Random directed trees and forest - drainage networks with dependence

Athreya, Siva R ; Roy, Rahul ; Sarkar, Anish (2008) Random directed trees and forest - drainage networks with dependence Electronic Journal of Probability, 13 . pp. 2160-2189. ISSN 1083-6489

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Official URL: http://ejp.ejpecp.org/article/view/580

Related URL: http://dx.doi.org/10.1214/EJP.v13-580

Abstract

Consider the d-dimensional lattice Zd 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.

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Source:Copyright of this article belongs to Institute of Mathematical Statistics.
ID Code:99326
Deposited On:28 Mar 2016 10:28
Last Modified:28 Mar 2016 10:28

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