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Journal URL http://www.math.washington.edu/~ejpecp/ Paper URL http://www.math.washington.edu/~ejpecp/EjpVol5/paper12.abs.html COALESCENTS WITH SIMULTANEOUS MULTIPLE COLLISIONS

We consider a system of particles which perform branching Brownian motion with negative drift and are killed upon reaching zero, in the near-critical regime where the total population stays roughly constant with approximately N particles. We show that the characteristic time scale for the evolution of this population is of order $(\log N)^3$, in the sense that when time is measured in these units, the scaled number of particles converges to a variant of Neveu's continuous-state branching process. Furthermore, the genealogy of the particles is then governed by a coalescent process known as the Bolthausen-Sznitman coalescent. This validates the nonrigorous predictions by Brunet, Derrida, Muller and Munier for a closely related model.Comment: Published in at http://dx.doi.org/10.1214/11-AOP728 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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When an advantageous mutation occurs in a population, the favorable allele may spread to the entire population in a short time, an event known as a selective sweep. As a result, when we sample n individuals from a population and trace their ancestral lines backwards in time, many lineages may coalesce almost instantaneously at the time of a selective sweep. We show that as the population size goes to infinity, this process converges to a coalescent process called a coalescent with multiple collisions. A better approximation for finite populations can be obtained using a coalescent with simultaneous multiple collisions. We also show how these coalescent approximations can be used to get insight into how beneficial mutations affect the behavior of statistics that have been used to detect departures from the usual Kingman's coalescent. r 2005 Elsevier B.V. All rights reserved. MSC: primary 60J27; secondary 90D10; 90D15