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We consider a random, uniformly elliptic coefficient field $a(x)$ on the$d$-dimensional integer lattice $\mathbb{Z}^d$. We are interested in thespatial decay of the quenched elliptic Green function $G(a;x,y)$. Next tostationarity, we assume that the spatial correlation of the coefficient fielddecays sufficiently fast to the effect that a logarithmic Sobolev inequalityholds for the ensemble $\langle\cdot\rangle$. We prove that all stochasticmoments of the first and second mixed derivatives of the Green function, thatis, $\langle|\nabla_x G(x,y)|^p\rangle$ and $\langle|\nabla_x\nabla_yG(x,y)|^p\rangle$, have the same decay rates in $|x-y|\gg 1$ as for theconstant coefficient Green function, respectively. This result relies on andsubstantially extends the one by Delmotte and Deuschel \cite{DeuschelDelmotte},which optimally controls second moments for the first derivatives and firstmoments of the second mixed derivatives of $G$, that is, $\langle|\nabla_xG(x,y)|^2\rangle$ and $\langle|\nabla_x\nabla_y G(x,y)|\rangle$. As anapplication, we are able to obtain optimal estimates on the random part of thehomogenization error even for large ellipticity contrast.

Annealed estimates on the Green function