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2013 The critical branching random walk in a random environment dies out
Olivier Garet, Régine Marchand
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Electron. Commun. Probab. 18: 1-15 (2013). DOI: 10.1214/ECP.v18-2438


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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Olivier Garet. Régine Marchand. "The critical branching random walk in a random environment dies out." Electron. Commun. Probab. 18 1 - 15, 2013.


Accepted: 31 January 2013; Published: 2013
First available in Project Euclid: 7 June 2016

zbMATH: 1306.60153
MathSciNet: MR3019672
Digital Object Identifier: 10.1214/ECP.v18-2438

Primary: 60K35
Secondary: 82B43


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