Directed polymers interacting with quenched disorder localized on a 1-dimensional defect line are known to undergo a localization-delocalization phase transition. A similar transition occurs in random heteropolymer models at selective interfaces. I will present some recent mathematical results on the subject: in particular (a) some inequalities for critical exponents, and (b) some results on the path behavior, both in the critical region and far from the transition. I will also discuss some interesting open problems.

LPT ENS, Lyon