  In this contribution we prove optimal rates of convergence in stochastic
homogenization of monotone operators with $p$-growth for all $2\le p <\infty$
in dimensions $d\le 3$ (and in periodic homogenization for all $p\ge 2$ and
$d\ge 1$). Previous contributions were so far restricted to $p=2$. The main
issues to treat super-quadratic growth are the potential degeneracy and the
unboundedness of the coefficients when linearizing the equation. To deal with
this we develop a perturbative regularity theory in the large for random linear
operators with unbounded coefficients. From the probabilistic point of view we
rely on functional inequalities, and make crucial use of the central limit
theorem scaling to absorb nonlinear contributions. Combining these two
ingredients we obtain sharp bounds on the growth of correctors and of corrector
differences, from which the quantitative two-scale expansion follows. Our
approach bypasses the need of non-perturbative large-scale regularity theory --
which might not hold true in the general setting we consider here.
