## Electronic Communications in Probability

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

#### 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
Accepted: 20 July 2019
First available in Project Euclid: 13 September 2019

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

Digital Object Identifier
doi:10.1214/19-ECP258

#### 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

#### References

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