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The local circular law III: general case

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Abstract

In the first part (Bourgade et al., Local circular law for random matrices, preprint, arXiv:1206.1449, 2012) of this article series, Bourgade, Yau and the author of this paper 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 constant \(c>0\) independent of the size of the matrix. In the second part (Bourgade et al., The local circular law II: the edge case, preprint, arXiv:1206.3187, 2012), they extended this result to include the edge case \( |z|-1={{\mathrm{o}}}(1)\), under the main assumption that the third moments of the matrix elements vanish. (Without the vanishing third moment assumption, they proved that the circular law is valid near the spectral edge \( |z|-1={{\mathrm{o}}}(1)\) up to scale \(N^{-1/4+ {\varepsilon }}\).) In this paper, we will remove this assumption, i.e. we prove a local version of the circular law up to the finest scale \(N^{-1/2+ {\varepsilon }}\) for non-Hermitian random matrices at any point \(z \in \mathbb {C}\).

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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.

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Author information

Correspondence to Jun Yin.

Additional information

J. Yin was partially supported by NSF grant DMS-1001655 and DMS- 1207961.

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Yin, J. The local circular law III: general case. Probab. Theory Relat. Fields 160, 679–732 (2014). https://doi.org/10.1007/s00440-013-0539-3

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Keywords

  • Local circular law
  • Universality

Mathematics Subject Classification (2010)

  • 15B52
  • 82B44