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.