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2019 On the eigenvalues of truncations of random unitary matrices
Elizabeth Meckes, Kathryn Stewart
Electron. Commun. Probab. 24: 1-12 (2019). DOI: 10.1214/19-ECP258

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.

Citation

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Elizabeth Meckes. Kathryn Stewart. "On the eigenvalues of truncations of random unitary matrices." Electron. Commun. Probab. 24 1 - 12, 2019. https://doi.org/10.1214/19-ECP258

Information

Received: 6 December 2018; Accepted: 20 July 2019; Published: 2019
First available in Project Euclid: 13 September 2019

zbMATH: 1422.60017
MathSciNet: MR4003131
Digital Object Identifier: 10.1214/19-ECP258

Subjects:
Primary: 60B20

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