Spectral measures of powers of random matrices

Elizabeth S Meckes (Case Western Reserve University)
Mark W Meckes (Case Western Reserve University)


This paper considers the empirical spectral measure of a power of a random matrix drawn uniformly from one of the compact classical matrix groups. We give sharp bounds on the $L_p$-Wasserstein distances between this empirical measure and the uniform measure on the circle, which show a smooth transition in behavior when the power increases and yield rates on almost sure convergence when the dimension grows. Along the way, we prove the sharp logarithmic Sobolev inequality on the unitary group.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-13

Publication Date: September 23, 2013

DOI: 10.1214/ECP.v18-2551


  • Anderson, Greg W.; Guionnet, Alice; Zeitouni, Ofer. An introduction to random matrices. Cambridge Studies in Advanced Mathematics, 118. Cambridge University Press, Cambridge, 2010. xiv+492 pp. ISBN: 978-0-521-19452-5 MR2760897
  • Bakry, D.; Émery, Michel. Diffusions hypercontractives. (French) [Hypercontractive diffusions] Séminaire de probabilités, XIX, 1983/84, 177--206, Lecture Notes in Math., 1123, Springer, Berlin, 1985. MR0889476
  • Bhatia, Rajendra. Matrix analysis. Graduate Texts in Mathematics, 169. Springer-Verlag, New York, 1997. xii+347 pp. ISBN: 0-387-94846-5 MR1477662
  • Blower, Gordon. Random matrices: high dimensional phenomena. London Mathematical Society Lecture Note Series, 367. Cambridge University Press, Cambridge, 2009. x+437 pp. ISBN: 978-0-521-13312-8 MR2566878
  • Dallaporta, S. Eigenvalue variance bounds for Wigner and covariance random matrices. Random Matrices Theory Appl. 1 (2012), no. 3, 1250007, 28 pp. MR2967966
  • del Barrio, Eustasio; Giné, Evarist; Matrán, Carlos. Central limit theorems for the Wasserstein distance between the empirical and the true distributions. Ann. Probab. 27 (1999), no. 2, 1009--1071. MR1698999
  • Diaconis, Persi. Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture. Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 2, 155--178. MR1962294
  • Dyson, Freeman J. Correlations between eigenvalues of a random matrix. Comm. Math. Phys. 19 1970 235--250. MR0278668
  • Gustavsson, Jonas. Gaussian fluctuations of eigenvalues in the GUE. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 2, 151--178. MR2124079
  • Hiai, Fumio; Petz, Dénes; Ueda, Yoshimichi. A free logarithmic Sobolev inequality on the circle. Canad. Math. Bull. 49 (2006), no. 3, 389--406. MR2252261
  • Hough, J. Ben; Krishnapur, Manjunath; Peres, Yuval; Virág, Bálint. Determinantal processes and independence. Probab. Surv. 3 (2006), 206--229. MR2216966
  • Katz, Nicholas M.; Sarnak, Peter. Random matrices, Frobenius eigenvalues, and monodromy. American Mathematical Society Colloquium Publications, 45. American Mathematical Society, Providence, RI, 1999. xii+419 pp. ISBN: 0-8218-1017-0 MR1659828
  • Ledoux, Michel. The concentration of measure phenomenon. Mathematical Surveys and Monographs, 89. American Mathematical Society, Providence, RI, 2001. x+181 pp. ISBN: 0-8218-2864-9 MR1849347
  • Meckes, Elizabeth S.; Meckes, Mark W. Concentration and convergence rates for spectral measures of random matrices. Probab. Theory Related Fields 156 (2013), no. 1-2, 145--164. MR3055255
  • Rains, E. M. High powers of random elements of compact Lie groups. Probab. Theory Related Fields 107 (1997), no. 2, 219--241. MR1431220
  • Rains, Eric M. Images of eigenvalue distributions under power maps. Probab. Theory Related Fields 125 (2003), no. 4, 522--538. MR1974413
  • Rothaus, O. S. Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities. J. Funct. Anal. 42 (1981), no. 1, 102--109. MR0620581
  • Soshnikov, Alexander B. Gaussian fluctuation for the number of particles in Airy, Bessel, sine, and other determinantal random point fields. J. Statist. Phys. 100 (2000), no. 3-4, 491--522. MR1788476
  • Talagrand, Michel. The generic chaining. Upper and lower bounds of stochastic processes. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005. viii+222 pp. ISBN: 3-540-24518-9 MR2133757
  • Weissler, Fred B. Logarithmic Sobolev inequalities and hypercontractive estimates on the circle. J. Funct. Anal. 37 (1980), no. 2, 218--234. MR0578933

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.