Percolation on the stationary distributions of the voter model. (English) Zbl 1437.60076

Ráth, Balázs; Valesin, Daniel

Ann. Probab. 45, No. 3, 1899-1951 (2017).

Summary: 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 \(d\geq3\), 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 \(d\geq5\), 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 \(d\geq3\).

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82C22	Interacting particle systems in time-dependent statistical mechanics
82B43	Percolation

Keywords:

interacting particle systems; voter model; percolation