Coarsening Model on ${\mathbb{Z}^{d}}$ with Biased Zero-Energy Flips and an Exponential Large Deviation Bound for ASEP

Abstract

We study the coarsening model (zero-temperature Ising Glauber dynamics) on ${\mathbb{Z}^{d}}$ (for ${d \geq 2}$) with an asymmetric tie-breaking rule. This is a Markov process on the state space ${\{-1,+1\}^{{\mathbb{Z}}^d}}$ of “spin configurations” in which each vertex updates its spin to agree with a majority of its neighbors at the arrival times of a Poisson process. If a vertex has equally many +1 and −1 neighbors, then it updates its spin value to +1 with probability ${q \in [0,1]}$ and to −1 with probability 1 − q. The initial state of this Markov chain is distributed according to a product measure with probability p for a spin to be +1. In this paper, we show that for any given ${p > 0}$, there exist q close enough to 1 such that a.s. every spin has a limit of +1. This is of particular interest for small values of p, for which it is known that if ${q = 1/2}$, a.s. all spins have a limit of −1. For dimension d = 2, we also obtain near-exponential convergence rates for q sufficiently large, and for general d, we obtain stretched exponential rates independent of d. Two important ingredients in our proofs are refinements of block arguments of Fontes–Schonmann–Sidoravicius and a novel exponential large deviation bound for the Asymmetric Simple Exclusion Process.