Communications in Mathematical Physics

, Volume 349, Issue 1, pp 47–107 | Cite as

Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with \({1 \le q \le 4}\)

  • Hugo Duminil-CopinEmail author
  • Vladas Sidoravicius
  • Vincent Tassion


This article studies the planar Potts model and its random-cluster representation. We show that the phase transition of the nearest-neighbor ferromagnetic q-state Potts model on \({\mathbb{Z}^2}\) is continuous for \({q \in \{2,3,4\}}\), in the sense that there exists a unique Gibbs state, or equivalently that there is no ordering for the critical Gibbs states with monochromatic boundary conditions.

The proof uses the random-cluster model with cluster-weight \({q \ge 1}\) (note that q is not necessarily an integer) and is based on two ingredients:
  • The fact that the two-point function for the free state decays sub-exponentially fast for cluster-weights \({1\le q\le 4}\), which is derived studying parafermionic observables on a discrete Riemann surface.

  • A new result proving the equivalence of several properties of critical random-cluster models:
    • the absence of infinite-cluster for wired boundary conditions,

    • the uniqueness of infinite-volume measures,

    • the sub-exponential decay of the two-point function for free boundary conditions,

    • a Russo–Seymour–Welsh type result on crossing probabilities in rectangles with arbitrary boundary conditions.

