Abstract: We consider the quantitative stochastic homogenization of linear elliptic operators on the half-space with homogeneous Dirichlet boundary data. In this situation there is a well-known boundary layer phenomenon: In a neighborhood around the boundary the heterogeneous coefficient solution behaves qualitatively different than in the bulk of the domain. In this talk we are interested in quantifying this boundary layer, which we do by obtaining optimal decay rates (away from the boundary) for the "boundary layer correction" -the correction of the whole-space homogenization corrector that is required to enforce homogeneous Dirichlet boundary data. As an application of our decay rates, we are able to obtain almost-optimal error estimates for a slightly modified RVE method (without a screening term) for the approximation of the homogenized coefficients in d = 3, 4. This is joint work with Peter Bella, Julian Fischer, and Marc Josien.