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TMF, 1997, Volume 112, Number 1, Pages 170–175 (Mi tmf1037)  

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

Partition structures of the Cayley tree and applications for describing periodic Gibbs distributions

U. A. Rozikov

Romanovskii Mathematical Institute of the National Academy of Sciences of Uzbekistan

Abstract: The disposition order of partition elements into adjacent classes of the group representation of the Cayley tree on its finite index normal subgroups is described. For the inhomogeneous Ising model it is proved that there exist three $H_0$-periodic Gibbs distributions, where $H_0$ is a normal subgroup of finite index.

DOI: https://doi.org/10.4213/tmf1037

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English version:
Theoretical and Mathematical Physics, 1997, 112:1, 929–933

Bibliographic databases:

Received: 05.08.1996
Revised: 16.01.1997

Citation: U. A. Rozikov, “Partition structures of the Cayley tree and applications for describing periodic Gibbs distributions”, TMF, 112:1 (1997), 170–175; Theoret. and Math. Phys., 112:1 (1997), 929–933

Citation in format AMSBIB
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\paper Partition structures of the Cayley tree and applications for describing periodic Gibbs distributions
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\yr 1997
\vol 112
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\pages 170--175
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\transl
\jour Theoret. and Math. Phys.
\yr 1997
\vol 112
\issue 1
\pages 929--933
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. N. N. Ganikhodzhaev, U. A. Rozikov, “On unordered phases of certain models on a Cayley tree”, Sb. Math., 190:2 (1999), 193–203  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. U. A. Rozikov, “Construction of an uncountable number of limiting Gibbs measures in the inhomogeneous Ising model”, Theoret. and Math. Phys., 118:1 (1999), 77–84  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. U. A. Rozikov, “Periodic Gibbs measures of the inhomogeneous Ising model on trees”, Russian Math. Surveys, 56:1 (2001), 172–173  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    4. U. A. Rozikov, “Countably Periodic Gibbs Measures of the Ising Model on the Cayley Tree”, Theoret. and Math. Phys., 130:1 (2002), 92–100  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. N. N. Ganikhodzhaev, F. M. Mukhamedov, U. A. Rozikov, “$\mathbb {Z}$Existence of a Phase Transition for the Potts $p$-adic Model on the Set $\mathbb {Z}$”, Theoret. and Math. Phys., 130:3 (2002), 425–431  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. A. M. Rakhmatullaev, U. A. Rozikov, “Gibbs Measures and Markov Random Fields with Association $I$”, Math. Notes, 72:1 (2002), 83–89  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. U. A. Rozikov, “Representability of Trees and Some of Their Applications”, Math. Notes, 72:4 (2002), 479–488  mathnet  crossref  crossref  mathscinet  zmath  isi
    8. N. N. Ganikhodzhaev, U. A. Rozikov, “Group representation of the Cayley forest and some of its applications”, Izv. Math., 67:1 (2003), 17–27  mathnet  crossref  crossref  mathscinet  zmath  isi
    9. Kh. A. Nazarov, U. A. Rozikov, “Periodic Gibbs Measures for the Ising Model with Competing Interactions”, Theoret. and Math. Phys., 135:3 (2003), 881–888  mathnet  crossref  crossref  mathscinet  zmath  isi
    10. Mukhamedov, F, “On inhomogeneous (p)-adic Potts model on a Cayley tree”, Infinite Dimensional Analysis Quantum Probability and Related Topics, 8:2 (2005), 277  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    11. Mukhamedov, F, “On Gibbs measures of models with competing ternary and binary interactions and corresponding von Neumann algebras II”, Journal of Statistical Physics, 119:1–2 (2005), 427  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    12. Martin, J, “A three state hard-core model on a Cayley tree”, Journal of Nonlinear Mathematical Physics, 12:3 (2005), 432  crossref  mathscinet  adsnasa  isi  scopus  scopus  scopus
    13. U. A. Rozikov, Sh. A. Shoyusupov, “Gibbs measures for the SOS model with four states on a Cayley tree”, Theoret. and Math. Phys., 149:1 (2006), 1312–1323  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    14. É. P. Normatov, U. A. Rozikov, “A description of harmonic functions via properties of the group representation of the Cayley tree”, Math. Notes, 79:3 (2006), 399–407  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    15. Rozikov, UA, “Gibbs measures for SOS models on a Cayley tree”, Infinite Dimensional Analysis Quantum Probability and Related Topics, 9:3 (2006), 471  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    16. Rozikov, UA, “A constructive description of ground states and Gibbs measures for Ising model with two-step interactions on Cayley tree”, Journal of Statistical Physics, 122:2 (2006), 217  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    