The result has important consequences toward the study of the scaling limit of the random-cluster model with \({q \in [1,4]}\). It shows that the family of interfaces (for instance for Dobrushin boundary conditions) are tight when taking the scaling limit and that any sub-sequential limit can be parametrized by a Loewner chain. We also study the effect of boundary conditions on these sub-sequential limits. Let us mention that the result should be instrumental in the study of critical exponents as well.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. AB99.
    Aizenman M., Burchard A.: Hölder regularity and dimension bounds for random curves. Duke Math. J. 99(3), 419–453 (1999)zbMATHCrossRefGoogle Scholar
  2. ADS15.
    Aizenman M., Duminil-Copin H., Sidoravicius V.: Random currents and continuity of Ising model’s spontaneous magnetization. Commun. Math. Phys. 334, 719–742 (2015)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  3. AF86.
    Aizenman M., Fernández R.: On the critical behavior of the magnetization in high-dimensional Ising models. J. Stat. Phys. 44(3–4), 393–454 (1986)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  4. Ale98.
    Alexander Kenneth S.: On weak mixing in lattice models. Probab. Theory Relat. Fields 110(4), 441–471 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  5. Bax71.
    Baxter R.J.: Generalized ferroelectric model on a square lattice. Stud. Appl. Math. 50, 51–69 (1971)MathSciNetCrossRefGoogle Scholar
  6. Bax73.
    Baxter R.J.: Potts model at the critical temperature. J. Phys. C: Solid State Phys. 6(23), L445 (1973)ADSCrossRefGoogle Scholar
  7. Bax89.
    Baxter, R.J.: Exactly solved models in statistical mechanics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London (1989) (Reprint of the 1982 original)Google Scholar
  8. BD16.
    Beffara, V., Duminil-Copin, H.: Critical point in planar lattice models. In: Sidoravicius, V., Smirnov, S. (eds) Probability and Statistical Physics in St. Petersburg, Proceedings of Symposia in Pure Mathematics, vol. 91. AMS (2016)Google Scholar
  9. BD12.
    Beffara V., Duminil-Copin H.: Smirnov’s fermionic observable away from criticality. Ann. Probab. 40(6), 2667–2689 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  10. BDS15.
    Beffara, V., Duminil-Copin, H., Smirnov, S.: On the critical parameters of the \({q\geq}\) 4 random cluster model on isoradial graphs. J. Phys. A Math Theoretical 48(48), 484003 (2015). DOI: 10.1088/1751-8113/48/48/484003
  11. BPZ84a.
    Belavin A.A., Polyakov A.M., Zamolodchikov A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241(2), 333–380 (1984)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  12. BPZ84b.
    Belavin A.A., Polyakov A.M., Zamolodchikov A.B.: Infinite conformal symmetry of critical fluctuations in two dimensions. J. Stat. Phys. 34(5–6), 763–774 (1984)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  13. BDH14.
    Benoist, S., Duminil-Copin, H., Hongler, C.: Conformal Invariance of Crossing Probabilities for the Ising Model with Free Boundary Conditions. arXiv:1410.3715 (2014)
  14. BCC06.
    Biskup M., Chayes L., Crawford N.: Mean-field driven first-order phase transitions in systems with long-range interactions. J. Stat. Phys. 122(6), 1139–1193 (2006)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  15. CN07.
    Camia F., Newman C.M.: Critical percolation exploration path and SLE6: a proof of convergence. Probab. Theory Relat. Fields 139(3–4), 473–519 (2007)zbMATHCrossRefGoogle Scholar
  16. CDH16.
    Chelkak, D., Duminil-Copin, H., Hongler, C.: Crossing probabilities in topological rectangles for the critical planar FK-Ising model. Electron. J. Probab. 21(5), 1–28 (2016)Google Scholar
  17. CDH+14.
    Chelkak D., Duminil-Copin H., Hongler C., Kemppainen A., Smirnov S.: Convergence of Ising interfaces to Schramm’s SLE curves. C. R. Acad. Sci. Paris Math. 352(2), 157–161 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  18. CHI15.
    Chelkak, D., Hongler, C., Izyurov, K.: Conformal invariance of spin correlations in the planar Ising model. Ann. Math. (2). 181(3), 1087–1138 (2015)Google Scholar
  19. CI13.
    Chelkak D., Izyurov K.: Holomorphic spinor observables in the critical Ising model. Commun. Math. Phys. 322(2), 303–332 (2013)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  20. CS12.
    Chelkak D., Smirnov S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189(3), 515–580 (2012)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  21. Dum12.
    Duminil-Copin H.: Divergence of the correlation length for critical planar FK percolation with \({1\le q\le 4}\) via parafermionic observables. J. Phys. A: Math. Theor. 45(49), 494013 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  22. Dum13.
    Duminil-Copin, H.: Parafermionic Observables and Their Applications to Planar Statistical Physics Models, Ensaios Matemáticos [Mathematical Surveys], vol. 25, Sociedade Brasileira de Matemática, Rio de Janeiro, p. ii+371 (2013)Google Scholar
  23. Dum15.
    Duminil-Copin, H.: Geometric Representations of Lattice Spin Models. Book, Edition Spartacus (2015)Google Scholar
  24. DGP14.
    Duminil-Copin H., Garban C., Pete G.: The near-critical planar FK-Ising model. Commun. Math. Phys. 326(1), 1–35 (2014)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  25. DHN11.
    Duminil-Copin H., Hongler C., Nolin P.: Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model. Commun. Pure Appl. Math. 64(9), 1165–1198 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  26. DLM15.
    Duminil-Copin, H., Li, J.-H., Manolescu, I.: Universality for Random-Cluster Models on Isoradial Graphs. Preprint (2015)Google Scholar
  27. DM16.
    Duminil-Copin, H., Manolescu, I.: The phase transitions of the planar random-cluster and Potts models with q > 1 are sharp. Probab. Theory Relat. Fields 164(3), 865–892 (2016)Google Scholar
  28. DS12.
    Duminil-Copin, H., Smirnov, S.: The connective constant of the honeycomb lattice equals \({\sqrt{2+\sqrt{2}}}\). Ann. Math. (2). 175(3), 1653–1665 (2012)Google Scholar
