Square lattice site percolation at increasing ranges of neighbor interactions

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We report site percolation thresholds for square lattice with neighbor interactions at various increasing ranges. Using Monte Carlo techniques we found that nearest neighbors (N²), next nearest neighbors (N³), next next nearest neighbors (N⁴) and fifth nearest neighbors (N⁶) yield the same pc=0.592.... At odds, fourth nearest neighbors (N⁵) give pc=0.298.... These results are given an explanation in terms of symmetry arguments. We then consider combinations of various ranges of interactions with (N²+N³), (N²+N⁴), (N²+N³+N⁴) and (N²+N⁵). The calculated associated thresholds are respectively pc=0.407..., 0.337..., 0.288..., 0.234.... The existing Galam--Mauger universal formula for percolation thresholds does not reproduce the data showing dimension and coordination number are not sufficient to build a universal law which extends to complex lattices.