Percolation thresholds on triangular lattice for neighbourhoods containing sites up-to the fifth coordination zone

TL;DR

This study calculates percolation thresholds for various neighborhoods on a triangular lattice, introducing a new scalar index to relate thresholds to neighborhood geometry, and finds a power-law relationship.

Contribution

It introduces a novel scalar index $\xi$ to relate percolation thresholds to neighborhood geometry on a triangular lattice, covering complex neighborhoods up to the fifth coordination zone.

Findings

Percolation thresholds depend on the neighborhood's geometric properties.

The new index $\xi$ effectively differentiates neighborhoods and predicts thresholds.

Thresholds follow a power law with respect to $\xi$, with an exponent around 0.71.

Abstract

We determine thresholds $p_c$ for random-site percolation on a triangular lattice for all available neighborhoods containing sites from the first to the fifth coordination zones, including their complex combinations. There are 31 distinct neighbourhoods. The dependence of the value of the percolation thresholds $p_c$ on the coordination number $z$ are tested against various theoretical predictions. The newly proposed single scalar index $\xi=\sum_i z_ir_i^2/i$ (depending on the coordination zone number $i$, the neighbourhood coordination number $z$ and the square-distance $r^2$ to sites in $i$-th coordination zone from the central site) allows to differentiate among various neighbourhoods and relate $p_c$ to $\xi$. The thresholds roughly follow a power law $p_c\propto\xi^{-\gamma}$ with $\gamma\approx 0.710(19)$.