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diff --git a/frsb_kac_new.tex b/frsb_kac_new.tex index d8f7f87..4932050 100644 --- a/frsb_kac_new.tex +++ b/frsb_kac_new.tex @@ -10,27 +10,24 @@ or Full solution for the counting of saddles of mean-field glass models} \maketitle \begin{abstract} We derive the general solution for the computation of saddle points - of mean-field complex landscapes. The solution incorporates Parisi's solution - for the ground state, as it should. + of mean-field complex landscapes. The solution incorporates Parisi's solution for the ground state, as it should. \end{abstract} \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, 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{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 + the Sherrington--Kirkpatrick model, in 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{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. -These include most notably the $p$-spin model ($p>2$) \cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer}. +These include most notably the (pure) $p$-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} 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, including the Sherrington--Kirkpatrick model. It incorporates the Parisi solution as the limit of lowest states, as it should. +computation of the number of saddles of generic mean-field models, including the Sherrington--Kirkpatrick model. It reproduces the Parisi result in the limit +of small temperature for the lowest states, as it should. \section{The model} @@ -442,30 +439,36 @@ which is precisely \eqref{eq:ground.state.free.energy} with $R_d=z$ and $\hat\ep {\em We arrive at one of the main results of our paper: a $(k-1)$-RSB ansatz in Kac--Rice will predict the correct ground state energy for a model whose equilibrium state at small temperatures is $k$-RSB } -\section{Ultrametricity rediscovered} -(not sure) +\section{Ultrametricity rediscovered} +TENTATIVE BUT INTERESTING -Three states chosen at the same energy share some common information if there is some `frozen' element common to all. Suppose we choose randomly -these states but restrict to those whose overlaps -take values $Q_{12}$ and $Q_{13}$. Unlike an equilibrium situation, where the Gibbs measure allows us to find such pairs (in a FRSB case) the cost in probability of this in the present case will be exponential. -Once conditioned this way, we compute $Q_{23}= \min(Q_{12},Q_{13})$ +The frozen phase for a given index ${\cal{I}}$ is the one for values of $\hat \beta> \hat \beta_{freeze}^{\cal{I}}$. +[Jaron: does $\hat \beta^I_{freeze}$ have a relation to the largest $x$ of the ansatz? If so, it would give an interesting interpretation for everything] + + The complexity of that index is zero, and we are looking at the lowest saddles +in the problem, a question that to the best of our knowledge has not been discussed +in the Kac-Rice context -- for good reason, since the complexity - the original motivation - is zero. +However, our ansatz tells us something of the actual organization of the lowest saddles of each index in phase space. \section{Conclusion} -We have constructed a replica solution for the general Kac-Rice problem, including systems +We have constructed a replica solution for the general problem of finding saddles of random mean-field landscapes, including systems with many steps of RSB. -The main results of this paper are the ansatz \ref{ansatz} and the check that the lowest energy -is the correct one obtained with the usual Parisi ansatz. -For systems with full RSB, we find that minima are, at all energy densities above the ground state one, exponentially subdominant with respect to saddles. -It remains to exploit the construction to study general landscapes in more detail. +The main results of this paper are the ansatz (\ref{ansatz}) and the check that the lowest energy is the correct one obtained with the usual Parisi ansatz. +For systems with full RSB, we find that minima are, at all energy densities above the ground state, exponentially subdominant with respect to saddles. +The solution contains valuable geometric information that has yet to be +extracted in all detail. +\paragraph{Funding information} +J K-D and J K are supported by the Simons Foundation Grant No. 454943. +\begin{appendix} -\section{Appendix: RSB for the Gibbs-Boltzmann measure} +\section{RSB for the Gibbs-Boltzmann measure} \begin{equation} \beta F=-\frac12\lim_{n\to0}\frac1n\left(\beta^2\sum_{ab}f(Q_{ab})+\log\det Q\right)-\frac12\log S_\infty @@ -575,7 +578,7 @@ $F$ is a $k-1$ RSB ansatz with all eigenvalues scaled by $y$ and shifted by $z$. -\section{Appendix: RSB for the Kac-Rice integral} +\section{ RSB for the Kac-Rice integral} \subsection{Solution} @@ -721,7 +724,7 @@ complexity in the ground state are D_d=R_d\hat\epsilon \end{align} -\section{Appendix: a motivation for the ansatz} +\section{ A motivation for the ansatz} We may encode the original variables in a superspace variable: \begin{equation} @@ -776,7 +779,7 @@ Not surprisingly, and for the same reason as in the quantum case, this ansatz cl -i\bar\theta_1\theta_1\bar\theta_2\theta_2D_1R_2 \end{aligned} \end{equation} - +\end{appendix} \bibliographystyle{plain} |