Survival asymptotics for branching random walks in IID environments

We first study a model, introduced recently in cite{ES}, of a critical branching random walk in an IID random environment on the $d$-dimensional integer lattice. The walker performs critical (0-2) branching at a lattice point if and only if there is no `obstacle placed there. The obstacles appear at each site with probability $pin [0,1)$ independently of each other. We also consider a similar model, where the offspring distribution is subcritical. Let $S_n$ be the event of survival up to time $n$. We show that on a set of full $mathbb P_p$-measure, as $ntoinfty$, (i) Critical case: P^{omega}(S_n)simfrac{2}{qn}; (ii) Subcritical case: P^{omega}(S_n)= expleft[left( -C_{d,q}cdot frac{n}{(log n)^{2/d}} right)(1+o(1))right], where $C_{d,q}>0$ does not depend on the branching law. Hence, the model exhibits `self-averaging in the critical case but not in the subcritical one. I.e., in (i) the asymptotic tail behavior is the same as in a toy model where space is removed, while in (ii) the spatial survival probability is larger than in the corresponding toy model, suggesting spatial strategies. We utilize a spine decomposition of the branching process as well as some known results on random walks.