Random directed trees and forest - drainage networks with dependence

Abstract

Consider the d-dimensional lattice Z^d where each vertex is `open' or `closed' with probability p or 1−p respectively. An open vertex v is connected by an edge to the closest open vertex w in the 45∘ (downward) light cone generated at v. In case of non-uniqueness of such a vertex w, we choose any one of the closest vertices with equal probability and independently of the other random mechanisms. It is shown that this random graph is a tree almost surely for d = 2 and 3 and it is an infinite collection of distinct trees for d≥4. In addition, for any dimension, we show that there is no bi-infinite path in the tree.