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

Ching-Wei Ho

Published 2019-03-19Version 1

We propose a boundary regularity condition for the $M_n(\C)$-valued subordination functions in free probability to prove the local limit theorem and delocalization of eigenvectors for polynomials in two random matrices. We prove this through estimating the pair of $M_n(\C)$-valued approximate subordination functions for the sum of two $M_n(\C)$-valued random matrices $\gamma_1\tensor C_N+\gamma_2\tensor U_N^*D_NU_N$, where $C_N$, $D_N$ are deterministic diagonal matrices, and $U_N$ is Haar unitary.