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

The discrete-time parabolic Anderson model with heavy-tailed potential