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Journal Articles Electronic Communications in Probability Year : 2013

The critical branching random walk in a random environment dies out

Olivier Garet
Régine Marchand

Abstract

We study the possibility for branching random walks in random environment (BRWRE) to survive. 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. As noted by Comets and Yoshida, the BRWRE is naturally associated with the directed polymers in random environment (DPRE), for which the quantity $\Psi$ called the free energy is well studied. Comets and Yoshida proved that there is no survival when $\Psi<0$ and that survival is possible when $\Psi>0$. We proved here that, except for degenerate cases, the BRWRE always die when $\Psi=0$. This solves a conjecture of Comets and Yoshida.
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Dates and versions

hal-00752420 , version 1 (15-11-2012)

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Olivier Garet, Régine Marchand. The critical branching random walk in a random environment dies out. Electronic Communications in Probability, 2013, 18 (9), pp.1-15. ⟨hal-00752420⟩
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