Site percolation thresholds and universal formulas for the Archimedean lattices
The site percolation thresholds p_c are determined to high precision for eight Archimedean lattices, by the hull-walk gradient-percolation simulation technique, with the results p_c = 0.697043, honeycomb or (6^3), 0.807904 (3,12^{2}), 0.747806 (4,6,12), 0.729724 (4,8^{2}), 0.579498 (3^{4},6), 0.621819 (3,4,6,4), 0.550213 (3^{3},4^{2}), and 0.550806 (3^{2},4,3,4), and errors of about +/- 2\times10^{-6}. (The remaining Archimedean lattices are the square (4^4), triangular (3^6) 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^{2}) lattice is consistent with the exact value [1-2 sin(π/18)]^{1/2}, which we also derive. The values of p_c for all Archimedean lattices are found to be linearly related to the density of sites within an error of about 1%, which predicts p_c more accurately than correlations based just upon the coordination number can give. Comparison with anisotropic lattices is also made.