Branching Brownian motion with selection of the N right-most particles: An approximate model

Abstract

We present an approximation to the Brunet--Derrida model of supercritical branching Brownian motion on the real line with selection of the $N$ right-most particles, valid when the population size $N$ is large. It consists of introducing a random space-time barrier at which particles are instantaneously killed in such a way that the population size stays almost constant over time. We prove that the suitably recentered position of this barrier converges at the $\log^3 N$ timescale to a Lévy process, which we identify. This validates the physicists' predictions about the fluctuations in the Brunet--Derrida model.

No change in content from v2, only typesetting. The results of this article are essentially contained in arXiv:1304.0562 or Chapter 2 of arXiv:1210.3500, which we recommend to read instead