I.i.d. matrices and the circular law

Jiyun Park
Fri, Feb 23 2024, 4:00pm

Abstract: The circular law states that the spectral measure of a square matrix with i.i.d. entries of mean zero must converge to the uniform distribution on the unit disk in the complex plane. This result is analogous to the semicircular law for Wigner matrices, but the spectral instability of non-Hermitian matrices makes it impossible to use the same techniques (e.g. the moment method). We discuss limitations of the moment method and sketch a proof of the circular law. This talk will mostly follow Chapter 2.8 of Tao's Topics on Random Matrix Theory.