Correlated percolation: exact Bethe lattice analyses

Abstract

Certain classes of correlated site-percolation problems (or correlated spreading phenomena) on Bethe lattices are analysed exactly. The author's analysis of percolation of, e.g., occupied sites, requires that spreading of clusters of occupied sites is determined by a finite number of conditional probabilities. A condition specifying the percolation threshold is provided, as well as expressions for the percolation probability and average cluster size. Previous results for random and nearest-neighbour Ising-model distributions are recovered as special cases. Results are illustrated with examples for equilibrium and non-equilibrium distributions, the latter obtained via irreversible cooperative filling. The author also discuss 'two-phase percolation' for distributions with no occupied NN pairs of sites, correlated bond percolation and other problems.