## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 24 (2019), paper no. 57, 12 pp.

### On the eigenvalues of truncations of random unitary matrices

Elizabeth Meckes and Kathryn Stewart

#### Abstract

We consider the empirical eigenvalue distribution of an $m\times m$ principle submatrix of an $n\times n$ random unitary matrix distributed according to Haar measure. Earlier work of Petz and Réffy identified the limiting spectral measure if $\frac{m} {n}\to \alpha $, as $n\to \infty $; under suitable scaling, the family $\{\mu _{\alpha }\}_{\alpha \in (0,1)}$ of limiting measures interpolates between uniform measure on the unit disc (for small $\alpha $) and uniform measure on the unit circle (as $\alpha \to 1$). In this note, we prove an explicit concentration inequality which shows 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. The approach is via the theory of two-dimensional Coulomb gases and makes use of a new “Coulomb transport inequality” due to Chafaï, Hardy, and Maïda.

#### Article information

**Source**

Electron. Commun. Probab., Volume 24 (2019), paper no. 57, 12 pp.

**Dates**

Received: 6 December 2018

Accepted: 20 July 2019

First available in Project Euclid: 13 September 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1568361883

**Digital Object Identifier**

doi:10.1214/19-ECP258

**Subjects**

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

**Keywords**

random matrices truncations submatrices empirical spectral measure Coulomb gas concentration inequalities Haar measure

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Meckes, Elizabeth; Stewart, Kathryn. On the eigenvalues of truncations of random unitary matrices. Electron. Commun. Probab. 24 (2019), paper no. 57, 12 pp. doi:10.1214/19-ECP258. https://projecteuclid.org/euclid.ecp/1568361883