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

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 $\gamma\approx 0.710(19)$.

K. Malarz · 2021