In this note we consider a branching random walk in random environment
on $\Z^d$ where particles perform independent simple random walks and branch,
according to a given offspring distribution, at a random subset of sites
whose density tends to zero at infinity. Given that initially one particle
starts at the origin, we identify the critical rate of decay of this density
separating transience from recurrence, i.e., the progeny hits the origin
with probability $<1$ resp.\ $=1$. We show that for $d \geq 3$ there
is a dichotomy in the critical rate of decay depending on whether 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 different 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 various co-authors investigating the problem of transience vs.\
recurrence for random motions in random media.

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