Statistical mechanics of high-density bond percolation

High-density (HD) percolation describes the percolation over specific $\kappa$ -clusters, which are the compact sets of sites each connected to $\kappa$ 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 exsists. Hence, the study of series of $\kappa$ -type percolations amounts to the description of structure of classical clusters for which $\kappa$ -clusters constitute $\kappa$ -cores nested one into another. Such data are needed for description of a number of physical, biological information and other 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 arbitrary graph. It is shown that generating function for $\kappa$ -clusters' size distribution can be obtained from partition function of specific $q$-state Potts-Ising model in $q \to 1$ limit. Using this approach we find exact $\kappa$ -clusters' size distribution for Bethe lattice and Erdos Renyi graph. The application of the method to Euclidean lattices is also discussed.