On the eigenvalues of truncations of random unitary matrices

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 R\'effy 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 $\alpha\to1$). 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 Chafa\"i, Hardy, and Ma\"ida.