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