# Branching Random Walks in Space-Time Random Environment: Survival Probability, Global and Local Growth Rates

Abstract : We study the survival probability and the 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. We 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. We also show 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.
Keywords :
Document type :
Preprints, Working Papers, ...
2009
Domain :

https://hal.archives-ouvertes.fr/hal-00402147
Contributor : Francis Comets <>
Submitted on : Monday, July 6, 2009 - 5:19:51 PM
Last modification on : Wednesday, October 12, 2016 - 1:01:09 AM

### Identifiers

• HAL Id : hal-00402147, version 1
• ARXIV : 0907.0509

### Citation

Francis Comets, Nobuo Yoshida. Branching Random Walks in Space-Time Random Environment: Survival Probability, Global and Local Growth Rates. 2009. <hal-00402147>

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