Universality of random-site percolation thresholds for two-dimensional complex noncompact 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 pc 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 pc∝(ζ/K)^-γ with γ=0.526(11), 0.5439(63), and 0.5932(47), for a honeycomb, square, and triangular lattice, respectively, and pc∝ζ^-γ with γ=0.5546(67) independently on the underlying lattice. The index ζ=∑ᵢzᵢrᵢ stands for an average coordination number weighted by distance, that is, depending on the coordination zone number i, the neighborhood coordination number zᵢ, and the distance rᵢ to sites in the ith 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.