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

## TL;DR

This paper establishes a local limit theorem and eigenvector delocalization for polynomials in two random matrices using boundary regularity conditions for subordination functions in free probability.

## Contribution

It introduces a boundary regularity condition for matrix-valued subordination functions to analyze eigenvector behavior in polynomials of two random matrices.

## Key findings

- Proves a local limit theorem for polynomials in two matrices.
- Shows eigenvector delocalization under certain conditions.
- Estimates approximate subordination functions for matrix sums.

## Abstract

We propose a boundary regularity condition for the $M_n(\mathbb{C})$-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 $M_n(\mathbb{C})$-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.