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 µα, for α between 0 and 1. A configuration sampled from µα is a strongly correlated field of 0's and 1's on ℤd in which the density of 1's is α. 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 a. If the voter model is allowed to have sufficiently spread-out interactions, we prove the same result for d ≥ 3.