The genealogy of a cluster in the multitype voter model

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.