From 978b67870210e740848a4887a21ac9ef25a6a963 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 12 Jul 2022 20:12:13 +0200
Subject: Started doing little finishing edits.

---
 frsb_kac-rice.tex | 53 +++++++++++++++++++++++++++--------------------------
 1 file changed, 27 insertions(+), 26 deletions(-)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 13ea844..6994a48 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -45,34 +45,34 @@ The computation of the number of metastable states of mean field spin glasses
 goes back to the beginning of the field. Over forty years ago, Bray and Moore
 \cite{Bray_1980_Metastable} attempted the first calculation for the
 Sherrington--Kirkpatrick model, in a paper remarkable for being one of the
-first applications of a replica symmetry breaking (RSB) scheme. As became clear when
-the actual ground-state of the model was computed by Parisi
-\cite{Parisi_1979_Infinite} with a different scheme, the Bray--Moore result was
-not exact, and in fact  the problem has been open ever since.  To this date the
-program of computing the number of stationary points---minima, saddle points,
-and maxima---of a mean-field glass has been only carried out for a small subset
-of models, including most notably the (pure) $p$-spin model ($p>2$)
+first applications of a replica symmetry breaking (RSB) scheme. As was clear
+when the actual ground-state of the model was computed by Parisi with a
+different scheme, the Bray--Moore result was not exact, and the problem has
+been open ever since \cite{Parisi_1979_Infinite}.  To date, the program of
+computing the number of stationary points---minima, saddle points, and
+maxima---of a mean-field glass has been only carried out for a small subset of
+models, including most notably the (pure) $p$-spin model ($p>2$)
 \cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer} and for similar
 energy functions inspired by molecular biology, evolution, and machine learning
-\cite{Maillard_2020_Landscape, Ros_2019_Complex, Altieri_2021_Properties}.  In a parallel development,
-it has evolved into an active field in probability theory
-\cite{Auffinger_2012_Random, Auffinger_2013_Complexity,
+\cite{Maillard_2020_Landscape, Ros_2019_Complex, Altieri_2021_Properties}.  In
+a parallel development, it has evolved into an active field of probability
+theory \cite{Auffinger_2012_Random, Auffinger_2013_Complexity,
 BenArous_2019_Geometry}.
 
 In this paper we present what we argue is the general replica ansatz for the
-computation of the number of stationary points 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.
+number of stationary points 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
+finds that there is a transition at a given temperature to a one-step replica symmetry
 breaking (1RSB) phase at a Kauzmann temperature, and, at a lower temperature,
 another  transition to  a full RSB (FRSB) 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
+timescale, a common question is: what is the signature of 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,
@@ -80,8 +80,8 @@ 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-FRSB
-transition are the high energy (or low density) states reachable dynamically'.
+temperature, the question is posed as `on what side of the 1RSB-FRSB
+transition are 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
@@ -92,15 +92,16 @@ characteristics of those minima.
 Perhaps the most interesting application of this computation is in the context
 of optimization problems, see for example \cite{Gamarnik_2021_The,
 ElAlaoui_2022_Sampling, Huang_2021_Tight}. A question that appears there is how
-to define a `threshold level,' the lowest energy level that good algorithms can
-expect to reach. This notion was introduced  \cite{Cugliandolo_1993_Analytical}
-in the context of the pure $p$-spin models, as the energy at which the patches of the
-same  energy in phase-space percolate  - hence explaining why dynamics never go
-below that level.  The notion of a `threshold' for more complicated landscapes has
-later been invoked several times, never to our knowledge in a clear and
-unambiguous way. One of the purposes of this paper is to give a sufficiently
-detailed characterization of a general landscape so that a meaningful general
-notion of threshold may  be introduced - if this is at all possible.
+to define a `threshold' level, the lowest energy level that good algorithms can
+expect to reach. This notion was introduced in the context of the pure $p$-spin
+models, as the energy at which level sets of the energy in phase-space
+percolate, explaining why dynamics never go below that level
+\cite{Cugliandolo_1993_Analytical}.  The notion of a `threshold' for more
+complicated landscapes has later been invoked several times, never to our
+knowledge in a clear and unambiguous way. One of the purposes of this paper is
+to give a sufficiently detailed characterization of a general landscape so that
+a meaningful general notion of threshold may  be introduced -- if this is at all
+possible.
 
 The format of this paper is as follows. In \S\ref{sec:model}, we introduce the
 mean-field model of study, the mixed $p$-spin spherical model. In
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