U. A. Rozikov, M. M. Rakhmatullaev, “Weakly periodic ground states and Gibbs measures for the Ising model with competing interactions on the Cayley tree”, TMF, 160:3 (2009), 507–516; Theoret. and Math. Phys., 160:3 (2009), 1292

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Teoreticheskaya i Matematicheskaya Fizika, 2009, Volume 160, Number 3, Pages 507–516
DOI: https://doi.org/10.4213/tmf6412 (Mi tmf6412)

This article is cited in 45 scientific papers (total in 45 papers)

Weakly periodic ground states and Gibbs measures for the Ising model with competing interactions on the Cayley tree

U. A. Rozikov^a, M. M. Rakhmatullaev^b

^a Institute for Mathematics and Information Technologies, Academy of Sciences of the Republic of Uzbekistan, Tashkent, Uzbekistan
^b Namangan State University, Namangan, Uzbekistan

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DOI: https://doi.org/10.4213/tmf6412

Abstract: We introduce the notion of a weakly periodic configuration. For the Ising model with competing interactions, we describe the set of all weakly periodic ground states corresponding to normal divisors of indices

$2$ and

$4$ of the group representation of the Cayley tree. In addition, we study new Gibbs measures for the Ising model.

Keywords: Cayley tree, Gibbs measure, Ising model, weakly periodic ground state.

Received: 06.08.2008

English version:
Theoretical and Mathematical Physics, 2009, Volume 160, Issue 3, Pages 1292–1300
DOI: https://doi.org/10.1007/s11232-009-0116-1

Bibliographic databases:

Language: Russian

Citation: U. A. Rozikov, M. M. Rakhmatullaev, “Weakly periodic ground states and Gibbs measures for the Ising model with competing interactions on the Cayley tree”, TMF, 160:3 (2009), 507–516; Theoret. and Math. Phys., 160:3 (2009), 1292–1300

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Linking options:

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https://doi.org/10.4213/tmf6412

https://www.mathnet.ru/eng/tmf/v160/i3/p507

This publication is cited in the following 45 articles:

N. M. Khatamov, N. N. Malikov, “Holliday junctions in the set of DNA molecules for new translation-invariant Gibbs measures of the Potts model”, Theoret. and Math. Phys., 218:2 (2024), 346–356
M. M. Rahmatullaev, B. M. Isakov, “Translation-invariant Gibbs measures for the Ising–Potts model on a second-order Cayley tree”, Theoret. and Math. Phys., 219:3 (2024), 1048–1059
Begzod Isakov, Olimkhon Akhmedov, “FUNKTsIONALNYE URAVNENIYa DLYa PREDELNYKh MER GIBBSA MODELI IZINGA-POTTSA NA DEREVE KELI”, VOGUMFT, 2024, no. 1(4), 90
Muzaffar M. Rahmatullaev, Zulxumor A. Burxonova, “Constructive Gibbs measures for the Ising model on the Cayley tree”, Reports on Mathematical Physics, 93:3 (2024), 361
Muzaffar Rahmatullaev, Zulxumor Abdukaxorova, “$H_A$-Weakly Periodic $p$-Adic Generalized Gibbs Measures for the $p$-Adic Ising Model on the Cayley Tree of Order Two”, P-Adic Num Ultrametr Anal Appl, 16:3 (2024), 233
Muhtorjon Makhammadaliev, “Weakly periodic gibbs measures for the HC model with a countable set of spin values”, Reports on Mathematical Physics, 94:1 (2024), 83
Muzaffar M. Rahmatullaev, Bunyod U. Abraev, “Ground States and Gibbs Measures for the SOS Model with an External Field and Countable Set of Spin Values on a Cayley Tree”, Lobachevskii J Math, 45:11 (2024), 5888
M. M. Rahmatullaev, B. M. Isakov, “Ground states of Ising-Potts model on Cayley tree”, Ufa Math. J., 15:1 (2023), 43–55
M. M. Rakhmatullaev, M. A. Rasulova, “Opisanie slabo periodicheskikh osnovnykh sostoyanii dlya modeli Pottsa s vneshnim polem i schetnym mnozhestvom znachenii spina na dereve Keli”, Vladikavk. matem. zhurn., 25:4 (2023), 103–119
Dilshod O. Egamov, “Periodic and weakly periodic ground states corresponding to the subgroups of index three for the Ising model on the Cayley tree of order three”, Ukr. Mat. Zhurn., 75:6 (2023), 793
Dilshod O. Egamov, “Periodic and Weakly Periodic Ground States Corresponding to the Subgroups of Index Three for the Ising Model on the Cayley Tree of Order Three”, Ukr Math J, 75:6 (2023), 908
Muzaffar M. Rahmatullaev, Muhayyo A. Rasulova, Javohir N. Asqarov, “Ground States and Gibbs Measures of Ising Model with Competing Interactions and an External Field on a Cayley Tree”, J Stat Phys, 190:7 (2023)
Farrukh Mukhamedov, Muzaffar M. Rahmatullaev, Dilshodbek O. EgAMOV, “Periodic ground states for the mixed spin ising model with competing interactions on a Cayley tree”, Reports on Mathematical Physics, 91:3 (2023), 379
Muzaffar M. Rahmatullaev, Bunyod U. Abraev, “On ground states for the SOS model with competing interactions”, Zhurn. SFU. Ser. Matem. i fiz., 15:2 (2022), 162–175
M. M. Rahmatullaev, M. A. Rasulova, “Periodic Ground States for the Potts Model with External Field and a Countable Set of Spin Values on the Cayley Tree”, Math. Notes, 112:1 (2022), 116–125
J. D. Dekhkonov, “On $(k_0)$-translation-invariant and $(k_0)$-periodic Gibbs measures for Potts model on Cayley tree”, Ufa Math. J., 14:4 (2022), 42–55
Farhod Haydarov, Utkir Rozikov, “Invariance Property on Group Representations of the Cayley Tree and Its Applications”, Results Math, 77:6 (2022)
M. M. Rahmatullaev, Zh. D. Dekhkonov, “Weakly periodic Gibbs measures for the Ising model on the Cayley tree of order $k=2$”, Theoret. and Math. Phys., 206:2 (2021), 185–198
Rahmatullaev M.M. Egamov D.O. Haydarov F.H., “Periodic and Weakly Periodic Ground States Corresponding to Subgroups of Index Three For the Ising Model on Cayley Tree”, Rep. Math. Phys., 88:2 (2021), 247–257
Rahmatullaev M.M. Rafikov F.K. Azamov Sh.Kh., “On the Constructive Description of Gibbs Measures For the Potts Model on a Cayley Tree”, Ukr. Math. J., 73:7 (2021), 1092–1106

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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