The Annals of Applied Probability

Random oriented trees: A model of drainage networks

Sreela Gangopadhyay, Rahul Roy, and Anish Sarkar

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Abstract

Consider the d-dimensional lattice ℤd where each vertex is “open” or “closed” with probability p or 1p, 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 d4. 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


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References

  • Aldous, D. and Steele, J. M. (1992). Asymptotics for Euclidean minimal spanning trees on random points. Probab. Theory Related Fields 92 247–258.
  • Asmussen, S. (1987). Applied Probability and Queues. Wiley, Chichester.
  • Billingsley, P. (1979). Probability and Measure. Wiley, New York.
  • Burton, R. M. and Keane, M. S. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121 501–505.
  • Chow, Y. S. and Teicher, H. (1978). Probability Theory: Independence, Interchangeability, Martingales. Springer, New York.
  • Ferrari, P. A., Landim, C. and Thorisson, H. (2002). Poisson trees and coalescing random walks. Preprint.
  • Howard, A. D. (1971). Simulation of stream networks by headward growth and branching. Geogr. Anal. 3 29–50.
  • Leopold, L. B. and Langbein, W. B. (1962). The concept of entropy in landscape evolution. U.S. Geol. Surv. Prof. Paper 500-A.
  • Newman, C. M. and Stein, D. L. (1996). Ground-state structure in a highly disordered spin-glass model. J. Statist. Phys. 82 1113–1132.
  • Pemantle, R. (1991). Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19 1559–1574.
  • Rodriguez-Iturbe, I. and Rinaldo, A. (1997). Fractal River Basins: Chance and Self-Organization. Cambridge Univ. Press.
  • Scheidegger, A. E. (1967). A stochastic model for drainage pattern into an intramontane trench. Bull. Ass. Sci. Hydrol. 12 15–20.
  • Spitzer, F. (1964). Principles of Random Walk. Van Nostrand, Princeton, NJ.