  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
$p_c\propto(\zeta/K)^{-\gamma}$ 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.
