A manifold of pure Gibbs states of the Ising model on a Cayley tree

Translation-invariant and periodic Gibbs measures for Potts model on a Cayley tree

In this paper is studied ferromagnetic three states Potts model on a Cayley tree of order three and we give explicit formulas for translation-invariant Gibbs measures. Furthermore, we show that under some conditions on the parameter of the antiferromagnetic Potts model with q-states with zero external field on the Cayley tree of order $k>2$, there are exactly 2(2^q-1) periodic (non translation-invariant) Gibbs measures.

Description of all translation-invariant $p$-adic Gibbs measures for the Potts model on a Cayley tree

Recently it was proved that usual (real) Potts model on a Cayley tree has up to $2^q-1$ translation-invariant Gibbs measures. This paper is devoted to description of translation- invariant $p$-adic Gibbs measures (TIpGMs) of the $p$-adic Potts model. In particular, for the Cayley tree of order two we give exact number of such measures. Mereover we give criterion of boundedness of TIpGMs.

On periodic Gibbs measures for Potts-SOS model on the Cayley tree

In this paper we give a description of periodic Gibbs measures for Potts-SOS model on the Cayley tree of order $k\geq 1$ , i.e. a characterization of such measures with respect to any normal subgroup of finite index of the group representation of a Cayley tree.

Ising model on Cayley trees: a new class of Gibbs measures and their comparison with known ones

For the Ising model on Cayley trees we give a very wide class of new Gibbs measures. We show that these new measures are extreme under some conditions on the temperature. We give a review of all known Gibbs measures of the Ising model on trees and compare them with our new measures.

An extremality of the translation-invariant Gibbs measure for the HC-model on a Cayley tree

In this paper we study the extremality of translation-invariant Gibbs measure for the HC-model on a Cayley tree. It is known that for this model the translation-invariant measure is unique. We give a new proof of this statement and found regions of the extremality of this measure on the Cayley tree of order k.

Uniqueness of the translation-invariant Gibbs measure for Hard-Core models with four-states on the Cayley tree

We consider fertile HC-models with four-states and the parametre activity on a Cayley tree. It is known that three types of such models exist. For each of these models we prove uniqueness of the translation-invarinat Gibbs measure on a Cayley tree of order two.

New periodic Gibbs measures for Potts model with q-states on a Cayley tree

In this paper under some conditions on parameters of the Potts model with q-states on a Cayley tree of order k it is proved existence of the periodic (non translation-invariant)Gibbs measures. Also given a theorem about the number of these measures.

Extremality of the unique translation-invariant Gibbs measure for HC models on the Cayley tree of order three

We consider fertile three-state hard core models with the activity parameter on an order-three Cayley tree. It is known that there exist four types of such models: in two of them the regions of extremality of the unique translation-invariant Gibbs measure were found.

On a $p$-Adic Generalized Gibbs Measure for Ising Model on a Cayley Tree

In this paper we consider a $p$-adic Ising model on the Cayley tree of order $k\geq 2$. We give full description of all $p$-adic translation-invariant generalized Gibbs measures for $k=3$. Moreover, we show the existence of phase transition for $p$-adic Ising model for any $k\geq3$ when $p\equiv1(\operatorname{mod }4)$.

Gradient Gibbs measures for the SOS model with integer spin values on a Cayley tree

In the present paper we continue the investigation from [1] and consider the SOS (solid-on-solid) model on the Cayley tree of order $k \geq 2$. In the ferromagnetic SOS case on the Cayley tree, we find three solutions to a class of period-4 height-periodic boundary law equations and these boundary laws define up to three periodic gradient Gibbs measures.

Gradient Gibbs measures of a SOS model on Cayley trees: 4-periodic boundary laws

For SOS (solid-on-solid) model with external field and with spin values from the set of all integers, on a Cayley tree we give gradient Gibbs measures (GGMs). Such a measure corresponds to a boundary law (a function defined on vertices of Cayley tree) satisfying an infinite system of functional equations. We give several concrete GGMs which correspond to periodic boundary laws.

Cayley树上具有竞争相互作用和外场的Ising模型的基态与Gibbs测度

Ground states and Gibbs measures of Ising model with competing interactions and an external field on a Cayley tree

针对二阶Cayley树上带竞争相互作用与非零外场的Ising模型，研究其基态并验证Peierls条件，进而通过围道论证证明存在两种不同Gibbs测度。

Mirror symmetry of height-periodic gradient Gibbs measures of a SOS model on Cayley trees

For the solid-on-solid (SOS) model with spin values from the set of all integers on a Cayley tree we give gradient Gibbs measures (GGMs). Such a measure corresponds to a boundary law (which is an infinite-dimensional vector-valued function defined on vertices of the Cayley tree) satisfying an infinite system of functional equations. We give several concrete GGMs of boundary laws which are independent from vertices of the Cayley tree and (as an infinite-dimensional vector) have periodic, (non-)mirror symmetric coordinates.

