Almost sure scattering of the energy-critical NLS in $ d>6 $

We study the energy-critical nonlinear Schrödinger equation with randomised initial data in dimensions $ d>6 $. We prove that the Cauchy problem is almost surely globally well-posed with scattering for randomised super-critical initial data in $ H^s(\mathbb{R}^d) $ whenever $ s>\max\{\frac{4d-1}{3(2d-1)},\frac{d^2+6d-4}{(2d-1)(d+2)}\} $. The randomisation is based on a decomposition of the data in physical space, frequency space and the angular variable. This extends previously known results in dimension 4 [18]. The main difficulty in the generalisation to high dimensions is the non-smoothness of the nonlinearity.