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

Abstract

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 zp c ∼ 16η c = 2.086 for large z, and finite-size corrections are accounted for by forms p c ∼ 16η c /(z + b) and p c ∼ 1 − exp(−16η c /z) where η c ≈ 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, zp c − 1 ∼ a 1(ln z)/z, although the behavior zp c − 1 ∼ a 1 z −3/4 cannot be ruled out.