  In this article we are interested in quantitative homogenization results for
linear elliptic equations in the non-stationary situation of a straight
interface between two heterogenous media. This extends the previous work
[Josien, 2019] to a substantially more general setting, in which the
surrounding heterogeneous media may be periodic or random stationary and
ergodic. Our main result is a quantification of the sublinearity of a
homogenization corrector adapted to the interface, which we construct using an
improved version of the method developed in [Fischer and Raithel, 2017]. This
quantification is optimal up to a logarithmic loss and allows to derive
almost-optimal convergence rates.
