Communications in Mathematical Physics

, Volume 367, Issue 3, pp 771–833 | Cite as

Ising Model: Local Spin Correlations and Conformal Invariance

  • Reza GheissariEmail author
  • Clément Hongler
  • S. C. Park
Open Access


We study the 2-dimensional Ising model at critical temperature on a simply connected subset \({\Omega_{\delta}}\) of the square grid \({\delta\mathbb{Z}^{2}}\). The scaling limit of the critical Ising model is conjectured to be described by Conformal Field Theory; in particular, there is expected to be a precise correspondence between local lattice fields of the Ising model and the local fields of Conformal Field Theory. Towards the proof of this correspondence, we analyze arbitrary spin pattern probabilities (probabilities of finite spin configurations occurring at the origin), explicitly obtain their infinite-volume limits, and prove their conformal covariance at the first (non-trivial) order. We formulate these probabilities in terms of discrete fermionic observables, enabling the study of their scaling limits. This generalizes results of Hongler (Conformal invariance of Ising model correlations. Ph.D. thesis, [Hon10]), Hongler and Smirnov (Acta Math 211(2):191–225, [HoSm13]), Chelkak, Hongler, and Izyurov (Ann. Math. 181(3), 1087–1138, [CHI15]) to one-point functions of any local spin correlations. We introduce a collection of tools which allow one to exactly and explicitly translate any spin pattern probability (and hence any lattice local field correlation) in terms of discrete complex analysis quantities. The proof requires working with multipoint lattice spinors with monodromy (including construction of explicit formulae in the full plane), and refined analysis near their source points to prove convergence to the appropriate continuous conformally covariant functions.



Firstly, the authors thank the anonymous referees for their many helpful suggestions and comments. This research was initiated during the Research Experience for Undergraduates program at Mathematics Department of Columbia University, funded by the NSF under grant DMS-0739392. We would like to thank the T.A., Krzysztof Putyra, for his help during this program and the program coordinator Robert Lipshitz, as well as all the participants, in particular Adrien Brochard and Woo Chang Chung. R.G. would like to thank Chuck Newman and Eyal Lubetzky for interesting discussions. C.H. would like to thank Dmitry Chelkak and Stanislav Smirnov, for sharing many ideas and insights about the Ising model and conformal invariance; Itai Benjamini and Curtis McMullen for asking questions that suggested we look at this problem; Stéphane Benoist, John Cardy, Julien Dubédat, Hugo Duminil-Copin, Konstantin Izyurov, Kalle Kytölä and Wendelin Werner for interesting discussions, the NSF under grant DMS-1106588, the ERC under grant SG CONSTAMIS, the NCCR Swissmap, the Blavatnik family foundation, the Latsis family foundation, and the Minerva Foundation for financial support. S.P. would like to thank C.M. Munteanu for interesting discussions.


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Corrected Publication March/2019

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Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Ecole Polytechnique Fédérale de LausanneEPFL SB MATHAA CSFTLausanneSwitzerland

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