Universality of Wigner random matrices: A survey of recent results

Russian Mathematical Surveys

Volumn 66, Issue 3, 2011, Pages 507-626

  Erdos, L ^a  

^a UNIVERSITY OF MUNICH   (Germany)

Author keywords

Dyson Brownian motion; Semicircle law; Sine kernel; Wigner random matrices

Indexed keywords

EID: 80052768867     PISSN: 00360279     EISSN: 14684829     Source Type: Journal    
DOI: 10.1070/RM2011v066n03ABEH004749     Document Type: Article

References (111)

1. M. L. Mehta, Random matrices, Academic Press, Boston, MA 1991, xviii+562 pp.
2. P. A. Deift, Orthogonal polynomials and random matrices: A Riemann-Hilbert approach, Courant Lect. Notes Math., vol. 3, New York Univ., Courant Institute of Mathematical Sciences, New York; Amer. Math. Soc., Providence, RI 1999, viii+273 pp.
3. G. W. Anderson, A. Guionnet, and O. Zeitouni, An introduction to random matrices, Cambridge Stud. Adv. Math., vol. 118, Cambridge Univ. Press, Cambridge 2010, xiv+492 pp.
4. P. J. Forrester, Log-gases and random matrices, London Math. Soc. Monogr. Ser., vol. 34, Princeton Univ. Press, Princeton, NJ 2010, 791 pp.
5. A. Guionnet, Grandes matrices aléatoires et théorÈ mes d'universalité, Séminaire Bourbaki, 62Ème année, no. 1019, 2009-2010, 33 pp.
6. L. Erdos, B. Schlein, H.-T. Yau, and J. Yin, "The local relaxation flow approach to universality of the local statistics for random matrices", Ann. Inst. Henri Poincaré (to appear); arXiv: 0911.3687.
7. L. Erdos, H.-T. Yau, and J. Yin, Rigidity of eigenvalues of generalized Wigner matrices, arXiv: 1007.4652.
8. L. Erdos, H.-T. Yau, and J. Yin, Universality for generalized Wigner matrices with Bernoulli distribution, arXiv: 1003.3813.
9. L. Erdos, H.-T. Yau, and J. Yin, Bulk universality for generalized Wigner matrices, arXiv: 1001.3453.
10. F. J. Dyson, "Correlations between eigenvalues of a random matrix", Comm. Math. Phys. 19:3 (1970), 235-250.
11. M. L. Mehta, "A note on correlations between eigenvalues of a random matrix", Comm. Math. Phys. 20:3 (1971), 245-250.
12. E. Brézin and S. Hikami, "Correlations of nearby levels induced by a random potential", Nuclear Phys. B 479:3 (1996), 697-706; (Pubitemid 126161217)
13. "Spectral form factor in a random matrix theory", Phys. Rev. E (3) 55:4 (1997), 4067-4083.
14. K. J. Johansson, "Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices", Comm. Math. Phys. 215:3 (2001), 683-705.
15. L. Erdos, S. Péché, J. A. Ramírez, B. Schlein, and H.-T. Yau, "Bulk universality for Wigner matrices", Comm. Pure Appl. Math. 63:7 (2010), 895-925; arXiv: 0905.4176.
16. F. J. Dyson, "A Brownian-motion model for the eigenvalues of a random matrix", J. Math. Phys. 3:6 (1962), 1191-1198.
17. L. Erdos, B. Schlein, and H.-T. Yau, Universality of random matrices and local relaxation flow, arXiv: 0907.5605.
18. T. Tao and V. Vu, "Random matrices: Universality of local eigenvalue statistics", Acta Math. 206:1 (2011), 127-204; arXiv: 0906.0510.
19. L. Erdos, J. Ramírez, B. Schlein, T. Tao, V. Vu, and H.-T. Yau, "Bulk universality for Wigner Hermitian matrices with subexponential decay", Math. Res. Lett. 17:4 (2010), 667-674, http://www.mathjournals.org/ mrl/2010-017-004/2010-017-004-007.html.
20. English transl., V. A. Marčenko and L. A. Pastur, "Distribution of eigenvalues for some sets of random matrices", Math. USSR-Sb. 1:4 (1967), 457-483.
