Eigenvector distribution of random matrices under critical finite-rank deformations

We investigate the eigenvector distribution at the soft edge for Gaussian random matrices with finite-rank deformations, in the critical regime of BBP transition. For finite-rank deformations of the GOE and GUE with critical spikes, we find that the squared overlap between a leading eigenvector and a spike, rescaled by \(N^{1/3}\), converges weakly to the negative reciprocal of the derivative of an Airy-Green function evaluated at the corresponding soft-edge root. For the rank-one critically spiked Gaussian \(\beta\)-ensemble, \(\beta>0\), we obtain an analogous result involving an Airy-Green function. In both cases, the Airy-Green functions are generalizations of the one introduced by Bykhovskaya--Gorin--Sodin \cite{Bykhovskaya-Gorin-Sodin25}. The proofs are both based on an eigenvector--eigenvalue identity and a resolvent-differentiation mechanism.