## 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 (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_{d}t)^{-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.

Original language | English (US) |
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Pages (from-to) | 183-196 |

Number of pages | 14 |

Journal | Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete |

Volume | 53 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 1980 |