We prove a central limit theorem for the length of the minimal
spanning tree of the set of sites of a Poisson process of intensity $\lambda$
in $[0, 1]^2$ as $\lambda \to \infty$. As observed previously by Ramey, the
main difficulty is the dependency between the contributions to this length from
different regions of $[0, 1]^2$; a percolation-theoretic result on circuits
surrounding a fixed site can be used to control this dependency. We prove such
a result via a continuum percolation version of the Russo-Seymour-Welsh theorem
for occupied crossings of a rectangle. This RSW theorem also yields a variety
of results for two-dimensional fixed-radius continuum percolation already well
known for lattice models, including a finite-box criterion for percolation and
absence of percolation at the critical point.
Publié le : 1996-05-14
Classification:
Central limit theorem,
occupied crossing,
continuum percolation,
minimal spanning tree,
60D05,
60K35,
90C27
@article{1034968140,
author = {Alexander, Kenneth S.},
title = {The RSW theorem for continuum percolation and the CLT for
Euclidean minimal spanning trees},
journal = {Ann. Appl. Probab.},
volume = {6},
number = {1},
year = {1996},
pages = { 466-494},
language = {en},
url = {http://dml.mathdoc.fr/item/1034968140}
}
Alexander, Kenneth S. The RSW theorem for continuum percolation and the CLT for
Euclidean minimal spanning trees. Ann. Appl. Probab., Tome 6 (1996) no. 1, pp. 466-494. http://gdmltest.u-ga.fr/item/1034968140/