The Annals of Probability

Percolation on the stationary distributions of the voter model

Balázs Ráth and Daniel Valesin

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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 $d\geq3$.

Article information

Ann. Probab., Volume 45, Number 3 (2017), 1899-1951.

Received: February 2015
Revised: February 2016
First available in Project Euclid: 15 May 2017

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C22: Interacting particle systems [See also 60K35] 82B43: Percolation [See also 60K35]

Interacting particle systems voter model percolation


Ráth, Balázs; Valesin, Daniel. Percolation on the stationary distributions of the voter model. Ann. Probab. 45 (2017), no. 3, 1899--1951. doi:10.1214/16-AOP1104.

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