Powers of large random unitary matrices and Toeplitz determinants

Duits, Maurice; Johansson, Kurt

doi:10.1090/S0002-9947-09-04542-5

Powers of large random unitary matrices and Toeplitz determinants
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by Maurice Duits and Kurt Johansson

Trans. Amer. Math. Soc. 362 (2010), 1169-1187

DOI: https://doi.org/10.1090/S0002-9947-09-04542-5

Published electronically: October 15, 2009

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Abstract:

We study the limiting behavior of $\operatorname {Tr}U^{k(n)}$, where $U$ is an $n\times n$ 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 of this result, we find that for each fixed $m\in \mathbb {N}$, the random variables $\operatorname {Tr}U^{k_j(n)}/\sqrt {\min (k_j(n),n)}$, $j=1,\ldots ,m$, converge to independent standard complex normals.

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