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.
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/
@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},
year = {2014},
volume = {180},
number = {3},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2014_180_3_a2/}
}
TY - JOUR
AU - N. M. Khatamov
TI - Nonuniqueness of a Gibbs measure for the Ising ball model
JO - Teoretičeskaâ i matematičeskaâ fizika
PY - 2014
SP - 318
EP - 328
VL - 180
IS - 3
UR - http://geodesic.mathdoc.fr/item/TMF_2014_180_3_a2/
LA - ru
ID - TMF_2014_180_3_a2
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%0 Journal Article
%A N. M. Khatamov
%T Nonuniqueness of a Gibbs measure for the Ising ball model
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2014
%P 318-328
%V 180
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2014_180_3_a2/
%G ru
%F TMF_2014_180_3_a2
