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

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.

**[BGT87]**N. H. Bingham, C. M. Goldie, and J. L. Teugels,*Regular variation*, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. MR**898871****[BS99]**Albrecht Böttcher and Bernd Silbermann,*Introduction to large truncated Toeplitz matrices*, Universitext, Springer-Verlag, New York, 1999. MR**1724795****[CL95]**O. Costin and J.L. Lebowitz,*Gaussian fluctuation in random matrices*, Phys. Rev. Lett.**75**(1995), 69-72.**[Dia87]**Persi Diaconis,*Application of the method of moments in probability and statistics*, Moments in mathematics (San Antonio, Tex., 1987) Proc. Sympos. Appl. Math., vol. 37, Amer. Math. Soc., Providence, RI, 1987, pp. 125–142. MR**921087**, 10.1090/psapm/037/921087**[DS94]**Persi Diaconis and Mehrdad Shahshahani,*On the eigenvalues of random matrices*, J. Appl. Probab.**31A**(1994), 49–62. Studies in applied probability. MR**1274717****[Dur96]**Richard Durrett,*Probability: theory and examples*, 2nd ed., Duxbury Press, Belmont, CA, 1996. MR**1609153****[DWH99]**William F. Doran IV, David B. Wales, and Philip J. Hanlon,*On the semisimplicity of the Brauer centralizer algebras*, J. Algebra**211**(1999), no. 2, 647–685. MR**1666664**, 10.1006/jabr.1998.7592**[Fel71]**William Feller,*An introduction to probability theory and its applications. Vol. II.*, Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR**0270403****[FH91]**William Fulton and Joe Harris,*Representation theory*, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR**1153249****[FOT94]**Masatoshi Fukushima, Yōichi Ōshima, and Masayoshi Takeda,*Dirichlet forms and symmetric Markov processes*, de Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 1994. MR**1303354****[HKO00]**C.P. Hughes, J.P. Keating, and N. O'Connell,*On the characteristic polynomial of a random unitary matrix*, Preprint, 2000.**[HKOS00]**B.M. Hambly, P. Keevash, N. O'Connell, and D. Stark,*The characteristic polynomial of a random permutation matrix*, To appear, 2000.**[HW89]**Phil Hanlon and David Wales,*On the decomposition of Brauer’s centralizer algebras*, J. Algebra**121**(1989), no. 2, 409–445. MR**992775**, 10.1016/0021-8693(89)90076-8**[Joh97]**Kurt Johansson,*On random matrices from the compact classical groups*, Ann. of Math. (2)**145**(1997), no. 3, 519–545. MR**1454702**, 10.2307/2951843**[Kah85]**Jean-Pierre Kahane,*Some random series of functions*, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR**833073****[KS00]**J.P. Keating and N.C. Snaith,*Random matrix theory and*, To appear, 2000.**[Lit58]**Dudley E. Littlewood,*The Theory of Group Characters and Matrix Representations of Groups*, Oxford University Press, New York, 1940. MR**0002127****[Mac79]**I. G. Macdonald,*Symmetric functions and Hall polynomials*, The Clarendon Press, Oxford University Press, New York, 1979. Oxford Mathematical Monographs. MR**553598****[Off72]**A. C. Offord,*The distribution of the values of a random function in the unit disk*, Studia Math.**41**(1972), 71–106. MR**0307393****[Rai97]**E. M. Rains,*High powers of random elements of compact Lie groups*, Probab. Theory Related Fields**107**(1997), no. 2, 219–241. MR**1431220**, 10.1007/s004400050084**[Ram95]**Arun Ram,*Characters of Brauer’s centralizer algebras*, Pacific J. Math.**169**(1995), no. 1, 173–200. MR**1346252****[Ram97]**Arun Ram,*A “second orthogonality relation” for characters of Brauer algebras*, European J. Combin.**18**(1997), no. 6, 685–706. MR**1468338**, 10.1006/eujc.1996.0132**[Sos00]**A. Soshnikov,*The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities*, Ann. Probab.**28**(2000), 1353-1370. CMP**2001:05****[ST87]**H.-J. Schmeisser and H. Triebel,*Topics in Fourier analysis and function spaces*, Mathematik und ihre Anwendungen in Physik und Technik [Mathematics and its Applications in Physics and Technology], vol. 42, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987. MR**900143****[Wie98]**K.L. Wieand,*Eigenvalue distributions of random matrices in the permutation group and compact Lie groups*, Ph.D. thesis, Harvard University, 1998.

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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