Borel theorems for random matrices from the classical compact symmetric spaces

Abstract

We study random vectors of the form $(\mathrm{Tr}(A^{(1)}V),...,\mathrm{Tr}(A^{(r)}V))$, where $V$ is a uniformly distributed element of a matrix version of a classical compact symmetric space, and the $A^{(\nu )}$ 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.

Published in at http://dx.doi.org/10.1214/07-AOP341 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
• Symmetry classes of disordered fermions
• Large deviations for random matrix ensembles in mesoscopic physics

Cited by in corpus (8)

• On some properties of orthogonal Weingarten functions
• Weingarten calculus for matrix ensembles associated with compact symmetric spaces
• Rate of convergence of linear functions on the unitary group
• General moments of matrix elements from circular orthogonal ensembles
• Lyapunov spectra for all symmetry classes of quasi-one-dimensional disordered systems of non-interacting Fermions
• Stein's Method and Characters of Compact Lie Groups
• Matrix Group Integrals, Surfaces, and Mapping Class Groups II: $\mathrm{O}\left(n\right)$ and $\mathrm{Sp}\left(n\right)$
• Euclidean distance between Haar orthogonal and gaussian matrices