##
**Wigner theorems for random matrices with dependent entries: ensembles associated to symmetric spaces and sample covariance matrices.**
*(English)*
Zbl 1189.15044

Summary: It is a classical result of E. P. Wigner [Ann. Math. (2) 67, 325–327 (1958; Zbl 0085.13203)] 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 [loc. cit.] and F. J. Dyson [J. Math. Phys. 3, 1199–1215 (1962; Zbl 0134.45703)], 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 J. H. Schenker and H. Schulz-Baldes [Math. Res. Lett. 12, No. 4, 531–542 (2005; Zbl 1095.82004)]. As a byproduct, we obtain a universality result for sample covariance matrices with dependent entries.

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 J. H. Schenker and H. Schulz-Baldes [Math. Res. Lett. 12, No. 4, 531–542 (2005; Zbl 1095.82004)]. As a byproduct, we obtain a universality result for sample covariance matrices with dependent entries.

### MSC:

15B52 | Random matrices (algebraic aspects) |

15A18 | Eigenvalues, singular values, and eigenvectors |

15B57 | Hermitian, skew-Hermitian, and related matrices |