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Local Law of Addition of Random Matrices on Optimal Scale

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  • Published: 18 November 2016
  • volume 349, pages 947–990 (2017)
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Local Law of Addition of Random Matrices on Optimal Scale
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  • Zhigang Bao1,2,
  • László Erdős2 &
  • Kevin Schnelli2,3 
  • 943 Accesses

  • 18 Citations

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Abstract

The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by Haar orthogonal matrix.

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Authors and Affiliations

  1. Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

    Zhigang Bao

  2. IST Austria, Am Campus 1, 3400, Klosterneuburg, Austria

    Zhigang Bao, László Erdős & Kevin Schnelli

  3. Department of Mathematics, KTH Royal Institute of Technology, Lindstedtsvägen 25, 100 44, Stockholm, Sweden

    Kevin Schnelli

Authors
  1. Zhigang Bao
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  2. László Erdős
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  3. Kevin Schnelli
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Corresponding author

Correspondence to László Erdős.

Additional information

Communicated by H.-T. Yau

Z. Bao, L. Erdős and K.Schnelli were supported by ERC Advanced Grant RANMAT No. 338804.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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Bao, Z., Erdős, L. & Schnelli, K. Local Law of Addition of Random Matrices on Optimal Scale. Commun. Math. Phys. 349, 947–990 (2017). https://doi.org/10.1007/s00220-016-2805-6

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  • Received: 08 December 2015

  • Accepted: 19 October 2016

  • Published: 18 November 2016

  • Issue Date: February 2017

  • DOI: https://doi.org/10.1007/s00220-016-2805-6

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