Abstract
We establish universality of local eigenvalue correlations in unitary random matrix ensembles \({{\frac{{1}}{{Z_n}}|\det M|^{{2\alpha}} e^{{-n{{\rm{ tr}}}\, V(M)}} dM}}\) near the origin of the spectrum. If V is even, and if the recurrence coefficients of the orthogonal polynomials associated with |x|2α e − nV(x) have a regular limiting behavior, then it is known from work of Akemann et al., and Kanzieper and Freilikher that the local eigenvalue correlations have universal behavior described in terms of Bessel functions. We extend this to a much wider class of confining potentials V. Our approach is based on the steepest descent method of Deift and Zhou for the asymptotic analysis of Riemann-Hilbert problems. This method was used by Deift et al. to establish universality in the bulk of the spectrum. A main part of the present work is devoted to the analysis of a local Riemann-Hilbert problem near the origin.
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References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. New York: Dover Publications, 1968
Akemann, G., Damgaard, P.H., Magnea, U., Nishigaki, S.: Universality of random matrices in the microscopic limit and the Dirac operator spectrum. Nucl. Phys. B 487(3), 721–738 (1997)
Akemann, G., Damgaard, P.H., Magnea, U., Nishigaki, S.: Multicritical microscopic spectral correlators of Hermitian and complex matrices. Nucl. Phys. B 519(3), 682–714 (1998)
Akemann, G., Vernizzi, G.: New critical matrix models and generalized universality. Nucl. Phys. B 631(3), 471–499 (2002)
Baik, J., Kriecherbauer, T., McLaughlin, K.T-R., Miller, P.: Uniform asymptotics for polynomials orthogonal with respect to a general class of discrete weights and universality results for associated ensembles: announcement of results. Int. Math. Res. Notices 2003(15), 821–858 (2003)
Bleher, P., Its, A.: Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. Math. 150(1), 185–266 (1999)
Bleher, P., Its, A.: Double scaling limit in the random matrix model: the Riemann-Hilbert approach. Comm. Pure Appl. Math. 56, 433–516 (2003)
Bowick, M.J., Brézin, E.: Universal scaling of the tail of the density of eigenvalues in random matrix models. Phys. Lett. B 268, 21–28 (1991)
Brézin, E., Zee, A.: Universality of the correlations between eigenvalues of large random matrices. Nucl. Phys. B 402(3), 613–627 (1993)
Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes 3, New York University, 1999
Deift, P., Its, A.R., Zhou, X.: A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics. Ann. Math. 146, 149–235 (1997)
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, 388–475 (1998)
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. Comm. Pure Appl. Math 52, 1335–1425 (1999)
Deift, P., Kriecherbauer, T., McLaughlin, K.T-R., Venakides, S., Zhou, X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math 52, 1491–1552 (1999)
Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. 137, 295–368 (1993)
Farkas, H.M., Kra, I.: Riemann Surfaces. Graduate Texts in Mathematics. New York–Berlin, Springer-Verlag, 1992
Fokas, A.S., Its, A.R., Kitaev, A.V.: The isomonodromy approach to matrix models in 2D quantum gravity. Commun. Math. Phys. 147, 395–430 (1992)
Forrester, P.J.: The spectrum edge of random matrix ensembles. Nucl. Phys. B 402(3), 709–728 (1993)
Forrester, P.J., Snaith, N.C., Verbaarschot, J.J.M.: Developments in random matrix theory. J. Phys. A.: Math. Gen. 36, R1–R10 (2003)
Gakhov, F.D.: Boundary value problems. New York: Dover Publications, 1990
Janik, R.A.: New multicritical random matrix ensembles. Nucl. Phys. B 635(3), 492–504 (2002)
Kanzieper, E., Freilikher, V.: Random matrix models with log-singular level confinement: method of fictitious fermions. Philos. Magazine B 77(5), 1161–1172 (1998)
Kriecherbauer, T., McLaughlin, K.T-R.: Strong asymptotics of polynomials orthogonal with respect to Freud weights. Int. Math. Res. Notices 1999(6), 299–333 (1999)
Kuijlaars, A.B.J.: Riemann-Hilbert analysis for orthogonal polynomials. In: Koelink, E., Van Assche, W. (eds), Orthogonal Polynomials and Special Functions: Leuven 2002. Lect. Notes Math. 1817, Springer-Verlag, 2003, pp. 167–210
Kuijlaars, A.B.J., McLaughlin, K.T-R.: Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields. Comm. Pure Appl. Math. 53, 736–785 (2000)
Kuijlaars, A.B.J., McLaughlin, K.T-R.: Riemann-Hilbert analysis for Laguerre polynomials with large negative parameter. Comput. Meth. Funct. Theory 1(1), 205–233 (2001)
Kuijlaars, A.B.J., McLaughlin, K.T-R., Van Assche, W., Vanlessen, M.: The Riemann–Hilbert approach to strong asymptotics for orthogonal polynomials. To appear in Adv. Math., Preprint math.CA/0111252
Kuijlaars, A.B.J., Vanlessen, M.: Universality for eigenvalue correlations from the modified Jacobi unitary ensemble. Int. Math. Res. Notices 2002(30), 1575–1600 (2002)
Mehta, M.L.: Random Matrices. 2nd ed., San Diego: Academic Press, 1991
Moore, G.: Matrix models of 2D gravity and isomonodromic deformation. Progr. Theor. Phys. Suppl. No. 102, 255–285 (1990)
Nagao, T., Wadati, M.: Eigenvalue distribution of random matrices at the spectrum edge. J. Phys. Soc. Japan 62, 3845–3856 (1993)
Nishigaki, S.: Microscopic universality in random matrix models of QCD. In: Damgaard, P.H., Jurkiewicz, J. (eds.) New developments in quantum field theory. Proceedings Zakopane 1997. New York: Plenum Press, 1998, pp. 287–295
Pastur, L., Shcherbina, M.: Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles. J. Stat. Phys. 86, 109–147 (1997)
Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. New York: Springer-Verlag, 1997
Strahov, E., Fyodorov, Y.V.: Universal results for correlations of characteristic polynomials: Riemann-Hilbert approach. To appear in Commun. Math. Phys., Preprint math-ph/0210010
Szegő, G.: Orthogonal Polynomials. 4th ed., Providence RI: Amer. Math. Soc. 1975
Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994)
Tracy, C.A., Widom, H.: Level-spacing distributions and the Bessel kernel. Commun. Math. Phys. 161, 289–309 (1994)
Vanlessen, M.: Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight. Preprint math.CA/0212014
Verbaarschot, J.J.M., Zahed, I.: Random matrix theory and three-dimensional QCD. Phys. Rev. Lett. 73(17), 2288–2291 (1994)
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Communicated by B. Simon
Supported by FWO research project G.0176.02 and by INTAS project 00-272 and by the Ministry of Science and Technology (MCYT) of Spain, project code BFM2001-3878-C02-02
Research Assistant of the Fund for Scientific Research – Flanders (Belgium)
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Kuijlaars, A., Vanlessen, M. Universality for Eigenvalue Correlations at the Origin of the Spectrum. Commun. Math. Phys. 243, 163–191 (2003). https://doi.org/10.1007/s00220-003-0960-z
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DOI: https://doi.org/10.1007/s00220-003-0960-z