This talk presents a probabilistic approach to the cubic nonlinear Schr¨odinger equation (NLS) on the two-dimensional sphere, focusing on the collective be-havior of random initial data supported below the deterministic threshold for the Cauchy theory, at positive regularities. We begin with a brief overview of the Cauchy problem for NLS on com-pact surfaces. Then, we introduce the Gibbs measure problem and Bourgain’s resolution scheme in the case of the flat torus. On the sphere, we show that strong instabilities arise due to the concentra-tion of spherical harmonics around great circles, preventing a direct extension of Bourgain’s method. To address these instabilities, we develop a non-perturbative resolution scheme, building on recent works by Bringmann, Deng, Nahmod and Yue. This talk is based on joint work with Nicolas Burq, Chenmin Sun, and Nikolay Tzvetkov.