Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods

Abstract

The asymptotic behavior of the percolation threshold $p_c$ and its dependence upon coordination number $z$ is investigated for both site and bond percolation on four-dimensional lattices with compact extended neighborhoods. Simple hypercubic lattices with neighborhoods up to 9th nearest neighbors are studied to high precision by means of Monte-Carlo simulations based upon a single-cluster growth algorithm. For site percolation, an asymptotic analysis confirms the predicted behavior $z{p}_c\sim 16{\eta }_c=2.086$ for large $z$, and finite-size corrections are accounted for by forms $p_c\sim 16{\eta }_c/(z+b)$ and $p_c\sim 1-\exp (-16{\eta }_c/z)$ where ${\eta }_c\approx 0.1304$ is the continuum percolation threshold of four-dimensional hyperspheres. For bond percolation, the finite-$z$ correction is found to be consistent with the prediction of Frei and Perkins, $z{p}_c-1\sim a_1(\ln z)/z$, although the behavior $z{p}_c-1\sim a_1{z}^{-3/4}$ cannot be ruled out.

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Cited by in corpus (5)

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• Random site percolation thresholds on square lattice for complex neighborhoods containing sites up to the sixth coordination zone
• Extended-range percolation in five dimensions