Eigenvector convergence for minors of unitarily invariant infinite random matrices

TL;DR

This paper proves weak convergence of eigenvectors of minors of unitarily invariant infinite random matrices, extending previous results on eigenvalue convergence and connecting to classical ensembles.

Contribution

It establishes the weak convergence of eigenvectors for minors of unitarily invariant infinite matrices, a novel extension beyond eigenvalue convergence results.

Findings

Eigenvector convergence under invariant measures

Extension of eigenvalue convergence results

Connection to Circular Unitary Ensemble

Abstract

Pickrell has fully characterized the unitarily invariant probability measures on infinite Hermitian matrices, and an alternative proof of this classification has been found by Olshanski and Vershik. Borodin and Olshanski deduced from this proof that under any of these invariant measures, the extreme eigenvalues of the minors, divided by the dimension, converge almost surely. In this paper, we prove that one also has a weak convergence for the eigenvectors, in a sense which is made precise. After mapping Hermitian to unitary matrices via the Cayley transform, our result extends a convergence proven in our paper with Maples and Nikeghbali, for which a coupling of the Circular Unitary Ensemble of all dimensions is considered.