  We prove quantitative decay estimates for the boundary layer corrector in
stochastic homogenization in the case of a half-space boundary. Our estimates
are of optimal order and show that the gradient of the boundary layer corrector
features nearly fluctuation-order decay; its expected value decays even one
order faster. As a corollary, we deduce estimates on the accuracy of the
representative volume element method for the computation of effective
coefficients: our understanding of the decay of boundary layers enables us to
improve the order of convergence of the RVE method for $d\geq 3$.
