Abstract. 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.

Key words: topology and combinatorics; classical phase transitions (theory); percolation problems (theory)

E-print number: 0708.3250
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Received 9 October 2007, accepted for publication 9 November 2007
Published 11 January 2008