Universality of Wigner random matrices: a Survey of Recent Results

Understand

We study the universality of spectral statistics of large random matrices.

• We consider $N\times N$ symmetric, hermitian or quaternion self-dual random matrices with independent, identically distributed entries (Wigner matrices) where the probability distribution for each matrix element is given by a measure $\nu$ with a subexponential decay.
• Our main result is that the correlation functions of the local eigenvalue statistics in the bulk of the spectrum coincide with those of the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE) and the Gaussian Symplectic Ensemble (GSE), respectively, in the limit $N\to \infty$.
• Our approach is based on the study of the Dyson Brownian motion via a related new dynamics, the local relaxation flow.