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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 b1c25ab..23a41d4 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -38,21 +38,29 @@ 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 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 symmetry breaking (1RSB) phase at a Kauzmann temperature, -and, at a lower temperature, -another transition to a full RSB 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, an often addressed question -to ask is: what is the Gardner transition line for higher than equilibrium -energy-densities? (see, for a review \cite{Berthier_2019_Gardner}) 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'. 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. +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 symmetry +breaking (1RSB) phase at a Kauzmann temperature, and, at a lower temperature, +another transition to a full RSB 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, an often addressed question to ask is: what is the Gardner +transition line for higher than equilibrium energy-densities? This is a +question whose answers are 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, +Hammond_2020_Experimental, Albert_2021_Searching} (see, for a review +\cite{Berthier_2019_Gardner}). 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'. +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. Perhaps the most interesting application of this computation is in the context of @@ -589,7 +597,10 @@ Understanding that $R$ is diagonal, this implies \mu^*=\frac1{r_d}+r_df''(1) \end{equation} which is precisely the condition \eqref{eq:mu.minima}. Therefore, \emph{the -supersymmetric solution only counts the most common minima} \cite{Annibale_2004_Coexistence}. +supersymmetric solution counts the most common minima} +\cite{Annibale_2004_Coexistence}. When minima are not the most common type of +stationary point, the supersymmetric solution correctly counts minima that +satisfy \eqref{eq:mu.minima}, but these do not have any special significance. Inserting the supersymmetric ansatz $D=\hat\beta R$ and $R=r_dI$, one gets \begin{equation} \label{eq:diagonal.action} |