2012/06/14 by Bourgade, Paul, Yau, Horng-Tzer, Yin, Jun
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.1206.3187
In the first part of this article, we proved a local version of the circular law up to the finest scale N-1/2+ \e 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=\oo(1). Without the vanishing third moment assumption, we prove that the circular law is valid near the spectral edge |z|-1=\oo(1) up to scale N-1/4+ \e.