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arXiv · 1205.6612

The Ising magnetization exponent on Z^2 is 1/15

Federico Camia·Christophe Garban·Charles M. Newman
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Abstract

We prove that for the Ising model defined on the plane $\Z^2$ at $β=β_c$, the average magnetization under an external magnetic field $h>0$ behaves exactly like \[{σ_0}_{β_c, h} \asymp h^{\frac 1 {15}}\,. \] The proof, which is surprisingly simple compared to an analogous result for percolation (i.e. that $θ(p)=(p-p_c)^{5/36+o(1)}$ on the triangular lattice \cite{\SmirnovWerner,\KestenScaling}) relies on the GHS inequality as well as the RSW theorem for FK percolation from \cite{\RSWfk}. 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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Federico Camia, Christophe Garban, Charles M. Newman. 2013-06-17. The Ising magnetization exponent on Z^2 is 1/15. https://arxiv.org/abs/1205.6612

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