  We explore the boundaries of sine kernel universality for the eigenvalues of
Gaussian perturbations of large deterministic Hermitian matrices. Equivalently,
we study for deterministic initial data the time after which Dyson's Brownian
motion exhibits sine kernel correlations. We explicitly describe this time span
in terms of the limiting density and rigidity of the initial points. Our main
focus lies on cases where the initial density vanishes at an interior point of
the support. We show that the time to reach universality becomes larger if the
density vanishes faster or if the initial points show less rigidity.
