Linear functionals of eigenvalues of random matrices

Diaconis, Persi; Evans, Steven

doi:10.1090/S0002-9947-01-02800-8

Linear functionals of eigenvalues of random matrices
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by Persi Diaconis and Steven N. Evans

Trans. Amer. Math. Soc. 353 (2001), 2615-2633

DOI: https://doi.org/10.1090/S0002-9947-01-02800-8

Published electronically: March 14, 2001

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Abstract:

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 $n \rightarrow \infty$. 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.

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