Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the large deviations of traces of random matrices

Fanny Augeri

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We present large deviations principles for the moments of the empirical spectral measure of Wigner matrices and empirical measure of $\beta $-ensembles in three cases: the case of $\beta $-ensembles associated with a convex potential with polynomial growth, the case of Gaussian Wigner matrices, and the case of Wigner matrices without Gaussian tails, that is Wigner matrices whose entries have tail distributions decreasing as $e^{-ct^{\alpha }}$, for some constant $c>0$ and with $\alpha \in (0,2)$.


Nous proposons des principes de grandes déviations pour les moments de la mesure spectrale empirique de matrices de Wigner et de la mesure empirique de $\beta $-ensembles dans trois cas : celui des $\beta $-ensembles associés à un potentiel convexe à croissance polynomiale, le cas des matrices de Wigner Gaussiennes, et le cas des matrices de Wigner sans queues Gaussiennes, c’est-à-dire dont les entrées ont une queue de distribution ayant le même comportement que $e^{-ct^{\alpha }}$, pour une certaine constante $c>0$ et $\alpha \in (0,2)$.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 4 (2018), 2239-2285.

Received: 23 May 2016
Revised: 6 September 2017
Accepted: 22 October 2017
First available in Project Euclid: 18 October 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Large deviations Wigner matrices $\beta $-ensembles


Augeri, Fanny. On the large deviations of traces of random matrices. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 4, 2239--2285. doi:10.1214/17-AIHP870.

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  • [1] G. W. Anderson, A. Guionnet and O. Zeitouni. An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge University Press, Cambridge, 2010.
  • [2] F. Augeri. Large deviations principle for the largest eigenvalue of Wigner matrices without Gaussian tails. Electron. J. Probab. 21 (2016) Paper No. 32, 49.
  • [3] Z. Bai and J. W. Silverstein. Spectral Analysis of Large Dimensional Random Matrices, 2nd edition. Springer Series in Statistics. Springer, New York, 2010.
  • [4] G. Ben Arous and A. Guionnet. Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy. Probab. Theory Related Fields 108 (1997) 517–542.
  • [5] R. Bhatia. Matrix Analysis. Graduate Texts in Mathematics 169. Springer-Verlag, New York, 1997.
  • [6] S. G. Bobkov and M. Ledoux. From Brunn–Minkowski to Brascamp–Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10 (5) (2000) 1028–1052.
  • [7] C. Bordenave and P. Caputo. A large deviation principle for Wigner matrices without Gaussian tails. Ann. Probab. 42 (6) (2014) 2454–2496.
  • [8] C. Borell. On polynomial chaos and integrability. Probab. Math. Statist. 3 (2) (1984) 191–203.
  • [9] S. Chatterjee and S. R. S. Varadhan. The large deviation principle for the Erdős–Rényi random graph. European J. Combin. 32 (7) (2011) 1000–1017.
  • [10] F. Clarke. Functional Analysis, Calculus of Variations and Optimal Control. Graduate Texts in Mathematics 264. Springer, London, 2013.
  • [11] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability 38. Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition.
  • [12] R. Dobrushin, P. Groeneboom and M. Ledoux Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics 1648. Springer-Verlag, Berlin, 1996. Lectures from the 24th Saint-Flour Summer School held July 7–23, 1994. Edited by P. Bernard.
  • [13] H. Döring and P. Eichelsbacher. Moderate deviations in a random graph and for the spectrum of Bernoulli random matrices. Electron. J. Probab. 14 (92) (2009) 2636–2656.
  • [14] P. Eichelsbacher and J. Sommerauer. Moderate deviations for traces of words in a multi-matrix model. Electron. Commun. Probab. 14 (2009) 572–586.
  • [15] 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.
  • [16] N. Gantert, K. Ramanan and F. Rembart. Large deviations for weighted sums of stretched exponential random variables. Electron. Commun. Probab. 19 (41) (2014) 14.
  • [17] B. Groux. Asymptotic freeness for rectangular random matrices and large deviations for sample covariance matrices with sub-Gaussian tails. Available at arXiv:1505.05733 [math.PR].
  • [18] A. Guionnet. Large Random Matrices: Lectures on Macroscopic Asymptotics. Lecture Notes in Mathematics 1957. Springer-Verlag, Berlin, 2009. Lectures from the 36th Probability Summer School held in Saint-Flour, 2006.
  • [19] D. Jonsson. Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12 (1) (1982) 1–38.
  • [20] R. Latała. Estimates of moments and tails of Gaussian chaoses. Ann. Probab. 34 (6) (2006) 2315–2331.
  • [21] M. Ledoux. A note on large deviations for Wiener chaos. In Séminaire de Probabilités XXIV, 1988/89 1–14. Lecture Notes in Math. 1426. Springer, Berlin, 1990.
  • [22] M. Ledoux. The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. American Mathematical Society, Providence, RI, 2001.
  • [23] M. Ledoux and M. Talagrand. Probability in Banach Spaces. Isoperimetry and processes. Classics in Mathematics. Springer-Verlag, Berlin, 2011. Reprint of the 1991 edition.
  • [24] P. Massart, G. Lugosi and S. Boucheron. Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford University Press, Oxford, 2013.
  • [25] M. W. Meckes and S. J. Szarek. Concentration for noncommutative polynomials in random matrices. Proc. Amer. Math. Soc. 140 (5) (2012) 1803–1813.
  • [26] S. V. Nagaev. Large deviations of sums of independent random variables. Ann. Probab. 7 (5) (1979) 745–789.
  • [27] Ya. Sinai and A. Soshnikov. A refinement of Wigner’s semicircle law in a neighborhood of the spectrum edge for random symmetric matrices. Funktsional. Anal. i Prilozhen. 32 (2) (1998) 56–79, 96.
  • [28] Ya. Sinai and A. Soshnikov. Central limit theorem for traces of large random symmetric matrices with independent matrix elements. Bull. Braz. Math. Soc. (N.S.) 29 (1) (1998) 1–24.
  • [29] A. Soshnikov. Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 (3) (1999) 697–733.
  • [30] E. P. Wigner. On the distribution of the roots of certain symmetric matrices. Ann. of Math. (2) 67 (1958) 325–327.
  • [31] X. Zhan. Matrix Inequalities. Lecture Notes in Mathematics 1790. Springer-Verlag, Berlin, 2002.