Directed polymer in random media with a defect

We investigate a directed polymer in random media with an attractive defect at the center of the one dimensional substrate. Without the defect, end to end distance $\Delta x$ of the polymer follows $\Delta x \sim t^{1/z}$ with $z=3/2$ which is related to the value of the dynamic exponent in Kardar- Parisi-Zhang equation. When the defect strength $\epsilon$ is weak, its contribution to $\Delta x$ is negligible. If $\epsilon > \epsilon_c$ then $\Delta x$ becomes constant. This kind of transition is related to a queueing phenomena in the asymmetric simple exclusion process. Various critical exponents near the transiton point are also discussed.