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Continuity of the phase transition for planar random-cluster and Potts models with \({1 \leq q \leq 4}\). (English) Zbl 1357.82011
Summary: 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 \({q \geq 1}\) (note that \(q\) is not necessarily an integer) and is based on two ingredients:
\(\bullet\) The fact that the two-point function for the free state decays sub-exponentially fast for cluster-weights \({1\leq q\leq 4}\), which is derived studying parafermionic observables on a discrete Riemann surface.
\(\bullet\) 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.

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B26 Phase transitions (general) in equilibrium statistical mechanics
82D40 Statistical mechanical studies of magnetic materials
82B27 Critical phenomena in equilibrium statistical mechanics
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References:
[1] Aizenman, M.; Burchard, A., Hölder regularity and dimension bounds for random curves, Duke Math. J., 99, 419-453, (1999) · Zbl 0944.60022
[2] Aizenman, M.; Duminil-Copin, H.; Sidoravicius, V., Random currents and continuity of Ising model’s spontaneous magnetization, Commun. Math. Phys., 334, 719-742, (2015) · Zbl 1315.82004
[3] Aizenman, M.; Fernández, R., On the critical behavior of the magnetization in high-dimensional Ising models, J. Stat. Phys., 44, 393-454, (1986) · Zbl 0629.60106
[4] Alexander Kenneth, S., On weak mixing in lattice models, Probab. Theory Relat. Fields, 110, 441-471, (1998) · Zbl 0906.60074
[5] Baxter, R.J., Generalized ferroelectric model on a square lattice, Stud. Appl. Math., 50, 51-69, (1971)
[6] Baxter, R.J., Potts model at the critical temperature, J. Phys. C: Solid State Phys., 6, l445, (1973)
[7] Baxter, R.J.: Exactly solved models in statistical mechanics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London (1989) (Reprint of the 1982 original) · Zbl 0723.60120
[8] 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) · Zbl 1257.82014
[9] Beffara, V.; Duminil-Copin, H., Smirnov’s fermionic observable away from criticality, Ann. Probab., 40, 2667-2689, (2012) · Zbl 1339.60136
[10] Beffara, V., Duminil-Copin, H., Smirnov, S.: On the critical parameters of the \({q≥}\) 4 random cluster model on isoradial graphs. J. Phys. A Math Theoretical 48(48), 484003 (2015). DOI:10.1088/1751-8113/48/48/484003 · Zbl 1342.82018
[11] Belavin, A.A.; Polyakov, A.M.; Zamolodchikov, A.B., Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B, 241, 333-380, (1984) · Zbl 0661.17013
[12] Belavin, A.A.; Polyakov, A.M.; Zamolodchikov, A.B., Infinite conformal symmetry of critical fluctuations in two dimensions, J. Stat. Phys., 34, 763-774, (1984) · Zbl 0661.17013
[13] Benoist, S., Duminil-Copin, H., Hongler, C.: Conformal Invariance of Crossing Probabilities for the Ising Model with Free Boundary Conditions. arXiv:1410.3715 (2014) · Zbl 1355.60119
[14] Biskup, M.; Chayes, L.; Crawford, N., Mean-field driven first-order phase transitions in systems with long-range interactions, J. Stat. Phys., 122, 1139-1193, (2006) · Zbl 1142.82329
[15] Camia, F.; Newman, C.M., Critical percolation exploration path and SLE_{6}: a proof of convergence, Probab. Theory Relat. Fields, 139, 473-519, (2007) · Zbl 1126.82007
[16] 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) · Zbl 1341.60124
[17] 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, 157-161, (2014) · Zbl 1294.82007
[18] Chelkak, D., Hongler, C., Izyurov, K.: Conformal invariance of spin correlations in the planar Ising model. Ann. Math. (2). 181(3), 1087-1138 (2015) · Zbl 1318.82006
[19] Chelkak, D.; Izyurov, K., Holomorphic spinor observables in the critical Ising model, Commun. Math. Phys., 322, 303-332, (2013) · Zbl 1277.82010
[20] Chelkak, D.; Smirnov, S., Universality in the 2D Ising model and conformal invariance of fermionic observables, Invent. Math., 189, 515-580, (2012) · Zbl 1257.82020
[21] 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, 494013, (2012) · Zbl 1257.82051
[22] 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) · Zbl 1298.82001
[23] Duminil-Copin, H.: Geometric Representations of Lattice Spin Models. Book, Edition Spartacus (2015) · Zbl 1315.82004
[24] Duminil-Copin, H.; Garban, C.; Pete, G., The near-critical planar FK-Ising model, Commun. Math. Phys., 326, 1-35, (2014) · Zbl 1286.82003
[25] 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, 1165-1198, (2011) · Zbl 1227.82015
