发表机构
Qiuzhen College, Tsinghua University; Yau Mathematical Sciences Center, Tsinghua University, and Beijing Institute of Mathematical Sciences and Applications（清华大学丘成桐数学科学中心; 清华大学姚班及北京国际数学研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明二维随机带矩阵在带宽 $W \geq (\log N)^{55}$ 时体特征向量高概率离域，得到局域化长度的拉伸指数下界 $\exp(W^{1/55})$，改进了先前超多项式下界。
AI 中文摘要
我们考虑 $N \times N$ 随机带矩阵 $H = (H_{xy})$，其元素为中心复高斯变量，由二维离散环面 $(\mathbb{Z} / \sqrt{N} \mathbb{Z})^2$ 上的点 $x,y$ 索引。当 $x$ 与 $y$ 之间的距离超过带宽参数 $W$ 时，矩阵元素 $H_{xy}$ 为零。我们证明，若 $W \geq (\log N)^{55}$，则高概率下所有体特征向量都是离域的。等价地，这给出了二维随机带矩阵局域化长度的下界为 $\exp(W^{1/55})$ 量级，改进了 arXiv:2503.07606 中建立的对于任意固定常数 $C>0$ 的超多项式下界 $W^C$。
英文摘要
We consider $N \times N$ random band matrices $H = (H_{xy})$ with centered complex Gaussian entries, indexed by points $x,y$ on the two-dimensional discrete torus $(\mathbb{Z} / \sqrt{N} \mathbb{Z})^2$. The matrix entries $H_{xy}$ vanish whenever the distance between $x$ and $y$ exceeds the bandwidth parameter $W$. We prove that if $W \geq (\log N)^{7}$, then, with high probability, all bulk eigenvectors are delocalized. Equivalently, this yields a lower bound of order $\exp(W^{1/7})$ for the localization lengths of two-dimensional random band matrices, improving the superpolynomial lower bound $W^C$ for every fixed constant $C>0$ established in arXiv:2503.07606.
Comments46 pages. The results are improved by sharper propagator estimates, a better choice of profile function, and, most importantly, a new method that closes the flow analysis at order 2. We release this version before fully exploring the method's potential and plan to pursue further improvements, possibly with the assistance of AI tools, in future versions
