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 | References | Similar Articles | Additional Information

Let be a random 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 to converge to a Gaussian limit as . 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 . 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 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