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 $p_c=0.592\dots$. The fourth-nearest neighbors (5N) give $p_c=0.298\dots$. This equality is proved to be mathematically exact using symmetry argument. We then consider combinations of various kinds of neighborhoods with $(\mathrm{NN}+\mathrm{NNN})$, $(\mathrm{NN}+4\mathrm{N})$, $(\mathrm{NN}+\mathrm{NNN}+4\mathrm{N})$, and $(\mathrm{NN}+5\mathrm{N})$. The calculated associated thresholds are respectively $p_c=0.407\dots$, 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.

• Received 17 August 2004

DOI:https://doi.org/10.1103/PhysRevE.71.016125