A Study on the Three-Step Strategy on Bulk Universality of Wigner Matrices

Abstract

This report summarizes the study of the book: A Dynamical Approach to Random Matrix Theory [14], which aims to prove the Wigner-Dyson-Metha universality conjecture, the universality of local eigenvalue statistics for high-dimensional random matrices with independent entries. The core content in this report focuses on the generalized Wigner matrices, and the concerned universal distribution is indeed Gaussian ensembles which merely depend on the symmetry class. The Three-Step Approach is used to demonstrate the averaged energy bulk universality through a succession of studies and analyses on the microscopic size of eigenvalues and spectral spacing. This step-by-step method is streamlined with the local semicircle law, Dyson’s conjecture, Gaussian divisible ensemble universality, Dyson Brownian Motion gap universality, and an approximation argument.