Bulk universality for generalized Wigner matrices

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Bulk universality for generalized Wigner matrices

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Title: Bulk universality for generalized Wigner matrices
Author: Erdos, Laszlo; Yau, Horng-Tzer; Yin, Jun

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Citation: Erdos, László, Horng-Tzer Yau, and Jun Yin. 2011. “Bulk Universality for Generalized Wigner Matrices.” Probab. Theory Relat. Fields 154 (1-2) (October 6): 341–407. doi:10.1007/s00440-011-0390-3. http://dx.doi.org/10.1007/s00440-011-0390-3.
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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 \(\nu_{ij}\) with a subexponential decay. Let \(\sigma_{ij}^2\) be the variance for the probability measure \(\nu_{ij}\) with the normalization property that \(\sum_i\sigma_{ij}^2 = 1\) for all j. Under essentially the only condition that \(c\leq N\sigma_{ij}^2 \leq c^{−1}\) for some constant \(c > 0\), we prove that, in the limit \(N \rightarrow \infty\), 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 \(M^{−1}\).
Published Version: doi:10.1007/s00440-011-0390-3
Other Sources: http://arxiv.org/abs/1001.3453
Terms of Use: This article is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAP
Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:25427234
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