Correlation length of the two-dimensional random field Ising model via greedy lattice animal

For the two-dimensional random field Ising model where the random field is given by i.i.d.\ mean zero Gaussian variables with variance $\epsilon^2$, we study (one natural notion of) the correlation length, which is the critical size of a box at which the influences of the random field and of the boundary condition on the spin magnetization are comparable. We show that as $\epsilon \to 0$, at zero temperature the correlation length scales as $e^{\Theta(\epsilon^{-4/3})}$ (and our upper bound applies for all positive temperatures).