  We consider a polymer with configuration modelled by the trajectory of a
Markov chain, interacting with a potential of form $u+V_n$ when it visits a
particular state 0 at time $n$, with $\{V_n\}$ representing i.i.d. quenched
disorder. There is a critical value of $u$ above which the polymer is pinned by
the potential. A particular case not covered in a number of previous studies is
that of loop exponent one, in which the probability of an excursion of length
$n$ takes the form $\phi(n)/n$ for some slowly varying $\phi$; this includes
simple random walk in two dimensions. We show that in this case, at all
temperatures, the critical values of $u$ in the quenched and annealed models
are equal, in contrast to all other loop exponents, for which these critical
values are known to differ, at least at low temperatures.
