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

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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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Authors and Affiliations

  1. Mathématiques, Case 7012, Bâtiment Chevaleret, Université Paris Diderot—Paris 7, 75205, Paris Cedex 13, France

    Francis Comets

  2. Division of Mathematics, Graduate School of Science, Kyoto University, Kyoto, 606-8502, Japan

    Nobuo Yoshida

Authors
  1. Francis Comets
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  2. Nobuo Yoshida
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Correspondence to Francis Comets.

Additional information

F. Comets partially supported by ANR Polintbio.

N. Yoshida partially supported by JSPS Grant-in-Aid for Scientific Research, Kiban (C) 21540125.

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Comets, F., Yoshida, N. Branching Random Walks in Space–Time Random Environment: Survival Probability, Global and Local Growth Rates. J Theor Probab 24, 657–687 (2011). https://doi.org/10.1007/s10959-009-0267-x

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  • DOI: https://doi.org/10.1007/s10959-009-0267-x

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