Scaling limits for nonlinear functionals of the discrete Gaussian free field with degenerate random conductances

TL;DR

This paper proves that nonlinear functionals of the discrete Gaussian free field with random conductances on certain subgraphs converge to continuum limits, using Green's function bounds for random walks in random environments.

Contribution

It establishes convergence of nonlinear functionals of the GFF with degenerate conductances to continuum limits, extending previous results to more general random environments.

Findings

Nonlinear functionals converge in $H^{-s}(D)$ for almost every environment.

Green's function bounds are valid for all dimensions $d \\geq 2$.

Results apply to supercritical percolation clusters with unbounded conductances.

Abstract

We consider nonlinear functionals of discrete Gaussian free fields with ergodic random conductances on a class of random subgraphs of $\mathbb{Z}^{2}$, including i.i.d. supercritical percolation clusters, where the conductances are possibly unbounded but satisfy an integrability condition. As our main result, we show that, for almost every realisation of the environment, the nonlinear functionals of the rescaled field converge to their continuum counterparts in the Sobolev space $H^{-s}(D)$ for suitable $s > 0$. To obtain the latter, we establish pointwise bounds for the Green's function of the associated random walk among random conductances with Dirichlet boundary conditions, which are valid for all $d \geq 2$.