Speed and fluctuations of N-particle branching Brownian motion with spatial selection

Abstract

We consider branching Brownian motion on the real line with the following selection mechanism: Every time the number of particles exceeds a (large) given number $N$, only the $N$ right-most particles are kept and the others killed. After rescaling time by $\log^3N$, we show that the properly recentred position of the $\lceil αN\rceil$-th particle from the right, $α\in(0,1)$, converges in law to an explicitly given spectrally positive Lévy process. This behaviour has been predicted to hold for a large class of models falling into the universality class of the FKPP equation with weak multiplicative noise [Brunet et al., Phys. Rev. E \textbf{73}(5), 056126 (2006)] and is proven here for the first time for such a model.

Continues and essentially replaces arXiv:1112.0266v1. Based on Chapter 2 of my PhD thesis at Université Pierre et Marie Curie, Paris, available at arXiv:1210.3500. Changes in v2 (74 pages): Reorganisation, simplifications in some places, typos corrected. Changes in v3 (84 pages): Many small corrections and additional details. Changes in v4 (87 pages): journal version, minor modifications