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. ‡ Present address: KITP, University of California, Santa Barbara. ar X iv :0 70 8. 32 50 v2 [ co nd -m at .s ta tm ec h] 8 O ct 2 00 7 Topological estimation of percolation thresholds 2 Consider a regular d-dimensional lattice where a fraction of sites is selected independently with probability p and deemed ’black’, with the complementary vertices said to be white. The aggregate of these spatial lattice elements forms a random pattern, which we may partition into clusters after specifying a neighborhood. This simple set-up constitutes the standard model of Bernoullian percolation theory, and is applied to problems as diverse as transport in disordered media, epidemics, and the quark confinement transition in the early universe [1, 2, 3]. The central result of this theory is the existence of a sharp threshold value 0 < pc < 1 in an infinite lattice of dimension d > 1: When p increases across pc, a single infinite cluster appears almost surely and grows in mass with increasing p beyond pc [4, 5, 6]. Research in percolation theory focussed predominantly on the universal critical phenomena showing up in the vicinity of the threshold. On the other hand, for practical application of percolation concepts, it is the specific and non-universal value of pc which is of primary importance. Exact values of pc are known only for special classes of 2-d lattices [7, 8, 9, 10, 11, 12]. In all other cases, values of pc are estimated numerically with computer simulations, which often are time consuming, in particular in three or higher dimensional lattices. Here, we investigate the signature of the percolation transition in the Euler Characteristic of the spatial pattern formed by the percolating clusters. Its mean value per site, χ(p), provides a topological descriptor, which for a lattice Λ turns out to be an exactly calculable finite polynomial in p. For 2d-lattices the polynomials χ(p) have one non-trivial zero 0 < p0(Λ) < 1. From the comparison with known threshold values, we find that p0(Λ) yields a tight upper bound for pc(Λ). In the case of 3d-lattices, each χ(p)-polynomial has two distinct nontrivial zeros which are again slightly larger than the thresholds values of two distinct percolation transitions of black and white clusters. For 2d-lattices, we explain this peculiar ordering of p0 and pc using the known scaling expression for the critical percolation clusters at pc. Moreover, this approach leads to a surprisingly simple relation which combines via χ(p) the specific lattice geometry with universal critical percolation features into an accurate parameter-free estimate of percolation thresholds of all 2d-lattices considered in this note. Our work also applies to bond percolation problems when they are reformulated as the equivalent site percolation problem on the covering lattice. 1. The Euler characteristic in percolation theory The percolation transition is a paradigm of a non-thermal phase transition, where the local merging of black clusters causes an abrupt change in the large scale-connectivity of black vertices. Since the Euler Characteristic (EC) is a prominent descriptor of global aspects of spatial patterns, we may expect it to be also a valuable tool in the study of the percolation transition. In this section, we introduce the EC descriptively and discuss its salient features; a more technical but elementary outline can be found in the supplementary notes. For the time being we consider planar lattices with cyclic boundary conditions. The basic object in site percolation are clusters of vertices, naturally defined by the connectivity of the host lattice: Two black vertices belong to the same cluster if they are joined by a path of black nearest-neighbors on the lattice. Moreover, each configuration of black clusters specifies in a natural way an aggregate of white clusters with a complementary neighborhood, which is in general distinct from the black one; Topological estimation of percolation thresholds 3