Stability of the uniqueness regime for ferromagnetic Glauber dynamics under non-equilibrium perturbations

TL;DR

This paper proves that for ferromagnetic Glauber dynamics, the uniqueness of the stationary measure is stable under small non-equilibrium perturbations, with exponential convergence maintained across the entire uniqueness phase.

Contribution

It establishes the stability of the uniqueness regime for Glauber dynamics under non-equilibrium perturbations, extending known results to the entire phase.

Findings

Exponential convergence to the unique measure under perturbations.

Stability of the uniqueness regime for Glauber dynamics with external field.

Extension of convergence results beyond the high-temperature phase.

Abstract

In this paper, we prove a general result concerning finite-range, attractive interacting particle systems on $\{-1, 1\}^{\mathbb{Z}^d}$. If the particle system has a unique stationary measure and, in a precise sense, relaxes to this stationary measure at an exponential rate then any small perturbation of the dynamics also has a unique stationary measure to which it relaxes at an exponential rate. To augment this result, we study the particular case of Glauber dynamics for the Ising model. We show that for any non-zero external field the dynamics converges to its unique invariant measure at an exponential rate. Previously, this was only known for $\beta<\beta_c$ and $\beta$ sufficiently large. As a consequence, Glauber dynamics is stable to small, non-equilibrium perturbations in the entire uniqueness phase.