Optimal Delocalization for Generalized Wigner Matrices

Understand

We study the eigenvectors of generalized Wigner matrices with subexponential entries and prove that they delocalize at the optimal rate with overwhelming probability.

• We also prove high probability delocalization bounds with sharp constants.
• Our proof uses an analysis of the eigenvector moment flow introduced by Bourgade and Yau (2017) to bound logarithmic moments of eigenvector entries for random matrices with small Gaussian components.
• We then extend this control to all generalized Wigner matrices by comparison arguments based on a framework of regularized eigenvectors, level repulsion, and the observable employed by Landon, Lopatto, and Marcinek (2018) to compare extremal eigenvalue statistics.