  In this paper, we study high order correctors in stochastic homogenization.
We consider elliptic equations in divergence form on $\mathbb{Z}^d$, with the
random coefficients constructed from i.i.d. random variables. We prove moment
bounds on the high order correctors and their gradients under dimensional
constraints. It implies the existence of stationary correctors and stationary
gradients in high dimensions. As an application, we prove a two-scale expansion
of the solutions to the random PDE, which identifies the first and higher order
random fluctuations in a strong sense.
