We consider a Hermitian random band matrix H on the d-dimensional lattice of linear size L. Its entries are independent centered complex Gaussian random variables with variances s_{xy}, that are negligible if |x-y| exceeds the band width W. In dimensions eight or higher, we prove that, as long as W> L^\epsilon for a small constant \epsilon>0, with high probability, most bulk eigenvectors of H are delocalized in the sense that their localization lengths are comparable to L. Moreover, we also prove a quantum diffusion result of this model in terms of the Green's function of H. Joint work with Horng-Tzer Yau and Jun Yin.

### Probability and Combinatorics

Tuesday, September 7, 2021 - 3:30pm

#### Fan Yang

University of Pennsylvania