## On the discrete Gaussian Free Field with disordered pinning on Z^d, d\geq 2    [PDF]

Loren Coquille, Piotr Miłoś
We study the discrete massless Gaussian Free Field on Z^d, d\geq2, in presence of two types of random environments : (1) \delta-pinning at height 0 of inhomogenous i.i.d. Bernoulli strengths; (2) square-well potential supported on a finite strip with i.i.d. Bernoulli reward/penalty coefficients e. We prove that the quenched free energy associated to these models exists in R^+, is self-averaging, and strictly smaller than the annealed free energy (whenever the latter is strictly positive). Moreover, for model (2), we prove that in the plane (Var(e),E(e)), the quenched critical line (separating the phases of positive and zero free energy) lies strictly below the line E(e)=0, showing in particular that there exists a non trivial region where the field is localized though repulsed on average by the environment.
View original: http://arxiv.org/abs/1207.5983