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

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