### 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.

Original language | English |
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Pages (from-to) | 659-688 |

Number of pages | 29 |

Journal | Advances in Applied Probability |

Volume | 42 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2010 |

### Keywords

- random spatial graph
- spanning tree
- weak convergence
- phase transition
- nearest-neighbour graph
- Dickman distribution
- distributional fixed-point equation

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## Cite this

Penrose, M. D., & Wade, A. R. (2010). Limit theorems for random spatial drainage networks.

*Advances in Applied Probability*,*42*(3), 659-688. https://doi.org/10.1239/aap/1282924058