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The Abel Symposium

Abelsymposium 2016: Computation and Combinatorics in Dynamics, Stochastics and Control pp 415–427Cite as

Heavy Tailed Random Matrices: How They Differ from the GOE, and Open Problems

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Part of the book series: Abel Symposia ((ABEL,volume 13))

Abstract

Since the pioneering works of Wishart and Wigner on random matrices, matrices with independent entries with finite moments have been intensively studied. Not only it was shown that their spectral measure converges to the semi-circle law, but fluctuations both global and local were analyzed in fine details. More recently, the domain of universality of these results was investigated, in particular by Erdos-Yau et al and Tao-Vu et al. This survey article takes the opposite point of view by considering matrices which are not in the domain of universality of Wigner matrices: they have independent entries but with heavy tails. We discuss the properties of these matrices. They are very different from Wigner matrices: the limit law of the spectral measure is not the semi-circle distribution anymore, the global fluctuations are stronger and the local fluctuations may undergo a transition and remain rather mysterious.

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References

  1. Aggarwal, A.: Bulk universality for generalized wigner matrices with few moments. arXiv:1612.00421 (2016)

    Google Scholar 

  2. Aggarwal, A., Lopatto, P., Yau, H.-T.: GOE statistics for Levy matrices. https://arxiv.org/abs/1806.07363 (2018)

  3. Aizenman, M., Warzel, S.: Disorder-induced delocalization on tree graphs. In: Exner, P. (ed.) Mathematical Results in Quantum Physics, pp. 107–109. World Scientific Publication, Hackensack (2011). MR 2885163

    Google Scholar 

  4. Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge University Press, Cambridge (2010). MR 2760897

    Google Scholar 

  5. Anderson, G.W., Zeitouni, O.: A CLT for a band matrix model. Probab. Theory Rel. Fields 134, 283–338 (2005). MR 2222385

    Article  MathSciNet  MATH  Google Scholar 

  6. Belinschi, S., Dembo, A., Guionnet, A.: Spectral measure of heavy tailed band and covariance random matrices. Commun. Math. Phys. 289(3), 1023–1055 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ben Arous, G., Guionnet, A.: The spectrum of heavy tailed random matrices. Commun. Math. Phys. 278(3), 715–751 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Benaych-Georges, F., Guionnet, A.: Central limit theorem for eigenvectors of heavy tailed matrices. Electron. J. Probab. 19(54), 27 (2014). MR 3227063

    Google Scholar 

  9. Benaych-Georges, F., Guionnet, A., Male, C.: Central limit theorems for linear statistics of heavy tailed random matrices. Commun. Math. Phys. 329(2), 641–686 (2014). MR 3210147

    Article  MathSciNet  MATH  Google Scholar 

  10. Benaych-Georges, F., Maltsev, A.: Fluctuations of linear statistics of half-heavy-tailed random matrices. Stoch. Process. Appl. 126(11), 3331–3352 (2016). MR 3549710

    Article  MathSciNet  MATH  Google Scholar 

  11. Bordenave, C., Caputo, P., Chafaï, D.: Spectrum of large random reversible Markov chains: heavy-tailed weights on the complete graph. Ann. Probab. 39(4), 1544–1590 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bordenave, C., Guionnet, A.: Localization and delocalization of eigenvectors for heavy-tailed random matrices. Probab. Theory Relat. Fields 157(3–4), 885–953 (2013). MR 3129806

    Article  MathSciNet  MATH  Google Scholar 

  13. Bordenave, C., Guionnet, A.: Delocalization at small energy for heavy-tailed random matrices. Commun. Math. Phys. 354(1), 115–159 (2017). MR 3656514

    Article  MathSciNet  MATH  Google Scholar 

  14. Bouchaud, J.-P., Cizeau, P.: Theory of Lévy matrices. Phys. Rev. E 3, 1810–1822 (1994)

    Google Scholar 

  15. Bourgade, P., Erdos, L., Yau, H.-T., Yin, J.: Universality for a class of random band matrices. Adv. Theor. Math. Phys. 21(3), 739–800 (2017). MR 3695802

    Article  MathSciNet  MATH  Google Scholar 

  16. Cébron, G., Dahlqvist, A., Male, C: Universal constructions for space of traffics (2016). arXiv:1601.00168

    Google Scholar 

  17. Bordenave, C., Sen, A., Virág, B.: Mean quantum percolation. J. Eur. Math. Soc. (JEMS) 19(12), 3679–3707 (2017). MR 3730511

