Domain walls and droplet-like excitation of the random-field Ising magnet are studied in $d=\{3,4,5,6,7\}$ dimensions by means of exact numerical ground-state calculations. They are obtained using the established mapping to the graph-theoretical maximum-flow problem. This allows us to study large system sizes of more than 5 $\times {10}^6$ spins in exact thermal equilibrium. All simulations are carried out at the critical point for the strength $h$ of the random fields, $h=h_c\left(d\right)$. Using finite-size scaling, energetic and geometric properties like stiffness exponents and fractal dimensions are calculated. Using these results, we test (hyper)scaling relations, which seem to be fulfilled below the upper critical dimension $d_u=6$. Also, for $d<d_u$, the stiffness exponent can be obtained from the scaling of the ground-state energy.

• Received 2 January 2012

DOI:https://doi.org/10.1103/PhysRevB.85.224421