Abstract
We consider an SOS (solid-on-solid) model, with spin values from the set of all integers, on a Cayley tree of order k and are interested in translation-invariant gradient Gibbs measures (GGMs) of the model. Such a measure corresponds to a boundary law (a function defined on vertices of the Cayley tree) satisfying a functional equation. In the ferromagnetic SOS case on the binary tree we find up to five solutions to a class of 4-periodic boundary law equations (in particular, some 2-periodic ones). We show that these boundary laws define up to four distinct GGMs. Moreover, we construct some 3-periodic boundary laws on the Cayley tree of arbitrary order k, which define GGMs different from the 4-periodic ones.
Highlights
We consider models where an infinite-volume spin-configuration ω is a function from the vertices of the Cayley tree to a local configuration space E ⊆ Z.A solid-on-solid (SOS) model is a spin system with spins taking values in the integers, and formal HamiltonianH(ω) = −J |ωx − ωy|, {x,y}where J ∈ R is a coupling constant
The procedure of constructing a Gibbs measures from boundary laws described in [25] can not be applied to elements of that class, we are still able to assign a tree-automorphism invariant gradient Gibbs measure (GGM) on the space of gradient configurations to each such spatially homogeneous heightperiodic boundary law, compare [15]. This motivates the study of spatially homogeneous height-periodic boundary laws as useful finite-dimensional objects which are are easier to handle than the non-periodic ones required to fulfill the normalisability condition
The main goal of this paper consists in the description of a class of boundary solutions for the Z-valued SOS-model which have periods of 2, 3 and 4 with respect to shift in the height direction on the local state space Z, and their associated GGMs
Summary
We consider models where an infinite-volume spin-configuration ω is a function from the vertices of the Cayley tree to a local configuration space E ⊆ Z. The procedure of constructing a Gibbs measures from boundary laws described in [25] can not be applied to elements of that class, we are still able to assign a tree-automorphism invariant gradient Gibbs measure (GGM) on the space of gradient configurations to each such spatially homogeneous heightperiodic boundary law, compare [15] This motivates the study of spatially homogeneous height-periodic boundary laws as useful finite-dimensional objects which are are easier to handle than the non-periodic ones required to fulfill the normalisability condition. The main goal of this paper consists in the description of a class of boundary solutions for the Z-valued SOS-model which have periods of 2, 3 and 4 with respect to shift in the height direction on the local state space Z, and their associated GGMs. Our motivation is to present closed-form solutions, prove identifiability, and demonstrate the richness of transitions even inside these families. In the last subsection we construct GGMs for period-3 height-periodic boundary laws on the k-regular tree for arbitrary k ≥ 2
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