Directed polymers in a random environment with a defect line

Abstract

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