Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with 1≤q≤4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt}

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 Z 2 is continuous for q ∈ {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 ≥ 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 ≤ q ≤ 4, which is derived studying parafermionic observables on a discrete Riemann surface. • A new result proving the equivalence of several properties of critical randomcluster models:

[1]  L. Onsager.  Crystal statistics. I. A two-dimensional model with an order-disorder transition , 1944 .

[2]  R. B. Potts.  Some generalized order-disorder transformations , 1952, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  Chen Ning Yang,et al.  The Spontaneous Magnetization of a Two-Dimensional Ising Model , 1952 .

[4]  R. Baxter.  Generalized Ferroelectric Model on a Square Lattice , 1971 .

[5]  C. Fortuin,et al.  On the random-cluster model: I. Introduction and relation to other models , 1972 .

[6]  R. Baxter.  Potts model at the critical temperature , 1973 .

[7]  L. Russo.  A note on percolation , 1978 .

[8]  R. Baxter,et al.  Solvable eight-vertex model on an arbitrary planar lattice , 1978, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[9]  D. Welsh,et al.  Percolation probabilities on the square lattice , 1978 .

[10]  L. Kadanoff,et al.  Disorder variables and para-fermions in two-dimensional statistical mechanics , 1980 .

[11]  Barry Simon,et al.  Correlation inequalities and the decay of correlations in ferromagnets , 1980 .

[12]  Senya Shlosman,et al.  First-order phase transitions in large entropy lattice models , 1982 .

[13]  R. Baxter.  Exactly solved models in statistical mechanics , 1982 .

[14]  F. Y. Wu.  The Potts model , 1982 .

[15]  A. Polyakov,et al.  Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory - Nucl. Phys. B241, 333 (1984) , 1984 .

[16]  Alexander M. Polyakov,et al.  Infinite conformal symmetry of critical fluctuations in two dimensions , 1984 .

[17]  Jean Ruiz,et al.  Phases coexistence and surface tensions for the potts model , 1986 .

[18]  Michael Aizenman,et al.  On the critical behavior of the magnetization in high-dimensional Ising models , 1986 .

[19]  A. Sokal,et al.  Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. , 1988, Physical review. D, Particles and fields.

[20]  Lahoussine Laanait,et al.  Interfaces in the Potts model I: Pirogov-Sinai theory of the Fortuin-Kasteleyn representation , 1991 .

[21]  Almut Burchard,et al.  Holder Regularity and Dimension Bounds for Random Curves , 1998 .

[22]  K. Alexander.  On weak mixing in lattice models , 1998 .

[23]  Richard Kenyon,et al.  Conformal invariance of domino tiling , 1999 .

[24]  R. Kenyon,et al.  Dominos and the Gaussian Free Field , 2000, math-ph/0002027.

[25]  Thermodynamic limit of the six-vertex model with domain wall boundary conditions , 2000, cond-mat/0004250.

[26]  Wendelin Werner,et al.  Conformal invariance of planar loop-erased random walks and uniform spanning trees , 2001 .

[27]  Nicholas Crawford,et al.  Mean-Field Driven First-Order Phase Transitions in Systems with Long-Range Interactions , 2005, math-ph/0501067.

[28]  C. Newman,et al.  Critical percolation exploration path and SLE6: a proof of convergence , 2006, math/0604487.

[29]  Stanislav Smirnov,et al.  Towards conformal invariance of 2D lattice models , 2007, 0708.0032.

[30]  J. Cardy,et al.  Holomorphic parafermions in the Potts model and stochastic Loewner evolution , 2006 .

[31]  Conformally invariant scaling limits: an overview and a collection of problems , 2006, math/0602151.

[32]  Geoffrey Grimmett.  The Random-Cluster Model , 2002, math/0205237.

[33]  S. Smirnov.  Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model , 2007, 0708.0039.

[34]  S. Smirnov,et al.  Critical percolation: the expected number of clusters in a rectangle , 2009, 0909.4490.

[35]  Percolation et modèle d'ising , 2009 .

[36]  S. Smirnov,et al.  Universality in the 2D Ising model and conformal invariance of fermionic observables , 2009, 0910.2045.

[37]  N. Reshetikhin.  Lectures on the integrability of the 6-vertex model , 2010, 1010.5031.

[38]  Allan Sly,et al.  Critical Ising on the Square Lattice Mixes in Polynomial Time , 2010, 1001.1613.

[39]  C. Hongler,et al.  Ising interfaces and free boundary conditions , 2011, 1108.0643.

[40]  Pierre Nolin,et al.  Connection probabilities and RSW‐type bounds for the two‐dimensional FK Ising model , 2011 .

[41]  Holomorphic Spinor Observables in the Critical Ising Model , 2011, 1105.5709.

[42]  H. Duminil-Copin,et al.  The Near-Critical Planar FK-Ising Model , 2011, 1111.0144.

[43]  H. Duminil-Copin,et al.  Smirnov's fermionic observable away from criticality , 2010, 1010.0526.

[44]  H. Duminil-Copin.  Divergence of the correlation length for critical planar FK percolation with 1 ⩽ q ⩽ 4 via parafermionic observables , 2012, 1208.3787.

[45]  C. Hongler.  Conformal Invariance of Ising Model Correlations , 2012 .

[46]  G. Grimmett,et al.  Bond percolation on isoradial graphs: criticality and universality , 2012, 1204.0505.

[47]  Dmitry Chelkak,et al.  Conformal invariance of spin correlations in the planar Ising model , 2012, 1202.2838.

[48]  H. Duminil-Copin,et al.  Convergence of Ising interfaces to Schramm's SLE curves , 2013, 1312.0533.

[49]  H. Duminil-Copin,et al.  Crossing probabilities in topological rectangles for the critical planar FK-Ising model , 2013, 1312.7785.

[50]  H. Duminil-Copin,et al.  Critical point and duality in planar lattice models , 2014 .

[51]  The phase transitions of the planar random-cluster and Potts models with q larger than 1 are sharp , 2014 .

[52]  H. Duminil-Copin,et al.  A New Proof of the Sharpness of the Phase Transition for Bernoulli Percolation and the Ising Model , 2015, Communications in Mathematical Physics.

[53]  H. Duminil-Copin,et al.  On the critical parameters of the q ≤ 4 random-cluster model on isoradial graphs , 2015, 1507.01356.

[54]  Vladas Sidoravicius,et al.  Random Currents and Continuity of Ising Model’s Spontaneous Magnetization , 2013, 1311.1937.

[55]  H. Duminil-Copin,et al.  A new proof of the sharpness of the phase transition for Bernoulli percolation on $\mathbb Z^d$ , 2015, 1502.03051.

[56]  K. Kozlowski.  On Condensation Properties of Bethe Roots Associated with the XXZ Chain , 2015, 1508.05741.

[57]  H. Duminil-Copin,et al.  A new computation of the critical point for the planar random-cluster model with $q\ge1$ , 2016, 1604.03702.

[58]  Conformal invariance of crossing probabilities for the Ising model with free boundary conditions , 2014, 1410.3715.

[59]  H. Duminil-Copin,et al.  The Bethe ansatz for the six-vertex and XXZ models: An exposition , 2016, 1611.09909.