  We study the corrector equation in stochastic homogenization for a simplified
Bernoulli percolation model on $\mathbb{Z}^d$, $d>2$. The model is obtained
from the classical $\{0,1\}$-Bernoulli bond percolation by conditioning all
bonds parallel to the first coordinate direction to be open. As a main result
we prove (in fact for a slightly more general model) that stationary correctors
exist and that all finite moments of the corrector are bounded. This extends a
previous result in [GO1], where uniformly elliptic conductances are treated, to
the degenerate case. With regard to the associated random conductance model, we
obtain as a side result that the corrector not only grows sublinearly, but
slower than any polynomial rate. Our argument combines a quantification of
ergodicity by means of a Spectral Gap on Glauber dynamics with regularity
estimates on the gradient of the elliptic Green's function.
