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Universality in the 2D Ising model and conformal invariance of fermionic observables

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It is widely believed that the celebrated 2D Ising model at criticality has a universal and conformally invariant scaling limit, which is used in deriving many of its properties. However, no mathematical proof has ever been given, and even physics arguments support (a priori weaker) Möbius invariance. We introduce discrete holomorphic fermions for the 2D Ising model at criticality on a large family of planar graphs. We show that on bounded domains with appropriate boundary conditions, those have universal and conformally invariant scaling limits, thus proving the universality and conformal invariance conjectures.

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Authors and Affiliations

  1. St.Petersburg Department of Steklov Mathematical Institute (PDMI RAS), Fontanka 27, 191023, St. Petersburg, Russia

    Dmitry Chelkak

  2. Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, Case postale 64, 1211, Genève 4, Switzerland

    Stanislav Smirnov

  3. Chebyshev Laboratory, Department of Mathematics and Mechanics, Saint-Petersburg State University, 14th Line, 29b, 199178, Saint-Petersburg, Russia

    Dmitry Chelkak & Stanislav Smirnov

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  1. Dmitry Chelkak
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  2. Stanislav Smirnov
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Correspondence to Dmitry Chelkak.

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Chelkak, D., Smirnov, S. Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. math. 189, 515–580 (2012). https://doi.org/10.1007/s00222-011-0371-2

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