    Article  MathSciNet  MATH  Google Scholar 

  18. Erdős, L., Schlein, B., Yau, H.-T.: Wegner estimate and level repulsion for Wigner random matrices. Int. Math. Res. Not. (IMRN) (3), 436–479 (2010). MR 2587574

    Google Scholar 

  19. Erdős, L.: Universality of Wigner random matrices: a survey of recent results. Uspekhi Mat. Nauk 66(3)(399), 67–198 (2011). MR 2859190

    Google Scholar 

  20. Erdős, L., Knowles, A., Yau, H.-T., Yin, J.: The local semicircle law for a general class of random matrices. Electron. J. Probab. 18(59), 58 (2013). MR 3068390

    Google Scholar 

  21. Erdős, L., Péché, S., Ramírez, J.A., Schlein, B., Yau, H.-T.: Bulk universality for Wigner matrices. Commun. Pure Appl. Math. 63(7), 895–925 (2010). MR 2662426

    Google Scholar 

  22. Erdős, L., Schlein, B., Yau, H.-T.: Local semicircle law and complete delocalization for Wigner random matrices. Commun. Math. Phys. 287(2), 641–655 (2009)

    Google Scholar 

  23. Erdős, L., Schlein, B., Yau, H.-T.: Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. Ann. Probab. 37(3), 815–852 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Johansson, K.: On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91, 151–204 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  25. Johansson, K.: Universality for certain Hermitian Wigner matrices under weak moment conditions. Ann. Inst. Henri Poincaré Probab. Stat. 48(1), 47–79 (2012). MR 2919198

    Article  MathSciNet  MATH  Google Scholar 

  26. Lee, J.O., Yin, J.: A necessary and sufficient condition for edge universality of Wigner matrices. Duke Math. J. 163(1), 117–173 (2014). MR 3161313

    Article  MathSciNet  MATH  Google Scholar 

  27. Lytova, A., Pastur, L.: Central limit theorem for linear eigenvalue statistics of random matrices with independent entries. Ann. Probab. 37(5), 1778–1840 (2009). MR 2561434

    Article  MathSciNet  MATH  Google Scholar 

  28. Male, C.: Traffics distributions and independence: the permutation invariant matrices and the notions of independence. arXiv:1111.4662 (2011)

    Google Scholar 

  29. Male, C.: The limiting distributions of large heavy Wigner and arbitrary random matrices. J. Funct. Anal. 272(1), 1–46 (2017). MR 3567500

    Article  MathSciNet  MATH  Google Scholar 

  30. Soshnikov, A.: Universality at the edge of the spectrum in Wigner random matrices. Commun. Math. Phys. 207(3), 697–733 (1999). MR 1727234

    Article  MathSciNet  MATH  Google Scholar 

  31. Sosoe, P., Wong, P.: Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices. Adv. Math. 249, 37–87 (2013). MR 3116567

    Article  MathSciNet  MATH  Google Scholar 

  32. Tao, T., Vu, V.: Random matrices: universality of local eigenvalue statistics up to the edge. Commun. Math. Phys. 298(2), 549–572 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tao, T., Vu, V.: The Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices. Electron. J. Probab. 16(77), 2104–2121 (2011). MR 2851058

    Article  MathSciNet  MATH  Google Scholar 

  34. Tarquini, E., Biroli, G., Tarzia, M.: Level statistics and localization transitions of levy matrices. Phys. Rev. Lett. 116, 010601 (2015)

    Article  MATH  Google Scholar 

  35. Wigner, E.P.: On the distribution of the roots of certain symmetric matrices. Ann. Math. 67, 325–327 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wishart, J.: The generalized product moment distribution in samples from a normal multivariate population. Biometrika 20A, 32–52 (1928)

    Article  MATH  Google Scholar 

  37. Zakharevich, I.: A generalization of Wigner’s law. Commun. Math. Phys. 268, 403–414 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Université de Lyon, CNRS, ENSL, Lyon, France

    Alice Guionnet

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  1. Alice Guionnet
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Correspondence to Alice Guionnet .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Elena Celledoni

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Giulia Di Nunno

  3. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  4. Department of Mathematics, University of Bergen, Bergen, Norway

    Hans Zanna Munthe-Kaas

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Guionnet, A. (2018). Heavy Tailed Random Matrices: How They Differ from the GOE, and Open Problems. In: Celledoni, E., Di Nunno, G., Ebrahimi-Fard, K., Munthe-Kaas, H. (eds) Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-030-01593-0_15

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