NDLI: The local circular law III: general case

Content Provider	Springer Nature Link
Author	Yin, Jun
Copyright Year	2013
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 $$z \in \mathbb {C}$$ .
Ending Page	732
Page Count	54
Starting Page	679
File Format	PDF
ISSN	01788051
e-ISSN	14322064
Journal	Probability Theory and Related Fields
Issue Number	3
Volume Number	160
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2013-11-24
Publisher Place	Berlin/Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Local circular law Random matrices Disordered systems (random Ising models, random SchrÃ¶dinger operators, etc.) Mathematical and Computational Biology Statistics for Business/Economics/Mathematical Finance/Insurance Theoretical, Mathematical and Computational Physics Operations Research/Decision Theory Probability Theory and Stochastic Processes Universality Quantitative Finance
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Analysis Statistics, Probability and Uncertainty

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