## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 7, Number 2 (1997), 340-361.

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

#### 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 $M'_n$ .

#### Article information

**Source**

Ann. Appl. Probab., Volume 7, Number 2 (1997), 340-361.

**Dates**

First available in Project Euclid: 14 October 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1034625335

**Digital Object Identifier**

doi:10.1214/aoap/1034625335

**Mathematical Reviews number (MathSciNet)**

MR1442317

**Zentralblatt MATH identifier**

0884.60042

**Subjects**

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G70: Extreme value theory; extremal processes

Secondary: 05C05: Trees 90C27: Combinatorial optimization

**Keywords**

Geometric probability minimal spanning tree nearest neighbor graph extreme values Poisson process Chen-Stein method continuum percolation

#### Citation

Penrose, Mathew D. The longest edge of the random minimal spanning tree. Ann. Appl. Probab. 7 (1997), no. 2, 340--361. doi:10.1214/aoap/1034625335. https://projecteuclid.org/euclid.aoap/1034625335