Abstract
Consider fixed and bounded trace Gaussian orthogonal, unitary and symplectic ensembles, closely related to Gaussian ensembles without any constraint. For three restricted trace Gaussian ensembles, we prove universal limits of correlation functions at zero and at the edge of the spectrum edge. Our argument also applies to restricted trace ensembles with monomial potentials. In addition, by using the universal result in the bulk for fixed trace Gaussian unitary ensemble, which has been obtained by Götze and Gordin, we also prove the universal limits of correlation functions everywhere in the bulk for bounded trace Gaussian unitary ensemble.
Article PDF
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.
Avoid common mistakes on your manuscript.
References
Akemann, G., Vernizzi, G.: Macroscopic and microscopic (non-)universality of compact support random matrix theory. Nucl. Phys. B 583(3), 739–757 (2000)
Akemann, G., Cicuta, G.M., Molinari, L., Vernizzi, G.: Compact support probability distributions in random matrix theory. Phys. Rev. E 59(2), 1489–1497 (1999)
Akemann, G., Cicuta, G.M., Molinari, L., Vernizzi, G.: Nonuniversality of compact support probability distributions in random matrix theory. Phys. Rev. E 60(5), 5287–5292 (1999)
Balian, R.: Random matrices and information theory. Nuovo Cimento B 57, 183–193 (1968)
Bleher, P., Its, A.: Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. Math. 150, 185–266 (1999)
Brezin, E., Zee, A.: Universality of the correlations between eigenvalues of large random matrices. Nucl. Phys. B 402, 613–627 (1993)
Bronk, B.V.: Topics in the theory of Random Matrices. Thesis, Princeton University (unpublished), a quote in Chapter 27 of Mehta’s book “Random Matrices”, 3rd edn.
Deift, P., Gioev, D.: Universality in random matrix theory for orthogonal and symplectic ensembles. Int. Math. Res. Pap. 2007, rpm004 (2007)
Deift, P., Gioev, D.: Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices. Commun. Pure Appl. Math. 60, 867–910 (2007)
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)
Delannay, R., LeCaër, G.: Exact densities of states of fixed trace ensembles of random matrices. J. Phys. A 33, 2611–2630 (2000)
Dyson, F.J.: Statistical theory of the energy levels of complex systems III. J. Math. Phys. 3, 166–175 (1962)
Erdős, L., Péché, S., Ramírez, J.A., Schlein, B., Yau, H.T.: Bulk universality for Wigner matrices. arXiv:0905.4176 [math-ph]
Erdős, L., Ramírez, J., Schlein, B., Tao, T., Vu, V., Yau, H.-T.: Bulk universality for Wigner hermitian matrices with subexponential decay. arXiv:0906.4400
Erdős, L., Schlein, B., Yau, H.-T.: Universality of random matrices and local relaxation flow. arXiv:0907.5605
Forrester, P.J.: The spectrum edge of random matrix ensembles. Nucl. Phys. B 402, 709–728 (1993)
Götze, F., Gordin, M.: Limit correlation functions for fixed trace random matrix ensembles. Commun. Math. Phys. 281, 203–229 (2008)
Götze, F., Gordin, M., Levina, A.: Limit correlation function at zero for fixed trace random matrix ensembles. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 341, 68–80 (2007) (Russian). Translation to appear in J. Math. Sci. (N.Y.) 145(3) (2007)
Guhr, T.: Norm-dependent random matrix ensembles in external field and supersymmetry. J. Phys. A, Math. Gen. 39, 12327–12342 (2006)
Guhr, T.: Arbitrary rotation invariant matrix ensembles and supersymmetry. J. Phys. A, Math. Gen. 39, 13191–13223 (2006)
Johansson, K.: Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Commun. Math. Phys. 215(3), 683–705 (2001)
LeCaër, G., Delannay, R.: The fixed-trace β-Hermite ensemble of random matrices and the low temperature distribution of the determinant of an N×N β-Hermite matrix. J. Phys. A 40, 1561–1584 (2007)
Liu, D.-Z., Zhou, D.-S.: Local statistical properties of Schmidt eigenvalues of bipartite entanglement for a random pure state. Int. Math. Res. Not. doi:10.1093/imrn/rnq091, arXiv:0912.3999v2
Mehta, M.L.: Random Matrices, 3rd edn. Pure and Applied Mathematics, vol. 142. Elsevier/Academic Press, Amsterdam (2004)
Pastur, L., Shcherbina, M.: Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles. J. Stat. Phys. 86(1–2), 109–147 (1997)
Rosenzweig, N.: Statistical mechanics of equally likely quantum systems. In: Statistical Physics (Brandeis Summer Institute, 1962), vol. 3, pp. 91–158. Benjamin, Elmsford (1963)
Soshnikov, A.: Universality at the edge of the spectrum in Wigner random matrices. Commun. Math. Phys. 207, 697–733 (1999)
Soshnikov, A.: Determinantal point random fields. Russ. Math. Surv. 55(5), 923–975 (2000)
Szegö, G.: Orthogonal Polynomials, 1st edn. Am. Math. Soc., New York (1939)
Tao, T., Vu, V.: Random matrices: Universality of local eigenvalue statistics. arXiv:0906.0510
Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994)
Tracy, C.A., Widom, H.: Fredholm determinants, differential equations and matrix models. Commun. Math. Phys. 163, 33–72 (1994)
Tracy, C.A., Widom, H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177, 727–754 (1996)
Zhou, D.-S., Liu, D.-Z., Qian, T.: Fixed trace β-Hermite ensembles: Asymptotic eigenvalue density and the edge of the density. J. Math. Phys. 51, 033301 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, DZ., Zhou, DS. Some Universal Properties for Restricted Trace Gaussian Orthogonal, Unitary and Symplectic Ensembles. J Stat Phys 140, 268–288 (2010). https://doi.org/10.1007/s10955-010-9993-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-010-9993-9