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).