  Ground states and domain walls are investigated with exact combinatorial
optimization in two-dimensional random field Ising magnets. The ground states
break into domains above a length scale that depends exponentially on the
random field strength squared. For weak disorder, this paramagnetic structure
has remnant long-range order of the percolation type. The domain walls are
super-rough in ordered systems with a roughness exponent $\zeta$ close to 6/5.
The interfaces exhibit rare fluctuations and multiscaling reminiscent of some
models of kinetic roughening and hydrodynamic turbulence.
