From 50a9abd97a9a7fbe88a499a9c300bf4a42864510 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 10 Dec 2020 16:49:31 +0100
Subject: More small changes.

---
 bezout.tex | 40 +++++++++++++++++++++-------------------
 1 file changed, 21 insertions(+), 19 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index 67a764b..57e48ae 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -108,9 +108,9 @@ z$ would only be satisfied for $\epsilon=0$.
 
 The critical points are given by the solutions to the set of equations
 \begin{equation}
-  \frac{p}{p!}\sum_{j_2\cdots j_p}^NJ_{ij_2\cdots j_p}z_{j_2}\cdots z_{j_p} = \epsilon z_i
+  \frac{p}{p!}\sum_{j_1\cdots j_{p-1}}^NJ_{ij_1\cdots j_{p-1}}z_{j_1}\cdots z_{j_{p-1}} = \epsilon z_i
 \end{equation}
-for all $i=\{1,\ldots,N\}$, which for given $\epsilon$ are a set of $N$
+for all $i=\{1,\ldots,N\}$, which for given $\epsilon$ is a set of $N$
 equations of degree $p-1$, to which one must add the constraint.  In this sense
 this study also provides a complement to the work on the distribution of zeroes
 of random polynomials \cite{Bogomolny_1992_Distribution}, which are for $N=1$
@@ -143,7 +143,7 @@ $\partial_y\operatorname{Re}H=-\operatorname{Im}\partial H$. Carrying
 these transformations through, we have
 \begin{equation} \label{eq:complex.kac-rice}
   \begin{aligned}
-    \mathcal N_J&(\kappa,\epsilon)
+    &\mathcal N_J(\kappa,\epsilon)
       = \int dx\,dy\,\delta(\operatorname{Re}\partial H)\delta(\operatorname{Im}\partial H) \\
         &\qquad\qquad\qquad\times\left|\det\begin{bmatrix}
             \operatorname{Re}\partial\partial H & -\operatorname{Im}\partial\partial H \\
@@ -169,10 +169,10 @@ singular values of $\partial\partial H$,  the distribution of square-rooted
 eigenvalues of $(\partial\partial H)^\dagger\partial\partial H$.
 
 The expression \eqref{eq:complex.kac-rice} is to be averaged over the $J$'s as
-$N \Sigma= \overline{\ln \mathcal N_J} = \int dJ \; \ln N_J$, a calculation
+$N \Sigma= \overline{\log\mathcal N} = \int dJ \, \log \mathcal N_J$, a calculation
 that involves the replica trick. In most the parameter-space that we shall
-study here, the {\em annealed approximation} $N \Sigma \sim \ln \overline{
-\mathcal N_J} = \ln \int dJ \; N_J$ is exact. 
+study here, the {\em annealed approximation} $N \Sigma \sim \log \overline{
+\mathcal N} = \log\int dJ \, \mathcal N_J$ is exact. 
 
 A useful property of the Gaussian distributions is that gradient and Hessian
 for given $\epsilon$ may be seen to be independent \cite{Bray_2007_Statistics,
@@ -256,12 +256,14 @@ that of the same ellipse 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 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 in the literature.  We have worked out an implicit form for
+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 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.
 
@@ -348,7 +350,7 @@ 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}
-  \ln \overline{\mathcal N}(\kappa,\epsilon)
+  \log\overline{\mathcal N}(\kappa,\epsilon)
   ={N\log(p-1)}
 \end{equation}
 This is precisely the Bézout bound, the maximum number of solutions that $N$
@@ -383,14 +385,14 @@ $a$-dependence gives a cumulative count, this implies a $\delta$-function
 density of critical points along the line $y=0$.  The number of critical points
 contained within is
 \begin{equation}
-  \lim_{a\to1}\lim_{\kappa\to1}\overline{\mathcal N}(\kappa,0,a)
-  \sim (p-1)^{N/2},
+  \lim_{a\to1}\lim_{\kappa\to1}\log\overline{\mathcal N}(\kappa,0,a)
+  = \frac12N\log(p-1),
 \end{equation}
-the square root of \eqref{eq:bezout} and precisely the number of critical
+half of \eqref{eq:bezout} and corresponding precisely to the number of critical
 points of the real pure spherical $p$-spin model. (note the role of conjugation
-symmetry, already underlined in Ref\cite{Bogomolny_1992_Distribution}). In
-fact, the full $\epsilon$-dependence of the real pure spherical $p$-spin is
-recovered by this limit as $\epsilon$ is varied.
+symmetry, already underlined in \cite{Bogomolny_1992_Distribution}). The full
+$\epsilon$-dependence of the real $p$-spin is recovered by this limit as
+$\epsilon$ is varied.
 
 \begin{figure}[htpb]
   \centering
-- 
cgit v1.2.3-54-g00ecf

