Bond percolation on simple cubic lattices with extended neighborhoods

Zhipeng Xun, Robert M. Ziff

Published 2020-01-02Version 1

We study bond percolation on the simple cubic (SC) lattice with combinations of nearest-neighbors (NN), second nearest-neighbors (2NN), third nearest-neighbors (3NN), and fourth nearest-neighbors (4NN) by Monte Carlo simulation. Using a single-cluster growth algorithm, we find precise values of the bond thresholds, which, to our knowledge, were not determined previously. The calculated thresholds are $0.1068263(7)$, $0.1012133(7)$, $0.0920213(7)$, $0.0752326(6)$, $0.0751589(9)$, $0.0629283(7)$, $0.0624379(9)$, $0.0533056(6)$, $0.0497080(10)$, $0.0474609(9)$, and $0.0392312(8)$ for (NN+4NN), (3NN+4NN), (NN+3NN), (NN+2NN), (2NN+4NN), (2NN+3NN), (NN+3NN+4NN), (NN+2NN+4NN), (NN+2NN+3NN), (2NN+3NN+4NN), and (NN+2NN+3NN+4NN) neighborhoods, respectively, where numbers in parentheses around the values of $p_c$ represent errors in the last one or two digits. Correlations between percolation thresholds and lattice properties are also 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 be bounded from below by the Bethe lattice result $p_c = 1/(z-1)$, implying that $a$ is in fact bounded from above by 1. For our extended lattices with large $z$, we find $p_c \approx 1.227/(z-1)$.