  We introduce a new method for obtaining quantitative results in stochastic
homogenization for linear elliptic equations in divergence form. Unlike
previous works on the topic, our method does not use concentration inequalities
(such as Poincar\'e or logarithmic Sobolev inequalities in the probability
space) and relies instead on a higher ($C^{k}$, $k \geq 1$) regularity theory
for solutions of the heterogeneous equation, which is valid on length scales
larger than a certain specified mesoscopic scale. This regularity theory, which
is of independent interest, allows us to, in effect, localize the dependence of
the solutions on the coefficients and thereby accelerate the rate of
convergence of the expected energy of the cell problem by a bootstrap argument.
The fluctuations of the energy are then tightly controlled using subadditivity.
The convergence of the energy gives control of the scaling of the spatial
averages of gradients and fluxes (that is, it quantifies the weak convergence
of these quantities) which yields, by a new "multiscale" Poincar\'e inequality,
quantitative estimates on the sublinearity of the corrector.
