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

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

Automated summary

The asymptotic behavior of the percolation threshold and its dependence on the coordination number is investigated for both site and bond percolation on four-dimensional lattices with compact extended neighborhoods. The study provides insights into the behavior of the percolation threshold for different lattice structures.