Gibbs Measures for HC-Model with a Cuountable Set of Spin Values on a Cayley Tree

IF 0.9 3区 数学 Q3 MATHEMATICS, APPLIED
R. M. Khakimov, M. T. Makhammadaliev, U. A. Rozikov
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引用次数: 2

Abstract

In this paper, we study the HC-model with a countable set \(\mathbb Z\) of spin values on a Cayley tree of order \(k\ge 2\). This model is defined by a countable set of parameters (that is, the activity function \(\lambda _i>0\), \(i\in \mathbb Z\)). A functional equation is obtained that provides the consistency condition for finite-dimensional Gibbs distributions. Analyzing this equation, the following results are obtained:

  • Let \(\Lambda =\sum _i\lambda _i\). For \(\Lambda =+\infty \) there is no translation-invariant Gibbs measure (TIGM) and no two-periodic Gibbs measure (TPGM);

  • For \(\Lambda <+\infty \), the uniqueness of TIGM is proved;

  • Let \(\Lambda _\textrm{cr}(k)=\frac{k^k}{(k-1)^{k+1}}\). If \(0<\Lambda \le \Lambda _\textrm{cr}\), then there is exactly one TPGM that is TIGM;

  • For \(\Lambda >\Lambda _\textrm{cr}\), there are exactly three TPGMs, one of which is TIGM.

Abstract Image

Cayley树上具有可计数自旋值集的hc模型的Gibbs测度
本文研究了\(k\ge 2\)阶Cayley树上具有自旋值的可数集\(\mathbb Z\)的hc模型。该模型由一组可计数的参数(即活动函数\(\lambda _i>0\), \(i\in \mathbb Z\))定义。得到了有限维吉布斯分布的一致性条件的泛函方程。分析该方程,得到如下结果:设\(\Lambda =\sum _i\lambda _i\)。对于\(\Lambda =+\infty \)不存在平移不变吉布斯测度(TIGM)和双周期吉布斯测度(TPGM);对于\(\Lambda <+\infty \),证明了TIGM的唯一性;让\(\Lambda _\textrm{cr}(k)=\frac{k^k}{(k-1)^{k+1}}\)。如果\(0<\Lambda \le \Lambda _\textrm{cr}\),那么只有一个TPGM是TIGM;对于\(\Lambda >\Lambda _\textrm{cr}\),有三种tpgm,其中一种是TIGM。
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来源期刊
Mathematical Physics, Analysis and Geometry
Mathematical Physics, Analysis and Geometry 数学-物理:数学物理
CiteScore
2.10
自引率
0.00%
发文量
26
审稿时长
>12 weeks
期刊介绍: MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas. The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process. The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed. The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.
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