#
The Longest Edge of the Random Minimal Spanning Tree

##
By Mathew D. Penrose.

For $n$ points placed uniformly at random on the unit square,
suppose $M_n$ (respectively $M'_n$)
denotes the longest edge-length of the nearest
neighbour graph (respectively the minimal spanning tree)
on these points.
It is known that the distribution of $ n \pi M_n^2 - \log n $ converges
weakly to the double exponential; we give a new proof of this.
We show that $M'_n = M_n$ with probability approaching 1 as $n$ becomes large,
so that the same weak convergence holds for $M'_n$.
Annals of Applied Probability
7, 340-361 (1997).