  We study the random field Ising model in a two-dimensional box with side
length $N$ where the external field is given by independent normal variables
with mean $0$ and variance $\epsilon^2$. Our primary result is the following
phase transition at $T = T_c$: for $\epsilon \ll N^{-7/8}$ the boundary
influence (i.e., the difference between the spin averages at the center of the
box with the plus and the minus boundary conditions) decays as $N^{-1/8}$ and
thus the disorder essentially has no effect on the boundary influence; for
$\epsilon \gg N^{-7/8}$, the boundary influence decays as
$N^{-\frac{1}{8}}e^{-\Theta(\epsilon^{8/7}\, N)}$ (i.e., the disorder
contributes a factor of $e^{-\Theta(\epsilon^{8/7}\, N)}$ to the decay rate).
For a natural notion of the correlation length, i.e., the minimal size of the
box where the boundary influence shrinks by a factor of $2$ from that with no
external field, we also prove the following: as $\epsilon\downarrow 0$ the
correlation length transits from $\Theta(\epsilon^{-8/7})$ at $T_c$ to
$e^{\Theta(\epsilon^{-4/3}\,\,)}$ for $T < T_c$.
