Coexistence of localized Gibbs measures and delocalized gradient Gibbs measures on trees

Abstract

We study gradient models for spins taking values in the integers (or an integer lattice), which interact via a general potential depending only on the differences of the spin values at neighboring sites, located on a regular tree with d + 1 neighbors. We first provide general conditions in terms of the relevant p-norms of the associated transfer operator Q which ensure the existence of a countable family of proper Gibbs measures. Next we prove existence of delocalized gradient Gibbs measures, under natural conditions on Q. This implies coexistence of both types of measures for large classes of models including the SOS-model, and heavy-tailed models arising for instance for potentials of logarithmic growth.

33 pages, 4 figures

References in corpus (3)

• Nonexistence of random gradient Gibbs measures in continuous interface models in $d=2$
• Gradient Gibbs measures for the SOS model with countable values on a Cayley tree
• Gradient Gibbs measures and fuzzy transformations on trees

Cited by in corpus (7)

• Extremal inhomogeneous Gibbs states for SOS-models and finite-spin models on trees
• Existence of gradient Gibbs measures on regular trees which are not translation invariant
• A HC model with countable set of spin values: uncountable set of Gibbs measures
• Gradient Gibbs measures of a SOS model on Cayley trees: 4-periodic boundary laws
• Mirror symmetry of height-periodic gradient Gibbs measures of a SOS model on Cayley trees
• Gibbs measures for HC-model with a countable set of spin values on a Cayley tree
• Fixed points of an infinite dimensional operator related to Gibbs measures