  In this article, I study the localization transition of an hydrophobic
homopolymer in interaction with an interface between oil and water. To that aim
I consider a model in which the trajectories of a simple random walk play the
role of the possible configurations of the polymer. The chain gains an
energetic factor for every monomer it puts in the oil and receives a random
price of positive average each time it touches the origin. So a competition
arises between this two effects to know wether the chain is delocalized in the
oil or if it stays localized in the neighborhood of the interface. As usual a
critical curve divides the phase spaces in a localized and a delocalized area,
and the point of my paper is to develop a new strategy to bound this critical
curve by below. To achieve this result I take into account the fact that the
polymer can target the sites where it comes back to the origin to receive
prices as high as possible.

Probability