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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


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.


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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.


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

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

Primary: 60D05
Secondary: 60K35 , 90C27

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

Rights: Copyright © 1996 Institute of Mathematical Statistics


Vol.6 • No. 2 • May 1996
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