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

TL;DR

This paper establishes the existence of non-translation-invariant gradient Gibbs measures on regular trees for Z-valued spin models, using a novel approach involving hidden Markov models and dynamical systems analysis.

Contribution

It introduces a new existence theory for non-invariant gradient Gibbs measures on trees, employing a two-layer hidden Markov model framework and dynamical systems techniques.

Findings

Existence of non-translation-invariant gradient Gibbs measures on regular trees.

Construction based on hidden Markov models and internal q-spin models.

Proofs leverage properties of local pseudo-unstable manifolds.

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.