RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



TMF:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


TMF, 1997, Volume 111, Number 1, Pages 109–117 (Mi tmf993)  

This article is cited in 76 scientific papers (total in 76 papers)

Discription of periodic extreme Gibbs measures of some lattice models on the Cayley tree

N. N. Ganikhodzhaev, U. A. Rozikov

Romanovskii Mathematical Institute of the National Academy of Sciences of Uzbekistan

Abstract: It is proved that the translational invariant extreme Gibbs measure is unique for the antiferromagnetic Potts model with the external field. The existence of uncountable numbers of the extreme Gibbs measures for the Ising model with the external field on the Cayley tree is proved. The classes of normal subgroups of finite index of group representation of the Cayley tree is constructed. The periodic extreme Gibbs measures invariant with respect to subgroups of index two for the Ising model are constructed and the existence of uncountable numberes of the nonperiodic extreme Gibbs measures for the antiferromagnetic Ising model is proved.

DOI: https://doi.org/10.4213/tmf993

Full text: PDF file (253 kB)
References: PDF file   HTML file

English version:
Theoretical and Mathematical Physics, 1997, 111:1, 480–486

Bibliographic databases:

Received: 16.04.1996

Citation: N. N. Ganikhodzhaev, U. A. Rozikov, “Discription of periodic extreme Gibbs measures of some lattice models on the Cayley tree”, TMF, 111:1 (1997), 109–117; Theoret. and Math. Phys., 111:1 (1997), 480–486

Citation in format AMSBIB
\Bibitem{GanRoz97}
\by N.~N.~Ganikhodzhaev, U.~A.~Rozikov
\paper Discription of periodic extreme Gibbs measures of some lattice models on the Cayley tree
\jour TMF
\yr 1997
\vol 111
\issue 1
\pages 109--117
\mathnet{http://mi.mathnet.ru/tmf993}
\crossref{https://doi.org/10.4213/tmf993}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1473427}
\zmath{https://zbmath.org/?q=an:0964.82504}
\transl
\jour Theoret. and Math. Phys.
\yr 1997
\vol 111
\issue 1
\pages 480--486
\crossref{https://doi.org/10.1007/BF02634202}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1997XX96900008}


Linking options:
  • http://mi.mathnet.ru/eng/tmf993
  • https://doi.org/10.4213/tmf993
  • http://mi.mathnet.ru/eng/tmf/v111/i1/p109

