Random site percolation on honeycomb lattices with complex neighborhoods

Abstract

We present a rough estimation—up to four significant digits, based on the scaling hypothesis and the probability of belonging to the largest cluster vs the occupation probability—of the critical occupation probabilities for the random site percolation problem on a honeycomb lattice with complex neighborhoods containing sites up to the fifth coordination zone. There are 31 such neighborhoods with a radius ranging from one to three and containing 3–24 sites. For two-dimensional regular lattices with compact extended-range neighborhoods, in the limit of the large number [Formula: see text] of sites in the neighborhoods, the site percolation thresholds [Formula: see text] follow the dependency [Formula: see text], as recently shown by Xun et al. [Phys. Rev. E 105, 024105 (2022)]. On the contrary, non-compact neighborhoods (with holes) destroy this dependence due to the degeneracy of the percolation threshold (several values of [Formula: see text] corresponding to the same number [Formula: see text] of sites in the neighborhoods). An example of a single-value index [Formula: see text]—where [Formula: see text] and [Formula: see text] are the number of sites and radius of the [Formula: see text]th coordination zone, respectively—characterizing the neighborhood and allowing avoiding the above-mentioned degeneracy is presented. The percolation threshold obtained follows the inverse square root dependence [Formula: see text]. The functions boundaries() (written in C) for basic neighborhoods (for the unique coordination zone) for the Newman and Ziff algorithm [Phys. Rev. E 64, 016706 (2001)] are also presented. The latter may be useful for computer physicists dealing with solid-state physics and interdisciplinary statistical physics applications, where the honeycomb lattice is the underlying network topology.