Non-reversible stationary states for majority voter and Ising dynamics on trees

TL;DR

This paper demonstrates the existence of non-reversible stationary states in certain Markov processes on infinite trees, contrasting with the behavior on regular lattices, and introduces new non-Gibbsian measures for the Ising model.

Contribution

It proves the existence of non-reversible stationary measures for the Ising and voter models on trees, a phenomenon not observed on regular lattices like d.

Findings

Existence of non-reversible stationary measures on trees

Non-Gibbsian stationary states for the Ising model on trees

Contrast with no such states on d

Abstract

We study three Markov processes on infinite, unrooted, regular trees: the stochastic Ising model (also known as the Glauber heat bath dynamics of the Ising model), a majority voter dynamic, and a coalescing particle model. In each of the three cases the tree exhibits a preferred direction encoded into the model. For all three models, our main result is the existence of a stationary but non-reversible measure. For the Ising model, this requires imposing that the inverse temperature is large and choosing suitable non-uniform couplings, and our theorem implies the existence of a stationary measure which looks nothing like a low-temperature Gibbs measure. The interesting aspect of our results lies in the fact that the analogous processes do not have non-Gibbsian stationary measures on $\mathbb Z^d$, owing to the amenability of that graph. In fact, no example of a stochastic Ising model with a non-reversible stationary state was known to date.