Wetting Transition on Trees I: Percolation With Clustering

A new “Percolation with Clustering” (PWC) model is introduced, where (the probabilities of) site percolation configurations on the leaf set of a binary tree are rewarded exponentially according to a generic function, which measures the degree of clustering in the configuration. Conditions on such “clustering function” are given for the existence of a limiting free energy and a wetting transition, namely the existence of a non-trivial percolation parameter threshold above and only above which the set of “dry” (open) sites have an asymptotic density. Several examples of clustering functions are given and studied using the general theory. The results here will be used in a sequel paper to study the wetting transition for the discrete Gaussian free field on the tree subject to a hard wall constraint.