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.
- Local circular law
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty