We prove that for the Ising model defined on the plane (Formula presented.) the average magnetization under an external magnetic field (Formula presented.) behaves exactly like (Formula presented.) The proof, which is surprisingly simple compared to an analogous result for percolation [i.e. that (Formula presented.) 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.
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty