Skip to main content
SpringerLink
Log in

Dyson Processes Associated with Associative Algebras: The Clifford Case

  • Chapter
  • First Online:
Geometric Aspects of Functional Analysis

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2116))

Abstract

We consider Brownian motions and other processes (Ornstein-Uhlenbeck processes, spherical Brownian motions) on various sets of symmetric matrices constructed from algebra structures, and look at their associated spectral measure processes. This leads to the identification of the multiplicity of the eigenvalues, together with the identification of the spectral measures. For Clifford algebras, we thus recover Bott’s periodicity.

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

Access this chapter

Log in via an institution
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 EPUB and 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

References

  1. G.W. Anderson, A. Guionnet, O. Zeitouni, An introduction to random matrices, in Cambridge Studies in Advanced Mathematics, vol. 118 (Cambridge University Press, Cambridge, 2010)

    Google Scholar 

  2. M.F. Atiyah, R. Bott, A. Shapiro, Clifford modules. Topology 3, 3–38 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  3. M.F. Atiyah, I.M. Singer, Index theory for skew-adjoint fredholm operators. Inst. Hautes Études Sci. Publ. Math. 37, 5–26 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Bakry, I. Gentil, M. Ledoux, Analysis and Geometry of Markov Diffusion Operators. Grundlehren der mathematischen Wissenschaften, vol. 348 (Springer, Berlin, 2013)

    Google Scholar 

  5. F. Chapon, M. Defosseux, Quantum random walks and minors of hermitian brownian motion. Can. J. Math. 64(4), 805–821 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Y. Doumerc, Matrices aléatoires, processus stochastiques et groupes de réflexions. Ph.D. thesis, Université Toulouse 3, 2005

    Google Scholar 

  7. F.J. Dyson, A brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191–1198 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Erdős, S. Péché, J.A. Ramírez, B. Schlein, H.-T. Yau, Bulk universality for wigner matrices. Commun. Pure Appl. Math. 63, 895–925 (2010)

    Google Scholar 

  9. L. Erdős, J. Ramírez, B. Schlein, T. Tao, V. Vu, H.-T. Yau, Bulk universality for wigner hermitian matrices with subexponential decay. Math. Res. Lett. 17, 667–674 (2010)

    Article  MathSciNet  Google Scholar 

  10. L. Erdős, A. Knowles, H.-T. Yau, J. Yin, The local semicircle law for a general class of random matrices. Electron. J. Probab. 18, 1–58 (2013)

    MathSciNet  Google Scholar 

  11. P.J. Forrester, Log-gases and random matrices. London Mathematical Society Monographs Series, vol. 34 (Princeton University Press, Princeton, 2010)

    Google Scholar 

  12. A. Gamburd, Some applications of symmetric functions theory in random matrix theory, in Ranks of Elliptic Curves and Random Matrix Theory, ed. by J.B. Conrey, D.W. Farmer, F. Mezzadri, N.C. Snaith. London Mathematical Society, Lecture Notes Series, vol. 341 (Cambridge University Press, Cambridge, 2007), pp. 143–169

    Google Scholar 

  13. R. Jagannathan, On generalized clifford algebras and their physical applications, in The legacy of Alladi Ramakrishnan in the Mathematical Sciences (Springer, New York, 2010), pp. 465–489

    Google Scholar 

  14. M. Karoubi, Algèbres de clifford et K-théorie. Ann. Sci. École Norm. Sup. (4) 1, 161–270 (1968)

    Google Scholar 

  15. M. Ledoux, Differential operators and spectral distributions of invariant ensembles from the classical orthogonal polynomials. The continuous case. Electron. J. Probab. 9, 177–208 (2004)

    MathSciNet  Google Scholar 

  16. M. Ledoux, Differential operators and spectral distributions of invariant ensembles from the classical orthogonal polynomials. the discrete case. Electron. J. Probab. 10, 1116–1146 (2005)

    Google Scholar 

  17. N. Lehmann, H.-J. Sommers, Eigenvalue statistics of random real matrices. Phys. Rev. Lett. 67(8), 941–944 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. P. Lounesto, Clifford Algebras and Spinors. London Mathematical Society Lecture Note Series, 2nd edn., vol. 286 (Cambridge University Press, Cambridge, 2001)

    Google Scholar 

  19. L.A. Marčenko, V.A. Pastur, Distribution of eigenvalues in certain sets of random matrices. Matematicheskiĭ Sbornik (N.S.) 72(114), 507–536 (1967)

    Google Scholar 

  20. M.L. Mehta, Random Matrices. Pure and Applied Mathematics (Amsterdam), vol. 142 (Elsevier/Academic, Amsterdam, 2004)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. E.P. Wigner, On the distribution of the roots of certain symmetric matrices. Ann. Math. 67, 325–327 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  23. M.R. Zirnbauer, Symmetry classes (2010). arXiv:1001.0722

    Google Scholar 

  24. M.R. Zirnbauer, Symmetry classes in random matrix theory (2004). arXiv:0404058

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse, France

    Dominique Bakry

  2. Laboratoire d’Analyse et Mathématiques appliquées, UMR CNRS 8050, Université Paris-Est -Créteil, 61 av du Général de Gaulle, 94010, Créteil Cedex, France

    Marguerite Zani

Authors
  1. Dominique Bakry
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marguerite Zani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dominique Bakry .

Editor information

Editors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Bo'az Klartag

  2. Technion - Israel Institute of Technology, Haifa, Israel

    Emanuel Milman

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Bakry, D., Zani, M. (2014). Dyson Processes Associated with Associative Algebras: The Clifford Case. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2116. Springer, Cham. https://doi.org/10.1007/978-3-319-09477-9_1

Download citation

Publish with us

Policies and ethics

Search

Navigation