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We establish an optimal, linear rate of convergence for the stochastichomogenization of discrete linear elliptic equations. We consider the modelproblem of independent and identically distributed coefficients on adiscretized unit torus. We show that the difference between the solution to therandom problem on the discretized torus and the first two terms of thetwo-scale asymptotic expansion has the same scaling as in the periodic case. Inparticular the $L^2$-norm in probability of the \mbox{$H^1$-norm} in space ofthis error scales like $\epsilon$, where $\epsilon$ is the discretizationparameter of the unit torus. The proof makes extensive use of previous resultsby the authors, and of recent annealed estimates on the Green's function byMarahrens and the third author.

An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations