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Random minimal directed spanning trees and Dickman-type distributions

Part of: Stochastic processes Geometric probability and stochastic geometry Graph theory Limit theorems

Published online by Cambridge University Press:  01 July 2016

Mathew D. Penrose  and
Andrew R. Wade
Mathew D. Penrose*
Affiliation:
University of Bath
Andrew R. Wade*
Affiliation:
University of Durham
*
∗ Postal address: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. Email address: m.d.penrose@bath.ac.uk
∗∗ Postal address: Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE, UK. Email address: a.r.wade@durham.ac.uk
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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.

Keywords

Spanning treeextreme valueweak convergenceDickman distributionPoisson-Dirichlet distribution

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G70: Extreme value theory; extremal processes
Secondary: 05C80: Random graphs 60F05: Central limit and other weak theorems
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 36 , Issue 3 , September 2004 , pp. 691 - 714
Copyright
Copyright © Applied Probability Trust 2004 

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