# Rates of Bulk Convergence for Ensembles of Classical Compact Groups

## TL;DR

This paper analyzes how quickly eigenvalue distributions of random matrices from classical compact groups converge to the sine point process as matrix size increases, providing explicit convergence rates for different groups.

## Contribution

It establishes precise convergence rates of eigenangle distributions to the sine process for various classical groups using determinantal point process techniques.

## Key findings

- Convergence rate of order N^{-2} for unitary groups.
- Convergence rate of order N^{-1} for orthogonal and symplectic groups.
- Quantitative bounds on eigenangle distribution convergence.

## Abstract

This paper considers random matrices distributed according to Haar measure in different classical compact groups. Utilizing the determinantal point structures of their nontrivial eigenangles, with respect to the $L_1$-Wasserstein distance, we obtain the rate of convergence for different ensembles to the sine point process when the dimension of matrices $N$ is sufficiently large. Specifically, the rate is roughly of order $N^{-2}$ on the unitary group and of order $N^{-1}$ on the orthogonal group and the compact symplectic group.