Fluctuations at the edges of the spectrum of the full rank deformed GUE

Abstract

We consider a full rank deformation of the GUE $W_N+A_N$ where $A_N$ is a full rank Hermitian matrix of size $N$ and $W_N$ is a GUE. The empirical eigenvalue distribution $\mu_{A_N}$ of $A_N$ converges to a probability distribution $\nu$. We identify all the possible limiting eigenvalue statistics at the edges of the spectrum, including outliers, edges and merging points of connected components of the limiting spectrum. The results are stated in terms of a deterministic equivalent of the empirical eigenvalue distribution of $W_N+A_N$, namely the free convolution of the semi-circle distribution and the empirical eigenvalues distribution of $A_N$.