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

On the almost sure scattering for the energy-critical cubic wave equation with supercritical data