Bond percolation on simple cubic lattices with extended neighborhoods

Abstract

We study bond percolation on the simple cubic lattice with various combinations of first, second, third, and fourth nearest neighbors by Monte Carlo simulation. Using a single-cluster growth algorithm, we find precise values of the bond thresholds. Correlations between percolation thresholds and lattice properties are discussed, and our results show that the percolation thresholds of these and other three-dimensional lattices decrease monotonically with the coordination number $z$ quite accurately according to a power-law $p_c\sim z^{-a}$ with exponent $a=1.111$. However, for large $z$, the threshold must approach the Bethe lattice result $p_c=1/(z-1)$. Fitting our data and data for additional nearest neighbors, we find $p_c(z-1)=1+1.224z^{-1/2}$.