  We consider the corrector equation from the stochastic homogenization of
uniformly elliptic finite-difference equations with random, possibly
non-symmetric coefficients. Under the assumption that the coefficients are
stationary and ergodic in the quantitative form of a Logarithmic Sobolev
inequality (LSI), we obtain optimal bounds on the corrector and its gradient in
dimensions $d \geq 2$. Similar estimates have recently been obtained in the
special case of diagonal coefficients making extensive use of the maximum
principle and scalar techniques. Our new method only invokes arguments that are
also available for elliptic systems and does not use the maximum principle. In
particular, our proof relies on the LSI to quantify ergodicity and on
regularity estimates on the derivative of the discrete Green's function in
weighted spaces.
