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diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index a285c97..1c2df68 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -37,6 +37,15 @@ 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 reproduces the Parisi result in the limit of small temperature for the lowest states, as it should. +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 one step symmetry breaking (1RSB) phase at a Kauzmann temperature. At a lower temperature, +the is a transition to a full RSB phase (see \cite{gross1985mean,gardner1985spin}, the `Gardner ' phase \cite{charbonneau2014fractal}. + +Now, this transition involves the lowest, equilibrium states. Because they are obviously unreachable at any reasonable timescale, an often addressed to ask "what is the Gardner transition line for higher than equilibrium energy-densities" (see, for a review \cite{berthier2019gardner})? For example, when studying `jamming' at zero temperature, the question is posed as to "on what side of the 1RSB-FRS transition +are the high energy (or low density) states reachable dynamically. Posed in this way, such a question does not have a clear definition. +In the present paper we give a concrete strategy to define unambiguously such an issue: 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. + + \section{The model} @@ -786,7 +795,10 @@ We have constructed a replica solution for the general problem of finding sadd with many steps of RSB. 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. +extracted in all detail. + +A first and very important application of the method here is to perform the calculation for high dimensional spheres, where it would give us +a clear understanding of what happens in a low-temperature realistic jamming dynamics \cite{maimbourg2016solution}. \paragraph{Funding information} J K-D and J K are supported by the Simons Foundation Grant No. 454943. |