Series expansions in a continuum percolation problem

Abstract

Power series in number density are used to study the distribution of cluster sizes in a continuum analogue of bond percolation on a lattice. The clusters are formed by overlapping of geometrical regions that are randomly distributed in space. The regions are circles and oriented squares in two dimensions, and spheres and oriented cubes in three dimensions. The power series are based on a graphical expansion, using topological weights from percolation theory and probabilities obtained from cluster integrals. Analysis of the mean cluster size series provides estimates of the critical percolation density, and the associated critical exponent.