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For the solid-on-solid (SOS) model with an external field and with spin values from the set of all integers on a Cayley tree, each (gradient) Gibbs measure corresponds to a boundary law (an infinite-dimensional vector function defined on vertices of the Cayley tree) satisfying a nonlinear functional equation. Recently some translation-invariant and height-periodic (non-normalizable) solutions to the equation are found. Here, our aim is to find non-height-periodic and non-normalizable boundary laws for the SOS model. By such a solution one can construct a non-probability Gibbs measure. We find explicitly several non-normalizable boundary laws. Moreover, we reduce the problem to solving of a nonlinear, second-order difference equation. We give analytic and numerical analyses of the difference equation. [ABSTRACT FROM AUTHOR]

In this paper, we study a two-state Hard-Core (HC) model with activity λ > 0 on a Cayley tree of order k ≥ 2. It is known that there are λcr, λ cr 0 , and λ cr ′ such that • for λ ≤ λcr this model has a unique Gibbs measure μ*, which is translation invariant. The measure μ* is extreme for λ < λ cr 0 and not extreme for λ > λ cr ′ ; • for λ > λcr there exist exactly three 2-periodic Gibbs measures, one of which is μ*, the other two are not translation invariant and are always extreme. The extremity of these periodic measures was proved using the maximality and minimality of the corresponding solutions of some equation, which ensures the consistency of these measures. In this paper, we give a brief overview of the known Gibbs measures for the HC-model and an alternative proof of the extremity of 2-periodic measures for k = 2, 3. Our proof is based on the tree reconstruction method. [ABSTRACT FROM AUTHOR]

We study the Blume–Capel model with a countable set of spin values and a force of interaction between the nearest neighbors on a Cayley tree of order . The following results are obtained. Let , , be the temperature. For , there exist no translation invariant Gibbs measures or -periodic Gibbs measures. For , we prove the uniqueness of a translation-invariant Gibbs measure. Let and . If , then there exists exactly one -periodic Gibbs measure that is translation invariant. For , there exist exactly three -periodic Gibbs measures, one of which is a translation-invariant Gibbs measure. [ABSTRACT FROM AUTHOR]

We consider models with four competing interactions ( external field, nearest neighbor, second neighbor, and three neighbors) and an uncountable set [0, 1] of spin values on the Cayley tree of order two. We reduce the problem of describing the splitting Gibbs measures of the model to the problem of analyzing solutions of a nonlinear integral equation and study some particular cases for Ising and Potts models. We also show that periodic Gibbs measures for the given models either are translation invariant or have the period two. We present examples where periodic Gibbs measures with the period two are not unique. [ABSTRACT FROM AUTHOR]

We consider translation-invariant splitting Gibbs measures (TISGMs) for the q-state Potts model on a Cayley tree of order two. Recently a full description of the TISGMs was obtained, and it was shown in particular that at sufficiently low temperatures their number is $$2^{q}-1$$ . In this paper for each TISGM $$\mu $$ we explicitly give the set of boundary conditions such that limiting Gibbs measures with respect to these boundary conditions coincide with $$\mu $$ . [ABSTRACT FROM AUTHOR]

For the Potts model on the Cayley tree, we obtain some explicit formulas for the free energies and entropies in the case of vector-valued boundary conditions. These formulas include translation-invariant, periodic, and Dobrushin-like boundary conditions and also those corresponding to weakly periodic Gibbs measures. [ABSTRACT FROM AUTHOR]

We consider the solid-on-solid model, with spin values $$0,1,2$$ , on the Cayley tree of order two (binary tree). We treat both ferromagnetic and antiferromagnetic coupling, with interactions which are proportional to the absolute value of the spin differences. We present a classification of all translation-invariant phases (splitting Gibbs measures) of the model. These measures are labeled by solutions to a nonlinear vector-valued functional equation. We show uniqueness in the case of antiferromagnetic interactions, and existence of up to seven phases in the case of ferromagnetic interactions, where the number of phases depends on the interaction strength. Next we investigate whether these states are extremal or non-extremal in the set of all Gibbs measures, when the coupling strength is varied, whenever they exist. We show that two states are always extremal, two states are always non-extremal, while three of the seven states make transitions between extremality and non-extremality. We provide explicit bounds on those transition values, making use of algebraic properties of the models, and an adaptation of the method of Martinelli, Sinclair, Weitz. [ABSTRACT FROM AUTHOR]

