  The aim of this paper is twofold:
  - To give an elementary and self-contained proof of an explicit formula for
the free energy for a general class of polymer chains interacting with an
environment through periodic potentials. This generalizes a result in
[Bolthausen and Giacomin, AAP 2005] in which the formula is derived by using
the Donsker-Varadhan Large Deviations theory for Markov chains. We exploit
instead tools from renewal theory.
  - To identify the infinite volume limits of the system. In particular, in the
different regimes we encounter transient, null recurrent and positive recurrent
processes (which correspond to delocalized, critical and localized behaviors of
the trajectories). This is done by exploiting the sharp estimates on the
partition function of the system obtained by the renewal theory approach.
  The precise characterization of the infinite volume limits of the system
exposes a non-uniqueness problem. We will however explain in detail how this
(at first) surprising phenomenon is instead due to the presence of a
first-order phase transition.
