Universal formulas for percolation thresholds. II. Extension to anisotropic and aperiodic lattices

Abstract

In a recent paper, we reported a universal power law for both site and bond percolation thresholds in any Bravais lattice with $q$ equivalent nearest neighbors in dimension $d$. We now extend it to three different classes of lattices which are, respectively, anisotropic lattices without equivalent nearest neighbors, non-Bravais lattices with two atom unit cells, and quasicrystals. The investigation is focused on $d=2$ and 3, due to the lack of experimental data at higher dimensions. The extension to these lattices requires the substitution of $q$ by an effective (non integer) value ${q}_{\mathrm{eff}}$ in the universal law. For each of the 17 lattices which constitute our sample, we argue for the existence of one ${q}_{\mathrm{eff}}$ which reproduces both the site and the percolation threshold, with a deviation with respect to numerical estimates which does not exceed $\ensuremath{\mp}0.01$.