Linear functionals of eigenvalues of random matrices
Authors:
Persi Diaconis and Steven N. Evans
Journal:
Trans. Amer. Math. Soc. 353 (2001), 26152633
MSC (2000):
Primary 15A52, 60B15, 60F05
Published electronically:
March 14, 2001
MathSciNet review:
1828463
Fulltext PDF Free Access
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Abstract: 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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 B.M. Hambly, P. Keevash, N. O'Connell, and D. Stark, The characteristic polynomial of a random permutation matrix, To appear, 2000.
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 J.P. Keating and N.C. Snaith, Random matrix theory and , To appear, 2000.
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 A. Soshnikov, The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities, Ann. Probab. 28 (2000), 13531370. CMP 2001:05
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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 947203860
Email:
evans@stat.Berkeley.edu
DOI:
http://dx.doi.org/10.1090/S0002994701028008
PII:
S 00029947(01)028008
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 DMS9504379
Research of second author supported in part by NSF grants DMS9504379, DMS9703845 and DMS0071468
Article copyright:
© Copyright 2001 American Mathematical Society
