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May 2017 Percolation on the stationary distributions of the voter model
Balázs Ráth, Daniel Valesin
Ann. Probab. 45(3): 1899-1951 (May 2017). DOI: 10.1214/16-AOP1104

Abstract

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 $d\geq3$.

Citation

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Balázs Ráth. Daniel Valesin. "Percolation on the stationary distributions of the voter model." Ann. Probab. 45 (3) 1899 - 1951, May 2017. https://doi.org/10.1214/16-AOP1104

Information

Received: 1 February 2015; Revised: 1 February 2016; Published: May 2017
First available in Project Euclid: 15 May 2017

zbMATH: 06754789
MathSciNet: MR3650418
Digital Object Identifier: 10.1214/16-AOP1104

Subjects:
Primary: 60K35, 82B43, 82C22

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 3 • May 2017
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