  We consider a discrete-time version of the parabolic Anderson model. This may
be described as a model for a directed (1+d)-dimensional polymer interacting
with a random potential, which is constant in the deterministic direction and
i.i.d. in the d orthogonal directions. The potential at each site is a positive
random variable with a polynomial tail at infinity. We show that, as the size
of the system diverges, the polymer extremity is localized almost surely at one
single point which grows ballistically. We give an explicit characterization of
the localization point and of the typical paths of the model.
