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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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