From random band matrices to block Anderson models: The localization/delocalization phase transition

In this talk, I will discuss recent progress on disordered quantum systems beyond the mean-field regime. Earlier this year (joint with H.-T. Yau), we resolved the delocalization conjecture for random band matrices by developing the loop hierarchy method and its tree approximation. In our new work (joint with S. Dubova, F. Yang, and H.-T. Yau), we combine this framework with nested diagrammatic techniques previously used in high-dimensional (d>7) analyses to construct a unified approach capable of handling non-mean-field operators. This allows us to prove the localization/delocalization phase transition in the block Anderson model for d≥3, and to identify the predicted critical coupling scale g = W^{-d/2} that separates the localized and delocalized phases of eigenvectors.