Conformal invariance of spin correlations in the planar Ising model. (English) Zbl 1318.82006

Summary: We rigorously prove the existence and the conformal invariance of scaling limits of the magnetization and multi-point spin correlations in the critical Ising model on arbitrary simply connected planar domains. This solves a number of conjectures coming from the physical and the mathematical literature. The proof relies on convergence results for discrete holomorphic spinor observables and probabilistic techniques.


82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82D40 Statistical mechanics of magnetic materials
82B27 Critical phenomena in equilibrium statistical mechanics
Full Text: DOI arXiv


[1] A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, ”Infinite conformal symmetry in two-dimensional quantum field theory,” Nuclear Phys. B, vol. 241, iss. 2, pp. 333-380, 1984. · Zbl 0661.17013
[2] A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, ”Infinite conformal symmetry of critical fluctuations in two dimensions,” J. Statist. Phys., vol. 34, iss. 5-6, pp. 763-774, 1984. · Zbl 0661.17013
[3] C. Boutillier and B. de Tilière, ”The critical \(Z\)-invariant Ising model via dimers: the periodic case,” Probab. Theory Related Fields, vol. 147, iss. 3-4, pp. 379-413, 2010. · Zbl 1195.82011
[4] C. Boutillier and B. de Tilière, ”The critical \(Z\)-invariant Ising model via dimers: locality property,” Comm. Math. Phys., vol. 301, iss. 2, pp. 473-516, 2011. · Zbl 1245.05027
[5] T. W. Burkhardt and I. Guim, ”Bulk, surface, and interface properties of the Ising model and conformal invariance,” Phys. Rev. B, vol. 36, iss. 4, pp. 2080-2083, 1987.
[6] T. W. Burkhardt and I. Guim, ”Conformal theory of the two-dimensional Ising model with homogeneous boundary conditions and with disordered boundary fields,” Phys. Rev. B, vol. 47, pp. 14306-14311, 1993.
[7] F. Camia, C. Garban, and C. M. Newman, ”Planar Ising magnetization field I. Uniqueness of the critical scaling limit,” Ann. Probab., vol. 43, iss. 2, pp. 528-571, 2015. · Zbl 1332.82012
[8] J. L. Cardy, ”Conformal invariance and surface critical behavior,” Nucl. Phys. B, vol. 240, pp. 514-532, 1984.
[9] D. Chelkak and C. Hongler.
[10] D. Chelkak and K. Izyurov, ”Holomorphic spinor observables in the critical Ising model,” Comm. Math. Phys., vol. 322, iss. 2, pp. 303-332, 2013. · Zbl 1277.82010
[11] D. Chelkak and S. Smirnov, ”Discrete complex analysis on isoradial graphs,” Adv. Math., vol. 228, iss. 3, pp. 1590-1630, 2011. · Zbl 1227.31011
[12] D. Chelkak, H. Duminil-Copin, C. Hongler, A. Kemppainen, and S. Smirnov, ”Convergence of Ising interfaces to Schramm’s SLE curves,” C. R. Acad. Sci. Paris, Ser. I, vol. 352, pp. 157-161, 2014. · Zbl 1294.82007
[13] D. Chelkak and S. Smirnov, ”Universality in the 2D Ising model and conformal invariance of fermionic observables,” Invent. Math., vol. 189, iss. 3, pp. 515-580, 2012. · Zbl 1257.82020
[14] A. Dembo and A. Montanari, ”Ising models on locally tree-like graphs,” Ann. Appl. Probab., vol. 20, iss. 2, pp. 565-592, 2010. · Zbl 1191.82025
[15] P. Di Francesco, H. Saleur, and J. -B. Zuber, ”Critical Ising correlation functions in the plane and on the torus,” Nuclear Phys. B, vol. 290, iss. 4, pp. 527-581, 1987.
[16] J. Dubédat, Dimers and analytic torsion I, 2011.
[17] J. Dubédat, Exact bosonization of the Ising model, 2011.
[18] H. Duminil-Copin, C. Hongler, and P. Nolin, ”Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model,” Comm. Pure Appl. Math., vol. 64, iss. 9, pp. 1165-1198, 2011. · Zbl 1227.82015
