Bulk universality and quantum unique ergodicity of random band matrices

Consider a general class of random band matrices $H$ on the $d$-dimensional lattice of linear size $L$. The entries of $H$ are independent centered complex Gaussian random variables with variances $s_{xy}$, which have a banded profile so that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. In dimensions $d\ge 7$, assuming that $W\geq L^\delta $ for a small constant $\delta>0$, we prove the deloclaization and quantum unique ergodicity (QUE) of the bulk eigenvectors of $H$. Furthermore, we prove the bulk universality of $H$ under the condition $W \gg L^{95/(d+95)}$. In the talk, I will discuss a new idea for the proof of the bulk universality through QUE, which verifies the conjectured connection between QUE and bulk universality. The proof of QUE is based on a local law for the Green's function of $H$ and a high-order $T$-expansion developed recently. Based on Joint work with Changji Xu, Horng-Tzer Yau and Jun Yin.

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