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Ground states and domain walls are investigated with exact combinatorialoptimization in two-dimensional random field Ising magnets. The ground statesbreak into domains above a length scale that depends exponentially on therandom field strength squared. For weak disorder, this paramagnetic structurehas remnant long-range order of the percolation type. The domain walls aresuper-rough in ordered systems with a roughness exponent $\zeta$ close to 6/5.The interfaces exhibit rare fluctuations and multiscaling reminiscent of somemodels of kinetic roughening and hydrodynamic turbulence.Comment: to be published in Phys.Rev.E/Rapid.Com

Disorder, order, and domain wall roughening in the two-dimensional random field Ising model