Percolation on the stationary distributions of the voter model

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