  We report studies of the behaviour of a single driven domain wall in the
2-dimensional non-equilibrium zero temperature random-field Ising model,
closely above the depinning threshold. It is found that even for very weak
disorder, the domain wall moves through the system in percolative fashion. At
depinning, the fraction of spins that are flipped by the proceeding avalanche
vanishes with the same exponent beta=5/36 as the infinite percolation cluster
in percolation theory. With decreasing disorder strength, however, the size of
the critical region decreases. Our numerical simulation data appear to reflect
a crossover behaviour to an exponent beta'=0 at zero disorder strength. The
conclusions of this paper strongly rely on analytical arguments. A scaling
theory in terms of the disorder strength and the magnetic field is presented
that gives the values of all critical exponent except for one, the value of
which is estimated from scaling arguments.
