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The survival probability of a branching random walk in presence of an absorbing wall. (English) Zbl 1244.82071

Summary: 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 \(t\) 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 \(v_{c}\). Moreover for \(v>v_{c}\), our data indicate that there is no quasi-stationary regime.

MSC:

82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
60G50 Sums of independent random variables; random walks
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