  With large-scale Monte Carlo simulations, we investigate the nonsteady
relaxation at the dynamic depinning transition in the two-dimensional Gaussian
random-field Ising model. The dynamic scaling behavior is carefully analyzed,
and the transition fields as well as static and dynamic exponents are
accurately determined based on the short-time dynamic scaling form. Different
from the usual assumption, two distinguished growth processes of spatial
correlation lengths for the velocity and height of the domain wall are found.
Thus, the universality class of the depinning transition is established, which
significantly differs from that of the quenched disorder equation but agrees
with that of the recent experiment as well as other simulations works. Under
the influence of the mesoscopic time regime, the crossover from the
second-order phase transition to the first-order one is confirmed in the
weak-disorder regime, yielding an abnormal disorder-dependent nature of the
criticality.

Statistical Mechanics