  A scaling description is obtained for the $d$--dimensional random field Ising
model from domains in a bar geometry. Wall roughening removes the marginality
of the $d=2$ case, giving the $T=0$ correlation length $\xi \sim \exp\left(A
h^{-\gamma}\right)$ in $d=2$, and for $d=2+\epsilon$ power law behaviour with
$\nu = 2/\epsilon \gamma$, $h^\star \sim \epsilon^{1/\gamma}$. Here, $\gamma =
2,4/3$ (lattice, continuum) is one of four rough wall exponents provided by the
theory. The analysis is substantiated by three different numerical techniques
(transfer matrix, Monte Carlo, ground state algorithm). These provide for
strips up to width $L=11$ basic ingredients of the theory, namely free energy,
domain size, and roughening data and exponents.
