On the Diaconis-Shahshahani Method in Random Matrix Theory
Michael Stolz
Ruhr-Universität Bochum Fakultät für Mathematik NA 4/30 D-44780 Bochum Germany NA 4/30 D-44780 Bochum Germany
DOI: 10.1007/s10801-005-4629-x
Abstract
If Γ is a random variable with values in a compact matrix group K, then the traces Tr(Γ j) ( j ϵ N) are real or complex valued random variables. As a crucial step in their approach to random matrix eigenvalues, Diaconis and Shahshahani computed the joint moments of any fixed number of these traces if Γ is distributed according to Haar measure and if K is one of U n, O n or Sp n, where n is large enough. In the orthogonal and symplectic cases, their proof is based on work of Ram on the characters of Brauer algebras.
The present paper contains an alternative proof of these moment formulae. It invokes classical invariant theory (specifically, the tensor forms of the First Fundamental Theorems in the sense of Weyl) to reduce the computation of matrix integrals to a counting problem, which can be solved by elementary means.
Pages: 471–491
Keywords: keywords random matrices; matrix integrals; classical invariant theory; tensor representations; Schur-Weyl duality
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References
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2. P. Diaconis and S.N. Evans, “Linear functionals of eigenvalues of random matrices,” Trans. Amer. Math. Soc. 353 (2001), 2615-2633.
3. P. Diaconis and M. Shahshahani, “On the eigenvalues of random matrices,” J. Appl. Probab. 31A (1994), 49-62.
4. R. Goodman and N.R. Wallach, Representations and Invariants of the Classical Groups, Cambridge UP, New York, 1998.
5. W.H. Greub, Multilinear Algebra, Springer, Berlin, 1967.
6. G. Hochschild, The Structure of Lie Groups, Holden Day, San Francisco, 1965.
7. C.P. Hughes and Z. Rudnick, “Mock-Gaussian behaviour for linear statistics of classical compact groups,” J. Phys. A 36 (2003), 2919-2932.
8. L. Pastur and V. Vasilchuk, “On the moments of traces of matrices of classical groups,” Commun. Math. Phys. 252 (2004), 149-166.
9. A. Ram, “Characters of Brauer's centralizer algebras,” Pacific J. Math. 169 (1995), 173-200.
10. A. Ram, “Second orthogonality relation for characters of Brauer algebras,” European J. Combin. 18 (1997), 685-706.
11. V.S. Varadarajan, Lie Groups, Lie Algebras, and their Representations, Springer, New York, 1984.
12. H. Weyl, The Classical Groups. Their Invariants and Representations, 2nd ed., Princeton UP, Princeton, 1953.
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