  We consider directed polymers in a random potential given by a deterministic
profile with a strong maximum at the origin taken with random sign at each
integer time.
  We study two main objects based on paths in this random potential. First, we
use the random potential and averaging over paths to define a parabolic model
via a random Feynman--Kac evolution operator. We show that for the resulting
cocycle, there is a unique positive cocycle eigenfunction serving as a forward
and pullback attractor. Secondly, we use the potential to define a Gibbs
specification on paths for any bounded time interval in the usual way and study
the thermodynamic limit and existence and uniqueness of an infinite volume
Gibbs measure. Both main results claim that the local structure of interaction
leads to a unique macroscopic object for almost every realization of the random
potential.
