Square lattice site percolation at increasing ranges of neighbor interactions

TL;DR

This paper investigates site percolation thresholds on square lattices with various neighbor interaction ranges using Monte Carlo simulations, revealing unexpected threshold patterns and challenging the universality of existing formulas.

Contribution

It provides new percolation threshold data for multiple neighbor interactions and demonstrates limitations of universal formulas for complex lattices.

Findings

Same threshold for N$^2$, N$^3$, N$^4$, N$^6$ interactions at 0.592

Different threshold for N$^5$ interaction at 0.298

Thresholds for combined interactions decrease with complexity

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.