Metastability in Glauber dynamics for heavy-tailed spin glasses

TL;DR

This paper rigorously analyzes the metastable behavior of Glauber dynamics in heavy-tailed spin glasses, revealing a detailed decomposition into wells and explicit transition dynamics, with implications for aging phenomena.

Contribution

It provides a rigorous description of metastability in heavy-tailed spin glasses, including a decomposition into wells and explicit transition rates, extending understanding beyond the Random Energy Model.

Findings

Decomposition of state space into sub-exponentially many wells.

Glauber dynamics projected onto wells behaves like a Markov chain.

Shorter mixing times within wells compared to between wells.

Abstract

We study the Glauber dynamics for heavy-tailed spin glasses, in which the couplings are in the domain of attraction of an $\alpha$-stable law for $\alpha\in (0,1)$. We show a sharp description of metastability on exponential timescales, in a form that is believed to hold for Glauber/Langevin dynamics for many mean-field spin glass models, but only known rigorously for the Random Energy Models. Namely, we establish a decomposition of the state space into sub-exponentially many wells, and show that the projection of the Glauber dynamics onto which well it resides in, asymptotically behaves like a Markov chain on wells with certain explicit transition rates. In particular, mixing inside wells occurs on much shorter timescales than transit times between wells, and the law of the next well the Glauber dynamics will fall into depends only on which well it currently resides in, not its full configuration. We can deduce consequences like an exact expression for the two-time autocorrelation functions that appear in the activated aging literature.