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Diffstat (limited to 'frsb_kac-rice.tex')
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diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index 3b06750..1b16410 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -15,17 +15,17 @@ 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{bray1980metastable} attempted the first calculation for +Bray and Moore \cite{Bray_1980_Metastable} attempted the first calculation for the Sherrington-Kirkpatrick model, a paper remarkable for being the first practical application of a replica symmetry breaking scheme. As became clear when the actual - ground-state of the model was computed by Parisi \cite{parisi1979infinite}, the Bray-Moore result + ground-state of the model was computed by Parisi \cite{Parisi_1979_Infinite}, the Bray-Moore result was not exact, and in fact the problem has been open ever since. Indeed, 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. -These include most notably the p-spin model ($p>2$) \cite{rieger1992number,crisanti1995thouless}. +These include most notably the p-spin model ($p>2$) \cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer}. The problem of studying the critical points of these landscapes -has evolved into an active field in probability theory \cite{Auffinger_2012_Random,Auffinger_2013_Complexity,BenArous_2019_Geometry} +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 believe is the general ansatz for the computation of saddles of generic mean-field models, including the Sherrington-Kirkpatrick model. It incorporates the Parisi solution as the limit of lowest states, as it should. @@ -34,7 +34,7 @@ computation of saddles of generic mean-field models, including the Sherrington- \section{The model} Here we consider, for definiteness, the mixed $p$-spin model, itself a particular case -of the `Toy Model' of M\'ezard and Parisi \cite{mezard1992manifolds} +of the `Toy Model' of M\'ezard and Parisi \cite{Mezard_1992_Manifolds} \begin{equation} H(s)=\sum_p\frac{a_p^{1/2}}{p!}\sum_{i_1\cdots i_p}J_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p} \end{equation} @@ -327,7 +327,7 @@ This will turn out to be important when we discriminate between counting all sol As noted by Bray and Dean \cite{Bray_2007_Statistics}, gradient and Hessian are independent for a Gaussian distribution, and the average over disorder breaks into a product of two independent averages, one for the gradient factor and one for the determinant. The integration of all variables, including the disorder in the last factor, may be restricted to the domain such that the matrix $\partial\partial H(s_a)-\mu I$ has a specified number of negative eigenvalues (the index {\cal{I}} of the saddle), -(see Fyodorov \cite{fyodorov2007replica} for a detailed discussion) } +(see Fyodorov \cite{Fyodorov_2007_Replica} for a detailed discussion) } \begin{equation} \begin{aligned} |