  We study the universality of the local eigenvalue statistics of Gaussian
divisible Hermitian Wigner matrices. These random matrices are obtained by
adding an independent GUE matrix to an Hermitian random matrix with independent
elements, a Wigner matrix. We prove that Tracy-Widom universality holds at the
edge in this class of random matrices under the optimal moment condition that
there is a uniform bound on the fourth moment of the matrix elements.
Furthermore, we show that universality holds in the bulk for Gaussian divisible
Wigner matrices if we just assume finite second moments.
