Comparison of analytic and numerical results for the mean cluster density in continuum percolation

Abstract

Recently a number of techniques have been developed for bounding and approximating the important quantities in a description of continuum percolation models, such as 〈nc〉/ρ̄, the mean number of clusters per particle. These techniques include Kirkwood–Salsburg bounds, and approximations from cluster enumeration series of Mayer–Montroll type, and the scaled-particle theory of percolation. In this paper, we test all of these bounds and approximations numerically by conducting the first systematic simulations of 〈nc〉/ρ̄ for continuum percolation. The rigorous Kirkwood–Salsburg bounds are confirmed numerically in both two and three dimensions. Although this class of bounds seems not to converge rapidly for higher densities, averaging an upper bound with the corresponding lower bound gives an exceptionally good estimate at all densities. The scaled-particle theory of percolation is shown to give extremely good estimates for the density of clusters in both two and three dimensions at all densities below the percolation threshold. Also, partial sums of the virial series for 〈nc〉 are shown numerically to give extremely tight upper and lower bounds for this quantity. We argue that these partial sums may have similar bounding properties for a general class of percolation models.