Loop equation analysis of the circular β ensembles
- 306 Downloads
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.
KeywordsMatrix Models Random Systems Integrable Hierarchies
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.
- 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
- .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].
- P.L. Duren, Univalent functions, in Grundlehren der Mathematischen Wissenschaften, volume 259, Springer-Verlag, New York (1983).Google Scholar
- 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
- 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
- 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
- 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).Google Scholar
- NIST Digital Library of Mathematical Functions, release 1.0.9 of 2014-08-29, http://dlmf.nist.gov/.