Site percolation thresholds for Archimedean lattices

Abstract

Precise thresholds for site percolation on eight Archimedean lattices are determined by the hull-walk gradient-percolation simulation method, with the results $p_c=0.697\mathrm{}043,$ honeycomb or ${(6\mathrm{}}^3),$ 0.807 904 ${(3,12\mathrm{}}^2),$ 0.747 806 (4,6,12), 0.729 724 ${(4,8\mathrm{}}^2),$ 0.579 498 ${(3\mathrm{}}^4,6),$ 0.621 819 (3,4,6,4), 0.550 213 ${(3\mathrm{}}^3{,4\mathrm{}}^2),$ and 0.550 806 ${(3\mathrm{}}^2,4,3,4),$ with errors of about $\pm 3\times {10}^{-6}.$ [The remaining Archimedean lattices are the square ${(4\mathrm{}}^4),$ triangular ${(3\mathrm{}}^6),$ and Kagomé (3,6,3,6), for which $p_c$ is already known exactly or to a high degree of accuracy.] The numerical result for the ${(3,12\mathrm{}}^2)$ lattice is consistent with the exact value $\left[1-2\sin \right(\pi /18){]}^{1/2}.$ The values of $p_c$ for all 11 Archimedean lattices, as well as a number of nonuniform lattices, are found to be well correlated by a nearly linear function of a generalized Scher-Zallen filling factor. This correlation is much more accurate than recently proposed correlations based solely upon coordination number.