From 572e2958fde0bdda61e8bc3b422c212c5381c167 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 7 Dec 2020 18:51:59 +0100
Subject: Tidied up notation.

---
 bezout.tex | 33 ++++++++++++++++++---------------
 1 file changed, 18 insertions(+), 15 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index de8ef6e..2b6512b 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -41,7 +41,7 @@ Spin-glasses have long been considered the paradigm of `complex landscapes' of m
 includes Neural Networks and optimization problems, most notably  Constraint Satisfaction ones.
 The most tractable family of these  are the mean-field spherical p-spin models defined by the energy:
 \begin{equation} \label{eq:bare.hamiltonian}
-  H_o = \sum_p \frac{c_p}{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p},
+  H_0 = \sum_p \frac{c_p}{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p},
 \end{equation}
 where the $J_{i_1\cdots i_p}$ are real Gaussian variables and the $z_i$ are real and constrained
 to a sphere $\sum_i z_i^2=N$. If there is a single  term of a given $p$, this is known as the `pure $p$-spin' model, the case we shall study here.
@@ -51,8 +51,8 @@ a Kac-Rice \cite{Kac,Fyodorov} procedure (similar to the Fadeev-Popov integral)
 Thanks to the relative simplicity of the energy, all these approaches are possible analytically in the large $N$ limit.
 
 In this paper we shall extend the study to the case  where $z\in\mathbb C^N$ are  and $J$ is a symmetric tensor
-whose elements are complex normal with $\langle|J|^2\rangle=p!/2N^{p-1}$ and
-$\langle J^2\rangle=\kappa\langle|J|^2\rangle$ for complex parameter $|\kappa|<1$. The constraint becomes $z^2=N$.
+whose elements are complex normal with $\overline{|J|^2}=p!/2N^{p-1}$ and
+$\overline{J^2}=\kappa\overline{|J|^2}$ for complex parameter $|\kappa|<1$. The constraint becomes $z^2=N$.
 
 The motivations for this paper are of two types. On the practical side, there are situations in which  complex variables
 have in a disorder problem appear naturally: such is the case in which they are {\em phases}, as in random laser problems \cite{Leuzzi-Crisanti, etc}. Another problem where a Hamiltonian very close to ours has been proposed is the Quiver Hamiltonians \cite{Anninos-Anous-Denef} modeling Black Hole horizons in the zero-temperature limit.  
@@ -84,12 +84,14 @@ $\mathop{\mathrm{Re}}H$. Writing $z=x+iy$, $\mathop{\mathrm{Re}}H$ can be
 interpreted as a real function of $2N$ real variables. The number of critical
 points it has is given by the usual Kac--Rice formula:
 \begin{equation} \label{eq:real.kac-rice}
-  \mathcal N_J(\kappa,\epsilon)
-    = \int dx\,dy\,\delta(\partial_x\mathop{\mathrm{Re}}H)\delta(\partial_y\mathop{\mathrm{Re}}H)
-      \left|\det\begin{bmatrix}
-        \partial_x\partial_x\mathop{\mathrm{Re}}H & \partial_x\partial_y\mathop{\mathrm{Re}}H \\
-        \partial_y\partial_x\mathop{\mathrm{Re}}H & \partial_y\partial_y\mathop{\mathrm{Re}}H
-      \end{bmatrix}\right|.
+  \begin{aligned}
+    \mathcal N_J(\kappa,\epsilon)
+      &= \int dx\,dy\,\delta(\partial_x\mathop{\mathrm{Re}}H)\delta(\partial_y\mathop{\mathrm{Re}}H) \\
+      &\qquad\times\left|\det\begin{bmatrix}
+          \partial_x\partial_x\mathop{\mathrm{Re}}H & \partial_x\partial_y\mathop{\mathrm{Re}}H \\
+          \partial_y\partial_x\mathop{\mathrm{Re}}H & \partial_y\partial_y\mathop{\mathrm{Re}}H
+        \end{bmatrix}\right|.
+  \end{aligned}
 \end{equation}
 {\color{red} {\bf perhaps not here} This expression is to be averaged over the $J$'s as 
 $N \Sigma= 
@@ -107,7 +109,8 @@ The Cauchy--Riemann relations imply that the matrix is of the form:
         \bar A & \bar B \\
         \bar B & -\bar A
       \end{bmatrix}
-\end{equation}with $\bar A$ and $\bar B$  Gaussian real symmetric matrices, {\em correlated}, as we shall see..
+\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
@@ -119,13 +122,13 @@ 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(\kappa,\epsilon)
+  \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 $\partial_i\partial_j
-H=\partial_i\partial_j H_0-p\epsilon\delta_{ij}$, or the Hessian of
+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
 eigenvalue distribution $\rho$ of the constrained Hessian is therefore related
 to the eigenvalue distribution $\rho_0$ of the unconstrained one by a similar
@@ -136,8 +139,8 @@ shift, or $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The Hessian of
   =\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. Given its variances
-$\langle|\partial_i\partial_j H_0|^2\rangle=p(p-1)a^{p-2}/2N$ and
-$\langle(\partial_i\partial_j H_0)^2\rangle=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is constant inside the ellipse
+$\overline{|\partial_i\partial_j H_0|^2}=p(p-1)a^{p-2}/2N$ and
+$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is constant inside the ellipse
 \begin{equation} \label{eq:ellipse}
   \left(\frac{\mathop{\mathrm{Re}}(\lambda e^{i\theta})}{a^{p-2}+|\kappa|}\right)^2+
   \left(\frac{\mathop{\mathrm{Im}}(\lambda e^{i\theta})}{a^{p-2}-|\kappa|}\right)^2
-- 
cgit v1.2.3-54-g00ecf

