Square lattice site percolation at increasing ranges of neighbor interactions

K. Malarz; S. Galam

doi:10.1103/physreve.71.016125

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arXiv · cond-mat/0408338

Square lattice site percolation at increasing ranges of neighbor interactions

K. Malarz·S. Galam

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Abstract

We report site percolation thresholds for square lattice with neighbor interactions at various increasing ranges. Using Monte Carlo techniques we found that nearest neighbors (N$^2$), next nearest neighbors (N$^3$), next next nearest neighbors (N$^4$) and fifth nearest neighbors (N$^6$) yield the same $p_c=0.592...$. At odds, fourth nearest neighbors (N$^5$) give $p_c=0.298...$. These results are given an explanation in terms of symmetry arguments. We then consider combinations of various ranges of interactions with (N$^2$+N$^3$), (N$^2$+N$^4$), (N$^2$+N$^3$+N$^4$) and (N$^2$+N$^5$). The calculated associated thresholds are respectively $p_c=0.407..., 0.337..., 0.288..., 0.234...$. 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.

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K. Malarz, S. Galam. 2004-08-15. Square lattice site percolation at increasing ranges of neighbor interactions. https://doi.org/10.1103/physreve.71.016125

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