### 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 language | English (US) |
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Pages (from-to) | 1588-1619 |

Number of pages | 32 |

Journal | Annals of Probability |

Volume | 28 |

Issue number | 4 |

State | Published - 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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## Cite this

*Annals of Probability*,

*28*(4), 1588-1619.