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 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.

28 pages

References in corpus (3)

• Extremal inhomogeneous Gibbs states for SOS-models and finite-spin models on trees
• Almost sure convergence and second moments of geometric functionals of fractal percolation
• Structural properties of Gibbsian point processes in abstract spaces