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

TL;DR

This study estimates percolation thresholds for simple cubic lattices with extended neighborhoods using Monte Carlo simulations and a fast algorithm, providing precise values and insights into lattice mappings.

Contribution

It introduces a low-sampling Monte Carlo method to efficiently determine percolation thresholds for complex neighborhoods in cubic lattices.

Findings

Percolation thresholds for various neighborhoods are precisely estimated.

The method is ten times faster than classical approaches.

4NN neighborhood percolation threshold equals that of NN due to lattice mapping.

Abstract

In the paper random-site percolation thresholds for simple cubic lattice with sites' 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 $p_C(\text{4NN})=0.31160(12)$, $p_C(\text{4NN+NN})=0.15040(12)$, $p_C(\text{4NN+2NN})=0.15950(12)$, $p_C(\text{4NN+3NN})=0.20490(12)$, $p_C(\text{4NN+2NN+NN})=0.11440(12)$, $p_C(\text{4NN+3NN+NN})=0.11920(12)$, $p_C(\text{4NN+3NN+2NN})=0.11330(12)$, $p_C(\text{4NN+3NN+2NN+NN})=0.10000(12)$, where 3NN, 2NN, NN stands 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 two times larger lattice constant the percolation threshold $p_C$(4NN) is exactly equal to $p_C$(NN). The simplified Bastas et al. method allows for reaching uncertainty of the percolation threshold value $p_C$ similar to those obtained with classical method but ten times faster.