Existence of gradient Gibbs measures on regular trees which are not translation invariant

Abstract

We provide an existence theory for gradient Gibbs measures for Z-valued spin models on regular trees which are not invariant under translations of the tree, assuming only summability of the transfer operator. The gradient states we obtain are delocalized. The construction we provide for them starts from a two-layer hidden Markov model representation in a setup which is not invariant under tree-automorphisms, involving internal q-spin models. The proofs of existence and lack of translation invariance of infinite-volume gradient states are based on properties of the local pseudo-unstable manifold of the corresponding discrete dynamical systems of these internal models, around the free state, at large q.

arXiv comment35 pages, 4 figures

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Cited by in corpus (4)

• Extremal inhomogeneous Gibbs states for SOS-models and finite-spin models on trees
• Gradient Gibbs measures of a SOS model on Cayley trees: 4-periodic boundary laws
• Infinite-volume states with irreducible localization sets for gradient models on trees
• Fixed points of an infinite dimensional operator related to Gibbs measures