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We consider a class of branching-selection particle systems on $\R$ similarto the one considered by E. Brunet and B. Derrida in their 1997 paper "Shift inthe velocity of a front due to a cutoff". Based on numerical simulations andheuristic arguments, Brunet and Derrida showed that, as the population size $N$of the particle system goes to infinity, the asymptotic velocity of the systemconverges to a limiting value at the unexpectedly slow rate $(\log N)^{-2}$. Inthis paper, we give a rigorous mathematical proof of this fact, for the classof particle systems we consider. The proof makes use of ideas and results by R.Pemantle, and by N. Gantert, Y. Hu and Z. Shi, and relies on a comparison ofthe particle system with a family of $N$ independent branching random walkskilled below a linear space-time barrier.

Brunet-Derrida behavior of branching-selection particle systems on the line