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 C0, 1 estimate for differences of solutions—with optimal stochastic integrability. Using this estimate, we prove a large‐scale C1, 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 C2, 1. © 2020 Wiley Periodicals LLC

We prove large-scale C ∞ regularity for solutions of nonlinear elliptic equations with random coefficients, thereby obtaining a version of the statement of Hilbert's 19th problem in the context of homogenization. The analysis proceeds by iteratively improving three statements together: (i) the regularity of the homogenized Lagrangian L, (ii) the commutation of higher-order linearization and homogenization, and (iii) large-scale C 0,1 -type regularity for higher-order linearization errors. We consequently obtain a quantitative estimate on the scaling of linearization errors, a Liouville-type theorem describing the polynomiallygrowing solutions of the system of higher-order linearized equations, and an explicit (heterogenous analogue of the) Taylor series for an arbitrary solution of the nonlinear equations-with the remainder term optimally controlled. These results give a complete generalization to the nonlinear setting of the large-scale regularity theory in homogenization for linear elliptic equations. 65 6. Sharp estimates of linearization errors 71 7. Liouville theorems and higher regularity 72 Appendix A. Deterministic regularity estimates 80 Appendix B. Differentiation of F m 82 Appendix C. Linearization errors 83 Appendix D. Regularity for constant coefficient linearized equations 87 Appendix E. C ∞ regularity for smooth constant-coefficient Lagrangians 92 References 95

We describe computer programs accompanying our paper [3] and show that running them suffices to verify Fuglede's conjecture in Z 5 2 and Z 6 2 .