Cayley树上可育三态硬核模型的替代Gibbs测度

Alternative Gibbs measure for fertile three-state Hard-Core models on a Cayley tree

本文研究Cayley树上可育三态硬核模型，引入替代Gibbs测度概念，在“wand”情形中给出其平移不变性条件，并证明存在非平移不变的替代Gibbs测度。

Boundary conditions for translation-invariant Gibbs measures of the Potts model on Cayley trees

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 $μ$ we explicitly give the set of boundary conditions such that limiting Gibbs measures with respect to these boundary conditions coincide with $μ$.

Non-translation-invariant Gibbs Measures for Models With Uncountable Set of Spin Values on a Cayley Tree

We consider models with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k\geq 1$. It is known that the "splitting Gibbs measures" of the model can be described by solutions of a nonlinear integral equation. Recently, solving this integral equation some periodic (in particular translation-invariant) splitting Gibbs measures were found. In this paper we give three constructions of new sets of non-translation-invariant splitting Gibbs measures. Our constructions are based on known solutions of the integral equation.

Weakly periodic Gibbs measures for two and three state HC models on a Cayley tree

In this paper we study two and three state HC-model on a Cayley tree. Under some conditions on parameters of the HC-model, in the case of normal divisor of the index four, we prove the existence of the weakly periodic (non periodic) Gibbs measures which are different from the known weakly periodic measures. Moreover, we find some regions for the $λ$ parameter ensuring that a extreme measure is not unique.

Gibbs measures and free energies of the Ising-Vannimenus Model on the Cayley tree

In this paper, we consider Ising-Vannimenus model on a Cayley tree for order two with competing nearest-neighbor, prolonged next-nearest neighbor interactions. We stress that the mentioned model was investigated only numerically, without rigorous (mathematical) proofs. One of the main point of this paper is to propose a measure-theoretical approach the considered model. We find certain conditions for the existence of Gibbs measures corresponding to the model. Then we establish the existence of the phase transition. Moreover, the free energies of the found Gibbs measures are calculated.

$p$-adic quasi Gibbs measure for Vannimenus model on a Cayley tree

In this paper we study the boundedness of the $p$-adic quasi Gibbs measures for the Vannimenus model on a Cayley tree of order two.

On Uniqueness of Gibbs Measures for $P$-Adic Nonhomogeneous $ł$-Model on the Cayley Tree

We consider a nearest-neighbor $p$-adic $ł$-model with spin values $\pm 1$ on a Cayley tree of order $k\geq 1$. We prove for the model there is no phase transition and as well as the unique $p$-adic Gibbs measure is bounded if and only if $p\geq 3$. If $p=2$ then we find a condition which guarantees nonexistence of a phase transition. Besides, the results are applied to the $p$-adic Ising model and we show that for the model there is a unique $p$-adic Gibbs measure.

Cayley树上SOS模型的一组新Gibbs测度

A new set of Gibbs measures for the SOS model on a Cayley tree

本文研究Cayley树上三态SOS模型，通过非线性泛函方程描述Gibbs测度，给出多测度存在充分条件，并证明所得新测度区别于已知测度。

Cayley树上具有交替磁性的SOS模型的梯度Gibbs测度

Gradient Gibbs measures of an SOS model with alternating magnetism on Cayley trees

本文研究Cayley树上具有交替磁性的SOS模型，利用Külske-Schriever论证构造了q=2,3,4时的多个q-高度周期平移不变梯度Gibbs测度。

Fuzzy transformations and extremality of Gibbs measures for the Potts model on a Cayley tree

We continue our study of the full set of translation-invariant splitting Gibbs measures (TISGMs, translation-invariant tree-indexed Markov chains) for the $q$-state Potts model on a Cayley tree. In our previous work \cite{KRK} we gave a full description of the TISGMs, and showed in particular that at sufficiently low temperatures their number is $2^{q}-1$. In this paper we find some regions for the temperature parameter ensuring that a given TISGM is (non-)extreme in the set of all Gibbs measures. In particular we show the existence of a temperature interval for which there are at least $2^{q-…

Almost Gibbsian Measures on a Cayley Tree

We consider the ferromagnetic n.n Ising model on Cayley trees in absence of external fields submitted to a modified majority rule transformation with overlapping cells already known to lead to non-Gibbsian measures. We describe the renormalized measures within the Generalized Gibbs framework and prove that they are Almost Gibbs at any temperature.

