On the depinning transition of the directed polymer in a random environment with a defect line

2013-07-03We study the depinning transition of the 1+1 dimensional directed polymer in a random environment with a defect line model. The monomer locations of the polymer is modeled by the space-time trajectory of a one dimensional simple symmetric random walk. Random environment is introduced by assigning each site of Z² an independent and identically distributed normal random variable that interacts with the polymer when it visits that site. The defect line is incorporated to the model by having an additional constant potential u at the origin which gives a reward or penalty to the polymer as it visits the origin. There is a critical value of u above which polymer is pinned, placing a positive fraction of its monomers at zero with high probability. ❧ Our first result is that the quenched free energy exists and the self-averaging holds, which implies that there is a nonrandom quenched critical point. To see the effect of disorder on the depinning transition, we compare the quenched free energy of the system as a function of u to the corresponding annealed system. Our main result is that the quenched and annealed free energies differ significantly only in a very small neighborhood of the critical point and we show that the size of this neighborhood scales as at most β as β → 0, where β is the inverse temperature