Eigenvector distribution in the critical regime of BBP transition

TL;DR

This paper analyzes the eigenvector distribution at the critical point of the BBP phase transition in spiked GUE matrices, revealing a determinantal point process with extended Airy kernel.

Contribution

It establishes the eigenvector distribution in the critical regime of the BBP transition using eigenvector-eigenvalue identities and determinantal processes, extending understanding of eigenvector behavior.

Findings

Eigenvector distribution described by extended Airy kernel

Results characterize eigenvector behavior at the BBP critical point

Method combines eigenvector-eigenvalue identities with GUE minor process

Abstract

In this paper, we study the random matrix model of Gaussian Unitary Ensemble (GUE) with fixed-rank (aka spiked) external source. We will focus on the critical regime of the Baik-Ben Arous-P\'ech\'e (BBP) phase transition and establish the distribution of the eigenvectors associated with the leading eigenvalues. The distribution is given in terms of a determinantal point process with extended Airy kernel. Our result can be regarded as an eigenvector counterpart of the BBP eigenvalue phase transition (arXiv:math/0403022). The derivation of the distribution makes use of the recently re-discovered eigenvector-eigenvalue identity, together with the determinantal point process representation of the GUE minor process with external source.