Local Stability of the Free Additive Convolution

TL;DR

This paper establishes the stability of the subordination equations defining free additive convolution, and applies this to analyze local spectral statistics of certain random matrix ensembles, showing spectral distribution concentration at small scales.

Contribution

It proves the stability of the subordination system away from singularities and applies this to demonstrate spectral distribution concentration in random matrices at fine scales.

Findings

Stability of subordination equations away from edges and singularities.

Spectral distribution of $A+UBU^*$ concentrates around free additive convolution.

Concentration holds down to scales of $N^{-2/3}$.

Abstract

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 $N^{-2/3}$.