Delocalized Phase of Random Band Matrices in High Dimensions
Time: 2023-03-07
Published By: He Liu
Speaker(s): Fan Yang(Tsinghua University)
Time: 14:00-15:30 March 12, 2023
Venue: Room 77201, Jingchunyuan 78, BICMR
In this talk, I will discuss some of our recent progresses in the study 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 $d\ge 7$, we prove that as long as $W\geq L^\delta $ for a small constant $\delta>0$, with high probability the bulk eigenvectors of $H$ are delocalized in the sense that their localization lengths are comparable to L. In addition, we prove a stronger quantum unique ergodicity (QUE) property of the bulk eigenvectors. Finally, with the QUE estimate, we prove that the local statistics of the bulk eigenvalues are universal, which verifies the conjectured connection between QUE and bulk universality. Our proof is based on a local law for the Green's function of $H$ and a high-order $T$-expansion. Joint work with Changji Xu, Horng-Tzer Yau and Jun Yin.
