Eigenvalue rigidity for truncations of random unitary matrices

We consider the empirical eigenvalue distribution of an $m\times m$ principal submatrix of an $n\times n$ random unitary matrix distributed according to Haar measure. For $n$ and $m$ large with $\frac{m}{n}=\alpha$, the empirical spectral measure is well-approximated by a deterministic measure ${\mu }_{\alpha }$ supported on the unit disc. In earlier work, we showed 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. In this paper, we consider eigenvalues on a microscopic scale, proving concentration inequalities for the eigenvalue counting function and for individual bulk eigenvalues.