Mathematics – Probability
Scientific paper
2009-09-26
Mathematics
Probability
45 pages, 1 figure. Some proofs corrected and simplified, and introduction modified
Scientific paper
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.
Angel Omer
Berestycki Nathanael
Limic Vlada
No associations
LandOfFree
Global divergence of spatial coalescents does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Global divergence of spatial coalescents, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Global divergence of spatial coalescents will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-155586