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

Abstract

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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Notes

  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 \(a \searrow 0.\)

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Acknowledgments

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). https://doi.org/10.1007/s00440-013-0526-8

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Mathematics Subject Classification

  • 82B20
  • 82B27
  • 60K35