From f63692f516c2b841ab86dd2e239cf158dbb657d1 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Thu, 8 Jun 2023 23:21:02 +0200 Subject: Lots of work. --- 2-point.tex | 69 +++++++++++++++++++++++++++++++++++++++++++++++-------------- 1 file changed, 53 insertions(+), 16 deletions(-) (limited to '2-point.tex') diff --git a/2-point.tex b/2-point.tex index 9d951e9..7c7d345 100644 --- a/2-point.tex +++ b/2-point.tex @@ -86,7 +86,7 @@ } \author{Jaron Kent-Dobias} -\affil{\textsc{DynSysMath}, Istituto Nazionale di Fisica Nucleare, Sezione di Roma} +\affil{\textsc{DynSysMath}, Istituto Nazionale di Fisica Nucleare, Sezione di Roma I} \maketitle \begin{abstract} @@ -113,19 +113,19 @@ over a short parameter interval. These include actual (structural) glasses, spin glasses, certain inference and optimization problems, and more \cite{lots}. Glassiness is qualitatively understood to arise from structure of an energy or cost landscape, whether due to the proliferation of metastable -states, to the raising of barriers which cause effective dynamic constraints. +states, or to the raising of barriers which cause effective dynamic constraints. However, in most models there is no quantitative correspondence between these -properties and the behavior they are purported to describe. +landscape properties and the dynamic behavior they are purported to describe. There is such a correspondence in one of the simplest mean-field model of glasses: in the pure spherical models, the dynamic transition corresponds -precisely with the energy level at which all marginal minima are concentrated. -At that level, called the \emph{threshold energy} $E_\mathrm{th}$, slices of -the landscape at fixed energy undergo a percolation transition. In fact, this -threshold energy is significant in other ways: it attracts the long-time -dynamics after quenches in temperature to below the dynamical transition from -any starting temperature. All of this can be understood in terms of the -landscape structure. \cite{Biroli_1999_Dynamical} +precisely with the energy level at which all marginal minima are concentrated +\cite{Castellani_2005_Spin-glass}. At that level, called the \emph{threshold +energy} $E_\mathrm{th}$, slices of the landscape at fixed energy undergo a +percolation transition. In fact, this threshold energy is significant in other +ways: it attracts the long-time dynamics after quenches in temperature to below +the dynamical transition from any starting temperature. All of this can be +understood in terms of the landscape structure. \cite{Biroli_1999_Dynamical} In slightly less simple models, the mixed spherical models, the story changes. There are now a range of energies with exponentially many marginal minima. It @@ -226,6 +226,41 @@ 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. +\begin{figure} + \includegraphics{figs/spectrum_saddle.pdf} + \hfill + \includegraphics{figs/spectrum_marginal.pdf} + \hfill + \includegraphics{figs/spectrum_minimum.pdf}\\ + + \vspace{1em} + + \includegraphics{figs/spectrum_saddle_2.pdf} + \hfill + \includegraphics{figs/spectrum_minimum_2.pdf} + \hfill + \includegraphics{figs/spectrum_saddle_3.pdf} + + \caption{ + Illustration of the interpretation of the stability $\mu$, which sets the + location of the center of the eigenvalue spectrum. In the top row we have + spectra without an isolated eigenvalue. \textbf{(a)} $\mu<\mu_\mathrm m$, + there are an extensive number of downward directions, and the associated + point is an \emph{extensive saddle}. \textbf{(b)} $\mu=\mu_\mathrm m$ and + we have a \emph{marginal minimum} with asymptotically flat directions. + \textbf{(c)} $\mu>\mu_\mathrm m$, all eigenvalues are positive, and the + point is a \emph{stable minimum}. On the bottom we show what happens in the + presence of an isolated eigenvalue. \textbf{(d)} One eigenvalue leaves the + bulk spectrum of a saddle point and it remains a saddle point, but now with + an eigenvector correlated with the orientation of the reference vector, so + we call this a \emph{oriented saddle}. \textbf{(e)} The same happens for + a minimum, and we can call it an \emph{oriented minimum}. \textbf{(f)} One + eigenvalue outside a positive bulk spectrum is negative, destabilizing what + would otherwise have been a stable minimum, producing an \emph{oriented + index-one saddle}. + } +\end{figure} + 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 @@ -247,13 +282,15 @@ and a $3+8$ model tuned to maximize the ``interesting'' region of the dynamics r \centering \includegraphics{figs/single_complexity.pdf} \caption{ - Plots of the complexity (logarithm of the number of stationary points) for - the mixed spherical models studied in this paper. Energies and stabilities + Plot of the complexity (logarithm of the number of stationary points) for + the $3+4$ mixed spherical model 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. + $E_\mathrm{gs}$, the marginal stability $\mu_\mathrm + m$, and the threshold energy $E_\mathrm{th}$. The blue line shows the location + of the most common type of stationary point at each energy level. The + highlighted red region shows the approximate range of minima which attract + aging dynamics from a quench to zero temperature found in + \cite{Folena_2020_Rethinking}. } \label{fig:complexities} \end{figure} -- cgit v1.2.3-70-g09d2