Percolation in Continuous Systems

Abstract

A rigorous proof of the existence of a percolation phase transition in a system of noninteracting discs in the plane is presented. In addition, bounds on the critical density and critical area fraction are derived. The lower bound makes use of Halperin's idea of a self-avoiding walk of discs. The upper bound is proved by relating the continuum model to the site percolation problem on a triangular lattice, whose critical probability is exactly known.