Gibbs measures and free energies of Ising-Vannimenus Model on the Cayley tree

In this paper, we consider the Ising-Vannimenus model on a Cayley tree for order two with competing nearest-neighbor and prolonged next-nearest neighbor interactions. We stress that the mentioned model was investigated only numerically, without rigorous (mathematical) proofs. One of the main points of this paper is to propose a measure-theoretical approach for the considered model. We find certain conditions for the existence of Gibbs measures corresponding to the model, which allowed to establish the existence of the phase transition. Moreover, the free energies and entropies, associated with…

Translation-invariant Gibbs measures for the fertile HC-models with three states on a Cayley tree

We consider fertile HC-models with three states on a Cayley tree. It is known that four types of such models exist. For these models we describe the translation-invarinat Gibbs measures on a Cayley tree of order three.

On some remarks on the Ising model with competing interactions on a Cayley tree

In the present paper the Ising model with competing binary $J$ and $J_1$ interactions with spin values $\pm 1$, on a Cayley tree is considered. We study translation-invatiant Gibbs measures and corresponding free energies ones.

Extremality of translation-invariant Gibbs measures for the Potts-SOS model on the Cayley tree

In this paper, we consider the Potts-SOS model where the spin takes values in the set $\{0, 1, 2\}$ on the Cayley tree of order two. We describe all the translation-invariant splitting Gibbs measures for this model in some conditions. Moreover, we investigate whether these Gibbs measures are extremal or non-extremal in the set of all Gibbs measures.

Non-uniqueness of Gibbs Measure for Models With Uncountable Set of Spin Values on a Cayley Tree

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 $k\geq 2$. We prove that each of the constructed model has at least two translational-invariant Gibbs measures.

Gibbs measures of Potts model on Cayley trees: a survey and applications

In this paper we give a systematic review of the theory of Gibbs measures of Potts model on Cayley trees (developed since 2013) and discuss many applications of the Potts model to real world situations: mainly biology, physics, and some examples of alloy behavior, cell sorting, financial engineering, flocking birds, flowing foams, image segmentation, medicine, sociology etc.

On Gibbs Measures of $P$-Adic Potts Model on the Cayley Tree

We consider a nearest-neighbor $p$-adic Potts (with $q\geq 2$ spin values and coupling constant $J\in \Q_p$) model on the Cayley tree of order $k\geq 1$. It is proved that a phase transition occurs at $k=2$, $q\in p\mathbb{N}$ and $p\geq 3$ (resp. $q\in 2^2\mathbb{N}$, $p=2$). It is established that for $p$-adic Potts model at $k\geq 3$ a phase transition may occur only at $q\in p\mathbb{N}$ if $p\geq 3$ and $q\in 2^2\mathbb{N}$ if $p=2$.

Cayley树上hardcore-SOS模型的Gibbs测度

Gibbs measures for hardcore-SOS models on Cayley trees

本文研究Cayley树上p=∞的有限状态p-SOS模型，通过泛函方程刻画分裂Gibbs测度，给出三状态下平移不变Gibbs测度随耦合强度变化的相图，并证明k=2时p→∞过渡连续。

New Phase transitions of the Ising model on Cayley trees

We show that the nearest neighbors Ising model on the Cayley tree exhibits new temperature driven phase transitions. These transitions holds at various inverse temperatures different from the critical one. They are depicted by a change in the number of Gibbs states as well as by a drastic change of the behavior of free energies at these new transition points.

Uniqueness of Gibbs Measure for Models With Uncountable Set of Spin Values on a Cayley Tree

We consider models with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k\geq 1$. It is known that the "splitting Gibbs measures" of the model can be described by solutions of a nonlinear integral equation. For arbitrary $k\geq 2$ 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.

