2017/05/17 by Ráth, Balázs, Valesin, Daniel
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1705.06244
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 \citeRV15, 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.