2012/11/16 by Garet, Olivier, Marchand, Régine
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1211.3827
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 Ψ called the free energy is well studied. Comets and Yoshida proved that there is no survival when Ψ<0 and that survival is possible when Ψ>0. We proved here that, except for degenerate cases, the BRWRE always die when Ψ=0. This solves a conjecture of Comets and Yoshida.