We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 1. We show that periodic Gibbs measures are either translation-invariant or periodic with period two. We describe two-periodic Gibbs measures of the model. For k = 1 we show that there is no any periodic Gibbs measure. In case k ≥ 2 we get a sufficient condition on Hamiltonian of the model with uncountable set of spin values under which the model has no periodic Gibbs measure. We construct several models which have at least two periodic Gibbs measures. [ABSTRACT FROM AUTHOR]

We study the Potts model on the Cayley tree. We demonstrate that for this model with a zero external field, periodic Gibbs measures on some invariant sets are translation invariant. Furthermore, we find the conditions under which the Potts model with a nonzero external field admits periodic Gibbs measures. [ABSTRACT FROM AUTHOR]

We present, for the Ising model on the Cayley tree, some explicit formulae of the free energies (and entropies) according to boundary conditions (b.c.). They include translation-invariant, periodic, Dobrushin-like b.c., as well as those corresponding to (recently discovered) weakly periodic Gibbs states. The weakly periodic measures are defined through a partition of the Cayley tree that induces a 4-edge-coloring on that tree. We compute the density of each color. We use these densities for computations of free energies corresponding to a weakly periodic b.c. [ABSTRACT FROM AUTHOR]

We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order $k\geqslant 1$. It is known that the 'splitting Gibbs measures' of the model can be described by solutions of a nonlinear integral equation. For arbitrary $k\geqslant 2$ we find a sufficient condition under which the integral equation has unique solution, hence under the condition the corresponding model has unique splitting Gibbs measure. [ABSTRACT FROM AUTHOR]

In this paper we construct several models with nearest-neighbor interactions and with the set [0,1] of spin values, on a Cayley tree of order k≥2. We prove that each of the constructed model has at least two translational-invariant Gibbs measures. [ABSTRACT FROM AUTHOR]

For the Ising model (with interaction constant J>0) on the Cayley tree of order k≥2 it is known that for the temperature T≥ T= J/arctan (1/ k) the limiting Gibbs measure is unique, and for T< T there are uncountably many extreme Gibbs measures. In the Letter we show that if $T\in(T_{c,\sqrt{k}}, T_{c,k_{0}})$, with $\sqrt{k}

We consider finite-range lattice models on Cayley trees with two basic properties: the existence of only a finite number of ground states and with a Peierls type condition. We define the notion of a contour for the model on the Cayley tree. By a contour argument we show the existence of s different (where s is the number of ground states) Gibbs measures. [ABSTRACT FROM AUTHOR]

We consider a nearest-neighbor Potts model, with countable spin values 0,1,..., and non zero external field, on a Cayley tree of order k (with k+1 neighbors). We study translation-invariant ‘splitting’ Gibbs measures. We reduce the problem to the description of the solutions of some infinite system of equations. For any k≥ 1 and any fixed probability measure ν with ν ( i)>0 on the set of all non negative integer numbers Φ={0,1,...} we show that the set of translation-invariant splitting Gibbs measures contains at most one point, independently on parameters of the Potts model with countable set of spin values on Cayley tree. Also we give description of the class of measures ν on Φ such that with respect to each element of this class our infinite system of equations has unique solution { a i , i=1,2,...}, where a ∈(0,1). [ABSTRACT FROM AUTHOR]

In the Letter we investigate aq-component models on a Cayley tree. The main goal of the Letter is to develop a ‘‘contour’’ argument on Cayley tree. We define contours and study some properties of these contours. Using a contour argument we show existence ofqdifferent Gibbs measures for severalq-component models. [ABSTRACT FROM AUTHOR]