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Quasi-stationary regime of a branching random walk in presence of an absorbing wall. (English) Zbl 1144.82321

Summary: 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 \(v \rightarrow v_{c}\). 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.

MSC:

82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
82B27 Critical phenomena in equilibrium statistical mechanics
92D25 Population dynamics (general)
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