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Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices

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Abstract

We study the fluctuations of the matrix entries of regular functions of Wigner random matrices in the limit when the matrix size goes to infinity. In the case of the Gaussian ensembles (GOE and GUE) this problem was considered by A. Lytova and L. Pastur (J. Stat. Phys. 134:147–159, 2009). Our results are valid provided the off-diagonal matrix entries have finite fourth moment, the diagonal matrix entries have finite second moment, and the test functions have four continuous derivatives in a neighborhood of the support of the Wigner semicircle law. Moreover, if the marginal distributions satisfy the Poincaré inequality our results are valid for Lipschitz continuous test functions.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Davis, One Shields Avenue, Davis, CA, 95616-8633, USA

    Alessandro Pizzo, David Renfrew & Alexander Soshnikov

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  1. Alessandro Pizzo
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  2. David Renfrew
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  3. Alexander Soshnikov
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Correspondence to Alexander Soshnikov.

Additional information

A.P. has been supported in part by the NSF grant DMS-0905988.

D.R. has been supported in part by the NSF grants VIGRE DMS-0636297, DMS-1007558, and DMS-0905988.

A.S. has been supported in part by the NSF grant DMS-1007558.

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Pizzo, A., Renfrew, D. & Soshnikov, A. Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices. J Stat Phys 146, 550–591 (2012). https://doi.org/10.1007/s10955-011-0404-7

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