Local laws for multiplication of random matrices

TL;DR

This paper establishes local laws and edge behavior for the eigenvalues and eigenvectors of a multiplicative random matrix model, extending techniques from additive models to the multiplicative case.

Contribution

It provides the first detailed local laws and edge eigenvalue analysis for the multiplicative free convolution model of random matrices.

Findings

Density and subordination functions have regular behavior at edges.

Optimal scale local laws for resolvent entries are proven.

Results extend techniques from additive to multiplicative models.

Abstract

Consider the random matrix model $A^{1/2} UBU^* A^{1/2},$ where $A$ and $B$ are two $N \times N$ deterministic matrices and $U$ is either an $N \times N$ Haar unitary or orthogonal random matrix. It is well-known that on the macroscopic scale, the limiting empirical spectral distribution (ESD) of the above model is given by the free multiplicative convolution of the limiting ESDs of $A$ and $B,$ denoted as $\mu_\alpha \boxtimes \mu_\beta,$ where $\mu_\alpha$ and $\mu_\beta$ are the limiting ESDs of $A$ and $B,$ respectively. In this paper, we study the asymptotic microscopic behavior of the edge eigenvalues and eigenvectors statistics. We prove that both the density of $\mu_A \boxtimes \mu_B,$ where $\mu_A$ and $\mu_B$ are the ESDs of $A$ and $B,$ respectively and the associated subordination functions have a regular behavior near the edges. Moreover, we establish the local laws near the edges on the optimal scale. In particular, we prove that the entries of the resolvent are close to some functionals depending only on the eigenvalues of $A, B$ and the subordination functions with optimal convergence rates. Our proofs and calculations are based on the techniques developed for the additive model $A+UBU^*$ in [3,5,6,8], and our results can be regarded as the counterparts of [8] for the multiplicative model.