# Limit theorems for random spatial drainage networks

Penrose, M.D. and Wade, A.R.
(2010)
*Limit theorems for random spatial drainage networks.*
Advances in Applied Probability, 42 (3).
pp. 659-688.
ISSN 0001-8678

## Abstract

Suppose that, under the action of gravity, liquid drains through the unit d-cube via a minimal-length network of channels constrained to pass through random sites and to flow with nonnegative component in one of the canonical orthogonal basis directions of Rd, d ≥ 2. The resulting network is a version of the so-called minimal directed spanning tree. We give laws of large numbers and convergence in distribution results on the large-sample asymptotic behaviour of the total power-weighted edge length of the network on uniform random points in (0, 1)d. The distributional results exhibit a weight-dependent phase transition between Gaussian and boundary-effect-derived distributions. These boundary contributions are characterized in terms of limits of the so-called on-line nearest-neighbour graph, a natural model of spatial network evolution, for which we also present some new results. Also, we give a convergence in distribution result for the length of the longest edge in the drainage network; when d = 2, the limit is expressed in terms of Dickman-type variables.

Item type: | Article |
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ID code: | 27356 |

Keywords: | random spatial graph, spanning tree, weak convergence, phase transition, nearest-neighbour graph, Dickman distribution, distributional fixed-point equation, Mathematics, Applied Mathematics, Statistics and Probability |

Subjects: | Science > Mathematics |

Department: | Faculty of Science > Mathematics and Statistics |

Depositing user: | Mrs Carolynne Westwood |

Date Deposited: | 03 Sep 2010 14:12 |

Last modified: | 06 Jan 2017 08:22 |

Related URLs: | |

URI: | http://strathprints.strath.ac.uk/id/eprint/27356 |

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