Abstract
This is a continuation of our earlier paper (Tao and Vu, http://arxiv.org/abs/0908.1982v4[math.PR], 2010) on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in Tao and Vu (http://arxiv.org/abs/0908.1982v4[math.PR], 2010) from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results of Soshnikov (Commun Math Phys 207(3):697–733, 1999) for the largest eigenvalues, assuming moment conditions rather than symmetry conditions. The main new technical observation is that there is a significant bias in the Cauchy interlacing law near the edge of the spectrum which allows one to continue ensuring the delocalization of eigenvectors.
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The authors thank the anonymous referee for helpful comments and references, and Horng-Tzer Yau for additional references.
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Communicated by H.-T. Yau
T. Tao is is supported by a grant from the MacArthur Foundation, by NSF grant DMS-0649473, and by the NSF Waterman award.
V. Vu is supported by research grants DMS-0901216 and AFOSAR-FA-9550-09-1-0167.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Tao, T., Vu, V. Random Matrices: Universality of Local Eigenvalue Statistics up to the Edge. Commun. Math. Phys. 298, 549–572 (2010). https://doi.org/10.1007/s00220-010-1044-5
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DOI: https://doi.org/10.1007/s00220-010-1044-5