  We consider a directed polymer of length $N$ interacting with a linear
interface. The monomers carry i.i.d. random charges $(\omega_i)_{i=1}^N$ taking
values in $\mathbb{R}$ with mean zero and variance one. Each monomer $i$
contributes an energy $(\beta\omega_i-h)\varphi(S_i)$ to the interaction
Hamiltonian, where $S_i \in \mathbb{Z}$ is the height of monomer $i$ with
respect to the interface, $\varphi: \mathbb{Z} \to [0,\infty)$ is the
interaction potential, $\beta \in [0,\infty)$ is the inverse temperature, and
$h \in \mathbb{R}$ is the charge bias parameter. The configurations of the
polymer are weighted according to the Gibbs measure associated with the
interaction Hamiltonian, where the reference measure is given by a Markov chain
on $\mathbb{Z}$. We study both the quenched and the annealed free energy per
monomer in the limit as $N\to\infty$. We show that each exhibits a phase
transition along a critical curve in the $(\beta, h)$-plane, separating a
localized phase (where the polymer stays close to the interface) from a
delocalized phase (where the polymer wanders away from the interface). We
derive variational formulas for the critical curves and we obtain upper and
lower bounds on the quenched critical curve in terms of the annealed critical
curve. In addition, for the special case where the reference measure is given
by a Bessel random walk, we derive the scaling limit of the annealed free
energy as $\beta, h \downarrow 0$ in three different regimes for the tail
exponent of $\varphi$.
