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On the eigenvalues of truncations of random unitary matrices. (English) Zbl 1422.60017
Summary: 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 D. Petz and J. Réffy [Probab. Theory Relat. Fields 133, No. 2, 175–189 (2005; Zbl 1076.60022)] 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 D. Chafaï et al. [J. Funct. Anal. 275, No. 6, 1447–1483 (2018; Zbl 1407.82045)].
60B20 Random matrices (probabilistic aspects)
15B52 Random matrices (algebraic aspects)
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