Geodesic
Nonuniqueness of a Gibbs measure for the Ising ball model
Teoretičeskaâ i matematičeskaâ fizika, Tome 180 (2014) no. 3, pp. 318-328
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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 $\lambda_{\rm cr}=\sqrt[4]{0.064}$ such that at least one translation-invariant Gibbs measure exists for $\lambda\ge\lambda_{\rm cr}$, at least three translation-invariant Gibbs measures exist for $0<\lambda<\lambda_{\rm cr}$, 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, Ising ball model, Gibbs measure.
Mots-clés : configuration 
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     title = {Nonuniqueness of {a~Gibbs} measure for {the~Ising} ball model},
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}
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N. M. Khatamov. Nonuniqueness of a Gibbs measure for the Ising ball model. Teoretičeskaâ i matematičeskaâ fizika, Tome 180 (2014) no. 3, pp. 318-328. http://geodesic.mathdoc.fr/item/TMF_2014_180_3_a2/

[1] N. N. Ganikhodzhaev, U. A. Rozikov, TMF, 111:1 (1997), 109–117 | DOI | DOI | MR | Zbl

[2] Kh.-O. Georgi, Gibbsovskie mery i fazovye perekhody, M., Mir, 1992 | MR | Zbl

[3] K. Preston, Gibbsovskie sostoyaniya na schetnykh mnozhestvakh, Mir, M., 1977 | MR

[4] Ya. G. Sinai, Teoriya fazovykh perekhodov. Strogie rezultaty, Nauka, M., 1980 | MR | MR | Zbl | Zbl

[5] U. A. Rozikov, G. T. Madgoziev, TMF, 167:2 (2011), 311–322 | DOI | DOI | MR | Zbl

[6] P. M. Blekher, N. N. Ganikhodzhaev, TVP, 35:2 (1990), 220–230 | DOI | MR | Zbl

[7] U. A. Rozikov, TMF, 118:1 (1999), 95–104 | DOI | DOI | MR | Zbl

[8] U. A. Rozikov, Gibbs Measures on Cayley Trees, World Sci., Singapore, 2013 | MR | Zbl