论文标题
散装普遍性和量子量的独特的牙本质矩阵，
Bulk universality and quantum unique ergodicity for random band matrices in high dimensions
论文作者
论文摘要
我们考虑$ d $ - 二维晶格$（\ Mathbb z/l \ Mathbb z）^d $上的隐性随机频段$ h =（h_ {xy}）$，其中条目$ h_ {xy} = \ edylline h_ {yx {yx} $是独立的中心复杂量的$ variances $ sbb e | h_ {xy} |^2 $。方差矩阵$ s =（s_ {xy}）$具有带状的配置文件，因此，如果$ | x-y | $超过频段宽度$ w $，则$ s_ {xy} $可以忽略不计。对于尺寸$ d \ ge 7 $，我们在条件$ w \ gg l^{95/（d+95）} $下证明了$ h $的批量特征性通用性。假设对于小常数$ε> 0 $，我们还证明了$ h $的批量特征向量的量子独特的牙齿和绿色函数$ g（z）=（h-z）=（h-z）^{ - 1} $ of $ g（h-z） w^{ - 5} l^{5-d} $。本地法律意味着$ h $的批量特征向量入口是$ {\ mathrm {o}}}（w^{ - 5/5/2} l^{ - d/2+5/2}）$，概率很高。
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 $W\geq L^ε$ for a small constant $ε>0$, 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.
