On the long term behavior of some finite particle systems

We consider the problem of comparing large finite and infinite systems with locally interacting components, and present a general comparison scheme for the case when the infinite system is nonergodic. We show that this scheme holds for some specific models. One of these is critical branching random walk on Z^d. Let η_t denote this system, and let η_t^N denote a finite version of η_t defined on the torus [-N, N]^d∩Z^d. For d≧3 we prove that for stationary, shift ergodic initial measures with density θ, that if T(N)→∞ and T(N)/(2 N+1)^d →s∈[0,∞] as N→∞, then[Figure not available: see fulltext.] {v_θ}, θ≧0 is the set of extremal invariant measures for the infinite system η_t and Q_s is the transition function of Feller's branching diffusion. We prove several extensions and refinements of this result. The other systems we consider are the voter model and the contact process.