Zero-temperature stochastic Ising model on planar quasi-transitive graphs

TL;DR

This paper investigates the zero-temperature stochastic Ising model on certain planar quasi-transitive graphs, demonstrating that with a specific initial distribution and graph property, all vertices flip infinitely often almost surely.

Contribution

It establishes conditions under which vertices flip infinitely often in the zero-temperature stochastic Ising model on planar quasi-transitive graphs.

Findings

Vertices flip infinitely often almost surely when p=1/2.

The planar shrink property ensures finite clusters vanish with positive probability.

Results apply to graphs invariant under rotation and translation.

Abstract

We study the zero-temperature stochastic Ising model on some connected planar quasi-transitive graphs, which are invariant under rotation and translation. The initial spin configuration is distributed according to a Bernoulli product measure with parameter $ p\in(0,1) $. In particular, we prove that if $ p=1/2 $ and the graph underlying the model satisfies the planar shrink property (which causes each finite cluster to shrink to a site and then vanish with positive probability) then all vertices flip infinitely often almost surely.