# 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).