## 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)

### Keywords:

random graph; martingale; random walk; central limit theorem
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### References:

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