We exhibit new functions of the eigenvectors of the Dyson Brownian motion which follow an equation similar to the Bourgade-Yau eigenvector moment flow. 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. The fluctuations $\sum_{ \alpha\in I}u_k(\alpha)^2-{ \vert I\vert}/{ N}$ decorrelate for distinct eigenvectors as the dimension $N$ grows and an optimal estimate on the partial inner product $\sum_{ \alpha\in I}u_k(\alpha)u_\ell(\alpha)$ 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.
Categories: math.PR math-ph math.MP