The Ising magnetization exponent on 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 \[{σ_0}_{β_c, h} \asymp h^{\frac 1 {15}}\,. \] The proof, which is surprisingly simple compared to an analogous result for percolation (i.e. that $\theta (p)=(p-p_c{)}^{5/36+o(1)}$ 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.

12 pages

Cited by in corpus (2)

• Revisiting the combinatorics of the 2D Ising model
• Polynomial chaos and scaling limits of disordered systems