From 8c8695de43dbc4d84c0441b2967b754d2a28dbda Mon Sep 17 00:00:00 2001
From: "kurchan.jorge" <kurchan.jorge@gmail.com>
Date: Thu, 10 Dec 2020 13:50:01 +0000
Subject: Update on Overleaf.

---
 bezout.tex | 27 +++++++++++++--------------
 1 file changed, 13 insertions(+), 14 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index 7da708b..51c1d99 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -161,7 +161,7 @@ partner. For each pair $\pm\lambda$ of the real matrix, $\lambda^2$ is an
 eigenvalue of the Hermitian matrix and $|\lambda|$ 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
+singular values of $\partial\partial H$,  the
 distribution of square-rooted eigenvalues of $(\partial\partial
 H)^\dagger\partial\partial H$.
 
@@ -172,7 +172,7 @@ study here, the {\em annealed approximation} $N \Sigma \sim \ln \overline{
 \mathcal N_J} = \ln \int dJ \; N_J$ is exact. 
 
 A useful property of the Gaussian distributions is that gradient and Hessian
-may be seen to be independent \cite{Bray_2007_Statistics,
+for given $\epsilon$ may be seen to be independent \cite{Bray_2007_Statistics,
 Fyodorov_2004_Complexity}, so that we may treat the $\delta$-functions and the
 Hessians as independent. We compute each by taking the saddle point. The
 $\delta$-functions are converted to exponentials by the introduction of
@@ -260,7 +260,7 @@ shift, or $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The Hessian of
   \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. Given its variances
+{\bf \color{red} restricting to  directions proportional to $z$, i.e. orthogonal to the constraint},  these makes its ensemble that of Gaussian complex symmetric matrices. Given its variances
 $\overline{|\partial_i\partial_j H_0|^2}=p(p-1)a^{p-2}/2N$ and
 $\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is
 constant inside the ellipse
@@ -274,17 +274,17 @@ spectrum of $\partial\partial H$ therefore is that of an ellipse in the complex
 plane whose center lies at $-p\epsilon$. Examples of these distributions are
 shown in the insets of Fig.~\ref{fig:spectra}.
 
-The eigenvalue spectrum of the Hessian of the real part, or equivalently the
-eigenvalue spectrum of $(\partial\partial H)^\dagger\partial\partial H$, is the
+The eigenvalue spectrum of the Hessian of the real part is the one we need for our Kac-Rice formula. It is  different from the spectrum $\partial\partial H$,  but rather equivalent  to the
+square-root eigenvalue spectrum of $(\partial\partial H)^\dagger\partial\partial H$,in other words, the
 singular value spectrum of $\partial\partial H$. When $\kappa=0$ and the
 elements of $J$ are standard complex normal, this corresponds to a complex
 Wishart distribution. For $\kappa\neq0$ the problem changes, and to our
-knowledge a closed form is not known.  We have worked out an implicit form for
+knowledge a closed form is not in the literature.  We have worked out an implicit form for
 this spectrum using the saddle point of a replica symmetric calculation for the
 Green function.
 
 Introducing replicas to bring the partition function to
-the numerator of the Green function gives
+the numerator of the Green function \cite{livan2018introduction} gives
 \begin{widetext}
   \begin{equation} \label{eq:green.replicas}
     G(\sigma)=\frac1N\lim_{n\to0}\int d\zeta\,d\zeta^*\,(\zeta_i^{(0)})^*\zeta_i^{(0)}
@@ -313,7 +313,7 @@ the numerator of the Green function gives
 \end{widetext}
 The argument of the exponential has several saddles. The solutions $\alpha_0$
 are the roots of a sixth-order polynomial, but the root with the
-smallest value of $\mathop{\mathrm{Re}}\alpha_0$ appears gives the correct
+smallest value of $\mathop{\mathrm{Re}}\alpha_0$ in all the cases we studied gives the correct
 solution. A detailed analysis of the saddle point integration is needed to
 understand why this is so. Given such $\alpha_0$, the density of singular
 values follows from the jump across the cut, or
@@ -345,19 +345,18 @@ may be preformed for arbitrary $a$. For all values of $\kappa$ and $\epsilon$,
 the resulting expression is always maximized for $a=\infty$. Taking this saddle
 gives
 \begin{equation} \label{eq:bezout}
-  \overline{\mathcal N}(\kappa,\epsilon)
-  =e^{N\log(p-1)}
-  =(p-1)^N.
+  \ln \overline{\mathcal N}(\kappa,\epsilon)
+  ={N\log(p-1)}
 \end{equation}
 This is precisely the Bézout bound, the maximum number of solutions that $N$
 equations of degree $p-1$ may have \cite{Bezout_1779_Theorie}. More insight is
 gained by looking at the count as a function of $a$, defined by
 \begin{equation} \label{eq:count.def.marginal}
-  \overline{\mathcal N}(\kappa,\epsilon)
-  =\int da\,\overline{\mathcal N}(\kappa,\epsilon,a)
+  {\mathcal N}(\kappa,\epsilon,a)
+  ={\mathcal N}(\kappa,\epsilon/ \sum_i y_i^2<Na)
 \end{equation}
 and likewise the $a$-dependant complexity
-$\Sigma(\kappa,\epsilon,a)=N\log\overline{\mathcal N}(\kappa,\epsilon,a)$. In
+$\Sigma(\kappa,\epsilon,a)=N\log\overline{\mathcal N}_a(\kappa,\epsilon,a)$
 the large-$N$ limit, the $a$-dependant expression may be considered the
 cumulative number of critical points up to the value $a$.
 
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