  We consider nonlinear, uniformly elliptic equations with random, highly
oscillating coefficients satisfying a finite range of dependence. We prove that
homogenization and linearization commute in the sense that the linearized
equation (linearized around an arbitrary solution) homogenizes to the
linearization of the homogenized equation (linearized around the corresponding
solution of the homogenized equation). We also obtain a quantitative estimate
on the rate of this homogenization. These results lead to a better
understanding of differences of solutions to the nonlinear equation, which is
of fundamental importance in quantitative homogenization. In particular, we
obtain a large-scale $C^{0,1}$ estimate for differences of solutions---with
optimal stochastic integrability. Using this estimate, we prove a large-scale
$C^{1,1}$ estimate for solutions, also with optimal stochastic integrability.
Each of these regularity estimates are new even in the periodic setting. As a
second consequence of the large-scale regularity for differences, we improve
the smoothness of the homogenized Lagrangian by showing that it has the same
regularity as the heterogeneous Lagrangian, up to $C^{2,1}$.
