## Electronic Journal of Probability

### 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.

#### Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 71, 2160-2189.

Dates
Accepted: 1 December 2008
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819144

Digital Object Identifier
doi:10.1214/EJP.v13-580

Mathematical Reviews number (MathSciNet)
MR2461539

Zentralblatt MATH identifier
1185.05124

Rights

#### Citation

Athreya, Siva; Roy, Rahul; Sarkar, Anish. Random directed trees and forest - drainage networks with dependence. Electron. J. Probab. 13 (2008), paper no. 71, 2160--2189. doi:10.1214/EJP.v13-580. https://projecteuclid.org/euclid.ejp/1464819144

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