Random oriented Trees: a Model of drainage networks

Details Random oriented Trees: a Model of drainage networks Random oriented Trees: a Model of drainage networks

2004-07-08

arxiv.org/abs/math/0407118v1

Annals of Probability 2004, Vol. 14, No. 3, 1242-1266

Mathematics

Probability

Scientific paper

10.1214/105051604000000288

Consider the d-dimensional lattice 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 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\geq4. 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 \nu and (b) the number of edges of a fixed length l.

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