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Bulk universality for generalized Wigner matrices with few moments. (English) Zbl 1442.15055

Summary: In this paper we consider \(N \times N\) real generalized Wigner matrices whose entries are only assumed to have finite \((2 + \varepsilon)\)th moment for some fixed, but arbitrarily small, \(\varepsilon > 0\). We show that the Stieltjes transforms \(m_N (z)\) of these matrices satisfy a weak local semicircle law on the nearly smallest possible scale, when \(\eta = \mathfrak {I}(z)\) is almost of order \(N^{-1}\). As a consequence, we establish bulk universality for local spectral statistics of these matrices at fixed energy levels, both in terms of eigenvalue gap distributions and correlation functions, meaning that these statistics converge to those of the Gaussian orthogonal ensemble in the large \(N\) limit.

MSC:

15B52 Random matrices (algebraic aspects)
60B20 Random matrices (probabilistic aspects)
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