  We consider the Wick ordered cubic Schr\"odinger 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.
