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The local circular law. II: The edge case. (English) Zbl 1342.15028
Summary: In the first part of this article [P. Bourgade et al., Local circular law for random matrices. 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 \(|z|-1 = \text o(1)\). Without the vanishing third moment assumption, we prove that the circular law is valid near the spectral edge \(|z|-1 = \text o(1)\) up to scale \(N^{-1/4 + \varepsilon}\).
Part III, see Probab. Theory Relat. Fields 160, No. 3–4, 679–732 (2014; Zbl 1342.15031).

MSC:
15B52 Random matrices (algebraic aspects)
60B20 Random matrices (probabilistic aspects)
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