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diff --git a/2-point.bib b/2-point.bib index 5ddacf3..4c08cca 100644 --- a/2-point.bib +++ b/2-point.bib @@ -111,6 +111,19 @@ doi = {10.1088/1742-5468/abe29f} } +@article{Folena_2023_On, + author = {Folena, Giampaolo and Zamponi, Francesco}, + title = {On weak ergodicity breaking in mean-field spin glasses}, + year = {2023}, + month = {feb}, + url = {http://arxiv.org/abs/2303.00026v2}, + date = {2023-02-28T19:02:47Z}, + eprint = {2303.00026v2}, + eprintclass = {cond-mat.dis-nn}, + eprinttype = {arxiv}, + urldate = {2023-05-20T17:09:49.352227Z} +} + @article{Franz_1995_Recipes, author = {Franz, Silvio and Parisi, Giorgio}, title = {Recipes for Metastable States in Spin Glasses}, diff --git a/2-point.tex b/2-point.tex index af0af87..d39f12c 100644 --- a/2-point.tex +++ b/2-point.tex @@ -159,7 +159,54 @@ The gradient and Hessian at a stationary point are then \end{align} where $\partial=\frac\partial{\partial\mathbf s}$ will always denote the derivative with respect to $\mathbf s$. +When we count stationary points, we classify them by certain properties. One of +these is the energy density $E=H/N$. We will also fix the \emph{stability} +$\mu=\frac1N\operatorname{Tr}\operatorname{Hess}H$, also known as the radial +reaction. In the mixed spherical models, all stationary points have a +semicircle law for the eigenvalue spectrum of their Hessians, each with the +same width $\mu_\mathrm m$, but whose center is shifted by different amounts. Fixing the +stability $\mu$ fixes this shift, and therefore fixes the spectrum of the +associated stationary point. When the stability is smaller than the width of +the spectrum, or $\mu<\mu_\mathrm m$, there are an extensive number of negative +eigenvalues, and the stationary point is a saddle with same large index whose +value is set by the stability. When the stability is greater than the width of +the spectrum, or $\mu>\mu_\mathrm m$, the semicircle distribution lies only +over positive eigenvalues, and unless an isolated eigenvalue leaves the +semicircle and becomes negative, the stationary point is a minimum. Finally, +when $\mu=\mu_\mathrm m$, the edge of the semicircle touches zero and we have +marginal minima. + +In the pure spherical models, $E$ and $\mu$ cannot be fixed separately: fixing +one uniquely fixes the other. This property leads to the great simplification +of these models: marginal minima exist \emph{only} at one energy level, and +therefore only that energy has the possibility of trapping the long-time +dynamics. + +\subsection{Models of focus} + +In this study, we focus exclusively on models whose complexity is replica symmetric. We study two models of interest, both with concave $f''(q)^{-1/2}$: a $3+4$ model whose dynamics were studied extensively in \cite{Folena_2020_Rethinking}, given by +\begin{equation} + f_{3+4}(q)=\frac12\big(q^3+q^4\big) +\end{equation} +and a $3+8$ model tuned to maximize the ``interesting'' region of the dynamics recently studied in \cite{Folena_2023_On} given by +\begin{equation} + f_{3+8}(q)=\frac12\big(\tfrac{76}{100}q^3+\tfrac{24}{100}q^8\big) +\end{equation} + +\begin{figure} + \caption{ + Plots of the complexity (logarithm of the number of stationary points) for + the mixed spherical models studied in this paper. Energies and stabilities + of interest are marked, including the ground state energy and stability + $E_\mathrm{gs}$ and $\mu_\mathrm{gs}$, the marginal stability $\mu_\mathrm + m$, and the threshold energy $E_\mathrm{th}$. The line shows the location + of the most common type of stationary point at each energy level. Estimated + locations of notable attractors of the dynamics are highlighted. + } \label{fig:complexities} +\end{figure} + \section{Results} +\label{sec:results} \begin{figure} \includegraphics{figs/gapped_min_energy.pdf} |