@article{ECP2438,
author = {Olivier Garet and Régine Marchand},
title = {The critical branching random walk in a random environment dies out},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {18},
year = {2013},
keywords = {branching random walk; random environment; survival; critical behavior; renormalization; block construction},
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.},
pages = {no. 9, 1-15},
issn = {1083-589X},
doi = {10.1214/ECP.v18-2438},
url = {http://ecp.ejpecp.org/article/view/2438}}