Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with 1≤q≤4documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt}
Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with 1≤q≤4documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt}
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1≤q≤4documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{ 平面随机簇和 Potts 模型相变的连续性amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt}
DOI：
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发表时间：
2016
影响因子：
2.4
通讯作者：
V. Tassion
中科院分区：
文献类型：
--
作者：
H. Duminil;V. Sidoravicius;V. 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 Z2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathbb{Z}^2}$$end{document} is continuous for q∈{2,3,4}documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${q in {2,3,4}}$$end{document}, 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≥1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${q ge 1}$$end{document} (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≤4documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${1le qle 4}$$end{document}, 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 fact that the two-point function for the free state decays sub-exponentially fast for cluster-weights 1≤q≤4documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${1le qle 4}$$end{document}, 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 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∈[1,4]documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${q in [1,4]}$$end{document}. 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.
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 Z2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${mathbb{Z}^2}$$end{document} is continuous for q∈{2,3,4}documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${q in {2,3,4}}$$end{document}, 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≥1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${q ge 1}$$end{document} (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≤4documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${1le qle 4}$$end{document}, 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 fact that the two-point function for the free state decays sub-exponentially fast for cluster-weights 1≤q≤4documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${1le qle 4}$$end{document}, 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 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∈[1,4]documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${q in [1,4]}$$end{document}. 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.