Uspekhi Matematicheskikh Nauk
General information
Latest issue
Impact factor
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Uspekhi Mat. Nauk:

Personal entry:
Save password
Forgotten password?

Uspekhi Mat. Nauk, 2011, Volume 66, Issue 3(399), Pages 67–198 (Mi umn9426)  

This article is cited in 62 scientific papers (total in 62 papers)

Universality of Wigner random matrices: a survey of recent results

L. Erdős

Ludwig-Maximilians-Universität München

Abstract: This is a study of the universality of spectral statistics for large random matrices. Considered are $N\times N$ symmetric, Hermitian, or quaternion self-dual random matrices with independent identically distributed entries (Wigner matrices), where the probability distribution of each matrix element is given by a measure $\nu$ with zero expectation and with subexponential decay. The main result is that the correlation functions of the local eigenvalue statistics in the bulk of the spectrum coincide with those of the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE), and the Gaussian Symplectic Ensemble (GSE), respectively, in the limit as $N\to\infty$. This approach is based on a study of the Dyson Brownian motion via a related new dynamics, the local relaxation flow. As a main input, it is established that the density of the eigenvalues converges to the Wigner semicircle law, and this holds even down to the smallest possible scale. Moreover, it is shown that the eigenvectors are completely delocalized. These results hold even without the condition that the matrix elements are identically distributed: only independence is used. In fact, for the matrix elements of the Green function strong estimates are given that imply that the local statistics of any two ensembles in the bulk are identical if the first four moments of the matrix elements match. Universality at the spectral edges requires matching only two moments. A Wigner-type estimate is also proved, and it is shown that the eigenvalues repel each other on arbitrarily small scales.
Bibliography: 108 titles.

Keywords: Wigner random matrices, Dyson Brownian motion, semicircle law, sine kernel.


Full text: PDF file (1544 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2011, 66:3, 507–626

Bibliographic databases:

UDC: 519.216+519.217+512.64
MSC: 15B52, 82B44
Received: 07.04.2010

Citation: L. Erdős, “Universality of Wigner random matrices: a survey of recent results”, Uspekhi Mat. Nauk, 66:3(399) (2011), 67–198; Russian Math. Surveys, 66:3 (2011), 507–626

Citation in format AMSBIB
\by L.~Erd{\H o}s
\paper Universality of Wigner random~matrices: a~survey of recent results
\jour Uspekhi Mat. Nauk
\yr 2011
\vol 66
\issue 3(399)
\pages 67--198
\jour Russian Math. Surveys
\yr 2011
\vol 66
\issue 3
\pages 507--626

Linking options:

