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Wigner theorems for random matrices with dependent entries: Ensembles associated
to symmetric spaces and sample covariance matrices
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Katrin Hofmann-Credner, Ruhr University Bochum
Michael Stolz, Ruhr University Bochum
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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.
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Full text: PDF
Pages: 401-414
Published on: July 1, 2008
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Electronic Communications in Probability. ISSN: 1083-589X |
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