An optimal error estimate in stochastic homogenization of discrete elliptic equations

This paper is the companion article to [Ann. Probab. 39 (2011) 779--856]. We consider a discrete elliptic equation on the $d$-dimensional lattice $\mathbb{Z}^d$ with random coefficients $A$ of the simplest type: They are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric "homogenized" matrix $A_{\mathrm{hom}}=a_{\mathrm{hom}}\mathrm{Id}$ is characterized by $\xi\cdot A_{\mathrm{hom}}\xi=<(\xi+\nabla\phi)\cdot A(\xi+\nabla\phi)>$ for any direction $\xi\in\mathbb{R}^d$, where the random field $\phi$ (the "corrector") is the unique solution of $-\nabla^*\cdot A(\xi+\nabla\phi)=0$ in $\mathbb{Z}^d$ such that $\phi(0)=0$, $\nabla\phi$ is stationary and $<\nabla\phi>=0$, $<\cdot>$ denoting the ensemble average (or expectation).