Extremes for the Minimal Spanning Tree
on Normally Distributed Points
By Mathew D. Penrose.
Let n points be placed independently in d-dimensional
to the standard d-dimensional normal distribution. Let
Mn be the
longest edge-length of the minimal spanning tree on these points;
equivalently let Mn be the infimum of those r
such that the union
of balls of radius r/2 centred at the points is connected. We show that
the distribution of
(2 log n)1/2 Mn - bn converges
weakly to the Gumbel (double exponential) distribution,
where bn are
explicit constants which are asymptotic to
(d-1) log log n
as n becomes large.
We also show the same result holds if Mn is the longest
for the nearest neighbour graph on the points.
Advances in Applied Probability 30, 628-639 (1998).