On the eigenvalues of truncations of random unitary matrices

Abstract

We consider the empirical eigenvalue distribution of an m × m principle submatrix of an n × n random unitary matrix distributed according to Haar measure. Earlier work of Petz and Réffy identified the limiting spectral measure if m/n → α, as n → ∞ under suitable scaling, the family {µα}αε(0;1) of limiting measures interpolates between uniform measure on the unit disc (for small α) and uniform measure on the unit circle (as α → 1). 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 µα is typically of order √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 Chafaï, Hardy, and Maïda.