Probability Theory and Related Fields

, Volume 158, Issue 1–2, pp 477–512 | Cite as

On the Gibbs states of the noncritical Potts model on \(\mathbb Z ^2\)

  • Loren Coquille
  • Hugo Duminil-Copin
  • Dmitry Ioffe
  • Yvan VelenikEmail author


We prove that all Gibbs states of the \(q\)-state nearest neighbor Potts model on \(\mathbb Z ^2\) below the critical temperature are convex combinations of the \(q\) pure phases; in particular, they are all translation invariant. To achieve this goal, we consider such models in large finite boxes with arbitrary boundary condition, and prove that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finite-volume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at any supercritical value of the inverse temperature \(\beta >\beta _c (q) = \log \left(1+\sqrt{q}\right)\).


Potts model Gibbs states DLR equation Aizenman–Higuchi theorem Translation invariance Interface fluctuations Pure phases 

Mathematics Subject Classification (2010)

60K35 82B20 82B24 



H. D.-C. was supported by the EU Marie-Curie RTN CODY, the ERC AG CONFRA, as well as by the Swiss National Science Foundation. The research of D.I. was supported by the Israeli Science Foundation (grant No 817/09). L.C. and Y.V. were partially supported by the Swiss National Science Foundation.


  1. 1.
    Aizenman, M.: Translation invariance and instability of phase coexistence in the two-dimensional Ising system. Commun. Math. Phys. 73(1), 83–94 (1980)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Aizenman, M., Chayes, J.T., Chayes, L., Newman, C.M.: Discontinuity of the magnetization in one-dimensional \(1/|x-y|^2\) Ising and Potts models. J. Stat. Phys. 50(1), 1–40 (1988)Google Scholar
  3. 3.
    Alexander, K.S.: On weak mixing in lattice models. Probab. Theory Relat. Fields 110(4), 441–471 (1998)CrossRefzbMATHGoogle Scholar
  4. 4.
    Alexander, K.S.: Mixing properties and exponential decay for lattice systems in finite volumes. Ann. Probab. 32(1A), 441–487 (2004)Google Scholar
  5. 5.
    Alfaro, M., Conger, M., Hodges, K., Levy, A., Kochar, R., Kuklinski, L., Mahmood, Z., Von Haam, K.: The structure of singularities in \(\Phi \)-minimizing networks in \({\mathbb{R}}^2\). Pacific J. Math. 149(2), 201–210 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Beffara, V., Duminil-Copin, H.: The self-dual point of the two-dimensional random-cluster model is critical for \(q\ge 1\). Probab. Theory Relat. Fields 153(3), 511–542 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bodineau, T.: Translation invariant Gibbs states for the Ising model. Probab. Theory Relat. Fields 135(2), 153–168 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Bricmont, J., Lebowitz, J.L., Pfister, C.E.: On the equivalence of boundary conditions. J. Stat. Phys. 21(5), 573–582 (1979)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Campanino, M., Gianfelice, M.: A local limit theorem for triple connections in subcritical Bernoulli percolation. Probab. Theory Relat. Fields 143(3–4), 353–378 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Campanino, M., Ioffe, D., Velenik, Y.: Fluctuation theory of connectivities for subcritical random cluster models. Ann. Probab. 36(4), 1287–1321 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Chayes, J.T., Chayes, L., Schonmann, R.H.: Exponential decay of connectivities in the two-dimensional Ising model. J. Stat. Phys. 49(3–4), 433–445 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Cockayne, E.J.: On the Steiner problem. Can. Math. Bull. 10, 431–450 (1967)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Coquille, L., Velenik, Y.: A finite-volume version of Aizenman-Higuchi theorem for the 2d Ising model. Probab. Theory Relat. Fields 153(1–2), 25–44 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Couronné, O.: A large deviation result for the subcritical Bernoulli percolation. Annales de la Faculté des Sciences de Toulouse 14(2), 201–2014 (2005)CrossRefzbMATHGoogle Scholar
  15. 15.
    Dobrushin, R.L., Shlosman, S.B.: The problem of translation invariance of Gibbs states at low temperatures. In: Mathematical Physics Reviews. Soviet Sci. Rev. Sect. C Math. Phys. Rev., vol. 5, pp. 53–195. Harwood Academic Publ. Chur (1985)Google Scholar
