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

Abstract : We consider a class of branching-selection particle systems on $\R$ similar to the one considered by E. Brunet and B. Derrida in their 1997 paper "Shift in the velocity of a front due to a cutoff". Based on numerical simulations and heuristic arguments, Brunet and Derrida showed that, as the population size $N$ of the particle system goes to infinity, the asymptotic velocity of the system converges to a limiting value at the unexpectedly slow rate $(\log N)^{-2}$. In this paper, we give a rigorous mathematical proof of this fact, for the class of 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 of the particle system with a family of $N$ independent branching random walks killed below a linear space-time barrier.
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Contributor : Jean Bérard <>
Submitted on : Wednesday, March 3, 2010 - 3:14:10 PM
Last modification on : Monday, June 28, 2021 - 2:26:03 PM
Long-term archiving on: : Thursday, September 23, 2010 - 6:16:41 PM


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  • HAL Id : hal-00339394, version 3
  • ARXIV : 0811.2782


Jean Bérard, Jean-Baptiste Gouéré. Brunet-Derrida behavior of branching-selection particle systems on the line. Communications in Mathematical Physics, Springer Verlag, 2010, 298 (2), pp.323-342. ⟨hal-00339394v3⟩



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