Quasi-stationary regime of a branching random walk in presence of an absorbing wall

Abstract

A branching random walk in presence of an absorbing wall moving at a constant velocity $v$ undergoes a phase transition as the velocity $v$ of the wall varies. Below the critical velocity $v_c$, the population has a non-zero survival probability and when the population survives its size grows exponentially. We investigate the histories of the population conditioned on having a single survivor at some final time $T$. We study the quasi-stationary regime for $v<v_c$ when $T$ is large. To do so, one can construct a modified stochastic process which is equivalent to the original process conditioned on having a single survivor at final time $T$. We then use this construction to show that the properties of the quasi-stationary regime are universal when $v\to v_c$. We also solve exactly a simple version of the problem, the exponential model, for which the study of the quasi-stationary regime can be reduced to the analysis of a single one-dimensional map.

arXiv comment2 figures, minor corrections, one reference added

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• Random Transverse Field Ising Model in dimension $d>1$ : scaling analysis in the disordered phase from the Directed Polymer model
• The critical random barrier for the survival of branching random walk with absorption
• Discontinuous Transition in a Boundary Driven Contact Process
• The role of noise and advection in absorbing state phase transitions
• Survival of homogeneous fragmentation processes with killing
• Total progeny in killed branching random walk
• Conformation-dependent sequence design of polymer chains in melts
• A note on the critical barrier for the survival of $\alpha -$stable branching random walk with absorption