  We study the defocusing energy-critical nonlinear wave equation in four
dimensions. Our main result proves the stability of the scattering mechanism
under random pertubations of the initial data. The random pertubation is
defined through a microlocal randomization, which is based on a unit-scale
decomposition in physical and frequency space. In contrast to the previous
literature, we do not require the spherical symmetry of the pertubation. The
main novelty lies in a wave packet decomposition of the random linear
evolution. Through this decomposition, we can adaptively estimate the
interaction between the rough and regular components of the solution. Our
argument relies on techniques from restriction theory, such as Bourgain's bush
argument and Wolff's induction on scales.
