Equality of critical points for polymer depinning transitions with loop exponent one

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.