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

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We consider a polymer with configuration modelled by the trajectory of a Markov chain, interacting with a potential of form u+Vₙ when it visits a particular state 0 at time n, with {Vₙ} 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 ϕ(n)/n for some slowly varying ϕ; 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.