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

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 $z{p}_c$ tends to Bethe-lattice behavior ($z{p}_c\sim 1$), and finite-$z$ correction is found to be consistent with $z{p}_c-1\sim a_1{z}^{-x}$, where $x=(d-1)\mathrm{/}d$. For site percolation, the value of $z{p}_c$ varies non-monotonically, exhibiting an initial increase followed by a decrease with the coordination number $z$. Linear fitting leads to $z{p}_c\sim 2.5987(1)$ for large $z$, predicting a value of continuum percolation threshold of aligned cubes ${\eta }_c=0.324838(12)$ using $z{p}_c=2^d{\eta }_c$. 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 $z$, $\xi ={\sum }_i{z}_i{r}_i^2\mathrm{/}i$, and $\zeta ={\sum }_i{z}_i{r}_i$, we find that $\zeta$ is effective in removing the degeneracy. The results obtained show power-law dependence $p_c\propto (\zeta \mathrm{/}K{)}^{-\gamma }$ with ${\gamma }_{\mathrm{S}\mathrm{C}}^{\zeta }=0.703(12)$, ${\gamma }_{\mathrm{F}\mathrm{C}\mathrm{C}}^{\zeta }=0.728(10)$, and ${\gamma }_{\mathrm{B}\mathrm{C}\mathrm{C}}^{\zeta }=0.699(12)$ for SC, FCC, and BCC lattices, and $p_c\propto {\zeta }^{-\gamma }$ with ${\gamma }^{\zeta }=0.719(9)$ independent of the underlying lattice. The number $K$ indicates lattice connectivity.