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We consider a discrete elliptic equation on the $d$-dimensional lattice$\mathbb{Z}^d$ with random coefficients $A$ of the simplest type: they areidentically distributed and independent from edge to edge. On scales largew.r.t. the lattice spacing (i.e., unity), the solution operator is known tobehave like the solution operator of a (continuous) elliptic equation withconstant deterministic coefficients. This symmetric ``homogenized'' matrix$A_{\mathrm {hom}}=a_{\mathrm {hom}}\operatorname {Id}$ is characterized by$\xi\cdot A_{\mathrm {hom}}\xi=\langle(\xi+\nabla\phi )\cdotA(\xi+\nabla\phi)\rangle$ for any direction $\xi\in\mathbb {R}^d$, where therandom field $\phi$ (the ``corrector'') is the unique solution of$-\nabla^*\cdot A(\xi+\nabla\phi)=0$ such that $\phi(0)=0$, $\nabla\phi$ isstationary and $\langle\nabla\phi\rangle=0$, $\langle\cdot\rangle$ denoting theensemble average (or expectation). It is known (``by ergodicity'') that theabove ensemble average of the energy density $\mathcal{E}=(\xi+\nabla\phi)\cdot A(\xi+\nabla\phi)$, which is a stationary randomfield, can be recovered by a system average. We quantify this by proving thatthe variance of a spatial average of $\mathcal {E}$ on length scales $L$satisfies the optimal estimate, that is, $\operatorname {var}[\sum \mathcal{E}\eta_L]\lesssim L^{-d}$, where the averaging function [i.e., $\sum\eta_L=1$,$\operatorname {supp}(\eta_L)\subset\{|x|\le L\}$] has to be smooth in thesense that $|\nabla\eta_L|\lesssim L^{-1-d}$. In two space dimensions (i.e.,$d=2$), there is a logarithmic correction. This estimate is optimal since itshows that smooth averages of the energy density $\mathcal {E}$ decay in $L$ asif $\mathcal {E}$ would be independent from edge to edge (which it is not for$d>1$). This result is of practical significance, since it allows to estimatethe dominant error when numerically computing $a_{\mathrm {hom}}$.

An optimal variance estimate in stochastic homogenization of discrete elliptic equations