Renormalizing the 3-Dimensional Voter Model

Abstract

It is shown that a discrete time voter model in equilibrium on $\mathbb{Z}_3$ approaches the 0-mass free field of 3-dimensional Euclidean field theory under appropriate renormalization. This result is of interest because the strong correlation between distant sites gives rise to the renormalization exponent $- \frac{5}{2}$ instead of the usual $- \frac{3}{2}.$ Dawson, Ivanoff, and Spitzer have examined models on $\mathbb{R}_3$ which exhibit precisely the same limit. Because the process we consider lives on a lattice, our method of proof is necessarily quite different from theirs. In particular, we make use of a "duality" between voter models and coalescing random walks which has been exploited effectively by Holley and Liggett.