The critical branching random walk in a random environment dies out
نویسندگان
چکیده
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.
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