Eigenvalue variance bounds for Wigner random matrices

Details Eigenvalue variance bounds for Wigner random matrices Eigenvalue variance bounds for Wigner random matrices

2012-03-07

arxiv.org/abs/1203.1597v2

Mathematics

Probability

Scientific paper

This work is concerned with finite range bounds on the variance of individual eigenvalues of Wigner random matrices, in the bulk and at the edge of the spectrum, as well as for some intermediate eigenvalues. Relying on the GUE example, which needs to be investigated first, the main bounds are extended to families of Hermitian Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erd\"os, Yau and Yin. The case of real Wigner matrices is obtained from interlacing formulas. As an application, bounds on the expected 2-Wasserstein distance between the empirical spectral measure and the semicircle law are derived.

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