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, the end to end distance Delta x of the polymer follows Delta x similar to t(1/z), with z=3/2 and t is the polymer length. When the defect strength epsilon is weak, its contribution to Delta x is negligible. If epsilon >=epsilon(c), then Delta x approaches a finite value Delta x(sat)(epsilon) in large t limit, and we find Delta x(sat)similar to(epsilon-epsilon(c))(-delta), with delta approximate to 3.0. Such transition is related to the queuing phenomena of the asymmetric simple exclusion process. The polymer energy fluctuation is also discussed.