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Borel theorems for random matrices from the classical compact symmetric spaces. (English) Zbl 1149.15016

The paper deals with Borel theorems for random matrices. The authors study random vectors of the form
\[ (\text{Tr}(A^{(1)}V), \text{Tr}(A^{(2)}V), \ldots, \text{Tr}(A^{(r)}V)), \]
where \(V\) is a uniformly distributed element of a matrix version of a classical compact symmetric space, and \(A^{(i)}\), \(i=1,2,\dots,r\), are deterministic parameter matrices. They show that for increasing matrix sizes these random vectors converge to a joint Gaussian limit, and compute its covariances. This generalizes previous results of P. Diaconis and S. N. Evans [Trans. Am. Math. Soc. 353, No. 7, 2615–2633 (2001; Zbl 1008.15013)] for Haar distributed matrices from the classical compact groups.
The authors review the extended Wick calculus and explain its connection with the Diaconis-Shahshahani integral [P. Diaconis and M. Shahshahani, Applied Probability Trust, 49–62 (1954; Zbl 0807.15015)]. The proof of the main results uses these tools and the integration formulas, due to B. Collins and P. Ṡniady [Commun. Math. Phys. 264, No. 3, 773–795 (2006; Zbl 1108.60004)], for polynomial functions on the classical compact groups.

MSC:

15B52 Random matrices (algebraic aspects)
60F05 Central limit and other weak theorems
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
43A75 Harmonic analysis on specific compact groups
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