Statistical mechanics of high-density bond percolation

High-density (HD) percolation describes the percolation of specific κ-clusters, which are the compact sets of sites each connected to κ nearest filled sites at least. It takes place in the classical patterns of independently distributed sites or bonds in which the ordinary percolation transition also exists. Hence, the study of series of κ-type HD percolations amounts to the description of classical clusters' structure for which κ-clusters constitute κ-cores nested one into another. Such data are needed for description of a number of physical, biological, and information properties of complex systems on random lattices, graphs, and networks. They range from magnetic properties of semiconductor alloys to anomalies in supercooled water and clustering in biological and social networks. Here we present the statistical mechanics approach to study HD bond percolation on an arbitrary graph. It is shown that the generating function for κ-clusters' size distribution can be obtained from the partition function of the specific q-state Potts-Ising model in the q→1 limit. Using this approach we find exact κ-clusters' size distributions for the Bethe lattice and Erdos-Renyi graph. The application of the method to Euclidean lattices is also discussed.