Continuum Percolation and Euclidean Minimal Spanning Trees in High Dimensions
By Mathew D. Penrose.
We prove that for continuum percolation in $d$ dimensions,
parametrised by the mean number $y$ of points connected to the origin,
as $d$ goes to infinity with $y$ fixed
the distribution of the number of points in the cluster at
the origin converges to that of the total number of progeny
of a branching process with a Poisson($y$) offspring distribution.
We also prove that for sufficiently large $d$,
the critical points for the existence of infinite occupied
and vacant regions are distinct.
Our results resolve conjectures made by Avram and
Bertsimas in connection with their formula for
the growth rate of the length of the Euclidean minimal
spanning tree on $n$ independent uniformly distributed
points in $d$ dimensions as $n$ becomes large.
Annals of Applied Probability 6, 528-544 (1996).