Percolation and variable-range hopping in random systems with spherical site symmetry

We show that the numerical work of Seager and Pike^1,2 suggests that the critical volume fraction (CVF) is a constant for sites of spherical symmetry in n dimensions, with $\text{CVF}\text{?nπ}{}^{\mathrm{1?n}}$ for small n. The average number of bonds per site, $\text{B}\text{?}{}_{\mathrm{c}}$, is calculated for a random distribution of site radii, and shown to agree with the Monte Carlo calculation. Analysis of a model having spherical site symmetry in (position-energy) space yields percolation constants C₂ = 2.1, C₃ = 2.6. This calculation indicates that there is an anomaly in some estimated values for the AHL percolation model. The physical significance of our model and its possible use in hard-core problems is discussed.