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

TL;DR

This paper analyzes the survival probability of a branching random walk with an absorbing wall, revealing a phase transition at a critical wall velocity and characterizing the behavior of survival probabilities and population sizes near this transition.

Contribution

It provides a detailed analysis of the phase transition in survival probability using the F-KPP equation, including the divergence of population size and relaxation times.

Findings

Survival probability vanishes at critical velocity v_c with an essential singularity.

Population size diverges exponentially as v approaches v_c from below.

No quasi-stationary regime exists for velocities above v_c.

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.