21. K. W. Wachter, "Strong limits of random matrix spectra for sample covariance matrices of independent elements", Ann. Probab. 6:1 (1978), 1-18.
22. E.P. Wigner, "Characteristic vectors of bordered matrices with infinite dimensions", Ann. of Math. (2) 62:3 (1955), 548-564.
23. J. Wishart, "The generalized product moment distribution in samples from a normal multivariate population", Biometrika 20A (1928), 32-52.
24. J. H. Schenker and H. Schulz-Baldes, "Semicircle law and freeness for random matrices with symmetries or correlations", Math. Res. Lett. 12:4 (2005), 531-542, http://www.mathjournals.org/mrl/2005-012-004/2005-012-004-007. html. (Pubitemid 41353383)
25. J. Schenker and H. Schulz-Baldes, "Gaussian fluctuations for random matrices with correlated entries", Int. Math. Res. Not. IMRN 15 (2007), rnm047, 36 pp.
26. P. W. Anderson, "Absences of diffusion in certain random lattices", Phys. Rev. 109:5 (1958), 1492-1505.
27. M. V. Berry and M. Tabor, "Level clustering in the regular spectrum", Proc. R. Soc. Lond. Ser. A 356:1686 (1977), 375-394.
28. O. Bohigas, M. J. Giannoni, and C. Schmit, "Characterization of chaotic quantum spectra and universality of level fluctuation laws", Phys. Rev. Lett. 52:1 (1984), 1-4.
29. J. Fr ohlich and T. Spencer, "Absence of diffusion in the Anderson tight binding model for large disorder or low energy", Comm. Math. Phys. 88:2 (1983), 151-184.
30. J. Fr ohlich, F. Martinelli, E. Scoppola, and T. Spencer, "Constructive proof of localization in the Anderson tight binding model", Comm. Math. Phys. 101:1 (1985), 21-46.
31. M. Aizenman and S. Molchanov, "Localization at large disorder and at extreme energies: an elementary derivation", Comm. Math. Phys. 157:2 (1993), 245-278.
32. N. Minami, "Local fluctuation of the spectrum of a multidimensional Anderson tight binding model", Comm. Math. Phys. 177:3 (1996), 709-725. (Pubitemid 126324595)
33. T. Spencer, Lifshitz tails and localization, Preprint, 1993.
34. A. Elgart, "Lifshitz tails and localization in the three-dimensional Anderson model", Duke Math. J. 146:2 (2009), 331-360.
35. A. Klein, "Absolutely continuous spectrum in the Anderson model on the Bethe lattice", Math. Res. Lett. 1:4 (1994), 399-407, http://www.mrlonline.org/mrl/1994-001-004/1994-001-004-001.html.
36. M. Aizenman, R. Sims, and S. Warzel, "Absolutely continuous spectra of quantum tree graphs with weak disorder", Comm. Math. Phys. 264:2 (2006), 371-389. (Pubitemid 43593297)
37. R. Froese, D. Hasler, and W. Spitzer, "Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schr odinger operators on graphs", J. Funct. Anal. 230:1 (2006), 184-221. (Pubitemid 41748477)
38. M. Aizenman and S. Warzel, "The canopy graph and level statistics for random operators on trees", Math. Phys. Anal. Geom. 9:4 (2007), 291-333.
39. I. Rodnianski and W. Schlag, "Classical and quantum scattering for a class of long range random potentials", Int. Math. Res. Not. 2003:5 (2003), 243-300.
40. J. Bourgain, "Random lattice Schr odinger operators with decaying potential: some higher dimensional phenomena", Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1807, Springer, Berlin 2003, pp. 70-98.
41. S. A. Denisov, "Absolutely continuous spectrum of multidimensional Schr odinger operator", Int. Math. Res. Not. 2004:74, 3963-3982; arXiv: math/0408376. (Pubitemid 40195076)
42. W. De Roeck and J. Fr ohlich, "Diffusion of a massive quantum particle coupled to a quasi-free thermal medium", Comm. Math. Phys. 303:3 (2011), 613-707; arXiv: 0906.5178.
