Square-lattice site percolation at increasing ranges of neighbor bonds

We report site percolation thresholds for square lattice with neighbor bonds at various increasing ranges. Using Monte Carlo techniques we found that nearest neighbors (NN), next-nearest neighbors (NNN), next-next-nearest neighbors (4N), and fifth-nearest neighbors (6N) yield the same p_c =0.592… . The fourth-nearest neighbors (5N) give p_c =0.298… . This equality is proved to be mathematically exact using symmetry argument. We then consider combinations of various kinds of neighborhoods with (NN+NNN) , (NN+4N) , (NN+NNN+4N) , and (NN+5N) . The calculated associated thresholds are respectively p_c =0.407… , 0.337…, 0.288…, and 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.