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Metastability in Glauber dynamics for heavy-tailed spin glasses

Reza Gheissari, Curtis Grant

TL;DR

The paper characterizes metastability for Glauber dynamics in heavy-tailed mean-field spin glasses with $α∈(0,1)$ by revealing a sharp decomposition of the configuration space into $t_{a,γ}$-metastable wells on timescales $t_{a,γ}=e^{aN^{γ}}$ (with $γ∈(γ_0,1/α)$). It shows that the dynamics projected onto wells converges to a Markov jump process on wells with rates governed by the largest bonds, while intra-well mixing is rapid, enabling a clean skeleton approximation. A rigorous coupling between the full dynamics and the reduced chain $Y(s)$ yields precise two-time autocorrelation limits and activated aging predictions, including an explicit expression for $q_{a,γ}^{(N)}$ and its behavior in high-temperature regimes. The work crucially leverages sparse structure of large bonds, block-dynamics techniques, and FK/random-cluster couplings to control mixing times and exit times, providing a framework that likely extends to broader mean-field spin-glass models with heavy-tailed couplings.

Abstract

We study the Glauber dynamics for heavy-tailed spin glasses, in which the couplings are in the domain of attraction of an $α$-stable law for $α\in (0,1)$. 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.

Metastability in Glauber dynamics for heavy-tailed spin glasses

TL;DR

The paper characterizes metastability for Glauber dynamics in heavy-tailed mean-field spin glasses with by revealing a sharp decomposition of the configuration space into -metastable wells on timescales (with ). It shows that the dynamics projected onto wells converges to a Markov jump process on wells with rates governed by the largest bonds, while intra-well mixing is rapid, enabling a clean skeleton approximation. A rigorous coupling between the full dynamics and the reduced chain yields precise two-time autocorrelation limits and activated aging predictions, including an explicit expression for and its behavior in high-temperature regimes. The work crucially leverages sparse structure of large bonds, block-dynamics techniques, and FK/random-cluster couplings to control mixing times and exit times, providing a framework that likely extends to broader mean-field spin-glass models with heavy-tailed couplings.

Abstract

We study the Glauber dynamics for heavy-tailed spin glasses, in which the couplings are in the domain of attraction of an -stable law for . 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.
Paper Structure (21 sections, 23 theorems, 231 equations)

This paper contains 21 sections, 23 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Metastability and a Markov jump process between wells
  3. Behavior of two-time auto-correlation functions
  4. Proof ideas
  5. Preliminary estimates on the coupling matrix
  6. Row sum estimates
  7. Gaps in the point process of large bonds
  8. Bounding the mixing time in a well
  9. Mixing time and spectral gap tools
  10. Inverse gap of the small bond block
  11. Inverse gap of the intermediate bond block
  12. Bounding the block dynamics
  13. Analagous mixing time bounds for $Y$
  14. Metastability of Glauber dynamics
  15. Limit of the auto-correlation function
  16. ...and 6 more sections

Key Result

Theorem 1.4

For any $\alpha\in (0,1)$, with $\mathbb{P}_{\mathbf{J}}$-probability $1-o(1)$, the following holds. Fix any $\beta>0$, any $a>0$, and $\gamma \in (\gamma_0, \frac{1}{\alpha})$ and recall that $K = |E_{a,\gamma}|$. Let $(X(t))_{t\ge 0}$ be Glauber dynamics initialized from $X(0) \sim \text{Unif}(\Si

Theorems & Definitions (55)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.5
  • Remark 1.3
  • Theorem 1.6
  • Lemma 2.1
  • ...and 45 more