Bond percolation thresholds on Archimedean lattices from critical polynomial roots

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 $p_{\mathrm{c}}=0.52440499916744820(1)$, whereas the best estimate using standard techniques is $p_{\mathrm{c}}=0.52440499(2)$. 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 $p_{\mathrm{c}}$ 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.

References in corpus (5)

• Exact bond percolation thresholds in two dimensions
• Estimation of Bond Percolation Thresholds on the Archimedean Lattices
• Critical surfaces for general bond percolation problems
• Rigorous confidence intervals for critical probabilities
• On the growth constant for square-lattice self-avoiding walks

Cited by in corpus (3)

• Exact percolation probabilities for a square lattice: Site percolation on a plane, cylinder, and torus
• Percolation thresholds of randomly rotating patchy particles on Archimedean lattices
• Site and bond percolation in linearly distorted triangular and square lattices