Exact site percolation thresholds using a site-to-bond transformation and the star-triangle transformation

Abstract

I construct a two-dimensional lattice on which the inhomogeneous site percolation threshold is exactly calculable and use this result to find two more lattices on which the site thresholds can be determined. The primary lattice studied here, the “martini lattice,” is a hexagonal lattice with every second site transformed into a triangle. The site threshold of this lattice is found to be 0.764826…, i.e., the solution to $p^4-3p^3+1=0$, while the others have $(\sqrt{5}-1)∕2$ (the inverse of the golden ratio) and $1∕\sqrt{2}$. This last solution suggests a possible approach to establishing the bound for the hexagonal site threshold, $p_c<1∕\sqrt{2}$. To derive these results, I solve a correlated bond problem on the hexagonal lattice by use of the star-triangle transformation and then, by a particular choice of correlations derived from a site-to-bond transformation, solve the site problem on the martini lattice.