Cayley树上可数态Potts模型的$p$ -adic Gibbs测度

Cayley树上Potts模型的自由能

On free energies of the Potts model on the Cayley tree

对于Cayley树上的Potts模型，给出了自由能和熵的显式表达式（根据向量值边界条件）。它们包括不变性，周期性，Dobrushin样的BC，以及对应于弱周期吉布斯措施。

Cayley树上Potts模型的所有平移不变$p$ -adic Gibbs测度的描述

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

最近证明了Cayley树上通常的（实）Potts模型具有最多$2^q-1$的不变性Gibbs测度.本文研究了$p$-adic Potts模型的平移不变$p$-adic Gibbs测度（TIpGM）.特别地，对于二阶Cayley树，我们给出了这种测度的确切个数。同时给出了TIpGM有界性的判据。

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.

Cayley树上q态Potts模型的新周期Gibbs测度

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

本文在k阶Cayley树上q态Potts模型的参数满足一定条件的情况下，证明了周期（非周期不变）Gibbs测度的存在性。同时给出了关于这类测度个数的一个定理。

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 $μ$.

On Ground States and Phase Transition for $λ$-Model with the Competing Potts Interactions on Cayley Trees

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…

Cayley树上$P$ -Adic Potts模型的Gibbs测度

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

考虑了$k\geq 1$阶Cayley树上的最近邻$p$-adic Potts（具有$q\geq 2$自旋值和耦合常数$J\in \Q_p$）模型.证明了在$k=2$、$q\in p\mathbb{N}$和$p\geq 3$处分别发生相变。$q\in 2^2\mathbb{N}$、$p=2$）。对于$p$-adic Potts模型，当$p\geq 3$为$q\in p\mathbb{N}$，当$p=2$为$q\in 2^2\mathbb{N}$时，在$k\geq 3$处只会发生相变.

Cayley树上的非齐次$p$ -Adic Potts模型

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

考虑了$k\geq 1$阶Cayley树上最近邻非齐次$p$-adic Potts（具有$q\geq 2$自旋值）模型.非均匀性意味着相互作用$J_{xy}$耦合依赖于Cayley树的最近邻点$x, y $。研究了模型的（$p-$ adic）Gibbs测度.证明了（i）若$q\notin p\mathbb{N}$，则对任何$k\geq 1$和含$|J_{xy}|<p^{-1/(p-1)}$的$\forall J_{xy}$都存在唯一的Gibbs测度. (ii)对于$q\in p\mathbb{N}, p\geq 3$，可以选择$J_{xy}$和$k\geq 1$，使得存在至少两个吉布斯测度，它们是平移不变的。

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.

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-…

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.

Cayley树上Potts模型的不变性分裂Gibbs测度的描述

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

对于阶为$k\geq 2$的Cayley树上的$q$ -状态Potts模型，众所周知，在足够低的温度下，至少存在$q+1$保留不变的吉布斯测度，它们也是树索引的马尔可夫链。这种测度被称为不变量分裂吉布斯测度（TISGM）。在本文中，我们发现所有的TISGM，并特别表明，在足够低的温度下，它们的数量是$2^{q}-1$。我们证明了存在$[q/2]$（其中$[a]$是$a$的整数部分）TISGM数发生变化的临界温度，并给出了每个中间温度下TISGM数的精确值.对于二叉树，我们给出了明确的临界温度和可能的TISGM的公式。虽然我们表明，这些措施从来没有凸组合的对方，这些措施的极值在所有吉布斯措施的问题将在未来的工作。

Cayley树上可数态Potts模型的$P$ -adic拟Gibbs测度的存在性

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

本文对Cayley树上$p$ -adic Potts模型的可数态给出了一种新的测度构造，称为$p$ -adic拟Gibbs测度.这种构造取决于参数$\frak{p}$和权重。在特定情况下，即，如果是$\frak{p}=\exp_p$，则所定义的度量与$p$ -adic Gibbs度量一致。本文在一定的权条件下，证明了与该模型相联系的$p$ -adic拟Gibbs测度的存在性.请注意，此条件不依赖于质数$p$的值。当自旋数是有限的时，类似的事实是无效的。

Cayley树上具有竞争作用的$P$ -Adic Potts模型的相变

On Phase Transitions for $P$-Adic Potts Model with Competing Interactions on a Cayley Tree

