Abstract
Consider N × N Hermitian or symmetric random matrices H where the distribution of the (i, j) matrix element is given by a probability measure ν ij with a subexponential decay. Let \({\sigma_{ij}^2}\) be the variance for the probability measure ν ij with the normalization property that \({\sum_{i} \sigma^2_{ij} = 1}\) for all j. Under essentially the only condition that \({c\le N \sigma_{ij}^2 \le c^{-1}}\) for some constant c > 0, we prove that, in the limit N → ∞, the eigenvalue spacing statistics of H in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth M the local semicircle law holds to the energy scale M −1.
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L. Erdős was partially supported by SFB-TR 12 Grant of the German Research Council; H-. T. Yau was partially supported by NSF Grants DMS-0602038, 0757425, 0804279; J. Yin was partially supported by NSF Grants DMS-100165.
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Erdős, L., Yau, HT. & Yin, J. Bulk universality for generalized Wigner matrices. Probab. Theory Relat. Fields 154, 341–407 (2012). https://doi.org/10.1007/s00440-011-0390-3
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DOI: https://doi.org/10.1007/s00440-011-0390-3