Local Circular Law for Random Matrices

2012/06/07 by Bourgade, Paul, Yau, Horng-Tzer, Yin, Jun · 3 citations
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1206.1449

Abstract

The circular law asserts that the spectral measure 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+ \e} for any \e > 0 under the assumption that the distributions of the matrix entries satisfy a uniform subexponential decay condition.

Cited by

• Circular law for non-Hermitian block band matrices with slowly growing bandwidth
• On the spectral radius of the ratio of Girko matrices
• The circular law for non-Hermitian random band matrices up to bandwidth N^1/2+c

Related

• The local circular law II: the edge case
• The Local Circular Law III: General Case
• Local inhomogeneous circular law
• The sparse circular law under minimal assumptions
• Random Matrices: The circular Law