The expected length of a Euclidean minimum spanning tree and the expected 1-norms of chromatic persistence diagrams

A classic result on Euclidean minimum spanning trees (EMSTs) is the existence of an asymptotic constant, c, such that the expected length of the EMST of n points sampled uniformly at random in the unit square is c times the square root of n, in the limit when n goes to infinity. However, the value of c is not known. Prior to this work, the known bounds were 0.6008 < c < 0.7072, and we improve the lower bound to 0.6289 < c.

The motivation for this work is the stochastic analysis of chromatic persistence diagrams. In particular, we show that similar asymptotic constants exist for the expected 1-norms of all diagrams in the 6-pack of a randomly 2-colored set of points in the unit square, in which we study the inclusion of the disjoint union of the sublevel sets of the two monochromatic distance functions into the sublevel set of the bichromatic distance function.

This is joint work with Ondrej Draganov, Sophie Rosenmeier, and Morteza Saghafian.