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Transactions of the American Mathematical Society

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Linear functionals of eigenvalues of random matrices


Authors: Persi Diaconis and Steven N. Evans
Journal: Trans. Amer. Math. Soc. 353 (2001), 2615-2633
MSC (2000): Primary 15A52, 60B15, 60F05
DOI: https://doi.org/10.1090/S0002-9947-01-02800-8
Published electronically: March 14, 2001
MathSciNet review: 1828463
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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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Additional Information

Persi Diaconis
Affiliation: Department of Mathematics, Stanford University, Building 380, MC 2125, Stanford, California 94305
Email: diaconis@math.Stanford.edu

Steven N. Evans
Affiliation: Department of Statistics #3860, University of California at Berkeley, 367 Evans Hall, Berkeley, California 94720-3860
Email: evans@stat.Berkeley.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02800-8
Keywords: Random matrix, central limit theorem, unitary, orthogonal, symplectic, trace, eigenvalue, characteristic polynomial, counting function, Schur function, character, Besov space, Bessel potential
Received by editor(s): July 6, 2000
Received by editor(s) in revised form: October 7, 2000
Published electronically: March 14, 2001
Additional Notes: Research of first author supported in part by NSF grant DMS-9504379
Research of second author supported in part by NSF grants DMS-9504379, DMS-9703845 and DMS-0071468
Article copyright: © Copyright 2001 American Mathematical Society

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