Abstract.
Let U be an n × n random matrix chosen from Haar measure on the unitary group. For a fixed arc of the unit circle, let X be the number of eigenvalues of M which lie in the specified arc. We study this random variable as the dimension n grows, using the connection between Toeplitz matrices and random unitary matrices, and show that (X -E [X])/(\Var (X))1/2 is asymptotically normally distributed. In addition, we show that for several fixed arcs I 1 , ..., I m , the corresponding random variables are jointly normal in the large n limit.
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Received: 15 November 2000 / Revised version: 27 September 2001 / Published online: 17 May 2002
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Wieand, K. Eigenvalue distributions of random unitary matrices. Probab. Theory Relat. Fields 123, 202–224 (2002). https://doi.org/10.1007/s004400100186
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DOI: https://doi.org/10.1007/s004400100186