Hierarchical Cubes: Gibbs Measures and Decay of Correlations

Abstract

We study a hierarchical model of non-overlapping cubes of sidelengths \(2^j\), \(j\in {\mathbb {Z}}\). The model allows for cubes of arbitrarily small size and the activities need not be translationally invariant. It can also be recast as a spin system on a tree with a long-range hard-core interaction. We prove necessary and sufficient conditions for the existence and uniqueness of Gibbs measures, discuss fragmentation and condensation, and prove bounds on the decay of two-point correlation functions.