We study a hierarchical model of non-overlapping cubes of sidelengths 2ʲ, j ∈ 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.
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Jansen et al. (2024) studied this question.
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