  We investigate the behavior of nonequilibrium phase transitions under the
influence of disorder that locally breaks the symmetry between two symmetrical
macroscopic absorbing states. In equilibrium systems such "random-field"
disorder destroys the phase transition in low dimensions by preventing
spontaneous symmetry breaking. In contrast, we show here that random-field
disorder fails to destroy the nonequilibrium phase transition of the one- and
two-dimensional generalized contact process. Instead, it modifies the dynamics
in the symmetry-broken phase. Specifically, the dynamics in the one-dimensional
case is described by a Sinai walk of the domain walls between two different
absorbing states. In the two-dimensional case, we map the dynamics onto that of
the well studied low-temperature random-field Ising model. We also study the
critical behavior of the nonequilibrium phase transition and characterize its
universality class in one dimension. We support our results by large-scale
Monte Carlo simulations, and we discuss the applicability of our theory to
other systems.
