From d1e1e449864f6daab9c9b9cedad26539f51ff01a Mon Sep 17 00:00:00 2001
From: "kurchan.jorge" <kurchan.jorge@gmail.com>
Date: Tue, 12 Jul 2022 14:20:01 +0000
Subject: Update on Overleaf.

---
 frsb_kac-rice.tex | 17 +++++++++++++----
 1 file changed, 13 insertions(+), 4 deletions(-)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 741cc80..9047178 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -26,7 +26,7 @@
 \maketitle
 \begin{abstract}
     We derive the general solution for the computation of stationary points of
-    mean-field complex landscapes. The solution incorporates Parisi's solution
+    mean-field complex landscapes. It incorporates Parisi's solution
     for the ground state, as it should.
 \end{abstract}
 
@@ -403,10 +403,19 @@ where the function $\mathcal D$ is defined by
   \right\}
   \end{aligned}
 \end{equation}
-It follows that by fixing the trace of the Hessian, we have effectively fixed
-the value of the stability $\mu$ in all replicas to the value $\mu^*$, and
-therefore the index of saddles in all replicas as well.
 
+{\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 $\mu^*$.\\
+
+$\bullet$ 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
-- 
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