The Ising magnetization exponent on 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 $${{\sigma }_0}_{{\beta }_c,h}\asymp h^{\frac{1}{15}}\mathrm{.}$$ The proof, which is surprisingly simple compared to an analogous result for percolation (i.e. that $\theta (p)=(p-p_c{)}^{5\mathrm{/}36+o(1)}$ 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.