Random minimal directed spanning trees and Dickman-type distributions

Abstract

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Limit theorems for random spatial drainage networks

“…A similar result (in d = 2 only) for the longest edge in the 'south-west' MDST was given in [23]. Let L d max (X) denote the length of the longest edge in the MDST (under ' * ') on the point set…”

“…We state and prove all of our main results in the present paper for the Poisson process P n . In all cases, the authors believe that analogous results hold for the binomial point process consisting of n independent uniform random points on (0, 1) d instead; it should be possible to use standard de-Poissonization arguments (such as applied in similar circumstances in [23] and [24]) to verify this.…”

“…Under this assumption, ' * ' is a partial order on X (in fact, a total order), and so the MDST that we shall construct fits into the theory of the MDST on partially ordered sets given in [23] and [27]. Let card(X) denote the cardinality (number of elements) of the set X.…”

“…Examples considered previously are the 'coordinatewise' (or 'south-west') partial ordering on point sets in (0, 1) 2 [7], [23], [24] or in (0, 1) d [5], and the radial spanning tree [4] on point sets in R 2 . Also, laws of large numbers for the MDST on a class of partial orders of R 2 were given in [34].…”

On the total length of the random minimal directed spanning tree

“…Here we briefly recall the relevant terminology. See [16] for more detail. Suppose V is a finite set endowed with a partial ordering .…”