Hierarchical Cubes: Gibbs Measures and Decay of Correlations

TL;DR

This paper investigates a hierarchical cube model with long-range interactions, establishing conditions for Gibbs measure existence, uniqueness, and analyzing correlation decay, applicable to spin systems on trees.

Contribution

It introduces a hierarchical cube model with non-uniform activities, providing new criteria for Gibbs measure properties and correlation decay analysis.

Findings

Necessary and sufficient conditions for Gibbs measure existence.

Bounds on decay of two-point correlations.

Analysis of fragmentation and condensation phenomena.

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.