On the threshold of spread-out voter model percolation

Abstract

In the R-spread out, d-dimensional voter model, each site x of Zd 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≥3, the set of (extremal) stationary measures of this model is given by a family μα,R, where α∈[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 α and the correlation weakens as R becomes larger. We study these configurations from the point of view of nearest neighbor site percolation on Zd, focusing on asymptotics as R→∞. In [RV15], we have shown that, if R is large, there is a critical value αc(R) such that there is percolation if α>αc(R) and no percolation if α<αc(R). Here we prove that, as R→∞, αc(R) converges to the critical probability for Bernoulli site percolation on Zd. Our proof relies on a new upper bound on the joint occurrence of events under μα,R which is of independent interest