Site percolation thresholds for Archimedean lattices

Percolation threshold is not a decreasing function of the average coordination number

It is commonly believed that the percolation critical probability is a monotonically decreasing function of the average coordination number for periodic lattice graphs in the same dimension. This paper provides counterexamples-a pair of planar lattices for which the bond percolation critical probabilities and average coordination numbers are in the same order, and a pair for which the site percolation critical probabilities and average coordination numbers are in the same order. These counterexamples confirm the existence of this counterintuitive phenomenon, which was observed in one case in numerical estimates by van der Marck.

Accuracy of universal formulas for percolation thresholds based on dimension and coordination number

Recent mathematical results regarding percolation thresholds are relevant to efforts to find universal formulas for the percolation threshold. This Brief Report uses exact solutions and recent rigorous bounds for site and bond percolation thresholds to demonstrate that any universal formula based on only the dimension and the coordination number must provide estimates differing substantially from the true threshold value for some lattices.