43. T. Chen, "Localization lengths and Boltzmann limit for the Anderson model at small disorders in dimension 3", J. Stat. Phys. 120:1-2 (2005), 279-337. (Pubitemid 41276800)
44. H. Spohn, "Derivation of the transport equation for electrons moving through random impurities", J. Statist. Phys. 17:6 (1977), 385-412.
45. L. Erd os and H.-T. Yau, "Linear Boltzmann equation as the weak coupling limit of the random Schr odinger equation", Comm. Pure Appl. Math. 53:6 (2000), 667-735.
46. L. Erd os, M. Salmhofer, and H.-T. Yau, "Quantum diffusion for the Anderson model in the scaling limit", Ann. Henri Poincar'e 8:4 (2007), 621-685.
47. K. B. Efetov, Supersymmetry in disorder and chaos, Cambridge Univ. Press, Cambridge 1997, xiv+441 pp.
48. Y. V. Fyodorov and A. D. Mirlin, "Scaling properties of localization in random band matrices: A σ-model approach", Phys. Rev. Lett. 67:18 (1991), 2405-2409.
49. J. Schenker, "Eigenvector localization for random band matrices with power law band width", Comm. Math. Phys. 290:3 (2009), 1065-1097.
50. L. Erdos and A. Knowles, "Quantum diffusion and eigenfunction delocalization in a random band matrix model", Comm. Math. Phys. 303:2 (2011), 509-554; arXiv: 1002.1695.
51. L. Erdos and A. Knowles, Quantum diffusion and delocalization for band matrices with general distribution, arXiv: 1005.1838.
52. M. Disertori, H. Pinson, and T. Spencer, "Density of states for random band matrices", Comm. Math. Phys. 232:1 (2002), 83-124.
53. M. Disertori and T. Spencer, "Anderson localization for a supersymmetric sigma model", Comm. Math. Phys. 300:3 (2010), 659-671; arXiv: 0910.3325.
54. M. Disertori, T. Spencer, and M. R. Zirnbauer, "Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model", Comm. Math. Phys. 300:2 (2010), 435-486; arXiv: 0901.1652.
55. T. Tao, V. Vu, and M. Krishnapur, "Random matrices: universality of ESD's and the circular law", Ann. Probab. 38:5 (2010), 2023-2065; arXiv: 0807.4898.
56. L. Erdos, B. Schlein, and H.-T. Yau, "Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices", Ann. Probab. 37:3 (2009), 815-852.
57. L. Erdos, B. Schlein, and H.-T. Yau, "Local semicircle law and complete delocalization for Wigner random matrices", Comm. Math. Phys. 287:2 (2009), 641-655.
58. L. Erdos, B. Schlein, and H.-T. Yau, "Wegner estimate and level repulsion for Wigner random matrices", Int. Math. Res. Not. IMRN 2010:3 (2010), 436-479; arXiv: 0811.2591.
59. Z. D. Bai and Y. Q. Yin, "Limit of the smallest eigenvalue of a large dimensional sample covariance matrix", Ann. Probab. 21:3 (1993), 1275-1294.
60. A. Guionnet, "Large deviations upper bounds and central limit theorems for non-commutative functionals of Gaussian large random matrices", Ann. Inst. H. Poincar'e Probab. Statist. 38:3 (2002), 341-384.
61. G. W. Anderson and O. Zeitouni, "A CLT for a band matrix model", Probab. Theory Related Fields 134:2 (2006), 283-338. (Pubitemid 41750690)
62. B. Simon, Trace ideals and their applications, 2nd ed., Math. Surveys Monogr., vol. 120, Amer. Math. Soc., Providence, RI 2005, viii+150 pp.
63. A. Soshnikov, "Universality at the edge of the spectrum in Wigner random matrices", Comm. Math. Phys. 207:3 (1999), 697-733.
64. English transl., Ya. G. Sinai and A. B. Soshnikov, "A refinement of Wigner's semicircle law in a neighbourhood of the spectrum edge for random symmetric matrices", Funct. Anal. Appl. 32:2 (1998), 114-131. (Pubitemid 128697748)
