Boolean percolation on digraphs and random exchange processes

Abstract

Abstract We study in a general graph-theoretic formulation a long-range percolation model introduced by Lamperti [27]. For various underlying digraphs, we discuss connections between this model and random exchange processes. We clarify, for all $n \in \mathbb{N}$ , 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.