Boolean percolation on digraphs and random exchange processes

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.
Comment: 14 pages