[26] Duminil-Copin, H., Li, J.-H., Manolescu, I.: Universality for Random-Cluster Models on Isoradial Graphs. Preprint (2015) · Zbl 1414.60076
[27] 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) · Zbl 1356.60167
[28] 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) · Zbl 1253.82012
[29] 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, 725-745, (2016) · Zbl 1342.82026
[30] 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) · Zbl 1359.60118
[31] Fortuin, C.M.; Kasteleyn, P.W., On the random-cluster model. I. introduction and relation to other models, Physica, 57, 536-564, (1972)
[32] Fradkin, E.; Kadanoff Leo, P., Disorder variables and para-fermions in two-dimensional statistical mechanics, Nucl. Phys. B, 170, 1-15, (1980)
[33] Gobron, T.; Merola, I., First-order phase transition in Potts models with finite-range interactions, J. Stat. Phys., 126, 507-583, (2007) · Zbl 1114.82007
[34] Geoffrey, G.: The random-cluster model, vol 333., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2006) · Zbl 1122.60087
[35] Grimmett Geoffrey, R.; Manolescu, I., Bond percolation on isoradial graphs: criticality and universality, Probab. Theory Relat. Fields, 159, 273-327, (2014) · Zbl 1296.60263
[36] Hongler, C.: Conformal invariance of Ising model correlations. In: XVIIth International Congress on Mathematical Physics, pp. 326-335. World Sci. Publ., Hackensack, NJ (2014) · Zbl 1304.82013
[37] Hongler, C.; Kytölä, K., Ising interfaces and free boundary conditions, J. Am. Math. Soc., 26, 1107-1189, (2013) · Zbl 1284.82021
[38] Hongler, C.; Smirnov, S., Critical percolation: the expected number of clusters in a rectangle, Probab. Theory Relat. Fields, 151, 735-756, (2011) · Zbl 1237.60077
[39] Kemppainen, A., Smirnov, S.: Random curves, scaling limits and loewner evolutions. arXiv:1212.6215 (2012) · Zbl 1393.60016
[40] Kenyon, R., Conformal invariance of domino tiling, Ann. Probab., 28, 759-795, (2000) · Zbl 1043.52014
[41] Kenyon, R., Dominos and the Gaussian free field, Ann. Probab., 29, 1128-1137, (2001) · Zbl 1034.82021
[42] Kesten, H., The critical probability of bond percolation on the square lattice equals \({{1\over 2}}\), Commun. Math. Phys., 74, 41-59, (1980) · Zbl 0441.60010
[43] Kotecký, R.; Shlosman, S.B., First-order phase transitions in large entropy lattice models, Commun. Math. Phys., 83, 493-515, (1982)
[44] Laanait, L.; Messager, A.; Ruiz, J., Phases coexistence and surface tensions for the Potts model, Commun. Math. Phys., 105, 527-545, (1986)
[45] 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, 81-91, (1991) · Zbl 0734.60108
[46] Lawler, G.F.: Conformally Invariant Processes in the Plane, vol. 114. Mathematical Surveys and MonographsAmerican Mathematical Society, Providence, RI (2005) · Zbl 1074.60002
[47] Lawler Gregory, F.; Schramm, O.; Werner, W., Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab., 32, 939-995, (2004) · Zbl 1126.82011
[48] Lubetzky, E.; Sly, A., Critical Ising on the square lattice mixes in polynomial time, Commun. Math. Phys., 313, 815-836, (2012) · Zbl 1250.82008
[49] Onsager, L., Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev., 2, 117-149, (1944) · Zbl 0060.46001
[50] 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) · Zbl 0048.45601
[51] 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) · Zbl 07078047
[52] Russo, L., A note on percolation, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 43, 39-48, (1978) · Zbl 0363.60120
[53] 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) · Zbl 1131.60088
[54] 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) · Zbl 0405.60015
[55] Simon, B., Correlation inequalities and the decay of correlations in ferromagnets, Commun. Math. Phys., 77, 111-126, (1980)
[56] Smirnov, S.: Towards conformal invariance of 2D lattice models. In: International Congress of Mathematicians. Vol. II, pp. 1421-1451. Eur. Math. Soc., Zürich (2006) · Zbl 1112.82014
[57] Smirnov, S.: Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. Math. (2). 172(2), 1435-1467 (2010) · Zbl 1200.82011
[58] Tassion, V.: Crossing probabilities for Voronoi percolation. Ann. Probab. 44(5), 3385-3398 (2016) · Zbl 1352.60130
[59] Werner, W.: Percolation et modèle d’Ising, volume 16 of Cours Spécialisés [Specialized Courses]. Société Mathématique de France, Paris (2009)
[60] Wu, F.Y., The Potts model, Rev. Mod. Phys., 54, 235-268, (1982)
[61] Yang, C.N., The spontaneous magnetization of a two-dimensional Ising model, Phys. Rev., 2, 808-816, (1952) · Zbl 0046.45304
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