Planar Ising magnetization field I. Uniqueness of the critical scaling limit

Abstract

The aim of this paper is to prove the following result. Consider the critical Ising model on the rescaled grid $a\mathbb{Z}^{2}$, then the renormalized magnetization field

\[\Phi^{a}:=a^{15/8}\sum_{x\in a\mathbb{Z}^{2}}\sigma_{x}\delta_{x},\]

seen as a random distribution (i.e., generalized function) on the plane, has a unique scaling limit as the mesh size $a\searrow0$. The limiting field is conformally covariant.