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Interpretations of some parameter dependent generalizations of classical matrix ensembles
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  • Published: 20 August 2004

Interpretations of some parameter dependent generalizations of classical matrix ensembles

  • Peter J. Forrester1 &
  • Eric M. Rains2,3 

Probability Theory and Related Fields volume 131, pages 1–61 (2005)Cite this article

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Abstract.

Two types of parameter dependent generalizations of classical matrix ensembles are defined by their probability density functions (PDFs). As the parameter is varied, one interpolates between the eigenvalue PDF for the superposition of two classical ensembles with orthogonal symmetry and the eigenvalue PDF for a single classical ensemble with unitary symmetry, while the other interpolates between a classical ensemble with orthogonal symmetry and a classical ensemble with symplectic symmetry. We give interpretations of these PDFs in terms of probabilities associated to the continuous Robinson-Schensted-Knuth correspondence between matrices, with entries chosen from certain exponential distributions, and non-intersecting lattice paths, and in the course of this probability measures on partitions and pairs of partitions are identified. The latter are generalized by using Macdonald polynomial theory, and a particular continuum limit – the Jacobi limit – of the resulting measures is shown to give PDFs related to those appearing in the work of Anderson on the Selberg integral, and also in some classical work of Dixon. By interpreting Anderson’s and Dixon’s work as giving the PDF for the zeros of a certain rational function, it is then possible to identify random matrices whose eigenvalue PDFs realize the original parameter dependent PDFs. This line of theory allows sampling of the original parameter dependent PDFs, their Dixon-Anderson-type generalizations and associated marginal distributions, from the zeros of certain polynomials defined in terms of random three term recurrences.

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References

  1. Aleiner, I.L., and Matveev, K.A.: Shifts of random energy levels by a local perturbation. Phys. Rev. Lett. 80, 814–816 (1998)

    Google Scholar 

  2. Anderson, G.W.: A short proof of Selberg’s generalized beta formula. Forum Math. 3, 415–417 (1991)

    MathSciNet  Google Scholar 

  3. Baik, J.: Painlevé expressions for LOE, LSE and interpolating ensembles. Int. Math. Res. Notices 33, 1739–1789 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baik, J., Rains, E.M.: Algebraic aspects of increasing subsequences. Duke Math. J. 109, 1–65 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baik, J., Rains, E.M.: Symmetrized random permutations. In: P.M. Bleher, A.R. Its (eds.), Random matrix models and their applications, volume 40 of Mathematical Sciences Research Institute Publications, Cambridge University Press, United Kingdom, 2001, pp. 171–208

  6. Baryshnikov, Yu.: GUES and queues. Probab. Theory Relat. Fields 119, 256–274 (2001)

    MathSciNet  MATH  Google Scholar 

  7. Beenakker, C.W.J.: Random-matrix theory of quantum transport. Rev. Mod. Phys. 69, 731–808 (1997)

    Article  Google Scholar 

  8. Bergére, M.C.: Proof of Serban’s conjecture. J. Math. Phys. 39, 30–46 (1998)

    Article  MathSciNet  Google Scholar 

  9. Bogomolny, E., Gerland, U., Schmit, C.: Singular statistics. Phys. Rev. E 63, 036206 (2000)

    Article  Google Scholar 

  10. Borodin, A., Olshanski, G.: Harmonic functions on multiplicative graphs and interpolation polynomials. math.CO/9912124, 1999

  11. Borodin, A.: Biorthogonal ensembles. Nucl. Phys. B 536, 704–732 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ciucu, M.: Enumeration of perfect matchings in graphs with reflective symmetry. J. Comb. Theory Ser. A 77, 67–97 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dixon, A.L.: Generalizations of Legendre’s formula Proc. London Math. Soc. 3, 206–224 (1905)

    Google Scholar 

  14. Dumitriu, I., Edelman, A.: Matrix models for beta ensembles. J. Math. Phys. 43, 5830–5847 (2002)

    Article  MathSciNet  Google Scholar 

  15. Evans, R.J.: Multidimensional q-beta integrals. SIAM J. Math. Anal. 23, 758–765 (1992)

    MathSciNet  MATH  Google Scholar 

  16. Evans, R.J.: Multidimensional beta and gamma integrals. Contemp. Math. 166, 341–357 (1994)

    MATH  Google Scholar 

  17. Fomin, S.: Schur operators and Knuth correspondences. J. Comb. Th. Ser. A 72, 277–292 (1995)

    Google Scholar 

  18. Forrester, P.J.: Log-gases and Random Matrices. http://www.ms.unimelb.edu.au/∼matpjf/ matpjf.html

  19. Forrester, P.J.: Exact integral formulas and asymptotics for the correlations in the 1/r2 quantum many body system. Phys. Lett. A, 179, 127–130 (1993)

