Universality of random-site percolation thresholds for two-dimensional complex non-compact neighborhoods

The phenomenon of percolation is one of the core topics in statistical mechanics. It allows one to study the phase transition known in real physical systems only in a purely geometrical way. In this paper, we determine thresholds $p_c$ for random site percolation in triangular and honeycomb lattices for all available neighborhoods containing sites from the sixth coordination zone. The results obtained (together with the percolation thresholds gathered from the literature also for other complex neighborhoods and also for a square lattice) show the power-law dependence $p_c\propto (\zeta \mathrm{/}K{)}^{-\gamma }$ with $\gamma =0.526(11)$, $0.5439(63)$ and $0.5932(47)$, for honeycomb, square, and triangular lattice, respectively, and $p_c\propto {\zeta }^{-\gamma }$ with $\gamma =0.5546(67)$ independently on the underlying lattice. The index $\zeta ={\sum }_i{z}_i{r}_i$ stands for an average coordination number weighted by distance, that is, depending on the coordination zone number $i$, the neighborhood coordination number $z_i$ and the distance $r_i$ to sites in $i$-th coordination zone from the central site. The number $K$ indicates lattice connectivity, that is, $K=3$, 4 and 6 for the honeycomb, square and triangular lattice, respectively.