A Local Limit Theorem and Delocalization of Eigenvectors for Polynomials in Two Matrices

We propose a boundary regularity condition for the $M_n({\mathbb{C}})$-valued subordination functions in free probability to prove a local limit theorem and delocalization of eigenvectors for self-adjoint polynomials in two random matrices. We prove this through estimating the pair of $M_n({\mathbb{C}})$-valued approximate subordination functions for the sum of two $M_n({\mathbb{C}})$-valued random matrices $\gamma _1\otimes C_N+\gamma _2\otimes U_N^*D_NU_N$, where $C_N$, $D_N$ are deterministic diagonal matrices, and $U_N$ is Haar unitary.

Limiting spectral distribution of the product of truncated Haar unitary matrices

Let [Formula: see text], [Formula: see text], be [Formula: see text] probabilistically independent matrices of order [Formula: see text] (with [Formula: see text]) which are the left-uppermost blocks of [Formula: see text] Haar unitary matrices. Suppose that [Formula: see text] as [Formula: see text], with [Formula: see text]. Using free probability and Brown measure techniques, we find the limiting spectral distribution of [Formula: see text].