Skip to main content

Free Probability Theory and Random Matrices

  • Chapter
  • First Online:
Asymptotic Combinatorics with Applications to Mathematical Physics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1815))

Abstract

Free probability theory originated in the context of operator algebras, however, one of the main features of that theory is its connection with random matrices.Indeed, free probability can be considered as the theory providing concepts and notations, without relying on random matrices, for dealing with the limit N → ∞ of N × N-random matrices.

Research supported by a grant of NSERC, Canada

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Hiai, F., Petz, D.: The semicircle law,free random variables and entropy. Mathematical Surveys and Monographs,Vol.77, AMS (2000)

    Google Scholar 

  2. Kreweras, G.: Sur les partitions non-croisees d’un cycle.Discrete Math.,1, 333–350 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  3. Nica, A.: R-transforms of free joint distributions,and non-crossing partitions.J.Funct.Anal.,135, 271–296 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  4. Nica, A., Speicher, R.: On the multiplication of free n-tuples of non-commutative random variables (with an appendix by D.Voiculescu).Amer.J.Math.,118.799–837 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  5. Nica, A., Speicher, R.: R-diagonal pairs—a common approach to Haar unitaries and circular elements.In: Voiculescu, D.-V.(ed) Free Probability Theory,AMS, 149–188 (1997)

    Google Scholar 

  6. Nica, A., Speicher, R.: Commutators of free random variables. Duke Math.J., 92, 553–592 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. Speicher, R.: Free convolution and the random sum of matrices.RIMS,29, 731–744 (1993)

    MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  9. Speicher, R.: Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Memoirs of the AMS,627 (1998)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Voiculescu, D.: Free probability theory:random matrices and von Neumann algebras.Proceedings of the ICM 1994, Birkhäuser, 227–241 (1995)

    Google Scholar 

  13. Voiculescu, D. (ed): Free Probability Theory.Fields Institute Communications, vol.12, AMS (1997)

    Google Scholar 

  14. Voiculescu, D.V., Dykema, K.J., Nica, A.: Free Random Variables.CRM Monograph Series,vol.1,AMS (1993)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Speicher, R. (2003). Free Probability Theory and Random Matrices. In: Vershik, A.M., Yakubovich, Y. (eds) Asymptotic Combinatorics with Applications to Mathematical Physics. Lecture Notes in Mathematics(), vol 1815. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44890-X_3

Download citation

  • DOI: https://doi.org/10.1007/3-540-44890-X_3

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40312-8

  • Online ISBN: 978-3-540-44890-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics