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

Abstract

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