GLOBAL DIVERGENCE OF SPATIAL COALECSENTS
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<br>
Omer Angel, Nathana&euml;l Berestycki and Vlada Limic
<br>
<br>
We study several fundamental properties of a class of stochastic processes
called spatial &Lambda;-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=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 $log^* N$.