Ising (Conformal) Fields and Cluster Area Measures

TL;DR

This paper represents the scaling limit of the 2D critical Ising magnetization field as a conformal random field using SLE clusters and renormalized area measures, with extensions to other models and dimensions.

Contribution

It introduces a novel representation of the critical Ising field via SLE clusters and area measures, extending to off-critical, 3D, and Potts models.

Findings

Conformal random field representation of 2D critical Ising magnetization.

Connection between FK cluster area measures and the magnetization field.

Extensions to off-critical, 3D, and Potts models.

Abstract

We provide a representation for the scaling limit of the d=2 critical Ising magnetization field as a (conformal) random field using SLE (Schramm-Loewner Evolution) clusters and associated renormalized area measures. The renormalized areas are from the scaling limit of the critical FK (Fortuin-Kasteleyn) clusters and the random field is a convergent sum of the area measures with random signs. Extensions to off-critical scaling limits, to d=3 and to Potts models are also considered.