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Global divergence of spatial coalescents

Research paper by Omer Angel, Nathanaël Berestycki, Vlada Limic

Indexed on: 24 Nov '10Published on: 24 Nov '10Published in: Probability Theory and Related Fields



Abstract

We study several fundamental properties of a class of stochastic processes called spatial Λ-coalescents. In these models, a number of particles perform independent random walks on some underlying graph G. In addition, particles on the same vertex merge randomly according to a given coalescing mechanism. A remarkable property of mean-field coalescent processes is that they may come down from infinity, meaning that, starting with an infinite number of particles, only a finite number remains after any positive amount of time, almost surely. We show here however that, in the spatial setting, on any infinite and bounded-degree graph, the total number of particles will always remain infinite at all times, almost surely. Moreover, if \({G\,=\,\mathbb{Z}^d}\), and the coalescing mechanism is Kingman’s coalescent, then starting with N particles at the origin, the total number of particles remaining is of order (log* N)d at any fixed positive time (where log* is the inverse tower function). At sufficiently large times the total number of particles is of order (log* N)d-2, when d > 2. We provide parallel results in the recurrent case d = 2. The spatial Beta-coalescents behave similarly, where log log N is replacing log* N.