Site percolation thresholds for Archimedean lattices.

Precise thresholds for site percolation on eight Archimedean lattices are determined by the hull-walk gradient-percolation simulation method, with the results p(c)=0.697 043, honeycomb or (6(3)), 0.807 904 (3,12(2)), 0.747 806 (4,6,12), 0.729 724 (4,8(2)), 0.579 498 (3(4),6), 0.621 819 (3,4,6,4), 0.550 213 (3(3),4(2)), and 0.550 806 (3(2),4,3,4), with errors of about +/- 3 x 10(-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(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.