## The Annals of Probability

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

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

#### 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