Random minimal directed spanning trees and Dickman-type distributions

Penrose, Mathew D.; Wade, Andrew R.

Abstract

In Bhatt and Roy's minimal directed spanning tree construction for n random points in the unit square, all edges must be in a south-westerly direction and there must be a directed path from each vertex to the root placed at the origin. We identify the limiting distributions (for large n) for the total length of rooted edges, and also for the maximal length of all edges in the tree. These limit distributions have been seen previously in analysis of the Poisson-Dirichlet distribution and elsewhere; they are expressed in terms of Dickman's function, and their properties are discussed in some detail.

Citation

Penrose, M. D., & Wade, A. R. (2004). Random minimal directed spanning trees and Dickman-type distributions. Advances in Applied Probability, 36(3), 691-714. https://doi.org/10.1239/aap/1093962229

• Journal Article Type: Article
• Publication Date: Jan 1, 2004
• 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: 36
• Issue: 3
• Pages: 691-714
• DOI: https://doi.org/10.1239/aap/1093962229
• Keywords: Spanning tree, Extreme value, Weak convergence, Dickman distribution, Poisson-Dirichlet distribution.
• Public URL: https://durham-repository.worktribe.com/output/1495724