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    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. U. A. Rozikov, “Structures of partition of the group representation of the Cayley tree into adjacent classes by finite index normal subgroups and their application for discription of periodic Gibbs distributions”, Theoret. and Math. Phys., 112:1 (1997), 929–933  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. 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
    4. F. M. Mukhamedov, “Von Neumann algebras generated by translation-invariant Gibbs states of the Ising model on a Bethe lattice”, Theoret. and Math. Phys., 123:1 (2000), 489–493  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. F. M. Mukhamedov, U. A. Rozikov, “The disordered phase of the inhomogeneous Potts model is extremal on the Cayley tree”, Theoret. and Math. Phys., 124:3 (2000), 1202–1210  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. Ganikhodjaev, N, “On disordered phase in the ferromagnetic Potts model on the Bethe lattice”, Osaka Journal of Mathematics, 37:2 (2000), 373  mathscinet  isi
    7. 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
    8. F. M. Mukhamedov, U. A. Rozikov, “Von Neumann algebra corresponding to one phase of the inhomogeneous Potts model on a Cayley tree”, Theoret. and Math. Phys., 126:2 (2001), 169–174  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. 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
    10. 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
    11. 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
    12. U. A. Rozikov, “Representability of Trees and Some of Their Applications”, Math. Notes, 72:4 (2002), 479–488  mathnet  crossref  crossref  mathscinet  zmath  isi
    13. 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
    14. 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
    15. Ganikhodjaev NN, Pah CH, Wahiddin MRB, “An Ising model with three competing interactions on a Cayley tree”, Journal of Mathematical Physics, 45:9 (2004), 3645–3658  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    16. Mukhamedov F, Rozikov U, “On Gibbs measures of models with competing ternary and binary interactions and corresponding von Neumann algebras”, Journal of Statistical Physics, 114:3–4 (2004), 825–848  crossref  mathscinet  zmath  adsnasa  isi
    17. Suhov, Y, “A hard-core model on a Cayley tree: An example of a loss network”, Queueing Systems, 46:1–2 (2004), 197  crossref  mathscinet  zmath  isi
    18. 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
    19. 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
    20. 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
    21. 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
    22. 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
    23. 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
    24. 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
    25. 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
    26. 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
    27. U. A. Rozikov, Sh. A. Shoyusupov, “Fertile HC models with three states on a Cayley tree”, Theoret. and Math. Phys., 156:3 (2008), 1319–1330  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    28. Rozikov, UA, “A contour method on Cayley trees”, Journal of Statistical Physics, 130:4 (2008), 801  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    29. 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
    30. 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
    31. 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  scopus
    32. 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
    33. Ganikhodjaev N., Uguz S., Akin H., Temir S., “On Pure Phases of the Vannimenus Model”, ICMS: International Conference on Mathematical Science, AIP Conference Proceedings, 1309, 2010, 288–293  crossref  isi  scopus  scopus  scopus
    34. 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
    35. Akin H., Rozikov U.A., Temir S., “A New Set of Limiting Gibbs Measures for the Ising Model on a Cayley Tree”, J Stat Phys, 142:2 (2011), 314–321  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    36. Ganikhodjaev N., Akin H., Uguz S., Temir S., “Phase diagram and extreme Gibbs measures of the Ising model on a Cayley tree in the presence of competing binary and ternary interactions”, Phase Transitions, 84:11–12 (2011), 1045–1063  crossref  mathscinet  isi  scopus  scopus  scopus
    37. Ganikhodjaev N., Akin H., Uguz S., Temir S., “On extreme Gibbs measures of the Vannimenus model”, J Stat Mech Theory Exp, 2011, P03025  crossref  isi  scopus  scopus  scopus
    38. 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
    39. 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
    40. 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
    41. U. A. Rozikov, R. M. Khakimov, “Periodic Gibbs measures for the Potts model on the Cayley tree”, Theoret. and Math. Phys., 175:2 (2013), 699–709  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    42. 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
    43. 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
    44. Mukhamedov F. Akin H., “Phase Transitions for P-Adic Potts Model on the Cayley Tree of Order Three”, J. Stat. Mech.-Theory Exp., 2013, P07014  crossref  mathscinet  isi  elib  scopus  scopus  scopus
    45. 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  scopus
    46. 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
    47. Mukhamedov F., “On Dynamical Systems and Phase Transitions for Q+1-State P-Adic Potts Model on the Cayley Tree”, Math. Phys. Anal. Geom., 16:1 (2013), 49–87  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    48. 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
    49. Ganikhodjaev N., Rahmatullaev A., “On Paramagnetic Phases of the Potts Model on a Bethe Lattice in the Presence of Competing Interactions”, International Conference on Advancement in Science and Technology 2012 (Icast): Contemporary Mathematics, Mathematical Physics and their Applications, Journal of Physics Conference Series, 435, eds. Ganikhodjaev N., Mukhamedov F., Hee P., IOP Publishing Ltd, 2013  crossref  mathscinet  isi  scopus  scopus  scopus
    50. [Anonymous], “A Multi-Dimensional-Time Dynamical System”, Qual. Theor. Dyn. Syst., 12:2 (2013), 361–375  crossref  mathscinet  isi  scopus  scopus  scopus
    51. R. M. Khakimov, “Localization of translation-invariant Gibbs measures for the Potts and “solid-on-solid” models on a Cayley tree”, Theoret. and Math. Phys., 179:1 (2014), 405–415  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    52. N. M. Khatamov, “New classes of ground states for the Potts model with random competing interactions on a Cayley tree”, Theoret. and Math. Phys., 180:1 (2014), 827–834  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    53. Rustamjon M. Khakimov, “New periodic Gibbs measures for $q$-state Potts model on a Cayley tree”, Zhurn. SFU. Ser. Matem. i fiz., 7:3 (2014), 297–304  mathnet
    54. M. M. Rakhmatullaev, “The existence of weakly periodic Gibbs measures for the Potts model on a Cayley tree”, Theoret. and Math. Phys., 180:3 (2014), 1019–1029  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    55. N. M. Khatamov, “Nonuniqueness of a Gibbs measure for the Ising ball model”, Theoret. and Math. Phys., 180:3 (2014), 1030–1039  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    56. 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
    57. R. M. Khakimov, “Translation-invariant Gibbs measures for fertile three-state “hard core” models on a Cayley tree”, Theoret. and Math. Phys., 183:3 (2015), 829–835  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    58. N. M. Khatamov, G. T. Madgoziev, “Gibbs measures for a generalized Potts model with the interaction radius two on a Cayley tree”, Theoret. and Math. Phys., 183:3 (2015), 836–845  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    59. 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
    60. M. M. Rahmatullaev, “A weakly periodic Gibbs measure for the ferromagnetic Potts model on a Cayley tree”, Siberian Math. J., 56:5 (2015), 929–935  mathnet  crossref  crossref  isi  elib
    61. R. M. Khakimov, “Gibbs measures for fertile hard-core models on the Cayley tree”, Theoret. and Math. Phys., 186:2 (2016), 294–305  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    62. O. N. Khakimov, “A $p$-adic hard-core model with three states on a Cayley tree”, Siberian Math. J., 57:4 (2016), 726–734  mathnet  crossref  crossref  isi  elib
    63. 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
    64. Haydarov F.H., “Characterization of the normal subgroups of finite index for the group representation of a Cayley tree”, Nanosyst.-Phys. Chem. Math., 7:5 (2016), 888–892  crossref  zmath  isi
    65. 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
    66. Rakhmatullaev M.M., “On Weakly Periodic Gibbs Measures for the Potts Model with External Field on the Cayley Tree”, Ukr. Math. J., 68:4 (2016), 598–611  crossref  mathscinet  isi  scopus
    67. Akin H., “Using new approaches to obtain Gibbs measures of Vannimenus model on a Cayley tree”, Chin. J. Phys., 54:4 (2016), 635–649  crossref  mathscinet  isi  elib  scopus
    68. 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
    69. 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
    70. Haydarov F. Khakimov R., “An improvement of extremality regions for Gibbs measures of the Potts model on a Cayley tree”, 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, 012019  crossref  mathscinet  isi  scopus
    71. 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, eds. Ayupov S., Chilin V., Ganikhodjaev N., Mukhamedov F., Rakhimov I., IOP Publishing Ltd, 2016, 012020  crossref  isi  scopus
    72. 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
    73. 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, eds. Accardi L., Mukhamedov F., Hee P., IOP Publishing Ltd, 2017, UNSP 012019  crossref  mathscinet  isi  scopus  scopus  scopus
    74. Kuelske C., Schriever P., “Non-Robust Phase Transitions in the Generalized Clock Model on Trees”, J. Stat. Phys., 170:1 (2018), 1–21  crossref  mathscinet  zmath  isi  scopus  scopus
    75. Haydarov F.H., “New Normal Subgroups For the Group Representation of the Cayley Tree”, Lobachevskii J. Math., 39:2, 3, SI (2018), 213–217  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    76. 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
    Number of views:
    This page:456
    Full text:83
    References:30
    First page:1

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019