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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} |