A Universal Formula for Percolation Thresholds II. Extension to Anisotropic and Aperiodic Lattices

TL;DR

This paper extends a universal percolation threshold formula to anisotropic, non-Bravais, and quasicrystal lattices in 2D and 3D by introducing an effective neighbor parameter, achieving high accuracy in predictions.

Contribution

It introduces a method to adapt the universal percolation law to complex lattices by using an effective neighbor count, broadening its applicability.

Findings

Effective q_{eff} reproduces thresholds within ±0.01

Universal law extended to anisotropic and quasicrystal lattices

High accuracy in 17 diverse lattice cases

Abstract

In a recent paper, we have reported a universal power law for both site and bond percolation thresholds in any Bravais lattice with q equivalent nearest neighbors in dimension d. We now extend it to three different classes of lattices which are, respectively, anisotropic lattices whith not equivalent nearest neighbors, non-Bravais lattices with two atom unit cells, and quasicrystals. The investigation is focussed on d=2 and d=3, due to the lack of experimental data at higher dimensions. The extension to these lattices requires the substitution of q by an effective (non integer) value $q_{eff}$ in the universal law. For each out of 17 lattices which constitute our sample, we argue the existence of one $q_{eff}$ which reproduces both the site and the percolation threshold, with a deviation with respect to numerical estimates which does not exceed $\mp 0.01$.