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Minimal directed spanning trees and Dickman-type distributions

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By Mathew D. Penrose and Andrew Wade

In Bhatt and Roy's minimal directed spanning tree construction for *n*
random points in the unit square,
all edges must be in a southwesterly 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.