Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information


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.

References [Enhancements On Off] (What's this?)

  • [BGT87] N.H. Bingham, C.M. Goldie, and J.L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. MR 88i:26004
  • [BS99] A. Böttcher and B. Silbermann, Introduction to Large Truncated Toeplitz Matrices, Springer-Verlag, New York, 1999. MR 2001b:47043
  • [CL95] O. Costin and J.L. Lebowitz, Gaussian fluctuation in random matrices, Phys. Rev. Lett. 75 (1995), 69-72.
  • [Dia87] P. Diaconis, Application of the method of moments in probability and statistics, Moments in mathematics (San Antonio, Tex., 1987), Amer. Math. Soc., Providence, RI, 1987, pp. 125-142. MR 89m:60006
  • [DS94] P. Diaconis and M. Shahshahani, On the eigenvalues of random matrices, J. Appl. Probab. 31A (1994), 49-62. MR 95m:60011
  • [Dur96] R. Durrett, Probability: Theory and Examples, 2nd ed., Duxbury, Belmont CA, 1996. MR 98m:60001
  • [DWH99] W.F. Doran, IV, D.B. Wales, and P.J. Hanlon, On the semisimplicity of the Brauer centralizer algebras, J. Algebra 211 (1999), 647-685. MR 99k:16034
  • [Fel71] W. Feller, An Introduction to Probability Theory and its Applications, 2nd ed., vol. 2, Wiley, New York, 1971. MR 42:5292
  • [FH91] W. Fulton and J. Harris, Representation Theory: a first course, Springer-Verlag, New York, 1991. MR 93a:20069
  • [FOT94] M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, Berlin, 1994. MR 96f:60126
  • [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] P. Hanlon and D. Wales, On the decomposition of Brauer's centralizer algebras, J. Algebra 121 (1989), 409-445. MR 91a:20041a
  • [Joh97] K. Johansson, On random matrices from the compact classical groups, Ann. of Math. (2) 145 (1997), 519-545. MR 98e:60016
  • [Kah85] J.-P. Kahane, Some Random Series of Functions, 2nd ed., Cambridge University Press, Cambridge, 1985. MR 87m:60119
  • [KS00] J.P. Keating and N.C. Snaith, Random matrix theory and $\zeta(\frac{1}{2} + it)$, To appear, 2000.
  • [Lit58] D.E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, 2nd ed., Clarendon Press, Oxford, 1958. MR 2,3a (1st ed.)
  • [Mac79] I.G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1979. MR 84g:05003
  • [Off72] A.C. Offord, The distribution of the values of a random function in the unit disk, Studia Math. 41 (1972), 71-106. MR 46:6513
  • [Rai97] E.M. Rains, High powers of random elements of compact Lie groups, Probab. Theory Related Fields 107 (1997), 219-241. MR 98b:15026
  • [Ram95] A. Ram, Characters of Brauer's centralizer algebras, Pacific J. Math. 169 (1995), 173-200. MR 96k:20020
  • [Ram97] A. Ram, A "second orthogonality relation" for characters of Brauer algebras, European J. Combin. 18 (1997), 685-706. MR 98m:20015
  • [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, Wiley, Chichester, 1987. MR 88k:42015a
  • [Wie98] K.L. Wieand, Eigenvalue distributions of random matrices in the permutation group and compact Lie groups, Ph.D. thesis, Harvard University, 1998.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 15A52, 60B15, 60F05

Retrieve articles in all journals with MSC (2000): 15A52, 60B15, 60F05

Additional Information

Persi Diaconis
Affiliation: Department of Mathematics, Stanford University, Building 380, MC 2125, Stanford, California 94305

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

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