Journal of High Energy Physics

, 2015:173 | Cite as

Loop equation analysis of the circular β ensembles

Open Access
Regular Article - Theoretical Physics

Abstract

We construct a hierarchy of loop equations for invariant circular ensembles. These are valid for general classes of potentials and for arbitrary inverse temperatures Re β > 0 and number of eigenvalues N. Using matching arguments for the resolvent functions of linear statistics f(ζ) = (ζ + z)/(ζz) in a particular asymptotic regime, the global regime, we systematically develop the corresponding large N expansion and apply this solution scheme to the Dyson circular ensemble. Currently we can compute the second resolvent function to ten orders in this expansion and also its general Fourier coefficient or moment mk to an equivalent length. The leading large N, large k, k/N fixed form of the moments can be related to the small wave-number expansion of the structure function in the bulk, scaled Dyson circular ensemble, known from earlier work. From the moment expansion we conjecture some exact partial fraction forms for the low k moments. For all of the forgoing results we have made a comparison with the exactly soluble cases of β = 1, 2, 4, general N and even, positive β, N = 2, 3.

Keywords

Matrix Models Random Systems Integrable Hierarchies 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    K. Aomoto, Jacobi polynomials associated with Selberg integrals, SIAM J. Math. Anal. 18 (1987) 545.CrossRefMATHMathSciNetGoogle Scholar
  2. [2]
    M. Bergere, B. Eynard, O. Marchal and A. Prats-Ferrer, Loop equations and topological recursion for the arbitrary-β two-matrix model, JHEP 03 (2012) 098 [arXiv:1106.0332] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  3. [3]
    G. Borot, B. Eynard, S.N. Majumdar and C. Nadal, Large deviations of the maximal eigenvalue of random matrices, J. Stat. Mech. Theory Exp. 11 (2011) P11024 [arXiv:1009.1945].CrossRefMathSciNetGoogle Scholar
  4. [4]
    G. Borot and A. Guionnet, Asymptotic expansion of beta matrix models in the one-cut regime, Commun. Math. Phys. 317 (2013) 447 [INSPIRE].CrossRefADSMATHMathSciNetGoogle Scholar
  5. [5]
    M.J. Bowick, A. Morozov and D. Shevitz, Reduced unitary matrix models and the hierarchy of τ-functions, Nucl. Phys. B 354 (1991) 496.CrossRefADSMathSciNetGoogle Scholar
  6. [6]
    E. Brézin and D.J. Gross, The External Field Problem in the Large-N Limit of QCD, Phys. Lett. B 97 (1980) 120 [INSPIRE].CrossRefADSGoogle Scholar
  7. [7]
    A. Brini, M. Mariño and S. Stevan, The Uses of the refined matrix model recursion, J. Math. Phys. 52 (2011) 052305 [arXiv:1010.1210] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  8. [8]
    R.C. Brower and M. Nauenberg, Group integration for lattice gauge theory at large N and at small coupling, Nucl. Phys. B 180 (1981) 221 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  9. [9]
    L.O. Chekhov, Logarithmic potential β-ensembles and Feynman graphs, arXiv:1009.5940 [INSPIRE].
  10. [10]
    L.O. Chekhov and B. Eynard, Matrix eigenvalue model: Feynman graph technique for all genera, JHEP 12 (2006) 026 [math-ph/0604014] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  11. [11]
    L.O. Chekhov, B. Eynard and O. Marchal, Topological expansion of β-ensemble model and quantum algebraic geometry in the sectorwise approach, Theor. Math. Phys. 166 (2011) 141 [arXiv:1009.6007] [INSPIRE].CrossRefMATHMathSciNetGoogle Scholar
  12. [12]
    W. Chu, Analytical formulae for extended 3 F 2 -series of Watson-Whipple-Dixon with two extra integer parameters, Math. Comp. 81 (2012) 467.CrossRefMATHMathSciNetGoogle Scholar
  13. [13]
    H. Cramér, Mathematical methods of statistics, in Princeton Landmarks in Mathematics, reprint of the 1946 original, Princeton University Press, Princeton NJ U.S.A. (1999)Google Scholar
  14. [14]
    P. Desrosiers and D.-Z. Liu, Asymptotics for products of characteristic polynomials in classical β-ensembles, Constr. Approx. 39 (2011) 273 [arXiv:1112.1119].CrossRefMathSciNetGoogle Scholar
  15. [15]
    .1093/imrn/rnu039 P. Desrosiers and D.-Z. Liu, Scaling limits of correlations of characteristic polynomials for the Gaussian β-ensemble with external source, Int. Math. Res. Not. 31 March 2014 [arXiv:1306.4058].
  16. [16]
    I. Dumitriu and A. Edelman, Global spectrum fluctuations for the β-Hermite and β-Laguerre ensembles via matrix models, J. Math. Phys. 47 (2006) 063302 [math-ph/0510043].CrossRefADSMathSciNetGoogle Scholar
  17. [17]
    P.L. Duren, Univalent functions, in Grundlehren der Mathematischen Wissenschaften, volume 259, Springer-Verlag, New York (1983).Google Scholar
  18. [18]
    F.J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Math. Phys. 3 (1962) 140 [INSPIRE].CrossRefADSMATHMathSciNetGoogle Scholar
  19. [19]
