Electronic Communications in Probability

On the threshold of spread-out voter model percolation

Balázs Ráth and Daniel Valesin

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Abstract

In the $R$-spread out, $d$-dimensional voter model, each site $x$ of $\mathbb{Z} ^d$ has state (or ‘opinion’) 0 or 1 and, with rate 1, updates its opinion by copying that of some site $y$ chosen uniformly at random among all sites within distance $R$ from $x$. If $d \geq 3$, the set of (extremal) stationary measures of this model is given by a family $\mu _{\alpha , R}$, where $\alpha \in [0,1]$. Configurations sampled from this measure are polynomially correlated fields of 0’s and 1’s in which the density of 1’s is $\alpha $ and the correlation weakens as $R$ becomes larger. We study these configurations from the point of view of nearest neighbor site percolation on $\mathbb{Z} ^d$, focusing on asymptotics as $R \to \infty $. In [RV15], we have shown that, if $R$ is large, there is a critical value $\alpha _c(R)$ such that there is percolation if $\alpha > \alpha _c(R)$ and no percolation if $\alpha < \alpha _c(R)$. Here we prove that, as $R \to \infty $, $\alpha _c(R)$ converges to the critical probability for Bernoulli site percolation on $\mathbb{Z} ^d$. Our proof relies on a new upper bound on the joint occurrence of events under $\mu _{\alpha ,R}$ which is of independent interest.

Article information

Source
Electron. Commun. Probab., Volume 22 (2017), paper no. 50, 12 pp.

Dates
Received: 6 June 2017
Accepted: 14 August 2017
First available in Project Euclid: 6 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1507255233

Digital Object Identifier
doi:10.1214/17-ECP80

Mathematical Reviews number (MathSciNet)
MR3710806

Zentralblatt MATH identifier
06797803

Subjects
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]

Keywords
interacting particle systems voter model percolation

Rights
Creative Commons Attribution 4.0 International License.

Citation

Ráth, Balázs; Valesin, Daniel. On the threshold of spread-out voter model percolation. Electron. Commun. Probab. 22 (2017), paper no. 50, 12 pp. doi:10.1214/17-ECP80. https://projecteuclid.org/euclid.ecp/1507255233


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