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Random oriented trees: a model of drainage networks. (English) Zbl 1047.60098

Summary: Consider the \(d\)-dimensional lattice \(\mathbb{Z}^d\) 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 \(d\)th coordinates 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 independent 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 \(d\geq 4\). 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 \(\nu\) and (b) the number of edges of a fixed length \(l\).

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
05C80 Random graphs (graph-theoretic aspects)
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