The genealogy of a cluster in the multitype voter model

J. Theodore Cox, Jochen Geiger

Research output: Contribution to journalArticle

3 Scopus citations

Abstract

The genealogy of a cluster in the multitype voter model can be defined in terms of a family of dual coalescing random walks. We represent the genealogy of a cluster as a point process in a size-time plane and show that in high dimensions the genealogy of the cluster at the origin has a weak Poisson limit. The limiting point process is the same as for the genealogy of the size-biased Galton-Watson tree. Moreover, our results show that the branching mechanism and the spatial effects of the voter model can be separated on a macroscopic scale. Our proofs are based on a probabilistic construction of the genealogy of the cluster at the origin derived from Harris' graphical representation of the voter model.

Original languageEnglish (US)
Pages (from-to)1588-1619
Number of pages32
JournalAnnals of Probability
Volume28
Issue number4
StatePublished - Oct 1 2000

Keywords

  • Poisson point process
  • Voter model
  • coalescing random walk

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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