65. T. Tao and V. Vu, "Random matrices: universality of local eigenvalue statistics up to the edge", Comm. Math. Phys. 298:2 (2010), 549-572; arXiv: 0908.1982.
66. C. A. Tracy and H. Widom, "Level-spacing distributions and the Airy kernel", Comm. Math. Phys. 159:1 (1994), 151-174.
67. C. A. Tracy and H. Widom, "On orthogonal and symplectic matrix ensembles", Comm. Math. Phys. 177:3 (1996), 727-754.
68. A. Ruzmaikina, "Universality of the edge distribution of eigenvalues of Wigner random matrices with polynomially decaying distributions of entries", Comm. Math. Phys. 261:2 (2006), 277-296. (Pubitemid 41697407)
69. S. P'ech'e and A. Soshnikov, "Wigner random matrices with non-symmetrically distributed entries", J. Stat. Phys. 129:5-6 (2007), 857-884.
70. T. Tao and V. Vu, "Random covariance matrices: Universality of local statistics of eigenvalues", Ann. Probab. (to appear); arXiv: 0912.0966.
71. F. J. Dyson, "Statistical theory of energy levels of complex systems. I", J. Math. Phys. 3:1 (1962), 140-156;
72. F. J. Dyson, "Statistical theory of energy levels of complex systems. II", J. Math. Phys. 3:1 (1962), 157-165;
73. F. J. Dyson, "Statistical theory of energy levels of complex systems. III", J. Math. Phys. 3:1 (1962), 166-175.
74. I. Dumitriu and A. Edelman, "Matrix models for beta ensembles", J. Math. Phys. 43:11 (2002), 5830-5847.
75. J. Ram'irez, B. Rider, and B. Virag, Beta ensembles, stochastic Airy spectrum, and a diffusion, arXiv: math/0607331.
76. J. A. Ram'irez and B. Rider, "Diffusion at the random matrix hard edge", Comm. Math. Phys. 288:3 (2009), 887-906.
77. A. Edelman and B. D. Sutton, "From random matrices to stochastic operators", J. Stat. Phys. 127:6 (2007), 1121-1165.
78. B. Valk'o and B. Vir'ag, "Continuum limits of random matrices and the Brownian carousel", Invent. Math. 177:3 (2009), 463-508.
79. B. Valk'o and B. Vir'ag, "Large gaps between random eigenvalues", Ann. Probab. 38:3 (2010), 1263-1279; arXiv: 0811.0007.
80. M. L. Mehta and M. Gaudin, "On the density of eigenvalues of a random matrix", Nuclear Phys. 18 (1960), 420-427.
81. P. Bleher and A. Its, "Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model", Ann. of Math. (2) 150:1 (1999), 185-266; arXiv: math-ph/9907025.
82. P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, "Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory", Comm. Pure Appl. Math. 52:11 (1999), 1335-1425.
83. P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, "Strong asymptotics of orthogonal polynomials with respect to exponential weights", Comm. Pure Appl. Math. 52:12 (1999), 1491-1552.
84. A. S. Fokas, A. R. Its, and A. V. Kitaev, "The isomonodromy approach to matric models in 2D quantum gravity", Comm. Math. Phys. 147:2 (1992), 395-430.
85. L. Pastur and M. Shcherbina, "Bulk universality and related properties of Hermitian matrix models", J. Stat. Phys. 130:2 (2008), 205-250.
86. D. S. Lubinsky, "A new approach to universality limits involving orthogonal polynomials", Ann. of Math. (2) 170:2 (2009), 915-939.
87. E. Levin and D. S. Lubinsky, "Universality limits in the bulk for varying measures", Adv. Math. 219:3 (2008), 743-779.
88. G. Ben Arous and S. P'ech'e, "Universality of local eigenvalue statistics for some sample covariance matrices", Comm. Pure Appl. Math. 58:10 (2005), 1316-1357.
