We consider the discrete Gaussian free field in random environment, where disorder is introduced through random edge conductances on the underlying graph. Such a model describes microscopic fluctuations of a crystal at positive temperature in the presence of inhomogeneities.

We focus on the integer lattice $\mathbb{Z}^d$ for $d\geq 3$, and analyse the maximal fluctuation of the field and its behaviour in the presence of a macroscopic hard wall constraint. First, we derive sharp quenched large deviation asymptotics for the hard wall event. The rate is governed by two key quantities: the homogenized capacity of the associated random conductance model, and the essential supremum of the on-site (random) variances of the field. Secondly, we investigate the law of the field conditioned on the hard wall. We prove that the conditioned field exhibits an entropic push away from the zero height, and identify its expected asymptotic profile. Lastly, we characterize the pathwise behaviour of the conditioned field. This is based on a joint work with Alberto Chiarini.

We conclude by discussing ongoing work with Alberto Chiarini and Alessandra Cipriani, where, still in the supercritical dimension, we replace the lattice $\mathbb{Z}^d$ with different underlying graphs, and study how their structure influences both the decay for the hard wall probability and the asymptotic profile for the expectation of the conditioned field.