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

Rozikov, U. A.

doi:10.48550/arxiv.2103.07391

Cited by 2 publications

(2 citation statements)

References 103 publications

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“…The property we call homogeneity is also known as translation invariance in various references, and Gibbs measures having the Markov property are also known as splitting Gibbs measures. In contrast with the present paper, the vast majority of this literature deals with a discrete state (spin) space instead of R. While we cannot do justice to this literature here, we refer to the monograph [Roz13] and the recent survey [Roz21] focused on Potts models for a thorough overview. Our use of the fixed point problem (Definition 1.3) is similar to the use of boundary laws in the study of homogeneous Markov Gibbs measures, recalled in Section 3.1 below, but we are not aware of any prior results analogous to our connection between the infinite SDE and local equation.…”

Section: Gibbs Measures On Cayley Treesmentioning

confidence: 98%

Stationary solutions and local equations for interacting diffusions on regular trees

Lacker¹,

Zhang²

2021

Preprint

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We study the invariant measures of infinite systems of stochastic differential equations (SDEs) indexed by the vertices of a regular tree. These invariant measures correspond to Gibbs measures associated with certain continuous specifications, and we focus specifically on those measures which are homogeneous Markov random fields. We characterize the joint law at any two adjacent vertices in terms of a new two-dimensional SDE system, called the "local equation," which exhibits an unusual dependence on a conditional law. Exploiting an alternative characterization in terms of an eigenfunction-type fixed point problem, we derive existence and uniqueness results for invariant measures of the local equation and infinite SDE system. This machinery is put to use in two examples. First, we give a detailed analysis of the surprisingly subtle case of linear coefficients, which yields a new way to derive the famous Kesten-McKay law for the spectral measure of the regular tree. Second, we construct solutions of tree-indexed SDE systems with nearest-neighbor repulsion effects, similar to Dyson's Brownian motion.

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Section: Gibbs Measures On Cayley Treesmentioning

confidence: 98%

Stationary solutions and local equations for interacting diffusions on regular trees

Lacker¹,

Zhang²

2021

Preprint

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“…The Gibbs measures, originate from Boltzmann and Gibbs, is a measure in probability theory and statistical mechanics in which a number is allocate to each acceptable attribute of a system, which indicates the outcome of the system's study [5]. It is a probability measure that is associated with the system's Hamiltonian where it gives a state of the system.…”

Section: Introductionmentioning

confidence: 99%

Translation-Invariant p-Adic Gibbs Measures for the Potts Model on the Cayley Tree of Order Four

Azahari¹,

Ahmad²,

Ali³

2022

Proceedings of the International Conference on Mathematical Sciences and Statistics 2022 (ICMSS 2022)

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Different topological structure between real and p-adic fields provides a distinct condition for solution of equations or system of equations. For example, the equation x 2 +1 = 0 does not have solution over real field but it has solution over p-adic field for p ≡ 1(mod4). Meanwhile, the equation x 3 = p has solution in real field but not in p-adic field. It is convenience to investigate the translation-invariant p-adic Gibbs measures of Potts model on Cayley trees in terms of zeros of a certain polynomial. The translation-invariant p-adic Gibbs measures of Potts model on Cayley trees of order two and three was described with respect to some respective conditions on the coefficient of certain quadratic and cubic polynomials. In this paper, the set of p-adic Gibbs measures of p-adic Potts model on the Cayley tree of order four is considered. For this case, it is possible to associate the existence of the translation-invariant p-adic Gibbs measures with zeros of quartic polynomial over p-adic field.

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

Cited by 2 publications

References 103 publications

Stationary solutions and local equations for interacting diffusions on regular trees

Stationary solutions and local equations for interacting diffusions on regular trees

Translation-Invariant p-Adic Gibbs Measures for the Potts Model on the Cayley Tree of Order Four

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