, Volume 23, Issue 2, pp 291–301 | Cite as

Phase transitions for a model with uncountable spin space on the Cayley tree: the general case

  • Golibjon BotirovEmail author
  • Benedikt Jahnel


In this paper we complete the analysis of a statistical mechanics model on Cayley trees of any degree, started in Botirov (Positivity 21(3):955–961, 2017), Eshkabilov et al. (J Stat Phys 147(4):779–794, 2012), Eshkabilov and Rozikov (Math Phys Anal Geom 13:275–286, 2010), Botirov et al. (Lobachevskii J Math 34(3):256–263 2013) and Jahnel et al. (Math Phys Anal Geom 17:323–331 2014). The potential is of nearest-neighbor type and the local state space is compact but uncountable. Based on the system parameters we prove existence of a critical value \(\theta _{\mathrm{c}}\) such that for \(\theta \le \theta _{\mathrm{c}}\) there is a unique translation-invariant splitting Gibbs measure. For \(\theta _{\mathrm{c}}<\theta \) there is a phase transition with exactly three translation-invariant splitting Gibbs measures. The proof rests on an analysis of fixed points of an associated non-linear Hammerstein integral operator for the boundary laws.


Cayley trees Hammerstein operators Splitting Gibbs measures Phase transitions 

Mathematics Subject Classification

82B05 82B20 (primary) 60K35 (secondary) 



Golibjon Botirov thanks the DAAD program for the financial support and the Weierstrass Institute Berlin for its hospitality. Benedikt Jahnel thanks the Leibniz program ’Probabilistic methods for mobile ad-hoc networks’ for the support.


  1. 1.
    Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic, London (1982)zbMATHGoogle Scholar
  2. 2.
    Botirov, G.I.: A model with uncountable set of spin values on a Cayley tree: phase transitions. Positivity 21(3), 955–961 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    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), 779–794 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Eshkabilov, Y.K., Rozikov, U.A.: On models with uncountable set of spin values on a Cayley tree: integral equations. Math. Phys. Anal. Geom. 13, 275–286 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Botirov, G.I., Eshkabilov, Y.K., Rozikov, U.A.: Phase transition for a model with uncountable set of spin values on Cayley tree. Lobachevskii J. Math. 34(3), 256–263 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ganikhodjaev, N.N.: Potts model on \(\mathbb{Z}^d\) with countable set of spin values. J. Math. Phys. 45, 1121–1127 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ganikhodjaev, N.N., Rozikov, U.A.: The Potts model with countable set of spin values on a Cayley tree. Lett. Math. Phys. 74, 99–109 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Georgii, H.-O.: Gibbs Measures and Phase Transitions. De Gruyter, New York (2011)CrossRefzbMATHGoogle Scholar
  9. 9.
    Jahnel, B., Külske, C., Botirov, G.I.: Phase transition and critical values of a nearest-neighbor system with uncountable local state space on Cayley trees. Math. Phys. Anal. Geom. 17, 323–331 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Krasnosel’skii, M.A.: Topological Methods in the Theory of Nonlinear Integral Equations. Macmillan, Basingstoke (1964)zbMATHGoogle Scholar
  11. 11.
    Krasnosel’skii, M.A., Zabreiko, P.P.: Geometrical Methods of Nonlinear Analysis. Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
  12. 12.
    Rozikov, U.A.: Gibbs Measures on Cayley Trees. World Scientific, Singapore (2013)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Weierstrass Institute BerlinBerlinGermany
  2. 2.National University of UzbekistanTashkentUzbekistan

Personalised recommendations