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

TL;DR

This paper proves that the magnetization exponent for the 2D Ising model at critical temperature is exactly 1/15, using a simple proof that combines the GHS inequality and RSW theorem.

Contribution

It provides a new, straightforward proof of the exact magnetization exponent for the 2D Ising model at criticality, combining GHS inequality and RSW theorem.

Findings

Magnetization behaves like h^{1/15} at criticality.

The proof is simpler than previous analogous results.

Uses GHS inequality and RSW theorem for FK percolation.

Abstract

We prove that for the Ising model defined on the plane $\Z^2$ at $\beta=\beta_c$, the average magnetization under an external magnetic field $h>0$ behaves exactly like \[{\sigma_0}_{\beta_c, h} \asymp h^{\frac 1 {15}}\,. \] 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.