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The Ising magnetization exponent on \(\mathbb{Z }^2\) is \(1/15\)


We prove that for the Ising model defined on the plane \(\mathbb Z ^2\) at \(\beta \,{=}\,\beta _c,\) the average magnetization under an external magnetic field \(h>0\) behaves exactly like

$$\begin{aligned} \langle \sigma _0\rangle _{\beta _c, h} \asymp h^{\frac{1}{15}}. \end{aligned}$$

The proof, which is surprisingly simple compared to an analogous result for percolation [i.e. that \(\theta (p)=(p-p_c)^{5/36+o(1)}\) on the triangular lattice (Kesten in Commun Math Phys 109(1):109–156, 1987; Smirnov and Werner in Math Res Lett 8(5–6):729–744, 2001)] relies on the GHS inequality as well as the RSW theorem for FK percolation from Duminil-Copin et al. (Commun Pure Appl Math 64:1165–1198, 2011). The use of GHS to obtain inequalities involving critical exponents is not new; in this paper we show how it can be combined with RSW to obtain matching upper and lower bounds for the average magnetization.

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  1. 1.

    In this paper \(f(a) \asymp g(a)\) as \(a \searrow 0\) means that \(f(a)/g(a)\) is bounded away from zero and \(\infty \) while \(f(a) \sim g(a)\) means that \(f(a)/g(a) \rightarrow 1\) as \(a \searrow 0.\)


  1. 1.

    Aizenman, M., Barsky, D.J., Fernández, R.: The phase transition in a general class of Ising-type models is sharp. J. Stat. Phys. 47, 343–374 (1987)

  2. 2.

    Baxter, R.J.: Onsager and Kaufman’s calculation of the spontaneous magnetization of the Ising model: II. arXiv:1211.2665 (2012)

  3. 3.

    Beffara, V., Duminil-Copin, H.: Smirnov’s fermionic observable away from criticality. Ann. Probab. 40, 2667–2689 (2012)

  4. 4.

    Camia, F.: Towards conformal invariance and a geometric representation of the 2D Ising magnetization field. Markov Process. Relat. Fields 18, 89–110 (2012)

  5. 5.

    Camia, F., Garban, C., Newman, C.M.: Planar Ising magnetization field I. Uniqueness of the critical scaling limit. arXiv:1205.6610 (2012)

  6. 6.

    Camia, F., Garban, C., Newman, C.M.: Planar Ising magnetization field II. Properties of the critical and near-critical scaling limits. arXiv:1307.3926 (2013)

  7. 7.

    Camia, F., Newman, C.M.: Ising (conformal) fields and cluster area measures. Proc. Natl. Acad. Sci. USA 106(14), 5457–5463 (2009)

  8. 8.

    Chayes, J.T., Puha, A.L., Sweet, T.: Independent and dependent percolation. In: Probability Theory and Applications (Princeton, NJ, 1996). IAS/Park City Mathematical Series, vol. 6, pp. 49–166. American Mathematical Society, Providence (1999)

  9. 9.

    Deift, P., Its, A., Krasovsky, I.: Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model. Some history and some recent results. arXiv1207.4990 (2012)

  10. 10.

    Duminil-Copin, H., Hongler, C., Nolin, P.: Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model. Commun. Pure Appl. Math. 64, 1165–1198 (2011)

  11. 11.

    Fernández, R., Fröhlich, J., Sokal, A.D.: Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory. Springer, Berlin (1992)

  12. 12.

    Fisher, M.E.: Rigorous inequalities for critical-point correlation exponents. Phys. Rev. 180, 594–600 (1969)

  13. 13.

    Garban, C., Pete, G., Schramm, O.: The Fourier spectrum of critical percolation. Acta Math. 205(1), 19–104 (2010)

  14. 14.

    Griffiths, R.B.: Correlations in Ising ferromagnets. II. External magnetic fields. J. Math. Phys. 8, 484–489 (1967)

  15. 15.

    Griffiths, R.B., Hurst, C.A., Sherman, S.: Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys. 11, 790–795 (1970)

  16. 16.

    Grimmett, G.: The Random-Cluster Model, vol. 333. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (2006)

  17. 17.

    Gunton, J.D., Buckingham, M.J.: Behavior of the correlation function near the critical point. Phys. Rev. Lett. 20, 143–146 (1968)

  18. 18.

    Huang, K.: Statistical Mechanics. Wiley, New York (1987)

  19. 19.

    Kesten, H.: Scaling relations for 2D-percolation. Commun. Math. Phys. 109(1), 109–156 (1987)

  20. 20.

    McCoy, B.: The romance of the Ising model. In: Iohara, K., Morier-Genoud, S., Rémy, B.(eds.) Symmetries, Integrable Systems and Representations, pp. 263–295. Springer, London (2013)

  21. 21.

    McCoy, B.M., Wu, T.T.: The Two-Dimensional Ising Model. University Press, Cambridge (1973)

  22. 22.

    Newman, C.M.: Critical point inequalities and scaling limits. Commun. Math. Phys. 66, 181–196 (1979)

  23. 23.

    Newman, C.M.: Percolation theory: a selective survey of rigorous results. In: Papanicolaou, G. (ed) Advances in Multiphase Flow and Related Problems, pp. 163–167. SIAM, Philadelphia (1986)

  24. 24.

    Onsager, L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. (2) 65, 117–149 (1944)

  25. 25.

    Palmer, J.: Planar Ising Correlations. Birkhäuser, Boston (2007)

  26. 26.

    Smirnov, S., Werner, W.: Critical exponents for two-dimensional percolation. Math. Res. Lett. 8(5–6), 729–744 (2001)

  27. 27.

    Wu, T.T.: Theory of Toeplitz determinants and the spin correlations of the two-dimensional Ising model. I. Phys. Rev. 149, 380–401 (1966)

  28. 28.

    Yang, C.N.: The spontaneous magnetization of a two-dimensional Ising model. Phys. Rev. 85, 808–816 (1952)

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The authors thank Douglas Abraham, Hugo Duminil-Copin and Roberto Fernández for useful discussions, and an anonymous referee for useful comments. They also thank Barry McCoy for correcting a misattribution of Theorem 1.2.

Author information

Correspondence to Federico Camia.

Additional information

F. Camia’s research was supported in part by NWO Grant Vidi 639.032.916. C. Garban was partially supported by ANR Grant BLAN06-3-134462. C. M. Newman’s research supported in part by NSF Grants OISE-0730136 and DMS-1007524.

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Camia, F., Garban, C. & Newman, C.M. The Ising magnetization exponent on \(\mathbb{Z }^2\) is \(1/15\) . Probab. Theory Relat. Fields 160, 175–187 (2014).

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Mathematics Subject Classification

  • 82B20
  • 82B27
  • 60K35