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.