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Linear functionals of eigenvalues of random matrices. (English) Zbl 1008.15013
Summary: Let \(M_n\) be a random \(n\times n\) unitary matrix with distribution given by Haar measure on the unitary group. Using explicit moment calculations, a general criterion is given for linear combinations of traces of powers of \(M_n\) to converge to a Gaussian limit as \(n\to\infty\). By Fourier analysis, this result leads to central limit theorems for the measure on the circle that places a unit mass at each of the eigenvalues of \(M_n\). For example, the integral of this measure against a function with suitably decaying Fourier coefficients converges to a Gaussian limit without any normalisation.
Known central limit theorems for the number of eigenvalues in a circular arc and the logarithm of the characteristic polynomial of \(M_n\) are also derived from the criterion. Similar results are sketched for Haar distributed orthogonal and symplectic matrices.

MSC:
15B52 Random matrices (algebraic aspects)
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60F05 Central limit and other weak theorems
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