Journal of Theoretical Probability

, Volume 20, Issue 3, pp 505–533 | Cite as

Cumulants for Random Matrices as Convolutions on the Symmetric Group, II

Article

Abstract

In a previous paper we defined some “cumulants of matrices” which naturally converge toward the free cumulants of the limiting non commutative random variables when the size of the matrices tends to infinity. Moreover these cumulants satisfied some of the characteristic properties of cumulants whenever the matrix model was invariant under unitary conjugation. In this paper we present the fitting cumulants for random matrices whose law is invariant under orthogonal conjugation. The symplectic case could be carried out in a similar way.

Keywords

Cumulants Random matrices Matrix moments Free probability 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Biane, P.: Minimal factorizations of a cycle and central multiplicative functions on the infinite symmetric group. J. Comb. Theory Ser. A 76, 197–212 (1996) MATHCrossRefGoogle Scholar
  2. 2.
    Biane, P.: Some properties of crossings and partitions. Discret. Math. 175, 41–53 (1997) MATHCrossRefGoogle Scholar
  3. 3.
    Capitaine, M., Casalis, M.: Asymptotic freeness by generalized moments for Gaussian and Wishart matrices. Application to Beta random matrices. Indiana Univ. Math. J. 53(2), 397–431 (2004) MATHCrossRefGoogle Scholar
  4. 4.
    Capitaine, M., Casalis, M.: Cumulants for random matrices as convolutions on the symmetric group. Probab. Theory Relat. Fields 136(1), 19–36 (2006) MATHCrossRefGoogle Scholar
  5. 5.
    Collins, B., Sniady, P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264, 773–795 (2006) MATHCrossRefGoogle Scholar
  6. 6.
    Graczyk, P., Letac, G., Massam, H.: The hyperoctahedral group, symmetric group representations and the moments of the real Wishart distribution. J. Theor. Probab. 18, 1–42 (2005) MATHCrossRefGoogle Scholar
  7. 7.
    Krawczyk, B., Speicher, R.: Combinatorics of free cumulants. J. Comb. Theory Ser. A 90, 267–292 (2000) MATHCrossRefGoogle Scholar
  8. 8.
    Letac, G.: Problèmes classiques de probabilité sur un couple de Gelfand. In: Analytical Methods in Probability Theory, Proceedings Oberwolfach. Lecture Notes, vol. 861, pp. 93–116 (1980) Google Scholar
  9. 9.
    Nica, A., Speicher, R.: On the multiplication of free N-uples of noncommutative random variables. Am. J. Math. 118, 799–837 (1996) MATHCrossRefGoogle Scholar
  10. 10.
    Olkin, I., Rubin, H.: A characterization of the Wishart distribution. Ann. Math. Stat. 33, 1272–1280 (1962) Google Scholar
  11. 11.
    Olkin, I., Rubin, H.: Multivariate Beta distributions and independence properties of the Wishart distribution. Ann. Math. Stat. 35, 261–269 (1964) Google Scholar
  12. 12.
    Speicher, R.: Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math. Ann. 298, 611–628 (1994) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.CNRS, Institut de Mathématiques de Toulouse, Equipe de Statistique et ProbabilitésUniversité Paul SabatierToulouse Cedex 09France
  2. 2.Institut de Mathématiques de Toulouse, Equipe de Statistique et ProbabilitésUniversité Paul SabatierToulouse Cedex 09France

Personalised recommendations