17. G. I. Botirov, U. A. Rozikov, “Potts model with competing interactions on the Cayley tree: The contour method”, Theoret. and Math. Phys., 153:1 (2007), 1423–1433  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    18. Mukhamedov, F, “On contour arguments for the three state Potts model with competing interactions on a semi-infinite Cayley tree”, Journal of Mathematical Physics, 48:1 (2007), 013301  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    19. U. A. Rozikov, M. M. Rakhmatullaev, “Description of weakly periodic Gibbs measures for the Ising model on a Cayley tree”, Theoret. and Math. Phys., 156:2 (2008), 1218–1227  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    20. Ganikhodjaev, NN, “On Ising Model with Four Competing Interactions on Cayley Tree”, Mathematical Physics Analysis and Geometry, 12:2 (2009), 141  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    21. U. A. Rozikov, F. T. Ishankulov, “Description of $p$-harmonic functions on the Cayley tree”, Theoret. and Math. Phys., 162:2 (2010), 222–229  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    22. G. I. Botirov, “Periodic Ground States of a Hamiltonian on a Cayley Tree”, Math. Notes, 87:4 (2010), 582–585  mathnet  crossref  crossref  mathscinet  zmath  isi
    23. Rozikov U.A., Ishankulov F.T., “Description of periodic p-harmonic functions on Cayley tree”, Nodea-Nonlinear Differential Equations and Applications, 17:2 (2010), 153–160  crossref  mathscinet  zmath  isi  scopus  scopus
    24. Rahmatullaev M.M., “Description of Weak Periodic Ground States of Ising Model with Competing Interactions on Cayley Tree”, Applied Mathematics & Information Sciences, 4:2 (2010), 237–251  mathscinet  zmath  isi
    25. 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
    26. Gandolfo D. Rozikov U.A. Ruiz J., “On P-Adic Gibbs Measures for Hard Core Model on a Cayley Tree”, Markov Process. Relat. Fields, 18:4 (2012), 701–720  mathscinet  zmath  isi
    27. Eshkabilov Y.K. Haydarov F.H. Rozikov U.A., “Non-Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree”, J. Stat. Phys., 147:4 (2012), 779–794  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    28. R. M. Khakimov, “Uniqueness of Weakly Periodic Gibbs Measure for HC-Models”, Math. Notes, 94:5 (2013), 834–838  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    29. 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
    30. Eshkabilov Yu.Kh. Haydarov F.H. Rozikov U.A., “Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree”, Math. Phys. Anal. Geom., 16:1 (2013), 1–17  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    31. 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  scopus
    32. [Anonymous], “A Multi-Dimensional-Time Dynamical System”, Qual. Theor. Dyn. Syst., 12:2 (2013), 361–375  crossref  mathscinet  isi  scopus  scopus  scopus
    33. Rozikov U.A. Akin H. Uguz S., “Exact Solution of a Generalized Annni Model on a Cayley Tree”, Math. Phys. Anal. Geom., 17:1-2 (2014), 103–114  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    34. Rozikov U.A., Haydarov F.H., “Periodic Gibbs Measures For Models With Uncountable Set of Spin Values on a Cayley Tree”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 18:1 (2015), 1550006  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    35. 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
    36. Albeverio S., Omirov B.A., Rozikov U.A., “Periodic Algebras Generated By Groups”, Algebr. Colloq., 22:4 (2015), 541–554  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    37. Eshkabilov Yu.Kh. Nodirov Sh.D. Haydarov F.H., “Positive fixed points of quadratic operators and Gibbs measures”, Positivity, 20:4 (2016), 929–943  crossref  mathscinet  zmath  isi  scopus
    38. Eshkabilov Yu.Kh. Bobonazarov Sh.P. Teshaboev R.I., “Translation-invariant Gibbs measures for a model with logarithmic potential on a Cayley tree”, Nanosyst.-Phys. Chem. Math., 7:5 (2016), 893–899  crossref  zmath  isi
    39. Khakimov R.M., “Weakly Periodic Gibbs Measures in the HC-Model for a Normal Divisor of Index Four”, Ukr. Math. J., 67:10 (2016), 1584–1598  crossref  mathscinet  zmath  isi  elib  scopus
    40. Gandolfo D. Rozikov U.A. Ruiz J., “on Four State Hard Core Models on the Cayley Tree”, Markov Process. Relat. Fields, 22:2 (2016), 359–377  mathscinet  zmath  isi
    41. 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
    42. U. A. Rozikov, F. Kh. Khaidarov, “Four competing interactions for models with an uncountable set of spin values on a Cayley Tree”, Theoret. and Math. Phys., 191:3 (2017), 910–923  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    43. 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  scopus
    44. 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  scopus
    45. M. A. Rasulova, “Periodic Gibbs measures for the Potts–SOS model on a Cayley tree”, Theoret. and Math. Phys., 199:1 (2019), 586–592  mathnet  crossref  crossref  adsnasa  elib
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