From 9dd25213556e7ba9dffca5a873d77369b4978a9a Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 8 Dec 2020 19:59:22 +0100
Subject: Work.

---
 bezout.tex | 19 ++++++++++---------
 1 file changed, 10 insertions(+), 9 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index d39a2e9..5ea3a33 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -131,15 +131,16 @@ 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 \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 that involves the replica trick. In most, but not all,  of the parameter-space that we shall study here, the {\em annealed approximation} $N \Sigma \sim 
-\ln \overline{ \mathcal N_J} = \ln \int dJ \; N_J$ is exact.
-
-A useful property of the Gaussian distributions is that gradient and Hessian may be seen to be independent \cite{Bray_2007_Statistics, Fyodorov_2004_Complexity},
-so that we may treat the delta-functions and the Hessians as independent.
-
-}
+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
+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. 
+
+A useful property of the Gaussian distributions is that gradient and Hessian
+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 Hessian of \eqref{eq:constrained.hamiltonian} is $\partial\partial
 H=\partial\partial H_0-p\epsilon I$, or the Hessian of
-- 
cgit v1.2.3-54-g00ecf

