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

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 $\mathbb{Z}^d$.