Description of the translation-invariant splitting Gibbs measures for the Potts model on a Cayley tree

For the $q$-state Potts model on a Cayley tree of order $k\geq 2$ it is well-known that at sufficiently low temperatures there are at least $q+1$ translation-invariant Gibbs measures which are also tree-indexed Markov chains. Such measures are called translation-invariant splitting Gibbs measures (TISGMs). In this paper we find all TISGMs, and show in particular that at sufficiently low temperatures their number is $2^{q}-1$. We prove that there are $[q/2]$ (where $[a]$ is the integer part of $a$) critical temperatures at which the number of TISGMs changes and give the exact number of TISGMs f…

On $p$-adic Gibbs Measures for Hard Core Model on a Cayley Tree

In this paper we consider a nearest-neighbor $p$-adic hard core (HC) model, with fugacity $λ$, on a homogeneous Cayley tree of order $k$ (with $k + 1$ neighbors). We focus on $p$-adic Gibbs measures for the HC model, in particular on $p$-adic "splitting" Gibbs measures generating a $p$-adic Markov chain along each path on the tree. We show that the $p$-adic HC model is completely different from real HC model: For a fixed $k$ we prove that the $p$-adic HC model may have a splitting Gibbs measure only if $p$ divides $2^k-1$. Moreover if $p$ divides $2^k-1$ but does not divide $k+2$ then there ex…

Phase transition and Gibbs Measures of Vannimenus model on semi-infinite Cayley tree of order three

Ising model with competing nearest-neighbors and prolonged next-nearest-neighbors interactions on a Cayley tree has long been studied but there are still many problems untouched. This paper tackles new Gibbs measures of Ising-Vannimenus model with competing nearest-neighbors and prolonged next-nearest-neighbors interactions on a Cayley tree (or Bethe lattice) of order three. By using a new approach, we describe the translation-invariant Gibbs measures for the model. We show that some of the measures are extreme Gibbs distributions. In this paper we take up with trying to determine when phase t…

Gibbs measures of the Ising model with mixed spin-1 and spin-1/2 on a Cayley tree

In the present paper, the Ising model with mixed spin-(1,1/2) is considered on the second order Cayley tree. A construction of splitting Gibbs measures corresponding the model is given which allows to establish the existence of the phase transition (non-uniqueness of Gibbs measures). We point out that, in the phase transition region, the considered model has three translation-invariant Gibbs measures in the ferromagnetic and anti-ferromagnetic regimes, while the classical Ising model does not possesses such Gibbs measures in the anti-ferromagnetic regime. It turns out that the considered model…

On free energies of the Ising model on the Cayley tree

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 later are defined through a partition of the tree that induces a 4-edge-coloring. We compute the density of each color.

On $p$-adic Gibbs measures of countable state Potts model on the Cayley tree

In the present paper we consider countable state $p$-adic Potts model on the Cayley tree. A construction of $p$-adic Gibbs measures which depends on weights $ł$ is given, and an investigation of such measures is reduced to examination of an infinite-dimensional recursion equation. Studying of the derived equation under some condition on weights, we prove absence of the phase transition. Note that the condition does not depend on values of the prime $p$, and an analogues fact is not true when the number of spins is finite. For homogeneous model it is shown that the recursive equation has only o…

Periodic Gibbs Measures for Models with Uncountable Set of Spin Values on a Cayley Tree

We consider models with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k\geq 1$. We study periodic Gibbs measures of the model with period two. For $k=1$ we show that there is no any periodic Gibbs measure. In case $k\geq 2$ we get a sufficient condition on Hamiltonian of the model with uncountable set of spin values under which the model have not any periodic Gibbs measure with period two. We construct several models which have at least two periodic Gibbs measures.

Four Competing interactions for models with uncountable set of spin values on a Cayley Tree

In this paper we consider four competing interactions (external field, nearest neighbor, second neighbors and triples of neighbors) of models with uncountable (i.e. $[0,1]$) set 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 analysis of solutions to some nonlinear integral equation and study some particular cases for Ising and Potts models. Also we show that periodic Gibbs measures for given models are either translation-invariant or periodic with period two and we give examples of the non-uniqueness of tra…

Kittel's molecular zipper model on Cayley trees

Kittel's 1D model represents a natural DNA with two strands as a (molecular) zipper, which may separated as the temperature is varied. We define multidimensional version of this model on a Cayley tree and study the set of Gibbs measures. We reduce description of Gibbs measures to solving of a non-linear functional equation, with unknown functions (called boundary laws) defined on vertices of the Cayley tree. Each boundary law defines a Gibbs measure. We give general formula of free energy depending on the boundary law. Moreover, we find some concrete boundary laws and corresponding Gibbs measu…

On Inhomogeneous $p$-Adic Potts Model on a Cayley Tree

We consider a nearest-neighbor inhomogeneous $p$-adic Potts (with $q\geq 2$ spin values) model on the Cayley tree of order $k\geq 1$. The inhomogeneity means that the interaction $J_{xy}$ couplings depend on nearest-neighbors points $x, y $ of the Cayley tree. We study ($p-$ adic) Gibbs measures of the model. We show that (i) if $q\notin p\mathbb{N}$ then there is unique Gibbs measure for any $k\geq 1$ and $\forall J_{xy}$ with $|J_{xy}|<p^{-1/(p-1)}$. (ii) For $q\in p\mathbb{N}, p\geq 3$ one can choose $J_{xy}$ and $k\geq 1$ such that there exist at least two Gibbs measures which are translat…

HC-Blume-Capel模型在Cayley树上

Gibbs measure for the HC-Blume-Capel model in the case of a "wand" type graph on a Cayley tree

研究HC-Blume-Capel模型在Cayley树上

wand情形下三态硬心模型的周期Gibbs测度

Periodic Gibbs measures for three-state hard-core models in the case Wand

本文研究Cayley树上三态HC模型的wand情形，求出两周期Gibbs测度不唯一的精确临界活度值，并分析二阶树上这些测度的极值性。

Cayley树上具有可数自旋值集合的HC模型的Gibbs测度

Gibbs measures for HC-model with a countable set of spin values on a Cayley tree

本文研究了Cayley树上具有可数自旋值的HC模型，通过推导泛函方程分析Gibbs分布的一致性条件，证明了平移不变和双周期Gibbs测度在不同参数下的存在性与唯一性。

Exact description of paramagnetic and ferromagnetic phases of an Ising model on a third-order Cayley tree

In this paper we analytically study the recurrence equations of an Ising model with three competing interactions on a Cayley tree of order three. We exactly describe paramagnetic and ferromagnetic phases of the Ising model. We obtain some rigorous results: critical temperatures and curves, number of phases, partition function. Ganikhodjaev et al. [J. Concrete and Applicable Mathematics, 9 (1), 26-34 (2011)] have numerically studied the Ising model on a second-order Cayley tree. We compare the numerical results to exact solutions of mentioned model.

Existence of $P$-adic quasi Gibbs measure for countable state Potts model on the Cayley tree

In the present paper we provide a new construction of measure, called $p$-adic quasi Gibbs measure, for countable state of $p$-adic Potts model on the Cayley tree. Such a construction depends on a parameter $\frak{p}$ and wights. In particular case, i.e. if $\frak{p}=\exp_p$, the defined measure coincides with $p$-adic Gibbs measure. In this paper, under some condition on weights we establish the existence of $p$-adic quasi Gibbs measures associated with the model. Note that this condition does not depend on values of the prime $p$. An analogues fact is not valid when the number of spins is fi…

On Gibbs Measures of Models with Competing Ternary and Binary Interactions and Corresponding Von Neumann Algebras II

In the present paper the Ising model with competing binary ($J$) and binary ($J_1$) interactions with spin values $\pm 1$, on a Cayley tree of order 2 is considered. The structure of Gibbs measures for the model considered is studied. We completely describe the set of all periodic Gibbs measures for the model with respect to any normal subgroup of finite index of a group representation of the Cayley tree. Types of von Neumann algebras, generated by GNS-representation associated with diagonal states corresponding to the translation invariant Gibbs measures, are determined. It is proved that the…

Description of Weak Periodic Ground States of Ising Model with Competing Interactions on Cayley Tree

Recently by Rozikov an Ising model with competing interactions and spin values $\pm 1$, on a Cayley tree of order $k\geq 1$ has been considered and the ground states of the model are described. In this paper we describe some weak periodic ground states of the model.

Phase Transitions for a model with uncountable set of spin values on a Cayley tree

In this paper we consider a model with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k \geq 2$. To study translation-invariant Gibbs measures of the model we drive an nonlinear functional equation. For $k=2$ and 3 under some conditions on parameters of the model we prove non-uniqueness of translation-invariant Gibbs measures (i.e. there are phase transitions).

Gibbs measures for SOS models with external field on a Cayley tree

We consider a nearest-neighbor solid-on-solid (SOS) model, with several spin values $0,1,\ldots,m,$ $m\geq2,$ and nonzero external field, on a Cayley tree of degree $k$ (with $k+1$ neighbors). We are aiming to extend the results of \cite{rs} where the SOS model is studied with (mainly) three spin values and zero external field. The SOS model can be treated as a natural generalization of the Ising model (obtained for $m=1$). We mainly assume that $m=2$ (three spin values) and study translation-invariant (TI) and splitting (S) Gibbs measures (GMs). (Splitting GMs have a particular Markov-type pr…

Fertile Three State Hard-Core Models on a Cayley Tree

We consider nearest-neighbor fertile hard-core models, with three states, on a homogeneous Cayley tree. It is known that there are four type of such models. We investigate all of them and describe translation-invariant and periodic hard-core Gibbs measures. Also we construct a continuum set of non-periodic Gibbs measures.