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 
@article{TMF_2014_180_3_a2,
     author = {N. M. Khatamov},
     title = {Nonuniqueness of {a~Gibbs} measure for {the~Ising} ball model},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {318--328},
     publisher = {mathdoc},
     volume = {180},
     number = {3},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2014_180_3_a2/}
}
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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/