  We consider quenched random perturbations of skew products of rotations on
the unit circle over uniformly expanding maps on the unit circle. It is known
that if the skew product satisfies a certain condition (shown to be generic in
the case of linear expanding maps), then the transfer operator of the skew
product has a spectral gap. Using semiclassical analysis we show that the
spectral gap is preserved under small random perturbations. This implies
exponential decay of quenched random correlation functions for smooth
observables at small noise levels.

Dynamical Systems