Infinite-volume states with irreducible localization sets for gradient models on trees

Abstract

We consider general classes of gradient models on regular trees with values in a countable Abelian group S such as Z or Zq, in regimes of strong coupling (or low temperature). This includes unbounded spin models like the p-SOS model and finite-alphabet clock models. We prove the existence of families of distinct homogeneous tree-indexed Markov chain Gibbs states μA whose single-site marginals concentrate on a given finite subset A ⊂ S of spin values, under a strong coupling condition for the interaction, depending only on the cardinality |A| of A. The existence of such states is a new and robust phenomenon which is of particular relevance for infinite spin models. These states are not convex combinations of each other, and in particular the states with |A| ≥ 2 can not be decomposed into homogeneous Markov-chain Gibbs states with a single-valued concentration center. As a further application of the method we obtain moreover the existence of new types of Z-valued gradient Gibbs states, whose single-site marginals do not localize, but whose correlation structure depends on the finite set A. MSC2020 subject classifications: 82B26 (primary); 60K35 (secondary)

Gradient Gibbs measures with periodic boundary laws of a generalized SOS model on a Cayley tree

“…In [15], general classes of gradient models on regular trees are described with values in a countable Abelian group S, such as Z or Z q , in regimes of strong coupling (or low temperature). This includes unbounded spin models like the p-SOS model and finite-alphabet clock models.…”