Eigenvalue rigidity for truncations of random unitary matrices

We consider the empirical eigenvalue distribution of an $mtimes m$ principal submatrix of an $ntimes n$ 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.