  We study the existence of uniformly bounded extension and trace operators for
$W^{1,p}$-functions on randomly perforated domains, where the geometry is
assumed to be stationary ergodic. Such extension and trace operators are
important for compactness in stochastic homogenization. In contrast to former
approaches and results, we use very weak assumptions on the geometry which we
call local $(\delta,M)$-regularity, isotropic cone mixing and bounded average
connectivity. The first concept measures local Lipschitz regularity of the
domain while the second measures the mesoscopic distribution of void space. The
third is the most tricky part and measures the "mesoscopic" connectivity of the
geometry. In contrast to former approaches we do not require a minimal distance
between the inclusions and we allow for globally unbounded Lipschitz constants
and percolating holes. We will illustrate our method by applying it to the
Boolean model based on a Poisson point process and to a Delaunay pipe process.
We finally introduce suitable Sobolev spaces on $\mathbb{R}^d$ and $\Omega$ in
order to construct a stochastic two-scale convergence method and apply the
resulting theory to the homogenization of a $p$-Laplace problem on a randomly
perforated domain.
