Universality of local spectral statistics of random matrices

Bulletin of the American Mathematical Society

Volumn 49, Issue 3, 2012, Pages 377-414

  Erdos, László ^a   Yau, Horng Tzer ^b  

^a UNIVERSITY OF MUNICH   (Germany)

^b HARVARD UNIVERSITY   (United States)

Author keywords

Dyson Brownian motion; Local semicircle law; Random matrix; Tracy Widom distribution

Indexed keywords

EID: 84863099462     PISSN: 02730979     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0273-0979-2012-01372-1     Document Type: Article

References (78)

1. Anderson, G., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Studies in advanced mathematics, 118, Cambridge University Press, 2009.
2. Albeverio, S., Pastur, L., Shcherbina, M.: On the 1/n expansion for some unitary invariant ensembles of random matrices, Commun. Math. Phys. 224, 271-305 (2001).
3. Auffinger, A., Ben Arous, G., Péché, S.: Poisson Convergence for the largest eigenvalues of heavy-tailed matrices. Ann. Inst. Henri Poincaŕe Probab. Stat. 45 (2009), no. 3, 589-610.
4. Bai, Z. D., Miao, B., Tsay, J.: Convergence rates of the spectral distributions of large Wigner matrices. Int. Math. J. 1 (2002), no. 1, 65-90.
5. Bakry, D., Émery, M.: Diffusions hypercontractives. In: Séminaire de probabilités, XIX, 1983/84, vol. 1123 of Lecture Notes in Math. Springer, Berlin, 1985, pp. 177-206.
6. Ben Arous, G., Péché, S.: Universality of local eigenvalue statistics for some sample covariance matrices. Comm. Pure Appl. Math. LVIII. (2005), 1-42.
7. Berry, M.V., Tabor, M.: Level clustering in the regular spectrum, Proc. Roy. Soc. A 356 (1977) 375-394.
8. Bleher, P., Its, A.: Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. of Math. 150 (1999), 185-266.
9. Bohigas, O.; Giannoni, M.-J.; Schmit, C.: Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys. Rev. Lett. 52 (1984), no. 1, 1-4.
10. Bourgade, P., Erdo{double acute}s, Yau, H.-T.: Universality of General β-Ensembles, arXiv:1104.2272
11. Bourgade, P., Erdo{double acute}s, Yau, H.-T.: Bulk Universality of General β-Ensembles with Non-convex Potential, arxiv:1201.2283
12. Brézin, E., Hikami, S.: Correlations of nearby levels induced by a random potential. Nucl. Phys. B 479 (1996), 697-706; and Spectral form factor in a random matrix theory. Phys. Rev. E 55 (1997), 4067-4083.
13. Chatterjee, S.: A generalization of the Lindeberg principle. Ann. Probab. 34 (2006), no. 6, 2061-2076.
14. Deift, P.: Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Courant Lecture Notes in Mathematics 3, American Mathematical Society, Providence, RI, 1999.
15. Deift, P., Gioev, D.: Universality in random matrix theory for orthogonal and symplectic ensembles. Int. Math. Res. Pap. IMRP 2007, no. 2, Art. ID rpm004, 116 pp.
16. Deift, P., Gioev, D.: Random Matrix Theory: Invariant Ensembles and Universality. Courant Lecture Notes in Mathematics 18, American Mathematical Society, Providence, RI, 2009.
17. 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 (1999):1335-1425.
18. 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 (1999): 1491-1552.
19. Dumitriu, I., Edelman, A.: Matrix Models for Beta Ensembles, Journal of Mathematical Physics 43 (11) (2002), 5830-5847.
20. Dyson, F.J.: Statistical theory of energy levels of complex systems, I, II, and III. J. Math. Phys. 3, 140-156, 157-165, 166-175 (1962).
21. Dyson, F.J.: A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191-1198 (1962).
22. Dyson, F.J.: Correlations between eigenvalues of a random matrix. Commun. Math. Phys. 19, 235-250 (1970).
23. Erdo{double acute}s, L., Knowles, A.: Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model. Preprint. Arxiv:1002.1695
