Bond percolation on simple cubic lattices with extended neighborhoods

TL;DR

This study uses Monte Carlo simulations to precisely determine bond percolation thresholds on simple cubic lattices with extended neighborhoods, revealing a power-law relationship with coordination number and approaching Bethe lattice limits.

Contribution

It provides new precise threshold values for various cubic lattices with extended neighborhoods and establishes a power-law relation between thresholds and coordination number.

Findings

Percolation thresholds decrease monotonically with coordination number.

Thresholds follow a power law $p_c \\sim z^{-a}$ with $a=1.111$.

Thresholds approach Bethe lattice results for large $z$.

Abstract

We study bond percolation on the simple cubic (SC) 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 lattices with additional nearest neighbors, we find $p_c(z-1)=1+1.224 z^{-1/2}$.