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We consider uniformly elliptic coefficient fields that are randomlydistributed according to a stationary ensemble of a finite range of dependence.We show that the gradient and flux $(\nabla\phi,a(\nabla \phi+e))$ of thecorrector $\phi$, when spatially averaged over a scale $R\gg 1$ decay like theCLT scaling $R^{-\frac{d}{2}}$. We establish this optimal rate on the level ofsub-Gaussian bounds in terms of the stochastic integrability, and alsoestablish a suboptimal rate on the level of optimal Gaussian bounds in terms ofthe stochastic integrability. The proof unravels and exploits theself-averaging property of the associated semi-group, which provides a naturaland convenient disintegration of scales, and culminates in a propagatorestimate with strong stochastic integrability. As an application, wecharacterize the fluctuations of the homogenization commutator, and prove sharpbounds on the spatial growth of the corrector, a quantitative two-scaleexpansion, and several other estimates of interest in homogenization.

The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations