  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.
