  Quantitative stochastic homogenization of linear elliptic operators is by now
well-understood. In this contribution we move forward to the nonlinear setting
of monotone operators with $p$-growth. This work is dedicated to a quantitative
two-scale expansion result. By treating the range of exponents $2\le p <\infty$
in dimensions $d\le 3$, we are able to consider genuinely nonlinear elliptic
equations and systems such as $-\nabla \cdot A(x)(1+|\nabla u|^{p-2})\nabla
u=f$ (with $A$ random, non-necessarily symmetric) for the first time. When
going from $p=2$ to $p>2$, the main difficulty is to analyze the associated
linearized operator, whose coefficients are degenerate, unbounded, and depend
on the random input $A$ via the solution of a nonlinear equation. One of our
main achievements is the control of this intricate nonlinear dependence,
leading to annealed Meyers' estimates for the linearized operator, which are
key to the optimal quantitative two-scale expansion result we derive (this is
also new in the periodic setting).
