# Local Correlation and Gap Statistics under Dyson Brownian Motion for Covariance Matrices

## TL;DR

This paper proves the universality of local bulk statistics for linearized covariance matrices evolving under Dyson Brownian motion, connecting Wigner flow results to covariance matrices and establishing level repulsion estimates.

## Contribution

It extends universality results from Wigner flow to linearized covariance matrices under Dyson Brownian motion, providing new insights into their local spectral statistics.

## Key findings

- Universality of bulk correlation functions for covariance matrices
- Weak level repulsion estimate for Dyson Brownian motion
- Bulk statistics convergence under Dyson flow

## Abstract

This paper is the third chapter of three of the author's undergraduate thesis. In this paper, we study the convergence of local bulk statistics for linearized covariance matrices under Dyson's Brownian motion. We consider deterministic initial data $V$ approximate the Dyson Brownian motion for linearized covariance matrices by the Wigner flow. Using universality results for the Wigner flow, we deduce universality for the linearized covariance matrices. We deduce bulk universality of averaged bulk correlation functions for both biregular bipartite graphs and honest covariance matrices. We also deduce a weak level repulsion estimate for the Dyson Brownian motion of linearized covariance matrices.