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TMF, 2008, Volume 156, Number 2, Pages 292–302 (Mi tmf6248)  

This article is cited in 16 scientific papers (total in 16 papers)

Description of weakly periodic Gibbs measures for the Ising model on a Cayley tree

U. A. Rozikova, M. M. Rakhmatullaevb

a Romanovskii Mathematical Institute of the National Academy of Sciences of Uzbekistan
b Namangam State University

Abstract: We introduce the concept of a weakly periodic Gibbs measure. For the Ising model, we describe a set of such measures corresponding to normal subgroups of indices two and four in the group representation of a Cayley tree. In particular, we prove that for a Cayley tree of order four, there exist critical values $T_{\mathrm{c}}<T_{\mathrm{cr}}$ of the temperature $T>0$ such that there exist five weakly periodic Gibbs measures for $0<T<T_{\mathrm{c}}$ or $T>T_{\mathrm{cr}}$, three weakly periodic Gibbs measures for $T=T_{\mathrm{c}}$, and one weakly periodic Gibbs measure for $T_{\mathrm{c}}<T\le T_{\mathrm{cr}}$.

Keywords: Cayley tree, Gibbs measure, Ising model, weakly periodic measure

DOI: https://doi.org/10.4213/tmf6248

Full text: PDF file (416 kB)
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English version:
Theoretical and Mathematical Physics, 2008, 156:2, 1218–1227

Bibliographic databases:

Received: 26.07.2007
Revised: 23.10.2007

Citation: U. A. Rozikov, M. M. Rakhmatullaev, “Description of weakly periodic Gibbs measures for the Ising model on a Cayley tree”, TMF, 156:2 (2008), 292–302; Theoret. and Math. Phys., 156:2 (2008), 1218–1227

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. U. A. Rozikov, M. M. Rakhmatullaev, “Weakly periodic ground states and Gibbs measures for the Ising model with competing interactions on the Cayley tree”, Theoret. and Math. Phys., 160:3 (2009), 1292–1300  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. U. A. Rozikov, G. T. Madgoziev, “Nonuniqueness of a Gibbs measure for a model on the Cayley tree”, Theoret. and Math. Phys., 167:2 (2011), 668–679  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    3. U. A. Rozikov, R. M. Khakimov, “The uniqueness condition for a weakly periodic Gibbs measure for the hard-core model”, Theoret. and Math. Phys., 173:1 (2012), 1377–1386  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    4. Gandolfo D., Ruiz J., Shlosman S., “A Manifold of Pure Gibbs States of the Ising Model on a Cayley Tree”, J. Stat. Phys., 148:6 (2012), 999–1005  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    5. M. M. Rakhmatullaev, “Weakly periodic Gibbs measures and ground states for the Potts model with competing interactions on the Cayley tree”, Theoret. and Math. Phys., 176:3 (2013), 1236–1251  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. Gandolfo D. Rakhmatullaev M.M. Rozikov U.A. Ruiz J., “On Free Energies of the Ising Model on the Cayley Tree”, J. Stat. Phys., 150:6 (2013), 1201–1217  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    7. Rozikov U.A., “Gibbs Measures on Cayley Trees: Results and Open Problems”, Rev. Math. Phys., 25:1 (2013), 1330001  crossref  mathscinet  isi  elib  scopus  scopus
    8. Gandolfo D., Ruiz J., Shlosman S., “A Manifold of Pure Gibbs States of the Ising Model on the Lobachevsky Plane”, Commun. Math. Phys., 334:1 (2015), 313–330  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    9. M. M. Rakhmatullaev, “Weakly periodic Gibbs measures of the Ising model with an external field on the Cayley tree”, Theoret. and Math. Phys., 183:3 (2015), 822–828  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    10. M. M. Rakhmatullaev, “New weakly periodic Gibbs measures of Ising model on Cayley tree”, Russian Math. (Iz. VUZ), 59:11 (2015), 45–53  mathnet  crossref
    11. M. M. Rahmatullaev, “On weakly periodic Gibbs measures of the Potts model with a special external field on a Cayley tree”, Zhurn. matem. fiz., anal., geom., 12:4 (2016), 302–314  mathnet  crossref  mathscinet
    12. Rahmatullaev M., “On new weakly periodic Gibbs measures of the Ising model on the Cayley tree of order ? 5”, Algebra, Analysis and Quantum Probability, Journal of Physics Conference Series, 697, ed. Ayupov S. Chilin V. Ganikhodjaev N. Mukhamedov F. Rakhimov I., IOP Publishing Ltd, 2016, 012020  crossref  isi  scopus
    13. Ganikhodjaev N. Rahmatullaev M. Rodzhan Mohd Hirzie Bin Mohd, “Weakly Periodic Gibbs Measures of the Ising Model on the Cayley Tree of Order Five and Six”, Math. Phys. Anal. Geom., 21:1 (2017), 2  crossref  mathscinet  isi  scopus  scopus
    14. Rahmatullaev M.M., Rozikov U.A., “Ising Model on Cayley Trees: a New Class of Gibbs Measures and Their Comparison With Known Ones”, J. Stat. Mech.-Theory Exp., 2017, 093205  crossref  mathscinet  isi  scopus  scopus
    15. Rahmatullaev M., “Ising Model on Trees: (K(0)) - Non Translation-Invariant Gibbs Measures”, 37Th International Conference on Quantum Probability and Related Topics (Qp37), Journal of Physics Conference Series, 819, ed. Accardi L. Mukhamedov F. Hee P., IOP Publishing Ltd, 2017, UNSP 012019  crossref  mathscinet  isi  scopus  scopus
    16. R. M. Khakimov, “Weakly periodic Gibbs measures for HC-models on Cayley trees”, Siberian Math. J., 59:1 (2018), 147–156  mathnet  crossref  crossref  isi  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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