Abstract.
In [HT2] Haagerup and Thorbjo rnsen prove the following extension of Voiculescu’s random matrix model (cf. [V2, Theorem 2.2]): For each n ∈ ℕ, let X1(n),..., X r (n) be r independent complex self-adjoint random matrices from the class and let x1,...,x r be a semicircular system in a C*-probability space. Then for any polynomial p in r non-commuting variables the convergence
holds almost surely. We generalize this result to sets of independent Gaussian random matrices with real or symplectic entries (the GOE- and the GSE-ensembles) and random matrix ensembles related to these.
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This work was partially supported by MaPhySto – A Network in Mathematical Physics and Stochastics, funded by The Danish National Research Foundation.
As a student of the PhD-school OP-ALG-TOP-GEO the author is partially supported by the Danish Research Training Council.
Acknowledgement I would like to thank my advisor, Uffe Haagerup, with whom I had many enlightening discussions, and who made some important contributions to this paper. Also, thanks to Steen Thorbjørnsen who took time to answer several questions.
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Schultz, H. Non-commutative polynomials of independent Gaussian random matrices. The real and symplectic cases.. Probab. Theory Relat. Fields 131, 261–309 (2005). https://doi.org/10.1007/s00440-004-0366-7
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DOI: https://doi.org/10.1007/s00440-004-0366-7