  We consider models of directed random polymers interacting with a defect
line, which are known to undergo a pinning/depinning (or
localization/delocalization) phase transition. We are interested in critical
properties and we prove, in particular, finite--size upper bounds on the order
parameter (the {\em contact fraction}) in a window around the critical point,
shrinking with the system size. Moreover, we derive a new inequality relating
the free energy $\tf$ and an annealed exponent $\mu$ which describes extreme
fluctuations of the polymer in the localized region. For the particular case of
a $(1+1)$--dimensional interface wetting model, we show that this implies an
inequality between the critical exponents which govern the divergence of the
disorder--averaged correlation length and of the typical one. Our results are
based on on the recently proven smoothness property of the depinning transition
in presence of quenched disorder and on concentration of measure ideas.
