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On the eigenvalues of random matrices

Published online by Cambridge University Press:  05 September 2017

Persi Diaconis  and
Mehrdad Shahshahani
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Abstract

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Let M be a random matrix chosen from Haar measure on the unitary group Un. Let Z = X + iY be a standard complex normal random variable with X and Y independent, mean 0 and variance ½ normal variables. We show that for j = 1, 2, …, Tr(Mj) are independent and distributed as √jZ asymptotically as n →∞. This result is used to study the set of eigenvalues of M. Similar results are given for the orthogonal and symplectic and symmetric groups.

Keywords

HAAR MEASUREORTHOGONAL GROUPSSYMPLECTIC GROUPSSYMMETRIC GROUPS
Type
Part 2 Probabilistic Methods
Information
Journal of Applied Probability , Volume 31 , Issue A , 1994 , pp. 49 - 62
Copyright
Copyright © Applied Probability Trust 1994 

References

Arratia, R. and Tavare, S. (1992) The cycle structure of random permutations. Ann. Prob. 20, 1567–1591.CrossRefGoogle Scholar
Brauer, R. (1937) On algebras which are connected with the semisimple continuous groups. Ann. Math. 38, 854–872.CrossRefGoogle Scholar
Diaconis, P. (1990) Applications of the method of moments in probability and statistics. In Moments in Mathematics , ed.Landau, H. J., American Mathematical Society, Providence, RI.Google Scholar
Diaconis, P. and Freedman, P. (1987) A dozen de Finetti theorems in search of a theory. Ann. Inst. H. Poincaré Sup. 23, 397–423.Google Scholar
Diaconis, P. and Mallows, C. (1986) On the trace of random orthogonal matrices. Unpublished manuscript. Results summarized in Diaconis (1990).Google Scholar
Diaconis, P. and Shahshahani, M. (1986) Products of random matrices as they arise in the study of random walks on groups. Contemp. Math. 50, 183–195.Google Scholar
Diaconis, P. and Shahshahani, M. (1987) The subgroup algorithm for generating uniform random variables. Prob. Eng. Inf. Sci. 1, 15–32.Google Scholar
Diaconis, P., Eaton, J. and Lauritzen, S. L. (1992) de Finetti style theorems for linear models. Scand. J. Statist. 19, 289–315.Google Scholar
Erdös, P. and Turan, P. (1967a, 1967b, 1968) On some problems of statistical group theory II, III, IV, Acta. Math. Acad. Sci. Hung. 18, 151–163; 18, 309-320; 19, 413-435.CrossRefGoogle Scholar
Goh, W. and Schmutz, E. (1991) A central limit theorem on GLn (𝔽 q ). Preprint, Department of Mathematics, Drexel University.Google Scholar
Goncharov, V. (1944) On the domain of combinatory analysis. Amer. Math. Soc. Transl. 2 (1962), 1–46.Google Scholar
Hanlon, P. and Wales, D. B. (1989a) On the decomposition of Brauer's centralizer algebras. J. Alg. 121, 409–445.Google Scholar
Hanlon, P. and Wales, D. B. (1989b) Eigenvalues connected with Brauer's centralizer algebras. J . Alg. 121, 446–476.Google Scholar
Hanlon, P., Stanley, R. and Stembridge, J. (1992) Some combinatorial aspects of the spectra of normally distributed random matrices. Contemp. Math. 138, 151–174.Google Scholar
Hansen, J. and Schmutz, E. (1991) How random is the characteristic polynomial of a random matrix. Preprint, Department of Mathematics, Drexel University.Google Scholar
Irwin, J. O. (1955) A unified derivation of some well-known frequency distribution of interest in biometry and statistics. J. R. Statist. Soc. 118, 389–404.Google Scholar
Larsen, M. (1993) The normal distribution as a limit of generalized Sato-Take measures. Technical report, Department of Mathematics, University of Pennsylvania.Google Scholar
Macdonald, I. (1979) Symmetric Functions and Hall Polynomials. Oxford University Press.Google Scholar
Mehta, M. L. (1991) Random Matrices. Academic Press, San Diego.Google Scholar
Pólya, G. and Read, R. (1987) Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds. Springer-Verlag, New York.Google Scholar
Ram, A. (1991) Characters of Brauer's centralizer algebras. Technical report, Department of Mathematics, University of Wisconsin.Google Scholar
Rudvalis, A. and Shinoda, K. (1991) An enumeration in finite classical groups. Preprint. Department of Mathematics, University of Massachusetts, Amherst.Google Scholar
Sagan, B. (1991) The Symmetric Group. Wadsworth, Pacific Grove, CA.Google Scholar
Shepp, L. and Lloyd, S. (1966) Ordered cycle length in a random permutation. Trans Amer. Math. Soc. 121, 340–357.Google Scholar
Stong, R. (1988) Some asymptotic results on finite vector spaces. Adv. Appl. Math. 9, 167–199.Google Scholar
Stong, R. (1992) The average order of a matrix. Preprint, Department of Mathematics, UCLA.Google Scholar
Takács, L. (1980) The problem of coincidences. Arch. Hist. Exact Sci. 21, 229–244.Google Scholar
Tracy, C. and Widom, H. (1992) Introduction to random matrices. Technical report, Department of Mathematics, University of California, Davis.Google Scholar
Vershik, A. M. and Kerov, S. V. (1981) Asymptotic theory of characters of the symmetric group. Funct. Anal. 15, 246–255.CrossRefGoogle Scholar
Watterson, G. (1974) Models for the logarithmic species abundance distribution. Theoret. Popn Biol. 6, 217–250.Google Scholar
Wenzl, H. (1988) On the structure of Brauer's centralizer algebras. Ann. Math. 128, 173–193.Google Scholar
Weyl, H. (1946) The Classical Groups. Princeton University Press.Google Scholar