Local circular law for random matrices

Paul Bourgade, Horng Tzer Yau, Jun Yin

Research output: Contribution to journalArticlepeer-review


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
Issue number3-4
StatePublished - Aug 2014


  • Local circular law
  • Universality

ASJC Scopus subject areas

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


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