Volume 78, Number 6, June 2007
|Number of page(s)||6|
|Published online||01 June 2007|
The survival probability of a branching random walk in presence of an absorbing wall
Laboratoire de Physique Statistique, École Normale Supérieure - 24 rue Lhomond, 75231 Paris cedex 05, France
Accepted: 30 April 2007
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 and . At 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 . Our numerical results indicate that the size of the population diverges like the exponential of in the quasi-stationary regime below vc. Moreover for , 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
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