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

Abstract

We study the discrete massless Gaussian Free Field on $\Z^d$, $d\geq2$, in the presence of a disordered square-well potential supported on a finite strip around zero. The disorder is introduced by reward/penalty interaction coefficients, which are given by i.i.d. random variables. Under minimal assumptions on the law of the environment, we prove that the quenched free energy associated to this model exists in $\R^+$, is deterministic, and strictly smaller than the annealed free energy whenever the latter is strictly positive.