Nathanaël Berestycki

Beta-coalescents and continuous stable random trees

Coalescents with multiple collisions, also known as A-coalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case that the measure A is the Beta(2 - a, a) distribution, they are also known to describe the genealogies of large populations where a single individual can produce a large number of offspring. Here, we use a recent result of Birkner et al. to prove that Beta-coalescents can be embedded in continuous stable random trees, about which much is known due to the recent progress of Duquesne and Le Gall. Our proof is based on a construction of the Donnelly-Kurtz lookdown process using continuous random trees, which is of independent interest. This produces a number of results concerning the small-time behavior of Beta-coalescents. Most notably, we recover an almost sure limit theorem of the present authors for the number of blocks at small times and give the multifractal spectrum corresponding to the emergence of blocks with atypical size. Also, we are able to find exact asymptotics for sampling formulae corresponding to the site frequency spectrum and the allele frequency spectrum associated with mutations in the context of population genetics.

Small-time behavior of beta coalescents

For a finite measureon (0,1), the �-coalescent is a coalescent process such that, whenever there are b clusters, each k-tuple of clusters merges into one at rate R 1 0 x k 2 (1 x) b k �(dx). It has recently been shown that if 1 < � < 2, the �-coalescent in whichis the Beta(2 �,�) distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an �-stable branching mechanism. Here we use facts about CSBPs to establish new results about the small-time asymptotics of beta coalescents. We prove an a.s. limit theorem for the number of blocks at small times, and we establish results about the sizes of the blocks. We also calculate the Hausdorff and packing dimensions of a metric space associated with the beta coalescents, and we find the sum of the lengths of the branches in the coalescent tree, both of which are determined by the behavior of coalescents at small times. We extend most of these results to other �-coalescents for whichhas the same asymptotic behavior near zero as the Beta(2 �,�) distribution. This work complements recent work of Bertoin and Le Gall, who also used CSBPs to study small-time properties of �-coalescents.

An elementary approach to Gaussian multiplicative chaos

A completely elementary and self-contained proof of convergence of Gaussian multiplicative chaos is given. The argument shows further that the limiting random measure is nontrivial in the entire subcritical phase

The Λ-coalescent speed of coming down from infinity

(\gamma < \sqrt{2d})

Diffusion in planar Liouville quantum gravity

and that the limit is universal (i.e., the limiting measure is independent of the regularisation of the underlying field)

Mixing times for random k-cycles and coalescence-fragmentation chains

Consider a

Survival of Near-Critical Branching Brownian Motion

\Lambda

KPZ formula derived from Liouville heat kernel

-coalescent that comes down from infinity (meaning that it starts from a configuration containing infinitely many blocks at time 0, yet it has a finite number

A small-time coupling between -coalescents and branching processes

N_t

Asymptotic sampling formulae for

of blocks at any positive time