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Exponential decay of correlations in the two-dimensional random field Ising model

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We study the random field Ising model on \({\mathbb {Z}}^2\) where the external field is given by i.i.d. Gaussian variables with mean zero and positive variance. We show that the effect of boundary conditions on the magnetization in a finite box decays exponentially in the distance to the boundary.

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We thank Tom Spencer for introducing the problem to us, thank Steve Lalley for many interesting discussions and thank Subhajit Goswami, Steve Lalley for a careful reading of an earlier version of the manuscript. We also thank Michael Aizenman and Ron Peled for helpful conversations. We thank two anonymous referees for many helpful suggestions on exposition.

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Correspondence to Jian Ding.

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Partially supported by NSF Grant DMS-1757479 and an Alfred Sloan fellowship.

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Ding, J., Xia, J. Exponential decay of correlations in the two-dimensional random field Ising model. Invent. math. 224, 999–1045 (2021).

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