From b04af5e803fe393c3fc9f64637526d442e8e039e Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 12 Jul 2022 16:29:09 +0200
Subject: Cleaned up Jorge's addition.

---
 frsb_kac-rice.tex | 21 +++++++++++----------
 1 file changed, 11 insertions(+), 10 deletions(-)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 9047178..ec71085 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -404,18 +404,19 @@ where the function $\mathcal D$ is defined by
   \end{aligned}
 \end{equation}
 
-{\color{blue}
 By fixing the trace of the Hessian, we have effectively fixed
-the value of the stability $\mu$ in all replicas to the value $\mu^*$.\\
+the value of the stability $\mu$ in all replicas to the value $\mu^*$.
+\begin{itemize}
+  \item For $\mu^*<\mu_m$, this amounts to fixing the index density. Since the
+    overwhelming majority of saddles have a semicircle distribution, the
+    fluctuations are rarer than exponential.
+  \item For the gapped case $\mu^*>\mu_m$, there is a an exponentially small
+    probability that $r=1,2,...$ eigenvalues detach from the semicircle in such
+    a way that the index is in fact $N {\cal{I}}=r$.  We shall not discuss
+    these subextensive index fluctuations in this paper, the interested reader
+    may find what is needed in \cite{Auffinger_2013_Complexity}.
+\end{itemize}
 
-$\bullet$ For $\mu^*<\mu_m$, this amounts to fixing the index density. The argument is that
-the overwhelming majority of saddles have a semicircle distribution, the fluctuations are rarer than exponential.
-
-$\bullet$ For the gapped case $\mu^*>\mu_m$, there is a an exponentially small probability that $r=1,2,...$ eigenvalues detach
-from the semicircle in such a way that the index is in fact $N {\cal{I}}=r$.
-We shall not discuss these index fluctuations in this paper, the interested
-reader may find what is needed in \cite{Auffinger_2013_Complexity}
-}
 \subsubsection{The gradient factors}
 
 The $\delta$-functions in the remaining factor are treated by writing them in
-- 
cgit v1.2.3-70-g09d2

