一维随机带矩阵的泊松统计
Poisson statistics of one-dimensional random band matrices
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中文总结 AI 辅助
本文证明在带宽平方远小于N/log N时，一维随机带矩阵的体相局部统计渐近为泊松点过程，结合已有普适性结果确立了泊松-随机矩阵转变，并补充了特征向量局域化转变的结论。
中文摘要 AI 辅助
考虑一个$N\times N$随机带矩阵$H$，带宽为$W$，其元素为居中的实高斯分布。我们证明，在（几乎）尖锐的假设$W^2\ll N/\log N$下，$H$的体相局部统计量渐近地服从泊松点过程。结合arXiv:2501.01718和arXiv:2506.06441中的体相普适性结果，这确立了随机带矩阵局部统计量的泊松-随机矩阵理论转变，并补充了先前在arXiv:2501.01718、arXiv:2506.06441和arXiv:2508.05802中建立的体相特征向量局域化-去局域化转变的结果。关键技术输入包括：（1）来自arXiv:2508.05802的分数矩估计；（2）从arXiv:2501.01718和arXiv:2506.06441中的局部律获得的态密度的新估计。
英文摘要
Consider an $N\times N$ random band matrix $H$ with bandwidth $W$ with centered real Gaussian entries. We prove that, under the (almost) sharp assumption $W^2\ll N/\log N$, the bulk local statistics of $H$ are asymptotically a Poisson point process. Combined with the bulk universality results from arXiv:2501.01718 and arXiv:2506.06441, this establishes the Poisson-RMT transition of the local statistics of random band matrices, and complements the results about the localization-delocalization transition of the bulk eigenvectors previously established in arXiv:2501.01718 and arXiv:2506.06441, and arXiv:2508.05802. The key technical inputs are: (1) the fractional moment estimates from arXiv:2508.05802; (2) a novel estimate for the density of states, obtained from the local laws in arXiv:2501.01718 and arXiv:2506.06441.
发表机构
- Qiuzhen College, Tsinghua University（清华大学丘成桐数学科学中心）
- Department of Mathematics, University of California, Irvine（加州大学尔湾分校数学系）
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