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Linear functionals of eigenvalues of random matrices. (English) Zbl 1008.15013
Summary: 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 $$n\to\infty$$. 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.

##### MSC:
 15B52 Random matrices (algebraic aspects) 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 60F05 Central limit and other weak theorems
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##### References:
 [1] 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. · Zbl 0617.26001 [2] Albrecht Böttcher and Bernd Silbermann, Introduction to large truncated Toeplitz matrices, Universitext, Springer-Verlag, New York, 1999. · Zbl 0916.15012 [3] O. Costin and J.L. Lebowitz, Gaussian fluctuation in random matrices, Phys. Rev. Lett. 75 (1995), 69-72. [4] 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. · Zbl 0631.60018 [5] Persi Diaconis and Mehrdad Shahshahani, On the eigenvalues of random matrices, J. Appl. Probab. 31A (1994), 49 – 62. Studies in applied probability. · Zbl 0807.15015 [6] Richard Durrett, Probability: theory and examples, 2nd ed., Duxbury Press, Belmont, CA, 1996. · Zbl 0709.60002 [7] 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. · Zbl 0944.16002 [8] William Feller, An introduction to probability theory and its applications. Vol. II., Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. · Zbl 0077.12201 [9] William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. · Zbl 0744.22001 [10] 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. · Zbl 0838.31001 [11] C.P. Hughes, J.P. Keating, and N. O’Connell, On the characteristic polynomial of a random unitary matrix, Preprint, 2000. [12] B.M. Hambly, P. Keevash, N. O’Connell, and D. Stark, The characteristic polynomial of a random permutation matrix, To appear, 2000. · Zbl 1047.60013 [13] Phil Hanlon and David Wales, On the decomposition of Brauer’s centralizer algebras, J. Algebra 121 (1989), no. 2, 409 – 445. , https://doi.org/10.1016/0021-8693(89)90076-8 Phil Hanlon and David Wales, Eigenvalues connected with Brauer’s centralizer algebras, J. Algebra 121 (1989), no. 2, 446 – 476. · Zbl 0695.20027 [14] Kurt Johansson, On random matrices from the compact classical groups, Ann. of Math. (2) 145 (1997), no. 3, 519 – 545. · Zbl 0883.60010 [15] Jean-Pierre Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. · Zbl 0571.60002 [16] J.P. Keating and N.C. Snaith, Random matrix theory and $$\zeta(\frac{1}{2} + it)$$, To appear, 2000. · Zbl 1051.11048 [17] D.E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, 2nd ed., Clarendon Press, Oxford, 1958. [18] I. G. Macdonald, Symmetric functions and Hall polynomials, The Clarendon Press, Oxford University Press, New York, 1979. Oxford Mathematical Monographs. · Zbl 0487.20007 [19] A. C. Offord, The distribution of the values of a random function in the unit disk, Studia Math. 41 (1972), 71 – 106. · Zbl 0236.30037 [20] E. M. Rains, High powers of random elements of compact Lie groups, Probab. Theory Related Fields 107 (1997), no. 2, 219 – 241. · Zbl 0868.60012 [21] Arun Ram, Characters of Brauer’s centralizer algebras, Pacific J. Math. 169 (1995), no. 1, 173 – 200. · Zbl 0839.20019 [22] Arun Ram, A ”second orthogonality relation” for characters of Brauer algebras, European J. Combin. 18 (1997), no. 6, 685 – 706. · Zbl 0884.20009 [23] 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 · Zbl 1021.60018 [24] 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. Hans-Jürgen Schmeisser and Hans Triebel, Topics in Fourier analysis and function spaces, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1987. · Zbl 0661.46024 [25] 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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