    Google Scholar 

  20. Forrester, P.J., Rains, E.M.: Inter-relationships between orthogonal, unitary and symplectic matrix ensembles. In: P.M. Bleher, A.R. Its (eds.), Random matrix models and their applications, volume 40 of Mathematical Sciences Research Institute Publications, Cambridge University Press, United Kingdom, 2001, pp. 171–207

  21. Forrester, P.J., Rains, E.M.: Correlations for superpositions and decimations of Laguerre and Jacobi orthogonal matrix ensembles with a parameter. Probab. Theory Relat. Fields 130, 518–576 (2004)

    Article  Google Scholar 

  22. Fulton, W.: Young Tableaux. London Mathematical Society Student Texts. CUP, Cambridge, 1997

  23. Glynn, P.W., Whitt, W.: Departures form many queues in series. Ann. Appl. Probability 1, 546–572 (1991)

    MathSciNet  MATH  Google Scholar 

  24. Guhr, T., Kohler, H.: Recursive construction for a class of radial functions I. Ordinary space. J. Math. Phys. 43, 2707–2740 (2002)

    Article  Google Scholar 

  25. Johansson, K.: Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. Math. 153, 259–296 (2001)

    MathSciNet  MATH  Google Scholar 

  26. Johansson, K.: Non-intersecting paths, random tilings and random matrices. Probab. Theory Relat. Fields 123, 225–280 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kirillov, A.N.: Introduction to tropical combinatorics. In: A.N. Kirillov, N. Liskova (eds.), Physics and Combinatorics 2000, Proceedings of the Nagoya 2000 International Workshop, World Scientific, 2001, pp. 82–150

  28. Knuth, D.E.: Permutations, matrices and generalized Young tableaux. Pacific J. Math. 34, 709–727 (1970)

    MATH  Google Scholar 

  29. Krattenthaler, C.: Schur function identities and the number of perfect matchings of holey Aztec rectangles. Preprint arXive:math.CO/9712204

  30. Macdonald, I.G.: Hall Polynomials and Symmetric Functions. 2nd edition, Oxford University Press, Oxford, 1995

  31. Mehta, M.L.: Random Matrices. Academic Press, New York, 2nd edition, 1991

  32. Noumi, M., Yamada, Y.: Tropical Robinson-Schensted-Knuth correspondence and birational Weyl group actions. arXiv:math-ph/0203030, 2002

  33. Okounkov, A.: Shifted Macdonald polynomials: q-integral representation and combinatorial formula. Comp. Math. 12, 147–182 (1998)

    Article  MathSciNet  Google Scholar 

  34. Rao, R.P.: Advanced Statistical Methods in Biometric Research. John Wiley & Sons, New York, 1952

  35. Richards, D., Zheng, Q.: Determinants of period matrices and an application to Selberg’s multidimensional beta integral. Adv. Appl. Math. 28, 602–633 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sagan, B.E.: The Symmetric Group. 2nd edition, Springer-Verlag, New York, 2000

  37. Szegö, G.: Orthogonal Polynomials. 4th edition, American Mathematical Society, Providence RI, 1975

  38. van Leeuwen, M.: The Robinson-Schensted and Schützenberger algorithms, an elementary approach. Elect. J. Combin. 3, R15 (1996)

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Melbourne, Victoria, 3010, Australia

    Peter J. Forrester

  2. AT&T Research, Florham Park, NJ, 07932, USA

    Eric M. Rains

  3. Department of Mathematics, University of California, Davis, CA, 95616, USA

    Eric M. Rains

Authors
  1. Peter J. Forrester
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  2. Eric M. Rains
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Corresponding author

Correspondence to Peter J. Forrester.

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Supported by the Australian Research Council

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Forrester, P., Rains, E. Interpretations of some parameter dependent generalizations of classical matrix ensembles. Probab. Theory Relat. Fields 131, 1–61 (2005). https://doi.org/10.1007/s00440-004-0375-6

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  • Received: 26 November 2002

  • Revised: 07 May 2004

  • Published: 20 August 2004

  • Issue Date: January 2005

  • DOI: https://doi.org/10.1007/s00440-004-0375-6

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Keywords

  • Probability Density
  • Rational Function
  • Stochastic Process
  • Probability Measure
  • Probability Theory
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