Ising (Conformal) Fields and Cluster Area Measures

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.

17 pages, 1 figure

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• Planar Ising magnetization field I. Uniqueness of the critical scaling limit
• Conformal loop ensembles and the stress-energy tensor
• Stochastic Loewner Evolution Relates Anomalous Diffusion and Anisotropic Percolation
• Conformal loop ensembles and the stress-energy tensor. I. Fundamental notions of CLE
• Investigating structural and functional aspects of the brain's criticality in stroke
• Conformal Measure Ensembles and Planar Ising Magnetization: A Review
• Towards Conformal Invariance and a Geometric Representation of the 2D Ising Magnetization Field
• Probability Theory in Statistical Physics, Percolation, and Other Random Topics: The Work of C. Newman