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

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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.

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MD thanks Eric Vigoda and Antonio Blanca for discussions related to the erosion times of boxes. MD and DS are grateful to Rob Morris for lengthy email discussions of coarsening dynamics. LP thanks Timo Seppäläinen and Benedek Valkó for helpful discussions on large deviation estimates in particle systems, and Ivan Corwin for useful remarks. MD is supported by an NSF CAREER grant. LP is partially supported by NSF Grant DMS-1664617. DS is supported by NSF Grants DMS-1418265 and CCF-1740761. We are grateful to anonymous referees for valuable suggestions.

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Correspondence to Leonid Petrov.

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Communicated by H. Spohn

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Damron, M., Petrov, L. & Sivakoff, D. Coarsening Model on \({\mathbb{Z}^{d}}\) with Biased Zero-Energy Flips and an Exponential Large Deviation Bound for ASEP. Commun. Math. Phys. 362, 185–217 (2018).

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