On the long term behavior of some finite particle systems

J. T. Cox, A. Greven

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


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 Zd. Let ηt denote this system, and let ηtN denote a finite version of ηt defined on the torus [-N, N]d∩Zd. 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.] {vθ}, θ≧0 is the set of extremal invariant measures for the infinite system ηt and Qs 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.

Original languageEnglish (US)
Pages (from-to)195-237
Number of pages43
JournalProbability Theory and Related Fields
Issue number2
StatePublished - Jun 1990

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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