  We consider the model of a directed polymer pinned to a line of i.i.d. random
charges, and focus on the interior of the delocalized phase. We first show that
in this region, the partition function remains bounded. We then prove that for
almost every environment of charges, the probability that the number of contact
points in [0,n] exceeds c log(n) tends to 0 as n tends to infinity. Our proofs
rely on recent results of Birkner, Greven, den Hollander (2010) and Cheliotis,
den Hollander (2010).
