We exhibit new functions of the eigenvectors of the Dyson Brownian motion which follow an equation similar to the Bourgade-Yau eigenvector moment flow (Bourgade and Yau in Commun Math Phys 350(1):231–278, 2017). These observables can be seen as a Fermionic counterpart to the original (Bosonic) ones. By analyzing both Fermionic and Bosonic observables, we obtain new correlations between eigenvectors: (i) The fluctuations \(\sum _{\alpha \in I}\vert u_k(\alpha )\vert ^2-{\vert I\vert }/{N}\) decorrelate for distinct eigenvectors as the dimension N grows. (ii) An optimal estimate on the partial inner product \(\sum _{\alpha \in I}u_k(\alpha )\overline{u_\ell }(\alpha )\) between two eigenvectors is given. These static results obtained by integrable dynamics are stated for generalized Wigner matrices and should apply to wide classes of mean field models.

我们展示了戴森布朗运动特征向量的新函数，该函数遵循与Bourgade-Yau特征向量矩流相似的方程（Bourgade和Yau在Commun Math Phys 350（1）：231–278，2017）。这些可观察到的东西可以看作是与原始（Bosonic）的费米离子相对应的。通过分析费米离子和玻色子的可观测量，我们获得了特征向量之间的新关联：（i）波动\（\ sum _ {\\ I \}} vert u_k（\ alpha）\ vert ^ 2-{\ vert I \ vert } / {N} \）随着维度N的增长对不同的特征向量进行解相关。（ii）对部分内积\（\ sum _ {\\\\ I} u_k（\ alpha）\ overline {u_ \ ell}（\ alpha \\）的最优估计给出两个特征向量之间的关系。通过可积动力学获得的这些静态结果适用于广义Wigner矩阵，并且应适用于各种平均场模型。