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On the support of the free additive convolution

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Abstract

We consider the free additive convolution of two probability measures μ and ν on the real line and show that μ ⊞ v is supported on a single interval if μ and ν each has single interval support. Moreover, the density of μ ⊞ ν is proven to vanish as a square root near the edges of its support if both μ and ν have power law behavior with exponents between −1 and 1 near their edges. In particular, these results show the ubiquity of the conditions in our recent work on optimal local law at the spectral edges for addition of random matrices [5].

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Authors and Affiliations

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

    Zhigang Bao

  2. Institute of Science and Technology (IST) Austria, Am Campus 1, A-3400, Klosterneuburg, Austria

    László Erdős

  3. Department of Mathematics, KTH Royal Institute of Technology, SE-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 Kevin Schnelli.

Additional information

Supported in part by Hong Kong RGC Grant ECS 26301517.

Supported in part by ERC Advanced Grant RANMAT No. 338804.

Supported in part by the Knut and Alice Wallenberg Foundation and the Swedish Research Council Grant VR-2017-05195.

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Bao, Z., Erdős, L. & Schnelli, K. On the support of the free additive convolution. JAMA 142, 323–348 (2020). https://doi.org/10.1007/s11854-020-0135-2

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  • DOI: https://doi.org/10.1007/s11854-020-0135-2

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