Extended-range percolation on lattices with cubic and non-compact neighborhoods in three dimensions

Abstract

Using MC simulations based on an efficient single-cluster growth algorithm, we investigate two types of three-dimensional extended-range percolation models, namely cubic and complex (non-compact) neighborhoods. We obtain high-precision percolation thresholds $p_c$ for approximately 100 distinct structures and further explore the relationship between these thresholds and the topology of the lattices, which includes the total coordination number $z$, the number of sites $z_i$ and these sites' distance to the central site $r_i$ of the given neighborhood in the $i$th coordination zone. Both bond and site percolation are analyzed for cubic neighborhoods on the SC lattice. For bond percolation, the asymptotic value of $zp_c$ tends to Bethe-lattice behavior ($zp_c\sim1$), and finite-$z$ correction is found to be consistent with $zp_{c}-1\sim a_{1}z^{-x}$, where $x=(d-1)/d$. For site percolation, the value of $zp_c$ varies non-monotonically, exhibiting an initial increase followed by a decrease with the coordination number $z$. Linear fitting leads to $zp_c\sim2.5987(1)$ for large $z$, predicting a value of continuum percolation threshold of aligned cubes $η_c=0.324838(12)$ using $zp_c=2^{d}η_c$. We then examine site percolation for non-compact neighborhoods extending to the fifth coordination zone on SC, FCC, and BCC lattices. Through a comparison of the three indices of $z$, $ξ=\sum_iz_ir_i^2/i$, and $ζ=\sum_iz_ir_i$, we find that $ζ$ is effective in removing the degeneracy. The results obtained show power-law dependence $p_c\propto(ζ/K)^{-γ}$ with $γ_{\mathrm{SC}}^ζ=0.703(12)$, $γ_{\mathrm{FCC}}^ζ=0.728(10)$, and $γ_{\mathrm{BCC}}^ζ=0.699(12)$ for SC, FCC, and BCC lattices, and $p_c\proptoζ^{-γ}$ with $γ^ζ=0.719(9)$ independent of the underlying lattice. The number $K$ indicates lattice connectivity.