The critical branching random walk in a random environment dies out

Olivier Garet (1) , Régine Marchand (1)

1 IECL - Institut Élie Cartan de Lorraine

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.

Mots clés: critical behavior; branching random walk; random environment; renormalization; survival; block construction

Domaines: Probabilités [math.PR]

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⟩