Strong law of large numbers for a branching random walk among Bernoulli traps

Abstract

We study a $d$-dimensional branching random walk (BRW) in an i.i.d. random environment on $\mathbb{Z}^d$ in discrete time. A Bernoulli trap field is attached to $\mathbb{Z}^d$, where each site, independently of the others, is a trap with a fixed probability. The interaction between the BRW and the trap field is given by the hard killing rule. Given a realization of the environment, over each time step, each particle first moves according to a simple symmetric random walk to a nearest neighbor, and immediately afterwards, splits into two particles if the new site is not a trap or is killed instantly if the new site is a trap. Conditional on the ultimate survival of the BRW, we prove a strong law of large numbers for the total mass of the process. Our result is quenched, that is, it holds in almost every environment in which the starting point of the BRW is inside the infinite connected component of trap-free sites.