Stochastic homogenization of Gaussian fields on random media

TL;DR

This paper investigates the scaling limits of Gaussian free fields and bi-Laplacian fields in random media, demonstrating their convergence to homogenized Gaussian fields through stochastic homogenization techniques.

Contribution

It introduces a novel analysis of the homogenization process for Gaussian fields in random environments, extending previous results to both continuous and discrete settings.

Findings

Fields converge to homogenized Gaussian free fields

Results apply to both regular domains and discrete tori

Homogenized fields determined by the effective operator a8a0 a8a0 a8a0 a8a0

Abstract

In this article, we study stochastic homogenization of non-homogeneous Gaussian free fields $\Xi^{g,{\bf a}} $ and bi-Laplacian fields $\Xi^{b,{\bf a}}$. They can be characterized as follows: for $f=\delta$ the solution $u$ of $\nabla \cdot \mathbf{a} \nabla u =f$, ${\bf a}$ is a uniformly elliptic random environment, is the covariance of $\Xi^{g,{\bf a}}$. When $f$ is the white noise, the field $\Xi^{b,{\bf a}}$ can be viewed as the distributional solution of the same elliptic equation. Our results characterize the scaling limit of such fields on both, a sufficiently regular domain $D\subset \mathbb{R}^d$, or on the discrete torus. Based on stochastic homogenization techniques applied to the eigenfunction basis of the Laplace operator $\Delta$, we will show that such families of fields converge to an appropriate multiple of the GFF resp. bi-Laplacian. The limiting fields are determined by their respective homogenized operator $\ahom \Delta$, with constant $\ahom$ depending on the law of the environment ${\bf a}$. The proofs are based on the results found in \cite{Armstrong2019} and \cite{gloria2014optimal}.