Stationary Distributions for the Voter Model in $d\geq 3$ are Factors of IID

TL;DR

This paper proves that for the voter model in dimensions three and higher, the extremal stationary distributions are isomorphic to Bernoulli shifts and can be explicitly constructed as factors of IID processes, answering an open question.

Contribution

It establishes that stationary distributions of the voter model in high dimensions are factors of IID and provides explicit constructions, resolving an open problem.

Findings

Stationary distributions are isomorphic to Bernoulli shifts.

Explicit constructions of stationary distributions as factors of IID.

Answers an open question by Steif and Tykesson.

Abstract

For the Voter Model on $\mathbb{Z}^d$, $d\geq 3$, we show that the (extremal) stationary distributions are isomorphic to Bernoulli shifts, and answer an open question asked by Steif and Tykesson. The proof gives explicit constructions of the stationary distributions as factors of IID processes on $\mathbb{Z}^d$.