Eigenvalue distributions of random unitary matrices. (English) Zbl 1044.15016

Wieand, K.

Probab. Theory Relat. Fields 123, No. 2, 202-224 (2002).

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.

Reviewer: Franz Merkl (Leiden)

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

Keywords:

eigenvalue distribution; random unitary matrix; Haar measure; normal distribution; covariance matrix; Toeplitz matrices