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Limit laws for random matrices and free products. (English) Zbl 0736.60007
The starting point of this paper are known limit theorems for free products of non-commutative random variables and random selfadjoint matrices where the semicircle law arises as limit law; see author [J. Funct. Anal. 66, 323-346 (1986; Zbl 0651.46063) and Operator algebras and their connection with topology and ergodic theory, Proc. Conf., Buşteni/Rom. 1983, Lect. Notes Math. 1132, 556-588 (1985; Zbl 0618.46048)]. This paper is devoted to the connections of these results. For this, the author recapitulates the concept of free and asymptotically free non-commutative random variables and establishes a limit result for sums of free random variables. If one applies this result to selfadjoint random matrices with independent Gaussian random variables as entries, then these matrices form an asymptotically free family for which a semicircle law holds if the dimension tends to infinity. Applications to random unitary and orthogonal matrices as well as to Grassmannian manifolds are also given.
Reviewer: M.Voit (München)

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60B10 Convergence of probability measures
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
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