Empirical formula for site and bond percolation thresholds on Archimedean and 2-uniform lattices

Abstract

The problem of percolation on Archimedean and 2-uniform lattices is investigated. An empirical formula is postulated for the ratio between the site percolation threshold pcs, and the bond percolation threshold pcb. The expression is pcs∕pcb≈az̄s3∕z̄s−12, where a=0.16428 and z̄s is the mean number of nearest neighbors of a given lattice site. For each of the 11 Archimedean and 20 2-uniform lattices which constitute our sample, a good correspondence is obtained with previous calculations of pcs∕pcb. The formula is found to be also valid for standard percolation in three-dimensional lattices and for percolation of tortuous k-mers [objects occupying k adjacent lattice elements (sites or bonds)] on square lattices.