• DOI:10.1007/s00220-018-3180-2
• Corpus ID: 119610169

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

@article{Damron2018CoarseningMO,
  title={Coarsening Model on \$\$\{\mathbb\{Z\}^\{d\}\}\$\$Zd with Biased Zero-Energy Flips and an Exponential Large Deviation Bound for ASEP},
  author={Michael Damron and Leonid A. Petrov and David J Sivakoff},
  journal={Communications in Mathematical Physics},
  year={2018},
  volume={362},
  pages={185-217}
}

• M. Damron, L. Petrov, David J Sivakoff
• Published 19 August 2017
• Mathematics
• Communications in Mathematical Physics

We study the coarsening model (zero-temperature Ising Glauber dynamics) on $${\mathbb{Z}^{d}}$$Zd (for $${d \geq 2}$$d≥2) with an asymmetric tie-breaking rule. This is a Markov process on the state space $${\{-1,+1\}^{{\mathbb{Z}}^d}}$${-1,+1}Zd 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 $${q \in [0,1]}$$q∈[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 $${p > 0}$$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 $${q = 1/2}$$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.