# Universality and Delocalization of Random Band Matrices

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We consider N × N symmetric one-dimensional random band matrices with general distribution of the entries and band width $W$. The localization - delocalization conjecture predicts that there is a phase transition on the behaviors of eigenvectors and eigenvalues of the band matrices. It occurs at $W=N^{1/2}$. For wider-band matrix, the eigenvalues satisfied the so called sine-kernal distribution, and the eigenvectors are delocalized. With Bourgade, Yau and Fan, we proved that it holds when $W\gg N^{3/4}$. The previous best work required $W=\Omega(N).$