### Abstract

Language | English |
---|---|

Pages | 336-372 |

Number of pages | 37 |

Journal | Advances in Applied Probability |

Volume | 38 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 2006 |

### Fingerprint

### Keywords

- spanning tree
- nearest-neighbour graph
- weak convergence
- fixed-point equation
- phase transition
- fragmentation process

### Cite this

*Advances in Applied Probability*,

*38*(2), 336-372. https://doi.org/10.1239/aap/1151337075

}

*Advances in Applied Probability*, vol. 38, no. 2, pp. 336-372. https://doi.org/10.1239/aap/1151337075

**On the total length of the random minimal directed spanning tree.** / Penrose, M.D.; Wade, Andrew.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the total length of the random minimal directed spanning tree

AU - Penrose, M.D.

AU - Wade, Andrew

PY - 2006/6

Y1 - 2006/6

N2 - In Bhatt and Roy's minimal directed spanning tree construction for a random, partially ordered set of points in the unit square, all edges must respect the `coordinatewise' partial order and there must be a directed path from each vertex to a minimal element. We study the asymptotic behaviour of the total length of this graph with power-weighted edges. The limiting distribution is given by the sum of a normal component away from the boundary plus a contribution introduced by the boundary effects, which can be characterized by a fixed-point equation, and is reminiscent of limits arising in the probabilistic analysis of certain algorithms. As the exponent of the power weighting increases, the distribution undergoes a phase transition from the normal contribution being dominant to the boundary effects being dominant. In the critical case in which the weight is simple Euclidean length, both effects contribute significantly to the limit law.

AB - In Bhatt and Roy's minimal directed spanning tree construction for a random, partially ordered set of points in the unit square, all edges must respect the `coordinatewise' partial order and there must be a directed path from each vertex to a minimal element. We study the asymptotic behaviour of the total length of this graph with power-weighted edges. The limiting distribution is given by the sum of a normal component away from the boundary plus a contribution introduced by the boundary effects, which can be characterized by a fixed-point equation, and is reminiscent of limits arising in the probabilistic analysis of certain algorithms. As the exponent of the power weighting increases, the distribution undergoes a phase transition from the normal contribution being dominant to the boundary effects being dominant. In the critical case in which the weight is simple Euclidean length, both effects contribute significantly to the limit law.

KW - spanning tree

KW - nearest-neighbour graph

KW - weak convergence

KW - fixed-point equation

KW - phase transition

KW - fragmentation process

UR - http://projecteuclid.org/euclid.aap/1151337075

UR - http://arxiv.org/abs/math/0409201

U2 - 10.1239/aap/1151337075

DO - 10.1239/aap/1151337075

M3 - Article

VL - 38

SP - 336

EP - 372

JO - Advances in Applied Probability

T2 - Advances in Applied Probability

JF - Advances in Applied Probability

SN - 0001-8678

IS - 2

ER -