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The hydrodynamic limit of beta coalescents that come down from infinity

Mathematical Physics & Probability

Speaker: Helmut Pitters, UC Berkeley
Location: 2112 MSB
Start time: Wed, Jan 20 2016, 4:10PM

Among the most celebrated mathematical models in population genetics is Kingman's coalescent Pi(t), with time parameter t. This Markov process takes its values in the partitions of non-negative integers and its restriction Pi^{n}(t) to {1, …, n} models the genealogy of a sample of size n drawn from a large neutral population of haploid individuals. More precisely, individuals i and j have a common ancestor at time t in the past if i, j are in a common block of Pi(t). In recent years natural generalizations of Kingman's coalescent, so-called multiple merger coalescents, have been suggested as better null models for genealogies in highly fecund populations or populations undergoing selection. We study an important subclass of multiple merger coalescents, namely the beta coalescents. We present two closely related results: The first result is a law of large numbers for the beta n-coalescents Pi^{n}(t) as n tends to infinity. The second result quantifies the phenomenon of coming down from infinity of Pi(t). In both cases the rescaling limits are characterized by a system of ordinary differential equations. We work out the solutions of these ordinary differential equations in terms of Bell polynomials. This is joint work with Luke Miller (University of Oxford).