The genealogy of a cluster in the multitype voter model
نویسندگان
چکیده
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. ∗Research supported in part by the National Science Foundation †Mathematics Department, Syracuse University, Syracuse, NY 13244, USA ‡Research supported by the German Research Foundation DFG §Fachbereich Mathematik, Postfach 11 19 32, 60054 Frankfurt, Germany
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