On the total length of the random minimal directed spanning tree

Penrose, Mathew D.; Wade, Andrew R.

Abstract

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.

Citation

Penrose, M. D., & Wade, A. R. (2006). On the total length of the random minimal directed spanning tree. Advances in Applied Probability, 38(2), 336-372. https://doi.org/10.1239/aap/1151337075

• Journal Article Type: Article
• Publication Date: Jan 1, 2006
• Deposit Date: Oct 4, 2012
• Publicly Available Date: Feb 13, 2013
• Journal: Advances in Applied Probability
• Print ISSN: 0001-8678
• Electronic ISSN: 1475-6064
• Publisher: Applied Probability Trust
• Peer Reviewed: Peer Reviewed
• Volume: 38
• Issue: 2
• Pages: 336-372
• DOI: https://doi.org/10.1239/aap/1151337075
• Keywords: Spanning tree, Nearest-neighbour graph, Weak convergence, Fixed-point equation, Phase transition, Fragmentation process.
• Public URL: https://durham-repository.worktribe.com/output/1473356