Abstract
We describe the distribution of the first finite number of eigenvalues in a newly-forming band of the spectrum of the random Hermitian matrix model. The method is rigorously based on the Riemann–Hilbert analysis of the corresponding orthogonal polynomials. We provide an analysis with an error term of order N −2γ where 1/γ=2ν+2 is the exponent of non-regularity of the effective potential, thus improving even in the usual case the analysis of the pertinent literature.
The behavior of the first finite number of zeroes (eigenvalues) appearing in the new band is analyzed and connected with the location of the zeroes of certain Freud polynomials. In general, all these newborn zeroes approach the point of nonregularity at the rate N −γ, whereas one (a stray zero) lags behind at a slower rate of approach. The kernels for the correlator functions in the scaling coordinate near the emerging band are provided together with the subleading term. In particular, the transition between K and K+1 eigenvalues is analyzed in detail.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Bertola, M.: Boutroux curves with external field: equilibrium measures without a minimization problem. arXiv:0705.1283, 2007
Bertola, M., Mo, M.Y.: Commuting difference operators, spinor bundles and the asymptotics of pseudo-orthogonal polynomials with respect to varying complex weights. ArXiv Math. Phys. e-prints, May 2006
Bleher, P., Its, A.: Semiclassical asymptotics of orthogonal polynomials, Riemann–Hilbert problem, and universality in the matrix model. Ann. Math. (2) 150(1), 185–266 (1999)
Claeys, T.: The birth of a cut in unitary random matrix ensembles. ArXiv Math. Phys. e-prints, arXiv:0711.2609, 2007
Deift, P.A.: Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. Courant Lecture Notes in Mathematics, vol. 3. New York University Courant Institute of Mathematical Sciences, New York (1999)
Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R.: New results on the equilibrium measure for logarithmic potentials in the presence of an external field. J. Approx. Theory 95(3), 388–475 (1998)
Deift, P., Kriecherbauer, T., McLaughlin, T.K.-R., Venakides, S., Zhou, X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math. 52(12), 1491–1552 (1999)
Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Commun. Pure Appl. Math. 52(11), 1335–1425 (1999)
Eynard, B.: Universal distribution of random matrix eigenvalues near the “birth of a cut” transition. J. Stat. Mech.: Theory Exp. 2006(07), P07005 (2006)
Farkas, H.M., Kra, I.: Riemann Surfaces, 2nd edn. Graduate Texts in Mathematics, vol. 71. Springer, New York (1992)
Fay, J.D.: Theta Functions on Riemann Surfaces. Lecture Notes in Mathematics, vol. 352. Springer, Berlin (1973)
Fokas, A.S., Its, A.R., Kitaev, A.V.: Discrete Painlevé equations and their appearance in quantum gravity. Commun. Math. Phys. 142(2), 313–344 (1991)
Fokas, A.S., Its, A.R., Kitaev, A.V.: The isomonodromy approach to matrix models in 2D quantum gravity. Commun. Math. Phys. 147(2), 395–430 (1992)
Freud, G.: On polynomial approximation with the weight \(\exp \{-{1\over 2}x^{2k}\}\) . Acta Math. Acad. Sci. Hung. 24, 363–371 (1973)
Its, A.R., Kitaev, A.V., Fokas, A.S.: An isomonodromy approach to the theory of two-dimensional quantum gravity. Usp. Mat. Nauk 45(6)(276), 135–136 (1990)
Its, A.R., Kitaev, A.V., Fokas, A.S.: Matrix models of two-dimensional quantum gravity, and isomonodromic solutions of Painlevé “discrete equations”. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 187 (Differentsialnaya Geom. Gruppy Li i Mekh. 12), 3–30, 171, 174 (1991)
Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory. Physica D 2(2), 306–352 (1981)
Kuijlaars, A.B.J., McLaughlin, K.T.-R.: Asymptotic zero behavior of Laguerre polynomials with negative parameter. Constr. Approx. 20(4), 497–523 (2004)
Mehta, M.L.: Random Matrices. 3rd edn., Pure and Applied Mathematics, vol. 142, Elsevier/Academic, Amsterdam (2004)
Mo, M.Y.: The Riemann–Hilbert approach to double scaling limit of random matrix eigenvalues near the “birth of a cut” transition. arxiv:0711.3208, 2007
Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316, Springer, Berlin (1997). Appendix B by Thomas Bloom
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Percy A. Deift.
Rights and permissions
About this article
Cite this article
Bertola, M., Lee, S.Y. First Colonization of a Spectral Outpost in Random Matrix Theory. Constr Approx 30, 225–263 (2009). https://doi.org/10.1007/s00365-008-9026-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00365-008-9026-y