2006/07/11 by Duits, Maurice, Johansson, Kurt
#FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.math-ph/0607017
We study the limiting behavior of \Tr Uk(n), where U is a n× n random unitary matrix and k(n) is a natural number that may vary with n in an arbitrary way. Our analysis is based on the connection with Toeplitz determinants. The central observation of this paper is a strong Szegö limit theorem for Toeplitz determinants associated to symbols depending on n in a particular way. As a consequence to this result, we find that for each fixed m∈ \N, the random variables \Tr Ukj(n)/√(min(kj(n),n)), j=1,..., m, converge to independent standard complex normals.