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. 
Kenneth S. Alexander, Nikos Zygouras