The critical branching random walk in a random environment dies out

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 $\mathrm{\Psi }$ called the free energy is well studied. Comets and Yoshida proved that there is no survival when $\mathrm{\Psi }<0$ and that survival is possible when $\mathrm{\Psi }>0$. We proved here that, except for degenerate cases, the BRWRE always die when $\mathrm{\Psi }=0$. This solves a conjecture of Comets and Yoshida.