Skip to main content
SpringerLink
Log in

Free Probability Theory

And Its Avatars in Representation Theory, Random Matrices, and Operator Algebras; also Featuring: Non-commutative Distributions

  • Survey Article
  • Published:
Jahresbericht der Deutschen Mathematiker-Vereinigung Aims and scope Submit manuscript

Abstract

This article is an invitation to the world of free probability theory. This theory was introduced by Dan Voiculescu at the beginning of the 1980’s and has developed since then into a vibrant and very active theory which connects with many different branches of mathematics. We will motivate Voiculescu’s basic notion of “freeness”, and relate it with problems in representation theory, random matrices, and operator algebras. The notion of “non-commutative distributions” is one of the central objects of the theory and awaits a deeper understanding.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Akemann, G., Baik, J., Di Francesco, P.: The Oxford Handbook of Random Matrix Theory. Oxford University Press, London (2011)

    MATH  Google Scholar 

  2. Anderson, G., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  3. Banica, T., Speicher, R.: Liberation of orthogonal Lie groups. Adv. Math. 222(4), 1461–1501 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Belinschi, S.T., Bercovici, H.: A new approach to subordination results in free probability. J. Anal. Math. 101(1), 357–365 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Belinschi, S.T., Mai, T., Speicher, R.: Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem. Journal für die reine und angewandte Mathematik (Crelles Journal) (2013)

  6. Bercovici, H., Voiculescu, D.: Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42(3), 733–774 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Biane, P.: Representations of symmetric groups and free probability. Adv. Math. 138(1), 126–181 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Biane, P.: Free probability and combinatorics. In: Proc. ICM 2002, Beijing, vol. II, pp. 765–774 (2002)

    Google Scholar 

  9. Cébron, G., Dahlqvist, A., Male, C.: Universal constructions for spaces of traffics. arXiv preprint (2016). arXiv:1601.00168

  10. Chistyakov, G., Götze, F.: The arithmetic of distributions in free probability theory. Open Math. 9(5), 997–1050 (2011)

    MathSciNet  MATH  Google Scholar 

  11. Couillet, R., Debbah, M.: Random Matrix Methods for Wireless Communications. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  12. Dykema, K.: Interpolated free group factors. Pac. J. Math. 163(1), 123–135 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Féray, V., Śniady, P.: Asymptotics of characters of symmetric groups related to Stanley character formula. Ann. Math. 173(2), 887–906 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Figa-Talamanca, A., Steger, T.: Harmonic analysis for anisotropic random walks on homogeneous trees. Mem. Am. Math. Soc. 110(531), 531 (1994)

    MathSciNet  MATH  Google Scholar 

  15. Guionnet, A., Shlyakhtenko, D.: Free monotone transport. Invent. Math. 197(3), 613–661 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Haagerup, U.: On Voiculescu’s R- and S-transforms for free non-commuting random variables. In: Free Probability Theory, Fields Institute Communications, vol. 12, pp. 127–148 (1997)

    Google Scholar 

  17. Haagerup, U.: Random matrices, free probability and the invariant subspace problem relative to a von Neumann algebra. In: Proc. ICM 2002, Beijing, vol. I, pp. 273–290 (2002)

    Google Scholar 

  18. Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V.: Foundations of Free Noncommutative Function Theory, vol. 199. Am. Math. Soc., Providence (2014)

    MATH  Google Scholar 

  19. Kerov, S.V.: Transition probabilities for continual young diagrams and the Markov moment problem. Funct. Anal. Appl. 27(2), 104–117 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lehner, F.: On the computation of spectra in free probability. J. Funct. Anal. 183(2), 451–471 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Maassen, H.: Addition of freely independent random variables. J. Funct. Anal. 106(2), 409–438 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mingo, J., Speicher, R.: Free Probability and Random Matrices. Fields Institute Monographs. Springer (2017, to appear)

  23. Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series, vol. 335. Cambridge University Press, Cambridge (2006). doi:10.1017/CBO9780511735127

    Book  MATH  Google Scholar 

  24. Novak, J., LaCroix, M.: Three lectures on free probability. arXiv preprint (2012). arXiv:1205.2097

  25. Novak, J., Sniady, P.: What is …a free cumulant? Not. Am. Math. Soc. 58(2), 300–301 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Radulescu, F.: Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index. Invent. Math. 115(1), 347–389 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  27. Raum, S., Weber, M.: The full classification of orthogonal easy quantum groups. Commun. Math. Phys. 341(3), 751–779 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Shlyakhtenko, D.: Free probability, planar algebras, subfactors and random matrices. In: Proc. ICM 2010, Hyderabad, vol. III, pp. 1603–1623 (2010)

    Google Scholar 

  29. Speicher, R.: Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math. Ann. 298(1), 611–628 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  30. Speicher, R.: Free probability and random matrices. In: Proc. ICM 2014, vol. III, pp. 477–501 (2014)

    Google Scholar 

  31. Tulino, A.M., Verdú, S.: Random Matrix Theory and Wireless Communications. Foundations and Trends® in Communications and Information Theory: Vol. 1. Now Publishers, Hanover (2004)

    MATH  Google Scholar 

  32. Vershik, A.M., Kerov, S.V.: Asymptotic theory of characters of the symmetric group. Funct. Anal. Appl. 15(4), 246–255 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  33. Voiculescu, D.: Symmetries of some reduced free product C*-algebras. In: Operator Algebras and Their Connections with Topology and Ergodic Theory, pp. 556–588 (1985)

    Chapter  Google Scholar 

  34. Voiculescu, D.: Addition of certain non-commuting random variables. J. Funct. Anal. 66(3), 323–346 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  35. Voiculescu, D.: Circular and semicircular systems and free product factors. In: Operator Algebras, Unitary Representations, Enveloping Algebras and Invariant Theory, Paris, 1989, vol. 92, pp. 45–60 (1990)

    Google Scholar 

  36. Voiculescu, D.: Limit laws for random matrices and free products. Invent. Math. 104(1), 201–220 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  37. Voiculescu, D.: The analogues of entropy and of Fisher’s information measure in free probability theory III: the absence of Cartan subalgebras. Geom. Funct. Anal. 6(1), 172–199 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  38. Voiculescu, D.: A strengthened asymptotic freeness result for random matrices with applications to free entropy. Int. Math. Res. Not. 1998, 41–63 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  39. Voiculescu, D.V.: Aspects of free analysis. Jpn. J. Math. 3(2), 163–183 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. Williams, J.D.: Analytic function theory for operator-valued free probability. Journal für die reine und angewandte Mathematik (Crelles Journal) (2013)

  41. Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics, vol. 138. Cambridge University Press, Cambridge (2000)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

I would like to thank Philippe Biane, Franz Lehner, Hannah Markwig and Jonathan Novak, as well as an anonymous referee, for suggestions to improve the presentation.

Author information

Authors and Affiliations

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

    Roland Speicher

Authors
  1. Roland Speicher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roland Speicher.

Additional information

Supported by the ERC Advanced Grant “Non-commutative Distributions in Free Probability”.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Speicher, R. Free Probability Theory. Jahresber. Dtsch. Math. Ver. 119, 3–30 (2017). https://doi.org/10.1365/s13291-016-0150-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1365/s13291-016-0150-5

Keywords

Mathematics Subject Classification (2000)

Search

Navigation