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2008 Random directed trees and forest - drainage networks with dependence
Siva Athreya, Rahul Roy, Anish Sarkar
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Electron. J. Probab. 13: 2160-2189 (2008). DOI: 10.1214/EJP.v13-580

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.

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Siva Athreya. Rahul Roy. Anish Sarkar. "Random directed trees and forest - drainage networks with dependence." Electron. J. Probab. 13 2160 - 2189, 2008. https://doi.org/10.1214/EJP.v13-580

Information

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

zbMATH: 1185.05124
MathSciNet: MR2461539
Digital Object Identifier: 10.1214/EJP.v13-580

Subjects:
Primary: 0505C80
Secondary: 60K35

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