  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. We assume that probability of an excursion
of length $n$ is given by $n^{-c}\phi(n)$ for some $1<c<2$ and slowly varying
$\phi$. Disorder is introduced by 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 (called the contact fraction) of its monomers at 0 with high
probability. To see the effect of disorder on the depinning transition, we
compare the contact fraction and free energy (as functions of $u$) to the
corresponding annealed system. We show that for $c>3/2$, at high temperature,
the quenched and annealed curves differ significantly only in a very small
neighborhood of the critical point--the size of this neighborhood scales as
$\beta^{1/(2c-3)}$ where $\beta$ is the inverse temperature. For $c<3/2$, given
$\epsilon>0$, for sufficiently high temperature the quenched and annealed
curves are within a factor of $1-\epsilon$ for all $u$ near the critical point;
in particular the quenched and annealed critical points are equal. For $c=3/2$
the regime depends on the slowly varying function $\phi$.
