Scaling limit of the discrete Gaussian free field with degenerate random conductances

TL;DR

This paper proves that discrete Gaussian free fields with certain random conductances on complex subgraphs of b^{d} converge to a continuum Gaussian free field, extending understanding of scaling limits in random environments.

Contribution

It establishes the convergence of the rescaled discrete Gaussian free field to a continuum Gaussian free field in environments with degenerate conductances, including percolation clusters.

Findings

Rescaled fields converge to continuum Gaussian free fields

Provides a scaling limit for covariances of the field

Establishes a quenched local limit theorem for the Green's function

Abstract

We consider discrete Gaussian free fields with ergodic random conductances on a class of random subgraphs of $\mathbb{Z}^{d}$, $d \geq 2$, including i.i.d.\ supercritical percolation clusters, where the conductances are possibly unbounded but satisfy a moment condition. As our main result, we show that, for almost every realization of the environment, the rescaled field converges in law towards a continuum Gaussian free field. We also present a scaling limit for the covariances of the field. To obtain the latter, we establish a quenched local limit theorem for the Green's function of the associated random walk among random conductances with Dirichlet boundary conditions.