  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.
