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Eigenvalue distributions of random unitary matrices. (English) Zbl 1044.15016
The main result of the paper concerns the eigenvalue distribution of a \(n\times n\) random unitary matrix \(U\), chosen from the Haar measure, as \(n\to \infty\). Let \(I_1=(e^{i\alpha_1},e^{i\beta_1}),\ldots, I_m=(e^{i\alpha_m},e^{i\beta_m})\) be intervals on the unit circle, and let \(X_{nk}\) denote the number of eigenvalues of \(U\) in \(I_k\). Set \(Y_{nk}=\pi (\log n)^{-1/2}(X_{nk}-E[X_{nk}])\). The main theorem of the article states that \((Y_{n1},\ldots, Y_{nm})\) converges jointly in distribution to a normal distribution with a specified covariance matrix. The theorem is proven using the connection between Toeplitz matrices and random unitary matrices.

15B52 Random matrices (algebraic aspects)
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60F05 Central limit and other weak theorems
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
15A18 Eigenvalues, singular values, and eigenvectors
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