2 What Is A Random Matrix?

• Definition (Wigner Matrix): A symmetric $n \times n$ matrix $A_n$ where the upper-triangular entries $a_{ij}$ (for $i \le j$) are independent, identically distributed random variables.
• Standard Assumptions: We typically assume the entries have mean zero ($\mathbb{E}[a_{ij}] = 0$) and unit variance ($\mathbb{E}[a_{ij}^2] = 1$). A common example is the Gaussian Orthogonal Ensemble (GOE).
• Goal: We study the statistical properties of the eigenvalues of $A_n$ as the dimension $n \to \infty$. Remarkably, the distribution of eigenvalues becomes universal, regardless of the specific distribution of entries!