DH 3rd floor SR

Lambda-coalescents were introduced by Pitman in (1999) and Sagitov (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 Lambda has the Beta$(2-\alpha,\alpha)$ they are also known to describe the genealogies of large populations where a single individual can produce a large number of offsprings. Here we use a recent result of Birkner et al. (2005) to prove that Beta-coalescents can be embedded in continuous stable random trees, for which much is known due to recent progress of Duquesne and Le Gall. This produces a number of results concerning the small-time behaviour of Beta-coalescents. Most notably, we recover an almost sure limit theorem 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 infinite site frequency spectrum associated with mutations in the context of population genetics.