  29. DT15a.
    Duminil-Copin H., Tassion V.: A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. Commun. Math. Phys. 343(2), 725–745 (2016)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  30. DT15b.
    Duminil-Copin, H., Tassion, V.: A new proof of the sharpness of the phase transition for Bernoulli percolation on \({\mathbb{Z}^d}\). arXiv:1502.03051 (2015)
  31. FK72.
    Fortuin C.M., Kasteleyn P.W.: On the random-cluster model. I. Introduction and relation to other models. Physica 57, 536–564 (1972)ADSMathSciNetCrossRefGoogle Scholar
  32. FK80.
    Fradkin E., Kadanoff Leo P.: Disorder variables and para-fermions in two-dimensional statistical mechanics. Nucl. Phys. B 170(1), 1–15 (1980)ADSCrossRefGoogle Scholar
  33. GM07.
    Gobron T., Merola I.: First-order phase transition in Potts models with finite-range interactions. J. Stat. Phys. 126(3), 507–583 (2007)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  34. Gri06.
    Geoffrey, G.: The random-cluster model, vol 333., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2006)Google Scholar
  35. GM14.
    Grimmett Geoffrey R., Manolescu I.: Bond percolation on isoradial graphs: criticality and universality. Probab. Theory Relat. Fields 159(1–2), 273–327 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  36. Hon14.
    Hongler, C.: Conformal invariance of Ising model correlations. In: XVIIth International Congress on Mathematical Physics, pp. 326–335. World Sci. Publ., Hackensack, NJ (2014)Google Scholar
  37. HK13.
    Hongler C., Kytölä K.: Ising interfaces and free boundary conditions. J. Am. Math. Soc. 26(4), 1107–1189 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  38. HS11.
    Hongler C., Smirnov S.: Critical percolation: the expected number of clusters in a rectangle. Probab. Theory Relat. Fields 151(3–4), 735–756 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  39. KS12.
    Kemppainen, A., Smirnov, S.: Random curves, scaling limits and loewner evolutions. arXiv:1212.6215 (2012)
  40. Ken00.
    Kenyon R.: Conformal invariance of domino tiling. Ann. Probab. 28(2), 759–795 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  41. Ken01.
    Kenyon R.: Dominos and the Gaussian free field. Ann. Probab. 29(3), 1128–1137 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  42. Kes80.
    Kesten H.: The critical probability of bond percolation on the square lattice equals \({{1\over 2}}\). Commun. Math. Phys. 74(1), 41–59 (1980)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  43. KS82.
    Kotecký R., Shlosman S.B.: First-order phase transitions in large entropy lattice models. Commun. Math. Phys. 83(4), 493–515 (1982)ADSMathSciNetCrossRefGoogle Scholar
  44. LMR86.
    Laanait L., Messager A., Ruiz J.: Phases coexistence and surface tensions for the Potts model. Commun. Math. Phys. 105(4), 527–545 (1986)ADSMathSciNetCrossRefGoogle Scholar
  45. LMMS+91.
    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)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  46. Law05.
    Lawler, G.F.: Conformally Invariant Processes in the Plane, vol. 114. Mathematical Surveys and MonographsAmerican Mathematical Society, Providence, RI (2005)Google Scholar
  47. LSW04.
    Lawler Gregory F., Schramm O., Werner W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32(1B), 939–995 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  48. LS12.
    Lubetzky E., Sly A.: Critical Ising on the square lattice mixes in polynomial time. Commun. Math. Phys. 313(3), 815–836 (2012)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  49. Ons44.
    Onsager L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 2(65), 117–149 (1944)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  50. Pot52.
    Potts, R.B.: Some generalized order-disorder transformations. In: Proceedings of the Cambridge Philosophical Society, vol. 48, pp. 106–109. Cambridge Univ Press, Cambridge (1952)Google Scholar
  51. RC06.
    Riva, V., Cardy, J.: Holomorphic parafermions in the Potts model and stochastic Loewner evolution. J. Stat. Mech. Theory Exp. (12):P12001, p. 19 (electronic) (2006)Google Scholar
  52. Rus78.
    Russo L.: A note on percolation. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 43(1), 39–48 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  53. Sch07.
    Schramm, O.: Conformally invariant scaling limits: an overview and a collection of problems. In: International Congress of Mathematicians. Vol. I, pp. 513–543. Eur. Math. Soc., Zürich (2007)Google Scholar
  54. SW78.
    Seymour, P.D., Welsh, D.J.A.: Percolation probabilities on the square lattice. Ann. Discr. Math. 3, 227–245 (1978). Advances in graph theory (Cambridge Combinatorial Conf., Trinity College, Cambridge, 1977)Google Scholar
  55. Sim80.
    Simon B.: Correlation inequalities and the decay of correlations in ferromagnets. Commun. Math. Phys. 77(2), 111–126 (1980)ADSMathSciNetCrossRefGoogle Scholar
  56. Smi06.
    Smirnov, S.: Towards conformal invariance of 2D lattice models. In: International Congress of Mathematicians. Vol. II, pp. 1421–1451. Eur. Math. Soc., Zürich (2006)Google Scholar
  57. Smi10.
    Smirnov, S.: Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. Math. (2). 172(2), 1435–1467 (2010)Google Scholar
  58. Tas16.
    Tassion, V.: Crossing probabilities for Voronoi percolation. Ann. Probab. 44(5), 3385–3398 (2016)Google Scholar
  59. Wer09.
    Werner, W.: Percolation et modèle d’Ising, volume 16 of Cours Spécialisés [Specialized Courses]. Société Mathématique de France, Paris (2009)Google Scholar
  60. Wu82.
    Wu F.Y.: The Potts model. Rev. Mod. Phys. 54(1), 235–268 (1982)ADSMathSciNetCrossRefGoogle Scholar
  61. Yan52.
    Yang C.N.: The spontaneous magnetization of a two-dimensional Ising model. Phys. Rev. 2(85), 808–816 (1952)ADSzbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Hugo Duminil-Copin
    • 1
    Email author
  • Vladas Sidoravicius
    • 2
  • Vincent Tassion
    • 1
  1. 1.Universite de GenevaGenevaSwitzerland
  2. 2.IMPARio de JaneiroBrazil

Personalised recommendations