Electronic Communications in Probability

Wigner theorems for random matrices with dependent entries: Ensembles associated to symmetric spaces and sample covariance matrices

Katrin Hofmann-Credner and Michael Stolz

Full-text: Open access


It is a classical result of Wigner that for an hermitian matrix with independent entries on and above the diagonal, the mean empirical eigenvalue distribution converges weakly to the semicircle law as matrix size tends to infinity. In this paper, we prove analogs of Wigner's theorem for random matrices taken from all infinitesimal versions of classical symmetric spaces. This is a class of models which contains those studied by Wigner and Dyson, along with seven others arising in condensed matter physics. Like Wigner's, our results are universal in that they only depend on certain assumptions about the moments of the matrix entries, but not on the specifics of their distributions. What is more, we allow for a certain amount of dependence among the matrix entries, in the spirit of a recent generalization of Wigner's theorem, due to Schenker and Schulz-Baldes. As a byproduct, we obtain a universality result for sample covariance matrices with dependent entries.

Article information

Electron. Commun. Probab., Volume 13 (2008), paper no. 39, 401-414.

Accepted: 1 July 2008
First available in Project Euclid: 6 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A52
Secondary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

random matrices symmetric spaces semicircle law Wigner Marcenko-Pastur Wishart sample covariance matrices dependent random variables density of states universality

This work is licensed under aCreative Commons Attribution 3.0 License.


Hofmann-Credner, Katrin; Stolz, Michael. Wigner theorems for random matrices with dependent entries: Ensembles associated to symmetric spaces and sample covariance matrices. Electron. Commun. Probab. 13 (2008), paper no. 39, 401--414. doi:10.1214/ECP.v13-1395. https://projecteuclid.org/euclid.ecp/1465233466

Export citation


  • A. Altland and B. Simons, Condensed matter field theory, Cambridge UP, Cambridge, 2006.
  • A. Altland and M. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal/superconducting hybrid structures, Physical Review B 55 (1997), no. 2, 1142–1161.
  • G. Anderson and O. Zeitouni, A law of large numbers for finite-range dependent random matrices, arXiv.org:math/0609364.
  • Z. D. Bai and J. Yao, On the convergence of the spectral empirical process of Wigner matrices, Bernoulli 11 (2005), 1059 – 1092
  • B. Collins and M. Stolz, Borel theorems for random elements of the classical compact symmetric spaces, Ann. of Probab. 36 (2008), 876 – 895
  • E. Dueñez, Random matrix ensembles associated to compact symmetric spaces, Comm. Math. Phys. 244 (2004), no. 1, 29–61.
  • F. J. Dyson, The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics, J. Math. Phys. 3 (1962), 1199–1215.
  • P. Eichelsbacher and M. Stolz, Large deviations for random matrix ensembles in mesoscopic physics, arXiv.org:math.PR/0610811, to appear in Markov Process. Related Fields
  • P. Forrester, Log-gases and random matrices, book in progress.
  • U. Haagerup and S. Thorbjørnsen, Random matrices with complex Gaussian entries, Expo. Math. 21 (2003), no. 4, 293–337.
  • P. Heinzner, A. Huckleberry and M. R. Zirnbauer, Symmetry classes of disordered fermions, Commun. Math. Phys. 257 (2005), 725–771.
  • F. Hiai and D. Petz, The semicircle law, free random variables and entropy, Mathematical Surveys and Monographs, vol. 77, American Mathematical Society, Providence, RI, 2000.
  • N. M. Katz and P. Sarnak, Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. 36 (1999), 1–26.
  • A. Nica and R. Speicher, Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, vol. 335, Cambridge University Press, Cambridge, 2006.
  • F. Oravecz and D. Petz, On the eigenvalue distribution of some symmetric random matrices, Acta Sci. Math. (Szeged) 63 (1997), no. 3-4, 383–395.
  • J. H. Schenker and H. Schulz-Baldes, Semicircle law and freeness for random matrices with symmetries or correlations, Math. Res. Lett. 12 (2005), no. 4, 531–542.
  • J. H. Schenker and H. Schulz-Baldes, Gaussian fluctuations for random matrices with correlated entries, Int. Math. Res. Not. 2007, Article ID rnm047, doi:10.1093/i/rnm047.
  • E. P. Wigner, On the distribution of the roots of certain symmetric matrices, Ann. of Math. (2) 67 (1958), 325–327.