  In this article we study the defocusing energy-critical nonlinear wave
equation on $\mathbb{R}^4$ with scaling supercritical data. We prove almost
sure scattering for randomized initial data in $H^s(\mathbb{R}^4) \times
H^{s-1}(\mathbb{R}^4)$ with $\frac{5}{6} < s < 1$. The proof relies on new
probabilistic estimates for the linear flow of the wave equation with
randomized data, where the randomization is based on a unit-scale decomposition
in frequency space, a decomposition in the angular variable, and a unit-scale
decomposition of physical space. In particular, we show that the solution to
the linear wave equation with randomized data almost surely belongs to $L^1_t
L^\infty_x$.
