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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)
##### Keywords:
local circular law; universality
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##### References:
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