  In this paper we study the quenched distributions of hitting times for a
class of random dynamical systems. We prove that hitting times to dynamically
defined cylinders converge to a Poisson point process under the law of random
equivariant measures with super-polynomial decay of correlations. In
particular, we apply our results to uniformly aperiodic random subshifts of
finite type equipped with random invariant Gibbs measures. We emphasize that we
make no assumptions about the mixing property of the marginal measure.
