### Abstract

Original language | English |
---|---|

Pages (from-to) | 659-688 |

Number of pages | 29 |

Journal | Advances in Applied Probability |

Volume | 42 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2010 |

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### Keywords

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

### Cite this

*Advances in Applied Probability*,

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

}

*Advances in Applied Probability*, vol. 42, no. 3, pp. 659-688. https://doi.org/10.1239/aap/1282924058

**Limit theorems for random spatial drainage networks.** / Penrose, M.D.; Wade, A.R.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Limit theorems for random spatial drainage networks

AU - Penrose, M.D.

AU - Wade, A.R.

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

KW - random spatial graph

KW - spanning tree

KW - weak convergence

KW - phase transition

KW - nearest-neighbour graph

KW - Dickman distribution

KW - distributional fixed-point equation

UR - http://www.scopus.com/inward/record.url?scp=78650953931&partnerID=8YFLogxK

UR - http://projecteuclid.org/euclid.aap/1282924058

UR - http://arxiv.org/abs/0901.3297

U2 - 10.1239/aap/1282924058

DO - 10.1239/aap/1282924058

M3 - Article

VL - 42

SP - 659

EP - 688

JO - Advances in Applied Probability

JF - Advances in Applied Probability

SN - 0001-8678

IS - 3

ER -