Geodesic
Gibbs measures for fertile models with hard-core interactions
Teoretičeskaâ i matematičeskaâ fizika, Tome 219 (2024) no. 2, pp. 335-351

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We consider fertile models with hard interactions, four states, and an activity parameter $\lambda>0$ on a Cayley tree. We show that there are three types of such models: “stick,” “key,” and “generalized key.” For the “generalized key” model on a Cayley tree of order $k=4$, the uniqueness of the translation-invariant Gibbs measure is proved, and conditions for the existence of double-periodic Gibbs measures other than the translation-invariant ones are found. Moreover, in the case of a fertile graph of the “stick” type, the translation invariance of double-periodic Gibbs measures on a Cayley tree of orders $k=2,3,4$ is shown and conditions for the existence of double-periodic Gibbs measures other than the translation-invariant ones on a Cayley tree of order $k\geq5$ are found.
Keywords: Cayley tree, fertile HC model, Gibbs measure, translation-invariant measures, periodic measures.
Mots-clés : configuration 
@article{TMF_2024_219_2_a9,
     author = {R. M. Khakimov and B. Z. Tozhiboev},
     title = {Gibbs measures for fertile models with hard-core interactions},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {335--351},
     publisher = {mathdoc},
     volume = {219},
     number = {2},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2024_219_2_a9/}
}
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R. M. Khakimov; B. Z. Tozhiboev. Gibbs measures for fertile models with hard-core interactions. Teoretičeskaâ i matematičeskaâ fizika, Tome 219 (2024) no. 2, pp. 335-351. http://geodesic.mathdoc.fr/item/TMF_2024_219_2_a9/