Asymptotics for Euclidean minimal spanning trees on random points

Abstract

Asymptotic results for the Euclidean minimal spanning tree on n random vertices in Rd can be obtained from consideration of a limiting infinite forest whose vertices form a Poisson process in all Rd. In particular we prove a conjecture of Robert Bland: the sum of the d'th powers of the edge-lengths of the minimal spanning tree of a random sample of n points from the uniform distribution in the unit cube of Rd tends to a constant as n→∞. Whether the limit forest is in fact a single tree is a hard open problem, relating to continuum percolation. © 1992 Springer-Verlag.