We study unitary random matrix ensembles using the theory of
orthogonal polynomials on the unit circle. In particular we explicitly
compute the joint eigenvalue statistics of their rank-one truncations.
We prove that this eigenvalue point process is universal under the
natural scaling limit for a class of subunitary operators. Putting it
differently, we compute the limiting density of zeros of orthogonal
polynomials on the unit circle with random Verblunsky coefficients.
Joint work with Rowan Killip (UCLA).