本文考虑二阶Cayley树上具有竞争作用的三态$p$ -adic Potts模型。我们将描述$p$ -adic Gibbs测度的问题归结为求解某个递归方程，并利用它证明了相变发生的充要条件是对任意的相互作用值（非零）都有$p=3$.此外，我们完全解决了所考虑的模型在$p$ -adic上下文中的唯一性问题。也就是说，如果$p\neq 3$，那么模型中只有唯一的Gibbs测度。

关于凯莱树上Potts模型的自由态要么是极值态要么是玻璃态

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。

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。

On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field

The paper concerns the $q$-state Potts model (i.e., with spin values in $\{1,\dots,q\}$) on a Cayley tree $\mathbb{T}^k$ of degree $k\geq 2$ (i.e., with $k+1$ edges emanating from each vertex) in an external (possibly random) field. We construct the so-called splitting Gibbs measures (SGM) using generalized boundary conditions on a sequence of expanding balls, subject to a suitable compatibility criterion. Hence, the problem of existence/uniqueness of SGM is reduced to solvability of the corresponding functional equation on the tree. In particular, we introduce the notion of translation-invari…

三阶凯莱树上$P$ -adic Potts模型的相变

Phase transitions for $P$-adic Potts model on the Cayley tree of order three

本文研究了三阶Cayley树上$q$ -态$p$-adic Potts模型的相变问题。我们考虑一个更一般的$p$ -adic Gibbs测度的概念，它依赖于参数$ρ\in\bq_p$ .这种测度称为{ \it广义$p$ -adic拟吉布斯测度}。当$ρ$等于$p$ -adic指数时，它与$p$ -adic Gibbs测度一致.当$ρ=p$时，则它与$p$ -adic拟Gibbs测度一致.因此，我们调查两个制度的价值$|ρ|_p$。也就是说，在第一种状态下，人们用$ρ=\exp_p(J)$代替一些$J\in\bq_p$，在第二种状态下，用$|ρ|_p<1$代替。在每一个政权，我们首先找到广义$p$ -adic拟吉布斯措施的存在条件。此外，在第一个政权，我们建立了在一定条件下的相变的存在。在第二个区域，当$|\r|_p,|q|_p\leq p^{-2}$时，我们证明了准相变的存在。结果表明，如果$|\r|_p<|q-1|_p^2<1$和$\sqrt{-3}\in\bq_p$，则发现存在强相变。

混合自旋与混合类型模型的Gibbs测度

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。

一维可数态$P$ -adic Potts模型的相变

On phase transition for one dimensional countable state $P$-adic Potts model

本文考虑$Z_+$上的可数态$p$-adic Potts模型.一个主要目的是建立该模型的相变的存在。在我们的研究中，我们基本上使用模型的一维。为了证明这一点，我们减少了问题，一个无限维的非线性方程的调查。我们找到了一个关于权的条件，证明了导出的方程有两个解，从而得到了相变的存在性。我们证明了第一和第二个解决方案对应的措施是一个$p$ -adic吉布斯和广义$p$ -adic吉布斯措施，分别。注意，发现条件不依赖于素数$p$的值，因此，当自旋的数量有限时，类似的事实不成立。注意，在通常的真实情况下，如果考虑具有最近邻相互作用的一维双折射不变模型，则这样的模型不表现出相变。然而，我们应该强调的是，在我们的模型中没有发生强相变，这意味着只有一个$p$ -adic Gibbs测度。在这里，我们可以看到一些类似的真实情况。此外，我们还证明了$p$ -adic Gibbs测度是有界的，而广义Gibbs测度是无界的.

关于Cayley树上的$q$ -构件模型：一般情况

On $q$- Component Models on Cayley Tree: The General Case

本文推广了文献[12]关于$k\geq 2$阶Cayley树上的$q$ -分量模型的结果.我们从两个方向推广它们：（1）从$k=2$到任何$k\geq 2;$;（2）从$q-$组件模型的具体例子（Potts和SOS模型）到任何$q$ -组件模型（具有最近邻交互）。我们给出了该模型的一组周期性基态。利用[12]中提出的轮廓论证，我们证明了$k\geq 2$阶Cayley树上$q$分量模型的$q$不同Gibbs测度的存在性.

The Dynamics of The Potts-Bethe Mapping over $\mathbb{Q}_p:$ The Case $p\equiv 2 \ (mod \ 3) $

In this paper, we study the dynamics of the Potts-Bethe mapping associated with the $p-$adic $q-$state Potts model over the Cayley tree of order three. Namely, we establish the regularity of the Potts-Bethe mapping for the case $p\equiv 2 \ (mod \ 3)$ with $p\geq5$.

