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

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 L\'evy process, which we identify. This validates the physicists' predictions about the fluctuations in the Brunet--Derrida model.