Infinite-volume states with irreducible localization sets for gradient models on trees (Q6564698)

In the paper the authors study some gradient models on regular trees with spin values in a countable abelian group \(S\) such as \(\mathbb Z\) or \(\mathbb Z_q\). For low temperatures and some conditions on the interaction, it is shown that existence of families of distinct homogeneous tree-indexed Markov chain Gibbs states (MCGS) \(\mu_A\) whose single-site marginals concentrate on a given finite subset \(A\subset S\). Moreover, it is shown that these states are extremal in the set of homogeneous Gibbs states, and cannot be decomposed into homogeneous MCGS with a single-valued concentration center. The existence of new types of gradient Gibbs states with \(\mathbb Z\)-valued spins is proved, whose single-site marginals do not localize, but whose correlation structure depends on the finite set \(A\), in this case explicit expressions for the correlation between the height-increments along disjoint edges are given.