## The circular law: Proof of the replacement principle

##### Abstract

It was conjectured in the early 1950¡¯s that the empirical
spectral distribution (ESD) of an $n \times n$ matrix whose entries
are independent and identically distributed with mean zero and
variance one, normalized by a factor of $\frac{1}{\sqrt{n}}$,
converges to the uniform distribution over the unit disk on the
complex plane, which is called the circular law. The goal of this
thesis is to prove the so called Replacement Principle introduced by
Tao and Vu which is a crucial step in their recent proof of the
circular law in full generality. It gives a general criterion for
the difference of the ESDs of two normalised random matrices
$\frac{1}{\sqrt{n}}A_n$, $\frac{1}{\sqrt{n}}B_n$ to converge to 0.