## 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

#### 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

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
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