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

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465233466

Digital Object Identifier
doi:10.1214/ECP.v13-1395

Mathematical Reviews number (MathSciNet)
MR2415147

Zentralblatt MATH identifier
1189.15044

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

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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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


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