From 9ecc2e2aba085c2624db01a1c1f55dfbb2424451 Mon Sep 17 00:00:00 2001
From: "kurchan.jorge" <kurchan.jorge@gmail.com>
Date: Fri, 8 Jul 2022 10:21:21 +0000
Subject: Update on Overleaf.

---
 frsb_kac-rice.tex | 17 ++++++++++++-----
 1 file changed, 12 insertions(+), 5 deletions(-)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index c65c55d..eeac95e 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -36,22 +36,29 @@ computation of the number of saddles of generic mean-field models,  which we exp
 of small temperature for the lowest states, as it should.
 
 To understand the importance of this computation, consider the following situation. When one solves the problem of spheres in large dimensions, one finds that there is
-a transition at a given temperature to a one-step one step symmetry breaking (1RSB) phase at a Kauzmann temperature,
+a transition at a given temperature to a one-step symmetry breaking (1RSB) phase at a Kauzmann temperature,
 and, at a lower temperature,
-another  transition to  a full RSB phase (see \cite{Gross_1985_Mean-field, Gardner_1985_Spin}, the `Gardner' phase \cite{Charbonneau_2014_Fractal}. 
+another  transition to  a full RSB phase (see \cite{Gross_1985_Mean-field, Gardner_1985_Spin}, the o-called `Gardner' phase \cite{Charbonneau_2014_Fractal}). 
 Now, this transition involves the lowest, equilibrium states. Because they are
 obviously unreachable at any reasonable timescale, an often addressed question
 to ask is: what is the Gardner transition line for higher than equilibrium
 energy-densities? (see, for a review \cite{Berthier_2019_Gardner})  For example,
-when studying `jamming' at zero temperature, the question is posed as to "on
+when studying `jamming' at zero temperature, the question is posed as to`on
 what side of the 1RSB-FRS transition are the high energy (or low density)
-states reachable dynamically.  Posed in this way, such a question does not have
-a clear definition.  In the present paper we give a concrete strategy to define
+states reachable dynamically'.   In the present paper we give a concrete strategy to define
 unambiguously such an issue: we consider the local energy minima at a given
 energy and study their number and other properties: the solution involves a
 replica-symmetry breaking scheme that is well-defined, and corresponds directly
 to the topological characteristics of those minima.
 
+
+A more general question, of interest in optimization problems, is how to define a `threshold level'. This notion was introduced in Ref \cite{cugliandolo1993analytical} in the context of the $p$-spin model, as the energy at which the constant energy patches of phase-space percolate - hence 
+explaining why dynamics should relax to that level.
+The notion of a `threshold' for more complex landscapes has later been 
+attempted several times, never to our knowledge in a clear and unambiguous 
+way. One of the purposes of this paper is to
+
+
 \section{The model}
 \label{sec:model}
 
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