Corrector estimates for elliptic systems with random periodic coefficients

We consider an elliptic system of equations on the torus $\left[ -\frac{L}{2}, \frac{L}{2} \right)^d$withrandomcoefficients$A$, 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.