Abstract
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:
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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.
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A new result proving the equivalence of several properties of critical random-cluster models:
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the absence of infinite-cluster for wired boundary conditions,
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the uniqueness of infinite-volume measures,
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the sub-exponential decay of the two-point function for free boundary conditions,
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a Russo–Seymour–Welsh type result on crossing probabilities in rectangles with arbitrary boundary conditions.
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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.
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References
Aizenman M., Burchard A.: Hölder regularity and dimension bounds for random curves. Duke Math. J. 99(3), 419–453 (1999)
Aizenman M., Duminil-Copin H., Sidoravicius V.: Random currents and continuity of Ising model’s spontaneous magnetization. Commun. Math. Phys. 334, 719–742 (2015)
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)
Alexander Kenneth S.: On weak mixing in lattice models. Probab. Theory Relat. Fields 110(4), 441–471 (1998)
Baxter R.J.: Generalized ferroelectric model on a square lattice. Stud. Appl. Math. 50, 51–69 (1971)
Baxter R.J.: Potts model at the critical temperature. J. Phys. C: Solid State Phys. 6(23), L445 (1973)
Baxter, R.J.: Exactly solved models in statistical mechanics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London (1989) (Reprint of the 1982 original)
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)
Beffara V., Duminil-Copin H.: Smirnov’s fermionic observable away from criticality. Ann. Probab. 40(6), 2667–2689 (2012)
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
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)
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)
Benoist, S., Duminil-Copin, H., Hongler, C.: Conformal Invariance of Crossing Probabilities for the Ising Model with Free Boundary Conditions. arXiv:1410.3715 (2014)
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)
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)
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)
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)
Chelkak, D., Hongler, C., Izyurov, K.: Conformal invariance of spin correlations in the planar Ising model. Ann. Math. (2). 181(3), 1087–1138 (2015)
Chelkak D., Izyurov K.: Holomorphic spinor observables in the critical Ising model. Commun. Math. Phys. 322(2), 303–332 (2013)
Chelkak D., Smirnov S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189(3), 515–580 (2012)
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)
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)
Duminil-Copin, H.: Geometric Representations of Lattice Spin Models. Book, Edition Spartacus (2015)
Duminil-Copin H., Garban C., Pete G.: The near-critical planar FK-Ising model. Commun. Math. Phys. 326(1), 1–35 (2014)
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)
Duminil-Copin, H., Li, J.-H., Manolescu, I.: Universality for Random-Cluster Models on Isoradial Graphs. Preprint (2015)
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)
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)
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)
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)
Fortuin C.M., Kasteleyn P.W.: On the random-cluster model. I. Introduction and relation to other models. Physica 57, 536–564 (1972)
Fradkin E., Kadanoff Leo P.: Disorder variables and para-fermions in two-dimensional statistical mechanics. Nucl. Phys. B 170(1), 1–15 (1980)
Gobron T., Merola I.: First-order phase transition in Potts models with finite-range interactions. J. Stat. Phys. 126(3), 507–583 (2007)
Geoffrey, G.: The random-cluster model, vol 333., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2006)
Grimmett Geoffrey R., Manolescu I.: Bond percolation on isoradial graphs: criticality and universality. Probab. Theory Relat. Fields 159(1–2), 273–327 (2014)
Hongler, C.: Conformal invariance of Ising model correlations. In: XVIIth International Congress on Mathematical Physics, pp. 326–335. World Sci. Publ., Hackensack, NJ (2014)
Hongler C., Kytölä K.: Ising interfaces and free boundary conditions. J. Am. Math. Soc. 26(4), 1107–1189 (2013)
Hongler C., Smirnov S.: Critical percolation: the expected number of clusters in a rectangle. Probab. Theory Relat. Fields 151(3–4), 735–756 (2011)
Kemppainen, A., Smirnov, S.: Random curves, scaling limits and loewner evolutions. arXiv:1212.6215 (2012)
Kenyon R.: Conformal invariance of domino tiling. Ann. Probab. 28(2), 759–795 (2000)
Kenyon R.: Dominos and the Gaussian free field. Ann. Probab. 29(3), 1128–1137 (2001)
Kesten H.: The critical probability of bond percolation on the square lattice equals \({{1\over 2}}\). Commun. Math. Phys. 74(1), 41–59 (1980)
Kotecký R., Shlosman S.B.: First-order phase transitions in large entropy lattice models. Commun. Math. Phys. 83(4), 493–515 (1982)
Laanait L., Messager A., Ruiz J.: Phases coexistence and surface tensions for the Potts model. Commun. Math. Phys. 105(4), 527–545 (1986)
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)
Lawler, G.F.: Conformally Invariant Processes in the Plane, vol. 114. Mathematical Surveys and MonographsAmerican Mathematical Society, Providence, RI (2005)
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)
Lubetzky E., Sly A.: Critical Ising on the square lattice mixes in polynomial time. Commun. Math. Phys. 313(3), 815–836 (2012)
Onsager L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 2(65), 117–149 (1944)
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)
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)
Russo L.: A note on percolation. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 43(1), 39–48 (1978)
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)
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)
Simon B.: Correlation inequalities and the decay of correlations in ferromagnets. Commun. Math. Phys. 77(2), 111–126 (1980)
Smirnov, S.: Towards conformal invariance of 2D lattice models. In: International Congress of Mathematicians. Vol. II, pp. 1421–1451. Eur. Math. Soc., Zürich (2006)
Smirnov, S.: Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. Math. (2). 172(2), 1435–1467 (2010)
Tassion, V.: Crossing probabilities for Voronoi percolation. Ann. Probab. 44(5), 3385–3398 (2016)
Werner, W.: Percolation et modèle d’Ising, volume 16 of Cours Spécialisés [Specialized Courses]. Société Mathématique de France, Paris (2009)
Wu F.Y.: The Potts model. Rev. Mod. Phys. 54(1), 235–268 (1982)
Yang C.N.: The spontaneous magnetization of a two-dimensional Ising model. Phys. Rev. 2(85), 808–816 (1952)
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Duminil-Copin, H., Sidoravicius, V. & Tassion, V. Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with \({1 \le q \le 4}\) . Commun. Math. Phys. 349, 47–107 (2017). https://doi.org/10.1007/s00220-016-2759-8
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DOI: https://doi.org/10.1007/s00220-016-2759-8