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We consider branching Brownian motion on the real line with the followingselection mechanism: Every time the number of particles exceeds a (large) givennumber $N$, only the $N$ right-most particles are kept and the others killed.After rescaling time by $\log^3N$, we show that the properly recentred positionof 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 fallinginto 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 herefor the first time for such a model.

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