lim_{n\to0}\int d\alpha_0\,d\chi_0\,d\chi_0^*\,\alpha_0
   \exp\left\{nN\left[
-    1+\frac{p(p-1)}{16}r^{p-2}\alpha_0^2-\frac{\alpha_0\sigma}2+\frac12\log(\alpha_0^2-|\chi_0|^2)
-    +\frac p4\operatorname{Re}\left(\frac{(p-1)}8\kappa^*\chi_0^2-\epsilon^*\chi_0\right)
+    1+\frac18A_0\alpha_0^2-\frac{\alpha_0\sigma}2+\frac12\log(\alpha_0^2-|\chi_0|^2)
+    +\frac14\operatorname{Re}\left(\frac14C_0^*\chi_0^2+\lambda_0^*\chi_0\right)
   \right]\right\}.
 \end{equation}
 
@@ -899,18 +943,15 @@ $\chi_{\alpha\beta}=\chi_0\delta_{\alpha\beta}$. The result is
   \includegraphics{figs/spectra_1.5.pdf}
 
   \caption{
-    Eigenvalue and singular value spectra of the Hessian $\partial\partial H$
-    of the $3$-spin model with $\kappa=\frac34e^{-i3\pi/4}$. Pictured
-    distributions are for critical points at `radius' $r=\sqrt{5/4}$ and with
-    energy per spin (a) $\epsilon=0$, (b)
-    $\epsilon=-\frac12|\epsilon_{\mathrm{th}}|$, (c)
-    $\epsilon=-|\epsilon_{\mathrm{th}}|$, and (d)
-    $\epsilon=-\frac32|\epsilon_{\mathrm{th}}|$. The shaded region of each
+    Eigenvalue and singular value spectra of a random matrix $A+\lambda_0I$, where the entries of $A$ are complex-normal distributed with $\overline{|A|^2}=A_0=\sqrt{5/4}$ and $\overline{A^2}=C_0=\frac34e^{-i3\pi/4}$.
+    The diaginal shifts differ in each plot, with (a) $\lambda_0=0$, (b)
+    $\lambda_0=\frac12|\lambda_{\mathrm{th}}|$, (c)
+    $\lambda_0=|\lambda_{\mathrm{th}}|$, and (d)
+    $\lambda_0=\frac32|\lambda_{\mathrm{th}}|$. The shaded region of each
     inset shows the support of the eigenvalue distribution \eqref{eq:ellipse}.
     The solid line on each plot shows the distribution of singular values
     \eqref{eq:spectral.density}, while the overlaid histogram shows the
-    empirical distribution from $2^{10}\times2^{10}$ complex normal matrices
-    with the same covariance and diagonal shift as $\partial\partial H$.
+    empirical distribution from $2^{10}\times2^{10}$ complex normal matrices.
   } \label{fig:spectra}
 \end{figure}
 
@@ -939,12 +980,11 @@ of singular values becomes gapped---is reduced to the geometry problem of
 determining when the boundary of the ellipse defined in \eqref{eq:ellipse}
 intersects the origin, and yields
 \begin{equation} \label{eq:threshold.energy}
-  |\epsilon_{\mathrm{th}}|^2
-  =\frac{p-1}{2p}\frac{(1-|\delta|^2)^2r^{p-2}}
-  {1+|\delta|^2-2|\delta|\cos(\arg\kappa+2\arg\epsilon)}
+  |\lambda_{\mathrm{th}}|^2
+  =\frac{(1-|\delta|^2)^2}
+  {1+|\delta|^2-2|\delta|\cos(\arg C_0+2\arg\lambda_0)}
 \end{equation}
-for $\delta=\kappa r^{-(p-2)}$. Notice that the threshold depends on both the
-energy per spin $\epsilon$ on the `radius' $r$ of the saddle.
+for $\delta=C_0/A_0$.
 
 \section{The \textit{p}-spin spherical models}
 
@@ -1314,13 +1354,7 @@ physical dynamics, are a problem we hope to address in future work.
   }
 \end{figure}
 
-\subsection{(2 + 4)-spin}
-
-\begin{equation}
-  H_2+H_4
-\end{equation}
-
-\section{Numerics}
+\section{The $p$-spin spherical models: numerics}
 
 To confirm the presence of Stokes lines under certain processes in the $p$-spin, we studied the problem numerically.
 
-- 
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