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August 2004 Random oriented trees: A model of drainage networks
Sreela Gangopadhyay, Rahul Roy, Anish Sarkar
Ann. Appl. Probab. 14(3): 1242-1266 (August 2004). DOI: 10.1214/105051604000000288

Abstract

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.

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Sreela Gangopadhyay. Rahul Roy. Anish Sarkar. "Random oriented trees: A model of drainage networks." Ann. Appl. Probab. 14 (3) 1242 - 1266, August 2004. https://doi.org/10.1214/105051604000000288

Information

Published: August 2004
First available in Project Euclid: 13 July 2004

zbMATH: 1047.60098
MathSciNet: MR2071422
Digital Object Identifier: 10.1214/105051604000000288

Subjects:
Primary: 05C80 , 60K35

Keywords: central limit theorem , martingale , random graph , Random walk

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.14 • No. 3 • August 2004
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