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Singularity Dominated Strong Fluctuations for Some Random Matrix Averages

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Abstract

The circular and Jacobi ensembles of random matrices have their eigenvalue support on the unit circle of the complex plane and the interval (0,1) of the real line respectively. The averaged value of the modulus of the corresponding characteristic polynomial raised to the power 2 μ diverges, for 2μ ≤ −1, at points approaching the eigenvalue support. Using the theory of generalized hypergeometric functions based on Jack polynomials, the functional form of the leading asymptotic behaviour is established rigorously. In the circular ensemble case this confirms a conjecture of Berry and Keating.

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References

  1. Berry, M.V.: Focusing and twinkling: critical exponents from catastrophes in non-Gaussian random short waves. J. Phys. A 10, 2061–2081 (1977)

    Article  Google Scholar 

  2. Berry, M.V.: Universal power-law tails for singularity-dominated strong fluctuations. J. Phys. A 15, 2735–2749 (1977)

    Google Scholar 

  3. Berry, M.V., Keating, J.P.: Clusters of near degenerate levels dominate negative moments of spectral determinants. J. Phys. A 35, L1–L6 (2002)

    Google Scholar 

  4. Berry, M.V., Keating, J.P., Schomerus, H.: Universal twinkling exponents for spectral fluctuations associated with mixed chaology. Proc. R. Soc. A 456, 1659–1668 (2000)

    Article  Google Scholar 

  5. Conrey, J.B., Farmer, D.W.: Mean values of L-functions and symmetry. Int. Math. Res. Notices 17, 883–908 (2000)

    Article  MathSciNet  Google Scholar 

  6. Conrey, J.B., Farmer, D.W., Keating, J.P., Rubinstein, M.O., Snaith, N.C.: Integral moments of zeta- and L-functions. Submitted to Proc. of the London Math.Soc.

  7. Forrester, P.J.: Log-gases and Random Matrices. http:// www.ms.unimelb.edu.au/~matpjf/matpjf.html, 2003

  8. Forrester, P.J.: Addendum to Selberg correlation integrals and the 1/r2 quantum many body system. Nucl. Phys. B 416, 377–385 (1994)

    Article  MathSciNet  Google Scholar 

  9. Forrester, P.J.: Integration formulas and exact calculations in the Calogero-Sutherland model. Mod. Phys. Lett B 9, 359–371 (1995)

    MathSciNet  Google Scholar 

  10. Forrester, P.J., Witte, N.S.: Discrete Painlevé equations, orthogonal polynomials on the unit circle and n-recurrences for averages over U(N) – PVI τ-functions. http:// arXiv.org/abs/math-ph/0308036, 2003

  11. Fyodorov, Y.V., Keating, J.P.: Negative moments of characteristic polynomials of random GOE matrices and singularity dominated strong fluctuations. J. Phys. A 36, 4035–4046 (2003)

    Article  MathSciNet  Google Scholar 

  12. Hughes, C.P., Keating, J.P., O’Connell, N.: On the characteristic polynomial of a random unitary matrix. Commun. Math. Phys. 220, 429–451 (2001)

    Article  MathSciNet  Google Scholar 

  13. Johansson, K.: On Szegö’s formula for Toeplitz determinants and generalizations. Bull. Sc. Math., 2e série 112, 257–304 (1988)

    Google Scholar 

  14. Johansson, K.: On random matrices from the compact classical groups. Ann. of Math. 145, 519–545 (1997)

    MathSciNet  Google Scholar 

  15. Johansson, K.: On fluctuation of eigenvalues of random Hermitian matrices. Duke Math. J. 91, 151–204 (1998)

    MathSciNet  Google Scholar 

  16. Kadell, K.W.J.: The Selberg-Jack symmetric functions. Adv. Math. 130, 33–102 (1997)

    Article  MathSciNet  Google Scholar 

  17. Kaneko, J.: Selberg integrals and hypergeometric functions associated with Jack polynomials. SIAM J. Math Anal. 24, 1086–1110 (1993)

    MathSciNet  Google Scholar 

  18. Keating, J.P., Prado, S.D.: Orbit bifurcations and the scarring of wave functions. Proc. R. Soc. A 457, 1855–1872 (2001)

    Article  Google Scholar 

  19. Keating, J.P., Snaith, N.C.: Random matrix theory and L-functions at s=1/2. Commun. Math. Phys. 214, 91–110 (2000)

    Article  MathSciNet  Google Scholar 

  20. Keating, J.P., Snaith, N.C.: Random matrix theory and ζ(1/2 + it). Commun. Math. Phys. 214, 57–89 (2001)

    Article  Google Scholar 

  21. Keating, J.P., Snaith, N.C.: Random matrices and L-functions. J. Phys. A 36, 2859–2881 (2003)

    Article  MathSciNet  Google Scholar 

  22. Macdonald, I.G.: Commuting differential operators and zonal spherical functions. In A.M. Cohen et al., editor, Algebraic Groups, Utrecht 1986, Vol. 1271 of Lecture Notes in Math., Heidelberg: Springer-Verlag, 1987, pp. 189–200

  23. Macdonald, I.G.: Hall polynomials and symmetric functions. 2nd edition, Oxford: Oxford University Press, 1995

  24. Szegö, G.: On certain Hermitian forms associated with the Fourier series of a positive function. Comm. Seminaire Math. de l’Univ. de Lund, tome supplémentaire, dédié á Marcel Riesz, 1952, pp. 228–237

  25. Yan, Z.: A class of generalized hypergeometric functions in several variables. Can J. Math. 44, 1317–1338 (1992)

    MathSciNet  Google Scholar 

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Authors and Affiliations

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

    P. J Forrester

  2. School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    J. P Keating

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  1. P. J Forrester
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  2. J. P Keating
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Communicated by P. Sarnak

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Forrester, P., Keating, J. Singularity Dominated Strong Fluctuations for Some Random Matrix Averages. Commun. Math. Phys. 250, 119–131 (2004). https://doi.org/10.1007/s00220-004-1121-8

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  • DOI: https://doi.org/10.1007/s00220-004-1121-8

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