The second Picard iteration of NLS on the $2d$ sphere does not regularize Gaussian random initial data

We consider the Wick-ordered cubic Schrödinger equation (NLS) posed on the two-dimensional sphere, with initial data distributed according to a Gaussian measure. We show that the second Picard iteration does not improve the regularity of the initial data in the scale of the classical Sobolev spaces. This is in sharp contrast with the Wick-ordered NLS on the two-dimensional tori, a model for which we know from the work of Bourgain that the second Picard iteration gains one half-derivative. Our proof relies on identifying a singular part of the nonlinearity. We show that this singular part is responsible for a concentration phenomenon on a large circle (i.e., a stable closed geodesic), which prevents any regularization in the second Picard iteration.