We consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS) on , , with random initial data and prove almost sure well-posedness results below the scaling-critical regularity . More precisely, given a function on , we introduce a randomization adapted to the Wiener decomposition, and, intrinsically, to the so-called modulation spaces. Our goal in this paper is three-fold.(i) We prove almost sure local well-posedness of the cubic NLS below the scaling-critical regularity along with small data global existence and scattering.(ii) We implement a probabilistic perturbation argument and prove ‘conditional’almost sure global well-posedness for in the defocusing case, assuming an a priori energy bound on the critical Sobolev norm of the nonlinear part of a solution; when , we show that conditional almost sure global well-posedness in the defocusing case also holds under an additional assumption of global well-posedness of solutions to the defocusing cubic NLS with deterministic initial data in the critical Sobolev regularity.(iii) Lastly, we prove global well-posedness and scattering with a large probability for initial data randomized on dilated cubes. References