Hierarchical Cubes: Gibbs Measures and Decay of Correlations

Abstract We study a hierarchical model of non-overlapping cubes of sidelengths $$2^j$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>j</mml:mi> </mml:msup> </mml:math> , $$j\in {\mathbb {Z}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> . 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.