Update on Overleaf.

1 files changed, 53 insertions, 26 deletions

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 875d5a9..3d59148 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -36,8 +36,8 @@
 \begin{abstract}
   We derive the general solution for counting the stationary points of
   mean-field complex landscapes. It incorporates Parisi's solution
-  for the ground state, as it should. Using this solution, we count the
-  stationary points of two models: one with multi-step replica symmetry
+  for the ground state, as it should. Using this solution, we count 
+   {\color{red} and discuss the distribution of the stability indices } of  stationary points of two {\color{red} representative} models: one with multi-step replica symmetry
   breaking, and one with full replica symmetry breaking.
 \end{abstract}
 
@@ -52,33 +52,56 @@ Sherrington--Kirkpatrick model, in a paper remarkable for being one of the
 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  mean-field complex landscapes 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, Cavagna_1997_An, Cavagna_1998_Stationary} 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 of probability
+been open ever since \cite{Parisi_1979_Infinite}.  
+Many other interesting aspects of the problem have been treated, and the subject has
+evolved into an active field of probability
 theory \cite{Auffinger_2012_Random, Auffinger_2013_Complexity,
-BenArous_2019_Geometry}.
+BenArous_2019_Geometry} and  has been applied to
+energy functions inspired by molecular biology, evolution, and machine learning
+\cite{Maillard_2020_Landscape, Ros_2019_Complex, Altieri_2021_Properties}.
+
+To date, however, the program of
+computing the statistics of stationary points---minima, saddle points, and
+maxima---of  mean-field complex landscapes has been only carried out in an exact form 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, Cavagna_1997_An, Cavagna_1998_Stationary}.
+
+{\color{red} Not having  a full, exact  (`quenched') solution of the generic problem  is not 
+primarily a matter of {\em accuracy} of the actual numbers  involved.
+In the same spirit (but in a geometrically distinct way) as in the case of the equilibrium properties of glasses, 
+much more basic structural questions are omitted in the approximate `annealed' solution. What is lost  is  the nature, at any given
+energy (or free energy) level, of the stationary points in a generic energy function: at low energies are they basically all minima, with  an exponentially small number of saddles, or
+-- as we show here -- do they consist of a mixture of saddles whose index -- the number of unstable directions -- is a smoothly distributed number?  Also, in an energy
+level where almost all saddle points are unstable, are there still a few  stable ones?
+
+These questions need to be answered for the understanding of the relevance of more complex objects such as
+barrier crossing (which barriers?) \cite{Ros_2021_Dynamical}, or the fate of long-time dynamics
+(which are the target states?). 
+
+
+
+
+
 
 In this paper we present what we argue is the general replica ansatz for 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.
+the limit of small temperature for the lowest states, as it should. For this kind of situation
+it clarifies the structure of lowest saddles: there is a continuous distribution of them,
+with stability characterized by a continuous distribution of indices.
+}
 
-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 replica symmetry
+From the point of view of glassy systems, consider the following
+situation. Generically, we now know \cite{Charbonneau_2014_Fractal}
+ 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, 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
+Gardner_1985_Spin}, the so-called `Gardner' phase.
+Now, this transition involves the lowest
+equilibrium states which are obviously unreachable at any reasonable
+timescale.  We should rather ask the question of what is the signature of the Gardner
+transition line for states with higher  energy-densities: the answer 
+will then be  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,
@@ -89,11 +112,9 @@ transition are high energy (or low density) states reachable dynamically?' One
 approach to answering such questions makes use of `state following,'
 which tracks metastable thermodynamic configurations to their zero temperature
 limit \cite{Rainone_2015_Following, Biroli_2016_Breakdown,
-Rainone_2016_Following, Biroli_2018_Liu-Nagel, Urbani_2017_Shear}.  In the
-present paper we give a purely geometric appoarch: 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.
+Rainone_2016_Following, Biroli_2018_Liu-Nagel, Urbani_2017_Shear}.  
+The present paper we provide a purely geometric approach, since we shall address the local energy
+minima at any given energy and study their number and stability properties.
 
 
 Perhaps the most interesting application of this computation is in the context
@@ -126,6 +147,12 @@ $3+16$ model with a 2RSB ground state and a 1RSB complexity, and a $2+4$ with a
 FRSB ground state and a FRSB complexity. Finally \S\ref{sec:interpretation}
 provides some interpretation of our results.
 
+{\color{red}   A final remark is in order here: for simplicity we have concentrated on the energy, rather
+than the {\em free-energy} landscape. Clearly, in the presence of thermal fluctuations, the latter is
+more appropriate. However, from the technical point of view, this makes no fundamental difference, it suffices 
+to apply the same computation to the Thouless-Andreson-Palmer \cite{Crisanti_1995_Thouless-Anderson-Palmer} (TAP) free energy, instead of the energy. No new 
+complications arise.}
+
 \section{The model}
 \label{sec:model}