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High Dimensional Probability VI pp 261–275Cite as

Birkhäuser

Edge Fluctuations of Eigenvalues of Wigner Matrices

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Part of the book series: Progress in Probability ((PRPR,volume 66))

Abstract

We establish a moderate deviation principle (MDP) for the number of eigenvalues of a Wigner matrix in an interval close to the edge of the spectrum. Moreover we prove a MDP for the jth largest eigenvalue close to the edge. The proof relies on fine asymptotics of the variance of the eigenvalue counting function of GUE matrices due to Gustavsson. The extension to large families of Wigner matrices is based on the Tao and Vu Four Moment Theorem. Possible extensions to other random matrix ensembles are commented.

Mathematics Subject Classification (2010). Primary 60B20; Secondary 60F10, 15A18.

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Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780, Bochum, Germany

    Hanna Döring & Peter Eichelsbacher

Authors
  1. Hanna Döring
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  2. Peter Eichelsbacher
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Corresponding author

Correspondence to Hanna Döring .

Editor information

Editors and Affiliations

  1. Georgia Institute of Technology, Atlanta, 30332, USA

    Christian Houdré

  2. University of Delaware, Newark, 19716, USA

    David M. Mason

  3. University of Tennessee, Knoxville, 37934, USA

    Jan Rosiński

  4. University of Washington, Seattle, 98195, USA

    Jon A. Wellner

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Döring, H., Eichelsbacher, P. (2013). Edge Fluctuations of Eigenvalues of Wigner Matrices. In: Houdré, C., Mason, D., Rosiński, J., Wellner, J. (eds) High Dimensional Probability VI. Progress in Probability, vol 66. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0490-5_17

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