The Toom Interface Via Coupling

TL;DR

This paper studies a one-dimensional particle system modeling interface dynamics of the Toom model at low temperature, establishing properties like mixing time bounds and characterizing invariant measures.

Contribution

It introduces a non-Feller process with infinite range interactions, proves bounds on mixing time, and characterizes invariant measures as Bernoulli measures.

Findings

Mixing time on finite interval bounded by 2N

Invariant measures are Bernoulli measures under certain conditions

Stationary measures on semi-infinite intervals converge to Bernoulli measures

Abstract

We consider a one dimensional interacting particle system which describes the effective interface dynamics of the two dimensional Toom model at low temperature and noise. We prove a number of basic properties of this model. First we consider the dynamics on a half open finite interval $[1, N)$, bounding the mixing time from above by $2N$. Then we consider the model defined on the integers. Due to infinite range interaction, this is a non-Feller process that we can define starting from product Bernoulli measures with density $p \in (0, 1)$, but not from arbitrary measures. We show, under a modest technical condition, that the only possible invariant measures are those product Bernoulli measures. We further show that the unique stationary measure on $[-k, \infty)$ converges weakly to a product Bernoulli measure on $\mathbb{Z}$ as $k \rightarrow \infty$.