    SHARE: FaceBook Twitter Livejournal

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Charlier Ch., “Large Gap Asymptotics For the Generating Function of the Sine Point Process”, Proc. London Math. Soc.  crossref  isi
    2. A. I. Aptekarev, V. I. Buslaev, A. Martínez-Finkelshtein, S. P. Suetin, “Padé approximants, continued fractions, and orthogonal polynomials”, Russian Math. Surveys, 66:6 (2011), 1049–1131  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. Shcherbina T., “On the correlation function of the characteristic polynomials of the Hermitian Wigner ensemble”, Comm. Math. Phys., 308:1 (2011), 1–21  crossref  mathscinet  zmath  adsnasa  isi
    4. Astrauskas A., “Extremal theory for spectrum of random discrete Schrödinger operator. II. Distributions with heavy tails”, J. Stat. Phys., 146:1 (2012), 98–117  crossref  mathscinet  zmath  adsnasa  isi
    5. A. V. Komlov, S. P. Suetin, “Widom's formula for the leading coefficient of a polynomial which is orthonormal with respect to a varying weight”, Russian Math. Surveys, 67:1 (2012), 183–185  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. A. V. Komlov, S. P. Suetin, “An asymptotic formula for polynomials orthonormal with respect to a varying weight”, Trans. Moscow Math. Soc., 73 (2012), 139–159  mathnet  crossref  mathscinet  zmath  elib
    7. Litvak A.E., Rivasplata O., “Smallest singular value of sparse random matrices”, Studia Math., 212:3 (2012), 195–218  crossref  mathscinet  zmath  isi  elib
    8. Metcalfe A.P., “Universality properties of Gelfand–Tsetlin patterns”, Probab. Theory Relat. Fields, 155:1-2 (2013), 303–346  crossref  mathscinet  zmath  isi  elib
    9. Th. C. Bachlechner, D. Marsh, L. McAllister, T. Wrase, “Supersymmetric vacua in random supergravity”, J. High Energ. Phys., 2013:1 (2013), 136, 23 pp.  crossref  mathscinet  zmath  isi
    10. M. Golshani, A. R. Bahrampour, A. Langari, A. Szameit, “Transverse localization in nonlinear photonic lattices with second-order coupling”, Phys. Rev. A, 87:3 (2013), 033817  crossref  mathscinet  adsnasa  isi  elib
    11. D. Lubinsky, “A variational principle for correlation functions for unitary ensembles, with applications”, Anal. PDE, 6:1 (2013), 109–130  crossref  mathscinet  zmath  isi
    12. M. C. David Marsh, L. McAllister, E. Pajer, T. Wrase, “Charting an inflationary landscape with random matrix theory”, J. Cosmol. Astropart. Phys., 2013:11 (2013), 040  crossref  isi
    13. A. Lytova, “On Non-Gaussian Limiting Laws for Certain Statistics of Wigner Matrices”, Zhurn. matem. fiz., anal., geom., 9:4 (2013), 536–581  mathnet  mathscinet
    14. Su Zh., “Fluctuations of deformed Wigner random matrices”, Front. Math. China, 8:3 (2013), 609–641  crossref  mathscinet  zmath  isi
    15. Friesen O., Löwe M., “A phase transition for the limiting spectral density of random matrices”, Electron. J. Probab., 18:17 (2013), 17, 17 pp.  crossref  mathscinet  zmath  isi  elib
    16. Bordenave Ch., Guionnet A., “Localization and delocalization of eigenvectors for heavy-tailed random matrices”, Probab. Theory Related Fields, 157:3-4 (2013), 885–953  crossref  mathscinet  zmath  isi
    17. Benaych-Georges F., Peche S., “Localization and Delocalization For Heavy Tailed Band Matrices”, Ann. Inst. Henri Poincare-Probab. Stat., 50:4 (2014), 1385–1403  crossref  mathscinet  zmath  isi
    18. Gurau R., “Universality For Random Tensors”, Ann. Inst. Henri Poincare-Probab. Stat., 50:4 (2014), 1474–1525  crossref  mathscinet  zmath  isi
    19. Addario-Berry L., Eslava L., “Hitting Time Theorems For Random Matrices”, Comb. Probab. Comput., 23:5 (2014), 635–669  crossref  mathscinet  zmath  isi
    20. Van Vu, Ke Wang, “Random weighted projections, random quadratic forms and random eigenvectors”, Random Struct. Alg., 47:4 (2015), 792–821  crossref  mathscinet  zmath  isi
    21. C. Cacciapuoti, A. Maltsev, B. Schlein, “Bounds for the Stieltjes transform and the density of states of Wigner matrices”, Probab. Theory Related Fields, 163:1 (2015), 1–59  crossref  mathscinet  zmath  isi
