Gradient Gibbs Measures and Fuzzy Transformations on Trees

#### C. Kuelske, P. Schriever

2017, v.23, Issue 4, 553-590

ABSTRACT

We study Gibbsian models of unbounded integer-valued spins on trees which
possess a symmetry under height-shift. We develop a theory relating boundary laws
to gradient Gibbs measures, which applies also in cases
where the corresponding Gibbs measures do not exist.
Our results extend the classical theory of Zachary \cite{Z83}
beyond the case of normalizable boundary laws, which implies existence of Gibbs measures,
to periodic boundary laws.
We provide a construction for classes of
tree-automorphism invariant gradient Gibbs measures in terms of mixtures of pinned measures,
whose marginals to infinite paths on the tree are random walks in a $q$-periodic environment.
Here the mixture measure is
the invariant measure of a finite state Markov chain
which arises as a mod-$q$ fuzzy transform, and which
governs the correlation decay.
The construction applies for example to SOS-models and discrete Gaussian models and delivers
a large number of gradient Gibbs measures.
We also discuss relations of certain gradient Gibbs measures to Potts and Ising models.

Keywords: Gibbs measure, gradient Gibbs measure, pinned gradient measure, fuzzy transform, tree