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(4 NN ) =0.311 60 (12 ) ,p_C(4 NN +NN ) =0.150 40 (12 ) ,p_C(4 NN +2 NN ) =0.159 50 (12 ) ,p_C(4 NN +3 NN ) =0.204 90 (12 ) ,p_C(4 NN +2 NN +NN ) =0.114 40 (12 ) ,p_C(4 NN +3 NN +NN ) =0.119 20 (12 ) ,p_C(4 NN +3 NN +2 NN ) =0.113 30 (12 ) , and p_C(4 NN +3 NN +2 NN +NN ) =0.100 00 (12 ) , 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.