The Ising magnetization exponent on Z2 is 1/15

We prove that for the Ising model defined on the plane {Mathematical expression} at {Mathematical expression} the average magnetization under an external magnetic field {Mathematical expression} behaves exactly like {Mathematical expression} The proof, which is surprisingly simple compared to an analogous result for percolation [i.e. that {Mathematical expression} 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. © 2013 Springer-Verlag Berlin Heidelberg