On the localization transition of random copolymers near selective interfaces

In this note we consider the (de) localization transition for random directed (1+1)-dimensional copolymers in the proximity of an interface separating selective solvents. We derive a rigorous lower bound on the free energy. This yields a substantial improvement on the bounds from below on the critical line that were known so far. Our result implies that the critical curve does not lie below the critical curve set forth by Monthus (Eur. Phys. J. B 13, 111-130, 2000) on the base of a renormalization group analysis. We discuss this result in the light of the ( rigorous and non rigorous) approaches present in the literature and, by making an analogy with a particular asymptotics of a disordered wetting model, we propose a simplified framework in which the question of identifying the critical curve, as well as understanding the nature of the transition, may be approached.