  Disordered pinning models deal with the (de)localization tran- sition of a
polymer in interaction with a heterogeneous interface. In this paper, we focus
on two models where the inhomogeneities at the interface are not independent
but given by an irreducible Markov chain on a finite state space. In the first
model, using Markov renewal tools, we give an expression for the annealed
critical curve in terms of a Perron-Frobenius eigenvalue, and provide examples
where exact computations are possible. In the second model, the transition
matrix vary with the size of the system so that, roughly speaking, disorder is
more and more correlated. In this case we are able to give the limit of the
averaged quenched free energy, therefore providing the full phase diagram
picture, and the number of critical points is related to the number of states
of the Markov chain. We also mention that the question of pinning in correlated
disorder appears in the context of DNA denaturation.
