Random oriented trees: a model of drainage networks

Gangopadhyay, Sreela ; Roy, Rahul ; Sarkar, Anish (2004) Random oriented trees: a model of drainage networks The Annals of Applied Probability, 14 (3). pp. 1242-1266. ISSN 1050-5164

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Official URL: http://projecteuclid.org/euclid.aoap/1089736284

Related URL: http://dx.doi.org/10.1214/105051604000000288


Consider the d-dimensional lattice Ζd 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 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 d≥4. 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.

Item Type:Article
Source:Copyright of this article belongs to Institute of Mathematical Statistics.
Keywords:Random Graph; Martingale; Random Walk; Central Limit Theorem
ID Code:72338
Deposited On:23 Jun 2012 14:06
Last Modified:23 Jun 2012 14:06

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