  We consider the nonlinear Schr\"odinger equation set on a flat torus, in the
regime which is conjectured to lead to the kinetic wave equation; in
particular, the data are random, and spread up to high frequency in a weakly
nonlinear regime. We pursue the investigations of our previous paper, and show
that, in the case where the torus is the standard one, only the scaling
considered there allows convergence of the Dyson series up to the kinetic time
scale. We also show that, for generic quadratic dispersion relations (non
rectangular tori), the Dyson series converges on significantly longer time
scales; we are able to reach the kinetic time up to an arbitrarily small
polynomial error for a larger set of scalings. These results show the
importance of the exact structure of the dispersion relation, more specifically
of equidistribution properties of some bilinear quantities akin to pair
correlations derived from it.
