# A Power-Law Upper Bound on the Correlations in the 2*D* Random Field Ising Model

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## Abstract

As first asserted by Y. Imry and S-K Ma, the famed discontinuity of the magnetization as function of the magnetic field in the two dimensional Ising model is eliminated, for all temperatures, through the addition of quenched random magnetic field of uniform variance, even if that is small. This statement is quantified here by a power-law upper bound on the decay rate of the effect of boundary conditions on the magnetization in finite systems, as function of the distance to the boundary. Unlike exponential decay which is only proven for strong disorder or high temperature, the power-law upper bound is established here for all field strengths and at all temperatures, including zero, for the case of independent Gaussian random field. Our analysis proceeds through a streamlined and quantified version of the Aizenman–Wehr proof of the Imry–Ma rounding effect.

## Notes

### Acknowledgements

The work of MA was supported in part by the NSF grant DMS-1613296 and the Weston Visiting Professorship at the Weizmann Institute. The work of RP was supported in part by Israel Science Foundation grant 861/15 and the European Research Council starting grant 678520 (LocalOrder). We thank the Faculty of Mathematics and Computer Science and the Faculty of Physics at WIS for the hospitality enjoyed there during work on this project. We also thank two anonymous referees for constructive comments by which the paper’s presentation was further improved.

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