On the threshold of spread-out voter model percolation

Ráth, Balázs and Valesin, Daniel (2017) On the threshold of spread-out voter model percolation. Electronic Communications in Probability. ISSN 1083-589X, ESSN: 1083-589X (In Press)

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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 [Ráth, Valesin, AoP, 2017] 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.

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