The Zigzag strategy for random band matrices

Random band matrices have entries concentrated in a narrow band of width W around the main diagonal, modeling systems with spatially localized interactions. We consider one-dimensional random band matrices with bandwidth W >> N^½, general variance profile, and arbitrary entry distributions. We establish complete isotropic delocalization, quantum unique ergodicity (eigenstate thermalization), and Wigner–Dyson universality in the bulk of the spectrum. The key technical input is a family of local laws capturing the spatial decay of resolvent entries, established using a combination of Ornstein–Uhlenbeck dynamics and Green function comparison: the Zigzag strategy.

This is based on joint work with László Erdős.