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Free Probability, Random Matrices, and Representations of Non-commutative Rational Functions

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Computation and Combinatorics in Dynamics, Stochastics and Control (Abelsymposium 2016)

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Abstract

A fundamental problem in free probability theory is to understand distributions of “non-commutative functions” in freely independent variables. Due to the asymptotic freeness phenomenon, which occurs for many types of independent random matrices, such distributions can describe the asymptotic eigenvalue distribution of corresponding random matrix models when their dimension tends to infinity. For non-commutative polynomials and rational functions, an algorithmic solution to this problem is presented. It relies on suitable representations for these functions.

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Acknowledgements

This work was supported by the ERC Advanced Grant “Non-commutative Distributions in Free Probability” (grant no. 339760).

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Authors and Affiliations

  1. Fachrichtung Mathematik, Universität des Saarlandes, Saarbrücken, Germany

    Tobias Mai & Roland Speicher

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  1. Tobias Mai
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  2. Roland Speicher
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Correspondence to Roland Speicher .

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Editors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Elena Celledoni

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Giulia Di Nunno

  3. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  4. Department of Mathematics, University of Bergen, Bergen, Norway

    Hans Zanna Munthe-Kaas

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Mai, T., Speicher, R. (2018). Free Probability, Random Matrices, and Representations of Non-commutative Rational Functions. In: Celledoni, E., Di Nunno, G., Ebrahimi-Fard, K., Munthe-Kaas, H. (eds) Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-030-01593-0_19

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