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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Aggarwal, A.: Bulk universality for generalized wigner matrices with few moments. arXiv:1612.00421 (2016)
Aggarwal, A., Lopatto, P., Yau, H.-T.: GOE statistics for Levy matrices. https://arxiv.org/abs/1806.07363 (2018)
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
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
Anderson, G.W., Zeitouni, O.: A CLT for a band matrix model. Probab. Theory Rel. Fields 134, 283–338 (2005). MR 2222385
Belinschi, S., Dembo, A., Guionnet, A.: Spectral measure of heavy tailed band and covariance random matrices. Commun. Math. Phys. 289(3), 1023–1055 (2009)
Ben Arous, G., Guionnet, A.: The spectrum of heavy tailed random matrices. Commun. Math. Phys. 278(3), 715–751 (2008)
Benaych-Georges, F., Guionnet, A.: Central limit theorem for eigenvectors of heavy tailed matrices. Electron. J. Probab. 19(54), 27 (2014). MR 3227063
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
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
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)
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
Bordenave, C., Guionnet, A.: Delocalization at small energy for heavy-tailed random matrices. Commun. Math. Phys. 354(1), 115–159 (2017). MR 3656514
Bouchaud, J.-P., Cizeau, P.: Theory of Lévy matrices. Phys. Rev. E 3, 1810–1822 (1994)
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
Cébron, G., Dahlqvist, A., Male, C: Universal constructions for space of traffics (2016). arXiv:1601.00168
Bordenave, C., Sen, A., Virág, B.: Mean quantum percolation. J. Eur. Math. Soc. (JEMS) 19(12), 3679–3707 (2017). MR 3730511
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
Erdős, L.: Universality of Wigner random matrices: a survey of recent results. Uspekhi Mat. Nauk 66(3)(399), 67–198 (2011). MR 2859190
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
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
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)
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)
Johansson, K.: On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91, 151–204 (1998)
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
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
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
Male, C.: Traffics distributions and independence: the permutation invariant matrices and the notions of independence. arXiv:1111.4662 (2011)
Male, C.: The limiting distributions of large heavy Wigner and arbitrary random matrices. J. Funct. Anal. 272(1), 1–46 (2017). MR 3567500
Soshnikov, A.: Universality at the edge of the spectrum in Wigner random matrices. Commun. Math. Phys. 207(3), 697–733 (1999). MR 1727234
Sosoe, P., Wong, P.: Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices. Adv. Math. 249, 37–87 (2013). MR 3116567
Tao, T., Vu, V.: Random matrices: universality of local eigenvalue statistics up to the edge. Commun. Math. Phys. 298(2), 549–572 (2010)
Tao, T., Vu, V.: The Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices. Electron. J. Probab. 16(77), 2104–2121 (2011). MR 2851058
Tarquini, E., Biroli, G., Tarzia, M.: Level statistics and localization transitions of levy matrices. Phys. Rev. Lett. 116, 010601 (2015)
Wigner, E.P.: On the distribution of the roots of certain symmetric matrices. Ann. Math. 67, 325–327 (1958)
Wishart, J.: The generalized product moment distribution in samples from a normal multivariate population. Biometrika 20A, 32–52 (1928)
Zakharevich, I.: A generalization of Wigner’s law. Commun. Math. Phys. 268, 403–414 (2006)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-030-01593-0_15
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-01592-3
Online ISBN: 978-3-030-01593-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)