Critical properties and finite--size estimates for the depinning transition of directed random polymers

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 $\text{tf}$ 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.