BETA-COALESCENTS AND CONTINUOUS STABLE RANDOM TREES

By J. Berestycki, Nathanaël Berestycki, Jason Schweinsberg. 2006

116 citations recorded by the source corpus.

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.

Subfields: Condensed Matter Physics, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/35dcd856600a28bc7ebd001f55bc4b60330fe408

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Citations in the corpus

Recent progress in coalescent theory (2009, unknown): “(1) The convergence in probability was first obtained in [20] in the case of Beta-coalescents with parameter (2 − α, α), using an exact embedding within the so-called stable Continuum Random Tree and an analysis of the mutation process using queues.”
The coalescent structure of continuous-time Galton–Watson trees (2017, background): “1) This random variable has a special connection with heavy-tailed critical processes which we explore shortly, but it is worth mentioning here it also has cameo appearences in [12] and in [8] in the analysis of Levy processes and beta coalescents respectively.”
The Λ-coalescent speed of coming down from infinity (2008, background): “For a generalization of -coalescents where multiple mergers are possible, see Schweinsberg [29].”
Small-time behavior of beta coalescents (2006, unknown): “This is made more precise in [3] where we analyze the precise multifractal nature of the Beta-coalescent.”
Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws ∗ (2007, background): “In many situations [22, 24, 32, 5 ] one encounters a relation”