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).
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.
"The two-dimensional continuum random field Ising model." Ann. Probab. 50 (2) 419 - 454, March 2022. https://doi.org/10.1214/21-AOP1536