Electronic Journal of Probability

Large deviations principle for the largest eigenvalue of Wigner matrices without Gaussian tails

Fanny Augeri

Full-text: Open access


We prove a large deviations principle for the largest eigenvalue of Wigner matrices without Gaussian tails, that is, the distribution tails of the diagonal entries $\mathbb{P} ( |X_{1,1}|>t)$ and off-diagonal entries $\mathbb{P} (|X_{1,2}|>t)$ behave like $e^{-bt^{\alpha }}$ and $e^{-at^{\alpha }}$ respectively, for some $a,b\in (0,+\infty )$ and $\alpha \in (0,2)$. The large deviations principle is of speed $N^{\alpha /2}$, and with a good rate function depending only on the distribution tail of the entries.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 32, 49 pp.

Received: 27 February 2015
Accepted: 27 January 2016
First available in Project Euclid: 18 April 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60F10: Large deviations

random matrices large deviations

Creative Commons Attribution 4.0 International License.


Augeri, Fanny. Large deviations principle for the largest eigenvalue of Wigner matrices without Gaussian tails. Electron. J. Probab. 21 (2016), paper no. 32, 49 pp. doi:10.1214/16-EJP4146. https://projecteuclid.org/euclid.ejp/1461007173

Export citation


  • [1] G. W. Anderson, A. Guionnet, and O. Zeitouni. An introduction to random matrices, volume 118 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010.
  • [2] G. Ben Arous and A. Guionnet. Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy. Probabilty theory and related fields, 108:517–542, 1997.
  • [3] Z. Bai and J. W. Silverstein. Spectral analysis of large dimensional random matrices. Springer Series in Statistics. Springer, New York, second edition, 2010.
  • [4] Z. D. Bai and Y. Q. Yin. Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix. Ann. Probab., 16(4):1729–1741, 1988.
  • [5] G. Ben Arous, A. Dembo, and A. Guionnet. Aging of spherical spin glasses. Probab. Theory Related Fields, 120(1):1–67, 2001.
  • [6] F. Benaych-Georges, A. Guionnet, and M. Maida. Large deviations of the extreme eigenvalues of random deformations of matrices. Probab. Theory Related Fields, 154(3-4):703–751, 2012.
  • [7] F. Benaych-Georges and R. Rao Nadakuditi. The eigenvalues and eigenvectors of finite, low rank pertubation of large random matrices. Advances in Mathematics, 227:494–521, 2011.
  • [8] R. Bhatia. Matrix analysis, volume 169 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997.
  • [9] C. Bordenave and P. Caputo. A large deviation principle for Wigner matrices without Gaussian tails. Ann. Probab., 42(6):2454–2496, 2014.
  • [10] T. Cabanal Duvillard and A. Guionnet. Large deviations upper bounds for the laws of matrix-valued processes and non-communicative entropies. Ann. Probab., 29(3):1205–1261, 2001.
  • [11] F. Clarke. Functional analysis, calculus of variations and optimal control, volume 264 of Graduate Texts in Mathematics. Springer, London, 2013.
  • [12] A. Dembo and O. Zeitouni. Large deviations techniques and applications, volume 38 of Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition.
  • [13] D. Féral and S. Péché. The largest eigenvalue of rank one deformation of large Wigner matrices. Comm. Math. Phys., 272(1):185–228, 2007.
  • [14] Z. Füredi and J. Komlós. The eigenvalues of random symmetric matrices. Combinatorica, 1(3):233–241, 1981.
  • [15] B. Groux. Asymptotic freeness for rectangular random matrices and large deviations for sample covariance matrices with sub-gaussian tails. arXiv:1505.05733 [math.PR].
  • [16] A. Guionnet, M. Maïda, and F. Benaych-Georges. Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices. Electronic Journal of Probability, 16(60):1621–1662, 2011.
  • [17] A. Guionnet and O. Zeitouni. Large deviations asymptotics for spherical integrals. J. Funct. Anal., 188(2):461–515, 2002.
  • [18] A. Hardy. A note on large deviations for 2D Coulomb gas with weakly confining potential. Electron. Commun. Probab., 17(19):12, 2012.
  • [19] M. Maïda. Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles. Electron. J. Probab., 12:1131–1150 (electronic), 2007.
  • [20] P. Massart, G. Lugosi, and S. Boucheron. Concentration Inequalities : A Nonasymptotic Theory of Independence. Oxford University Press, 2013.
  • [21] S. Péché. The largest eigenvalue of small rank perturbations of Hermitian random matrices. Probab. Theory Related Fields, 134(1):127–173, 2006.
  • [22] A. Pizzo, D. Renfrew, and A. Soshnikov. Fluctuations of matrix entries of regular functions of Wigner matrices. Journal of Statistical Physics, 146(3):550–591, 2012.
  • [23] A. Pizzo, D. Renfrew, and A. Soshnikov. On finte rank deformations of Wigner matrices. Ann. Inst. H. Poincaré Probab. Statist., 49(120):64–94, 2013.
  • [24] R. T. Rockafellar. Convex analysis. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1997.
  • [25] Eugene P. Wigner. On the distribution of the roots of certain symmetric matrices. Ann. of Math. (2), 67:325–327, 1958.
  • [26] X. Zhan. Matrix inequalities, volume 1790 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2002.