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Local Law of Addition of Random Matrices on Optimal Scale

Communications in Mathematical Physics volume 349, pages 947–990 (2017)Cite this article

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

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

Communicated by H.-T. Yau

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