Open Access
Translator Disclaimer
May 1996 The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees
Kenneth S. Alexander
Ann. Appl. Probab. 6(2): 466-494 (May 1996). DOI: 10.1214/aoap/1034968140

Abstract

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.

Citation

Download Citation

Kenneth S. Alexander. "The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees." Ann. Appl. Probab. 6 (2) 466 - 494, May 1996. https://doi.org/10.1214/aoap/1034968140

Information

Published: May 1996
First available in Project Euclid: 18 October 2002

zbMATH: 0855.60009
MathSciNet: MR1398054
Digital Object Identifier: 10.1214/aoap/1034968140

Subjects:
Primary: 60D05
Secondary: 60K35 , 90C27

Keywords: central limit theorem , continuum percolation , Minimal spanning tree , occupied crossing

Rights: Copyright © 1996 Institute of Mathematical Statistics

JOURNAL ARTICLE
29 PAGES


SHARE
Vol.6 • No. 2 • May 1996
Back to Top