Eigenvector distribution in the critical regime of BBP transition

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–Péché (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 [6]. 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.