The Annals of Probability

The Russo-Seymour-Welsh Theorem and the Equality of Critical Densities and the "Dual" Critical Densities for Continuum Percolation on $|mathbb{R}^2$

Rahul Roy

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Abstract

A Russo-Seymour-Welsh (RSW) theorem is established for continuum percolation on $\mathbb{R}^2$. The equality of various definitions of critical densities for the continuum percolation on $\mathbb{R}^2$ is deduced as an application of the RSW theorem. It is also shown that various notions of the size of a cluster yield the same notion of critical density.

Article information

Source
Ann. Probab., Volume 18, Number 4 (1990), 1563-1575.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990632

Digital Object Identifier
doi:10.1214/aop/1176990632

Mathematical Reviews number (MathSciNet)
MR1071809

Zentralblatt MATH identifier
0719.60119

JSTOR
links.jstor.org

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82A43 82A68 60K10: Applications (reliability, demand theory, etc.)

Keywords
Poisson process continuum percolation critical densities FKG inequality RSW theorem

Citation

Roy, Rahul. The Russo-Seymour-Welsh Theorem and the Equality of Critical Densities and the "Dual" Critical Densities for Continuum Percolation on $|mathbb{R}^2$. Ann. Probab. 18 (1990), no. 4, 1563--1575. doi:10.1214/aop/1176990632. https://projecteuclid.org/euclid.aop/1176990632


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