Eigenvalue rigidity for truncations of random unitary matrices

Abstract

We consider the empirical eigenvalue distribution of an [Formula: see text] principal submatrix of an [Formula: see text] random unitary matrix distributed according to Haar measure. For [Formula: see text] and [Formula: see text] large with [Formula: see text], the empirical spectral measure is well approximated by a deterministic measure [Formula: see text] supported on the unit disc. In earlier work, we showed that for fixed [Formula: see text] and [Formula: see text], the bounded-Lipschitz distance [Formula: see text] between the empirical spectral measure and the corresponding [Formula: see text] is typically of order [Formula: see text] 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.