Predictions of bond percolation thresholds for the kagomé and Archimedean $(3,12^2)$ lattices

Abstract

Here we show how the recent exact determination of the bond percolation threshold for the martini lattice can be used to provide approximations to the unsolved kagomé and (3,12^2) lattices. We present two different methods, one of which provides an approximation to the inhomogeneous kagomé and (3,12^2) bond problems, and the other gives estimates of $p_c$ for the homogeneous kagomé (0.5244088...) and (3,12^2) (0.7404212...) problems that respectively agree with numerical results to five and six significant figures.