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.
Publié le : 2000-10-14
Classification:
Voter model,
coalescing random walk,
Poisson point process,
60K35,
60F05,
60J80
@article{1019160499,
author = {Cox, J. Theodore and Geiger, Jochen},
title = {The genealogy of a cluster in the multitype voter model},
journal = {Ann. Probab.},
volume = {28},
number = {1},
year = {2000},
pages = { 1588-1619},
language = {en},
url = {http://dml.mathdoc.fr/item/1019160499}
}
Cox, J. Theodore; Geiger, Jochen. The genealogy of a cluster in the multitype voter model. Ann. Probab., Tome 28 (2000) no. 1, pp. 1588-1619. http://gdmltest.u-ga.fr/item/1019160499/