Global divergence of spatial coalescents

Abstract

We study several fundamental properties of a class of stochastic processes called spatial -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 .

Coalescing Brownian flows: A new approach

“…Our second lemma is an induction scheme which is inspired by an argument in [11] for the fact that coalescing Brownian particles come down from infinity. See also [2] where a similar argument is used. Lemma 6.5.…”

Some large deviations in Kingman's coalescent

“…The main goal of the present paper is to study the corresponding large deviations results. To the best of our knowledge, except for Angel et al (2012), cf. Remark 2.6, results in this direction are not present in the literature.…”

“…The figure on the left shows the rate function I from Corollary 2.4. The figure on the right is a comparison of I with the lower bound obtained fromAngel et al (2012).Remark 2.6 (The rate function I and comparison withAngel et al (2012)). Although the main goal ofAngel et al (2012) was the analysis of spatial Λ-coalescents, they also provide some large deviations bounds on Kingman's coalescent.…”

“…These bounds are mainly based on Markov inequality. Precisely, in Lemma 2.2 inAngel et al (2012) it is shown that for…”

The Effect of Single Recombination Events on Coalescent Tree Height and Shape

“…It constitutes the basis of genealogical models, of statistical tests of the neutral evolution hypothesis [4] as well as of many simulation tools [5]–[7]. Besides application in population genetics, coalescent models and their various generalizations became an object of study in their own right in probability, graph theory and combinatorics [8]–[12].…”