  We study, in a general graph-theoretic formulation, a long-range percolation
model introduced by Lamperti. For various underlying directed graphs, we
discuss connections between this model and random exchange processes. We
clarify, for $n \in \mathbb{N}$, under which conditions the lattices
$\mathbb{N}_0^n$ and $\mathbb{Z}^n$ are essentially covered in this model.
Moreover, for all $n \geq 2$, we establish that it is impossible to cover the
directed $n$-ary tree in our model.
