## Electronic Communications in Probability

### The critical branching random walk in a random environment dies out

#### 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.

#### Article information

Source
Electron. Commun. Probab., Volume 18 (2013), paper no. 9, 15 pp.

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

https://projecteuclid.org/euclid.ecp/1465315548

Digital Object Identifier
doi:10.1214/ECP.v18-2438

Mathematical Reviews number (MathSciNet)
MR3019672

Zentralblatt MATH identifier
1306.60153

Rights

#### Citation

Garet, Olivier; Marchand, Régine. The critical branching random walk in a random environment dies out. Electron. Commun. Probab. 18 (2013), paper no. 9, 15 pp. doi:10.1214/ECP.v18-2438. https://projecteuclid.org/euclid.ecp/1465315548

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