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arXiv · 1501.04746

The Toom Interface Via Coupling

Nick Crawford·Gady Kozma·Wojciech de Roeck
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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 $k \rightarrow \infty$.

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Nick Crawford, Gady Kozma, Wojciech de Roeck. 2015-01-20. The Toom Interface Via Coupling. https://arxiv.org/abs/1501.04746

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