    22. Bao Zh., Pan G., Zhou W., “Universality For the Largest Eigenvalue of Sample Covariance Matrices With General Population”, Ann. Stat., 43:1 (2015), 382–421  crossref  mathscinet  zmath  isi
    23. Tao T., Vu V., “Random Matrices: Universality of Local Spectral Statistics of Non-Hermitian Matrices”, Ann. Probab., 43:2 (2015), 782–874  crossref  mathscinet  zmath  isi
    24. Zdzisław Burda, Giacomo Livan, Pierpaolo Vivo, “Invariant sums of random matrices and the onset of level repulsion”, J. Stat. Mech., 2015:6 (2015), P06024  crossref  mathscinet  isi
    25. Theory Probab. Appl., 60:3 (2016), 407–443  mathnet  crossref  crossref  mathscinet  isi  elib
    26. Schubert K., “Spacings in Orthogonal and Symplectic Random Matrix Ensembles”, 42, no. 3, 2015, 481–518  crossref  mathscinet  zmath  isi
    27. Kumar S., “Random Matrix Ensembles Involving Gaussian Wigner and Wishart Matrices, and Biorthogonal Structure”, 92, no. 3, 2015, 032903  crossref  mathscinet  isi
    28. Kargin V., “Subordination For the Sum of Two Random Matrices”, 43, no. 4, 2015, 2119–2150  crossref  mathscinet  zmath  isi
    29. Xie J., “Limit Theorems For the Counting Function of Eigenvalues Up To Edge in Covariance Matrices”, 65, no. 1, 2015, 199–214  crossref  mathscinet  zmath  isi
    30. Schlein B., “Spectral Properties of Wigner Matrices”, Correlated Random Systems: Five Different Methods: Cirm Jean-Morlet Chair, Spring 2013, Lecture Notes in Mathematics, 2143, eds. Gayrard V., Kistler N., Springer Int Publishing Ag, 2015, 179–205  crossref  mathscinet  zmath  isi
    31. Yanqing Yin, Zhidong Bai, Jiang Hu, “On the Semicircular Law of Large-Dimensional Random Quaternion Matrices”, J. Theor. Probab., 2016  crossref  mathscinet
    32. Doron S. Lubinsky, “An Update on Local Universality Limits for Correlation Functions Generated by Unitary Ensembles”, SIGMA, 12 (2016), 078, 36 pp.  mathnet  crossref
    33. O'Rourke S., Vu V., Wang K., “Eigenvectors of random matrices: A survey”, J. Comb. Theory Ser. A, 144:SI (2016), 361–442  crossref  mathscinet  zmath  isi  scopus
    34. Manrique P., Perez-Abreu V., Roy R., “On the universality of the non-singularity of general Ginibre and Wigner random matrices”, Random Matrices-Theor. Appl., 5:1 (2016), 1650002  crossref  mathscinet  zmath  isi
    35. Schubert K., “Spectral Density for Random Matrices with Independent Skew-Diagonals”, Electron. Commun. Probab., 21 (2016), 40  crossref  mathscinet  zmath  isi  scopus
    36. Wegner F., “Random Matrix Theory”: Wegner, F, Supermathematics and its Applications in Statistical Physics, Lecture Notes in Physics, 920, Springer-Verlag Berlin, 2016, 227–259  crossref  mathscinet  isi  scopus
    37. Couillet R., Benaych-Georges F., “Kernel spectral clustering of large dimensional data”, Electron. J. Stat., 10:1 (2016), 1393–1454  crossref  mathscinet  zmath  isi  scopus
    38. F. Götze, A. A. Naumov, A. N. Tikhomirov, “Local semicircle law under moment conditions: Stieltjes transform, rigidity and delocalization”, Theory Probab. Appl., 62:1 (2018), 58–83  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    39. Bun J., Bouchaud J.-Ph., Potters M., “Cleaning large correlation matrices: Tools from Random Matrix Theory”, Phys. Rep.-Rev. Sec. Phys. Lett., 666 (2017), 1–109  crossref  mathscinet  zmath  isi  scopus
    40. Kusmierz L., Toyoizumi T., “Emergence of Levy Walks From Second-Order Stochastic Optimization”, Phys. Rev. Lett., 119:25 (2017), 250601  crossref  isi
    41. Soloveychik I., Xiang Yu., Tarokh V., “Explicit Symmetric Pseudo-Random Matrices”, 2017 IEEE Information Theory Workshop (ITW), Information Theory Workshop, IEEE, 2017, 424–428  isi
    42. Lubinsky D.S., “Scaling Limits of Polynomials and Entire Functions of Exponential Type”, Approximation Theory Xv, Springer Proceedings in Mathematics & Statistics, 201, eds. Fasshauer G., Schumaker L., Springer, 2017, 219–238  crossref  mathscinet  zmath  isi
    43. Bordenave Ch., Guionnet A., “Delocalization At Small Energy For Heavy-Tailed Random Matrices”, Commun. Math. Phys., 354:1 (2017), 115–159  crossref  mathscinet  zmath  isi
