Wetting Transition on Trees I: Percolation With Clustering

TL;DR

This paper introduces a new percolation model on trees that incorporates clustering effects, establishing conditions for phase transitions and providing a foundation for future studies on related stochastic fields.

Contribution

It develops a novel 'Percolation with Clustering' model on trees, analyzing conditions for phase transitions based on clustering functions.

Findings

Existence of a limiting free energy under certain clustering conditions

Identification of a wetting transition threshold for percolation

Examples of clustering functions analyzed within the model

Abstract

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.