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

TL;DR

This paper proves that the critical Ising model's magnetization field on a rescaled grid converges to a unique, conformally covariant scaling limit as the mesh size approaches zero.

Contribution

It establishes the uniqueness and conformal covariance of the critical Ising magnetization field's scaling limit, advancing understanding of critical phenomena in statistical physics.

Findings

The renormalized magnetization field converges to a unique limit.

The limiting field is conformally covariant.

The result applies to the critical Ising model on rescaled grids.

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.