Boolean percolation on digraphs and random exchange processes

TL;DR

This paper investigates a general graph-theoretic long-range percolation model, exploring its connections to random exchange processes and conditions for coverage in various directed graphs, including lattices and trees.

Contribution

It provides a unified framework for understanding Boolean percolation on directed graphs and clarifies coverage conditions for lattices and trees within this model.

Findings

Lattices $ abla_0^n$ and $ abla^n$ can be covered under certain conditions.

Coverage of the directed $n$-ary tree is impossible for all $n  2.

Connections between percolation models and random exchange processes are established.

Abstract

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.