  We derive exact asymptotics of time correlation functions for the parabolic
Anderson model with homogeneous initial condition and time-independent tails
that decay more slowly than those of a double exponential distribution and have
a finite cumulant generating function. We use these results to give precise
asymptotics for statistical moments of positive order. Furthermore, we show
what the potential peaks that contribute to the intermittency picture look like
and how they are distributed in space. We also investigate for how long
intermittency peaks remain relevant in terms of ageing properties of the model.
