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

TL;DR

This paper studies a particle system where only the rightmost N particles survive, revealing that after rescaling, the position of a specific particle converges to a spectrally positive Lévy process, confirming a universality prediction.

Contribution

It proves for the first time that a branching Brownian motion with spatial selection converges to a Lévy process after rescaling, supporting universality in FKPP models.

Findings

Convergence of particle position to a Lévy process after rescaling

Validation of universality class predictions for spatially selected branching Brownian motion

Explicit description of the limiting Lévy process

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 $\lceil \alpha N\rceil$-th particle from the right, $\alpha\in(0,1)$, converges in law to an explicitly given spectrally positive L\'evy 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.