Issue
EPL
Volume 78, Number 6, June 2007
Article Number 60006
Number of page(s) 6
Section General
DOI http://dx.doi.org/10.1209/0295-5075/78/60006
Published online 01 June 2007
EPL, 78 (2007) 60006
DOI: 10.1209/0295-5075/78/60006

The survival probability of a branching random walk in presence of an absorbing wall

B. Derrida and D. Simon

Laboratoire de Physique Statistique, École Normale Supérieure - 24 rue Lhomond, 75231 Paris cedex 05, France

bernard.derrida@lps.ens.fr
damien.simon@lps.ens.fr

received 12 March 2007; accepted in final form 30 April 2007; published June 2007
published online 1 June 2007

Abstract
A branching random walk in presence of an absorbing wall moving at a constant velocity v undergoes a phase transition as v varies. The problem can be analyzed using the properties of the Fisher-Kolmogorov-Petrovsky-Piscounov (F-KPP) equation. We find that the survival probability of the branching random walk vanishes at a critical velocity vc of the wall with an essential singularity and we characterize the divergences of the relaxation times for v < vc and v > vc. At v = vc the survival probability decays like a stretched exponential. Using the F-KPP equation, one can also calculate the distribution of the population size at time t conditioned by the survival of one individual at a later time T > t. Our numerical results indicate that the size of the population diverges like the exponential of (vc-v)-1/2 in the quasi-stationary regime below vc. Moreover for v > vc, our data indicate that there is no quasi-stationary regime.

PACS
02.50.-r - Probability theory, stochastic processes, and statistics.
05.40.-a - Fluctuation phenomena, random processes, noise, and Brownian motion.
05.70.Jk - Critical point phenomena.

© Europhysics Letters Association 2007