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Random matrices: Universality of local eigenvalue statistics

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Acta Mathematica

Abstract

In this paper, we consider the universality of the local eigenvalue statistics of random matrices. Our main result shows that these statistics are determined by the first four moments of the distribution of the entries. As a consequence, we derive the universality of eigenvalue gap distribution and k-point correlation, and many other statistics (under some mild assumptions) for both Wigner Hermitian matrices and Wigner real symmetric matrices.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, 90095–1555, U.S.A.

    Terence Tao

  2. Department of Mathematics, Rutgers, Piscataway, NJ, 08854, U.S.A.

    Van Vu

Authors
  1. Terence Tao
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  2. Van Vu
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Correspondence to Terence Tao.

Additional information

T. Tao 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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Tao, T., Vu, V. Random matrices: Universality of local eigenvalue statistics. Acta Math 206, 127–204 (2011). https://doi.org/10.1007/s11511-011-0061-3

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  • DOI: https://doi.org/10.1007/s11511-011-0061-3

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