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

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

2007-03-13

arxiv.org/abs/cond-mat/0703353v3

EPL, 78 (2007) 60006

Physics

Condensed Matter

Statistical Mechanics

References and one figure added

Scientific paper

10.1209/0295-5075/78/60006

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 vv_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.

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