Fermionic eigenvector moment flow

TL;DR

This paper introduces Fermionic eigenvector observables for Dyson Brownian motion, revealing new eigenvector correlations and fluctuation behaviors, with results applicable to generalized Wigner matrices and mean field models.

Contribution

It develops Fermionic eigenvector observables following a similar flow to the Bosonic case, providing new insights into eigenvector correlations and fluctuations.

Findings

Eigenvector fluctuations decorrelate as dimension grows

Optimal estimate for partial inner product between eigenvectors

Results apply to generalized Wigner matrices and mean field models

Abstract

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.