Hard wall repulsion for the discrete Gaussian free field in random environment on Z^d, d≥3
The discrete Gaussian free field (also called the harmonic crystal) can be used to model the microscopic fluctuations in homogeneous crystals at positive temperature. In this talk we investigate how impurities (that is, disorder), modeled by random conductances, affect the field by studying its maximal fluctuations and its behavior in the presence of a macroscopic hard wall constraint. We will focus on the supercritical regime (i.e., dimension d>2). First, we establish exact quenched large deviation asymptotics for the hard wall event. These asymptotics are governed by two quantities: the homogenized capacity of a random walk among random conductances, and the essential supremum of the field on-site (random) variances. Then we prove that, conditioning on the hard wall, the field exhibits an entropic push away from zero. Finally, we describe the pathwise behavior of the field under the hard wall constraint. This is a joint work with Alberto Chiarini (Università degli Studi di Padova).