Advances in Applied Probability

Random minimal directed spanning trees and Dickman-type distributions

Mathew D. Penrose and Andrew R. Wade

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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.

Article information

Adv. in Appl. Probab. Volume 36, Number 3 (2004), 691-714.

First available: 31 August 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G70: Extreme value theory; extremal processes
Secondary: 05C80: Random graphs [See also 60B20] 60F05: Central limit and other weak theorems

Spanning tree extreme value weak convergence Dickman distribution Poisson-Dirichlet distribution


Penrose, Mathew D.; Wade, Andrew R. Random minimal directed spanning trees and Dickman-type distributions. Advances in Applied Probability 36 (2004), no. 3, 691--714. doi:10.1239/aap/1093962229.

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