Fermionic eigenvector moment flow

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 ${\sum }_{\alpha \in I}{u}_k(\alpha {)}^2-|I|/N$ 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.

31 pages, 5 figures

References in corpus (3)

• Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices
• Eigenvectors distribution and quantum unique ergodicity for deformed Wigner matrices
• Random band matrices in the delocalized phase, I: Quantum unique ergodicity and universality

Cited by in corpus (4)

• Fluctuations in local quantum unique ergodicity for generalized Wigner matrices
• Equipartition principle for Wigner matrices
• Efficient circular Dyson Brownian motion algorithm
• Eigenvector distribution in the critical regime of BBP transition