@article{EJP580,
author = {Siva Athreya and Rahul Roy and Anish Sarkar},
title = {Random directed trees and forest - drainage networks with dependence},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {13},
year = {2008},
keywords = {Random Graph, Random Oriented Trees, Random Walk},
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.},
pages = {no. 71, 2160-2189},
issn = {1083-6489},
doi = {10.1214/EJP.v13-580},
url = {http://ejp.ejpecp.org/article/view/580}}