Local Stability of the Free Additive Convolution

We prove that the system of subordination equations, defining the free additive convolution of two probability measures, is stable away from the edges of the support and blow-up singularities by showing that the recent smoothness condition of Kargin is always satisfied. As an application, we consider the local spectral statistics of the random matrix ensemble $A+UBU^*$, where $U$ is a Haar distributed random unitary or orthogonal matrix, and $A$ and $B$ are deterministic matrices. In the bulk regime, we prove that the empirical spectral distribution of $A+UBU^*$ concentrates around the free additive convolution of the spectral distributions of $A$ and $B$ on scales down to $N^{-2/3}$.