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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.

MSC:
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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