  We consider a random walk in a random potential, which models a situation of
a random polymer and we study the annealed and quenched costs to perform long
crossings from a point to a hyperplane. These costs are measured by the so
called Lyapounov norms. We identify situations where the point-to-hyperplane
annealed and quenched Lyapounov norms are different. We also prove that in
these cases the polymer path exhibits localization.
