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We study the discrete massless Gaussian Free Field on Z^d, d \geq 2, in thepresence of a disordered square-well potential supported on a finite striparound zero. The disorder is introduced by reward/penalty interactioncoefficients, which are given by i.i.d. random variables. In the previous note,we proved under minimal assumptions on the law of the environment, that thequenched free energy associated to this model exists in R^+, is deterministic,and strictly smaller than the annealed free energy whenever the latter isstrictly positive. Here we consider Bernoulli reward/penalty coefficients b e_x+ h with P(e_x=-1)=P(e_x=+1)=1/2 for all x in Z^d, and b > 0, h in R. We provethat in the plane (b,h), the quenched critical line (separating the phases ofpositive and zero free energy) lies strictly below the line h = 0, showing inparticular that there exists a non trivial region where the field is localizedthough repulsed on average by the environment.

A second note on the discrete Gaussian Free Field with disordered pinning on Z^d, d\geq 2