Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Product of random projections, Jacobi ensembles and universality problems arising from free probability
Download PDF
Download PDF
  • Published: 15 March 2005

Product of random projections, Jacobi ensembles and universality problems arising from free probability

  • Benoît Collins1 

Probability Theory and Related Fields volume 133, pages 315–344 (2005)Cite this article

  • 335 Accesses

  • 70 Citations

  • Metrics details

Abstract.

We consider the product of two independent randomly rotated projectors. The square of its radial part turns out to be distributed as a Jacobi ensemble. We study its global and local properties in the large dimension scaling relevant to free probability theory. We establish asymptotics for one point and two point correlation functions, as well as properties of largest and smallest eigenvalues.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Abramowitz, M., Stegun, I.A. (eds.): Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publications Inc., New York, 1992. Reprint of the 1972 edition

  2. Askey, R.: Orthogonal polynomials and special functions. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975

  3. Bosbach, C., Gawronski, W.: Strong asymptotics for Jacobi polnomials with varying weights. Methods Appl. Anal. 6 (1), 39–54 (1999) Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part I

    Google Scholar 

  4. Capitaine, M., Casalis, M.: Asymptotic freeness by generalized moments for Gaussian and Wishart matrices. Applications to Beta random matrices. To appear in Indiana University Mathematics Journal November 2002

  5. Chen, L.-C., Ismail, M.E.H.: On asymptotics of Jacobi polynomials. SIAM J. Math. Anal. 22 (5), 1442–1449 (1991)

    Article  Google Scholar 

  6. Collins, B.: Intégrales matricielles et probabilités non-commutatives. Thése de doctorat de l'Université Paris 6, 2003

  7. Collins, B.: Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability. IMRN 17, 953–982 (2003)

    Article  Google Scholar 

  8. Deift, P.A.: Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Volume 3 of Courant Lecture Notes in Mathematics. New York University Courant Institute of Mathematical Sciences, New York, 1999

  9. Diaconis, P.W., Eaton, M.L., Lauritzen, S.L.: Finite de Finetti theorems in linear models and multivariate analysis. Scand. J. Statist. 19 (4), 289–315 (1992)

    Google Scholar 

  10. Doumerc, Y.: Matrix Jacobi process. Work in progress. 2003

  11. Forrester, P.: Log-gases and Random matrices, Chapter 2. http://www.ms.unimelb.edu.au/ matpjf/matpjf.html, 2002

  12. Gawronski, W., Shawyer, B.: Strong asymptotics and the limit distribution of the zeros of Jacobi polynomials In: Progress in approximation theory. Academic Press, Boston, MA, 1991, pp. 379–404

  13. Jiang, T.: Maxima of entries of Haar distributed matrices. preprint, available at http://www.stat.umn.edu/ tjiang/papers/haar1.pdf, 2003

  14. Johansson, K.: Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Commun. Math. Phys. 215 (3), 683–705 (2001)

    Article  Google Scholar 

  15. Kuijlaars, A., Vanlessen, M.: Universality for eigenvalue correlations from the modified Jacobi unitary ensemble. IMRN 30, 1575–1600 (2002)

    Article  Google Scholar 

  16. Ledoux, M.: Differential operators and spectral distributions of invariant ensembles from the classical orthogonal polynomials part I: the continuous case. To appear in Elect. J. Probab. November 2002

  17. Mehta, M.L.: Random matrices. Academic Press Inc., Boston, MA, second edition, 1991

  18. Moak, D.S., Saff, E.B., Varga, R.S.: On the zeros of Jacobi polynomials Trans. Am. Math. Soc. 249 (1), 159–162 (1979)

    Google Scholar 

  19. Nica, A., Speicher, R.: Lectures notes of the free probability semester at IHP. 2000

  20. Ol'shanskij, G.I.: Unitary representations of infinite dimensional pairs (g, k) and the formalism of R. Howe. Representation of Lie groups and related topics, Adv. Stud. Contemp. Math. 7, 269–463 (1990)

    Google Scholar 

  21. Soshnikov, A.: Universality at the edge of the spectrum in Wigner random matrices. Commun. Math. Phys. 207 (3), 697–733 (1999)

    Article  Google Scholar 

  22. Szegő, G.: Orthogonal polynomials. American Mathematical Society, Providence, R.I., fourth edition, 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII

  23. Voiculescu, D.V., Dykema, K.J., Nica, A.: Free random variables. American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups

  24. Voiculescu, D.: A strengthened asymptotic freeness result for random matrices with applications to free entropy. Internat. Math. Res. Notices (1), 41–63 (1998)

  25. Xu, F.: A random matrix model from two-dimensional Yang-Mills theory. Commun. Math. Phys. 190 (2), 287–307 (1997)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science Kyoto University, Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto, 606-8502, Japan

    Benoît Collins

Authors
  1. Benoît Collins
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Benoît Collins.

Additional information

B.C. is currently a JSPS postdoctoral fellow

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Collins, B. Product of random projections, Jacobi ensembles and universality problems arising from free probability. Probab. Theory Relat. Fields 133, 315–344 (2005). https://doi.org/10.1007/s00440-005-0428-5

Download citation

  • Received: 06 January 2004

  • Revised: 22 December 2004

  • Published: 15 March 2005

  • Issue Date: November 2005

  • DOI: https://doi.org/10.1007/s00440-005-0428-5

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Correlation Function
  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Local Property
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature