The Annals of Probability

Borel theorems for random matrices from the classical compact symmetric spaces

Benoît Collins and Michael Stolz

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Abstract

We study random vectors of the form (Tr(A(1)V), …, Tr(A(r)V)), where V is a uniformly distributed element of a matrix version of a classical compact symmetric space, and the A(ν) are deterministic parameter matrices. We show that for increasing matrix sizes these random vectors converge to a joint Gaussian limit, and compute its covariances. This generalizes previous work of Diaconis et al. for Haar distributed matrices from the classical compact groups. The proof uses integration formulas, due to Collins and Śniady, for polynomial functions on the classical compact groups.

Article information

Source
Ann. Probab., Volume 36, Number 3 (2008), 876-895.

Dates
First available in Project Euclid: 9 April 2008

Permanent link to this document
https://projecteuclid.org/euclid.aop/1207749084

Digital Object Identifier
doi:10.1214/07-AOP341

Mathematical Reviews number (MathSciNet)
MR2408577

Zentralblatt MATH identifier
1149.15016

Subjects
Primary: 15A52 60F05: Central limit and other weak theorems
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 43A75: Analysis on specific compact groups

Keywords
Random matrices symmetric spaces central limit theorem matrix integrals classical invariant theory

Citation

Collins, Benoît; Stolz, Michael. Borel theorems for random matrices from the classical compact symmetric spaces. Ann. Probab. 36 (2008), no. 3, 876--895. doi:10.1214/07-AOP341. https://projecteuclid.org/euclid.aop/1207749084


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References

  • [1] Altland, A. and Zirnbauer, M. (1997). Nonstandard symmetry classes in mesoscopic normal/superconducting hybrid structures. Phys. Rev. B 55 1142–1161.
  • [2] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
  • [3] Collins, B. (2003). Moments and cumulants of polynomial random variables on unitary groups, the Itzykson–Zuber integral, and free probability. Int. Math. Res. Not. 17 953–982.
  • [4] Collins, B. and Śniady, P. (2006). Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Comm. Math. Phys. 264 773–795.
  • [5] D’Aristotile, A., Diaconis, P. and Newman, C. (2003). Brownian motion and the classical groups. Probability, Statistics and Their Applications: Papers in Honor of Rabi Bhattacharya 97–116. IMS, Beachwood, OH.
  • [6] Diaconis, P., Eaton, M. and Lauritzen, S. (1992). Finite de Finetti theorems in linear models and multivariate analysis. Scand. J. Statist. 19 289–315.
  • [7] Diaconis, P. and Evans, S. (2001). Linear functionals of eigenvalues of random matrices. Trans. Amer. Math. Soc. 353 2615–2633.
  • [8] Diaconis, P. and Shahshahani, M. (1994). On the eigenvalues of random matrices. J. Appl. Probab. 31A 49–62.
  • [9] Dueñez, E. (2004). Random matrix ensembles associated to compact symmetric spaces. Comm. Math. Phys. 244 29–61.
  • [10] Dyson, F. (1962). The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics. J. Math. Phys. 3 1199–1215.
  • [11] Eichelsbacher, P. and Stolz, M. (2006). Large deviations for random matrix ensembles in mesoscopic physics. Markov Process. Related Fields. To appear.
  • [12] Goodman, R. and Wallach, N. (1998). Representations and Invariants of the Classical Groups. Cambridge Univ. Press.
  • [13] Heinzner, P., Huckleberry, A. and Zirnbauer, M. (2005). Symmetry classes of disordered fermions. Comm. Math. Phys. 257 725–771.
  • [14] Helgason, S. (2001). Differential Geometry, Lie Groups, and Symmetric Spaces. Amer. Math. Soc., Providence, RI.
  • [15] Meckes, E. (2006). An infinitesimal version of Stein’s method of exchangeable pairs. Ph.D. thesis, Stanford Univ.
  • [16] Stolz, M. (2005). On the Diaconis–Shahshahani method in random matrix theory. J. Algebraic Combin. 22 471–491.
  • [17] Weingarten, D. (1978). Asymptotic behavior of group integrals in the limit of infinite rank. J. Math. Phys. 19 999–1001.