Hard wall repulsion for the discrete Gaussian free field in random environment on $\mathbb{Z}^d$, $d\geq 3$
Alberto Chiarini, Emanuele Pasqui
TL;DR
This work analyzes the discrete Gaussian free field on $\mathbb Z^d$, $d\ge 3$, with uniformly elliptic random conductances drawn from a stationary ergodic environment, under a hard wall constraint on a macroscopic domain. The authors derive quenched large-deviation asymptotics for the hard-wall probability, showing a rate governed by the homogenized capacity $\mathrm{Cap}^{\mathrm{hom}}(V)$ and the essential supremum of on-site variances $\overline{g}$, with speed $N^{d-2}\log N$. They also characterize the conditioned field: its first-order mean profile matches a homogenized harmonic potential, and after recentering by a diverging amount, the field converges in law to the quenched discrete GFF, revealing a pathwise entropic repulsion despite the lack of translation invariance. The analysis combines coarse-graining, solidification, Gaussian functional bounds, and stochastic homogenization to handle the environment-induced inhomogeneity and to connect microscopic variance effects with macroscopic capacity. The results show that impurities in the random environment can enhance both the maximum and the entropic repulsion compared to homogeneous conductances, and they establish a framework for pathwise conditioning in non-translation-invariant gradient models.
Abstract
We study the discrete Gaussian free field (harmonic crystal) on $\mathbb{Z}^d$, $d\geq 3$, with uniformly elliptic and bounded random conductances sampled according to a sufficiently mixing environment measure. We consider the hard wall event that the field is non-negative on the discrete blow-up of a bounded regular domain $V\subseteq\mathbb{R}^d$, and establish a quenched large deviation asymptotic for its probability. The asymptotic rate is characterized by the essential supremum of the on-site variances and the homogenized capacity of $V$, which arises from a quenched invariance principle. We then analyze the law of the field conditioned on the hard wall event. We determine the first-order asymptotic profile for its expectation and demonstrate that an entropic push-away of the field from the origin occurs. Furthermore, we characterize the field pathwise behavior under the constraint, showing that, when properly recentered, the field converges weakly to the (quenched) discrete Gaussian free field. A major challenge is the lack of translation invariance in the model. We exploit the superharmonicity of the conditioned expectation of the field to obtain a sharp uniform lower bound for it. We then use this bound to estimate the harmonicity defect of the conditioned expectation. This is a key step that allows us to prove a corresponding sharp upper bound and establish a pathwise entropic repulsion phenomenon.