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)/(ζ − z) 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.
Article PDF
Similar content being viewed by others
References
K. Aomoto, Jacobi polynomials associated with Selberg integrals, SIAM J. Math. Anal. 18 (1987) 545.
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].
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].
G. Borot and A. Guionnet, Asymptotic expansion of beta matrix models in the one-cut regime, Commun. Math. Phys. 317 (2013) 447 [INSPIRE].
M.J. Bowick, A. Morozov and D. Shevitz, Reduced unitary matrix models and the hierarchy of τ-functions, Nucl. Phys. B 354 (1991) 496.
E. Brézin and D.J. Gross, The External Field Problem in the Large-N Limit of QCD, Phys. Lett. B 97 (1980) 120 [INSPIRE].
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].
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].
L.O. Chekhov, Logarithmic potential β-ensembles and Feynman graphs, arXiv:1009.5940 [INSPIRE].
L.O. Chekhov and B. Eynard, Matrix eigenvalue model: Feynman graph technique for all genera, JHEP 12 (2006) 026 [math-ph/0604014] [INSPIRE].
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].
W. Chu, Analytical formulae for extended 3 F 2 -series of Watson-Whipple-Dixon with two extra integer parameters, Math. Comp. 81 (2012) 467.
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)
P. Desrosiers and D.-Z. Liu, Asymptotics for products of characteristic polynomials in classical β-ensembles, Constr. Approx. 39 (2011) 273 [arXiv:1112.1119].
.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].
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].
P.L. Duren, Univalent functions, in Grundlehren der Mathematischen Wissenschaften, volume 259, Springer-Verlag, New York (1983).
F.J. Dyson, Statistical theory of the energy levels of complex systems. I, J. Math. Phys. 3 (1962) 140 [INSPIRE].
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].
B. Eynard, Asymptotics of skew orthogonal polynomials, J. Phys. A 34 (2001) 7591 [cond-mat/0012046].
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].
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).
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].
A.Z. Grinshpan, The Bieberbach conjecture and Milin’s functionals, Am. Math. Mon. 106 (1999) 203.
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].
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].
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].
M. Hisakado, Unitary matrix models and Painlevé III, Mod. Phys. Lett. A 11 (1996) 3001 [hep-th/9609214] [INSPIRE].
M. Hisakado, Unitary matrix models with a topological term and discrete time Toda equation, Phys. Lett. B 395 (1997) 208 [hep-th/9611177] [INSPIRE].
M. Hisakado, Unitary matrix models and phase transition, Phys. Lett. B 416 (1998) 179 [hep-th/9705121] [INSPIRE].
M.G. Kendall and A. Stuart, The advanced theory of statistics. Vol. 1: Distribution theory, Third edition, Hafner Publishing Co., New York (1969).
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.
D.-Z. Liu, Limits for circular Jacobi beta-ensembles, arXiv:1408.0486.
M.L. Mehta, Random Matrices, in Pure and Applied Mathematics (Amsterdam), volume 142, third edition, Elsevier/Academic Press, Amsterdam (2004).
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].
S. Mizoguchi, On unitary/hermitian duality in matrix models, Nucl. Phys. B 716 (2005) 462 [hep-th/0411049] [INSPIRE].
R.C. Myers and V. Periwal, Exact Solution of Critical Selfdual Unitary Matrix Models, Phys. Rev. Lett. 65 (1990) 1088 [INSPIRE].
R.C. Myers and V. Periwal, Exactly solvable self-dual strings, Phys. Rev. Lett. 64 (1990) 3111 [INSPIRE].
V. Periwal and D. Shevitz, Exactly Solvable Unitary Matrix Models: Multicritical Potentials and Correlations, Nucl. Phys. B 344 (1990) 731 [INSPIRE].
V. Periwal and D. Shevitz, Unitary-matrix models as exactly solvable string theories, Phys. Rev. Lett. 64 (1990) 1326 [INSPIRE].
NIST Digital Library of Mathematical Functions, release 1.0.9 of 2014-08-29, http://dlmf.nist.gov/.
M.M. Robinson, The Orthogonal circular emsemble, Phys. Rev. D 45 (1992) 2872 [INSPIRE].
P.J. Smith, A recursive formulation of the old problem of obtaining moments from cumulants and vice versa, Am. Stat. 49 (1995) 217.
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].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1409.6038
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Witte, N.S., Forrester, P.J. Loop equation analysis of the circular β ensembles. J. High Energ. Phys. 2015, 173 (2015). https://doi.org/10.1007/JHEP02(2015)173
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2015)173