The random minimal spanning tree in high dimensions

Abstract

For the minimal spanning tree on n independent uniform points in the d-dimensional unit cube, the proportionate number of points of degree k is known to converge to a limit $\alpha_{k,d}$ as $n \to \infty$. We show that $\alpha_{k,d}$ converges to a limit $\alpha_k$ as $d \to \infty$ for each k. The limit $\alpha_k$ arose in earlier work by Aldous, as the asymptotic proportionate number of vertices of degree k in the minimum-weight spanning tree on k vertices, when the edge weights are taken to be independent, identically distributed random variables. We give a graphical alternative to Aldous's characterization of the $\alpha_k$.