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 $p_c$. From this relation, we deduce a simple rule to estimate $p_c$, which is remarkably accurate. We present some evidence that similar relations might hold for continuum percolation and percolation in higher dimensions.