Random directed trees and forest - drainage networks with dependence

Abstract

Consider the $d$-dimensional lattice $\mathbb Z^d$ 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 $45^\circ$ (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 $d \geq 4$. In addition, for any dimension, we show that there is no bi-infinite path in the tree.