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On multidimensional branching random walks in random environment. (English) Zbl 1114.60084

Summary: We study branching random walks in random i.i.d. environment in \(\mathbb{Z}^d\), \(d\geq 1\). For this model, the population size cannot decrease, and a natural ural definition of recurrence is introduced. We prove a dichotomy for recurrence/transience, depending only on the support of the environmental law. We give sufficient conditions for recurrence and for transience. In the recurrent case, we study the asymptotics of the tail of the distribution of the hitting times and prove a shape theorem for the set of lattice sites which are visited up to a large time.

MSC:

60K37 Processes in random environments
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
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