Boolean percolation on digraphs and random exchange processes

Abstract

We study, in a general graph-theoretic formulation, a long-range percolation model introduced by Lamperti in [27]. For various underlying directed graphs, we discuss connections between this model and random exchange processes. We clarify, for n ∈ N, under which conditions the lattices N n 0 and Z n are essentially covered in this model. Moreover, for all n ≥ 2, we establish that it is impossible to cover the directed n-ary tree in our model.