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Quasi-Stationary Regime of a Branching Random Walk in Presence of an Absorbing Wall

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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→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.

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  1. Laboratoire de Physique Statistique, École Normale Supérieure, 24 rue Lhomond, 75231, Paris Cedex 05, France

    Damien Simon & Bernard Derrida

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  1. Damien Simon
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  2. Bernard Derrida
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Correspondence to Damien Simon.

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Simon, D., Derrida, B. Quasi-Stationary Regime of a Branching Random Walk in Presence of an Absorbing Wall. J Stat Phys 131, 203–233 (2008). https://doi.org/10.1007/s10955-008-9504-4

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  • DOI: https://doi.org/10.1007/s10955-008-9504-4

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