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

We consider general classes of gradient models on regular trees with values in a countable Abelian group $S$ such as $\mathbb{Z}$ or $\mathbb{Z}_q$, in regimes of strong coupling (or low temperature). This includes unbounded spin models like the p-SOS model and finite-alphabet clock models. We prove the existence of families of distinct homogeneous tree-indexed Markov chain Gibbs states $\mu_A$ whose single-site marginals concentrate on a given finite subset $A \subset S$ of spin values, under a strong coupling condition for the interaction, depending only on the cardinality $\vert A \vert$ 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 $\vert A \vert \geq 2$ 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 $\mathbb{Z}$-valued gradient Gibbs states, whose single-site marginals do not localize, but whose correlation structure depends on the finite set $A$.