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In this paper, we prove a general result concerning finite-range, attractiveinteracting particle systems on $\{-1, 1\}^{\mathbb{Z}^d}$. If the particlesystem has a unique stationary measure and, in a precise sense, relaxes to thisstationary measure at an exponential rate then any small perturbation of thedynamics also has a unique stationary measure to which it relaxes at anexponential rate. To augment this result, we study the particular case ofGlauber dynamics for the Ising model. We show that for any non-zero externalfield the dynamics converges to its unique invariant measure at an exponentialrate. 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.

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