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. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2009-07-11 14:57:44.225