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The local circular law II: the edge case

Abstract

In the first part of this article (Bourgade et al. arXiv:1206.1449, 2012), we proved a local version of the circular law up to the finest scale \(N^{-1/2+ {\varepsilon }}\) for non-Hermitian random matrices at any point \(z \in \mathbb C \) with \(||z| - 1| > c \) for any \(c>0\) independent of the size of the matrix. Under the main assumption that the first three moments of the matrix elements match those of a standard Gaussian random variable after proper rescaling, we extend this result to include the edge case \( |z|-1={{\mathrm{o}}}(1)\). Without the vanishing third moment assumption, we prove that the circular law is valid near the spectral edge \( |z|-1={{\mathrm{o}}}(1)\) up to scale \(N^{-1/4+ {\varepsilon }}\).

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Fig. 1

Notes

  1. 1.

    For the sake of notational simplicity we do not consider complex entries in this paper, but the statements and proofs are similar.

  2. 2.

    Although these bounds in [8] are stated under the assumption \(||z|-1|\geqslant \tau \), the same argument holds also for \(z\) close to the unit circle, under the extra assumption \(|w|>{\varepsilon }\).

  3. 3.

    All the expansions considered here converge with high probability. Anyways, they aim at proving identities between rational functions, which just need to be checked for small values of the perturbation.

  4. 4.

    \(\vert \vert \vert {f}\vert \vert \vert _{\infty }:= \max (\Vert f^{\prime }\Vert _\infty ^3,\Vert f^{\prime }\Vert _\infty \Vert f^{\prime \prime }\Vert _\infty ,\Vert f^{(3)}\Vert _\infty )\).

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Author information

Correspondence to Paul Bourgade.

Additional information

P. Bourgade was partially supported by NSF grant DMS-1208859. H.-T. Yau was partially supported by NSF grants DMS-0757425, 0804279. J. Yin was partially supported by NSF grants DMS-1001655,1207961.

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Bourgade, P., Yau, H. & Yin, J. The local circular law II: the edge case. Probab. Theory Relat. Fields 159, 619–660 (2014). https://doi.org/10.1007/s00440-013-0516-x

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Keywords

  • Local circular law
  • Universality

Mathematics Subject Classification (2010)

  • 15B52
  • 82B44