Percolation on the stationary distributions of the voter model

The voter model on ${\mathbb{Z}}^d$ is a particle system that serves as a rough model for changes of opinions among social agents or, alternatively, competition between biological species occupying space. When $d\ge 3$, the set of (extremal) stationary distributions is a family of measures ${\mu }_{\alpha }$, for $\alpha$ between 0 and 1. A configuration sampled from ${\mu }_{\alpha }$ is a strongly correlated field of 0’s and 1’s on ${\mathbb{Z}}^d$ in which the density of 1’s is $\alpha$. We consider such a configuration as a site percolation model on ${\mathbb{Z}}^d$. We prove that if $d\ge 5$, the probability of existence of an infinite percolation cluster of 1’s exhibits a phase transition in $\alpha$. If the voter model is allowed to have sufficiently spread-out interactions, we prove the same result for $d\ge 3$.