The longest edge of the random minimal spanning tree
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 $M'_n$ .
Permanent link to this document: http://projecteuclid.org/euclid.aoap/1034625335
Mathematical Reviews number (MathSciNet): MR1442317
Digital Object Identifier: doi:10.1214/aoap/1034625335
Zentralblatt MATH identifier: 0884.60042