We prove that for continuum percolation in $\mathbb{R}^d$,
parametrized by the mean number y of points connected to the origin, as
$d \to \infty$ 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 $n \to \infty$.
Publié le : 1996-05-14
Classification:
Geometric probability,
continuum percolation,
phase transitions,
minimal spanning tree constant,
high dimensions,
Poisson process,
branching process,
60K35,
60D05,
60J80,
82B43
@article{1034968142,
author = {Penrose, Mathew D.},
title = {Continuum percolation and Euclidean minimal spanning trees in high
dimensions},
journal = {Ann. Appl. Probab.},
volume = {6},
number = {1},
year = {1996},
pages = { 528-544},
language = {en},
url = {http://dml.mathdoc.fr/item/1034968142}
}
Penrose, Mathew D. Continuum percolation and Euclidean minimal spanning trees in high
dimensions. Ann. Appl. Probab., Tome 6 (1996) no. 1, pp. 528-544. http://gdmltest.u-ga.fr/item/1034968142/