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Local laws for multiplication of random matrices

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 $μ_α\boxtimes μ_β,$ where $μ_α$ and $μ_β$ 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 $μ_A \boxtimes μ_B,$ where $μ_A$ and $μ_B$ 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.