Scaling limits of energies and correctors

TL;DR

This paper investigates the large-scale behavior of correctors in stochastic homogenization of elliptic equations, showing they converge to a Gaussian free field variant without requiring product structure assumptions.

Contribution

It introduces a new approach based on energy additivity to establish central limit theorems and large-scale limits for correctors, bypassing previous structural restrictions.

Findings

Correctors converge to a Gaussian free field variant at large scales

Energy quantities satisfy central limit theorems under finite dependence

Method does not rely on product measure assumptions

Abstract

In stochastic homogenization of elliptic equations, the corrector plays a central role. Under a finite range of dependence assumption on the coefficient field, we show that the large-scale spatial averages of the corrector approach those of a variant of the Gaussian free field. In contrast to previous work, the argument does not rely on an underlying product structure of the probability measure. Instead, we rely on the additivity of energy quantities to show central limit theorems for these, and derive the large-scale behavior of the corrector as a consequence.