Topological estimation of percolation thresholds

Global physical properties of random media change qualitatively at a percolation threshold, where isolated clusters merge to form one infinite connected component. The precise knowledge of percolation thresholds is thus of paramount importance. For two-dimensional lattice graphs, we use the universal scaling form of the cluster size distributions to derive a relation between the mean Euler characteristic of the critical percolation patterns and the threshold density pc. From this relation, we deduce a simple rule to estimate pc, which is remarkably accurate. We present some evidence that similar relations might hold for continuum percolation and percolation in higher dimensions.