89. K. Johansson, Universality for certain Hermitian Wigner matrices under weak moment conditions, arXiv: 0910.4467.
90. C. Itzykson and J.-B. Zuber, "The planar approximation. II", J. Math. Phys. 21:3 (1980), 411-421.
91. E. Br'ezin and S. Hikami, "An extension of the Harish-Chandra- Itzykson-Zuber integral", Comm. Math. Phys. 235:1 (2003), 125-137.
92. L. Erdos, J. Ram'irez, B. Schlein, and H.-T. Yau, "Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation", Electron. J. Probab. 15 (2010), 526-603, Paper 18, http://www.math.washington. edu/-ejpecp/viewarticle.php?id=2084&layout=abstract; arXiv: 0905.2089.
93. M. Shcherbina, "Edge universality for orthogonal ensembles of random matrices", J. Stat. Phys. 136:1 (2009), 35-50; arXiv: 0812.3228.
94. D. Bakry and M. 'Emery, "Diffusions hypercontractives", S'eminaire de probabilit'es XIX, Lecture Notes in Math., vol. 1123, Springer, Berlin 1985, pp. 177-206.
95. J. Gustavsson, "Gaussian fluctuations of eigenvalues in the GUE", Ann. Inst. H. Poincar'e Probab. Statist. 41:2 (2005), 151-178. (Pubitemid 40300386)
96. S. O'Rourke, "Gaussian fluctuations of eigenvalues in Wigner random matrices", J. Stat. Phys. 138:6 (2010), 1045-1066.
97. R. Curto and L. Fialkow, "Recursiveness, positivity and truncated moment problems", Houston J. Math. 17:4 (1991), 603-635.
98. O. N. Feldheim and S. Sodin, "A universality result for the smallest eigenvalues of certain sample covariance matrices", Geom. Funct. Anal. 20:1 (2010), 88-123; arXiv: 0812.1961.
99. S. Sodin, "The spectral edge of some random band matrices", Ann. of Math. (2) 172:3 (2010), 2223-2251; arXiv: 0906.4047.
100. S. Sodin, "The Tracy-Widom law for some sparse random matrices", J. Stat. Phys. 136:5 (2010), 834-841; arXiv: 0903.4295.
101. S. P'ech'e and A. Soshnikov, "On the lower bound of the spectral norm of symmetric random matrices with independent entries", Electron. Commun. Probab. 13 (2008), 280-290, http://www.math.washington.edu/-ejpecp/ECP/ viewarticle.php?id=2004&layout=abstract.
102. A. Auffinger, G. Ben Arous, and S. P'ech'e, "Poisson convergence for the largest eigenvalues of heavy tailed random matrices", Ann. Inst. Henri Poincar'e Probab. Statist. 45:3 (2009), 589-610.
103. G. Biroli, J.-P. Bouchaud, and M. Potters, "On the top eigenvalue of heavy-tailed random matrices", Europhys. Lett. EPL 78:1 (2007), Art. 10001, 5 pp.
104. D. L. Hanson and F. T. Wright, "A bound on tail probabilities for quadratic forms in independent random variables", Ann. Math. Statist. 42:3 (1971), 1079-1083.
105. E. B. Davies, "The functional calculus", J. London Math. Soc. (2) 52:1 (1995), 166-176.
106. T. Spencer, "Random banded and sparse matrices (Chapter 23)", Oxford handbook of random matrix theory (G. Akemann, J. Baik, P. Di Francesco, eds.), Oxford Univ. Press, Oxford 2011, 960 pp.
107. Z. D. Bai, B. Miao, and J. Tsay, "Convergence rates of the spectral distributions of large Wigner matrices", Int. Math. J. 1:1 (2002), 65-90.
108. T. Tao and V. Vu, Random matrices: localization of the eigenvalues and the necessity of four moments, arXiv: 1005.2901.
109. H.-T. Yau, "Relative entropy and the hydrodynamics of Ginzburg-Landau models", Lett. Math. Phys. 22:1 (1991), 63-80.
110. D. W. Stroock, Probability theory: an analytic view, Cambridge Univ. Press, Cambridge 1993, xvi+512 pp.
111. English transl. of 1st ed., A. N. Shiryaev, Probability, Grad. Texts in Math., vol. 95, Springer-Verlag, New York 1984, xi+577 pp.