## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 14, Number 3 (2004), 1242-1266.

### Random oriented trees: A model of drainage networks

Sreela Gangopadhyay, Rahul Roy, and Anish Sarkar

#### 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 *d*th 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*.

#### Article information

**Source**

Ann. Appl. Probab., Volume 14, Number 3 (2004), 1242-1266.

**Dates**

First available in Project Euclid: 13 July 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1089736284

**Digital Object Identifier**

doi:10.1214/105051604000000288

**Mathematical Reviews number (MathSciNet)**

MR2071422

**Zentralblatt MATH identifier**

1047.60098

**Subjects**

Primary: 05C80: Random graphs [See also 60B20] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Keywords**

Random graph martingale random walk central limit theorem

#### Citation

Gangopadhyay, Sreela; Roy, Rahul; Sarkar, Anish. Random oriented trees: A model of drainage networks. Ann. Appl. Probab. 14 (2004), no. 3, 1242--1266. doi:10.1214/105051604000000288. https://projecteuclid.org/euclid.aoap/1089736284