Coalescing Random Walks and Voter Model Consensus Times on the Torus in Zᵈ

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.