The Annals of Probability

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

Federico Camia, Christophe Garban, and Charles M. Newman

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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.

Article information

Ann. Probab., Volume 43, Number 2 (2015), 528-571.

First available in Project Euclid: 2 February 2015

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Primary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B27: Critical phenomena 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G20: Generalized stochastic processes 60G60: Random fields

Planar Ising model critical Ising model continuum scaling limit magnetization field Euclidean field theory conformal invariance FK clusters


Camia, Federico; Garban, Christophe; Newman, Charles M. Planar Ising magnetization field I. Uniqueness of the critical scaling limit. Ann. Probab. 43 (2015), no. 2, 528--571. doi:10.1214/13-AOP881.

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