Issue
EPL
Volume 87, Number 6, September 2009
Article Number 60010
Number of page(s) 5
Section General
DOI http://dx.doi.org/10.1209/0295-5075/87/60010
Published online 12 October 2009
EPL, 87 (2009) 60010
DOI: 10.1209/0295-5075/87/60010

Statistics at the tip of a branching random walk and the delay of traveling waves

É. Brunet and B. Derrida

Laboratoire de Physique Statistique, École Normale Supérieure, UPMC Paris 6, Université Paris Diderot, CNRS 24 rue Lhomond, 75005 Paris, France, EU

Eric.Brunet@lps.ens.fr
Bernard.Derrida@lps.ens.fr

received 24 July 2009; accepted in final form 16 September 2009; published September 2009
published online 12 October 2009

Abstract
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