  16. 16.
    Dobrušin, R.L.: Gibbsian random fields for lattice systems with pairwise interactions. In Funkcional. Anal. i Priložen. 4(2), 31–43 (1968)Google Scholar
  17. 17.
    Dobrušin, R.L.: The Gibbs state that describes the coexistence of phases for a three-dimensional Ising model. Teor. Verojatnost. i Primenen. 17, 619–639 (1972)MathSciNetGoogle Scholar
  18. 18.
    Gallavotti, G.: The phase separation line in the two-dimensional Ising model. Commun. Math. Phys. 27, 103–136 (1972)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Georgii, H.-O.: Gibbs measures and phase transitions. de Gruyter Studies in Mathematics, vol. 9. Walter de Gruyter & Co., Berlin (1988)Google Scholar
  20. 20.
    Georgii, H.-O., Häggström, O., Maes, C.: The random geometry of equilibrium phases. In: Phase transitions and critical phenomena Phase Transit. Crit. Phenom., vol. 18, pp. 1–142. Academic Press, San Diego (2001)Google Scholar
  21. 21.
    Georgii, H.-O., Higuchi, Y.: Percolation and number of phases in the two-dimensional Ising model. J. Math. Phys. 41(3), 1153–1169 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Greenberg, L., Ioffe, D.: On an invariance principle for phase separation lines. Ann. Inst. H. Poincaré Probab. Statist. 41(5), 871–885 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Grimmett, G.R.: The random-cluster model. Grundlehren der Mathematischen Wissenschaften, vol. 333. Springer, Berlin (2006)Google Scholar
  24. 24.
    Higuchi, Y.: On some limit theorems related to the phase separation line in the two-dimensional Ising model. Z. Wahrsch. Verw. Gebiete 50(3), 287–315 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Higuchi, Y.: On the absence of non-translation invariant Gibbs states for the two-dimensional Ising model. In: Random Fields, vol. I, II (Esztergom, 1979). Colloq. Math. Soc. János Bolyai, vol. 27, pp. 517–534. North-Holland, Amsterdam (1981)Google Scholar
  26. 26.
    Ioffe, D.: Large deviations for the 2D Ising model: a lower bound without cluster expansions. J. Stat. Phys. 74(1–2), 411–432 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Laanait, L., Messager, A., Miracle-Solé, S., Ruiz, J., Shlosman, S.: Interfaces in the Potts model. I. Pirogov-Sinai theory of the Fortuin-Kasteleyn representation. Commun. Math. Phys. 140(1), 81–91 (1991)CrossRefzbMATHGoogle Scholar
  28. 28.
    Laanait, L., Messager, A., Ruiz, J.: Phases coexistence and surface tensions for the Potts model. Commun. Math. Phys. 105(4), 527–545 (1986)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Lanford, O.E., Ruelle, D.: Observables at infinity and states with short range correlations in statistical mechanics. Commun. Math. Phys. 13, 194–215 (1969)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Martirosian, D.H.: Translation invariant Gibbs states in the q-state Potts model. Commun. Math. Phys. 105(2), 281–290 (1986)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Messager, A., Miracle-Sole, S.: Correlation functions and boundary conditions in the Ising ferromagnet. J. Stat. Phys. 17(4), 245–262 (1977)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Pirogov, S.A.: Sinaĭ, Ja G.: Phase diagrams of classical lattice systems. Teoret. Mat. Fiz. 25(3), 358–369 (1975)MathSciNetGoogle Scholar
  33. 33.
    Pfister, C.-E., Velenik, Y.: Interface, surface tension and reentrant pinning transition in the 2D ising model. Commun. Math. Phys. 204(2), 269–312 (1998)CrossRefMathSciNetGoogle Scholar
  34. 34.
    Russo, L.: The infinite cluster method in the two-dimensional Ising model. Commun. Math. Phys. 67(3), 251–266 (1979)CrossRefGoogle Scholar
  35. 35.
    Zahradník, M.: An alternate version of Pirogov-Sinaĭ theory. Commun. Math. Phys. 93(4), 559–581 (1984)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Loren Coquille
    • 1
  • Hugo Duminil-Copin
    • 1
  • Dmitry Ioffe
    • 2
  • Yvan Velenik
    • 1
    Email author
  1. 1.Département de MathématiquesUniversité de GenèveGenevaSwitzerland
  2. 2.Faculty of IE&MTechnionHaifaIsrael

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