Directed polymers in a random environment with a defect line

We study the depinning transition of the 1 + 1 dimensional directed polymer in a random environment with a defect line. The random environment consists of i.i.d. potential values assigned to each site of Z(2); sites on the positive axis have the potential enhanced by a deterministic value u. We show that for small inverse temperature beta the quenched and annealed free energies differ significantly at most in a small neighborhood (of size of order beta) of the annealed critical point u(c)(a) = 0. For the case u = 0, we show that the difference between quenched and annealed free energies is of order beta(4) as beta -> 0, assuming only finiteness of exponential moments of the potential values, improving existing results which required stronger assumptions.