Abstract
We consider N × 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. Under suitable assumptions on the distribution of the single matrix element, we first prove that, away from the spectral edges, the empirical density of eigenvalues concentrates around the Wigner semicircle law on energy scales η ≫ N −1 . This result establishes the semicircle law on the optimal scale and it removes a logarithmic factor from our previous result [6]. We then show a Wegner estimate, i.e., that the averaged density of states is bounded. Finally, we prove that the eigenvalues of a Wigner matrix repel each other, in agreement with the universality conjecture.
Highlights
Let H = be an N × N hermitian matrix, N ≥ 2, and let μ1 ≤ μ2 ≤ . . . ≤ μN denote its eigenvalues
We assume that zij (i < j) all have a common distribution ν with variance C |z|2dν(z) = 1 and with a strictly positive density function h : R2 → R+, i.e
In this paper we prove the strong level repulsion and a subexponential estimate for the large distance behavior of the gap distribution
Summary
The large distance behavior, f (x) ∼ exp(−x2), x ≫ 1, expresses a strong supression of large eigenvalue gaps These properties of the eigenvalue statistics are conjectured to hold for much more general matrix ensembles beyond the invariant ensembles, in particular for general Wigner matrices. Apart from the constant, this upper bound coincides with the prediction obtained from the universality conjecture on the level spacing distribution and it proves that the level repulsion in Wigner matrices is as strong as in the GUE ensemble. We remark that the original paper [8] assumed that dν was symmetric; this conditon was later removed by Wright [16] Another assumption we made in [5, 6] states that either the Hessian of g = − log h is bounded from above or the distribution is compactly supported. We assume condition C1) throughout the paper and every constant may depend on the constants δ0, D, D from (1.3) without further notice
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