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Quasi-stationary regime of a branching random walk in presence of an absorbing wall
A branching random walk in presence of an absorbing wall moving at a constant
velocity undergoes a phase transition as the velocity of the wall
varies. Below the critical velocity , 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 . We study the quasi-stationary
regime for when 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 . We then use this construction to
show that the properties of the quasi-stationary regime are universal when
. 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.Comment: 2 figures, minor corrections, one reference adde