  Simulations with more than $10^{12}$ spins are used to study the motion of a
domain wall driven through a three-dimensional random-field Ising magnet (RFIM)
by an external field $H$. The interface advances in a series of avalanches
whose size diverges at a critical external field $H_c$. Finite-size scaling is
applied to determine critical exponents and test scaling relations. Growth is
intrinsically anisotropic with the height of an avalanche normal to the
interface $\ell_\perp$ scaling as the width along the interface $\ell_\|$ to a
power $\chi=0.85 \pm 0.01$. The total interface roughness is consistent with
self-affine scaling with a roughness exponent $\zeta \approx \chi$ that is much
larger than values found previously for the RFIM and related models that
explicitly break orientational symmetry by requiring the interface to be
single-valued. Because the RFIM maintains orientational symmetry, the interface
develops overhangs that may surround unfavorable regions to create uninvaded
bubbles. Overhangs complicate measures of the roughness exponent but decrease
in importance with increasing system size.
