M. M. Rahmatullaev
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.
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We study the Ising model on a Cayley tree. A wide class of new Gibbs states is exhibited.
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.
For Ising model on the Cayley tree of order five and six we present new weakly periodic (non-periodic) Gibbs measures corresponding to normal subgroups of indices two in the group representation of the Cayley tree.
We consider the Potts model with two-step interactions and spin values 1,2,3,4 on a Cayley tree. We describe periodic ground states and verify the Peierls condition for the model.
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 $p\equiv1(\operatorname{mod }4)$.
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.
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。
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.
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.
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.
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.
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…
In this paper, we consider the $λ$-model with nearest neighbor interactions and with competing Potts interactions on the Cayley tree of order-two. We notice that if $λ$-function is taken as a Potts interaction function, then this model contains as a particular case of Potts model with competing interactions on Cayley tree. In this paper, we first describe all ground states of the model. We point out that the Potts model with considered interactions was investigated only numerically, without rigorous (mathematical) proofs. One of the main points of this paper is to propose a measure-theoretical…
In this paper, we consider the classical Ising model on the Cayley tree of order k and show the existence of the phase transition in the following sense: there exists two quantum Markov states which are not quasi-equivalent. It turns out that the found critical temperature coincides with usual critical temperature.
In the paper the Ising model with competing $J_1$ and $J_2$ interactions with spin values $\pm 1$, on a Cayley tree of order 2 (with 3 neighbors) is considered . We study the structure of the ground states and verify the Peierls condition for the model. Our second result gives description of Gibbs measures for ferromagnetic Ising model with $J_1<0$ and $J_2=0$, using a contour argument which we also develop in the paper. By the argument we also study Gibbs measures for a natural generalization of the Ising model. We discuss some open problems and state several conjectures.\
We consider the ferromagnetic Ising model with spatially dependent external fields on a Cayley tree, and we investigate the conditions for the existence of the phase transition for a class of external fields, asymptotically approaching a homogeneous critical external field. Our results extend earlier results by Rozikov and Ganikhodjaev.
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…
The main aim of the present paper is to establish the existence of a phase transition for the quantum Ising model with competing XY interactions within the quantum Markov chain (QMC) scheme. In this scheme, we employ the $C^*$-algebraic approach to the phase transition problem. Note that these kinde of models do not have one-dimensional analogues, i.e. the considered model persists only on trees. It turns out that if the Ising part interactions vanish then the model with only competing XY -interactions on the Cayley tree of order two does not have a phase transition. By phase transition we mea…
In this paper, we continue an investigation of the $p$-adic Ising-Vannimenus model on the Cayley tree of an arbitrary order $k$ $(k\geq 2$). We prove the existence of $p$-adic quasi Gibbs measures by analyzing fixed points of multi-dimensional $p$-adic system of equations. We are also able to show the uniqueness of translation-invariant $p$-adic Gibbs measure. Finally, it is established the existence of the phase transition for the Ising-Vannimenus model depending on the order $k$ of the Cayley tree and the prime $p$. Note that the methods used in the paper are not valid in the real setting, s…
Some results on the two-point function and on the analytic structure of the momenta of the effective fugacity at the origin for a class of diluted ferromagnetic Ising models on the Cayley tree are presented.
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…
The main aim of the present paper is to prove the existence of a phase transition in quantum Markov chain (QMC) scheme for the Ising type models on a Cayley tree. Note that this kind of models do not have one-dimensional analogous, i.e. the considered model persists only on trees. In this paper, we provide a more general construction of forward QMC. In that construction, a QMC is defined as a weak limit of finite volume states with boundary conditions, i.e. QMC depends on the boundary conditions. Our main result states the existence of a phase transition for the Ising model with competing inte…
In the paper we considere three state $p$-adic Potts model with competing interactions on a Cayley tree of order two. We reduce a problem of describing of the $p$-adic Gibbs measures to the solution of certain recursive equation, and using it we will prove that a phase transition occurs if and only if $p=3$ for any value (non zero) of interactions. As well, we completely solve the uniqueness problem for the considered model in a $p$-adic context. Namely, if $p\neq 3$ then there is only a unique Gibbs measure the model.
We consider Potts model, with competing interactions and countable spin values $Φ=\{0,1,\dots \}$ on a Cayley tree of order three. We study periodic ground states for this model.
The main purpose of this review paper is to give systematically all the known results on phase diagrams corresponding to lattice models (Ising and Potts) on Cayley tree (or Bethe lattice) and chandelier networks. A detailed survey of various modelling applications of lattice models is reported. By using Vannimenus's approach, the recursive equations of Ising and Potts models associated to a given Hamiltonian on the Cayley tree are presented and analyzed. The corresponding phase diagrams with programming codes in different programming languages are plotted. To detect the phase transitions in th…
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…
It is known that any locally faithful quantum Markov state (QMS) on one dimensional setting can be considered as a Gibbs state associated with Hamiltonian with commuting nearest-neighbor interactions. In our previous results, we have investigated quantum Markov states (QMS) associated with Ising type models with competing interactions, which are expected to be QMS, but up to now, there is no any characterization of QMS over trees. We notice that these QMS do not have one-dimensional analogues, hence results of related to one dimensional QMS are not applicable. Therefore, the main aim of the pr…
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…
Phase Transitions, Non-Extremality (Reconstruction), and Markov Entropy Rate for the Mixed Spin-$(s,\tfrac12)$ Ising Model on a Cayley Tree of Order Three
研究三阶凯莱树上混合自旋伊辛模型的相变、非极端性及马尔可夫熵率特性。
Quantum Markov Chains for an Asymmetric Mixed Ising-XY Model on a Cayley Tree
研究二阶半无限有根凯莱树上的混合量子伊辛 - XY 模型，利用树索引量子马尔可夫链相容性准则等推导边界方程、计算局部转移算子，证明有唯一正平移不变解及相关结论，还得出约化边界律动力学性质并计算局部两体纠缠。
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.
