Abstract
In this work, we develop an orthogonal-polynomials approach for random matrices with orthogonal or symplectic invariant laws, called one-matrix models with polynomial potential in theoretical physics, which are a generalization of Gaussian random matrices. The representation of the correlation functions in these matrix models, via the technique of quaternion determinants, makes use of matrix kernels. We get new formulas for matrix kernels, generalizing the known formulas for Gaussian random matrices, which essentially express them in terms of the reproducing kernel of the theory of orthogonal polynomials. Finally, these formulas allow us to prove the universality of the local statistics of eigenvalues, both in the bulk and at the edge of the spectrum, for matrix models with two-band quartic potential by using the asymptotics given by Bleher and Its for the corresponding orthogonal polynomials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M. and Stegun, I. A. (eds): Handbook of Mathematical Functions, Dover, New York, 1968.
Bleher, P. and Its, A.: Semi-classical asymptotics of orthogonal polynomials, Riemann-Hilbert problem and universality in the matrix model, Ann. of Math. 150(1999), 185–266.
Bleistein, N. and Handelsman, R. A.: Asymptotic Expansions of Integrals, Dover, New York, 1986.
Boutet de Monvel, A., Pastur, L. and Shcherbina, M.: On the statistical mechanics approach to the random matrix theory: the integrated density of states, J. Statist. Phys. 79(1995), 585–611.
Deift, P., Kriecherbauer, T., McLaughlin, K. T.-R., Venakides, S. and Zhou, X.: Asymptotics for polynomial orthogonal with respect to varying exponential weight, Internat. Math. Res. Notes 16(1997), 759–782.
DiFrancesco, P., Ginsparg, P. and Zinn-Justin, J.: 2D gravity and random matrices, Phys. Rep. 254(1995), 1–133.
Forrester, P. J., Nagao, T. and Honner, G.: Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges, Nuclear Phys. B 553[PM] (1999), 601–643.
Johansson, K.: On fluctuation of eigenvalues of random Hermitian matrices, Duke Math. J. 91(1998), 151–204.
Mehta, M. L.: Random Matrices, and the Statistical Theory of Energy Levels, Academic Press, New York, 1967.
Mehta, M. L.: Matrix Theory, Selected Topics and Useful Results, Les Éditions de physique, France, 1989.
Mehta, M. L.: Random Matrices, Academic Press, New York, 1991.
Pastur, L. and Shcherbina, M.: Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles, J. Statist. Phys. 86(1997), 109–147.
Stojanovic, A.: Une approche par les polynômes orthogonaux pour des classes de matrices aléatoires orthogonalement et symplectiquement invariantes: application à l'universalité de la statistique locale des valeurs propres, Preprint, www.physik.uni-bielefeld.de/bibos/preprints, 00–01–06.
Szego, G.: Orthogonal Polynomials, Amer. Math. Soc., Providence, 1939.
Tracy, C. A. and Widom, H.: Orthogonal and symplectic matrix ensembles, Comm. Math. Phys. 177(1996), 727–754.
Tracy, C. A. and Widom, H.: Correlation functions, cluster functions and spacing distributions for random matrices, J. Statist. Phys. 92(1998), 809–835.
Widom, H.: On the relation between orthogonal, symplectic and unitary matrix ensembles, J. Statist. Phys. 94(1999), 347–364.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Stojanovic, A. Universality in Orthogonal and Symplectic Invariant Matrix Models with Quartic Potential. Mathematical Physics, Analysis and Geometry 3, 339–373 (2000). https://doi.org/10.1023/A:1011457714198
Issue Date:
DOI: https://doi.org/10.1023/A:1011457714198