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Branching random walks in space-time random environment: Survival probability, global and local growth rates. (English) Zbl 1235.60146

The authors are concerned with the survival probability and growth rate for branching random walks in random environment (BRWRE). The particles perform simple symmetric random walks on the \(d\)-dimensional integer lattice, while at each time unit, they split into independent copies according to time-space i.i.d. offspring distributions. The BRWRE is naturally associated with the directed polymers in random environment (DPRE) for which the quantity called the free energy is well studied. The authors discuss the survival probability (both global and local) for BRWRE and give a criterion for its positivity in terms of the free energy of the associated DPRE. It is also shown that the global growth rate for the number of particles in BRWRE is given by the free energy of the associated DPRE, though the local growth rate is given by the directional free energy.

MSC:

60K37 Processes in random environments
60F20 Zero-one laws
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
82D60 Statistical mechanics of polymers
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References:

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