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

TL;DR

This paper proves bulk eigenvalue universality and quantum unique ergodicity for high-dimensional random band matrices with certain band width conditions, extending understanding of spectral properties in complex quantum systems.

Contribution

It establishes bulk eigenvalue universality and quantum unique ergodicity for random band matrices in high dimensions, with explicit conditions on band width and local spectral laws.

Findings

Bulk eigenvalue universality for $d \,\geq\, 7$ under $W \gg L^{95/(d+95)}$.

Quantum unique ergodicity for eigenvectors assuming $W \geq L^\epsilon$.

Sharp local law for Green's function up to ${\mathrm{Im}} \, z \gg W^{-5}L^{5-d}$.

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\geq 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.