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

Abstract

Abstract 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.