Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data

We consider the energy-critical defocusing nonlinear wave equation on ${\Bbb R}^4$ and establish almost sure global existence and scattering for randomized radially symmetric initial data in $H^s_x({\Bbb R}^4)\times H^{s-1}_x({\Bbb R}^4)$ for ${1\over 2}<s<1$. This is the first almost sure scattering result for an energy-critical dispersive or hyperbolic equation with scaling super-critical initial data. The proof is based on the introduction of an approximate Morawetz estimate to the random data setting and new large deviation estimates for the free wave evolution of randomized radially symmetric data.