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 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 $v v_{c}$. At $v = v_{c}$ 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 $(v_{c}-v)^{-1/2}$ in the quasi-stationary regime below vc. Moreover for $v > v_{c}$, our data indicate that there is no quasi-stationary regime