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Monatshefte für Mathematik

, Volume 166, Issue 1, pp 121–144 | Cite as

Ensemble averages when β is a square integer

  • Christopher D. SinclairEmail author
Article

Abstract

We give a hyperpfaffian formulation of partition functions and ensemble averages for Hermitian and circular ensembles when L is an arbitrary integer and β = L 2 and when L is an odd integer and β = L 2 + 1.

Keywords

Random matrix theory Partition function Pfaffian Hyperpfaffian Selberg integral 

Mathematics Subject Classification (2010)

15B52 82C22 60G55 

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Supplementary material

605_2011_371_MOESM1_ESM.tex (48 kb)
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References

  1. 1.
    Adler M., Forrester P.J., Nagao T., van Moerbeke P.: Classical skew orthogonal polynomials and random matrices. J. Stat. Phys. 99(1–2), 141–170 (2000)zbMATHCrossRefGoogle Scholar
  2. 2.
    Askey R.: Some basic hypergeometric extensions of integrals of Selberg and Andrews. SIAM J. Math. Anal. 11(6), 938–951 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Barvinok A.I.: New algorithms for linear k-matroid intersection and matroid k-parity problems. Math. Program. 69(3, Ser. A), 449–470 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Borodin, A.: Determinantal point processes. In: Oxford Handbook of Random Matrix Theory. Oxford University Press, Oxford (2010)Google Scholar
  5. 5.
    Borodin A., Sinclair C.D.: The Ginibre ensemble of real random matrices and its scaling limits. Commun. Math. Phys. 291(1), 177–224 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bostan, A., Dumas, P.: Wronskians and linear independence. Am. Math. Mon. (to appear)Google Scholar
  7. 7.
    Redelmeier, D.: Hyperpfaffians in algebraic combinatorics. Master’s thesis, University of Waterloo. http://uwspace.uwaterloo.ca/handle/10012/1055 (2006)
  8. 8.
    de Bruijn, N.G.: On some multiple integrals involving determinants. J. Indian Math. Soc. (N.S.) 19, 133–151 (1955)Google Scholar
  9. 9.
    Dyson F.J.: Statistical theory of the energy levels of complex systems. I. J. Math. Phys. 3, 140–156 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Dyson F.J.: Correlations between eigenvalues of a random matrix. Commun. Math. Phys. 19, 235–250 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Forrester P.J., Warnaar S.O.: The importance of the Selberg integral. Bull. Am. Math. Soc. (N.S.) 45(4), 489–534 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Good I.J.: Short proof of a conjecture by Dyson. J. Math. Phys. 11, 1884 (1970). doi: 10.1063/1.1665339 MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gunson J.: Proof of a conjecture by Dyson in the statistical theory of energy levels. J. Math. Phys. 3(4), 752–753 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Zeros of Gaussian analytic functions and determinantal point processes. University Lecture Series, vol. 51. American Mathematical Society, Providence, RI (2009)Google Scholar
  15. 15.
    Ishikawa M., Okada S., Tagawa H., Zeng J.: Generalizations of Cauchy’s determinant and Schur’s Pfaffian. Adv. Appl. Math. 36(3), 251–287 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Luque J.-G., Thibon J.-Y.: Pfaffian and Hafnian identities in shuffle algebras. Adv. Appl. Math. 29(4), 620–646 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Mahoux G., Mehta M.L.: A method of integration over matrix variables. IV. J. Phys. I 1(8), 1093–1108 (1991)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Matsumoto S.: Hyperdeterminantal expressions for Jack functions of rectangular shapes. J. Algebra 320(2), 612–632 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Mehta M.L.: Random matrices. Pure and Applied Mathematics, vol. 142 (Amsterdam). 3rd edn. Elsevier/Academic Press, Amsterdam (2004)Google Scholar
  20. 20.
    Mehta M.L., Dyson F.J.: Statistical theory of the energy levels of complex systems. V. J. Math. Phys. 4, 713–719 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Meray C.: Sur un determinant dont celui de Vandermonde n’est qu’un particulier. Revue de Mathématiques Spéciales 9, 217–219 (1899)Google Scholar
  22. 22.
    Rains E.M.: Correlation functions for symmetrized increasing subsequences. http://arXiv.org:math/0006097 (2000)
  23. 23.
    Selberg A.: Bemerkninger om et multipelt integral. Norsk Mat. Tidsskr 26, 71–78 (1944)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Selberg, A.: Collected papers, vol. I. Springer-Verlag, Berlin (1989) (with a foreword by K. Chandrasekharan)Google Scholar
  25. 25.
    Sinclair C.D.: Correlation functions for β = 1 ensembles of matrices of odd size. J. Stat. Phys. 136(1), 17–33 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Tracy C.A., Widom H.: Correlation functions, cluster functions, and spacing distributions for random matrices. J. Stat. Phys. 92(5–6), 809–835 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Zeilberger D.: A combinatorial proof of Dyson’s conjecture. Discret. Math. 41(3), 317–321 (1982)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OregonEugeneUSA

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