Fermionic eigenvector moment flow

We exhibit new functions of the eigenvectors Dyson Brownian motion which follow an equation similar to 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 original (Bosonic) ones. By analyzing both Bosonic observables, we obtain correlations between eigenvectors: (i) The fluctuations $$\sum _{\alpha \in I}\vert u_k(\alpha )\vert ^2-{\vert I\vert }/{N}$$ decorrelate for distinct dimension N grows. (ii) An optimal estimate on partial inner product I}u_k(\alpha )\overline{u_\ell }(\alpha )$$ two is given. static results obtained by integrable dynamics are stated generalized Wigner matrices should apply wide classes mean field models.