Optimal delocalization for generalized Wigner matrices (Q2071630)

The authors study the eigenvectors of generalized Wigner matrices with subexponential entries and their delocalization at the optimal rate with overwhelming probability. The high probability delocalization bounds with sharp constants are also proved. The proof relies on the analysis of the eigenvector moment flow, as introduced in [\textit{P. Bourgade} and \textit{H.-T. Yau}, Commun. Math. Phys. 350, No. 1, 231--278 (2017; Zbl 1379.58014)]. One of the main results of the paper captures the optimal delocalization constant for the \(l_\infty\) norm of an eigenvector. Besides, for generalized Wigner matrices with smoother entry distributions, it is possible to give an optimal form for the delocalization of the maximal entry of the whole eigenbasis. The obtained delocalization constants are sharp (see Proposition 1 in [\textit{T. Jiang}, Probab. Theory Relat. Fields 131, No. 1, 121--144 (2005; Zbl 1067.15021)]). Additionally, the level repulsion and eigenvalue overcrowding estimates for the entire spectrum are proved, which may be of independent interest.