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Bond percolation on simple cubic lattices with extended neighborhoods

Zhipeng Xun,
Published: 1 July 2020

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 pc∼z−a with exponent a=1.111. However, for large z, the threshold must approach the Bethe lattice result pc=1/(z−1). Fitting our data and data for additional nearest neighbors, we find pc(z−1)=1+1.224z−1/2.
Keywords: lattice / percolation / bond / thresholds / neighbors / nearest / simple cubic
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