In this talk, we present a new proof of the delocalization conjecture for random band matrices. The conjecture asserts that for an $N\times N$ random matrix with bandwidth $W$, the eigenvectors in the bulk of the spectrum are delocalized provided that $W \gg N^{1/2}$. Moreover, in this regime, the eigenvalue distribution aligns with that of Gaussian random ensembles (i.e., GOE or GUE). Our proof employs a novel loop hierarchy method and leverages the sum-zero property, a concept that was instrumental in the previous work on high-dimensional random matrices. 

This work is a joint collaboration with H.T. Yau.