Queueing Systems

, Volume 81, Issue 1, pp 49–69 | Cite as

Gibbs measures for the fertile three-state hard-core models on a Cayley tree

  • U. A. RozikovEmail author
  • R. M. Khakimov


We study translation-invariant splitting Gibbs measures (TISGMs, tree-indexed Markov chains) for the fertile three-state hard-core models with activity \(\lambda >0\) on the Cayley tree of order \(k\ge 1\). There are four such models: wrench, wand, hinge, and pipe. These models arise as simple examples of loss networks with nearest-neighbor exclusion. It is known that (i) for the wrench and pipe cases \(\forall \lambda >0\) and \(k\ge 1\), there exists a unique TISGM; (ii) for hinge (resp. wand) case at \(k=2\) if \(\lambda <\lambda _\mathrm{cr}=9/4\) (resp. \(\lambda <\lambda _\mathrm{cr}=1\)), there exists a unique TISGM, and for \(\lambda > 9/4\) (resp. \(\lambda >1\)), there exist three TISGMs. In this paper we generalize the result (ii) for any \(k\ge 2\), i.e., for hinge and wand cases we find the exact critical value \(\lambda _\mathrm{cr}(k)\) with properties mentioned in (ii). Moreover, we find some regions for the \(\lambda \) parameter ensuring that a given TISGM is extreme or non-extreme in the set of all Gibbs measures. For the Cayley tree of order two, we give explicit formulae and some numerical values.


Fertile Hard-core model Critical temperature  Cayley tree  Gibbs measure Extreme measure Reconstruction problem 

Mathematics Subject Classification

82B26 (Primary) 60K35 (Secondary) 



U. A. Rozikov thanks the Université du Sud Toulon Var, the Centre de Physique Théorique for support of his many visits, and Aix-Marseille University Institute for Advanced Study IMéRA (Marseille, France) for support by a residency scheme. He also thanks the Ruhr-University Bochum (Germany) for financial support and hospitality. We thank both referees for careful reading of the manuscript and for useful suggestions.


  1. 1.
    Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic, London (1982)Google Scholar
  2. 2.
    Brightwell, G.R., Winkler, P.: Graph homomorphisms and phase transitions. J. Comb. Theory Ser. B 77(2), 221–262 (1999)CrossRefGoogle Scholar
  3. 3.
    Brightwell, G.R., Winkler, P.: Hard constraints and the Bethe lattice: adventures at the interface of combinatorics and statistical physics. In: Proceedings of the ICM 2002, vol. IIIi, pp. 605–624. Higher Education Press, Beijing (2002)Google Scholar
  4. 4.
    Brightwell, G.R., Häggström, O., Winkler, P.: Nonmonotonic behavior in hard-core and Widom–Rowlinson models. J. Stat. Phys. 94(3–4), 415–435 (1999)CrossRefGoogle Scholar
  5. 5.
    Formentin, M., Külske, C.: A symmetric entropy bound on the non-reconstruction regime of Markov chains on Galton–Watson trees. Electron. Commun. Probab. 14, 587–596 (2009)Google Scholar
  6. 6.
    Galvin, D., Kahn, J.: On phase transition in the hard-core model on \(Z^d\). Comb. Prob. Comp. 13, 137–164 (2004)CrossRefGoogle Scholar
  7. 7.
    Galvin, D., Martinelli, F., Ramanan, K., Tetali, P.: The multistate hard core model on a regular tree. SIAM J. Discret. Math. 25(2), 894–915 (2011)CrossRefGoogle Scholar
  8. 8.
    Kelly, F.: Loss networks. Ann. Appl. Probab. 1(3), 319–378 (1991)CrossRefGoogle Scholar
  9. 9.
    Kesten, H.: Quadratic transformations: a model for population growth. I. Adv. Appl. Probab. 2, 1–82 (1970)CrossRefGoogle Scholar
  10. 10.
    Kesten, H., Stigum, B.P.: Additional limit theorem for indecomposable multi-dimensional Galton–Watson processes. Ann. Math. Stat. 37, 1463–1481 (1966)CrossRefGoogle Scholar
  11. 11.
    Khakimov, R.M.: Translation-invariant Gibbs measures for the fertile HC-models with three state on a Cayley tree. ArXiv:1406.0473v1
  12. 12.
    Louth, G.: Stochastic networks: complexity, dependence and routing. Cambridge University (thesis) (1990)Google Scholar
  13. 13.
    Luen, B., Ramanan, K., Ziedins, I.: Nonmonotonicity of phase transitions in a loss network with controls. Ann. Appl. Probab. 16(3), 1528–1562 (2006)CrossRefGoogle Scholar
  14. 14.
    Mazel, A.E., Suhov, YuM: Random surfaces with two-sided constraints: an application of the theory of dominant ground states. J. Stat. Phys. 64, 111–134 (1991)CrossRefGoogle Scholar
  15. 15.
    Mitra, P., Ramanan, K., Sengupta, A., Ziedins, I.: Markov random field models of multicasting in tree networks. Adv. Appl. Probab. 34(1), 1–27 (2002)CrossRefGoogle Scholar
  16. 16.
    Martin, J.B., Rozikov, U.A., Suhov, Y.M.: A three state hard-core model on a Cayley tree. J. Nonlinear Math. Phys. 12(3), 432–448 (2005)CrossRefGoogle Scholar
  17. 17.
    Martinelli, F., Sinclair, A., Weitz, D.: Fast mixing for independent sets, coloring and other models on trees. Random Struct. Algoritms 31, 134–172 (2007)CrossRefGoogle Scholar
  18. 18.
    Mossel, E.: Survey: Information Flow on Trees. In: Graphs, morphisms and statistical physics. DIMACS Ser. Discrete Mathematics Theoretical Computer Science, vol. 63, pp. 155–170. American Mathematical Society, Providence (2004)Google Scholar
  19. 19.
    Rozikov, U.A.: Gibbs measures on Cayley trees, p. 404. World Sci. Publ., Singapore (2013)CrossRefGoogle Scholar
  20. 20.
    Rozikov, U.A., Shoyusupov, ShA: Fertile three state HC models on Cayley tree. Theor. Math. Phys. 156(3), 1319–1330 (2008)CrossRefGoogle Scholar
  21. 21.
    Suhov, YuM, Rozikov, U.A.: A hard-core model on a Cayley tree: an example of a loss network. Queueing Syst. 46(1/2), 197–212 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institute of MathematicsTashkentUzbekistan

Personalised recommendations