Inventiones mathematicae

, Volume 189, Issue 3, pp 515–580

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


DOI: 10.1007/s00222-011-0371-2

Cite this article as:
Chelkak, D. & Smirnov, S. Invent. math. (2012) 189: 515. doi:10.1007/s00222-011-0371-2


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.

Mathematics Subject Classification

82B20 60K35 30C35 81T40 

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.St.Petersburg Department of Steklov Mathematical Institute (PDMI RAS)St. PetersburgRussia
  2. 2.Section de MathématiquesUniversité de GenèveGenève 4Switzerland
  3. 3.Chebyshev Laboratory, Department of Mathematics and MechanicsSaint-Petersburg State UniversitySaint-PetersburgRussia

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