Random site percolation on honeycomb lattices with complex neighborhoods

TL;DR

This paper estimates critical occupation probabilities for site percolation on honeycomb lattices with complex neighborhoods, revealing a universal inverse square root relation with a neighborhood index, and provides computational tools for analysis.

Contribution

It introduces a neighborhood index to avoid degeneracy in percolation thresholds and demonstrates a universal $p_c o 1/\sqrt{ ext{index}}$ relation for complex neighborhoods.

Findings

Percolation thresholds follow $p_c o 1/z$ for large compact neighborhoods.

Noncompact neighborhoods break the $p_c$ dependence on $z$ due to degeneracy.

A new index $\zeta$ predicts $p_c o 1/\sqrt{\zeta}$ for 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 their radius ranging from one to three and containing from three to 24 sites. For two-dimensional regular lattices with compact extended-range neighborhoods, in the limit of the large number $z$ of sites in the neighborhoods, the site percolation thresholds $p_c$ follow the dependency $p_c\propto 1/z$, as recently shown by Xun, Hao and Ziff [Physical Review E 105, 024105 (2022)]. On the contrary, noncompact neighborhoods (with holes) destroy this dependence due to the degeneracy of the percolation threshold (several values of $p_c$ corresponding to the same number $z$ of sites in the neighborhoods). An example of a single-value index $\zeta=\sum_i z_i r_i$ -- where $z_i$ and $r_i$ are the number of sites and radius of the $i$-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 $p_c\propto 1/\sqrt\zeta$. The functions boundaries() (written in C) for basic neighborhoods (for the unique coordination zone) for the Newman and Ziff algorithm [Physical Review E 64, 016706 (2001)] are also presented.