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

A branching random walk in presence of an absorbing wall moving at a constant velocity upsilon undergoes a phase transition as upsilon 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 upsilon(c) of the wall with an essential singularity and we characterize the divergences of the relaxation times for upsilon < upsilon(c) and upsilon > upsilon(c). At upsilon = upsilon(c) 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 (upsilon(c) - upsilon)(-1/2) in the quasi-stationary regime below upsilon(c). Moreover for upsilon > upsilon(c), our data indicate that there is no quasi-stationary regime. Copyright (C) EPLA, 2007