Advertisement

Journal of Theoretical Probability

, Volume 31, Issue 3, pp 1759–1778 | Cite as

Moments of the Hermitian Matrix Jacobi Process

  • Luc Deleaval
  • Nizar Demni
Article

Abstract

In this paper, we compute the expectation of traces of powers of the Hermitian matrix Jacobi process for a large enough but fixed size. To proceed, we first derive the semi-group density of its eigenvalues process as a bilinear series of symmetric Jacobi polynomials. Next, we use the expansion of power sums in the Schur polynomial basis and the integral Cauchy–Binet formula in order to determine the partitions having nonzero contributions after integration. It turns out that these are hooks of bounded weight and the sought expectation results from the integral of a product of two Schur functions with respect to a generalized beta distribution. For special values of the parameters on which the matrix Jacobi process depends, the last integral reduces to the Cauchy determinant and we close the paper with the investigation of the asymptotic behavior of the resulting formula as the matrix size tends to infinity.

Keywords

Hermitian matrix Jacobi process Schur polynomial Symmetric Jacobi polynomial Hook 

Mathematics Subject Classification (2010)

15B52 33C45 60H15 

References

  1. 1.
    Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)CrossRefzbMATHGoogle Scholar
  2. 2.
    Beerends, R.J., Opdam, E.M.: Certain hypergeometric series related to the root system \(BC\). Trans. Am. Math. Soc. 339(2), 581–607 (1993)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Berezin, F.A., Karpelevic, F.I.: Zonal spherical functions and Laplace operators on some symmetric spaces. Dokl. Akad. Nauk SSSR (N.S.) 118, 9–12 (1958)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Biane, P.: Free Brownian Motion, Free Stochastic Calculus and Random Matrices. Fields. Inst. Commun., 12, Amer. Math. Soc. Providence, RI, 1–19 (1997)Google Scholar
  5. 5.
    Capitaine, M., Casalis, M.: Asymptotic freeness by generalized moments for Gaussian and Wishart matrices. Appl. Beta Random Matrices. Indiana Univ. Math. J. 53(2), 397–431 (2004)CrossRefzbMATHGoogle Scholar
  6. 6.
    Carré, C., Deneufchatel, M., Luque, J.G., Vivo, P.: Asymptotics of Selberg-like integrals: the unitary case and Newton’s interpolation formula. J. Math. Phys. 51(12), 19 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Collins, B.: Product of random projections, Jacobi ensembles and universality problems arising from free probability. Probab. Theory Relat. Fields 133(3), 315–344 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dahlqvist, A., Collins, B., Kemp, T.: The hard edge of unitary Brownian motion. Probab. Theory Relat. Fields (2017)Google Scholar
  9. 9.
    Débiard, A.: Système Différentiel Hypergéométrique et Parties Radiales des Opérateurs Invariants des Espaces Symétriques de Type \(BC_p\). Lecture Notes in Math., vol. 1296. Springer, Berlin (1987)zbMATHGoogle Scholar
  10. 10.
    Deift, P., Gioev, D.: Random Matrix Theory: Invariant Ensembles and Universality. Courant Lecture Notes in Mathematics, vol. 18. Courant Institute of Mathematical Sciences, New York (2009)zbMATHGoogle Scholar
  11. 11.
    Demni, N.: Free Jacobi process. J. Theory Probab. 21(1), 118–143 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Demni, N.: \(\beta \)-Jacobi processes. Adv. Pure Appl. Math. 1(3), 325–344 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Demni, N.: Inverse of the flow and moments of the free Jacobi process associated with one projection. Available on ArXivGoogle Scholar
  14. 14.
    Demni, N., Hamdi, T., Hmidi, T.: Spectral distribution of the free Jacobi process. Indiana Univ. J. 61(3) (2012)Google Scholar
  15. 15.
    Demni, N., Hmidi, T.: Spectral distribution of the free Jacobi process associated with one projection. Colloq. Math. 137(2), 271–296 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Doumerc, Y.: Matrices aléatoires, processus stochastiques et groupes de réflexions. Ph.D. thesis, Paul Sabatier Univ. Available at http://perso.math.univ-toulouse.fr/ledoux/doctoral-students/
  17. 17.
    Hoogenboom, B.: Spherical functions and invariant differential operators on complex Grassmann manifolds. Ark. Mat. 20(1), 69–85 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Katori, M., Tanemura, H.: Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems. J. Math. Phys. 45(8), 3058–3085 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lascoux, A.: Square-ice enumeration. Sém. Lothar. Combin. 42, Art. B42p, 15 pp (1999)Google Scholar
  20. 20.
    Lassalle, M.: Une formule du binôme généralisée pour les polynômes de Jack. C. R. Acad. Sci. Paris Sér. I Math. 310, 253–256 (1990)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Lassalle, M.: Coefficients du binôme généralisés. C. R. Acad. Sci. Paris. Sér. I Math. 310, 257–260 (1990)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Lassalle, M.: Polynômes de Jacobi. C. R. Acad. Sci. Paris. t. 312, Série I. pp. 425–428 (1991)Google Scholar
  23. 23.
    Lévy, T.: Schur–Weyl duality and the heat kernel measure on the unitary group. Adv. Math. 218(2), 537–575 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Liao, M.: Lévy Processes in Lie Groups. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  25. 25.
    MacDonald, I.G.: Symmetric Functions and Hall Polynomials. Second edition, Mathematical Monographs, Oxford (1995)Google Scholar
  26. 26.
    Olshanski, G., Okounkov, A.: Limits of \(BC\)-type orthogonal polynomials as the number of variables goes too infinity. Jack, Hall-Littlewood and Macdonald polynomials, 281–318, Contemp. Math. 417, Amer. Math. Soc., Providence, RI (2006)Google Scholar
  27. 27.
    Olshanski, G.I., Osinenko, A.A.: Multivariate Jacobi polynomial and the Selberg integral. Funct. Anal. Appl. 46(4), 262–278 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Rains, E.M.: Combinatorial properties of Brownian motion on the compact classical groups. J. Theor. Probab. 10(3), 659–679 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.LAMAUniversité Marne la ValléeMarne la Valle Cedex 2France
  2. 2.IRMARUniversité de Rennes 1Rennes CedexFrance

Personalised recommendations