Local circular law for random matrices

Paul Bourgade, Horng Tzer Yau, Jun Yin

Research output: Contribution to journalArticle

Abstract

The circular law asserts that the spectralmeasure of eigenvalues of rescaled random matrices without symmetry assumption converges to the uniform measure on the unit disk. We prove a local version of this law at any point z away from the unit circle. More precisely, if ||z| − 1| ≥ τ for arbitrarily small τ > 0, the circular law is valid around z up to scale N−1/2+ε for any ε > 0 under the assumption that the distributions of the matrix entries satisfy a uniform subexponential decay condition.

Original languageEnglish (US)
Pages (from-to)545-595
Number of pages51
JournalProbability Theory and Related Fields
Volume159
Issue number3-4
DOIs
StatePublished - 2014

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Random Matrices
Unit circle
Unit Disk
Decay
Valid
Eigenvalue
Converge
Symmetry

Keywords

  • Local circular law
  • Universality

ASJC Scopus subject areas

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

Cite this

Local circular law for random matrices. / Bourgade, Paul; Yau, Horng Tzer; Yin, Jun.

In: Probability Theory and Related Fields, Vol. 159, No. 3-4, 2014, p. 545-595.

Research output: Contribution to journalArticle

Bourgade, Paul ; Yau, Horng Tzer ; Yin, Jun. / Local circular law for random matrices. In: Probability Theory and Related Fields. 2014 ; Vol. 159, No. 3-4. pp. 545-595.
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