  We consider an elliptic system of equations on the torus $\left[
-\frac{L}{2}, \frac{L}{2} \right)^d$ with random coefficients $A$, that are
assumed to be coercive and stationary. Using two different approaches we obtain
moment bounds on the gradient of the corrector, independent of the domain size
$L$. In the first approach we use Green function representation. For that we
require $A$ to be locally H\"older continuous and distribution of $A$ to
satisfy Logarithmic Sobolev inequality. The second method works for non-smooth
(possibly discontinuous) coefficients, and it requires that statistics of $A$
satisfies Spectral Gap estimate.
