On the total length of the random minimal directed spanning tree

Abstract

In Bhatt and Roy's minimal directed spanning tree (MDST) construction for a random partially ordered set of points in the unit square, all edges must respect the "coordinatewise" partial order and there must be a directed path from each vertex to a minimal element. We study the asymptotic behaviour of the total length of this graph with power weighted edges. The limiting distribution is given by the sum of a normal component away from the boundary and a contribution introduced by the boundary effects, which can be characterized by a fixed point equation, and is reminiscent of limits arising in the probabilistic analysis of certain algorithms. As the exponent of the power weighting increases, the distribution undergoes a phase transition from the normal contribution being dominant to the boundary effects dominating. In the critical case where the weight is simple Euclidean length, both effects contribute significantly to the limit law. We also give a law of large numbers for the total weight of the graph.

Limit theorems for random spatial drainage networks

“…[17]). This relationship was first seen in our previous work [24], [34] on limit theorems for the length of the 'south-west' MDST in the unit square. The present work adds to this by considering the 'south' MDST, for which the fixed-point distributions that arise are different.…”

“…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].…”

“…In [24] and [34] only the case d = 2 of the 'south-west'MDST was studied. In the present paper, for the 'south' MDST, we deal not only with d = 2 but also with higher dimensions.…”

“…In the present paper, for the 'south' MDST, we deal not only with d = 2 but also with higher dimensions. With fairly straightforward modifications, the method used in [24] could be adapted to prove the d = 2 case of our Theorem 2.2 below. However, at several points the proofs used in [24] are not easily adapted to higher dimensions, and, thus, we have adopted different proofs; sometimes these improve or extend ideas from [24] and sometimes we use entirely different techniques.…”

“…With fairly straightforward modifications, the method used in [24] could be adapted to prove the d = 2 case of our Theorem 2.2 below. However, at several points the proofs used in [24] are not easily adapted to higher dimensions, and, thus, we have adopted different proofs; sometimes these improve or extend ideas from [24] and sometimes we use entirely different techniques. Another difference is that in [24] and [34] we made use of general results of Penrose and Yukich [28], [29], while in the present paper we instead use the results of Penrose [22], [21] (see also [19]) which are in several ways more convenient for the current application.…”