Percolation on the stationary distributions of the voter model

Abstract

The voter model on ZdZd 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≥3d≥3, the set of (extremal) stationary distributions is a family of measures μαμα, for αα between 0 and 1. A configuration sampled from μαμα is a strongly correlated field of 0’s and 1’s on ZdZd in which the density of 1’s is αα. We consider such a configuration as a site percolation model on ZdZd. We prove that if d≥5d≥5, the probability of existence of an infinite percolation cluster of 1’s exhibits a phase transition in αα. If the voter model is allowed to have sufficiently spread-out interactions, we prove the same result for d≥3d≥3.