Asymptotics for interacting particle systems on Z^d

Two of the simplest interacting particle systems are the coalescing random walks and the voter model. We are interested here in the asymptotic density and growth of these systems as t→∞. More specifically, let (ζ_t^Zd) be a system of coalescing random walks with initial state Z^d, and (ζ_t^O) a voter model with a single individual originating at O. We analyse {Mathematical expression}, and show that {Mathematical expression} as t→∞ for d=2, and p_t∼(γ_dt)^-1 as t→∞ for d≧3 for some γ_d. As a consequence, conditioned on non-extinction of ζ_t^O, P_t|ζ_t^O| approaches an exponential distribution. Results of a recent paper by Sawyer are applied.