Non-commutative polynomials of independent Gaussian random matrices. The real and symplectic cases.

In HT2] Haagerup and Thorbjo rnsen prove the following extension of Voiculescus random matrix model (cf. V2, Theorem 2.2]): For each 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 convergenceholds 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.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.