From 075a6c7feff4aa5b5a7636482addafa4269893fa Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 9 Feb 2022 18:32:38 +0100
Subject: More work on updating the matrix ensemble section.

---
 stokes.tex | 45 +++++++++++++++++++++++++--------------------
 1 file changed, 25 insertions(+), 20 deletions(-)

diff --git a/stokes.tex b/stokes.tex
index 286b3d9..1715160 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -841,28 +841,19 @@ function is defined by
   \right\}
 \end{equation}
 
-The Hessian $\partial\partial H=\partial\partial H_0-p\epsilon I$ is equal to
-the unconstrained Hessian with a constant added to its diagonal. The eigenvalue
-distribution $\rho$ is therefore related to the unconstrained distribution
-$\rho_0$ by a similar shift: $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The
-Hessian of the unconstrained Hamiltonian is
-\begin{equation} \label{eq:bare.hessian}
-  \partial_i\partial_jH_0
-  =\frac{p(p-1)}{p!}\sum_{k_1\cdots k_{p-2}}^NJ_{ijk_1\cdots k_{p-2}}z_{k_1}\cdots z_{k_{p-2}},
-\end{equation}
-which makes its ensemble that of Gaussian complex symmetric matrices, when the
-anomalous direction normal to the constraint surface is neglected. Given its variances
-$\overline{|\partial_i\partial_j H_0|^2}=p(p-1)r^{p-2}/2N$ and
-$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is
-constant inside the ellipse
+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
 \begin{equation} \label{eq:ellipse}
-  \left(\frac{\operatorname{Re}(\lambda e^{i\theta})}{r^{p-2}+|\kappa|}\right)^2+
-  \left(\frac{\operatorname{Im}(\lambda e^{i\theta})}{r^{p-2}-|\kappa|}\right)^2
-  <\frac{p(p-1)}{2r^{p-2}}
+  \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
 \end{equation}
-where $\theta=\frac12\arg\kappa$ \cite{Nguyen_2014_The}. The eigenvalue
-spectrum of $\partial\partial H$ is therefore constant inside the same ellipse
-translated so that its center lies at $-p\epsilon$.  Examples of these
+where $\theta=\frac12\arg C$ \cite{Nguyen_2014_The}. The eigenvalue
+spectrum of $A$ 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
@@ -1117,6 +1108,20 @@ which can in turn be used to compute the determinant. Then we will treat the
 $\delta$-functions and the resulting saddle point equations. The results of
 these calculations begin around \eqref{eq:bezout}.
 
+The Hessian $\partial\partial H=\partial\partial H_0-p\epsilon I$ is equal to
+the unconstrained Hessian with a constant added to its diagonal. The eigenvalue
+distribution $\rho$ is therefore related to the unconstrained distribution
+$\rho_0$ by a similar shift: $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The
+Hessian of the unconstrained Hamiltonian is
+\begin{equation} \label{eq:bare.hessian}
+  \partial_i\partial_jH_0
+  =\frac{p(p-1)}{p!}\sum_{k_1\cdots k_{p-2}}^NJ_{ijk_1\cdots k_{p-2}}z_{k_1}\cdots z_{k_{p-2}},
+\end{equation}
+which makes its ensemble that of Gaussian complex symmetric matrices, when the
+anomalous direction normal to the constraint surface is neglected. Given its variances
+$\overline{|\partial_i\partial_j H_0|^2}=p(p-1)r^{p-2}/2N$ and
+$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is
+constant inside the ellipse
 
 We will now address the $\delta$-functions of \eqref{eq:complex.kac-rice}.
 These are converted to exponentials by the introduction of auxiliary fields
-- 
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