论文标题
广义Wigner矩阵的最佳离域化
Optimal Delocalization for Generalized Wigner Matrices
论文作者
论文摘要
我们研究具有次指数条目的广义Wigner矩阵的特征向量，并证明它们以压倒性的概率以最佳速率离居。我们还证明了与尖锐常数的高概率定位边界。我们的证明使用了Bourgade和Yau（2017）引入的特征向量矩流的分析，以结合具有小高斯组件的随机矩阵特征向量入口的对数力矩。然后，我们通过基于正规特征向量，级别排斥和Landon，Lopatto和Marcinek（2018）可观察到的框架的比较参数将此控制扩展到所有广义的Wigner矩阵，以比较极端特征值统计。此外，我们证明了整个频谱的水平排斥和特征值过度拥挤的估计值，这可能具有独立感兴趣。
We study the eigenvectors of generalized Wigner matrices with subexponential entries and prove that they delocalize at the optimal rate with overwhelming probability. We also prove high probability delocalization bounds with sharp constants. Our proof uses an analysis of the eigenvector moment flow introduced by Bourgade and Yau (2017) to bound logarithmic moments of eigenvector entries for random matrices with small Gaussian components. We then extend this control to all generalized Wigner matrices by comparison arguments based on a framework of regularized eigenvectors, level repulsion, and the observable employed by Landon, Lopatto, and Marcinek (2018) to compare extremal eigenvalue statistics. Additionally, we prove level repulsion and eigenvalue overcrowding estimates for the entire spectrum, which may be of independent interest.
