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TMF, 2014, Volume 180, Number 3, Pages 318–328 (Mi tmf8685)  

Nonuniqueness of a Gibbs measure for the Ising ball model

N. M. Khatamov

Namangan State University, Namangan, Uzbekistan

Abstract: We study a new model, the so-called Ising ball model on a Cayley tree of order $k\ge2$. We show that there exists a critical activity $\lambda_cr=\sqrt[4]{0.064}$ such that at least one translation-invariant Gibbs measure exists for $\lambda\ge\lambda_cr$, at least three translation-invariant Gibbs measures exist for $0<\lambda<\lambda_cr$, and for some $\lambda$, there are five translation-invariant Gibbs measures and a continuum of Gibbs measures that are not translation invariant. For any normal divisor $\widehat{G}$ of index $2$ of the group representation on the Cayley tree, we study $\widehat{G}$-periodic Gibbs measures. We prove that there exists an uncountable set of $\widehat{G}$-periodic (not translation invariant and “checkerboard” periodic) Gibbs measures.

Keywords: Cayley tree, configuration, Ising ball model, Gibbs measure

DOI: https://doi.org/10.4213/tmf8685

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English version:
Theoretical and Mathematical Physics, 2014, 180:3, 1030–1039

Bibliographic databases:

Received: 29.03.2014

Citation: N. M. Khatamov, “Nonuniqueness of a Gibbs measure for the Ising ball model”, TMF, 180:3 (2014), 318–328; Theoret. and Math. Phys., 180:3 (2014), 1030–1039

Citation in format AMSBIB
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\paper Nonuniqueness of a~Gibbs measure for the~Ising ball model
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\pages 1030--1039
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  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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