Random oriented trees: A model of drainage networks

Abstract

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 ≥ 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 v and (b) the number of edges of a fixed length l. © Institute of Mathematical Statistics, 2004.