Wigner Hermitian matrices have universal eigenvalue gap distribution

You need to know: Complex numbers, $n \times n$ matrix with complex entries, eigenvalues, Hermitian matrix, probability, random variable, mean, variance, notation P for probability, E for expectation, i.i.d. for independent identically distributed random variables, notation $|A|$ for the number of elements in finite set A.

Background: A Wigner Hermitian $n \times n$ matrix $M_n$ is a random Hermitian matrix with upper triangular complex entries $\psi_{ij}=\xi_{ij}+\tau_{ij}\sqrt{-1}$ , $1 \leq i < j \leq n$ , and diagonal real entries $\xi_{ii}$ , $1\leq i \leq n$ , where (i) for $1 \leq i < j \leq n$ , $\xi_{ij}$ and $\tau_{ij}$ are i.i.d. copies of a real random variable $\xi$ with mean zero and variance $1/2$ ; (ii) for $1\leq i \leq n$ , $\xi_{ii}$ are i.i.d. copies of a real random variable $\xi'$ with mean zero and variance 1; and (iii) $\xi$ and $\xi'$ have exponential decay, i.e., there exist constants $C$ and $C_0$ such that $P(|\xi|\geq t^C) \leq e^{-t}$ and $P(|\xi'|\geq t^C) \leq e^{-t}$ for all $t > C_0$ . $\xi$ and $\xi'$ are called the atom distributions of $M_n$ . Let $\lambda(M_n\sqrt{n})=(\lambda_1, \dots, \lambda_n)$ be the vector of eigenvalues $\lambda_i$ of matrix $M_n\sqrt{n}$ , arranged in the non-increasing order: $\lambda_1 \leq \lambda_2 \leq \dots \leq \lambda_n$ . Let $S_n(s; \lambda(M_n\sqrt{n})) = \frac{1}{n}|\{1\leq i \leq n-1: \, \lambda_{i+1}-\lambda_i \leq s\}|$ .

The Theorem: On 2nd June 2009, Terence Tao and Van Vu submitted to arxiv a paper in which they proved the following result. Let $M_n$ be a Wigner Hermitian matrix whose atom distributions $\xi, \xi'$ have support on at least three points, and $s>0$ . The the limit $G(s)=\lim\limits_{n\to\infty}E[S_n(s; \lambda(M_n\sqrt{n}))]$ exists and depends only on s but not on $\xi, \xi'$ .

Short context: The study of eigenvalues of large random matrices (largest eigenvalues, smallest, gaps between them, etc.) is one of the central themes in probability theory. In 1967, Mehta computed the limit $G(s)$ assuming that $\xi$ and $\xi'$ have normal distribution. The Theorem states that $E[S_n(s; \lambda(M_n\sqrt{n}))]$ converges to the same limit $G(s)$ , no matter how $\xi$ and $\xi'$ are distributed. In fact, the Theorem is just one out of many corollaries of a much more general “Four moment theorem” proved by authors, which, informally, states that many properties of the eigenvalues of a random Hermitian matrix are determined by the first four moments of the atom distributions. In a later work, the authors also treated a non-Hermitian case.