  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).
