March 2022 The two-dimensional continuum random field Ising model
Adam Bowditch, Rongfeng Sun
Author Affiliations +
Ann. Probab. 50(2): 419-454 (March 2022). DOI: 10.1214/21-AOP1536


In this paper, we construct the two-dimensional continuum random field Ising model via scaling limits of a random field perturbation of the critical two-dimensional Ising model with diminishing disorder strength. Furthermore, we show that almost surely with respect to the continuum random field given by a white noise, the law of the magnetisation field is singular with respect to that of the two-dimensional continuum pure Ising model constructed by Camia, Garban and Newman (Ann. Probab. 43 (2015) 528–571).

Funding Statement

R. Sun and A. Bowditch are supported by NUS grants R-146-000-260-114 and R-146-000-300-114.


We wish to thank in particular F. Caravenna for fruitful discussions during various stages of this project, which helped us to overcome some of the technical difficulties. We would also like to thank C. Garban, Y. Le Jan and N. Zygouras for helpful discussions, and an anonymous referee for suggesting a simplified proof of tightness.


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Adam Bowditch. Rongfeng Sun. "The two-dimensional continuum random field Ising model." Ann. Probab. 50 (2) 419 - 454, March 2022.


Received: 1 September 2020; Revised: 1 March 2021; Published: March 2022
First available in Project Euclid: 24 March 2022

MathSciNet: MR4399291
zbMATH: 1489.82040
Digital Object Identifier: 10.1214/21-AOP1536

Primary: 82B44
Secondary: 60G60 , 60K35 , 82B20 , 82B27

Keywords: continuum scaling limit , magnetisation field , random field Ising model

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 2 • March 2022
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