Site percolation thresholds and universal formulas for the Archimedean lattices

The site percolation thresholds pc are determined to high precision for eight Archimedean lattices, by the hull-walk gradient-percolation simulation technique, with the results pc = 0.697 043, honeycomb or (6 ), 0.807 904 (3, 12), 0.747 806 (4, 6, 12), 0.729 724 (4, 8), 0.579 498 (3, 6), 0.621 819 (3, 4, 6, 4), 0.550 213 (3, 4), and 0.550 806 (3, 4, 3, 4), and errors of about ±2 × 10. (The remaining Archimedean lattices are the square (4), triangular (3) and Kagomé (3, 6, 3, 6), for which pc is already known exactly or to a high degree of accuracy.) The numerical result for the (3, 12) lattice is consistent with the exact value [1 − 2 sin(π/18)]1/2, which we also derive. The values of pc for all Archimedean lattices are found to be linearly related to the density of sites within an error of about 1%, which predicts pc more accurately than correlations based just upon the coordination number can give. Comparison with anisotropic lattices is also made. PACS numbers(s): 64.60Ak, 05.70.Jk Typeset using REVTEX 1