From ca2c30ef85e8e9e9e42a03127cb23ef8f1b6dfbe Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 12 Mar 2021 17:29:02 +0100
Subject: Wording changes.

---
 bezout.tex | 32 +++++++++++++-------------------
 1 file changed, 13 insertions(+), 19 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index db14a52..f950b8c 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -128,8 +128,7 @@ whole space.) Where it might be a problem, we introduce the additional
 constraint $z^\dagger z\leq Nr^2$. The resulting configuration space is a complex
 manifold with boundary. We shall see that the `radius' $r$ proves an insightful
 knob in our present problem, revealing structure as it is varied. Note
-that---combined with the constraint $z^Tz=N$---taking $r=1$ reduces the problem
-to that of the ordinary $p$-spin.
+that taking $r=1$ reduces the problem to that of the ordinary $p$-spin.
 
 The critical points are of $H$ given by the solutions to the set of equations
 \begin{equation} \label{eq:polynomial}
@@ -143,18 +142,14 @@ of random polynomials \cite{Bogomolny_1992_Distribution}, which are for $N=1$
 and $p\to\infty$.  We see from \eqref{eq:polynomial} that at any critical
 point $\epsilon=H/N$, the average energy.
 
-Since $H$ is holomorphic, any critical point of $\operatorname{Re}H$ is also a
-critical point of $\operatorname{Im}H$. The number of critical points of $H$ is
-therefore the same as that of $\operatorname{Re}H$. From each saddle 
-emerge gradient lines of $\operatorname{Re}H$, which are also ones of constant
-$\operatorname{Im}H$ and therefore constant phase.
-
-Writing $z=x+iy$, $\operatorname{Re}H$ can be considered a real-valued function
-of $2N$ real variables. Its number of saddle-points is given by the usual
-Kac--Rice formula:
+Since $H$ is holomorphic, any critical point of $\operatorname{Re}H$ is also
+one of $\operatorname{Im}H$, and therefore of $H$ itself. Writing $z=x+iy$ for
+$x,y\in\mathbb R^N$, $\operatorname{Re}H$ can be considered a real-valued
+function of $2N$ real variables. The number of critical points of $H$ is thus given by the
+usual Kac--Rice formula applied to $\operatorname{Re}H$:
 \begin{equation} \label{eq:real.kac-rice}
   \begin{aligned}
-    \mathcal N_J&(\kappa,\epsilon,r)
+    \mathcal N&(\kappa,\epsilon,r)
       = \int dx\,dy\,\delta(\partial_x\operatorname{Re}H)\delta(\partial_y\operatorname{Re}H) \\
       &\hspace{6pc}\times\left|\det\begin{bmatrix}
           \partial_x\partial_x\operatorname{Re}H & \partial_x\partial_y\operatorname{Re}H \\
@@ -169,7 +164,7 @@ $\partial_y\operatorname{Re}H=-\operatorname{Im}\partial H$. Carrying these
 transformations through, we have
 \begin{equation} \label{eq:complex.kac-rice}
   \begin{aligned}
-    \mathcal N_J&(\kappa,\epsilon,r)
+    \mathcal N&(\kappa,\epsilon,r)
       = \int dx\,dy\,\delta(\operatorname{Re}\partial H)\delta(\operatorname{Im}\partial H) \\
                 &\hspace{6pc}\times\left|\det\begin{bmatrix}
             \operatorname{Re}\partial\partial H & -\operatorname{Im}\partial\partial H \\
@@ -183,7 +178,7 @@ transformations through, we have
 \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,
-i.e. the norm squared of that of an $N\times N$ complex symmetric 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 positive eigenvalue of the real matrix has a
@@ -195,11 +190,10 @@ $\partial\partial H$, or 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 $J$ to give
-the complexity $\Sigma$ as $N \Sigma= \overline{\log\mathcal N} = \int dJ \,
-\log \mathcal N_J$, a calculation that involves the replica trick. In most of the
-parameter-space that we shall study here, the \emph{annealed approximation} $N
-\Sigma \sim \log \overline{ \mathcal N} = \log\int dJ \, \mathcal N_J$ is
-exact. 
+the complexity $\Sigma$ as $N \Sigma= \overline{\log\mathcal N}$, a calculation
+that involves the replica trick. In most of the parameter-space that we shall
+study here, the \emph{annealed approximation} $N \Sigma \sim \log \overline{
+\mathcal N}$ is exact. 
 
 A useful property of the Gaussian $J$ is that gradient and Hessian at fixed
 $\epsilon$ are statistically independent \cite{Bray_2007_Statistics,
-- 
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