We study different models of polymers (discrete and continuous) in the neighborhood of an interface between two solvents(oil-water). These models give rise to a transition between a localized phase and a delocalized phase. We provefirst several convergence results of discrete models towards their associated continuous counterparts. These convergencehold when the coupling tends to $0$ (for high temperatures) and concerns the free energy and the slope of the criticalcurve at the origin. To that aim, we develop a method of coarse graining, introduced by Bolthausen and den Hollander,which we generalize to the case of a copolymer under the influence of a random pinning potential along theoil-water interface. We prove also a pathwise result in the case of a copolymer, which is pulled up and awayfrom the interface. We show in particular that inside the localized phase, the polymer comes back to the interfaceonly a finite number of times. Finally, we study the case of an hydrophobic homopolymer in the neighborhood of anoil-water interface, and also under the influence of a random potential when touching the interface. Through a methodconsisting of adapting the law of each excursion to its local random environment, we take into account the fact thatthe polymer can target the sites in which it comes back to the interface. This allows us to improve in a quantitative waythe lower bound of the quenched critical curve.
