On the threshold of spread-out voter model percolation

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 $d\ge 3$, 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 \cite{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.

Accepted for publication in Electronic Communications in Probability

References in corpus (3)

• On decoupling inequalities and percolation of excursion sets of the Gaussian free field
• A short proof of the phase transition for the vacant set of random interlacements
• Generalized Divide and Color models