On the mass of a branching random walk among hard Bernoulli traps

In this talk, we consider a model of branching random walk (BRW) in an i.i.d. random environment on $\mathbb{Z}^d$ in discrete time. Each site of $\mathbb{Z}^d$, independently of the others, is a trap with a fixed probability. 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. We call this random environment on $\mathbb{Z}^d$ along with its interaction with the BRW the model of hard Bernoulli traps. It is clear that the presence of traps tends to reduce the mass (population size) of the BRW compared to an ordinary BRW in a ‘free’ environment. We study the reduced mass of the BRW, with particular emphasis on a strong law of large numbers that is valid in almost every environment in which the starting point of the BRW is inside the infinite connected component of trap-free sites.