Asymptotics for interacting particle systems on Zd

Maury Bramson, David Griffeath

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105 Scopus citations

Abstract

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 (ζtZd) be a system of coalescing random walks with initial state Zd, and (ζtO) 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 pt∼(γdt)-1 as t→∞ for d≧3 for some γd. As a consequence, conditioned on non-extinction of ζtO, PttO| approaches an exponential distribution. Results of a recent paper by Sawyer are applied.

Original languageEnglish (US)
Pages (from-to)183-196
Number of pages14
JournalZeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete
Volume53
Issue number2
DOIs
StatePublished - Jun 1 1980

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