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

Abstract

We provide an existence theory for gradient Gibbs measures for Zvalued 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.

Gradient Gibbs measures of a SOS model on Cayley trees: 4-periodic boundary laws

“…Following [1], [2], [6], [7], [8], [9], [10], [11], [14], [15], let us give basic definitions and some known facts related to (gradient) Gibbs measures.…”

“…However, the Markov chain given by (1.5), in general, does not have an invariant probability measure. Therefore in [8], [9], [10], [11] some nonnormalisable boundary laws are used to give gradient Gibbs measures. Now we give some results of above mentioned papers.…”

“…The following is the main result of [9]: Theorem 6. For any summable Q and any degree k ≥ 2 there is a finite period q 0 (k) such that for all q ≥ q 0 (k) there are GGMs of period q which are not translation invariant.…”

Mirror symmetry of height-periodic gradient Gibbs measures of a SOS model on Cayley trees

“…In [8], [9], [10], [12] some non-normalisable boundary laws are used to give gradient Gibbs measures.…”

“…Because for systems with an infinite set of spin values the set of Gibbs measures may be empty. In such a situation some gradient Gibbs measures (GGMs) may exist (for detailed motivations and very recent results see [1], [2], [7], [8], [9], [10], [11], [12], [15], [16]).…”