The survival probability of a branching random walk in presence of an absorbing wall
B. Derrida, D. Simon,
https://arxiv.org/abs/cond-mat/0703353,
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 $v_c$ of the wall with an essential singularity and we characterize the divergences of the relaxation times for $v<v_c$ and $v>v_c$. 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 $t$ conditionned 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 $v_c$. Moreover for $v>v_c$, our data indicate that there is no quasi-stationary regime.