Critical surfaces for general inhomogeneous bond percolation problems

TL;DR

This paper introduces a systematic polynomial-based method to estimate bond percolation thresholds on various lattices, accurately predicting known solutions and closely approximating numerical results for complex cases.

Contribution

The authors develop a general polynomial association method that predicts percolation thresholds, matching exact solutions and providing highly accurate approximations for inhomogeneous lattices.

Findings

Predicted percolation thresholds for several Archimedean lattices.

Method achieves accuracy within 10^{-5} of numerical estimates.

Errors less than 10^{-6} for checkerboard and bowtie lattices.

Abstract

We present a method of general applicability for finding exact or accurate approximations to bond percolation thresholds for a wide class of lattices. To every lattice we sytematically associate a polynomial, the root of which in $[0,1]$ is the conjectured critical point. The method makes the correct prediction for every exactly solved problem, and comparison with numerical results shows that it is very close, but not exact, for many others. We focus primarily on the Archimedean lattices, in which all vertices are equivalent, but this restriction is not crucial. Some results we find are kagome: $p_c=0.524430...$, $(3,12^2): p_c=0.740423...$, $(3^3,4^2): p_c=0.419615...$, $(3,4,6,4):p_c=0.524821...$, $(4,8^2):p_c=0.676835...$, $(3^2,4,3,4)$: $p_c=0.414120...$ . The results are generally within $10^{-5}$ of numerical estimates. For the inhomogeneous checkerboard and bowtie lattices, errors in the formulas (if they are not exact) are less than $10^{-6}$.