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We prove almost sure global existence and scattering for the energy-criticalnonlinear Schr\"odinger equation with randomized spherically symmetric initialdata in $H^s(\mathbb{R}^4)$ with $\frac56<s<1$. We were inspired to considerthis problem by the recent work of Dodson--L\"uhrmann--Mendelson, which treatedthe analogous problem for the energy-critical wave equation.

Almost sure scattering for the energy-critical NLS with radial data below $H^1(\mathbb{R}^4)$