  We prove spatial decay estimates on disorder-averaged position-momentum
correlations in a gapless class of random oscillator models. First, we prove a
decay estimate on dynamic correlations for general eigenstates with a bound
that depends on the magnitude of the maximally excited mode. Then, we consider
the situation of a quantum quench. We prove that the full time-evolution of an
initially chosen (uncorrelated) product state has disorder-averaged
correlations which decay exponentially in space, uniformly in time.
