The competition in the pinning of a directed polymer by a columnar pin and a background of random point impurities is investigated systematically using the renormalization-group method. With the aid of the mapping to the noisy-Burgers equation and the use of the mode-coupling method, the directed polymer is shown to be marginally localized to an arbitrary weak columnar pin in 1+1 dimensions. This weak-localization effect is attributed to the existence of large scale, nearly degenerate optimal paths of the randomly pinned directed polymer. The critical behavior of the depinning transition above 1+1 dimensions is obtained via an {ε} expansion.
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Hwa et al. (1995) studied this question.