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diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index 23a41d4..6880717 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -15,7 +15,10 @@ \addbibresource{frsb_kac-rice.bib} \begin{document} -\title{Full solution for counting stationary points of mean-field complex energy landscapes} +\title{Full solution for counting stationary points of mean-field complex energy landscapes\\ + \textit{or (fun title)} \\ + How to count in hierarchical landscapes: a `full' solution to mean-field complexity +} \author{Jaron Kent-Dobias \& Jorge Kurchan} \maketitle \begin{abstract} @@ -25,14 +28,19 @@ \section{Introduction} 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 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 saddles 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}. -In a parallel development, it -has evolved into an active field in probability theory \cite{Auffinger_2012_Random, Auffinger_2013_Complexity, BenArous_2019_Geometry} +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 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 saddles 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}. In a parallel development, it has +evolved into an active field in 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 saddles of generic mean-field models, which we expect to include the Sherrington--Kirkpatrick model. It reproduces the Parisi result in the limit |