24. Erdo{double acute}s, L., A. Knowles, A.: Quantum Diffusion and Delocalization for Band Matrices with General Distribution. Annales Inst. H. Poincare, 12 (7), 1227-1319 (2011).
25. Erdo{double acute}s, L., Knowles, A., Yau, H.-T., Yin, J.: Spectral Statistics of Erdo{double acute}s-Rényi Graphs I: Local Semicircle Law. Preprint. Arxiv:1103.1919
26. Erdo{double acute}s, L., Knowles, A., Yau, H.-T., Yin, J.: Spectral Statistics of Erdo{double acute}s-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues. Preprint. Arxiv:1103.3869
27. Erdo{double acute}s, L., Péché, G., Ramírez, J., Schlein, B., and Yau, H.-T., Bulk universality for Wigner matrices. Commun. Pure Appl. Math. 63, No. 7, 895-925 (2010).
28. Erdo{double acute}s, L., Ramirez, J., Schlein, B., Tao, T., Vu, V., Yau, H.-T.: Bulk Universality for Wigner Hermitian matrices with subexponential decay. Math. Res. Lett. 17 (2010), no. 4, 667-674.
29. Erdo{double acute}s, L., Ramirez, J., Schlein, B., Yau, H.-T.: Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation. Electr. J. Prob. 15, Paper 18, 526-604 (2010).
30. Erdo{double acute}s, L., Schlein, B., Yau, H.-T.: Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. Ann. Probab. 37, No. 3, 815-852 (2009).
31. Erdo{double acute}s, L., Schlein, B., Yau, H.-T.: Local semicircle law and complete delocalization for Wigner random matrices. Commun. Math. Phys. 287, 641-655 (2009).
32. Erdo{double acute}s, L., Schlein, B., Yau, H.-T.: Wegner estimate and level repulsion for Wigner random matrices. Int. Math. Res. Notices. 2010, No. 3, 436-479 (2010).
33. Erdo{double acute}s, L., Schlein, B., Yau, H.-T.: Universality of random matrices and local relaxation flow. Invent. Math. 185 (2011), no.1, 75-119.
34. Erdo{double acute}s, L., Schlein, B., Yau, H.-T., Yin, J.: The local relaxation flow approach to universality of the local statistics for random matrices. Preprint arXiv:0911.3687
35. Erdo{double acute}s, L., Yau, H.-T.: A comment on the Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices. Preprint arXiv:1201.5619
36. Erdo{double acute}s, L., Yau, H.-T., Yin, J.: Bulk universality for generalized Wigner matrices. Preprint arXiv:1001.3453
37. Erdo{double acute}s, L., Yau, H.-T., Yin, J.: Universality for generalized Wigner matrices with Bernoulli distribution. J. of Combinatorics, 1 (2011), no. 2, 15-85.
38. Erdo{double acute}s, L., Yau, H.-T., Yin, J.: Rigidity of Eigenvalues of Generalized Wigner Matrices, preprint arXiv:1007.4652
39. Erdo{double acute}s, P., Rényi, A.: On Random Graphs. I. Publicationes Mathematicae 6, 290-297 (1959).
40. Erdo{double acute}s, P., Rényi, A.: The Evolution of Random Graphs. Magyar Tud. Akad. Mat. Kutat́o Int. K̈ozl. 5 17-61 (1960).
41. Eynard, B.: Master loop equations, free energy and correlations for the chain of matrices. J. High Energy Phys. 11 2003, 018.
42. Fokas, A. S., Its, A. R., Kitaev, A. V.: The isomonodromy approach to matrix models in 2D quantum gravity. Comm. Math. Phys. 147 (1992), 395-430.
43. Gaudin, M.: Sur la loi limit de l'espacement des valeurs propres d'une matrice aléatoire. Nucl. Phys. 25, 447-458.
44. Guionnet, A., Zeitouni, O.: Concentration of the spectral measure for large matrices. Electronic Comm. in Probability 5 (2000) Paper 14.
45. Itzykson, C., Zuber, J.B.: The planar approximation, II. J. Math. Phys. 21 411-421 (1980).
46. Johansson, K.: Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Comm. Math. Phys. 215 (2001), no.3. 683-705.
47. Johansson, K.: Universality for certain Hermitian Wigner matrices under weak moment conditions. Preprint arxiv.org/abs/0910.4467
48. Knowles, A., Yin, J.: Eigenvector distribution of Wigner matrices. Preprint arXiv:1102.0057.
