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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.

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