The Annals of Applied Probability

Random oriented trees: A model of drainage networks

Sreela Gangopadhyay, Rahul Roy, and Anish Sarkar

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Consider the d-dimensional lattice ℤd where each vertex is “open” or “closed” with probability p or 1p, respectively. An open vertex v is connected by an edge to the closest open vertex w such that the dth co-ordinates of v and w satisfy w(d)=v(d)1. In case of nonuniqueness 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 d4. In addition, for any dimension, we show that there is no bi-infinite path in the tree and we also obtain central limit theorems of (a) the number of vertices of a fixed degree ν and (b) the number of edges of a fixed length l.

Article information

Ann. Appl. Probab., Volume 14, Number 3 (2004), 1242-1266.

First available in Project Euclid: 13 July 2004

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Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Random graph martingale random walk central limit theorem


Gangopadhyay, Sreela; Roy, Rahul; Sarkar, Anish. Random oriented trees: A model of drainage networks. Ann. Appl. Probab. 14 (2004), no. 3, 1242--1266. doi:10.1214/105051604000000288.

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