49. Kriecherbauer, T., Shcherbina, M.: Fluctuations of eigenvalues of matrix models and their applications. Preprint arXiv:1003.6121
50. Lubinsky, D.S.: A New Approach to Universality Limits Involving Orthogonal Polynomials, Annals of Mathematics, 170, 915-939 (2009).
51. Mehta, M.L.: Random Matrices. Third Edition, Academic Press, New York, 1991.
52. Mehta, M.L.: A note on correlations between eigenvalues of a random matrix. Commun. Math. Phys. 20 no.3. 245-250 (1971).
53. Mehta, M.L., Gaudin, M.: On the density of eigenvalues of a random matrix. Nuclear Phys. 18, 420-427 (1960).
54. Montgomery, H.L.: The pair correlation of zeros of the zeta function. Analytic number theory, Proc. of Sympos. in Pure Math. 24, Amer. Math. Soc. Providence, R.I., 181-193 (1973).
55. 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).
56. Pastur, L., Shcherbina M.: Bulk universality and related properties of Hermitian matrix models. J. Stat. Phys. 130 (2008), no.2., 205-250.
57. Ramirez, J., Rider, B., Virág, B.: Beta ensembles, stochastic Airy spectrum, and a diffusion. arXiv:math/0607331. To appear in J. Amer. Math. Soc.
58. Schenker, J.: Eigenvector localization for random band matrices with power law band width. Commun. Math. Phys. 290, 1065-1097 (2009).
59. Péché, S., Soshnikov, A.: On the lower bound of the spectral norm of symmetric random matrices with independent entries. Electron. Commun. Probab. 13 (2008), 280-290.
60. Péché, S., Soshnikov, A.: Wigner random matrices with non-symmetrically distributed entries. J. Stat. Phys. 129 (2007), no. 5-6, 857-884.
61. Shcherbina, M.: Orthogonal and symplectic matrix models: universality and other properties. Preprint arXiv:1004.2765
62. Sinai, Y. and Soshnikov, A.: A refinement of Wigner's semicircle law in a neighborhood of the spectrum edge. Functional Anal. and Appl. 32 (1998), no. 2, 114-131.
63. Sodin, S.: The spectral edge of some random band matrices. Preprint. arXiv: 0906.4047
64. Sodin, S.: The Tracy-Widom law for some sparse random matrices. Preprint. arXiv:0903.4295
65. Soshnikov, A.: Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 (1999), no.3. 697-733.
66. Spencer, T.: Review article on random band matrices. Draft in preparation.
67. Tao, T. and Vu, V.: Random matrices: Universality of the local eigenvalue statistics, Acta Math., 206 (2011), no. 1, 127-204.
68. Tao, T. and Vu, V.: Random matrices: Universality of local eigenvalue statistics up to the edge. Commun. Math. Phys., 298 (2010), no. 2, 549-572.
69. Tao, T. and Vu, V.: Random covariance matrices: Universality of local statistics of eigenvalues. Preprint. arXiv:0912.0966
70. Tao, T. and Vu, V.: Random matrices: Universal properties of eigenvectors. Preprint. arXiv:1103.2801
71. Tao, T. and Vu, V.: The Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices. Preprint. arXiv:1101.5707
72. Tracy, C., Widom, H.: Level-Spacing Distributions and the Airy Kernel, Comm. Math. Phys. 159 (1994), 151-174.
73. Tracy, C., Widom, H.: On orthogonal and symplectic matrix ensembles, Comm. Math. Phys. 177 (1996), no. 3, 727-754.
74. Valkö, B.; Virág, B.: Continuum limits of random matrices and the Brownian carousel. Invent. Math. 177 (2009), no. 3, 463-508.
75. Widom H.: On the relation between orthogonal, symplectic and unitary matrix ensembles. J. Statist. Phys. 94 (1999), no. 3-4, 347-363.
76. Wigner, E.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. 62 (1955), 548-564.
77. Wishart, J.: The generalized product moment distribution in samples from a Normal multivariate population. Biometrika 20A, 32-52 (1928).
78. Yau, H. T.: Relative entropy and the hydrodynamics of Ginzburg-Landau models, Lett. Math. Phys. 22 (1991) 63-80.