# Eigenvalue rigidity for truncations of random unitary matrices

## TL;DR

This paper studies the eigenvalue distribution of submatrices of random unitary matrices, showing that eigenvalues concentrate around a deterministic measure and providing bounds on their fluctuations.

## Contribution

It extends previous results by establishing concentration inequalities for eigenvalues at a microscopic scale in large random unitary submatrices.

## Key findings

- Eigenvalues are tightly concentrated around the deterministic measure.
- Concentration inequalities are proved for eigenvalue counting functions.
- Results apply to individual bulk eigenvalues on a microscopic scale.

## Abstract

We consider the empirical eigenvalue distribution of an $m\times m$ principal submatrix of an $n\times n$ random unitary matrix distributed according to Haar measure. For $n$ and $m$ large with $\frac{m}{n}=\alpha$, the empirical spectral measure is well-approximated by a deterministic measure $\mu_\alpha$ supported on the unit disc. In earlier work, we showed that for fixed $n$ and $m$, the bounded-Lipschitz distance between the empirical spectral measure and the corresponding $\mu_\alpha$ is typically of order $\sqrt{\frac{\log(m)}{m}}$ or smaller. In this paper, we consider eigenvalues on a microscopic scale, proving concentration inequalities for the eigenvalue counting function and for individual bulk eigenvalues.