Predictions of bond percolation thresholds for the kagom\'e and Archimedean $(3,12^2)$ lattices

TL;DR

This paper introduces two methods to approximate bond percolation thresholds for the kagomé and (3,12^2) lattices, achieving high accuracy and leveraging recent exact results for related lattices.

Contribution

The paper presents novel approximation methods for percolation thresholds on complex lattices using recent exact solutions, improving accuracy over previous estimates.

Findings

Approximate thresholds for kagomé and (3,12^2) lattices match numerical results to five and six significant figures.

Two different methods are proposed, one for inhomogeneous and one for homogeneous lattice problems.

The approach leverages exact solutions from the martini lattice to estimate thresholds for other 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 kagom\'e and (3,12^2) lattices. We present two different methods, one of which provides an approximation to the inhomogeneous kagom\'e and (3,12^2) bond problems, and the other gives estimates of $p_c$ for the homogeneous kagom\'e (0.5244088...) and (3,12^2) (0.7404212...) problems that respectively agree with numerical results to five and six significant figures.