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-rw-r--r--frsb_kac-rice.tex79
1 files changed, 26 insertions, 53 deletions
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
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@@ -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
- {\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
+ for the ground state, as it should. Using this solution, we count the
+ stationary points of two models: one with multi-step replica symmetry
breaking, and one with full replica symmetry breaking.
\end{abstract}
@@ -50,56 +50,33 @@ 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}.
-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} 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
+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}.
-
-{\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?).
-
-
-
-
-
+\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
+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
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. 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.
-}
+the limit of small temperature for the lowest states, as it should.
-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
+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
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.
-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
+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
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,
@@ -110,9 +87,11 @@ 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}.
-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.
+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.
Perhaps the most interesting application of this computation is in the context
@@ -145,12 +124,6 @@ $3+16$ 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}