    44. He Yu., Knowles A., “Mesoscopic Eigenvalue Statistics of Wigner Matrices”, Ann. Appl. Probab., 27:3 (2017), 1510–1550  crossref  mathscinet  zmath  isi
    45. Ho W.W., Radicevic D., “The Ergodicity Landscape of Quantum Theories”, Int. J. Mod. Phys. A, 33:4 (2018), 1830004  crossref  mathscinet  zmath  isi
    46. I. Soloveychik, Yu. Xiang, V. Tarokh, “Symmetric pseudo-random matrices”, IEEE Trans. Inform. Theory, 64:4/2 (2018), 3179–3196  crossref  mathscinet  zmath  isi
    47. Lee J.O., Schnelli K., “Local Law and Tracy-Widom Limit For Sparse Random Matrices”, Probab. Theory Relat. Field, 171:1-2 (2018), 543–616  crossref  mathscinet  zmath  isi
    48. Forrester P.J., Trinh A.K., “Functional Form For the Leading Correction to the Distribution of the Largest Eigenvalue in the Gue and Lue”, J. Math. Phys., 59:5 (2018), 053302  crossref  mathscinet  zmath  isi
    49. Livan G., Novaes M., Vivo P., “Saddle-Point-of-View”: Livan, G Novaes, M Vivo, P, Introduction to Random Matrices: Theory and Practice, Springerbriefs in Mathematical Physics, 26, Springer International Publishing Ag, 2018, 33–43  crossref  mathscinet  isi
    50. von Soosten P., Warzel S., “The Phase Transition in the Ultrametric Ensemble and Local Stability of Dyson Brownian Motion”, Electron. J. Probab., 23 (2018), 70  crossref  mathscinet  zmath  isi  scopus
    51. Brinkman B.A.W., Rieke F., Shea-Brown E., Buice M.A., “Predicting How and When Hidden Neurons Skew Measured Synaptic Interactions”, PLoS Comput. Biol., 14:10 (2018), e1006490  crossref  isi  scopus
    52. von Soosten P., Warzel S., “Singular Spectrum and Recent Results on Hierarchical Operators”, Mathematical Problems in Quantum Physics, Contemporary Mathematics, 717, eds. Bonetto F., Borthwick D., Harrell E., Loss M., Amer Mathematical Soc, 2018, 215–225  crossref  mathscinet  isi  scopus
    53. “Abstracts of talks given at the 3rd International Conference on Stochastic Methods”, Theory Probab. Appl., 64:1 (2019), 124–169  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    54. Forrester P.J., Trinh A.K., “Optimal Soft Edge Scaling Variables For the Gaussian and Laguerre Even Beta Ensembles”, Nucl. Phys. B, 938 (2019), 621–639  crossref  mathscinet  zmath  isi  scopus
    55. Aggarwal A., “Bulk Universality For Generalized Wigner Matrices With Few Moments”, Probab. Theory Relat. Field, 173:1-2 (2019), 375–432  crossref  mathscinet  zmath  isi  scopus
    56. Acharya A., Kypraios T., Guta M., “A Comparative Study of Estimation Methods in Quantum Tomography”, J. Phys. A-Math. Theor., 52:23 (2019), 234001  crossref  isi
    57. Loewe M., Schubert K., “On the Limiting Spectral Density of Random Matrices Filled With Stochastic Processes”, Random Operators Stoch. Equ., 27:2 (2019), 89–105  crossref  isi
    58. Chan Ch.H., Xiong M., 2019 Ninth International Workshop on Signal Design and Its Applications in Communications (Iwsda), International Workshop on Signal Design and Its Applications in Communications, IEEE, 2019  isi
    59. Morrison M., Gabbay M., “Community Detectability and Structural Balance Dynamics in Signed Networks”, Phys. Rev. E, 102:1 (2020), 012304  crossref  mathscinet  isi
    60. Mohanty S., Raghavendra P., Xu J., “Lifting Sum-of-Squares Lower Bounds: Degree-2 to Degree-4”, Proceedings of the 52Nd Annual Acm Sigact Symposium on Theory of Computing (Stoc `20), Annual Acm Symposium on Theory of Computing, eds. Makarychev K., Makarychev Y., Tulsiani M., Kamath G., Chuzhoy J., Assoc Computing Machinery, 2020, 840–853  crossref  mathscinet  isi
    61. Tao Ya., “Exploring Anderson Localization in One-Dimensional Single-Electron Lattice Systems Via Randomness of Eigenvalue Spectra”, Phys. Lett. A, 385 (2021), 126955  crossref  isi
    62. Chan Ch.H., Tarokh V., Xiong M., “Convergence Rate of Empirical Spectral Distribution of Random Matrices From Linear Codes”, IEEE Trans. Inf. Theory, 67:2 (2021), 1080–1087  crossref  mathscinet  isi
  • Успехи математических наук Russian Mathematical Surveys
    Number of views:
    This page:1308
    Full text:532
    First page:41

    Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2021