Bond percolation thresholds on Archimedean lattices from critical polynomial roots

TL;DR

This paper introduces highly accurate numerical calculations of bond percolation thresholds on Archimedean lattices using an advanced eigenvalue method, providing reference values and insights into lattice classes and scaling exponents.

Contribution

It presents a new, highly precise numerical approach for calculating percolation thresholds and classifies lattices based on their finite-size scaling exponents.

Findings

Achieved highly accurate bond percolation thresholds for Archimedean lattices.

Identified two classes of lattices with distinct scaling exponents.

Provided evidence for the potential of the method to determine other critical points.

Abstract

We present percolation thresholds calculated numerically with the eigenvalue formulation of the method of critical polynomials; developed in the last few years, it has already proven to be orders of magnitude more accurate than traditional techniques. Here we report the result of large parallel calculations to produce what we believe may become the reference values of bond percolation thresholds on the Archimedean lattices for years to come. For example, for the kagome lattice we find $p_{\rm c}=0.524\,404\,999\,167\,448\,20 (1)$, whereas the best estimate using standard techniques is $p_{\rm c}=0.524\,404\,99(2)$. We further provide strong evidence that there are two classes of lattices: one for which the first three scaling exponents characterizing the finite-size corrections to $p_{\rm c}$ are $\Delta=6,7,8$, and another for which $\Delta=4,6,8$. We discuss the open questions related to the method, such as the full scaling law, as well as its potential for determining critical points of other models.