[19] R. Gheissari, C. Hongler, and S. Park, Conformal Invariance of Spin Pattern Probabilities in the Planar Ising Model, 2013.
[20] R. B. Griffiths, C. A. Hurst, and S. Sherman, ”Concavity of magnetization of an Ising ferromagnet in a positive external field,” J. Mathematical Phys., vol. 11, pp. 790-795, 1970.
[21] G. Grimmett, The Random-Cluster Model, New York: Springer-Verlag, 2006, vol. 333. · Zbl 1122.60087
[22] C. Hongler, Conformal invariance of Ising model correlations, 2010. · Zbl 1304.82013
[23] C. Hongler and S. Smirnov, ”The energy density in the planar Ising model,” Acta Math., vol. 211, iss. 2, pp. 191-225, 2013. · Zbl 1287.82007
[24] C. Hongler, ”Conformal invariance of Ising model correlations,” in XVIIth International Congress on Mathematical Physics, Hackensack, NJ: World Scientific Publ., 2014, pp. 326-335. · Zbl 1304.82013
[25] K. Izyurov, Smirnov’s observable for free boundary conditions, interfaces and crossing probabilities, 2014. · Zbl 1318.82010
[26] L. P. Kadanoff and H. Ceva, ”Determination of an operator algebra for the two-dimensional Ising model,” Phys. Rev. B, vol. 3, pp. 3918-3939, 1971.
[27] H. Kesten, ”Hitting probabilities of random walks on \({\mathbf Z}^d\),” Stochastic Process. Appl., vol. 25, iss. 2, pp. 165-184, 1987. · Zbl 0626.60067
[28] H. A. Kramers and G. H. Wannier, ”Statistics of the two-dimensional ferromagnet. Part I,” Phys. Rev., vol. 60, pp. 252-262, 1941. · Zbl 0027.28505
[29] G. F. Lawler and V. Limic, ”The Beurling estimate for a class of random walks,” Electron. J. Probab., vol. 9, p. no. 27, 846-861, 2004. · Zbl 1063.60066
[30] B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model, Cambridge, MA: Harvard Univ. Press, 1973. · Zbl 1094.82500
[31] L. Onsager, ”Crystal statistics. I. A two-dimensional model with an order-disorder transition,” Phys. Rev., vol. 65, pp. 117-149, 1944. · Zbl 0060.46001
[32] J. Palmer, Planar Ising Correlations, Boston: Birkhäuser, 2007, vol. 49. · Zbl 1136.82001
[33] J. Palmer and C. Tracy, ”Two-dimensional Ising correlations: the SMJ analysis,” Adv. in Appl. Math., vol. 4, iss. 1, pp. 46-102, 1983. · Zbl 0542.46044
[34] H. Pinson, ”Rotational invariance of the 2d spin-spin correlation function,” Comm. Math. Phys., vol. 314, iss. 3, pp. 807-816, 2012. · Zbl 1251.82022
[35] M. Sato, T. Miwa, and M. Jimbo, ”Studies on holonomic quantum fields. I,” Proc. Japan Acad. Ser. A Math. Sci., vol. 53, iss. 1, pp. 6-10, 1977. · Zbl 0383.35066
[36] M. Sato, T. Miwa, and M. Jimbo, ”Holonomic quantum fields. III,” Publ. Res. Inst. Math. Sci., vol. 15, iss. 2, pp. 577-629, 1979. · Zbl 0436.35076
[37] M. Sato, T. Miwa, and M. Jimbo, ”Holonomic quantum fields. IV,” Publ. Res. Inst. Math. Sci., vol. 15, iss. 3, pp. 871-972, 1979. · Zbl 0436.35077
[38] M. Sato, T. Miwa, and M. Jimbo, ”Holonomic quantum fields. V,” Publ. Res. Inst. Math. Sci., vol. 16, iss. 2, pp. 531-584, 1980. · Zbl 0479.35072
[39] S. Smirnov, ”Towards conformal invariance of 2D lattice models,” in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1421-1451. · Zbl 1112.82014
[40] S. Smirnov, ”Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model,” Ann. of Math., vol. 172, iss. 2, pp. 1435-1467, 2010. · Zbl 1200.82011
[41] C. N. Yang, ”The spontaneous magnetization of a two-dimensional Ising model,” Physical Rev., vol. 85, pp. 808-816, 1952. · Zbl 0046.45304
[42] T. T. Wu, B. M. McCoy, C. A. Tracy, and E. Barouch, ”Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region,” Phys. Rev. B, vol. 13, pp. 316-374, 1976.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.