Branching Random Walks in Space–Time Random Environment: Survival Probability,Global and Local Growth Rates |
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Authors: | Francis Comets Nobuo Yoshida |
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Institution: | 1.Mathématiques, Case 7012, Batiment Chevaleret,Université Paris Diderot—Paris 7,Paris Cedex 13,France;2.Division of Mathematics, Graduate School of Science,Kyoto University,Kyoto,Japan |
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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. |
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