Circular law

In probability theory, more specifically the study of random matrices, the circular law is the distribution of eigenvalues of an n × n random matrix with independent and identically distributed entries in the limit n → ∞.

It asserts that for any sequence of random n × n matrices whose entries are independent and identically distributed random variables, all with mean zero and variance equal to 1/n, the limiting spectral distribution is the uniform distribution over the unit disc.

Let ${\displaystyle (X_{n})_{n=1}^{\infty }}$ be a sequence of n × n matrix ensembles whose entries are i.i.d. copies of a complex random variable x with mean 0 and variance 1. Let ${\displaystyle \lambda _{1},\ldots ,\lambda _{n},1\leq j\leq n}$ denote the eigenvalues of ${\displaystyle \displaystyle {\frac {1}{\sqrt {n}}}X_{n}}$. Define the empirical spectral measure of ${\displaystyle \displaystyle {\frac {1}{\sqrt {n}}}X_{n}}$ as

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