Percolation on the stationary distributions of the voter model

2015/02/04 by Rath, Balazs, Valesin, Daniel
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1502.01306

Abstract

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

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