    N.M. Ercolani and K.D.T.-R. McLaughlin, Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques and applications to graphical enumeration, Int. Math. Res. Not. 14 (2003) 755 [math-ph/0211022].CrossRefMathSciNetGoogle Scholar
  20. [20]
    B. Eynard, Asymptotics of skew orthogonal polynomials, J. Phys. A 34 (2001) 7591 [cond-mat/0012046].ADSMathSciNetGoogle Scholar
  21. [21]
    P.J. Forrester, Normalization of the wavefunction for the Calogero-Sutherland model with internal degrees of freedom, Int. J. Mod. Phys. B 9 (1995) 1243 [cond-mat/9412058].CrossRefADSMathSciNetGoogle Scholar
  22. [22]
    P.J. Forreste, Log Gases and Random Matrices, in London Mathematical Society Monograph, volume 34, first edition, Princeton University Press, Princeton NJ U.S.A. (2010).Google Scholar
  23. [23]
    P.J. Forrester, B. Jancovici and D.S. McAnally, Analytic properties of the structure function for the one-dimensional one-component log-gas J. Stat. Phys. 102 (2001) 737 [cond-mat/0002060].CrossRefADSMATHMathSciNetGoogle Scholar
  24. [24]
    A.Z. Grinshpan, The Bieberbach conjecture and Milins functionals, Am. Math. Mon. 106 (1999) 203.CrossRefMATHMathSciNetGoogle Scholar
  25. [25]
    D.J. Gross and M.J. Newman, Unitary and Hermitian matrices in an external field. 2: The Kontsevich model and continuum Virasoro constraints, Nucl. Phys. B 380 (1992) 168 [hep-th/9112069] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  26. [26]
    D.J. Gross and E. Witten, Possible Third Order Phase Transition in the Large-N Lattice Gauge Theory, Phys. Rev. D 21 (1980) 446 [INSPIRE].ADSGoogle Scholar
  27. [27]
    U. Haagerup and S. Thorbjørnsen, Asymptotic expansions for the Gaussian unitary ensemble, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15 (2012) 1250003 [arXiv:1004.3479].CrossRefMathSciNetGoogle Scholar
  28. [28]
    M. Hisakado, Unitary matrix models and Painlevé III, Mod. Phys. Lett. A 11 (1996) 3001 [hep-th/9609214] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  29. [29]
    M. Hisakado, Unitary matrix models with a topological term and discrete time Toda equation, Phys. Lett. B 395 (1997) 208 [hep-th/9611177] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  30. [30]
    M. Hisakado, Unitary matrix models and phase transition, Phys. Lett. B 416 (1998) 179 [hep-th/9705121] [INSPIRE].CrossRefADSGoogle Scholar
  31. [31]
    M.G. Kendall and A. Stuart, The advanced theory of statistics. Vol. 1: Distribution theory, Third edition, Hafner Publishing Co., New York (1969).Google Scholar
  32. [32]
    W. Koepf, Power series, Bieberbach conjecture and the de Branges and Weinstein functions, in Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, ACM, New York (2003), pp. 169–175.Google Scholar
  33. [33]
    D.-Z. Liu, Limits for circular Jacobi beta-ensembles, arXiv:1408.0486.
  34. [34]
    M.L. Mehta, Random Matrices, in Pure and Applied Mathematics (Amsterdam), volume 142, third edition, Elsevier/Academic Press, Amsterdam (2004).Google Scholar
  35. [35]
    A. Mironov, A. Morozov, A.V. Popolitov and S. Shakirov, Resolvents and Seiberg-Witten representation for Gaussian β-ensemble, Theor. Math. Phys. 171 (2012) 505 [arXiv:1103.5470] [INSPIRE].CrossRefMATHMathSciNetGoogle Scholar
  36. [36]
    S. Mizoguchi, On unitary/hermitian duality in matrix models, Nucl. Phys. B 716 (2005) 462 [hep-th/0411049] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  37. [37]
    R.C. Myers and V. Periwal, Exact Solution of Critical Selfdual Unitary Matrix Models, Phys. Rev. Lett. 65 (1990) 1088 [INSPIRE].CrossRefADSMATHMathSciNetGoogle Scholar
  38. [38]
    R.C. Myers and V. Periwal, Exactly solvable self-dual strings, Phys. Rev. Lett. 64 (1990) 3111 [INSPIRE].CrossRefADSMATHMathSciNetGoogle Scholar
  39. [39]
    V. Periwal and D. Shevitz, Exactly Solvable Unitary Matrix Models: Multicritical Potentials and Correlations, Nucl. Phys. B 344 (1990) 731 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  40. [40]
    V. Periwal and D. Shevitz, Unitary-matrix models as exactly solvable string theories, Phys. Rev. Lett. 64 (1990) 1326 [INSPIRE].CrossRefADSGoogle Scholar
  41. [41]
    NIST Digital Library of Mathematical Functions, release 1.0.9 of 2014-08-29, http://dlmf.nist.gov/.
  42. [42]
    M.M. Robinson, The Orthogonal circular emsemble, Phys. Rev. D 45 (1992) 2872 [INSPIRE].ADSGoogle Scholar
  43. [43]
    P.J. Smith, A recursive formulation of the old problem of obtaining moments from cumulants and vice versa, Am. Stat. 49 (1995) 217.ADSGoogle Scholar
  44. [44]
    N.S. Witte and P.J. Forrester, Moments of the Gaussian β Ensembles and the large-N expansion of the densities, J. Math. Phys. 55 (2014) 083302 [arXiv:1310.8498] [INSPIRE].CrossRefADSGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of MelbourneMelbourneAustralia

Personalised recommendations