Calculation of Percolation Thresholds in High Dimensions for Fcc, Bcc and Diamond Lattices

Percolation problems have a wide range of applicability, and have therefore attracted a fair bit of attention over many years. Nevertheless the percolation thresholds, which are among the basic quantities for percolation on lattices, have been calculated exactly for only a few two-dimensional lattices. For many other lattices these thresholds have been calculated numerically. These numerical values can then be analysed, to determine regular behaviour, or trends, as a function of the lattice coordination number, dimensionality, etc. A limited review of such efforts has been given elsewhere. Recently, Galam and Mauger proposed an invariant for site and bond percolation thresholds, pcs and pcb respectively. The proposed invariant reads