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 $e^{\epsilon^{-4/3+o(1)}}$ (and our upper bound applies for all positive temperatures). As a proof ingredient, we establish a growth rate for the two-dimensional greedy lattice animal normalized by its boundary size, which may be of independent interest.