Universality of local eigenvalue statistics in random matrices with external source

Understand

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.