Cayley树上$Q+1$ -态$P$ -adic Potts模型的动力系统和相变

On dynamical systems and phase transitions for $Q+1$-state $P$-adic Potts model on the Cayley tree

本文对$q+1$ -态Potts模型引入了一种新的$p$ -adic测度，称为{ \it $p$ -adic quasi-Gibbs测度}.对于这样一个模型，我们推导出关于边界条件的递归关系。注意，我们考虑两种相互作用模式：铁磁和反铁磁。在这两种情况下，我们调查的相变现象从相关的动力学系统的观点。即利用导出的递推关系定义了一维分数阶$p$ -adic动力系统。在铁磁情形下，我们证明了如果$q$可被$p$整除，则这样的动力系统有两个排斥不动点和一个吸引不动点。我们找到了不动点的吸引域。这使我们能够描述非线性递归方程的所有解。此外，在这种情况下，存在着强烈的相变。如果$q$不能被$p$整除，则不动点是中性的，这就证明了准相变的存在.在反铁磁情形下，存在两个吸引不动点，找到了这两个不动点的吸引盆，并描述了非线性递归方程的解。在这种情况下，我们证明了准相变的存在。

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…

Phase diagrams of lattice models on Cayley tree and chandelier network: a review

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…

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。

Chaotic behavior of the $P$-adic Potts-Bethe mapping

In our previous investigations, we have developed the renormalization group method to $p$-adic models on Cayley trees, this method is closely related to the investigation of $p$-adic dynamical systems associated with a given model. In this paper, we study chaotic behavior of the Potts-Bethe mapping. We point out that a similar kind of result is not known in the case of real numbers (with rigorous proofs).

具有可数状态集的硬核模型的吉布斯测度

Gibbs measures for a Hard-Core model with a countable set of states

本文研究任意阶凯莱树上硬核模型的非概率吉布斯测度，通过边界律方法给出平移不变与周期情形的唯一性和非唯一性条件。

研究Cayley树上Potts模型Lee-Yang零点的动力学方法

A dynamical approach to studying the Lee-Yang zeros for the Potts Model on the Cayley Tree

本文采用重整化程序与复动力学方法，研究了Cayley树上$q$态Potts模型配分函数的Lee-Yang零点在物理值射线上的聚集行为，并给出了铁磁与反铁磁情况下聚集点的显式公式。

Chaotic behavior of the $p$-adic Potts-Bethe mapping II

In our previous investigations, we have developed the renormalization group method to $p$-adic $q$-state Potts model on the Cayley tree of order $k$. This method is closely related to the examination of dynamical behavior of the $p$-adic Potts-Bethe mapping which depends on parameters $q,k$. In \cite{MFKh18} we have considered the case when $q$ is not divisible by $p$, and under some conditions it was established that the mapping is conjugate to the full shift. The present paper is a continuation of the mentioned paper, but here we investigate the case when $q$ is divisible by $p$ and $k$ is a…

Fixed points of an infinite dimensional operator related to Gibbs measures

We describe fixed points of an infinite dimensional non-linear operator related to a hard core (HC) model with a countable set $\mathbb{N}$ of spin values on the Cayley tree. This operator is defined by a countable set of parameters $λ_{i}>0$, $a_{ij}\in \{0,1\}$, $ i,j \in \mathbb{N}$. We find a sufficient condition on these parameters under which the operator has unique fixed point. When this condition is not satisfied then we show that the operator may have up to five fixed points. Also, we prove that every fixed point generates a normalisable boundary law and therefore defines a Gibbs meas…

Cayley树上有限自旋模型自由态极分解的连续性

Continuity of the extremal decomposition of the free state for finite-spin models on Cayley trees

本文证明低温有限自旋模型（含Potts与钟模型）自由态极分解测度的连续性，其分解支撑于不可数玻璃态，且该现象不依赖哈密顿量对称性。

具有可数自旋值集合的Hard-Core模型的平移不变Gibbs测度

Tranlation-invariant Gibbs measures for the Hard-Core model with a countable set of spin values

本文研究阶数为2的Cayley树上具有可数自旋值的Hard-Core模型，推导出Gibbs分布一致性的泛函方程，并分析了不同参数与图约束下平移不变Gibbs测度的存在性、唯一性及非唯一性条件。