Volume 87, Number 6, September 2009
|Number of page(s)||5|
|Published online||12 October 2009|
Statistics at the tip of a branching random walk and the delay of traveling waves
Laboratoire de Physique Statistique, École Normale Supérieure, UPMC Paris 6, Université Paris Diderot, CNRS 24 rue Lhomond, 75005 Paris, France, EU
Accepted: 16 September 2009
We study the limiting distribution of particles at the frontier of a branching random walk. The positions of these particles can be viewed as the lowest energies of a directed polymer in a random medium in the mean-field case. We show that the average distances between these leading particles can be computed as the delay of a traveling wave evolving according to the Fisher-KPP front equation. These average distances exhibit universal behaviors, different from those of the probability cascades studied recently in the context of mean-field spin-glasses.
PACS: 02.50.-r – Probability theory, stochastic processes, and statistics / 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion / 89.75.Hc – Networks and genealogical trees
© EPLA, 2009
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