A note on transience versus recurrence for a branching random walk in random environment

We consider a branching random walk in random environment on Z(d) where par ticles perform independent simple random walks and branch, according to a g iven offspring distribution, at a random subset of sites whose density tend s to zero at infinity. Given that initially one particle starts at the orig in, we identify the critical rate of decay of the density of the branching sites separating transience from recurrence, i.e., the progeny hits the ori gin with probability <1 resp. =1. We show that for d greater than or equal to 3 there is a dichotomy in the critical rate of decay, depending on wheth er the mean offspring at a branching site is above or below a certain value related to the return probability of the simple random walk. The dichotomy marks a transition from local to global behavior in the progeny that hits the origin. We also consider the situation where the branching sites occur in two or more types, with different offspring distributions, and show that the classification is more subtle due to a possible interplay between the types. This note is part of a series of papers by the second author and var ious co-authors investigating the problem of transience versus recurrence f or random motions in random media.