Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors

In this paper, random-site percolation thresholds for a simple cubic (SC) lattice with site neighborhoods containing next-next-next-nearest neighbors (4NN) are evaluated with Monte Carlo simulations. A recently proposed algorithm with low sampling for percolation thresholds estimation (Bastas et al., arXiv:1411.5834) is implemented for the studies of the top-bottom wrapping probability. The obtained percolation thresholds are $p_C\left(4\mathrm{NN}\right)=0.31160\left(12\right),p_C\left(4\mathrm{NN}+\mathrm{NN}\right)=0.15040\left(12\right),p_C\left(4\mathrm{NN}+2\mathrm{NN}\right)=0.15950\left(12\right),p_C\left(4\mathrm{NN}+3\mathrm{NN}\right)=0.20490\left(12\right),p_C\left(4\mathrm{NN}+2\mathrm{NN}+\mathrm{NN}\right)=0.11440\left(12\right),p_C\left(4\mathrm{NN}+3\mathrm{NN}+\mathrm{NN}\right)=0.11920\left(12\right),p_C\left(4\mathrm{NN}+3\mathrm{NN}+2\mathrm{NN}\right)=0.11330\left(12\right)$, and $p_C\left(4\mathrm{NN}+3\mathrm{NN}+2\mathrm{NN}+\mathrm{NN}\right)=0.10000\left(12\right)$, where 3NN, 2NN, and NN stand for next-next-nearest neighbors, next-nearest neighbors, and nearest neighbors, respectively. As an SC lattice with 4NN neighbors may be mapped onto two independent interpenetrated SC lattices but with a lattice constant that is twice as large, the percolation threshold $p_C$(4NN) is exactly equal to $p_C$(NN). The simplified method of Bastas et al. allows for uncertainty of the percolation threshold value $p_C$ to be reached, similar to that obtained with the classical method but ten times faster.

Figure 1

Single sites from various neighborhoods of an SC lattice. The full neighborhoods contain $z=6$, 12, 8, and 6 sites for NN, 2NN, 3NN, and 4NN neighborhoods, respectively.

Figure 2

Wrapping probability $W(p;L)$ vs occupation probability $p$. The results are averaged over $N_{\text{run}}={10}^4$ runs. The symbols $(+,\times ,+\times ,\square )$ indicate the system linear sizes $(L=40$, 80, 120, 160), respectively.

Figure 3

Wrapping probability $W(p;L)$ and the pairwise sum $\lambda \left(p\right)$ vs occupation probability $p$. The results are averaged over $N_{\text{run}}={10}^4$ runs. The symbols $\left(+,\times ,+\times ,\square \right)$ indicate the system linear sizes $(L=40$, 80, 120, 160), respectively. The minima of $\lambda \left(p\right)$ correspond to the percolation thresholds $p_C$.