  We consider a polymer, with monomer locations modeled by the trajectory of a
Markov chain, in the presence of a potential that interacts with the polymer
when it visits a particular site 0. Disorder is introduced by, for example,
having the interaction vary from one monomer to another, as a constant $u$ plus
i.i.d. mean-0 randomness. There is a critical value of $u$ above which the
polymer is pinned, placing a positive fraction of its monomers at 0 with high
probability. This critical point may differ for the quenched, annealed and
deterministic cases. We show that self-averaging occurs, meaning that the
quenched free energy and critical point are nonrandom, off a null set. We
evaluate the critical point for a deterministic interaction ($u$ without added
randomness) and establish our main result that the critical point in the
quenched case is strictly smaller. We show that, for every fixed
$u\in\mathbb{R}$, pinning occurs at sufficiently low temperatures. If the
excursion length distribution has polynomial tails and the interaction does not
have a finite exponential moment, then pinning occurs for all $u\in\mathbb{R}$
at arbitrary temperature. Our results apply to other mathematically similar
situations as well, such as a directed polymer that interacts with a random
potential located in a one-dimensional defect, or an interface in two
dimensions interacting with a random potential along a wall.
