From 237092a616219c3fab36e9bbb78f15c74d60f3a3 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 8 Jul 2022 16:40:40 +0200
Subject: Lots more references.

---
 frsb_kac-rice.tex | 43 +++++++++++++++++++++++++++----------------
 1 file changed, 27 insertions(+), 16 deletions(-)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index b1c25ab..23a41d4 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -38,21 +38,29 @@ In this paper we present what we argue is the general replica ansatz for the
 computation of the number of saddles of generic mean-field models,  which we expect to include the Sherrington--Kirkpatrick model. It reproduces  the Parisi result in the limit
 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 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 so-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
-what side of the 1RSB-FRS transition are the high energy (or low density)
-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.
+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 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 so-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? This is a
+question whose answers are significant to interpreting the results of myriad
+experiments and simulations \cite{Xiao_2022_Probing, Hicks_2018_Gardner,
+Liao_2019_Hierarchical, Dennis_2020_Jamming, Charbonneau_2015_Numerical,
+Li_2021_Determining, Seguin_2016_Experimental, Geirhos_2018_Johari-Goldstein,
+Hammond_2020_Experimental, Albert_2021_Searching} (see, for a review
+\cite{Berthier_2019_Gardner}).  For example, 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'.
+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.
 
 
 Perhaps the most interesting application of this computation is in the context  of
@@ -589,7 +597,10 @@ Understanding that $R$ is diagonal, this implies
   \mu^*=\frac1{r_d}+r_df''(1)
 \end{equation}
 which is precisely the condition \eqref{eq:mu.minima}. Therefore, \emph{the
-supersymmetric solution only counts the most common minima} \cite{Annibale_2004_Coexistence}.
+supersymmetric solution counts the most common minima}
+\cite{Annibale_2004_Coexistence}. When minima are not the most common type of
+stationary point, the supersymmetric solution correctly counts minima that
+satisfy \eqref{eq:mu.minima}, but these do not have any special significance.
 
 Inserting the supersymmetric ansatz $D=\hat\beta R$ and $R=r_dI$, one gets
 \begin{equation} \label{eq:diagonal.action}
-- 
cgit v1.3

