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Asymptotic behaviour of near-critical branching Brownian motion

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Consider a system of particles that perform branching Brownian motion with negative drift \sqrt(2- \eps) and are killed upon hitting zero. Initially, there is just one particle at x. Kesten (1978) proved that the system survives if and only if \eps>0. In this talk I shall describe recent joint work with Julien Berestycki and Jason Schweinsberg concerning the limiting behaviour of this process as \eps tends to 0. In particular we establish sharp asymptotics for the limiting survival probability as a function of the starting point x. Moreover, the limiting genealogy between individuals from this population is shown to have a characteristic time scale of order \eps^{-3/2}. When time is measured in these units we show that the geometry of the genealogical tree converges to the Bolthausen-Sznitman coalescent. This is closely related to a set of conjectures by Brunet, Derrida and Simon.

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