  Consider a random matrix of the form $W_n = M_n + D_n$, where $M_n$ is a
Wigner matrix and $D_n$ is a real deterministic diagonal matrix ($D_n$ is
commonly referred to as an external source in the mathematical physics
literature). We study the universality of the local eigenvalue statistics of
$W_n$ for a general class of Wigner matrices $M_n$ and diagonal matrices $D_n$.
Unlike the setting of many recent results concerning universality, the global
semicircle law fails for this model. However, we can still obtain the universal
sine kernel formula for the correlation functions. This demonstrates the
remarkable phenomenon that local laws are more resilient than global ones. The
universality of the correlation functions follows from a four moment theorem,
which we prove using a variant of the approach used earlier by Tao and Vu.
