From caaeb327bca574b9833600f462c4ca58e7b33274 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 8 Dec 2020 15:54:13 +0100
Subject: Reorganized Cauchy-Riemann and matrix forms.

---
 bezout.tex | 58 ++++++++++++++++++++++++++++++++++------------------------
 1 file changed, 34 insertions(+), 24 deletions(-)

diff --git a/bezout.tex b/bezout.tex
index 96245af..880c490 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -97,6 +97,40 @@ points it has is given by the usual Kac--Rice formula:
         \end{bmatrix}\right|.
   \end{aligned}
 \end{equation}
+The Cauchy--Riemann equations may be used to write \eqref{eq:real.kac-rice} in
+a manifestly complex way.  Using the Wirtinger derivative
+$\partial=\partial_x-i\partial_y$, one can write
+$\partial_x\mathop{\mathrm{Re}}H=\mathop{\mathrm{Re}}\partial H$ and
+$\partial_y\mathop{\mathrm{Re}}H=-\mathop{\mathrm{Im}}\partial H$. Carrying
+these transformations through, we have
+\begin{equation} \label{eq:complex.kac-rice}
+  \begin{aligned}
+    \mathcal N_J&(\kappa,\epsilon)
+      = \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H) \\
+        &\qquad\qquad\qquad\times\left|\det\begin{bmatrix}
+            \mathop{\mathrm{Re}}\partial\partial H & -\mathop{\mathrm{Im}}\partial\partial H \\
+            -\mathop{\mathrm{Im}}\partial\partial H & -\mathop{\mathrm{Re}}\partial\partial H
+          \end{bmatrix}\right| \\
+      &= \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H)
+        \left|\det[(\partial\partial H)^\dagger\partial\partial H]\right| \\
+      &= \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H)
+        |\det\partial\partial H|^2.
+  \end{aligned}
+\end{equation}
+This gives three equivalent expressions for the determinant of the Hessian: as
+that of a $2N\times 2N$ real matrix, that of an $N\times N$ Hermitian matrix,
+or the norm squared of that of an $N\times N$ complex symmetric matrix.
+
+These equivalences belie a deeper connection between the spectra of the
+corresponding matrices: each eigenvalue of the real matrix has a negative
+partner. For each pair $\pm\lambda$ of the real matrix, $\lambda^2$ is an
+eigenvalue of the Hermitian matrix. Finally, $|\lambda|$ is a \emph{singular
+value} of the complex symmetric matrix. The distribution of positive
+eigenvalues of the Hessian is therefore the same as the distribution of
+singular values of $\partial\partial H$, while both are the same as the
+distribution of square-rooted eigenvalues of $(\partial\partial
+H)^\dagger\partial\partial H$.
+
 {\color{red} {\bf perhaps not here} This expression is to be averaged over the $J$'s as 
 $N \Sigma= 
 \overline{\ln \mathcal N_J} = \int dJ \; \ln N_J$, a calculation that involves the replica trick. In most, but not all,  of the parameter-space that we shall study here, the {\em annealed approximation} $N \Sigma \sim 
@@ -107,30 +141,6 @@ so that we may treat the delta-functions and the Hessians as independent.
 
 }
 
-The Cauchy--Riemann relations imply that the matrix is of the form:
-\begin{equation} \label{eq:real.kac-rice1}
-      \begin{bmatrix}
-        \bar A & \bar B \\
-        \bar B & -\bar A
-      \end{bmatrix}
-\end{equation}
-with $\bar A=-\mathop{\mathrm{Re}}\partial\partial H$ and $\bar B=-\mathop{\mathrm{Im}}\partial\partial H$ Gaussian real symmetric matrices, \emph{correlated}, as we shall see..
-Using the Wirtinger
-derivative $\partial=\partial_x-i\partial_y$, one can write
-$\partial_x\mathop{\mathrm{Re}}H=\mathop{\mathrm{Re}}\partial H$ and
-$\partial_y\mathop{\mathrm{Re}}H=-\mathop{\mathrm{Im}}\partial H$. With similar
-transformations, the eigenvalue spectrum of the Hessian of
-$\mathop{\mathrm{Re}}H$ can be shown to be equivalent to the singular value
-spectrum of the Hessian $\partial\partial H$ of $H$, and as a result the
-determinant that appears above is equivalent to $|\det\partial\partial H|^2$.
-This allows us to write \eqref{eq:real.kac-rice} in the manifestly complex
-form
-\begin{equation} \label{eq:complex.kac-rice}
-  \mathcal N_J(\kappa,\epsilon)
-    = \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H)
-      |\det\partial\partial H|^2.
-\end{equation}
-
 The Hessian of \eqref{eq:constrained.hamiltonian} is $\partial\partial
 H=\partial\partial H_0-p\epsilon I$, or the Hessian of
 \eqref{eq:bare.hamiltonian} with a constant added to its diagonal. The
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