  We derive optimal-order homogenization rates for random nonlinear elliptic
PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely,
for a random monotone operator on $\mathbb{R}^d$ with stationary law (i.e.
spatially homogeneous statistics) and fast decay of correlations on scales
larger than the microscale $\varepsilon>0$, we establish homogenization error
estimates of the order $\varepsilon$ in case $d\geq 3$, respectively of the
order $\varepsilon |\log \varepsilon|^{1/2}$ in case $d=2$. Previous results in
nonlinear stochastic homogenization have been limited to a small algebraic rate
of convergence $\varepsilon^\delta$. We also establish error estimates for the
approximation of the homogenized operator by the method of representative
volumes of the order $(L/\varepsilon)^{-d/2}$ for a representative volume of
size $L$. Our results also hold in the case of systems for which a
(small-scale) $C^{1,\alpha}$ regularity theory is available.
