### The longest edge of the random minimal spanning tree

Mathew D. Penrose
Source: Ann. Appl. Probab. Volume 7, Number 2 (1997), 340-361.

#### Abstract

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 neighbor 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 $P[M'_n = M_n] \to 1$, so that the same weak convergence holds for $M'_n$ .

First Page:
Primary Subjects: 60D05, 60G70
Secondary Subjects: 05C05, 90C27
Full-text: Open access