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

  • Tobias Mai
  • Roland SpeicherEmail author
Conference paper
Part of the Abel Symposia book series (ABEL, volume 13)

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.

Notes

Acknowledgements

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

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Fachrichtung MathematikUniversität des SaarlandesSaarbrückenGermany

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