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A branching random walk in presence of an absorbing wall moving at a constantvelocity v undergoes a phase transition as v varies. The problem can beanalyzed using the properties of the Fisher-Kolmogorov-Petrovsky-Piscounov(F-KPP) equation. We find that the survival probability of the branching randomwalk vanishes at a critical velocity v_c of the wall with an essentialsingularity and we characterize the divergences of the relaxation times forv<v_c and v>v_c. At v=v_c the survival probability decays like a stretchedexponential. Using the F-KPP equation, one can also calculate the distributionof the population size at time t conditionned by the survival of one individualat a later time T>t. Our numerical results indicate that the size of thepopulation diverges like the exponential of (v_c-v)^{-1/2} in thequasi-stationary regime below v_c. Moreover for v>v_c, our data indicate thatthere is no quasi-stationary regime.Comment: References and one figure adde

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