Probability and Mathematical Physics Seminar

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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 $d \ge 3$, and to identify the predicted critical coupling scale $g = W^{-d/2}$ that separates the localized and delocalized phases of eigenvectors.