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

We study 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 at zero temperature the effect of boundary conditions on the magnetization in a finite box decays exponentially in the distance to the boundary.
Comment: 11 pages, 1 figure. Version 2 fixes a mistake in Version 1; Version 3 incorporates minor update in the proof of Lemma 3.1