Recently, the quantification of errors in the stochastic homogenization of divergence-form operators has witnessed important progress. Our aim now is to go beyond error bounds, and give precise descriptions of the effect of the randomness, in the large-scale limit. This paper is a first step in this direction. Our main result is to identify the correlation structure of the corrector, in dimension $3$ and higher. This correlation structure is similar to, but different from that of a Gaussian free field.
"Correlation structure of the corrector in stochastic homogenization." Ann. Probab. 44 (5) 3207 - 3233, September 2016. https://doi.org/10.1214/15-AOP1045