Ising Models with Hidden Markov Structure: Applications to Probabilistic Inference in Machine Learning
本文提出耦合两层Ising层的Cayley树索引马尔可夫链模型，证明其在特定参数下最多存在3种平移不变吉布斯测度，为分层数据推断提供结构化方法，适用于去噪等任务且支持精确推断。
Gibbs measure for mixed spins and mixed types model
本文研究了Cayley树上的$(2,q)$-Ising-Potts模型，推导递推方程证明分裂Gibbs测度的存在性，并证明在特定条件下该模型存在多个平移不变的分裂Gibbs测度。
In this paper, we consider an Ising model with three competing interactions on a triangular chandelier-lattice (TCL). We describe the existence, uniqueness, and non-uniqueness of translation-invariant Gibbs measures associated with the Ising model. We obtain an explicit formula for Gibbs measures with a memory of length 2 satisfying consistency conditions. It is rigorously proved that the model exhibits phase transitions only for given values of the coupling constants. As a consequence of our approach, the dichotomy between alternative solutions of Hamiltonian models on TCLs is solved. Finally…
We investigate the location of zeros for the partition function of the anti-ferromagnetic Ising Model, focusing on the zeros lying on the unit circle. We give a precise characterization for the class of rooted Cayley trees, showing that the zeros are nowhere dense on the most interesting circular arcs. In contrast, we prove that when considering all graphs with a given degree bound, the zeros are dense in a circular sub-arc, implying that Cayley trees are in this sense not extremal. The proofs rely on describing the rational dynamical systems arising when considering ratios of partition functi…
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.
Splitting Gibbs Measures for a Periodic Triple Mixed-Spin Ising Model on a Cayley Tree
本文研究了凯莱树上三重混合异斯模型的分裂吉布斯测度，推导了边界定律兼容方程，并通过固定点方程分析相共存条件，揭示了周期不均匀性驱动的相变现象。
We consider the Ising model on a Cayley tree of order two with nearest neighbor interactions and competing next nearest neighbor interactions restricted to spins belonging to the same branch of the tree. This model was studied by Vannimenus and found a new modulated phase, in addition to the paramagnetic, ferromagnetic, antiferromagnetic phases and a (+ + - -) periodic phase. Vannimenus's results based on the recurrence equations (relating the partition function of an $n-$ generation tree to the partition function of its subsystems containing $(n-1)$ generations) and most results are obtained …
The phase transition for the three-state SOS model with one-level competing interactions on the binary tree
本文研究二叉树上带最近邻与单层次近邻竞争相互作用的三态SOS模型，证明其在特定参数下发生相变，并在一定条件下不存在反铁磁相。
We consider Ising models in two and three dimensions, with short range ferromagnetic and long range, power-law decaying, antiferromagnetic interactions. We let $J$ be the ratio between the strength of the ferromagnetic to antiferromagnetic interactions. The competition between these two kinds of interactions induces the system to form domains of minus spins in a background of plus spins, or vice versa. If the decay exponent $p$ of the long range interaction is larger than $d+1$, with $d$ the space dimension, this happens for all values of $J$ smaller than a critical value $J_c(p)$, beyond whic…
The free state for the Potts model on Cayley trees is either extremal or glassy
研究凯莱树\(\mathbb{T}^d\)（\(d\geq2\)）上Potts模型的自由态，通过极限吉布斯测度与自由边界条件得到。证明自由态要么是极值态要么是玻璃态，还给出了伊辛模型自由态的相关推论。
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…
This paper is devoted to an in-depth study of the limiting measure of Lee--Yang zeroes for the Ising Model on the Cayley Tree. We build on previous works of Müller-Hartmann-Zittartz (1974 and 1977), Barata--Marchetti (1997), and Barata--Goldbaum (2001), to determine the support of the limiting measure, prove that the limiting measure is not absolutely continuous with respect to Lebesgue measure, and determine the pointwise dimension of the measure at Lebesgue a.e. point on the unit circle and every temperature. The latter is related to the critical exponents for the phase transitions in the mo…
In this paper we study homogeneous Gibbs measures on a Cayley tree, subjected to an infinite-temperature Glauber evolution, and consider their (non-)Gibbsian properties. We show that the intermediate Gibbs state (which in zero field is the free-boundary-condition Gibbs state) behaves different from the plus and the minus state. E.g. at large times, all configurations are bad for the intermediate state, whereas the plus configuration never is bad for the plus state. Moreover, we show that for each state there are two transitions. For the intermediate state there is a transition from a Gibbsian …
Central limit theorems for linear statistics of lattice random fields (including spin models) are usually proven under suitable mixing conditions or quasi-associativity. Many interesting examples of spin models do not satisfy mixing conditions, and on the other hand, it does not seem easy to show central limit theorem for local statistics via quasi-associativity. In this work, we prove general central limit theorems for local statistics and exponentially quasi-local statistics of spin models on discrete Cayley graphs with polynomial growth. Further, we supplement these results by proving simil…
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