Bulk universality and quantum unique ergodicity for random band matrices in high dimensions

Abstract

We consider Hermitian random band matrices $H=(h_{xy})$ on the $d$-dimensional lattice $(\mathbb{Z}/L\mathbb{Z}{)}^d$, where the entries $h_{xy}={\overline{h}}_{yx}$ are independent centered complex Gaussian random variables with variances $s_{xy}=\mathbb{E}|h_{xy}{|}^2$. The variance matrix $S=(s_{xy})$ has a banded profile so that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. For dimensions $d\ge 7$, we prove the bulk eigenvalue universality of $H$ under the condition $W\gg L^{95/(d+95)}$. Assuming that $W\ge L^{\epsilon }$ for a small constant $\epsilon >0$, we also prove the quantum unique ergodicity for the bulk eigenvectors of $H$ and a sharp local law for the Green's function $G(z)=(H-z{)}^{-1}$ up to $\mathrm{Im}z\gg W^{-5}{L}^{5-d}$. The local law implies that the bulk eigenvector entries of $H$ are of order $\mathrm{O}(W^{-5/2}{L}^{-d/2+5/2})$ with high probability.

72 pages. Accepted by Annals of Probability