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Non-commutative polynomials of independent Gaussian random matrices. The real and symplectic cases.
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  • Published: 05 July 2004

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

  • Hanne Schultz1 

Probability Theory and Related Fields volume 131, pages 261–309 (2005)Cite this article

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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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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230, Odense M, Denmark

    Hanne Schultz

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  1. Hanne Schultz
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Correspondence to Hanne Schultz.

Additional information

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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Cite this article

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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  • Received: 14 November 2003

  • Revised: 06 April 2004

  • Published: 05 July 2004

  • Issue Date: February 2005

  • DOI: https://doi.org/10.1007/s00440-004-0366-7

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Keywords

  • Stochastic Process
  • Probability Theory
  • Matrix Model
  • Mathematical Biology
  • Probability Space
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