The local circular law II: the edge case

Paul Bourgade, Horng Tzer Yau, Jun Yin

Research output: Contribution to journalArticle

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+ε for non-Hermitian random matrices at any point z ∈ 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 = o(1). Without the vanishing third moment assumption, we prove that the circular lawis valid near the spectral edge |z| − 1 = o(1) up to scale N−1/4+ε.

Original languageEnglish (US)
Pages (from-to)619-660
Number of pages42
JournalProbability Theory and Related Fields
Volume159
Issue number3-4
DOIs
StatePublished - 2014

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Non-Hermitian Matrix
Moment
Rescaling
Random Matrices
Random variable
Valid
Standards
Random variables

Keywords

  • Local circular law
  • Universality

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

The local circular law II : the edge case. / Bourgade, Paul; Yau, Horng Tzer; Yin, Jun.

In: Probability Theory and Related Fields, Vol. 159, No. 3-4, 2014, p. 619-660.

Research output: Contribution to journalArticle

Bourgade, Paul ; Yau, Horng Tzer ; Yin, Jun. / The local circular law II : the edge case. In: Probability Theory and Related Fields. 2014 ; Vol. 159, No. 3-4. pp. 619-660.
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