  Since the seminal results by Avellaneda \& Lin it is known that elliptic
operators with periodic coefficients enjoy the same regularity theory as the
Laplacian on large scales. In a recent inspiring work, Armstrong \& Smart
proved large-scale Lipschitz estimates for such operators with random
coefficients satisfying a finite-range of dependence assumption. In the present
contribution, we extend the \emph{intrinsic large-scale} regularity of
Avellaneda \& Lin (namely, intrinsic large-scale Schauder and
Calder\'eron-Zygmund estimates) to elliptic systems with random coefficients.
The scale at which this improved regularity kicks in is characterized by a
stationary field $r_*$ which we call the minimal radius. This regularity theory
is \textit{qualitative} in the sense that $r_*$ is almost surely finite (which
yields a new Liouville theorem) under mere ergodicity, and it is
\textit{quantifiable} in the sense that $r_*$ has high stochastic integrability
provided the coefficients satisfy quantitative mixing assumptions. We
illustrate this by establishing \emph{optimal} moment bounds on $r_*$ for a
class of coefficient fields satisfying a multiscale functional inequality, and
in particular for Gaussian-type coefficient fields with arbitrary slow-decaying
correlations.
