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October 1, 2009: Jason
Schweinsberg (University of California, San Diego)
Title: The genealogy of branching Brownian motion with absorption.
Abstract: 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 version 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 non-rigorous predictions by Brunet, Derrida, Muller, and Munier
for a closely related model. This is joint work with Julien Berestycki and
Nathanael Berestycki.
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