Exact Site Percolation Thresholds Using the Site-to-Bond and Star-Triangle Transformations

TL;DR

This paper derives exact site percolation thresholds for specific two-dimensional lattices, including the martini lattice, using star-triangle transformations and correlated bond problems, advancing understanding of percolation thresholds.

Contribution

It introduces a method to exactly compute inhomogeneous site percolation thresholds on complex lattices using transformations and correlation techniques.

Findings

Site threshold of martini lattice: 0.764826...

Derived thresholds for related lattices: 0.618034... and 1/√2

Suggested a bound for the hexagonal site threshold, p_c < 1/√2

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 $0.764826...$, while the others have $0.618034...$ 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, solve the site problem on the martini lattice.