Stability of the Uniqueness Regime for Ferromagnetic Glauber Dynamics Under Non-reversible Perturbations

Abstract

We prove a general stability property 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 apply this result, we study the particular case of Glauber dynamics for the Ising model. We show that for any nonzero 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 of our stability property, we then conclude that Glauber dynamics for the Ising model is stable to small, non-reversible perturbations in the entire uniqueness phase, excluding only the critical point.