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Wegner estimate and level repulsion for Wigner random matrices. (English) Zbl 1204.15043

The authors consider \(N \times N\) Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing between consecutive eigenvalues is of order \(1/N\). In addition, the authors assume suitable conditions on the distributions of the matrix elements and on the density function.
Under these assumptions, the authors prove that, away from the spectral edges, the empirical density of eigenvalues concentrates around the Wigner semicircle law on energy scales \(\eta \gg 1/N\). This result establishes the semicircle law on the optimal scale and it removes a logarithmic factor from previous results published by the authors [Commun. Math. Phys. 287, No. 2, 641–655 (2009; Zbl 1186.60005)].
Another main result of this paper provides an upper bound on the tail distribution of the distance between consecutive eigenvalues. The authors also prove the Wegner estimate for Wigner matrices, i.e., that the averaged density of states is uniformly bounded.
Finally, the authors establish an upper bound on the level repulsion.

MSC:

15B52 Random matrices (algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
60B20 Random matrices (probabilistic aspects)

Citations:

Zbl 1186.60005
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