Coalescing Random Walks and Voter Model Consensus Times on the Torus in $\mathbb{Z}^d$

Abstract

Let ηt be the basic voter model on Zd and let η(N) t be the voter model on Λ(N), the torus of side N in Zd. Unlike ηt, η(N) t (for fixed N) gets trapped with probability 1 as t →∞ at all 0's or all 1's. We examine the asymptotic growth of these trapping or consensus times τ(N) as N →∞. To do this we obtain limit theorems for coalescing random walk systems on the torus Λ(N), including a new hitting time limit theorem for (noncoalescing) random walk on the torus.