The Annals of Applied Probability

The longest edge of the random minimal spanning tree

Mathew D. Penrose

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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

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

First available in Project Euclid: 14 October 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G70: Extreme value theory; extremal processes
Secondary: 05C05: Trees 90C27: Combinatorial optimization

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


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.

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