The Random Minimal Spanning Tree in High Dimensions

By Mathew D. Penrose.

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 $ goes to infinity. We show that $\alpha(k,d)$ converges to a limit $\alpha_k$ as $d$ goes to infinity, 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, indentically distributed random variables. We give a graphical alternative to Aldous's characterisation of the $\alpha_k$.

Annals of Probability 24, , 1903-1925 (1996).