  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.
