  In a companion article we have introduced a notion of multiscale functional
inequalities for functions $X(A)$ of an ergodic stationary random field $A$ on
the ambient space $\mathbb R^d$. These inequalities are multiscale weighted
versions of standard Poincar\'e, covariance, and logarithmic Sobolev
inequalities. They hold for all the examples of fields $A$ arising in the
modelling of heterogeneous materials in the applied sciences whereas their
standard versions are much more restrictive. In this contribution we first
investigate the link between multiscale functional inequalities and more
standard decorrelation or mixing properties of random fields. Next, we show
that multiscale functional inequalities imply fine concentration properties for
nonlinear functions $X(A)$. This constitutes the main stochastic ingredient to
the quenched large-scale regularity theory for random elliptic operators by the
second author, Neukamm, and Otto, and to the corresponding quantitative
stochastic homogenization results.
